Changeset 9407
- Timestamp:
- 2018-03-15T17:40:35+01:00 (6 years ago)
- Location:
- branches/2017/dev_merge_2017/DOC
- Files:
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- 3 added
- 25 edited
- 5 moved
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branches/2017/dev_merge_2017/DOC/HTML_htlatex.sh
r9394 r9407 1 1 #!/bin/bash 2 2 3 latex -shell-escape NEMO_manual 4 makeindex NEMO_manual 5 bibtex NEMO_manual 6 latex -shell-escape NEMO_manual 3 ./inc/clean.sh 4 ./inc/build.sh 7 5 8 6 mkdir -p html_htlatex 9 htlatex NEMO_manual "htlatex,2" "" "-dhtml_htlatex/" "-shell-escape" 7 cd tex_main 8 htlatex NEMO_manual "NEMO_manual,2" "" "-d../html_htlatex/" "-shell-escape" 9 cd - 10 10 11 11 exit 0 -
branches/2017/dev_merge_2017/DOC/HTML_latex2html.sh
r9394 r9407 1 1 #!/bin/bash 2 2 3 ./inc/clean.sh 4 ./inc/build.sh 5 6 cd tex_main 3 7 sed -i -e 's#\\documentclass#%\\documentclass#' -e '/{document}/ s/^/%/' \ 4 texfiles/chapters/*.tex5 sed -i ' 30,${s#\\subfile{#\\include{#g}' \8 ../tex_sub/*.tex 9 sed -i 's#\\subfile{#\\include{#g' \ 6 10 NEMO_manual.tex 7 8 latex -shell-escape NEMO_manual 9 makeindex NEMO_manual 10 bibtex NEMO_manual 11 12 latex2html -local_icons -no_footnode -split 4 -link 2 -mkdir -dir html_LaTeX2HTML \ 13 $* \ 11 latex2html -local_icons -no_footnode -split 4 -link 2 -mkdir -dir ../html_LaTeX2HTML \ 12 $* \ 14 13 NEMO_manual 15 16 14 sed -i -e 's#%\\documentclass#\\documentclass#' -e '/{document}/ s/^%//' \ 17 texfiles/chapters/*.tex18 sed -i ' 30,${s#\\include{#\\subfile{#g}' \15 ../tex_sub/*.tex 16 sed -i 's#\\include{#\\subfile{#g' \ 19 17 NEMO_manual.tex 18 cd - 20 19 21 20 exit 0 -
branches/2017/dev_merge_2017/DOC/inc/build.sh
r9394 r9407 11 11 latex ${latex_opts} ${latex_file} 12 12 13 pdflatex ${latex_opts} ${latex_file}14 15 mv ${latex_file}.pdf ..16 17 13 cd - 18 14 -
branches/2017/dev_merge_2017/DOC/inc/clean.sh
r9394 r9407 1 1 #!/bin/bash 2 2 3 rm -f $( ls -1 tex_main/NEMO_* | egrep -v "\.(bib| ist|sty|tex)$" )3 rm -f $( ls -1 tex_main/NEMO_* | egrep -v "\.(bib|cfg|ist|sty|tex)$" ) 4 4 #rm -rf _minted-* 5 5 #rm -rf html* -
branches/2017/dev_merge_2017/DOC/tex_main/NEMO_manual.cfg
r9394 r9407 1 \Preamble{html} 1 \Preamble{xhtml,mathml} 2 3 \Configure{@HEAD}{% 4 \HCode{<script type="text/javascript" 5 src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=MML_CHTML"> 6 </script>\Hnewline}} 2 7 3 8 \begin{document} -
branches/2017/dev_merge_2017/DOC/tex_sub/annex_A.tex
r9393 r9407 6 6 % ================================================================ 7 7 \chapter{Curvilinear $s-$Coordinate Equations} 8 \label{ Apdx_A}8 \label{apdx:A} 9 9 \minitoc 10 10 … … 16 16 % ================================================================ 17 17 \section{Chain rule for $s-$coordinates} 18 \label{ Apdx_A_continuity}18 \label{sec:A_continuity} 19 19 20 20 In order to establish the set of Primitive Equation in curvilinear $s$-coordinates 21 21 ($i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian 22 22 Eulerian (ALE) coordinate in the vertical), we start from the set of equations established 23 in \ S\ref{PE_zco_Eq} for the special case $k = z$ and thus $e_3 = 1$, and we introduce23 in \autoref{subsec:PE_zco_Eq} for the special case $k = z$ and thus $e_3 = 1$, and we introduce 24 24 an arbitrary vertical coordinate $a = a(i,j,z,t)$. Let us define a new vertical scale factor by 25 25 $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal 26 26 slope of $s-$surfaces by : 27 \begin{equation} \label{ Apdx_A_s_slope}27 \begin{equation} \label{apdx:A_s_slope} 28 28 \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s 29 29 \quad \text{and} \quad … … 33 33 The chain rule to establish the model equations in the curvilinear $s-$coordinate 34 34 system is: 35 \begin{equation} \label{ Apdx_A_s_chain_rule}35 \begin{equation} \label{apdx:A_s_chain_rule} 36 36 \begin{aligned} 37 37 &\left. {\frac{\partial \bullet }{\partial t}} \right|_z = … … 54 54 In particular applying the time derivative chain rule to $z$ provides the expression 55 55 for $w_s$, the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: 56 \begin{equation} \label{ Apdx_A_w_in_s}56 \begin{equation} \label{apdx:A_w_in_s} 57 57 w_s = \left. \frac{\partial z }{\partial t} \right|_s 58 58 = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t} … … 65 65 % ================================================================ 66 66 \section{Continuity equation in $s-$coordinates} 67 \label{ Apdx_A_continuity}68 69 Using (\ ref{Apdx_A_s_chain_rule}) and the fact that the horizontal scale factors67 \label{sec:A_continuity} 68 69 Using (\autoref{apdx:A_s_chain_rule}) and the fact that the horizontal scale factors 70 70 $e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of 71 71 the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows … … 131 131 Introducing the dia-surface velocity component, $\omega $, defined as 132 132 the volume flux across the moving $s$-surfaces per unit horizontal area: 133 \begin{equation} \label{ Apdx_A_w_s}133 \begin{equation} \label{apdx:A_w_s} 134 134 \omega = w - w_s - \sigma _1 \,u - \sigma _2 \,v \\ 135 135 \end{equation} 136 with $w_s$ given by \ eqref{Apdx_A_w_in_s}, we obtain the expression for136 with $w_s$ given by \autoref{apdx:A_w_in_s}, we obtain the expression for 137 137 the divergence of the velocity in the curvilinear $s-$coordinate system: 138 138 \begin{subequations} … … 167 167 \end{subequations} 168 168 169 As a result, the continuity equation \ eqref{Eq_PE_continuity} in the169 As a result, the continuity equation \autoref{eq:PE_continuity} in the 170 170 $s-$coordinates is: 171 \begin{equation} \label{ Apdx_A_sco_Continuity}171 \begin{equation} \label{apdx:A_sco_Continuity} 172 172 \frac{1}{e_3 } \frac{\partial e_3}{\partial t} 173 173 + \frac{1}{e_1 \,e_2 \,e_3 }\left[ … … 184 184 % ================================================================ 185 185 \section{Momentum equation in $s-$coordinate} 186 \label{ Apdx_A_momentum}186 \label{sec:A_momentum} 187 187 188 188 Here we only consider the first component of the momentum equation, … … 193 193 $\bullet$ \textbf{Total derivative in vector invariant form} 194 194 195 Let us consider \ eqref{Eq_PE_dyn_vect}, the first component of the momentum195 Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum 196 196 equation in the vector invariant form. Its total $z-$coordinate time derivative, 197 197 $\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain … … 258 258 \end{subequations} 259 259 % 260 Applying the time derivative chain rule (first equation of (\ ref{Apdx_A_s_chain_rule}))261 to $u$ and using (\ ref{Apdx_A_w_in_s}) provides the expression of the last term260 Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) 261 to $u$ and using (\autoref{apdx:A_w_in_s}) provides the expression of the last term 262 262 of the right hand side, 263 263 \begin{equation*} {\begin{array}{*{20}l} … … 269 269 leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, 270 270 $i.e.$ the total $s-$coordinate time derivative : 271 \begin{align} \label{ Apdx_A_sco_Dt_vect}271 \begin{align} \label{apdx:A_sco_Dt_vect} 272 272 \left. \frac{D u}{D t} \right|_s 273 273 = \left. {\frac{\partial u }{\partial t}} \right|_s … … 285 285 286 286 Let us start from the total time derivative in the curvilinear $s-$coordinate system 287 we have just establish. Following the procedure used to establish (\ ref{Eq_PE_flux_form}),287 we have just establish. Following the procedure used to establish (\autoref{eq:PE_flux_form}), 288 288 it can be transformed into : 289 289 %\begin{subequations} … … 356 356 which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, 357 357 $i.e.$ the total $s-$coordinate time derivative in flux form : 358 \begin{flalign}\label{ Apdx_A_sco_Dt_flux}358 \begin{flalign}\label{apdx:A_sco_Dt_flux} 359 359 \left. \frac{D u}{D t} \right|_s = \frac{1}{e_3} \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s 360 360 + \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s … … 365 365 It has the same form as in the $z-$coordinate but for the vertical scale factor 366 366 that has appeared inside the time derivative which comes from the modification 367 of (\ ref{Apdx_A_sco_Continuity}), the continuity equation.367 of (\autoref{apdx:A_sco_Continuity}), the continuity equation. 368 368 369 369 $\ $\newline % force a new ligne … … 381 381 \end{equation*} 382 382 Applying similar manipulation to the second component and replacing 383 $\sigma _1$ and $\sigma _2$ by their expression \ eqref{Apdx_A_s_slope}, it comes:384 \begin{equation} \label{ Apdx_A_grad_p}383 $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes: 384 \begin{equation} \label{apdx:A_grad_p} 385 385 \begin{split} 386 386 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z … … 394 394 \end{equation} 395 395 396 An additional term appears in (\ ref{Apdx_A_grad_p}) which accounts for the396 An additional term appears in (\autoref{apdx:A_grad_p}) which accounts for the 397 397 tilt of $s-$surfaces with respect to geopotential $z-$surfaces. 398 398 … … 408 408 \end{equation*} 409 409 Therefore, $p$ and $p_h'$ are linked through: 410 \begin{equation} \label{ Apdx_A_pressure}410 \begin{equation} \label{apdx:A_pressure} 411 411 p = \rho_o \; p_h' + g \, ( z + \eta ) 412 412 \end{equation} … … 416 416 \end{equation*} 417 417 418 Substituing \ eqref{Apdx_A_pressure} in \eqref{Apdx_A_grad_p} and using the definition of418 Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p} and using the definition of 419 419 the density anomaly it comes the expression in two parts: 420 \begin{equation} \label{ Apdx_A_grad_p}420 \begin{equation} \label{apdx:A_grad_p} 421 421 \begin{split} 422 422 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z … … 430 430 \end{equation} 431 431 This formulation of the pressure gradient is characterised by the appearance of a term depending on the 432 the sea surface height only (last term on the right hand side of expression \ eqref{Apdx_A_grad_p}).432 the sea surface height only (last term on the right hand side of expression \autoref{apdx:A_grad_p}). 433 433 This term will be loosely termed \textit{surface pressure gradient} 434 434 whereas the first term will be termed the … … 445 445 The coriolis and forcing terms as well as the the vertical physics remain unchanged 446 446 as they involve neither time nor space derivatives. The form of the lateral physics is 447 discussed in appendix~\ref{Apdx_B}.447 discussed in \autoref{apdx:B}. 448 448 449 449 … … 455 455 solved by the model has the same mathematical expression as the one in a curvilinear 456 456 $z-$coordinate, except for the pressure gradient term : 457 \begin{subequations} \label{ Apdx_A_dyn_vect}458 \begin{multline} \label{ Apdx_A_PE_dyn_vect_u}457 \begin{subequations} \label{apdx:A_dyn_vect} 458 \begin{multline} \label{apdx:A_PE_dyn_vect_u} 459 459 \frac{\partial u}{\partial t}= 460 460 + \left( {\zeta +f} \right)\,v … … 465 465 + D_u^{\vect{U}} + F_u^{\vect{U}} 466 466 \end{multline} 467 \begin{multline} \label{ Apdx_A_dyn_vect_v}467 \begin{multline} \label{apdx:A_dyn_vect_v} 468 468 \frac{\partial v}{\partial t}= 469 469 - \left( {\zeta +f} \right)\,u … … 477 477 whereas the flux form momentum equation differ from it by the formulation of both 478 478 the time derivative and the pressure gradient term : 479 \begin{subequations} \label{ Apdx_A_dyn_flux}480 \begin{multline} \label{ Apdx_A_PE_dyn_flux_u}479 \begin{subequations} \label{apdx:A_dyn_flux} 480 \begin{multline} \label{apdx:A_PE_dyn_flux_u} 481 481 \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t} = 482 482 \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) … … 487 487 + D_u^{\vect{U}} + F_u^{\vect{U}} 488 488 \end{multline} 489 \begin{multline} \label{ Apdx_A_dyn_flux_v}489 \begin{multline} \label{apdx:A_dyn_flux_v} 490 490 \frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}= 491 491 - \nabla \cdot \left( {{\rm {\bf U}}\,v} \right) … … 499 499 Both formulation share the same hydrostatic pressure balance expressed in terms of 500 500 hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: 501 \begin{equation} \label{ Apdx_A_dyn_zph}501 \begin{equation} \label{apdx:A_dyn_zph} 502 502 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 503 503 \end{equation} … … 516 516 % ================================================================ 517 517 \section{Tracer equation} 518 \label{ Apdx_A_tracer}518 \label{sec:A_tracer} 519 519 520 520 The tracer equation is obtained using the same calculation as for the continuity 521 521 equation and then regrouping the time derivative terms in the left hand side : 522 522 523 \begin{multline} \label{ Apdx_A_tracer}523 \begin{multline} \label{apdx:A_tracer} 524 524 \frac{1}{e_3} \frac{\partial \left( e_3 T \right)}{\partial t} 525 525 = -\frac{1}{e_1 \,e_2 \,e_3} -
branches/2017/dev_merge_2017/DOC/tex_sub/annex_B.tex
r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Appendix B : Diffusive Operators} 7 \label{ Apdx_B}7 \label{apdx:B} 8 8 \minitoc 9 9 … … 16 16 % ================================================================ 17 17 \section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators} 18 \label{ Apdx_B_1}18 \label{sec:B_1} 19 19 20 20 \subsubsection*{In z-coordinates} 21 21 In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator 22 22 is given by: 23 \begin{eqnarray} \label{ Apdx_B1}23 \begin{eqnarray} \label{apdx:B1} 24 24 &D^T = \frac{1}{e_1 \, e_2} \left[ 25 25 \left. \frac{\partial}{\partial i} \left( \frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z \right) \right|_z \right. … … 31 31 \subsubsection*{In generalized vertical coordinates} 32 32 In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and 33 $\sigma_2$ by \ eqref{Apdx_A_s_slope} and the vertical/horizontal ratio of diffusion33 $\sigma_2$ by \autoref{apdx:A_s_slope} and the vertical/horizontal ratio of diffusion 34 34 coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by: 35 35 36 \begin{equation} \label{ Apdx_B2}36 \begin{equation} \label{apdx:B2} 37 37 D^T = \left. \nabla \right|_s \cdot 38 38 \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ … … 56 56 \end{subequations} 57 57 58 Equation \ eqref{Apdx_B2} is obtained from \eqref{Apdx_B1} without any58 Equation \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any 59 59 additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$, 60 we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in Appendix~\ref{Apdx_A}61 and use \ eqref{Apdx_A_s_slope} and \eqref{Apdx_A_s_chain_rule}.60 we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A} 61 and use \autoref{apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}. 62 62 Since no cross horizontal derivative $\partial _i \partial _j $ appears in 63 \ eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) planes are independent.63 \autoref{apdx:B1}, the ($i$,$z$) and ($j$,$z$) planes are independent. 64 64 The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) 65 65 transformation without any loss of generality: … … 139 139 % ================================================================ 140 140 \section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators} 141 \label{ Apdx_B_2}141 \label{sec:B_2} 142 142 143 143 \subsubsection*{In z-coordinates} … … 147 147 formulated, takes the following form \citep{Redi_JPO82}: 148 148 149 \begin{equation} \label{ Apdx_B3}149 \begin{equation} \label{apdx:B3} 150 150 \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} 151 151 \left[ {{\begin{array}{*{20}c} … … 166 166 In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, so 167 167 $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}: 168 \begin{subequations} \label{ Apdx_B4}169 \begin{equation} \label{ Apdx_B4a}168 \begin{subequations} \label{apdx:B4} 169 \begin{equation} \label{apdx:B4a} 170 170 {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = 171 171 \left[ {{\begin{array}{*{20}c} … … 176 176 \end{equation} 177 177 and the iso/dianeutral diffusive operator in $z$-coordinates is then 178 \begin{equation}\label{ Apdx_B4b}178 \begin{equation}\label{apdx:B4b} 179 179 D^T = \left. \nabla \right|_z \cdot 180 180 \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T \right]. \\ … … 183 183 184 184 185 Physically, the full tensor \ eqref{Apdx_B3}185 Physically, the full tensor \autoref{apdx:B3} 186 186 represents strong isoneutral diffusion on a plane parallel to the isoneutral 187 187 surface and weak dianeutral diffusion perpendicular to this plane. 188 However, the approximate `weak-slope' tensor \ eqref{Apdx_B4a} represents strong188 However, the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong 189 189 diffusion along the isoneutral surface, with weak 190 190 \emph{vertical} diffusion -- the principal axes of the tensor are no 191 191 longer orthogonal. This simplification also decouples 192 192 the ($i$,$z$) and ($j$,$z$) planes of the tensor. The weak-slope operator therefore takes the same 193 form, \ eqref{Apdx_B4}, as \eqref{Apdx_B2}, the diffusion operator for geopotential193 form, \autoref{apdx:B4}, as \autoref{apdx:B2}, the diffusion operator for geopotential 194 194 diffusion written in non-orthogonal $i,j,s$-coordinates. Written out 195 195 explicitly, 196 196 197 \begin{multline} \label{ Apdx_B_ldfiso}197 \begin{multline} \label{apdx:B_ldfiso} 198 198 D^T=\frac{1}{e_1 e_2 }\left\{ 199 199 {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2}{e_1}\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]} … … 203 203 204 204 205 The isopycnal diffusion operator \ eqref{Apdx_B4},206 \ eqref{Apdx_B_ldfiso} conserves tracer quantity and dissipates its207 square. The demonstration of the first property is trivial as \ eqref{Apdx_B4} is the divergence205 The isopycnal diffusion operator \autoref{apdx:B4}, 206 \autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its 207 square. The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence 208 208 of fluxes. Let us demonstrate the second one: 209 209 \begin{equation*} … … 233 233 \subsubsection*{In generalized vertical coordinates} 234 234 235 Because the weak-slope operator \ eqref{Apdx_B4}, \eqref{Apdx_B_ldfiso} is decoupled235 Because the weak-slope operator \autoref{apdx:B4}, \autoref{apdx:B_ldfiso} is decoupled 236 236 in the ($i$,$z$) and ($j$,$z$) planes, it may be transformed into 237 generalized $s$-coordinates in the same way as \ eqref{Apdx_B_1} was transformed into238 \ eqref{Apdx_B_2}. The resulting operator then takes the simple form239 240 \begin{equation} \label{ Apdx_B_ldfiso_s}237 generalized $s$-coordinates in the same way as \autoref{sec:B_1} was transformed into 238 \autoref{sec:B_2}. The resulting operator then takes the simple form 239 240 \begin{equation} \label{apdx:B_ldfiso_s} 241 241 D^T = \left. \nabla \right|_s \cdot 242 242 \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ … … 258 258 \end{equation*} 259 259 260 To prove \ eqref{Apdx_B5} by direct re-expression of \eqref{Apdx_B_ldfiso} is260 To prove \autoref{apdx:B5} by direct re-expression of \autoref{apdx:B_ldfiso} is 261 261 straightforward, but laborious. An easier way is first to note (by reversing the 262 derivation of \ eqref{Apdx_B_2} from \eqref{Apdx_B_1} ) that the262 derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that the 263 263 weak-slope operator may be \emph{exactly} reexpressed in 264 264 non-orthogonal $i,j,\rho$-coordinates as 265 265 266 \begin{equation} \label{ Apdx_B5}266 \begin{equation} \label{apdx:B5} 267 267 D^T = \left. \nabla \right|_\rho \cdot 268 268 \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T \right] \\ … … 274 274 \end{equation} 275 275 Then direct transformation from $i,j,\rho$-coordinates to 276 $i,j,s$-coordinates gives \ eqref{Apdx_B_ldfiso_s} immediately.276 $i,j,s$-coordinates gives \autoref{apdx:B_ldfiso_s} immediately. 277 277 278 278 Note that the weak-slope approximation is only made in … … 282 282 the $s$-surfaces, in the same way as the transformation of 283 283 horizontal/vertical Laplacian diffusion in $z$-coordinates, 284 \ eqref{Apdx_B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.284 \autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. 285 285 286 286 … … 289 289 % ================================================================ 290 290 \section{Lateral/Vertical momentum diffusive operators} 291 \label{ Apdx_B_3}291 \label{sec:B_3} 292 292 293 293 The second order momentum diffusion operator (Laplacian) in the $z$-coordinate 294 is found by applying \ eqref{Eq_PE_lap_vector}, the expression for the Laplacian294 is found by applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian 295 295 of a vector, to the horizontal velocity vector : 296 296 \begin{align*} … … 329 329 \end{array} }} \right) 330 330 \end{align*} 331 Using \ eqref{Eq_PE_div}, the definition of the horizontal divergence, the third331 Using \autoref{eq:PE_div}, the definition of the horizontal divergence, the third 332 332 componant of the second vector is obviously zero and thus : 333 333 \begin{equation*} … … 336 336 337 337 Note that this operator ensures a full separation between the vorticity and horizontal 338 divergence fields (see Appendix~\ref{Apdx_C}). It is only equal to a Laplacian338 divergence fields (see \autoref{apdx:C}). It is only equal to a Laplacian 339 339 applied to each component in Cartesian coordinates, not on the sphere. 340 340 341 341 The horizontal/vertical second order (Laplacian type) operator used to diffuse 342 342 horizontal momentum in the $z$-coordinate therefore takes the following form : 343 \begin{equation} \label{ Apdx_B_Lap_U}343 \begin{equation} \label{apdx:B_Lap_U} 344 344 {\textbf{D}}^{\textbf{U}} = 345 345 \nabla _h \left( {A^{lm}\;\chi } \right) … … 360 360 \end{align*} 361 361 362 Note Bene: introducing a rotation in \ eqref{Apdx_B_Lap_U} does not lead to a362 Note Bene: introducing a rotation in \autoref{apdx:B_Lap_U} does not lead to a 363 363 useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. 364 364 Similarly, we did not found an expression of practical use for the geopotential 365 365 horizontal/vertical Laplacian operator in the $s$-coordinate. Generally, 366 \ eqref{Apdx_B_Lap_U} is used in both $z$- and $s$-coordinate systems, that is366 \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems, that is 367 367 a Laplacian diffusion is applied on momentum along the coordinate directions. 368 368 \end{document} -
branches/2017/dev_merge_2017/DOC/tex_sub/annex_C.tex
r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Discrete Invariants of the Equations} 7 \label{ Apdx_C}7 \label{apdx:C} 8 8 \minitoc 9 9 … … 20 20 % ================================================================ 21 21 \section{Introduction / Notations} 22 \label{ Apdx_C.0}22 \label{sec:C.0} 23 23 24 24 Notation used in this appendix in the demonstations : … … 69 69 \end{flalign*} 70 70 that is in a more compact form : 71 \begin{flalign} \label{ Eq_Q2_flux}71 \begin{flalign} \label{eq:Q2_flux} 72 72 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 73 73 =& \int_D { \frac{Q}{e_3} \partial_t \left( e_3 \, Q \right) dv } … … 83 83 \end{flalign*} 84 84 that is in a more compact form : 85 \begin{flalign} \label{ Eq_Q2_vect}85 \begin{flalign} \label{eq:Q2_vect} 86 86 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 87 87 =& \int_D { Q \,\partial_t Q \;dv } … … 94 94 % ================================================================ 95 95 \section{Continuous conservation} 96 \label{ Apdx_C.1}96 \label{sec:C.1} 97 97 98 98 … … 104 104 Let us first establish those constraint in the continuous world. 105 105 The total energy ($i.e.$ kinetic plus potential energies) is conserved : 106 \begin{flalign} \label{ Eq_Tot_Energy}106 \begin{flalign} \label{eq:Tot_Energy} 107 107 \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 + \rho \, g \, z \right) \;dv \right) = & 0 108 108 \end{flalign} … … 114 114 The transformation for the advection term depends on whether 115 115 the vector invariant form or the flux form is used for the momentum equation. 116 Using \ eqref{Eq_Q2_vect} and introducing \eqref{Apdx_A_dyn_vect} in \eqref{Eq_Tot_Energy}116 Using \autoref{eq:Q2_vect} and introducing \autoref{apdx:A_dyn_vect} in \autoref{eq:Tot_Energy} 117 117 for the former form and 118 Using \ eqref{Eq_Q2_flux} and introducing \eqref{Apdx_A_dyn_flux} in \eqref{Eq_Tot_Energy}118 Using \autoref{eq:Q2_flux} and introducing \autoref{apdx:A_dyn_flux} in \autoref{eq:Tot_Energy} 119 119 for the latter form leads to: 120 120 121 \begin{subequations} \label{ E_tot}121 \begin{subequations} \label{eq:E_tot} 122 122 123 123 advection term (vector invariant form): 124 \begin{equation} \label{ E_tot_vect_vor}124 \begin{equation} \label{eq:E_tot_vect_vor} 125 125 \int\limits_D \zeta \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ 126 126 \end{equation} 127 127 % 128 \begin{equation} \label{ E_tot_vect_adv}128 \begin{equation} \label{eq:E_tot_vect_adv} 129 129 \int\limits_D \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right) dv 130 130 + \int\limits_D \textbf{U}_h \cdot \nabla_z \textbf{U}_h \;dv … … 133 133 134 134 advection term (flux form): 135 \begin{equation} \label{ E_tot_flux_metric}135 \begin{equation} \label{eq:E_tot_flux_metric} 136 136 \int\limits_D \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1 \right)\; 137 137 \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ 138 138 \end{equation} 139 139 140 \begin{equation} \label{ E_tot_flux_adv}140 \begin{equation} \label{eq:E_tot_flux_adv} 141 141 \int\limits_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} 142 142 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ … … 146 146 147 147 coriolis term 148 \begin{equation} \label{ E_tot_cor}148 \begin{equation} \label{eq:E_tot_cor} 149 149 \int\limits_D f \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ 150 150 \end{equation} 151 151 152 152 pressure gradient: 153 \begin{equation} \label{ E_tot_pg}153 \begin{equation} \label{eq:E_tot_pg} 154 154 - \int\limits_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 155 155 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv … … 171 171 172 172 Vector invariant form: 173 \begin{subequations} \label{ E_tot_vect}174 \begin{equation} \label{ E_tot_vect_vor}173 \begin{subequations} \label{eq:E_tot_vect} 174 \begin{equation} \label{eq:E_tot_vect_vor} 175 175 \int\limits_D \textbf{U}_h \cdot \text{VOR} \;dv = 0 \\ 176 176 \end{equation} 177 \begin{equation} \label{ E_tot_vect_adv}177 \begin{equation} \label{eq:E_tot_vect_adv} 178 178 \int\limits_D \textbf{U}_h \cdot \text{KEG} \;dv 179 179 + \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 180 180 - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 \\ 181 181 \end{equation} 182 \begin{equation} \label{ E_tot_pg}182 \begin{equation} \label{eq:E_tot_pg} 183 183 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 184 184 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv … … 188 188 189 189 Flux form: 190 \begin{subequations} \label{ E_tot_flux}191 \begin{equation} \label{ E_tot_flux_metric}190 \begin{subequations} \label{eq:E_tot_flux} 191 \begin{equation} \label{eq:E_tot_flux_metric} 192 192 \int\limits_D \textbf{U}_h \cdot \text {COR} \; dv = 0 \\ 193 193 \end{equation} 194 \begin{equation} \label{ E_tot_flux_adv}194 \begin{equation} \label{eq:E_tot_flux_adv} 195 195 \int\limits_D \textbf{U}_h \cdot \text{ADV} \;dv 196 196 + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 \\ 197 197 \end{equation} 198 \begin{equation} \label{ E_tot_pg}198 \begin{equation} \label{eq:E_tot_pg} 199 199 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 200 200 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv … … 207 207 208 208 209 \ eqref{E_tot_pg} is the balance between the conversion KE to PE and PE to KE.210 Indeed the left hand side of \ eqref{E_tot_pg} can be transformed as follows:209 \autoref{eq:E_tot_pg} is the balance between the conversion KE to PE and PE to KE. 210 Indeed the left hand side of \autoref{eq:E_tot_pg} can be transformed as follows: 211 211 \begin{flalign*} 212 212 \partial_t \left( \int\limits_D { \rho \, g \, z \;dv} \right) … … 221 221 \end{flalign*} 222 222 where the last equality is obtained by noting that the brackets is exactly the expression of $w$, 223 the vertical velocity referenced to the fixe $z$-coordinate system (see \ eqref{Apdx_A_w_s}).223 the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{apdx:A_w_s}). 224 224 225 The left hand side of \ eqref{E_tot_pg} can be transformed as follows:225 The left hand side of \autoref{eq:E_tot_pg} can be transformed as follows: 226 226 \begin{flalign*} 227 227 - \int\limits_D \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv … … 325 325 % ================================================================ 326 326 \section{Discrete total energy conservation: vector invariant form} 327 \label{ Apdx_C.1}327 \label{sec:C.1} 328 328 329 329 % ------------------------------------------------------------------------------------------------------------- … … 331 331 % ------------------------------------------------------------------------------------------------------------- 332 332 \subsection{Total energy conservation} 333 \label{ Apdx_C_KE+PE}334 335 The discrete form of the total energy conservation, \ eqref{Eq_Tot_Energy}, is given by:333 \label{subsec:C_KE+PE} 334 335 The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: 336 336 \begin{flalign*} 337 337 \partial_t \left( \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v + \rho \, g \, z_t \,b_t \biggr\} \right) &=0 \\ 338 338 \end{flalign*} 339 339 which in vector invariant forms, it leads to: 340 \begin{equation} \label{ KE+PE_vect_discrete} \begin{split}340 \begin{equation} \label{eq:KE+PE_vect_discrete} \begin{split} 341 341 \sum\limits_{i,j,k} \biggl\{ u\, \partial_t u \;b_u 342 342 + v\, \partial_t v \;b_v \biggr\} … … 348 348 349 349 Substituting the discrete expression of the time derivative of the velocity either in vector invariant, 350 leads to the discrete equivalent of the four equations \ eqref{E_tot_flux}.350 leads to the discrete equivalent of the four equations \autoref{eq:E_tot_flux}. 351 351 352 352 % ------------------------------------------------------------------------------------------------------------- … … 354 354 % ------------------------------------------------------------------------------------------------------------- 355 355 \subsection{Vorticity term (coriolis + vorticity part of the advection)} 356 \label{ Apdx_C_vor}356 \label{subsec:C_vor} 357 357 358 358 Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$), or … … 364 364 % ------------------------------------------------------------------------------------------------------------- 365 365 \subsubsection{Vorticity term with ENE scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})} 366 \label{ Apdx_C_vorENE}366 \label{subsec:C_vorENE} 367 367 368 368 For the ENE scheme, the two components of the vorticity term are given by : … … 401 401 % ------------------------------------------------------------------------------------------------------------- 402 402 \subsubsection{Vorticity term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 403 \label{ Apdx_C_vorEEN}403 \label{subsec:C_vorEEN} 404 404 405 405 With the EEN scheme, the vorticity terms are represented as: 406 \begin{equation} \label{ Eq_dynvor_een}406 \begin{equation} \label{eq:dynvor_een} 407 407 \left\{ { \begin{aligned} 408 408 +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} … … 415 415 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 416 416 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 417 \begin{equation} \label{ Q_triads}417 \begin{equation} \label{eq:Q_triads} 418 418 _i^j \mathbb{Q}^{i_p}_{j_p} 419 419 = \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right) … … 471 471 % ------------------------------------------------------------------------------------------------------------- 472 472 \subsubsection{Gradient of kinetic energy / Vertical advection} 473 \label{ Apdx_C_zad}473 \label{subsec:C_zad} 474 474 475 475 The change of Kinetic Energy (KE) due to the vertical advection is exactly … … 480 480 + \frac{1}{2} \int_D { \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv } \\ 481 481 \end{equation*} 482 Indeed, using successively \ eqref{DOM_di_adj} ($i.e.$ the skew symmetry482 Indeed, using successively \autoref{eq:DOM_di_adj} ($i.e.$ the skew symmetry 483 483 property of the $\delta$ operator) and the continuity equation, then 484 \ eqref{DOM_di_adj} again, then the commutativity of operators485 $\overline {\,\cdot \,}$ and $\delta$, and finally \ eqref{DOM_mi_adj}484 \autoref{eq:DOM_di_adj} again, then the commutativity of operators 485 $\overline {\,\cdot \,}$ and $\delta$, and finally \autoref{eq:DOM_mi_adj} 486 486 ($i.e.$ the symmetry property of the $\overline {\,\cdot \,}$ operator) 487 487 applied in the horizontal and vertical directions, it becomes: … … 536 536 % 537 537 \intertext{The first term provides the discrete expression for the vertical advection of momentum (ZAD), 538 while the second term corresponds exactly to \ eqref{KE+PE_vect_discrete}, therefore:}538 while the second term corresponds exactly to \autoref{eq:KE+PE_vect_discrete}, therefore:} 539 539 \equiv& \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 540 540 + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t (e_3) \;dv } &&&\\ … … 568 568 \end{flalign*} 569 569 which is (over-)satified by defining the vertical scale factor as follows: 570 \begin{flalign} \label{e 3u-e3v}570 \begin{flalign} \label{eq:e3u-e3v} 571 571 e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2} \\ 572 572 e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2} … … 580 580 % ------------------------------------------------------------------------------------------------------------- 581 581 \subsection{Pressure gradient term} 582 \label{ Apdx_C.1.4}582 \label{subsec:C.1.4} 583 583 584 584 \gmcomment{ 585 585 A pressure gradient has no contribution to the evolution of the vorticity as the 586 586 curl of a gradient is zero. In the $z$-coordinate, this property is satisfied locally 587 on a C-grid with 2nd order finite differences (property \ eqref{Eq_DOM_curl_grad}).587 on a C-grid with 2nd order finite differences (property \autoref{eq:DOM_curl_grad}). 588 588 } 589 589 … … 611 611 % 612 612 \allowdisplaybreaks 613 \intertext{Using successively \eqref{DOM_di_adj}, $i.e.$ the skew symmetry property of614 the $\delta$ operator, \ eqref{Eq_wzv}, the continuity equation, \eqref{Eq_dynhpg_sco},613 \intertext{Using successively \autoref{eq:DOM_di_adj}, $i.e.$ the skew symmetry property of 614 the $\delta$ operator, \autoref{eq:wzv}, the continuity equation, \autoref{dynhpg_sco}, 615 615 the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $, 616 616 which comes from the definition of $z_t$, it becomes: } … … 657 657 % 658 658 \end{flalign*} 659 The first term is exactly the first term of the right-hand-side of \ eqref{KE+PE_vect_discrete}.659 The first term is exactly the first term of the right-hand-side of \autoref{eq:KE+PE_vect_discrete}. 660 660 It remains to demonstrate that the last term, which is obviously a discrete analogue of 661 $\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to the last term of \ eqref{KE+PE_vect_discrete}.661 $\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to the last term of \autoref{eq:KE+PE_vect_discrete}. 662 662 In other words, the following property must be satisfied: 663 663 \begin{flalign*} … … 733 733 % ================================================================ 734 734 \section{Discrete total energy conservation: flux form} 735 \label{ Apdx_C.1}735 \label{sec:C.1} 736 736 737 737 % ------------------------------------------------------------------------------------------------------------- … … 739 739 % ------------------------------------------------------------------------------------------------------------- 740 740 \subsection{Total energy conservation} 741 \label{ Apdx_C_KE+PE}742 743 The discrete form of the total energy conservation, \ eqref{Eq_Tot_Energy}, is given by:741 \label{subsec:C_KE+PE} 742 743 The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: 744 744 \begin{flalign*} 745 745 \partial_t \left( \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v + \rho \, g \, z_t \,b_t \biggr\} \right) &=0 \\ … … 763 763 % ------------------------------------------------------------------------------------------------------------- 764 764 \subsection{Coriolis and advection terms: flux form} 765 \label{ Apdx_C.1.3}765 \label{subsec:C.1.3} 766 766 767 767 % ------------------------------------------------------------------------------------------------------------- … … 769 769 % ------------------------------------------------------------------------------------------------------------- 770 770 \subsubsection{Coriolis plus ``metric'' term} 771 \label{ Apdx_C.1.3.1}771 \label{subsec:C.1.3.1} 772 772 773 773 In flux from the vorticity term reduces to a Coriolis term in which the Coriolis … … 783 783 Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. 784 784 It therefore conserves the total KE. The derivation is the same as for the 785 vorticity term in the vector invariant form (\ S\ref{Apdx_C_vor}).785 vorticity term in the vector invariant form (\autoref{subsec:C_vor}). 786 786 787 787 % ------------------------------------------------------------------------------------------------------------- … … 789 789 % ------------------------------------------------------------------------------------------------------------- 790 790 \subsubsection{Flux form advection} 791 \label{ Apdx_C.1.3.2}791 \label{subsec:C.1.3.2} 792 792 793 793 The flux form operator of the momentum advection is evaluated using a … … 797 797 the horizontal kinetic energy, that is : 798 798 799 \begin{equation} \label{ Apdx_C_ADV_KE_flux}799 \begin{equation} \label{eq:C_ADV_KE_flux} 800 800 - \int_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} 801 801 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ … … 856 856 which is the discrete form of 857 857 $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv $. 858 \ eqref{Apdx_C_ADV_KE_flux} is thus satisfied.858 \autoref{eq:C_ADV_KE_flux} is thus satisfied. 859 859 860 860 … … 877 877 % ================================================================ 878 878 \section{Discrete enstrophy conservation} 879 \label{ Apdx_C.1}879 \label{sec:C.1} 880 880 881 881 … … 884 884 % ------------------------------------------------------------------------------------------------------------- 885 885 \subsubsection{Vorticity term with ENS scheme (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})} 886 \label{ Apdx_C_vorENS}886 \label{subsec:C_vorENS} 887 887 888 888 In the ENS scheme, the vorticity term is descretized as follows: 889 \begin{equation} \label{ Eq_dynvor_ens}889 \begin{equation} \label{eq:dynvor_ens} 890 890 \left\{ \begin{aligned} 891 891 +\frac{1}{e_{1u}} & \overline{q}^{\,i} & {\overline{ \overline{\left( e_{1v}\,e_{3v}\; v \right) } } }^{\,i, j+1/2} \\ … … 896 896 The scheme does not allow but the conservation of the total kinetic energy but the conservation 897 897 of $q^2$, the potential enstrophy for a horizontally non-divergent flow ($i.e.$ when $\chi$=$0$). 898 Indeed, using the symmetry or skew symmetry properties of the operators ( Eqs \eqref{DOM_mi_adj}899 and \ eqref{DOM_di_adj}), it can be shown that:900 \begin{equation} \label{ Apdx_C_1.1}898 Indeed, using the symmetry or skew symmetry properties of the operators ( \autoref{eq:DOM_mi_adj} 899 and \autoref{eq:DOM_di_adj}), it can be shown that: 900 \begin{equation} \label{eq:C_1.1} 901 901 \int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0 902 902 \end{equation} 903 903 where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. Indeed, using 904 \ eqref{Eq_dynvor_ens}, the discrete form of the right hand side of \eqref{Apdx_C_1.1}904 \autoref{eq:dynvor_ens}, the discrete form of the right hand side of \autoref{eq:C_1.1} 905 905 can be transformed as follow: 906 906 \begin{flalign*} … … 944 944 % ------------------------------------------------------------------------------------------------------------- 945 945 \subsubsection{Vorticity Term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 946 \label{ Apdx_C_vorEEN}946 \label{subsec:C_vorEEN} 947 947 948 948 With the EEN scheme, the vorticity terms are represented as: 949 \begin{equation} \label{ Eq_dynvor_een}949 \begin{equation} \label{eq:dynvor_een} 950 950 \left\{ { \begin{aligned} 951 951 +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} … … 958 958 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 959 959 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 960 \begin{equation} \label{ Q_triads}960 \begin{equation} \label{eq:Q_triads} 961 961 _i^j \mathbb{Q}^{i_p}_{j_p} 962 962 = \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right) … … 968 968 Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $, 969 969 similar manipulation can be done for the 3 others. The discrete form of the right hand 970 side of \ eqref{Apdx_C_1.1} applied to this triad only can be transformed as follow:970 side of \autoref{eq:C_1.1} applied to this triad only can be transformed as follow: 971 971 972 972 \begin{flalign*} … … 1017 1017 % ================================================================ 1018 1018 \section{Conservation properties on tracers} 1019 \label{ Apdx_C.2}1019 \label{sec:C.2} 1020 1020 1021 1021 … … 1033 1033 % ------------------------------------------------------------------------------------------------------------- 1034 1034 \subsection{Advection term} 1035 \label{ Apdx_C.2.1}1035 \label{subsec:C.2.1} 1036 1036 1037 1037 conservation of a tracer, $T$: … … 1100 1100 % ================================================================ 1101 1101 \section{Conservation properties on lateral momentum physics} 1102 \label{ Apdx_dynldf_properties}1102 \label{sec:dynldf_properties} 1103 1103 1104 1104 … … 1114 1114 1115 1115 These properties of the horizontal diffusion operator are a direct consequence 1116 of properties \ eqref{Eq_DOM_curl_grad} and \eqref{Eq_DOM_div_curl}.1116 of properties \autoref{eq:DOM_curl_grad} and \autoref{eq:DOM_div_curl}. 1117 1117 When the vertical curl of the horizontal diffusion of momentum (discrete sense) 1118 1118 is taken, the term associated with the horizontal gradient of the divergence is … … 1123 1123 % ------------------------------------------------------------------------------------------------------------- 1124 1124 \subsection{Conservation of potential vorticity} 1125 \label{ Apdx_C.3.1}1125 \label{subsec:C.3.1} 1126 1126 1127 1127 The lateral momentum diffusion term conserves the potential vorticity : … … 1143 1143 \right\} \\ 1144 1144 % 1145 \intertext{Using \ eqref{DOM_di_adj}, it follows:}1145 \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} 1146 1146 % 1147 1147 \equiv& \sum\limits_{i,j,k} … … 1157 1157 % ------------------------------------------------------------------------------------------------------------- 1158 1158 \subsection{Dissipation of horizontal kinetic energy} 1159 \label{ Apdx_C.3.2}1159 \label{subsec:C.3.2} 1160 1160 1161 1161 The lateral momentum diffusion term dissipates the horizontal kinetic energy: … … 1209 1209 % ------------------------------------------------------------------------------------------------------------- 1210 1210 \subsection{Dissipation of enstrophy} 1211 \label{ Apdx_C.3.3}1211 \label{subsec:C.3.3} 1212 1212 1213 1213 The lateral momentum diffusion term dissipates the enstrophy when the eddy … … 1223 1223 + \delta_{j+1/2} \left[ \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right] \right\} &&&\\ 1224 1224 % 1225 \intertext{Using \ eqref{DOM_di_adj}, it follows:}1225 \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} 1226 1226 % 1227 1227 &\quad \equiv - A^{\,lm} \sum\limits_{i,j,k} … … 1234 1234 % ------------------------------------------------------------------------------------------------------------- 1235 1235 \subsection{Conservation of horizontal divergence} 1236 \label{ Apdx_C.3.4}1236 \label{subsec:C.3.4} 1237 1237 1238 1238 When the horizontal divergence of the horizontal diffusion of momentum 1239 1239 (discrete sense) is taken, the term associated with the vertical curl of the 1240 vorticity is zero locally, due to \ eqref{Eq_DOM_div_curl}.1240 vorticity is zero locally, due to \autoref{eq:DOM_div_curl}. 1241 1241 The resulting term conserves the $\chi$ and dissipates $\chi^2$ 1242 1242 when the eddy coefficients are horizontally uniform. … … 1251 1251 + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] \right\} \\ 1252 1252 % 1253 \intertext{Using \ eqref{DOM_di_adj}, it follows:}1253 \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} 1254 1254 % 1255 1255 &\equiv \sum\limits_{i,j,k} … … 1263 1263 % ------------------------------------------------------------------------------------------------------------- 1264 1264 \subsection{Dissipation of horizontal divergence variance} 1265 \label{ Apdx_C.3.5}1265 \label{subsec:C.3.5} 1266 1266 1267 1267 \begin{flalign*} … … 1277 1277 \right\} \; e_{1t}\,e_{2t}\,e_{3t} \\ 1278 1278 % 1279 \intertext{Using \ eqref{DOM_di_adj}, it turns out to be:}1279 \intertext{Using \autoref{eq:DOM_di_adj}, it turns out to be:} 1280 1280 % 1281 1281 &\equiv - A^{\,lm} \sum\limits_{i,j,k} … … 1289 1289 % ================================================================ 1290 1290 \section{Conservation properties on vertical momentum physics} 1291 \label{ Apdx_C_4}1291 \label{sec:C_4} 1292 1292 1293 1293 As for the lateral momentum physics, the continuous form of the vertical diffusion … … 1461 1461 % ================================================================ 1462 1462 \section{Conservation properties on tracer physics} 1463 \label{ Apdx_C.5}1463 \label{sec:C.5} 1464 1464 1465 1465 The numerical schemes used for tracer subgridscale physics are written such … … 1473 1473 % ------------------------------------------------------------------------------------------------------------- 1474 1474 \subsection{Conservation of tracers} 1475 \label{ Apdx_C.5.1}1475 \label{subsec:C.5.1} 1476 1476 1477 1477 constraint of conservation of tracers: … … 1507 1507 % ------------------------------------------------------------------------------------------------------------- 1508 1508 \subsection{Dissipation of tracer variance} 1509 \label{ Apdx_C.5.2}1509 \label{subsec:C.5.2} 1510 1510 1511 1511 constraint on the dissipation of tracer variance: -
branches/2017/dev_merge_2017/DOC/tex_sub/annex_D.tex
r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Coding Rules} 7 \label{ Apdx_D}7 \label{apdx:D} 8 8 \minitoc 9 9 … … 47 47 % ================================================================ 48 48 \section{Program structure} 49 \label{ Apdx_D_structure}49 \label{sec:D_structure} 50 50 51 51 Each program begins with a set of headline comments containing : … … 76 76 % ================================================================ 77 77 \section{Coding conventions} 78 \label{ Apdx_D_coding}78 \label{sec:D_coding} 79 79 80 80 - Use of the universal language \textsc{Fortran} 90, and try to avoid obsolescent … … 107 107 % ================================================================ 108 108 \section{Naming conventions} 109 \label{ Apdx_D_naming}109 \label{sec:D_naming} 110 110 111 111 The purpose of the naming conventions is to use prefix letters to classify … … 116 116 117 117 %--------------------------------------------------TABLE-------------------------------------------------- 118 \begin{table}[htbp] \label{ Tab_VarName}118 \begin{table}[htbp] \label{tab:VarName} 119 119 \begin{center} 120 120 \begin{tabular}{|p{45pt}|p{35pt}|p{45pt}|p{40pt}|p{40pt}|p{40pt}|p{40pt}|p{40pt}|} … … 187 187 \hline 188 188 \end{tabular} 189 \label{tab 1}189 \label{tab:tab1} 190 190 \end{center} 191 191 \end{table} … … 201 201 % ================================================================ 202 202 %\section{Program structure} 203 % label{Apdx_D_structure}203 %abel{sec:Apdx_D_structure} 204 204 205 205 %To be done.... -
branches/2017/dev_merge_2017/DOC/tex_sub/annex_E.tex
r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Note on some algorithms} 7 \label{ Apdx_E}7 \label{apdx:E} 8 8 \minitoc 9 9 … … 20 20 % ------------------------------------------------------------------------------------------------------------- 21 21 \section{Upstream Biased Scheme (UBS) (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})} 22 \label{ TRA_adv_ubs}22 \label{sec:TRA_adv_ubs} 23 23 24 24 The UBS advection scheme is an upstream biased third order scheme based on … … 26 26 QUICK scheme (Quadratic Upstream Interpolation for Convective 27 27 Kinematics). For example, in the $i$-direction : 28 \begin{equation} \label{ Eq_tra_adv_ubs2}28 \begin{equation} \label{eq:tra_adv_ubs2} 29 29 \tau _u^{ubs} = \left\{ \begin{aligned} 30 30 & \tau _u^{cen4} + \frac{1}{12} \,\tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ … … 33 33 \end{equation} 34 34 or equivalently, the advective flux is 35 \begin{equation} \label{ Eq_tra_adv_ubs2}35 \begin{equation} \label{eq:tra_adv_ubs2} 36 36 U_{i+1/2} \ \tau _u^{ubs} 37 37 =U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2} … … 61 61 scheme when \np{ln\_traadv\_ubs}\forcode{ = .true.}. 62 62 63 For stability reasons, in \ eqref{Eq_tra_adv_ubs}, the first term which corresponds63 For stability reasons, in \autoref{eq:tra_adv_ubs}, the first term which corresponds 64 64 to a second order centred scheme is evaluated using the \textit{now} velocity 65 65 (centred in time) while the second term which is the diffusive part of the scheme, … … 67 67 by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme. UBS and QUICK 68 68 schemes only differ by one coefficient. Substituting 1/6 with 1/8 in 69 (\ ref{Eq_tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.69 (\autoref{eq:tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. 70 70 This option is not available through a namelist parameter, since the 1/6 71 71 coefficient is hard coded. Nevertheless it is quite easy to make the … … 87 87 eight-order accurate conventional scheme. 88 88 89 NB 3 : It is straight forward to rewrite \ eqref{Eq_tra_adv_ubs} as follows:90 \begin{equation} \label{ Eq_tra_adv_ubs2}89 NB 3 : It is straight forward to rewrite \autoref{eq:tra_adv_ubs} as follows: 90 \begin{equation} \label{eq:tra_adv_ubs2} 91 91 \tau _u^{ubs} = \left\{ \begin{aligned} 92 92 & \tau _u^{cen4} + \frac{1}{12} \tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ … … 95 95 \end{equation} 96 96 or equivalently 97 \begin{equation} \label{ Eq_tra_adv_ubs2}97 \begin{equation} \label{eq:tra_adv_ubs2} 98 98 \begin{split} 99 99 e_{2u} e_{3u}\,u_{i+1/2} \ \tau _u^{ubs} … … 102 102 \end{split} 103 103 \end{equation} 104 \ eqref{Eq_tra_adv_ubs2} has several advantages. First it clearly evidence that104 \autoref{eq:tra_adv_ubs2} has several advantages. First it clearly evidence that 105 105 the UBS scheme is based on the fourth order scheme to which is added an 106 106 upstream biased diffusive term. Second, this emphasises that the $4^{th}$ order 107 107 part have to be evaluated at \emph{now} time step, not only the $2^{th}$ order 108 part as stated above using \ eqref{Eq_tra_adv_ubs}. Third, the diffusive term is108 part as stated above using \autoref{eq:tra_adv_ubs}. Third, the diffusive term is 109 109 in fact a biharmonic operator with a eddy coefficient with is simply proportional 110 110 to the velocity. 111 111 112 112 laplacian diffusion: 113 \begin{equation} \label{ Eq_tra_ldf_lap}113 \begin{equation} \label{eq:tra_ldf_lap} 114 114 \begin{split} 115 115 D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\; e_{3T} } &\left[ {\quad \delta _i … … 124 124 125 125 bilaplacian: 126 \begin{equation} \label{ Eq_tra_ldf_lap}126 \begin{equation} \label{eq:tra_ldf_lap} 127 127 \begin{split} 128 128 D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ … … 136 136 $i.e.$ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$ 137 137 it comes : 138 \begin{equation} \label{ Eq_tra_ldf_lap}138 \begin{equation} \label{eq:tra_ldf_lap} 139 139 \begin{split} 140 140 D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ … … 146 146 \end{equation} 147 147 if the velocity is uniform ($i.e.$ $|u|=cst$) then the diffusive flux is 148 \begin{equation} \label{ Eq_tra_ldf_lap}148 \begin{equation} \label{eq:tra_ldf_lap} 149 149 \begin{split} 150 150 F_u^{lT} = - \frac{1}{12} … … 157 157 beurk.... reverte the logic: starting from the diffusive part of the advective flux it comes: 158 158 159 \begin{equation} \label{ Eq_tra_adv_ubs2}159 \begin{equation} \label{eq:tra_adv_ubs2} 160 160 \begin{split} 161 161 F_u^{lT} … … 166 166 167 167 sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$): 168 \begin{equation} \label{ Eq_tra_adv_ubs2}168 \begin{equation} \label{eq:tra_adv_ubs2} 169 169 \begin{split} 170 170 F_u^{lT} … … 175 175 176 176 sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$ 177 \begin{equation} \label{ Eq_tra_adv_ubs2}177 \begin{equation} \label{eq:tra_adv_ubs2} 178 178 \begin{split} 179 179 F_u^{lT} … … 189 189 % ------------------------------------------------------------------------------------------------------------- 190 190 \section{Leapfrog energetic} 191 \label{ LF}191 \label{sec:LF} 192 192 193 193 We adopt the following semi-discrete notation for time derivative. Given the values of a variable $q$ at successive time step, the time derivation and averaging operators at the mid time step are: 194 \begin{subequations} \label{ dt_mt}194 \begin{subequations} \label{eq:dt_mt} 195 195 \begin{align} 196 196 \delta _{t+\rdt/2} [q] &= \ \ \, q^{t+\rdt} - q^{t} \\ … … 202 202 , respectively. 203 203 204 The Leap-frog time stepping given by \ eqref{Eq_DOM_nxt} can be defined as:205 \begin{equation} \label{ LF}204 The Leap-frog time stepping given by \autoref{eq:DOM_nxt} can be defined as: 205 \begin{equation} \label{eq:LF} 206 206 \frac{\partial q}{\partial t} 207 207 \equiv \frac{1}{\rdt} \overline{ \delta _{t+\rdt/2}[q]}^{\,t} 208 208 = \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} 209 209 \end{equation} 210 Note that \ eqref{LF} shows that the leapfrog time step is $\rdt$, not $2\rdt$210 Note that \autoref{chap:LF} shows that the leapfrog time step is $\rdt$, not $2\rdt$ 211 211 as it can be found sometime in literature. 212 212 The leap-Frog time stepping is a second order centered scheme. As such it respects 213 213 the quadratic invariant in integral forms, $i.e.$ the following continuous property, 214 \begin{equation} \label{ Energy}214 \begin{equation} \label{eq:Energy} 215 215 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 216 216 =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} … … 252 252 scheme, but is formulated within the \NEMO framework ($i.e.$ using scale 253 253 factors rather than grid-size and having a position of $T$-points that is not 254 necessary in the middle of vertical velocity points, see Fig.~\ref{Fig_zgr_e3}).255 256 In the formulation \ eqref{Eq_tra_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO,254 necessary in the middle of vertical velocity points, see \autoref{fig:zgr_e3}). 255 256 In the formulation \autoref{eq:tra_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO, 257 257 the off-diagonal terms of the small angle diffusion tensor contain several double 258 258 spatial averages of a gradient, for example $\overline{\overline{\delta_k \cdot}}^{\,i,k}$. … … 263 263 In other word, the operator applied to a tracer does not warranties the decrease of 264 264 its global average variance. To circumvent this, we have introduced a smoothing of 265 the slopes of the iso-neutral surfaces (see \ S\ref{LDF}). Nevertheless, this technique265 the slopes of the iso-neutral surfaces (see \autoref{chap:LDF}). Nevertheless, this technique 266 266 works fine for $T$ and $S$ as they are active tracers ($i.e.$ they enter the computation 267 267 of density), but it does not work for a passive tracer. \citep{Griffies_al_JPO98} introduce … … 270 270 with a derivative in the same direction by considering triads. For example in the 271 271 (\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows: 272 \begin{equation} \label{ Gf_triads}272 \begin{equation} \label{eq:Gf_triads} 273 273 _i^k \mathbb{T}_{i_p}^{k_p} (T) 274 274 = \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \left( … … 282 282 $A_i^k$ is the lateral eddy diffusivity coefficient defined at $T$-point, 283 283 and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad : 284 \begin{equation} \label{ Gf_slopes}284 \begin{equation} \label{eq:Gf_slopes} 285 285 _i^k \mathbb{R}_{i_p}^{k_p} 286 286 =\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac … … 288 288 {\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] } 289 289 \end{equation} 290 Note that in \ eqref{Gf_slopes} we use the ratio $\alpha / \beta$ instead of290 Note that in \autoref{eq:Gf_slopes} we use the ratio $\alpha / \beta$ instead of 291 291 multiplying the temperature derivative by $\alpha$ and the salinity derivative 292 292 by $\beta$. This is more efficient as the ratio $\alpha / \beta$ can to be 293 293 evaluated directly. 294 294 295 Note that in \ eqref{Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of295 Note that in \autoref{eq:Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of 296 296 ${b_{uw}}_{\,i+i_p}^{\,k+k_p}$. This choice has been motivated by the decrease 297 297 of tracer variance and the presence of partial cell at the ocean bottom 298 (see Appendix~\ref{Apdx_Gf_operator}).298 (see \autoref{apdx:Gf_operator}). 299 299 300 300 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 301 \begin{figure}[!ht] \label{Fig_ISO_triad} 302 \begin{center} 301 \begin{figure}[!ht] \begin{center} 303 302 \includegraphics[width=0.70\textwidth]{Fig_ISO_triad} 304 \caption{ \protect\label{ Fig_ISO_triad}303 \caption{ \protect\label{fig:ISO_triad} 305 304 Triads used in the Griffies's like iso-neutral diffision scheme for 306 305 $u$-component (upper panel) and $w$-component (lower panel).} … … 311 310 The four iso-neutral fluxes associated with the triads are defined at $T$-point. 312 311 They take the following expression : 313 \begin{flalign} \label{ Gf_fluxes}312 \begin{flalign} \label{eq:Gf_fluxes} 314 313 \begin{split} 315 314 {_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) … … 322 321 323 322 The resulting iso-neutral fluxes at $u$- and $w$-points are then given by the 324 sum of the fluxes that cross the $u$- and $w$-face ( Fig.~\ref{Fig_ISO_triad}):325 \begin{flalign} \label{ Eq_iso_flux}323 sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:ISO_triad}): 324 \begin{flalign} \label{eq:iso_flux} 326 325 \textbf{F}_{iso}(T) 327 326 &\equiv \sum_{\substack{i_p,\,k_p}} … … 353 352 resulting in a iso-neutral diffusion tendency on temperature given by the divergence 354 353 of the sum of all the four triad fluxes : 355 \begin{equation} \label{ Gf_operator}354 \begin{equation} \label{eq:Gf_operator} 356 355 D_l^T = \frac{1}{b_T} \sum_{\substack{i_p,\,k_p}} \left\{ 357 356 \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] … … 365 364 \item[$\bullet$ horizontal diffusion] The discretization of the diffusion operator 366 365 recovers the traditional five-point Laplacian in the limit of flat iso-neutral direction : 367 \begin{equation} \label{ Gf_property1a}366 \begin{equation} \label{eq:Gf_property1a} 368 367 D_l^T = \frac{1}{b_T} \ \delta_{i} 369 368 \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] … … 388 387 \item[$\bullet$ pure iso-neutral operator] The iso-neutral flux of locally referenced 389 388 potential density is zero, $i.e.$ 390 \begin{align} \label{ Gf_property2}389 \begin{align} \label{eq:Gf_property2} 391 390 \begin{matrix} 392 391 &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)} … … 398 397 \end{matrix} 399 398 \end{align} 400 This result is trivially obtained using the \ eqref{Gf_triads} applied to $T$ and $S$401 and the definition of the triads' slopes \ eqref{Gf_slopes}.399 This result is trivially obtained using the \autoref{eq:Gf_triads} applied to $T$ and $S$ 400 and the definition of the triads' slopes \autoref{eq:Gf_slopes}. 402 401 403 402 \item[$\bullet$ conservation of tracer] The iso-neutral diffusion term conserve the 404 403 total tracer content, $i.e.$ 405 \begin{equation} \label{ Gf_property1}404 \begin{equation} \label{eq:Gf_property1} 406 405 \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 407 406 \end{equation} … … 411 410 \item[$\bullet$ decrease of tracer variance] The iso-neutral diffusion term does 412 411 not increase the total tracer variance, $i.e.$ 413 \begin{equation} \label{ Gf_property1}412 \begin{equation} \label{eq:Gf_property1} 414 413 \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 415 414 \end{equation} 416 The property is demonstrated in the Appendix~\ref{Apdx_Gf_operator}. It is a415 The property is demonstrated in the \autoref{apdx:Gf_operator}. It is a 417 416 key property for a diffusion term. It means that the operator is also a dissipation 418 417 term, $i.e.$ it is a sink term for the square of the quantity on which it is applied. … … 422 421 \item[$\bullet$ self-adjoint operator] The iso-neutral diffusion operator is self-adjoint, 423 422 $i.e.$ 424 \begin{equation} \label{ Gf_property1}423 \begin{equation} \label{eq:Gf_property1} 425 424 \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 426 425 \end{equation} … … 428 427 operator. We just have to apply the same routine. This properties can be demonstrated 429 428 quite easily in a similar way the "non increase of tracer variance" property has been 430 proved (see Appendix~\ref{Apdx_Gf_operator}).429 proved (see \autoref{apdx:Gf_operator}). 431 430 \end{description} 432 431 … … 442 441 eddy induced velocity, the formulation of which depends on the slopes of iso- 443 442 neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used 444 here are referenced to the geopotential surfaces, $i.e.$ \ eqref{Eq_ldfslp_geo}445 is used in $z$-coordinate, and the sum \ eqref{Eq_ldfslp_geo}446 + \ eqref{Eq_ldfslp_iso} in $z^*$ or $s$-coordinates.443 here are referenced to the geopotential surfaces, $i.e.$ \autoref{eq:ldfslp_geo} 444 is used in $z$-coordinate, and the sum \autoref{eq:ldfslp_geo} 445 + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates. 447 446 448 447 The eddy induced velocity is given by: 449 \begin{equation} \label{ Eq_eiv_v}448 \begin{equation} \label{eq:eiv_v} 450 449 \begin{split} 451 450 u^* & = - \frac{1}{e_2\,e_{3}} \;\partial_k \left( e_2 \, A_e \; r_i \right) … … 467 466 A traditional way to implement this additional advection is to add it to the eulerian 468 467 velocity prior to compute the tracer advection. This allows us to take advantage of 469 all the advection schemes offered for the tracers (see \ S\ref{TRA_adv}) and not just468 all the advection schemes offered for the tracers (see \autoref{sec:TRA_adv}) and not just 470 469 a $2^{nd}$ order advection scheme. This is particularly useful for passive tracers 471 470 where \emph{positivity} of the advection scheme is of paramount importance. 472 % give here the expression using the triads. It is different from the one given in \ eqref{Eq_ldfeiv}471 % give here the expression using the triads. It is different from the one given in \autoref{eq:ldfeiv} 473 472 % see just below a copy of this equation: 474 %\begin{equation} \label{ Eq_ldfeiv}473 %\begin{equation} \label{eq:ldfeiv} 475 474 %\begin{split} 476 475 % u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ … … 479 478 %\end{split} 480 479 %\end{equation} 481 \begin{equation} \label{ Eq_eiv_vd}480 \begin{equation} \label{eq:eiv_vd} 482 481 \textbf{F}_{eiv}^T \equiv \left( \begin{aligned} 483 482 \sum_{\substack{i_p,\,k_p}} & … … 491 490 \end{equation} 492 491 493 \ ref{Griffies_JPO98} introduces another way to implement the eddy induced advection,492 \citep{Griffies_JPO98} introduces another way to implement the eddy induced advection, 494 493 the so-called skew form. It is based on a transformation of the advective fluxes 495 494 using the non-divergent nature of the eddy induced velocity. … … 522 521 and since the eddy induces velocity field is no-divergent, we end up with the skew 523 522 form of the eddy induced advective fluxes: 524 \begin{equation} \label{ Eq_eiv_skew_continuous}523 \begin{equation} \label{eq:eiv_skew_continuous} 525 524 \textbf{F}_{eiv}^T = \begin{pmatrix} 526 525 {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ … … 529 528 \end{equation} 530 529 The tendency associated with eddy induced velocity is then simply the divergence 531 of the \ eqref{Eq_eiv_skew_continuous} fluxes. It naturally conserves the tracer530 of the \autoref{eq:eiv_skew_continuous} fluxes. It naturally conserves the tracer 532 531 content, as it is expressed in flux form and, as the advective form, it preserve the 533 tracer variance. Another interesting property of \ eqref{Eq_eiv_skew_continuous}532 tracer variance. Another interesting property of \autoref{eq:eiv_skew_continuous} 534 533 form is that when $A=A_e$, a simplification occurs in the sum of the iso-neutral 535 534 diffusion and eddy induced velocity terms: 536 \begin{flalign} \label{ Eq_eiv_skew+eiv_continuous}535 \begin{flalign} \label{eq:eiv_skew+eiv_continuous} 537 536 \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &= 538 537 \begin{pmatrix} … … 554 553 has been used to reduce the computational time \citep{Griffies_JPO98}, but it is 555 554 not of practical use as usually $A \neq A_e$. Nevertheless this property can be used to 556 choose a discret form of \ eqref{Eq_eiv_skew_continuous} which is consistent with the557 iso-neutral operator \ eqref{Gf_operator}. Using the slopes \eqref{Gf_slopes}555 choose a discret form of \autoref{eq:eiv_skew_continuous} which is consistent with the 556 iso-neutral operator \autoref{eq:Gf_operator}. Using the slopes \autoref{eq:Gf_slopes} 558 557 and defining $A_e$ at $T$-point($i.e.$ as $A$, the eddy diffusivity coefficient), 559 558 the resulting discret form is given by: 560 \begin{equation} \label{ Eq_eiv_skew}559 \begin{equation} \label{eq:eiv_skew} 561 560 \textbf{F}_{eiv}^T \equiv \frac{1}{4} \left( \begin{aligned} 562 561 \sum_{\substack{i_p,\,k_p}} & … … 569 568 \end{aligned} \right) 570 569 \end{equation} 571 Note that \ eqref{Eq_eiv_skew} is valid in $z$-coordinate with or without partial cells.570 Note that \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells. 572 571 In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces 573 572 must be added to $\mathbb{R}$ for the discret form to be exact. … … 575 574 Such a choice of discretisation is consistent with the iso-neutral operator as it uses the 576 575 same definition for the slopes. It also ensures the conservation of the tracer variance 577 (see Appendix \ ref{Apdx_eiv_skew}), $i.e.$ it does not include a diffusive component576 (see Appendix \autoref{apdx:eiv_skew}), $i.e.$ it does not include a diffusive component 578 577 but is a "pure" advection term. 579 578 … … 586 585 % ================================================================ 587 586 \subsection{Discrete invariants of the iso-neutral diffrusion} 588 \label{ Apdx_Gf_operator}587 \label{subsec:Gf_operator} 589 588 590 589 Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane. … … 596 595 \int_D D_l^T \; T \;dv \leq 0 597 596 \end{align*} 598 The discrete form of its left hand side is obtained using \ eqref{Eq_iso_flux}597 The discrete form of its left hand side is obtained using \autoref{eq:iso_flux} 599 598 600 599 \begin{align*} … … 673 672 % 674 673 \allowdisplaybreaks 675 \intertext{Then outing in factor the triad in each of the four terms of the summation and substituting the triads by their expression given in \ eqref{Gf_triads}. It becomes: }674 \intertext{Then outing in factor the triad in each of the four terms of the summation and substituting the triads by their expression given in \autoref{eq:Gf_triads}. It becomes: } 676 675 % 677 676 &\equiv -\sum_{i,k} … … 739 738 % ================================================================ 740 739 \subsection{Discrete invariants of the skew flux formulation} 741 \label{ Apdx_eiv_skew}740 \label{subsec:eiv_skew} 742 741 743 742 … … 750 749 \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv \equiv 0 751 750 \end{align*} 752 The discrete form of its left hand side is obtained using \ eqref{Eq_eiv_skew}751 The discrete form of its left hand side is obtained using \autoref{eq:eiv_skew} 753 752 \begin{align*} 754 753 \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; -
branches/2017/dev_merge_2017/DOC/tex_sub/annex_iso.tex
r9393 r9407 4 4 % Iso-neutral diffusion : 5 5 % ================================================================ 6 \chapter{Iso-neutral diffusion and eddy advection using triads} 7 \label{sec:triad} 6 \chapter[Iso-Neutral Diffusion and Eddy Advection using Triads] 7 {\texorpdfstring{Iso-Neutral Diffusion and\\ Eddy Advection using Triads}{Iso-Neutral Diffusion and Eddy Advection using Triads}} 8 \label{apdx:triad} 8 9 \minitoc 9 10 \pagebreak … … 18 19 of iso-neutral diffusion and the eddy-induced advective skew (GM) fluxes. 19 20 If the namelist logical \np{ln\_traldf\_iso} is set true, 20 the filtered version of Cox's original scheme (the Standard scheme) is employed (\ S\ref{LDF_slp}).21 the filtered version of Cox's original scheme (the Standard scheme) is employed (\autoref{sec:LDF_slp}). 21 22 In the present implementation of the Griffies scheme, 22 23 the advective skew fluxes are implemented even if \np{ln\_traldf\_eiv} is false. 23 24 24 25 Values of iso-neutral diffusivity and GM coefficient are set as 25 described in \ S\ref{LDF_coef}. Note that when GM fluxes are used,26 described in \autoref{sec:LDF_coef}. Note that when GM fluxes are used, 26 27 the eddy-advective (GM) velocities are output for diagnostic purposes using xIOS, 27 28 even though the eddy advection is accomplished by means of the skew fluxes. … … 30 31 The options specific to the Griffies scheme include: 31 32 \begin{description}[font=\normalfont] 32 \item[\np{ln\_triad\_iso}] See \ S\ref{sec:triad:taper}. If this is set false (the default), then33 \item[\np{ln\_triad\_iso}] See \autoref{sec:taper}. If this is set false (the default), then 33 34 `iso-neutral' mixing is accomplished within the surface mixed-layer 34 35 along slopes linearly decreasing with depth from the value immediately below 35 the mixed-layer to zero (flat) at the surface (\ S\ref{sec:triad:lintaper}).36 This is the same treatment as used in the default implementation \ S\ref{LDF_slp_iso}; Fig.~\ref{Fig_eiv_slp}.36 the mixed-layer to zero (flat) at the surface (\autoref{sec:lintaper}). 37 This is the same treatment as used in the default implementation \autoref{subsec:LDF_slp_iso}; \autoref{fig:eiv_slp}. 37 38 Where \np{ln\_triad\_iso} is set true, the vertical skew flux is further reduced 38 39 to ensure no vertical buoyancy flux, giving an almost pure 39 40 horizontal diffusive tracer flux within the mixed layer. This is similar to 40 the tapering suggested by \citet{Gerdes1991}. See \ S\ref{sec:triad:Gerdes-taper}41 \item[\np{ln\_botmix\_triad}] See \ S\ref{sec:triad:iso_bdry}.41 the tapering suggested by \citet{Gerdes1991}. See \autoref{subsec:Gerdes-taper} 42 \item[\np{ln\_botmix\_triad}] See \autoref{sec:iso_bdry}. 42 43 If this is set false (the default) then the lateral diffusive fluxes 43 44 associated with triads partly masked by topography are neglected. … … 53 54 54 55 \section{Triad formulation of iso-neutral diffusion} 55 \label{sec: triad:iso}56 \label{sec:iso} 56 57 We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98}, 57 58 but formulated within the \NEMO framework, using scale factors rather than grid-sizes. … … 60 61 The iso-neutral second order tracer diffusive operator for small 61 62 angles between iso-neutral surfaces and geopotentials is given by 62 \ eqref{Eq_PE_iso_tensor}:63 \begin{subequations} \label{eq: triad:PE_iso_tensor}63 \autoref{eq:PE_iso_tensor}: 64 \begin{subequations} \label{eq:PE_iso_tensor} 64 65 \begin{equation} 65 66 D^{lT}=-\Div\vect{f}^{lT}\equiv … … 72 73 \end{equation} 73 74 \begin{equation} 74 \label{eq: triad:PE_iso_tensor:c}75 \label{eq:PE_iso_tensor:c} 75 76 \mbox{with}\quad \;\;\Re = 76 77 \begin{pmatrix} … … 92 93 % {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ 93 94 % \end{array} }} \right) 94 Here \ eqref{Eq_PE_iso_slopes}95 Here \autoref{eq:PE_iso_slopes} 95 96 \begin{align*} 96 97 r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i} … … 108 109 space; we write 109 110 \begin{equation} 110 \label{eq: triad:Fijk}111 \label{eq:Fijk} 111 112 \vect{F}_{\mathrm{iso}}=\left(f_1^{lT}e_2e_3, f_2^{lT}e_1e_3, f_3^{lT}e_1e_2\right). 112 113 \end{equation} … … 117 118 118 119 The off-diagonal terms of the small angle diffusion tensor 119 \ eqref{Eq_PE_iso_tensor}, \eqref{eq:triad:PE_iso_tensor:c} produce skew-fluxes along the120 \autoref{eq:PE_iso_tensor}, \autoref{eq:PE_iso_tensor:c} produce skew-fluxes along the 120 121 $i$- and $j$-directions resulting from the vertical tracer gradient: 121 122 \begin{align} 122 \label{eq: triad:i13c}123 \label{eq:i13c} 123 124 f_{13}=&+\Alt r_1\frac{1}{e_3}\frac{\partial T}{\partial k},\qquad f_{23}=+\Alt r_2\frac{1}{e_3}\frac{\partial T}{\partial k}\\ 124 125 \intertext{and in the k-direction resulting from the lateral tracer gradients} 125 \label{eq: triad:i31c}126 \label{eq:i31c} 126 127 f_{31}+f_{32}=& \Alt r_1\frac{1}{e_1}\frac{\partial T}{\partial i}+\Alt r_2\frac{1}{e_1}\frac{\partial T}{\partial i} 127 128 \end{align} … … 130 131 component of the small angle diffusion tensor is 131 132 \begin{equation} 132 \label{eq: triad:i33c}133 \label{eq:i33c} 133 134 f_{33}=-\Alt(r_1^2 +r_2^2) \frac{1}{e_3}\frac{\partial T}{\partial k}. 134 135 \end{equation} … … 141 142 142 143 There is no natural discretization for the $i$-component of the 143 skew-flux, \ eqref{eq:triad:i13c}, as144 skew-flux, \autoref{eq:i13c}, as 144 145 although it must be evaluated at $u$-points, it involves vertical 145 146 gradients (both for the tracer and the slope $r_1$), defined at 146 $w$-points. Similarly, the vertical skew flux, \ eqref{eq:triad:i31c}, is evaluated at147 $w$-points. Similarly, the vertical skew flux, \autoref{eq:i31c}, is evaluated at 147 148 $w$-points but involves horizontal gradients defined at $u$-points. 148 149 149 150 \subsection{Standard discretization} 150 151 The straightforward approach to discretize the lateral skew flux 151 \ eqref{eq:triad:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995152 into OPA, \ eqref{Eq_tra_ldf_iso}, is to calculate a mean vertical152 \autoref{eq:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 153 into OPA, \autoref{eq:tra_ldf_iso}, is to calculate a mean vertical 153 154 gradient at the $u$-point from the average of the four surrounding 154 155 vertical tracer gradients, and multiply this by a mean slope at the … … 159 160 $e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with 160 161 the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer 161 gradient, is then \ eqref{Eq_tra_ldf_iso}162 gradient, is then \autoref{eq:tra_ldf_iso} 162 163 \begin{equation*} 163 164 \left(F_u^{13} \right)_{i+\hhalf}^k = \Alts_{i+\hhalf}^k … … 181 182 operator to a tracer does not guarantee the decrease of its 182 183 global-average variance. To correct this, we introduced a smoothing of 183 the slopes of the iso-neutral surfaces (see \ S\ref{LDF}). This184 the slopes of the iso-neutral surfaces (see \autoref{chap:LDF}). This 184 185 technique works for $T$ and $S$ in so far as they are active tracers 185 186 ($i.e.$ they enter the computation of density), but it does not work … … 194 195 \begin{figure}[tb] \begin{center} 195 196 \includegraphics[width=1.05\textwidth]{Fig_GRIFF_triad_fluxes} 196 \caption{ \protect\label{fig: triad:ISO_triad}197 \caption{ \protect\label{fig:ISO_triad} 197 198 (a) Arrangement of triads $S_i$ and tracer gradients to 198 199 give lateral tracer flux from box $i,k$ to $i+1,k$ … … 205 206 slope calculated from the lateral density gradient across the $u$-point 206 207 divided by the vertical density gradient at the same $w$-point as the 207 tracer gradient. See Fig.~\ref{fig:triad:ISO_triad}a, where the thick lines208 tracer gradient. See \autoref{fig:ISO_triad}a, where the thick lines 208 209 denote the tracer gradients, and the thin lines the corresponding 209 210 triads, with slopes $s_1, \dotsc s_4$. The total area-integrated 210 211 skew-flux from tracer cell $i,k$ to $i+1,k$ 211 212 \begin{multline} 212 \label{eq: triad:i13}213 \label{eq:i13} 213 214 \left( F_u^{13} \right)_{i+\frac{1}{2}}^k = \Alts_{i+1}^k a_1 s_1 214 215 \delta _{k+\frac{1}{2}} \left[ T^{i+1} … … 225 226 stencil, and disallows the two-point computational modes. 226 227 227 The vertical skew flux \ eqref{eq:triad:i31c} from tracer cell $i,k$ to $i,k+1$ at the228 $w$-point $i,k+\hhalf$ is constructed similarly ( Fig.~\ref{fig:triad:ISO_triad}b)228 The vertical skew flux \autoref{eq:i31c} from tracer cell $i,k$ to $i,k+1$ at the 229 $w$-point $i,k+\hhalf$ is constructed similarly (\autoref{fig:ISO_triad}b) 229 230 by multiplying lateral tracer gradients from each of the four 230 231 surrounding $u$-points by the appropriate triad slope: 231 232 \begin{multline} 232 \label{eq: triad:i31}233 \label{eq:i31} 233 234 \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} = \Alts_i^{k+1} a_{1}' 234 235 s_{1}' \delta _{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1} … … 241 242 (appearing in both the vertical and lateral gradient), and the $u$- and 242 243 $w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the 243 triad as follows (see also Fig.~\ref{fig:triad:ISO_triad}):244 \begin{equation} 245 \label{eq: triad:R}244 triad as follows (see also \autoref{fig:ISO_triad}): 245 \begin{equation} 246 \label{eq:R} 246 247 _i^k \mathbb{R}_{i_p}^{k_p} 247 248 =-\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} … … 258 259 \begin{figure}[tb] \begin{center} 259 260 \includegraphics[width=0.80\textwidth]{Fig_GRIFF_qcells} 260 \caption{ \protect\label{fig: triad:qcells}261 \caption{ \protect\label{fig:qcells} 261 262 Triad notation for quarter cells. $T$-cells are inside 262 263 boxes, while the $i+\half,k$ $u$-cell is shaded in green and the … … 265 266 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 266 267 267 Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated ( Fig.~\ref{fig:triad:qcells}) with the quarter268 Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (\autoref{fig:qcells}) with the quarter 268 269 cell that is the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ $u$-cell and the $i,k+k_p$ $w$-cell. 269 Expressing the slopes $s_i$ and $s'_i$ in \ eqref{eq:triad:i13} and \eqref{eq:triad:i31} in this notation,270 Expressing the slopes $s_i$ and $s'_i$ in \autoref{eq:i13} and \autoref{eq:i31} in this notation, 270 271 we have $e.g.$ \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. 271 272 Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) … … 276 277 of a unique triad, and we notate these areas, similarly to the triad slopes, 277 278 as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, 278 where $e.g.$ in \ eqref{eq:triad:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$,279 and in \ eqref{eq:triad:i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$.279 where $e.g.$ in \autoref{eq:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, 280 and in \autoref{eq:i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. 280 281 281 282 \subsection{Full triad fluxes} … … 287 288 form 288 289 \begin{equation} 289 \label{eq: triad:i11}290 \label{eq:i11} 290 291 \left( F_u^{11} \right) _{i+\frac{1}{2}} ^{k} = 291 292 - \left( \Alts_i^{k+1} a_{1} + \Alts_i^{k+1} a_{2} + \Alts_i^k … … 293 294 \frac{\delta _{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}}, 294 295 \end{equation} 295 where the areas $a_i$ are as in \ eqref{eq:triad:i13}. In this case,296 separating the total lateral flux, the sum of \ eqref{eq:triad:i13} and297 \ eqref{eq:triad:i11}, into triad components, a lateral tracer296 where the areas $a_i$ are as in \autoref{eq:i13}. In this case, 297 separating the total lateral flux, the sum of \autoref{eq:i13} and 298 \autoref{eq:i11}, into triad components, a lateral tracer 298 299 flux 299 300 \begin{equation} 300 \label{eq: triad:latflux-triad}301 \label{eq:latflux-triad} 301 302 _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) = - \Alts_i^k{ \:}_i^k{\mathbb{A}_u}_{i_p}^{k_p} 302 303 \left( … … 312 313 density flux associated with each triad separately disappears. 313 314 \begin{equation} 314 \label{eq: triad:latflux-rho}315 \label{eq:latflux-rho} 315 316 {\mathbb{F}_u}_{i_p}^{k_p} (\rho)=-\alpha _i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (S)=0 316 317 \end{equation} … … 319 320 to $i+1,k$ must also vanish since it is a sum of four such triad fluxes. 320 321 321 The squared slope $r_1^2$ in the expression \ eqref{eq:triad:i33c} for the322 The squared slope $r_1^2$ in the expression \autoref{eq:i33c} for the 322 323 $_{33}$ component is also expressed in terms of area-weighted 323 324 squared triad slopes, so the area-integrated vertical flux from tracer 324 325 cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is 325 326 \begin{equation} 326 \label{eq: triad:i33}327 \label{eq:i33} 327 328 \left( F_w^{33} \right) _i^{k+\frac{1}{2}} = 328 329 - \left( \Alts_i^{k+1} a_{1}' s_{1}'^2 … … 332 333 \end{equation} 333 334 where the areas $a'$ and slopes $s'$ are the same as in 334 \ eqref{eq:triad:i31}.335 Then, separating the total vertical flux, the sum of \ eqref{eq:triad:i31} and336 \ eqref{eq:triad:i33}, into triad components, a vertical flux335 \autoref{eq:i31}. 336 Then, separating the total vertical flux, the sum of \autoref{eq:i31} and 337 \autoref{eq:i33}, into triad components, a vertical flux 337 338 \begin{align} 338 \label{eq: triad:vertflux-triad}339 \label{eq:vertflux-triad} 339 340 _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) 340 341 &= \Alts_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p} … … 345 346 \right) \\ 346 347 &= - \left(\left.{ }_i^k{\mathbb{A}_w}_{i_p}^{k_p}\right/{ }_i^k{\mathbb{A}_u}_{i_p}^{k_p}\right) 347 {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq: triad:vertflux-triad2}348 {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:vertflux-triad2} 348 349 \end{align} 349 350 may be associated with each triad. Each vertical density flux $_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)$ … … 355 356 fluxes. 356 357 357 We can explicitly identify ( Fig.~\ref{fig:triad:qcells}) the triads associated with the $s_i$, $a_i$, and $s'_i$, $a'_i$ used in the definition of358 We can explicitly identify (\autoref{fig:qcells}) the triads associated with the $s_i$, $a_i$, and $s'_i$, $a'_i$ used in the definition of 358 359 the $u$-fluxes and $w$-fluxes in 359 \ eqref{eq:triad:i31}, \eqref{eq:triad:i13}, \eqref{eq:triad:i11} \eqref{eq:triad:i33} and360 Fig.~\ref{fig:triad:ISO_triad} to write out the iso-neutral fluxes at $u$- and360 \autoref{eq:i31}, \autoref{eq:i13}, \autoref{eq:i11} \autoref{eq:i33} and 361 \autoref{fig:ISO_triad} to write out the iso-neutral fluxes at $u$- and 361 362 $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces: 362 %( Fig.~\ref{Fig_ISO_triad}):363 \begin{flalign} \label{ Eq_iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv363 %(\autoref{fig:ISO_triad}): 364 \begin{flalign} \label{eq:iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv 364 365 \sum_{\substack{i_p,\,k_p}} 365 366 \begin{pmatrix} … … 371 372 372 373 \subsection{Ensuring the scheme does not increase tracer variance} 373 \label{s ec:triad:variance}374 \label{subsec:variance} 374 375 375 376 We now require that this operator should not increase the … … 397 398 &= -T_{i+i_p-1/2}^k{\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \quad + \quad T_{i+i_p+1/2}^k 398 399 {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\ 399 &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq: triad:dvar_iso_i}400 &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:dvar_iso_i} 400 401 \end{aligned} 401 402 \end{multline} … … 404 405 $i,k+k_p+\half$ (below) of 405 406 \begin{equation} 406 \label{eq: triad:dvar_iso_k}407 \label{eq:dvar_iso_k} 407 408 _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. 408 409 \end{equation} 409 410 The total variance tendency driven by the triad is the sum of these 410 411 two. Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and 411 $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with \ eqref{eq:triad:latflux-triad} and412 \ eqref{eq:triad:vertflux-triad}, it is412 $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with \autoref{eq:latflux-triad} and 413 \autoref{eq:vertflux-triad}, it is 413 414 \begin{multline*} 414 415 -\Alts_i^k\left \{ … … 430 431 to be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by 431 432 \begin{equation} 432 \label{eq: triad:V-A}433 \label{eq:V-A} 433 434 _i^k\mathbb{V}_{i_p}^{k_p} 434 435 ={\;}_i^k{\mathbb{A}_u}_{i_p}^{k_p}\,{e_{1u}}_{\,i + i_p}^{\,k} … … 437 438 the variance tendency reduces to the perfect square 438 439 \begin{equation} 439 \label{eq: triad:perfect-square}440 \label{eq:perfect-square} 440 441 -\Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} 441 442 \left( … … 445 446 \right)^2\leq 0. 446 447 \end{equation} 447 Thus, the constraint \ eqref{eq:triad:V-A} ensures that the fluxes (\ref{eq:triad:latflux-triad}, \ref{eq:triad:vertflux-triad}) associated448 Thus, the constraint \autoref{eq:V-A} ensures that the fluxes (\autoref{eq:latflux-triad}, \autoref{eq:vertflux-triad}) associated 448 449 with a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase 449 450 the net variance. Since the total fluxes are sums of such fluxes from … … 452 453 increase. 453 454 454 The expression \ eqref{eq:triad:V-A} can be interpreted as a discretization455 The expression \autoref{eq:V-A} can be interpreted as a discretization 455 456 of the global integral 456 457 \begin{equation} 457 \label{eq: triad:cts-var}458 \label{eq:cts-var} 458 459 \frac{\partial}{\partial t}\int\!\half T^2\, dV = 459 460 \int\!\mathbf{F}\cdot\nabla T\, dV, … … 480 481 cells, defined in terms of the distances between $T$, $u$,$f$ and 481 482 $w$-points. This is the natural discretization of 482 \ eqref{eq:triad:cts-var}. The \NEMO model, however, operates with scale483 \autoref{eq:cts-var}. The \NEMO model, however, operates with scale 483 484 factors instead of grid sizes, and scale factors for the quarter 484 485 cells are not defined. Instead, therefore we simply choose 485 486 \begin{equation} 486 \label{eq: triad:V-NEMO}487 \label{eq:V-NEMO} 487 488 _i^k\mathbb{V}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k, 488 489 \end{equation} … … 492 493 $i+1,k$ reduces to the classical form 493 494 \begin{equation} 494 \label{eq: triad:lat-normal}495 \label{eq:lat-normal} 495 496 -\overline\Alts_{\,i+1/2}^k\; 496 497 \frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} … … 500 501 In fact if the diffusive coefficient is defined at $u$-points, so that 501 502 we employ $\Alts_{i+i_p}^k$ instead of $\Alts_i^k$ in the definitions of the 502 triad fluxes \ eqref{eq:triad:latflux-triad} and \eqref{eq:triad:vertflux-triad},503 triad fluxes \autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad}, 503 504 we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above. 504 505 … … 506 507 The iso-neutral fluxes at $u$- and 507 508 $w$-points are the sums of the triad fluxes that cross the $u$- and 508 $w$-faces \ eqref{Eq_iso_flux}:509 \begin{subequations}\label{eq: triad:alltriadflux}510 \begin{flalign}\label{eq: triad:vect_isoflux}509 $w$-faces \autoref{eq:iso_flux}: 510 \begin{subequations}\label{eq:alltriadflux} 511 \begin{flalign}\label{eq:vect_isoflux} 511 512 \vect{F}_{\mathrm{iso}}(T) &\equiv 512 513 \sum_{\substack{i_p,\,k_p}} … … 517 518 \end{pmatrix}, 518 519 \end{flalign} 519 where \ eqref{eq:triad:latflux-triad}:520 where \autoref{eq:latflux-triad}: 520 521 \begin{align} 521 \label{eq:triad :triadfluxu}522 \label{eq:triadfluxu} 522 523 _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - \Alts_i^k{ 523 524 \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}} … … 534 535 -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ 535 536 \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 536 \right),\label{eq:triad :triadfluxw}537 \right),\label{eq:triadfluxw} 537 538 \end{align} 538 with \ eqref{eq:triad:V-NEMO}539 with \autoref{eq:V-NEMO} 539 540 \begin{equation} 540 \label{eq: triad:V-NEMO2}541 \label{eq:V-NEMO2} 541 542 _i^k{\mathbb{V}}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k. 542 543 \end{equation} 543 544 \end{subequations} 544 545 545 The divergence of the expression \ eqref{Eq_iso_flux} for the fluxes gives the iso-neutral diffusion tendency at546 The divergence of the expression \autoref{eq:iso_flux} for the fluxes gives the iso-neutral diffusion tendency at 546 547 each tracer point: 547 \begin{equation} \label{eq: triad:iso_operator} D_l^T = \frac{1}{b_T}548 \begin{equation} \label{eq:iso_operator} D_l^T = \frac{1}{b_T} 548 549 \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k 549 550 {\mathbb{F}_u }_{i_p}^{k_p}} \right] + \delta_{k} \left[ … … 554 555 \begin{description} 555 556 \item[$\bullet$ horizontal diffusion] The discretization of the 556 diffusion operator recovers \ eqref{eq:triad:lat-normal} the traditional five-point Laplacian in557 diffusion operator recovers \autoref{eq:lat-normal} the traditional five-point Laplacian in 557 558 the limit of flat iso-neutral direction : 558 \begin{equation} \label{eq: triad:iso_property0} D_l^T = \frac{1}{b_T} \559 \begin{equation} \label{eq:iso_property0} D_l^T = \frac{1}{b_T} \ 559 560 \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; 560 561 \overline\Alts^{\,i} \; \delta_{i+1/2}[T] \right] \qquad … … 564 565 \item[$\bullet$ implicit treatment in the vertical] Only tracer values 565 566 associated with a single water column appear in the expression 566 \ eqref{eq:triad:i33} for the $_{33}$ fluxes, vertical fluxes driven by567 \autoref{eq:i33} for the $_{33}$ fluxes, vertical fluxes driven by 567 568 vertical gradients. This is of paramount importance since it means 568 569 that a time-implicit algorithm can be used to solve the vertical … … 582 583 \item[$\bullet$ pure iso-neutral operator] The iso-neutral flux of 583 584 locally referenced potential density is zero. See 584 \ eqref{eq:triad:latflux-rho} and \eqref{eq:triad:vertflux-triad2}.585 \autoref{eq:latflux-rho} and \autoref{eq:vertflux-triad2}. 585 586 586 587 \item[$\bullet$ conservation of tracer] The iso-neutral diffusion 587 588 conserves tracer content, $i.e.$ 588 \begin{equation} \label{eq: triad:iso_property1} \sum_{i,j,k} \left\{ D_l^T \589 \begin{equation} \label{eq:iso_property1} \sum_{i,j,k} \left\{ D_l^T \ 589 590 b_T \right\} = 0 590 591 \end{equation} … … 594 595 \item[$\bullet$ no increase of tracer variance] The iso-neutral diffusion 595 596 does not increase the tracer variance, $i.e.$ 596 \begin{equation} \label{eq: triad:iso_property2} \sum_{i,j,k} \left\{ T \ D_l^T597 \begin{equation} \label{eq:iso_property2} \sum_{i,j,k} \left\{ T \ D_l^T 597 598 \ b_T \right\} \leq 0 598 599 \end{equation} 599 600 The property is demonstrated in 600 \ S\ref{sec:triad:variance} above. It is a key property for a diffusion601 \autoref{subsec:variance} above. It is a key property for a diffusion 601 602 term. It means that it is also a dissipation term, $i.e.$ it 602 603 dissipates the square of the quantity on which it is applied. It … … 607 608 \item[$\bullet$ self-adjoint operator] The iso-neutral diffusion 608 609 operator is self-adjoint, $i.e.$ 609 \begin{equation} \label{eq: triad:iso_property3} \sum_{i,j,k} \left\{ S \ D_l^T610 \begin{equation} \label{eq:iso_property3} \sum_{i,j,k} \left\{ S \ D_l^T 610 611 \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 611 612 \end{equation} … … 614 615 routine. This property can be demonstrated similarly to the proof of 615 616 the `no increase of tracer variance' property. The contribution by a 616 single triad towards the left hand side of \ eqref{eq:triad:iso_property3}, can617 be found by replacing $\delta[T]$ by $\delta[S]$ in \ eqref{eq:triad:dvar_iso_i}618 and \ eqref{eq:triad:dvar_iso_k}. This results in a term similar to619 \ eqref{eq:triad:perfect-square},620 \begin{equation} 621 \label{eq: triad:TScovar}617 single triad towards the left hand side of \autoref{eq:iso_property3}, can 618 be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq:dvar_iso_i} 619 and \autoref{eq:dvar_iso_k}. This results in a term similar to 620 \autoref{eq:perfect-square}, 621 \begin{equation} 622 \label{eq:TScovar} 622 623 - \Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} 623 624 \left( … … 634 635 This is symmetrical in $T $ and $S$, so exactly the same term arises 635 636 from the discretization of this triad's contribution towards the 636 RHS of \ eqref{eq:triad:iso_property3}.637 RHS of \autoref{eq:iso_property3}. 637 638 \end{description} 638 639 639 \subsection{Treatment of the triads at the boundaries}\label{sec: triad:iso_bdry}640 \subsection{Treatment of the triads at the boundaries}\label{sec:iso_bdry} 640 641 The triad slope can only be defined where both the grid boxes centred at 641 642 the end of the arms exist. Triads that would poke up 642 643 through the upper ocean surface into the atmosphere, or down into the 643 ocean floor, must be masked out. See Fig.~\ref{fig:triad:bdry_triads}. Surface layer triads644 ocean floor, must be masked out. See \autoref{fig:bdry_triads}. Surface layer triads 644 645 $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and 645 646 $\triad{i+1}{1}{R}{-1/2}{-1/2}$ (blue) that require density to be 646 specified above the ocean surface are masked ( Fig.~\ref{fig:triad:bdry_triads}a): this ensures that lateral647 specified above the ocean surface are masked (\autoref{fig:bdry_triads}a): this ensures that lateral 647 648 tracer gradients produce no flux through the ocean surface. However, 648 649 to prevent surface noise, it is customary to retain the $_{11}$ contributions towards … … 650 651 $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$; this drives diapycnal tracer 651 652 fluxes. Similar comments apply to triads that would intersect the 652 ocean floor ( Fig.~\ref{fig:triad:bdry_triads}b). Note that both near bottom653 ocean floor (\autoref{fig:bdry_triads}b). Note that both near bottom 653 654 triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and 654 655 $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ … … 665 666 \begin{figure}[h] \begin{center} 666 667 \includegraphics[width=0.60\textwidth]{Fig_GRIFF_bdry_triads} 667 \caption{ \protect\label{fig: triad:bdry_triads}668 \caption{ \protect\label{fig:bdry_triads} 668 669 (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer 669 670 points (black dots), and $i+1/2,1$ $u$-point (blue square). Triad … … 678 679 or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point 679 680 is masked. The associated lateral fluxes (grey-black dashed 680 line) are masked if \ np{botmix\_triad}\forcode{ = .false.}, but left681 unmasked, giving bottom mixing, if \ np{botmix\_triad}\forcode{ = .true.}}681 line) are masked if \protect\np{botmix\_triad}\forcode{ = .false.}, but left 682 unmasked, giving bottom mixing, if \protect\np{botmix\_triad}\forcode{ = .true.}} 682 683 \end{center} \end{figure} 683 684 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 684 685 685 \subsection{ Limiting of the slopes within the interior}\label{sec: triad:limit}686 As discussed in \ S\ref{LDF_slp_iso}, iso-neutral slopes relative to686 \subsection{ Limiting of the slopes within the interior}\label{sec:limit} 687 As discussed in \autoref{subsec:LDF_slp_iso}, iso-neutral slopes relative to 687 688 geopotentials must be bounded everywhere, both for consistency with the small-slope 688 689 approximation and for numerical stability \citep{Cox1987, … … 692 693 It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to 693 694 geopotentials (here the $\sigma_i$ are the slopes of the coordinate surfaces relative to 694 geopotentials) \ eqref{Eq_PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate695 geopotentials) \autoref{eq:PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate 695 696 surfaces, so we require 696 697 \begin{equation*} … … 700 701 Each individual triad slope 701 702 \begin{equation} 702 \label{eq: triad:Rtilde}703 \label{eq:Rtilde} 703 704 _i^k\tilde{\mathbb{R}}_{i_p}^{k_p} = {}_i^k\mathbb{R}_{i_p}^{k_p} + \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}} 704 705 \end{equation} … … 709 710 is always downwards, and so acts to reduce gravitational potential energy. 710 711 711 \subsection{Tapering within the surface mixed layer}\label{sec:t riad:taper}712 \subsection{Tapering within the surface mixed layer}\label{sec:taper} 712 713 Additional tapering of the iso-neutral fluxes is necessary within the 713 714 surface mixed layer. When the Griffies triads are used, we offer two 714 715 options for this. 715 716 716 \subsubsection{Linear slope tapering within the surface mixed layer}\label{sec: triad:lintaper}717 \subsubsection{Linear slope tapering within the surface mixed layer}\label{sec:lintaper} 717 718 This is the option activated by the default choice 718 719 \np{ln\_triad\_iso}\forcode{ = .false.}. Slopes $\tilde{r}_i$ relative to 719 720 geopotentials are tapered linearly from their value immediately below the mixed layer to zero at the 720 surface, as described in option (c) of Fig.~\ref{Fig_eiv_slp}, to values721 surface, as described in option (c) of \autoref{fig:eiv_slp}, to values 721 722 \begin{subequations} 722 723 \begin{equation} 723 \label{eq: triad:rmtilde}724 \label{eq:rmtilde} 724 725 \rMLt = 725 726 -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for } z>-h, … … 728 729 adjusted to 729 730 \begin{equation} 730 \label{eq: triad:rm}731 \label{eq:rm} 731 732 \rML =\rMLt -\sigma_i \quad \text{ for } z>-h. 732 733 \end{equation} 733 734 \end{subequations} 734 735 Thus the diffusion operator within the mixed layer is given by: 735 \begin{equation} \label{eq: triad:iso_tensor_ML}736 \begin{equation} \label{eq:iso_tensor_ML} 736 737 D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 737 738 \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} … … 745 746 mixed-layer and in isopycnal layers immediately below, in the 746 747 thermocline. It is consistent with the way the $\tilde{r}_i$ are 747 tapered within the mixed layer (see \ S\ref{sec:triad:taperskew} below)748 tapered within the mixed layer (see \autoref{sec:taperskew} below) 748 749 so as to ensure a uniform GM eddy-induced velocity throughout the 749 750 mixed layer. However, it gives a downwards density flux and so acts so 750 751 as to reduce potential energy in the same way as does the slope 751 limiting discussed above in \ S\ref{sec:triad:limit}.752 limiting discussed above in \autoref{sec:limit}. 752 753 753 As in \ S\ref{sec:triad:limit} above, the tapering754 \ eqref{eq:triad:rmtilde} is applied separately to each triad754 As in \autoref{sec:limit} above, the tapering 755 \autoref{eq:rmtilde} is applied separately to each triad 755 756 $_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the 756 757 $_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted. For clarity, we assume 757 758 $z$-coordinates in the following; the conversion from 758 759 $\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as described 759 above by \ eqref{eq:triad:Rtilde}.760 above by \autoref{eq:Rtilde}. 760 761 \begin{enumerate} 761 762 \item Mixed-layer depth is defined so as to avoid including regions of weak 762 763 vertical stratification in the slope definition. 763 764 At each $i,j$ (simplified to $i$ in 764 Fig.~\ref{fig:triad:MLB_triad}), we define the mixed-layer by setting765 \autoref{fig:MLB_triad}), we define the mixed-layer by setting 765 766 the vertical index of the tracer point immediately below the mixed 766 767 layer, $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) … … 768 769 ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, where $i,k_{10}$ is 769 770 the tracer gridbox within which the depth reaches 10~m. See the left 770 side of Fig.~\ref{fig:triad:MLB_triad}. We use the $k_{10}$-gridbox771 side of \autoref{fig:MLB_triad}. We use the $k_{10}$-gridbox 771 772 instead of the surface gridbox to avoid problems e.g.\ with thin 772 773 daytime mixed-layers. Currently we use the same … … 784 785 ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are 785 786 representative of the thermocline. The four basal triads defined in the bottom part 786 of Fig.~\ref{fig:triad:MLB_triad} are then787 of \autoref{fig:MLB_triad} are then 787 788 \begin{align} 788 789 {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &= 789 {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}, \label{eq: triad:Rbase}790 {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}, \label{eq:Rbase} 790 791 \\ 791 792 \intertext{with e.g.\ the green triad} … … 797 798 so it is this depth 798 799 \begin{equation} 799 \label{eq: triad:zbase}800 \label{eq:zbase} 800 801 {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2} 801 802 \end{equation} 802 803 (one gridbox deeper than the 803 804 diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper 804 the slopes in \ eqref{eq:triad:rmtilde}.805 the slopes in \autoref{eq:rmtilde}. 805 806 \item Finally, we calculate the adjusted triads 806 807 ${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within the mixed … … 815 816 \intertext{and more generally} 816 817 {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p} &= 817 \frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}.\label{eq: triad:RML}818 \frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}.\label{eq:RML} 818 819 \end{align} 819 820 \end{enumerate} … … 822 823 \begin{figure}[h] 823 824 % \fcapside { 824 \caption{\protect\label{fig: triad:MLB_triad} Definition of825 \caption{\protect\label{fig:MLB_triad} Definition of 825 826 mixed-layer depth and calculation of linearly tapered 826 827 triads. The figure shows a water column at a given $i,j$ … … 846 847 847 848 \subsubsection{Additional truncation of skew iso-neutral flux components} 848 \label{s ec:triad:Gerdes-taper}849 \label{subsec:Gerdes-taper} 849 850 The alternative option is activated by setting \np{ln\_triad\_iso} = 850 851 true. This retains the same tapered slope $\rML$ described above for the … … 853 854 replaces the $\rML$ in the skew term by 854 855 \begin{equation} 855 \label{eq: triad:rm*}856 \label{eq:rm*} 856 857 \rML^*=\left.\rMLt^2\right/\tilde{r}_i-\sigma_i, 857 858 \end{equation} 858 859 giving a ML diffusive operator 859 \begin{equation} \label{eq: triad:iso_tensor_ML2}860 \begin{equation} \label{eq:iso_tensor_ML2} 860 861 D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 861 862 \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} … … 887 888 % Skew flux formulation for Eddy Induced Velocity : 888 889 % ================================================================ 889 \section{Eddy induced advection formulated as a skew flux}\label{sec: triad:skew-flux}890 891 \subsection{Continuous skew flux formulation}\label{sec: triad:continuous-skew-flux}890 \section{Eddy induced advection formulated as a skew flux}\label{sec:skew-flux} 891 892 \subsection{Continuous skew flux formulation}\label{sec:continuous-skew-flux} 892 893 893 894 When Gent and McWilliams's [1990] diffusion is used, … … 895 896 eddy induced velocity, the formulation of which depends on the slopes of iso- 896 897 neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used 897 here are referenced to the geopotential surfaces, $i.e.$ \ eqref{Eq_ldfslp_geo}898 is used in $z$-coordinate, and the sum \ eqref{Eq_ldfslp_geo}899 + \ eqref{Eq_ldfslp_iso} in $z^*$ or $s$-coordinates.898 here are referenced to the geopotential surfaces, $i.e.$ \autoref{eq:ldfslp_geo} 899 is used in $z$-coordinate, and the sum \autoref{eq:ldfslp_geo} 900 + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates. 900 901 901 902 The eddy induced velocity is given by: 902 \begin{subequations} \label{eq: triad:eiv}903 \begin{equation}\label{eq: triad:eiv_v}903 \begin{subequations} \label{eq:eiv} 904 \begin{equation}\label{eq:eiv_v} 904 905 \begin{split} 905 906 u^* & = - \frac{1}{e_{3}}\; \partial_i\psi_1, \\ … … 910 911 \end{equation} 911 912 where the streamfunctions $\psi_i$ are given by 912 \begin{equation} \label{eq: triad:eiv_psi}913 \begin{equation} \label{eq:eiv_psi} 913 914 \begin{split} 914 915 \psi_1 & = A_{e} \; \tilde{r}_1, \\ … … 924 925 default implementation, where \np{ln\_traldf\_triad} is set 925 926 false. This allows us to take advantage of all the advection schemes 926 offered for the tracers (see \ S\ref{TRA_adv}) and not just a $2^{nd}$927 offered for the tracers (see \autoref{sec:TRA_adv}) and not just a $2^{nd}$ 927 928 order advection scheme. This is particularly useful for passive 928 929 tracers where \emph{positivity} of the advection scheme is of … … 962 963 and since the eddy induced velocity field is non-divergent, we end up with the skew 963 964 form of the eddy induced advective fluxes per unit area in $ijk$ space: 964 \begin{equation} \label{eq: triad:eiv_skew_ijk}965 \begin{equation} \label{eq:eiv_skew_ijk} 965 966 \textbf{F}_\mathrm{eiv}^T = \begin{pmatrix} 966 967 {+ e_{2} \, \psi_1 \; \partial_k T} \\ … … 969 970 \end{equation} 970 971 The total fluxes per unit physical area are then 971 \begin{equation}\label{eq: triad:eiv_skew_physical}972 \begin{equation}\label{eq:eiv_skew_physical} 972 973 \begin{split} 973 974 f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T \\ … … 977 978 \end{split} 978 979 \end{equation} 979 Note that Eq.~ \eqref{eq:triad:eiv_skew_physical} takes the same form whatever the980 Note that \autoref{eq:eiv_skew_physical} takes the same form whatever the 980 981 vertical coordinate, though of course the slopes 981 $\tilde{r}_i$ which define the $\psi_i$ in \ eqref{eq:triad:eiv_psi} are relative to geopotentials.982 $\tilde{r}_i$ which define the $\psi_i$ in \autoref{eq:eiv_psi} are relative to geopotentials. 982 983 The tendency associated with eddy induced velocity is then simply the convergence 983 of the fluxes (\ ref{eq:triad:eiv_skew_ijk}, \ref{eq:triad:eiv_skew_physical}), so984 \begin{equation} \label{eq: triad:skew_eiv_conv}984 of the fluxes (\autoref{eq:eiv_skew_ijk}, \autoref{eq:eiv_skew_physical}), so 985 \begin{equation} \label{eq:skew_eiv_conv} 985 986 \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 } \left[ 986 987 \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right) … … 995 996 996 997 \subsection{Discrete skew flux formulation} 997 The skew fluxes in (\ ref{eq:triad:eiv_skew_physical}, \ref{eq:triad:eiv_skew_ijk}), like the off-diagonal terms998 (\ ref{eq:triad:i13c}, \ref{eq:triad:i31c}) of the small angle diffusion tensor, are best999 expressed in terms of the triad slopes, as in Fig.~\ref{fig:triad:ISO_triad}1000 and Eqs~(\ref{eq:triad:i13}, \ref{eq:triad:i31}); but now in terms of the triad slopes998 The skew fluxes in (\autoref{eq:eiv_skew_physical}, \autoref{eq:eiv_skew_ijk}), like the off-diagonal terms 999 (\autoref{eq:i13c}, \autoref{eq:i31c}) of the small angle diffusion tensor, are best 1000 expressed in terms of the triad slopes, as in \autoref{fig:ISO_triad} 1001 and (\autoref{eq:i13}, \autoref{eq:i31}); but now in terms of the triad slopes 1001 1002 $\tilde{\mathbb{R}}$ relative to geopotentials instead of the 1002 1003 $\mathbb{R}$ relative to coordinate surfaces. The discrete form of 1003 \ eqref{eq:triad:eiv_skew_ijk} using the slopes \eqref{eq:triad:R} and1004 \autoref{eq:eiv_skew_ijk} using the slopes \autoref{eq:R} and 1004 1005 defining $A_e$ at $T$-points is then given by: 1005 1006 1006 1007 1007 \begin{subequations}\label{eq: triad:allskewflux}1008 \begin{flalign}\label{eq: triad:vect_skew_flux}1008 \begin{subequations}\label{eq:allskewflux} 1009 \begin{flalign}\label{eq:vect_skew_flux} 1009 1010 \vect{F}_{\mathrm{eiv}}(T) &\equiv 1010 1011 \sum_{\substack{i_p,\,k_p}} … … 1016 1017 \end{flalign} 1017 1018 where the skew flux in the $i$-direction associated with a given 1018 triad is (\ ref{eq:triad:latflux-triad}, \ref{eq:triad:triadfluxu}):1019 triad is (\autoref{eq:latflux-triad}, \autoref{eq:triadfluxu}): 1019 1020 \begin{align} 1020 \label{eq: triad:skewfluxu}1021 \label{eq:skewfluxu} 1021 1022 _i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) &= + \quarter {A_e}_i^k{ 1022 1023 \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} … … 1024 1025 \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }, 1025 1026 \\ 1026 \intertext{and \ eqref{eq:triad:triadfluxw} in the $k$-direction, changing the sign1027 to be consistent with \ eqref{eq:triad:eiv_skew_ijk}:}1027 \intertext{and \autoref{eq:triadfluxw} in the $k$-direction, changing the sign 1028 to be consistent with \autoref{eq:eiv_skew_ijk}:} 1028 1029 _i^k {\mathbb{S}_w}_{i_p}^{k_p} (T) 1029 1030 &= -\quarter {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}} 1030 {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq: triad:skewfluxw}1031 {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:skewfluxw} 1031 1032 \end{align} 1032 1033 \end{subequations} … … 1040 1041 include a diffusive component but is a `pure' advection term. This can 1041 1042 be seen 1042 %either from Appendix \ ref{Apdx_eiv_skew} or1043 %either from Appendix \autoref{apdx:eiv_skew} or 1043 1044 by considering the 1044 1045 fluxes associated with a given triad slope 1045 1046 $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. For, following 1046 \ S\ref{sec:triad:variance} and \eqref{eq:triad:dvar_iso_i}, the1047 \autoref{subsec:variance} and \autoref{eq:dvar_iso_i}, the 1047 1048 associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$ 1048 1049 drives a net rate of change of variance, summed over the two 1049 1050 $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of 1050 1051 \begin{equation} 1051 \label{eq: triad:dvar_eiv_i}1052 \label{eq:dvar_eiv_i} 1052 1053 _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], 1053 1054 \end{equation} … … 1055 1056 $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of 1056 1057 \begin{equation} 1057 \label{eq: triad:dvar_eiv_k}1058 \label{eq:dvar_eiv_k} 1058 1059 _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. 1059 1060 \end{equation} 1060 Inspection of the definitions (\ ref{eq:triad:skewfluxu}, \ref{eq:triad:skewfluxw})1061 shows that these two variance changes (\ ref{eq:triad:dvar_eiv_i}, \ref{eq:triad:dvar_eiv_k})1061 Inspection of the definitions (\autoref{eq:skewfluxu}, \autoref{eq:skewfluxw}) 1062 shows that these two variance changes (\autoref{eq:dvar_eiv_i}, \autoref{eq:dvar_eiv_k}) 1062 1063 sum to zero. Hence the two fluxes associated with each triad make no 1063 1064 net contribution to the variance budget. … … 1072 1073 For the change in gravitational PE driven by the $k$-flux is 1073 1074 \begin{align} 1074 \label{eq: triad:vert_densityPE}1075 \label{eq:vert_densityPE} 1075 1076 g {e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) 1076 1077 &=g {e_{3w}}_{\,i}^{\,k+k_p}\left[-\alpha _i^k {\:}_i^k … … 1078 1079 {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\ 1079 1080 \intertext{Substituting ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from 1080 \ eqref{eq:triad:skewfluxw}, gives}1081 \autoref{eq:skewfluxw}, gives} 1081 1082 % and separating out 1082 1083 % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$, … … 1090 1091 \end{align} 1091 1092 using the definition of the triad slope $\rtriad{R}$, 1092 \ eqref{eq:triad:R} to express $-\alpha _i^k\delta_{i+ i_p}[T^k]+1093 \autoref{eq:R} to express $-\alpha _i^k\delta_{i+ i_p}[T^k]+ 1093 1094 \beta_i^k\delta_{i+ i_p}[S^k]$ in terms of $-\alpha_i^k \delta_{k+ 1094 1095 k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$. … … 1096 1097 Where the coordinates slope, the $i$-flux gives a PE change 1097 1098 \begin{multline} 1098 \label{eq: triad:lat_densityPE}1099 \label{eq:lat_densityPE} 1099 1100 g \delta_{i+i_p}[z_T^k] 1100 1101 \left[ … … 1106 1107 \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}, 1107 1108 \end{multline} 1108 (using \ eqref{eq:triad:skewfluxu}) and so the total PE change1109 \ eqref{eq:triad:vert_densityPE} + \eqref{eq:triad:lat_densityPE} associated with the triad fluxes is1109 (using \autoref{eq:skewfluxu}) and so the total PE change 1110 \autoref{eq:vert_densityPE} + \autoref{eq:lat_densityPE} associated with the triad fluxes is 1110 1111 \begin{multline} 1111 \label{eq:t riad:tot_densityPE}1112 \label{eq:tot_densityPE} 1112 1113 g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) + 1113 1114 g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\ … … 1119 1120 \beta_i^k\delta_{k+ k_p}[S^i]<0$, this PE change is negative. 1120 1121 1121 \subsection{Treatment of the triads at the boundaries}\label{sec: triad:skew_bdry}1122 \subsection{Treatment of the triads at the boundaries}\label{sec:skew_bdry} 1122 1123 Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes 1123 1124 are masked at the boundaries in exactly the same way as are the triad 1124 1125 slopes \rtriad{R} used for the iso-neutral diffusive fluxes, as 1125 described in \ S\ref{sec:triad:iso_bdry} and1126 Fig.~\ref{fig:triad:bdry_triads}. Thus surface layer triads1126 described in \autoref{sec:iso_bdry} and 1127 \autoref{fig:bdry_triads}. Thus surface layer triads 1127 1128 $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are 1128 1129 masked, and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ … … 1132 1133 no effect on the eddy-induced skew-fluxes. 1133 1134 1134 \subsection{Limiting of the slopes within the interior}\label{sec: triad:limitskew}1135 \subsection{Limiting of the slopes within the interior}\label{sec:limitskew} 1135 1136 Presently, the iso-neutral slopes $\tilde{r}_i$ relative 1136 1137 to geopotentials are limited to be less than $1/100$, exactly as in 1137 calculating the iso-neutral diffusion, \S \ ref{sec:triad:limit}. Each1138 calculating the iso-neutral diffusion, \S \autoref{sec:limit}. Each 1138 1139 individual triad \rtriadt{R} is so limited. 1139 1140 1140 \subsection{Tapering within the surface mixed layer}\label{sec:t riad:taperskew}1141 \subsection{Tapering within the surface mixed layer}\label{sec:taperskew} 1141 1142 The slopes $\tilde{r}_i$ relative to 1142 1143 geopotentials (and thus the individual triads \rtriadt{R}) are always tapered linearly from their value immediately below the mixed layer to zero at the 1143 surface \ eqref{eq:triad:rmtilde}, as described in \S\ref{sec:triad:lintaper}. This is1144 option (c) of Fig.~\ref{Fig_eiv_slp}. This linear tapering for the1144 surface \autoref{eq:rmtilde}, as described in \autoref{sec:lintaper}. This is 1145 option (c) of \autoref{fig:eiv_slp}. This linear tapering for the 1145 1146 slopes used to calculate the eddy-induced fluxes is 1146 1147 unaffected by the value of \np{ln\_triad\_iso}. … … 1148 1149 The justification for this linear slope tapering is that, for $A_e$ 1149 1150 that is constant or varies only in the horizontal (the most commonly 1150 used options in \NEMO: see \ S\ref{LDF_coef}), it is1151 used options in \NEMO: see \autoref{sec:LDF_coef}), it is 1151 1152 equivalent to a horizontal eiv (eddy-induced velocity) that is uniform 1152 within the mixed layer \ eqref{eq:triad:eiv_v}. This ensures that the1153 within the mixed layer \autoref{eq:eiv_v}. This ensures that the 1153 1154 eiv velocities do not restratify the mixed layer \citep{Treguier1997, 1154 1155 Danabasoglu_al_2008}. Equivantly, in terms … … 1158 1159 horizontal flux convergence is relatively insignificant within the mixed-layer). 1159 1160 1160 \subsection{Streamfunction diagnostics}\label{sec: triad:sfdiag}1161 \subsection{Streamfunction diagnostics}\label{sec:sfdiag} 1161 1162 Where the namelist parameter \np{ln\_traldf\_gdia}\forcode{ = .true.}, diagnosed 1162 1163 mean eddy-induced velocities are output. Each time step, … … 1164 1165 $uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$ 1165 1166 (integer $i$, integer +1/2 $j$, integer +1/2 $k$) points (see Table 1166 \ ref{Tab_cell}) respectively. We follow \citep{Griffies_Bk04} and1167 \autoref{tab:cell}) respectively. We follow \citep{Griffies_Bk04} and 1167 1168 calculate the streamfunction at a given $uw$-point from the 1168 1169 surrounding four triads according to: 1169 1170 \begin{equation} 1170 \label{eq: triad:sfdiagi}1171 \label{eq:sfdiagi} 1171 1172 {\psi_1}_{i+1/2}^{k+1/2}={\quarter}\sum_{\substack{i_p,\,k_p}} 1172 1173 {A_e}_{i+1/2-i_p}^{k+1/2-k_p}\:\triadd{i+1/2-i_p}{k+1/2-k_p}{R}{i_p}{k_p}. … … 1174 1175 The streamfunction $\psi_1$ is calculated similarly at $vw$ points. 1175 1176 The eddy-induced velocities are then calculated from the 1176 straightforward discretisation of \ eqref{eq:triad:eiv_v}:1177 \begin{equation}\label{eq: triad:eiv_v_discrete}1177 straightforward discretisation of \autoref{eq:eiv_v}: 1178 \begin{equation}\label{eq:eiv_v_discrete} 1178 1179 \begin{split} 1179 1180 {u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k}}\left({\psi_1}_{i+1/2}^{k+1/2}-{\psi_1}_{i+1/2}^{k+1/2}\right), \\ -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_ASM.tex
r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Apply Assimilation Increments (ASM)} 7 \label{ ASM}7 \label{chap:ASM} 8 8 9 9 Authors: D. Lea, M. Martin, K. Mogensen, A. Weaver, ... % do we keep … … 26 26 27 27 \section{Direct initialization} 28 \label{ ASM_DI}28 \label{sec:ASM_DI} 29 29 30 30 Direct initialization (DI) refers to the instantaneous correction … … 33 33 34 34 \section{Incremental analysis updates} 35 \label{ ASM_IAU}35 \label{sec:ASM_IAU} 36 36 37 37 Rather than updating the model state directly with the analysis increment, … … 83 83 \end{eqnarray} 84 84 where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even. 85 The weights described by \ eqref{eq:F2_i} provide a85 The weights described by \autoref{eq:F2_i} provide a 86 86 smoother transition of the analysis trajectory from one assimilation cycle 87 to the next than that described by \ eqref{eq:F1_i}.87 to the next than that described by \autoref{eq:F1_i}. 88 88 89 89 %========================================================================== 90 90 % Divergence damping description %%% 91 91 \section{Divergence damping initialisation} 92 \label{ ASM_details}92 \label{sec:ASM_details} 93 93 94 94 The velocity increments may be initialized by the iterative application of … … 110 110 +\delta _j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right). 111 111 \end{equation} 112 By the application of \ eqref{eq:asm_dmp} and \eqref{eq:asm_dmp} the divergence is filtered112 By the application of \autoref{eq:asm_dmp} and \autoref{eq:asm_dmp} the divergence is filtered 113 113 in each iteration, and the vorticity is left unchanged. In the presence of coastal boundaries 114 114 with zero velocity increments perpendicular to the coast the divergence is strongly damped. … … 125 125 126 126 \section{Implementation details} 127 \label{ ASM_details}127 \label{sec:ASM_details} 128 128 129 129 Here we show an example \ngn{namasm} namelist and the header of an example assimilation -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_CONFIG.tex
r9394 r9407 5 5 % ================================================================ 6 6 \chapter{Configurations} 7 \label{ CFG}7 \label{chap:CFG} 8 8 \minitoc 9 9 … … 15 15 % ================================================================ 16 16 \section{Introduction} 17 \label{ CFG_intro}17 \label{sec:CFG_intro} 18 18 19 19 … … 33 33 % ================================================================ 34 34 \section{C1D: 1D Water column model (\protect\key{c1d}) } 35 \label{ CFG_c1d}35 \label{sec:CFG_c1d} 36 36 37 37 $\ $\newline … … 81 81 % ================================================================ 82 82 \section{ORCA family: global ocean with tripolar grid} 83 \label{ CFG_orca}83 \label{sec:CFG_orca} 84 84 85 85 The ORCA family is a series of global ocean configurations that are run together with … … 95 95 \begin{figure}[!t] \begin{center} 96 96 \includegraphics[width=0.98\textwidth]{Fig_ORCA_NH_mesh} 97 \caption{ \protect\label{ Fig_MISC_ORCA_msh}97 \caption{ \protect\label{fig:MISC_ORCA_msh} 98 98 ORCA mesh conception. The departure from an isotropic Mercator grid start poleward of 20\degN. 99 99 The two "north pole" are the foci of a series of embedded ellipses (blue curves) … … 108 108 % ------------------------------------------------------------------------------------------------------------- 109 109 \subsection{ORCA tripolar grid} 110 \label{ CFG_orca_grid}110 \label{subsec:CFG_orca_grid} 111 111 112 112 The ORCA grid is a tripolar is based on the semi-analytical method of \citet{Madec_Imbard_CD96}. … … 116 116 computing the associated set of mesh meridians, and projecting the resulting mesh onto the sphere. 117 117 The set of mesh parallels used is a series of embedded ellipses which foci are the two mesh north 118 poles ( Fig.~\ref{Fig_MISC_ORCA_msh}). The resulting mesh presents no loss of continuity in118 poles (\autoref{fig:MISC_ORCA_msh}). The resulting mesh presents no loss of continuity in 119 119 either the mesh lines or the scale factors, or even the scale factor derivatives over the whole 120 120 ocean domain, as the mesh is not a composite mesh. … … 123 123 \includegraphics[width=1.0\textwidth]{Fig_ORCA_NH_msh05_e1_e2} 124 124 \includegraphics[width=0.80\textwidth]{Fig_ORCA_aniso} 125 \caption { \protect\label{ Fig_MISC_ORCA_e1e2}125 \caption { \protect\label{fig:MISC_ORCA_e1e2} 126 126 \textit{Top}: Horizontal scale factors ($e_1$, $e_2$) and 127 127 \textit{Bottom}: ratio of anisotropy ($e_1 / e_2$) … … 141 141 the Gulf Stream) and keeping the smallest scale factor in the northern hemisphere larger 142 142 than the smallest one in the southern hemisphere. 143 The resulting mesh is shown in Fig.~\ref{Fig_MISC_ORCA_msh} and \ref{Fig_MISC_ORCA_e1e2}143 The resulting mesh is shown in \autoref{fig:MISC_ORCA_msh} and \autoref{fig:MISC_ORCA_e1e2} 144 144 for a half a degree grid (ORCA\_R05). 145 145 The smallest ocean scale factor is found in along Antarctica, while the ratio of anisotropy remains close to one except near the Victoria Island … … 150 150 % ------------------------------------------------------------------------------------------------------------- 151 151 \subsection{ORCA pre-defined resolution} 152 \label{ CFG_orca_resolution}152 \label{subsec:CFG_orca_resolution} 153 153 154 154 … … 156 156 horizontal resolution. The value of the resolution is given by the resolution at the Equator 157 157 expressed in degrees. Each of configuration is set through the \textit{domain\_cfg} domain configuration file, 158 which sets the grid size and configuration name parameters. The NEMO System Team provides only ORCA2 domain input file "\ifile{ORCA\_R2\_zps\_domcfg}" file (Tab. \ ref{Tab_ORCA}).158 which sets the grid size and configuration name parameters. The NEMO System Team provides only ORCA2 domain input file "\ifile{ORCA\_R2\_zps\_domcfg}" file (Tab. \autoref{tab:ORCA}). 159 159 160 160 … … 175 175 \hline \hline 176 176 \end{tabular} 177 \caption{ \protect\label{ Tab_ORCA}177 \caption{ \protect\label{tab:ORCA} 178 178 Domain size of ORCA family configurations. 179 179 The flag for configurations of ORCA family need to be set in \textit{domain\_cfg} file. } … … 196 196 For ORCA\_R1 and R025, setting the configuration key to 75 allows to use 75 vertical levels, 197 197 otherwise 46 are used. In the other ORCA configurations, 31 levels are used 198 (see Tab.~\ref{Tab_orca_zgr} \sfcomment{HERE I need to put new table for ORCA2 values} and Fig.~\ref{Fig_zgr}).198 (see \autoref{tab:orca_zgr} \sfcomment{HERE I need to put new table for ORCA2 values} and \autoref{fig:zgr}). 199 199 200 200 Only the ORCA\_R2 is provided with all its input files in the \NEMO distribution. … … 204 204 205 205 This version of ORCA\_R2 has 31 levels in the vertical, with the highest resolution (10m) 206 in the upper 150m (see Tab.~\ref{Tab_orca_zgr} and Fig.~\ref{Fig_zgr}).206 in the upper 150m (see \autoref{tab:orca_zgr} and \autoref{fig:zgr}). 207 207 The bottom topography and the coastlines are derived from the global atlas of Smith and Sandwell (1997). 208 The default forcing uses the boundary forcing from \citet{Large_Yeager_Rep04} (see \ S\ref{SBC_blk_core}),208 The default forcing uses the boundary forcing from \citet{Large_Yeager_Rep04} (see \autoref{subsec:SBC_blk_core}), 209 209 which was developed for the purpose of running global coupled ocean-ice simulations 210 210 without an interactive atmosphere. This \citet{Large_Yeager_Rep04} dataset is available … … 222 222 % ------------------------------------------------------------------------------------------------------------- 223 223 \section{GYRE family: double gyre basin } 224 \label{ CFG_gyre}224 \label{sec:CFG_gyre} 225 225 226 226 The GYRE configuration \citep{Levy_al_OM10} has been built to simulate … … 234 234 The domain geometry is a closed rectangular basin on the $\beta$-plane centred 235 235 at $\sim$ 30\degN and rotated by 45\deg, 3180~km long, 2120~km wide 236 and 4~km deep ( Fig.~\ref{Fig_MISC_strait_hand}).236 and 4~km deep (\autoref{fig:MISC_strait_hand}). 237 237 The domain is bounded by vertical walls and by a flat bottom. The configuration is 238 238 meant to represent an idealized North Atlantic or North Pacific basin. … … 257 257 Obviously, the namelist parameters have to be adjusted to the chosen resolution, see the Configurations 258 258 pages on the NEMO web site (Using NEMO\/Configurations) . 259 In the vertical, GYRE uses the default 30 ocean levels (\jp{jpk}\forcode{ = 31}) ( Fig.~\ref{Fig_zgr}).259 In the vertical, GYRE uses the default 30 ocean levels (\jp{jpk}\forcode{ = 31}) (\autoref{fig:zgr}). 260 260 261 261 The GYRE configuration is also used in benchmark test as it is very simple to increase … … 267 267 \begin{figure}[!t] \begin{center} 268 268 \includegraphics[width=1.0\textwidth]{Fig_GYRE} 269 \caption{ \protect\label{ Fig_GYRE}269 \caption{ \protect\label{fig:GYRE} 270 270 Snapshot of relative vorticity at the surface of the model domain 271 271 in GYRE R9, R27 and R54. From \citet{Levy_al_OM10}.} … … 277 277 % ------------------------------------------------------------------------------------------------------------- 278 278 \section{AMM: atlantic margin configuration} 279 \label{ MISC_config_AMM}279 \label{sec:MISC_config_AMM} 280 280 281 281 The AMM, Atlantic Margins Model, is a regional model covering the -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_DIA.tex
r9394 r9407 5 5 % ================================================================ 6 6 \chapter{Output and Diagnostics (IOM, DIA, TRD, FLO)} 7 \label{ DIA}7 \label{chap:DIA} 8 8 \minitoc 9 9 … … 15 15 % ================================================================ 16 16 \section{Old model output (default)} 17 \label{ DIA_io_old}17 \label{sec:DIA_io_old} 18 18 19 19 The model outputs are of three types: the restart file, the output listing, … … 56 56 % ================================================================ 57 57 \section{Standard model output (IOM)} 58 \label{ DIA_iom}58 \label{sec:DIA_iom} 59 59 60 60 … … 595 595 596 596 \subsection{XML reference tables} 597 \label{ IOM_xmlref}597 \label{subsec:IOM_xmlref} 598 598 599 599 (1) Simple computation: directly define the computation when refering to the variable in the file definition. … … 998 998 \subsection{CF metadata standard compliance} 999 999 1000 Output from the XIOS-1.0 IO server is compliant with \href{http://cfconventions.org/Data/cf-conventions/cf-conventions-1.5/build/cf-conventions.html}{version 1.5} of the CF metadata standard. Therefore while a user may wish to add their own metadata to the output files (as demonstrated in example 4 of section \ ref{IOM_xmlref}) the metadata should, for the most part, comply with the CF-1.5 standard.1000 Output from the XIOS-1.0 IO server is compliant with \href{http://cfconventions.org/Data/cf-conventions/cf-conventions-1.5/build/cf-conventions.html}{version 1.5} of the CF metadata standard. Therefore while a user may wish to add their own metadata to the output files (as demonstrated in example 4 of section \autoref{subsec:IOM_xmlref}) the metadata should, for the most part, comply with the CF-1.5 standard. 1001 1001 1002 1002 Some metadata that may significantly increase the file size (horizontal cell areas and vertices) are controlled by the namelist parameter \np{ln\_cfmeta} in the \ngn{namrun} namelist. This must be set to true if these metadata are to be included in the output files. … … 1007 1007 % ================================================================ 1008 1008 \section{NetCDF4 support (\protect\key{netcdf4})} 1009 \label{ DIA_iom}1009 \label{sec:DIA_iom} 1010 1010 1011 1011 Since version 3.3, support for NetCDF4 chunking and (loss-less) compression has … … 1070 1070 respectively in the mono-processor case (i.e. global domain of {\small\tt 182x149x31}). 1071 1071 An illustration of the potential space savings that NetCDF4 chunking and compression 1072 provides is given in table \ ref{Tab_NC4} which compares the results of two short1072 provides is given in table \autoref{tab:NC4} which compares the results of two short 1073 1073 runs of the ORCA2\_LIM reference configuration with a 4x2 mpi partitioning. Note 1074 1074 the variation in the compression ratio achieved which reflects chiefly the dry to wet … … 1106 1106 ORCA2\_2d\_grid\_W\_0007.nc & 4416 & 1368 & 70\%\\ 1107 1107 \end{tabular} 1108 \caption{ \protect\label{ Tab_NC4}1108 \caption{ \protect\label{tab:NC4} 1109 1109 Filesize comparison between NetCDF3 and NetCDF4 with chunking and compression} 1110 1110 \end{table} … … 1126 1126 % ------------------------------------------------------------------------------------------------------------- 1127 1127 \section{Tracer/Dynamics trends (\protect\ngn{namtrd})} 1128 \label{ DIA_trd}1128 \label{sec:DIA_trd} 1129 1129 1130 1130 %------------------------------------------namtrd---------------------------------------------------- … … 1166 1166 % ------------------------------------------------------------------------------------------------------------- 1167 1167 \section{FLO: On-Line Floats trajectories (\protect\key{floats})} 1168 \label{ FLO}1168 \label{sec:FLO} 1169 1169 %--------------------------------------------namflo------------------------------------------------------- 1170 1170 \forfile{../namelists/namflo} … … 1274 1274 % ------------------------------------------------------------------------------------------------------------- 1275 1275 \section{Harmonic analysis of tidal constituents (\protect\key{diaharm}) } 1276 \label{ DIA_diag_harm}1276 \label{sec:DIA_diag_harm} 1277 1277 1278 1278 %------------------------------------------namdia_harm---------------------------------------------------- … … 1316 1316 % ------------------------------------------------------------------------------------------------------------- 1317 1317 \section{Transports across sections (\protect\key{diadct}) } 1318 \label{ DIA_diag_dct}1318 \label{sec:DIA_diag_dct} 1319 1319 1320 1320 %------------------------------------------namdct---------------------------------------------------- … … 1467 1467 % ================================================================ 1468 1468 \section{Diagnosing the steric effect in sea surface height} 1469 \label{ DIA_steric}1469 \label{sec:DIA_steric} 1470 1470 1471 1471 … … 1500 1500 1501 1501 A non-Boussinesq fluid conserves mass. It satisfies the following relations: 1502 \begin{equation} \label{ Eq_MV_nBq}1502 \begin{equation} \label{eq:MV_nBq} 1503 1503 \begin{split} 1504 1504 \mathcal{M} &= \mathcal{V} \;\bar{\rho} \\ … … 1507 1507 \end{equation} 1508 1508 Temporal changes in total mass is obtained from the density conservation equation : 1509 \begin{equation} \label{ Eq_Co_nBq}1509 \begin{equation} \label{eq:Co_nBq} 1510 1510 \frac{1}{e_3} \partial_t ( e_3\,\rho) + \nabla( \rho \, \textbf{U} ) = \left. \frac{\textit{emp}}{e_3}\right|_\textit{surface} 1511 1511 \end{equation} … … 1513 1513 exchanges with the other media of the Earth system (atmosphere, sea-ice, land). 1514 1514 Its global averaged leads to the total mass change 1515 \begin{equation} \label{ Eq_Mass_nBq}1515 \begin{equation} \label{eq:Mass_nBq} 1516 1516 \partial_t \mathcal{M} = \mathcal{A} \;\overline{\textit{emp}} 1517 1517 \end{equation} 1518 1518 where $\overline{\textit{emp}}=\int_S \textit{emp}\,ds$ is the net mass flux 1519 1519 through the ocean surface. 1520 Bringing \ eqref{Eq_Mass_nBq} and the time derivative of \eqref{Eq_MV_nBq}1520 Bringing \autoref{eq:Mass_nBq} and the time derivative of \autoref{eq:MV_nBq} 1521 1521 together leads to the evolution equation of the mean sea level 1522 \begin{equation} \label{ Eq_ssh_nBq}1522 \begin{equation} \label{eq:ssh_nBq} 1523 1523 \partial_t \bar{\eta} = \frac{\overline{\textit{emp}}}{ \bar{\rho}} 1524 1524 - \frac{\mathcal{V}}{\mathcal{A}} \;\frac{\partial_t \bar{\rho} }{\bar{\rho}} 1525 1525 \end{equation} 1526 The first term in equation \ eqref{Eq_ssh_nBq} alters sea level by adding or1526 The first term in equation \autoref{eq:ssh_nBq} alters sea level by adding or 1527 1527 subtracting mass from the ocean. 1528 1528 The second term arises from temporal changes in the global mean … … 1531 1531 In a Boussinesq fluid, $\rho$ is replaced by $\rho_o$ in all the equation except when $\rho$ 1532 1532 appears multiplied by the gravity ($i.e.$ in the hydrostatic balance of the primitive Equations). 1533 In particular, the mass conservation equation, \ eqref{Eq_Co_nBq}, degenerates into1533 In particular, the mass conservation equation, \autoref{eq:Co_nBq}, degenerates into 1534 1534 the incompressibility equation: 1535 \begin{equation} \label{ Eq_Co_Bq}1535 \begin{equation} \label{eq:Co_Bq} 1536 1536 \frac{1}{e_3} \partial_t ( e_3 ) + \nabla( \textbf{U} ) = \left. \frac{\textit{emp}}{\rho_o \,e_3}\right|_ \textit{surface} 1537 1537 \end{equation} 1538 1538 and the global average of this equation now gives the temporal change of the total volume, 1539 \begin{equation} \label{ Eq_V_Bq}1539 \begin{equation} \label{eq:V_Bq} 1540 1540 \partial_t \mathcal{V} = \mathcal{A} \;\frac{\overline{\textit{emp}}}{\rho_o} 1541 1541 \end{equation} … … 1553 1553 by the Boussinesq model, via the steric contribution to the sea level, $\eta_s$, a spatially 1554 1554 uniform variable, as follows: 1555 \begin{equation} \label{ Eq_M_Bq}1555 \begin{equation} \label{eq:M_Bq} 1556 1556 \mathcal{M}_o = \mathcal{M} + \rho_o \,\eta_s \,\mathcal{A} 1557 1557 \end{equation} … … 1559 1559 the ocean surface is converted into a mean change in sea level. Introducing the total density 1560 1560 anomaly, $\mathcal{D}= \int_D d_a \,dv$, where $d_a= (\rho -\rho_o ) / \rho_o$ 1561 is the density anomaly used in \NEMO (cf. \ S\ref{TRA_eos}) in \eqref{Eq_M_Bq}1561 is the density anomaly used in \NEMO (cf. \autoref{subsec:TRA_eos}) in \autoref{eq:M_Bq} 1562 1562 leads to a very simple form for the steric height: 1563 \begin{equation} \label{ Eq_steric_Bq}1563 \begin{equation} \label{eq:steric_Bq} 1564 1564 \eta_s = - \frac{1}{\mathcal{A}} \mathcal{D} 1565 1565 \end{equation} … … 1581 1581 (wetting and drying of grid point is not allowed). 1582 1582 1583 Third, the discretisation of \ eqref{Eq_steric_Bq} depends on the type of free surface1583 Third, the discretisation of \autoref{eq:steric_Bq} depends on the type of free surface 1584 1584 which is considered. In the non linear free surface case, $i.e.$ \key{vvl} defined, it is 1585 1585 given by 1586 \begin{equation} \label{ Eq_discrete_steric_Bq}1586 \begin{equation} \label{eq:discrete_steric_Bq} 1587 1587 \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t} e_{2t} e_{3t} } 1588 1588 { \sum_{i,\,j,\,k} e_{1t} e_{2t} e_{3t} } … … 1590 1590 whereas in the linear free surface, the volume above the \textit{z=0} surface must be explicitly taken 1591 1591 into account to better approximate the total ocean mass and thus the steric sea level: 1592 \begin{equation} \label{ Eq_discrete_steric_Bq}1592 \begin{equation} \label{eq:discrete_steric_Bq} 1593 1593 \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} d_a\; e_{1t}e_{2t} \eta } 1594 1594 {\sum_{i,\,j,\,k} e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} e_{1t}e_{2t} \eta } … … 1608 1608 In AR5 outputs, the thermosteric sea level is demanded. It is steric sea level due to 1609 1609 changes in ocean density arising just from changes in temperature. It is given by: 1610 \begin{equation} \label{ Eq_thermosteric_Bq}1610 \begin{equation} \label{eq:thermosteric_Bq} 1611 1611 \eta_s = - \frac{1}{\mathcal{A}} \int_D d_a(T,S_o,p_o) \,dv 1612 1612 \end{equation} … … 1622 1622 % ------------------------------------------------------------------------------------------------------------- 1623 1623 \section{Other diagnostics (\protect\key{diahth}, \protect\key{diaar5})} 1624 \label{ DIA_diag_others}1624 \label{sec:DIA_diag_others} 1625 1625 1626 1626 … … 1658 1658 as well as for the World Ocean. The sub-basin decomposition requires an input file 1659 1659 (\ifile{subbasins}) which contains three 2D mask arrays, the Indo-Pacific mask 1660 been deduced from the sum of the Indian and Pacific mask ( Fig~\ref{Fig_mask_subasins}).1660 been deduced from the sum of the Indian and Pacific mask (\autoref{fig:mask_subasins}). 1661 1661 1662 1662 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1663 1663 \begin{figure}[!t] \begin{center} 1664 1664 \includegraphics[width=1.0\textwidth]{Fig_mask_subasins} 1665 \caption{ \protect\label{ Fig_mask_subasins}1665 \caption{ \protect\label{fig:mask_subasins} 1666 1666 Decomposition of the World Ocean (here ORCA2) into sub-basin used in to compute 1667 1667 the heat and salt transports as well as the meridional stream-function: Atlantic basin (red), … … 1681 1681 A series of diagnostics has been added in the \mdl{diaar5}. 1682 1682 They corresponds to outputs that are required for AR5 simulations (CMIP5) 1683 (see also Section \ref{DIA_steric} for one of them).1683 (see also \autoref{sec:DIA_steric} for one of them). 1684 1684 Activating those outputs requires to define the \key{diaar5} CPP key. 1685 1685 -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_DIU.tex
r9394 r9407 6 6 % ================================================================ 7 7 \chapter{Diurnal SST Models (DIU)} 8 \label{ DIU}8 \label{chap:DIU} 9 9 10 10 \minitoc … … 54 54 %=============================================================== 55 55 \section{Warm layer model} 56 \label{ warm_layer_sec}56 \label{sec:warm_layer_sec} 57 57 %=============================================================== 58 58 … … 62 62 \frac{\partial{\Delta T_{\rm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p 63 63 \nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,} 64 \label{e cmwf1} \\65 L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{e cmwf2}64 \label{eq:ecmwf1} \\ 65 L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:ecmwf2} 66 66 \end{eqnarray} 67 67 where $\Delta T_{\rm{wl}}$ is the temperature difference between the top of the warm 68 68 layer and the depth $D_T=3$\,m at which there is assumed to be no diurnal signal. In 69 equation (\ ref{ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion69 equation (\autoref{eq:ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion 70 70 coefficient of water, $\kappa=0.4$ is von K\'{a}rm\'{a}n's constant, $c_p$ is the heat 71 71 capacity at constant pressure of sea water, $\rho_w$ is the … … 81 81 $u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is 82 82 the drag coefficient, and $\rho_a$ is the density of air. The symbol $Q$ in equation 83 (\ ref{ecmwf1}) is the instantaneous total thermal energy83 (\autoref{eq:ecmwf1}) is the instantaneous total thermal energy 84 84 flux into 85 85 the diurnal layer, $i.e.$ 86 86 \begin{equation} 87 Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,} \label{e _flux_eqn}87 Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,} \label{eq:e_flux_eqn} 88 88 \end{equation} 89 89 where $Q_{\rm{h}}$ is the sensible and latent heat flux, $Q_{\rm{lw}}$ is the long 90 90 wave flux, and $Q_{\rm{sol}}$ is the solar flux absorbed 91 91 within the diurnal warm layer. For $Q_{\rm{sol}}$ the 9 term 92 representation of \citet{Gentemann_al_JGR09} is used. In equation \ ref{ecmwf1}92 representation of \citet{Gentemann_al_JGR09} is used. In equation \autoref{eq:ecmwf1} 93 93 the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$, where $L_a=0.3$\footnote{This 94 94 is a global average value, more accurately $L_a$ could be computed as … … 103 103 4\zeta^2}{1+3\zeta+0.25\zeta^2} &(\zeta \ge 0) \\ 104 104 (1 - 16\zeta)^{-\frac{1}{2}} & (\zeta < 0) \mbox{,} 105 \end{array} \right. \label{ stab_func_eqn}105 \end{array} \right. \label{eq:stab_func_eqn} 106 106 \end{equation} 107 107 where $\zeta=\frac{D_T}{L}$. It is clear that the first derivative of 108 (\ ref{stab_func_eqn}), and thus of (\ref{ecmwf1}),109 is discontinuous at $\zeta=0$ ($i.e.$ $Q\rightarrow0$ in equation (\ ref{ecmwf2})).108 (\autoref{eq:stab_func_eqn}), and thus of (\autoref{eq:ecmwf1}), 109 is discontinuous at $\zeta=0$ ($i.e.$ $Q\rightarrow0$ in equation (\autoref{eq:ecmwf2})). 110 110 111 The two terms on the right hand side of (\ ref{ecmwf1}) represent different processes.111 The two terms on the right hand side of (\autoref{eq:ecmwf1}) represent different processes. 112 112 The first term is simply the diabatic heating or cooling of the 113 113 diurnal warm … … 121 121 122 122 \section{Cool skin model} 123 \label{ cool_skin_sec}123 \label{sec:cool_skin_sec} 124 124 125 125 %=============================================================== … … 131 131 Saunders equation for the cool skin temperature difference $\Delta T_{\rm{cs}}$ becomes 132 132 \begin{equation} 133 \label{ sunders_eqn}133 \label{eq:sunders_eqn} 134 134 \Delta T_{\rm{cs}}=\frac{Q_{\rm{ns}}\delta}{k_t} \mbox{,} 135 135 \end{equation} … … 138 138 skin layer and is given by 139 139 \begin{equation} 140 \label{ sunders_thick_eqn}140 \label{eq:sunders_thick_eqn} 141 141 \delta=\frac{\lambda \mu}{u^*_{w}} \mbox{,} 142 142 \end{equation} … … 144 144 proportionality which \citet{Saunders_JAS82} suggested varied between 5 and 10. 145 145 146 The value of $\lambda$ used in equation (\ ref{sunders_thick_eqn}) is that of146 The value of $\lambda$ used in equation (\autoref{eq:sunders_thick_eqn}) is that of 147 147 \citet{Artale_al_JGR02}, 148 148 which is shown in \citet{Tu_Tsuang_GRL05} to outperform a number of other 149 149 parametrisations at both low and high wind speeds. Specifically, 150 150 \begin{equation} 151 \label{ artale_lambda_eqn}151 \label{eq:artale_lambda_eqn} 152 152 \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,} 153 153 \end{equation} … … 155 155 $\gamma$ is a dimensionless function of wind speed $u$: 156 156 \begin{equation} 157 \label{ artale_gamma_eqn}157 \label{eq:artale_gamma_eqn} 158 158 \gamma = \left\{ \begin{matrix} 159 159 0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\ -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_DOM.tex
r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Space Domain (DOM)} 7 \label{ DOM}7 \label{chap:DOM} 8 8 \minitoc 9 9 … … 20 20 $\ $\newline % force a new line 21 21 22 Having defined the continuous equations in Chap.~\ref{PE} and chosen a time23 discretization Chap.~\ref{STP}, we need to choose a discretization on a grid,22 Having defined the continuous equations in \autoref{chap:PE} and chosen a time 23 discretization \autoref{chap:STP}, we need to choose a discretization on a grid, 24 24 and numerical algorithms. In the present chapter, we provide a general description 25 25 of the staggered grid used in \NEMO, and other information relevant to the main … … 32 32 % ================================================================ 33 33 \section{Fundamentals of the discretisation} 34 \label{ DOM_basics}34 \label{sec:DOM_basics} 35 35 36 36 % ------------------------------------------------------------------------------------------------------------- … … 38 38 % ------------------------------------------------------------------------------------------------------------- 39 39 \subsection{Arrangement of variables} 40 \label{ DOM_cell}40 \label{subsec:DOM_cell} 41 41 42 42 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 43 43 \begin{figure}[!tb] \begin{center} 44 44 \includegraphics[width=0.90\textwidth]{Fig_cell} 45 \caption{ \protect\label{ Fig_cell}45 \caption{ \protect\label{fig:cell} 46 46 Arrangement of variables. $t$ indicates scalar points where temperature, 47 47 salinity, density, pressure and horizontal divergence are defined. ($u$,$v$,$w$) … … 56 56 space directions. The arrangement of variables is the same in all directions. 57 57 It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector 58 points $(u, v, w)$ defined in the centre of each face of the cells ( Fig. \ref{Fig_cell}).58 points $(u, v, w)$ defined in the centre of each face of the cells (\autoref{fig:cell}). 59 59 This is the generalisation to three dimensions of the well-known ``C'' grid in 60 60 Arakawa's classification \citep{Mesinger_Arakawa_Bk76}. The relative and … … 66 66 by the transformation that gives ($\lambda$ ,$\varphi$ ,$z$) as a function of $(i,j,k)$. 67 67 The grid-points are located at integer or integer and a half value of $(i,j,k)$ as 68 indicated on Table \ref{Tab_cell}. In all the following, subscripts $u$, $v$, $w$,68 indicated on \autoref{tab:cell}. In all the following, subscripts $u$, $v$, $w$, 69 69 $f$, $uw$, $vw$ or $fw$ indicate the position of the grid-point where the scale 70 70 factors are defined. Each scale factor is defined as the local analytical value 71 provided by \ eqref{Eq_scale_factors}. As a result, the mesh on which partial71 provided by \autoref{eq:scale_factors}. As a result, the mesh on which partial 72 72 derivatives $\frac{\partial}{\partial \lambda}, \frac{\partial}{\partial \varphi}$, and 73 73 $\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size of unity. … … 78 78 from their analytical expression. This preserves the symmetry of the discrete set 79 79 of equations and therefore satisfies many of the continuous properties (see 80 Appendix~\ref{Apdx_C}). A similar, related remark can be made about the domain80 \autoref{apdx:C}). A similar, related remark can be made about the domain 81 81 size: when needed, an area, volume, or the total ocean depth must be evaluated 82 as the sum of the relevant scale factors (see \ eqref{DOM_bar}) in the next section).82 as the sum of the relevant scale factors (see \autoref{eq:DOM_bar}) in the next section). 83 83 84 84 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 95 95 fw & $i+1/2$ & $j+1/2$ & $k+1/2$ \\ \hline 96 96 \end{tabular} 97 \caption{ \protect\label{ Tab_cell}97 \caption{ \protect\label{tab:cell} 98 98 Location of grid-points as a function of integer or integer and a half value of the column, 99 99 line or level. This indexing is only used for the writing of the semi-discrete equation. 100 100 In the code, the indexing uses integer values only and has a reverse direction 101 in the vertical (see \ S\ref{DOM_Num_Index})}101 in the vertical (see \autoref{subsec:DOM_Num_Index})} 102 102 \end{center} 103 103 \end{table} … … 108 108 % ------------------------------------------------------------------------------------------------------------- 109 109 \subsection{Discrete operators} 110 \label{ DOM_operators}110 \label{subsec:DOM_operators} 111 111 112 112 Given the values of a variable $q$ at adjacent points, the differencing and 113 113 averaging operators at the midpoint between them are: 114 \begin{subequations} \label{ Eq_di_mi}114 \begin{subequations} \label{eq:di_mi} 115 115 \begin{align} 116 116 \delta _i [q] &= \ \ q(i+1/2) - q(i-1/2) \\ … … 120 120 121 121 Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and 122 $k+1/2$. Following \ eqref{Eq_PE_grad} and \eqref{Eq_PE_lap}, the gradient of a122 $k+1/2$. Following \autoref{eq:PE_grad} and \autoref{eq:PE_lap}, the gradient of a 123 123 variable $q$ defined at a $t$-point has its three components defined at $u$-, $v$- 124 124 and $w$-points while its Laplacien is defined at $t$-point. These operators have 125 125 the following discrete forms in the curvilinear $s$-coordinate system: 126 \begin{equation} \label{ Eq_DOM_grad}126 \begin{equation} \label{eq:DOM_grad} 127 127 \nabla q\equiv \frac{1}{e_{1u} } \delta _{i+1/2 } [q] \;\,\mathbf{i} 128 128 + \frac{1}{e_{2v} } \delta _{j+1/2 } [q] \;\,\mathbf{j} 129 129 + \frac{1}{e_{3w}} \delta _{k+1/2} [q] \;\,\mathbf{k} 130 130 \end{equation} 131 \begin{multline} \label{ Eq_DOM_lap}131 \begin{multline} \label{eq:DOM_lap} 132 132 \Delta q\equiv \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 133 133 \;\left( \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \;\delta_{i+1/2} [q] \right] … … 136 136 \end{multline} 137 137 138 Following \ eqref{Eq_PE_curl} and \eqref{Eq_PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$138 Following \autoref{eq:PE_curl} and \autoref{eq:PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$ 139 139 defined at vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$, 140 140 and $f$-points, and its divergence defined at $t$-points: 141 \begin{eqnarray} \label{ Eq_DOM_curl}141 \begin{eqnarray} \label{eq:DOM_curl} 142 142 \nabla \times {\rm{\bf A}}\equiv & 143 143 \frac{1}{e_{2v}\,e_{3vw} } \ \left( \delta_{j +1/2} \left[e_{3w}\,a_3 \right] -\delta_{k+1/2} \left[e_{2v} \,a_2 \right] \right) &\ \mathbf{i} \\ … … 145 145 +& \frac{1}{e_{1f} \,e_{2f} } \ \left( \delta_{i +1/2} \left[e_{2v}\,a_2 \right] -\delta_{j +1/2} \left[e_{1u}\,a_1 \right] \right) &\ \mathbf{k} 146 146 \end{eqnarray} 147 \begin{eqnarray} \label{ Eq_DOM_div}147 \begin{eqnarray} \label{eq:DOM_div} 148 148 \nabla \cdot \rm{\bf A} \equiv 149 149 \frac{1}{e_{1t}\,e_{2t}\,e_{3t}} \left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right] … … 153 153 The vertical average over the whole water column denoted by an overbar becomes 154 154 for a quantity $q$ which is a masked field (i.e. equal to zero inside solid area): 155 \begin{equation} \label{ DOM_bar}155 \begin{equation} \label{eq:DOM_bar} 156 156 \bar q = \frac{1}{H} \int_{k^b}^{k^o} {q\;e_{3q} \,dk} 157 157 \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} } … … 163 163 164 164 In continuous form, the following properties are satisfied: 165 \begin{equation} \label{ Eq_DOM_curl_grad}165 \begin{equation} \label{eq:DOM_curl_grad} 166 166 \nabla \times \nabla q ={\rm {\bf {0}}} 167 167 \end{equation} 168 \begin{equation} \label{ Eq_DOM_div_curl}168 \begin{equation} \label{eq:DOM_div_curl} 169 169 \nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0 170 170 \end{equation} … … 181 181 operators, $i.e.$ 182 182 \begin{align} 183 \label{ DOM_di_adj}183 \label{eq:DOM_di_adj} 184 184 \sum\limits_i { a_i \;\delta _i \left[ b \right]} 185 185 &\equiv -\sum\limits_i {\delta _{i+1/2} \left[ a \right]\;b_{i+1/2} } \\ 186 \label{ DOM_mi_adj}186 \label{eq:DOM_mi_adj} 187 187 \sum\limits_i { a_i \;\overline b^{\,i}} 188 188 & \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} } … … 192 192 $\delta_i^*=\delta_{i+1/2}$ and 193 193 ${(\overline{\,\cdot \,}^{\,i})}^*= \overline{\,\cdot\,}^{\,i+1/2}$, respectively. 194 These two properties will be used extensively in the Appendix~\ref{Apdx_C} to194 These two properties will be used extensively in the \autoref{apdx:C} to 195 195 demonstrate integral conservative properties of the discrete formulation chosen. 196 196 … … 199 199 % ------------------------------------------------------------------------------------------------------------- 200 200 \subsection{Numerical indexing} 201 \label{ DOM_Num_Index}201 \label{subsec:DOM_Num_Index} 202 202 203 203 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 204 204 \begin{figure}[!tb] \begin{center} 205 205 \includegraphics[width=0.90\textwidth]{Fig_index_hor} 206 \caption{ \protect\label{ Fig_index_hor}206 \caption{ \protect\label{fig:index_hor} 207 207 Horizontal integer indexing used in the \textsc{Fortran} code. The dashed area indicates 208 208 the cell in which variables contained in arrays have the same $i$- and $j$-indices} … … 211 211 212 212 The array representation used in the \textsc{Fortran} code requires an integer 213 indexing while the analytical definition of the mesh (see \ S\ref{DOM_cell}) is213 indexing while the analytical definition of the mesh (see \autoref{subsec:DOM_cell}) is 214 214 associated with the use of integer values for $t$-points and both integer and 215 215 integer and a half values for all the other points. Therefore a specific integer … … 222 222 % ----------------------------------- 223 223 \subsubsection{Horizontal indexing} 224 \label{ DOM_Num_Index_hor}225 226 The indexing in the horizontal plane has been chosen as shown in Fig.\ref{Fig_index_hor}.224 \label{subsec:DOM_Num_Index_hor} 225 226 The indexing in the horizontal plane has been chosen as shown in \autoref{fig:index_hor}. 227 227 For an increasing $i$ index ($j$ index), the $t$-point and the eastward $u$-point 228 (northward $v$-point) have the same index (see the dashed area in Fig.\ref{Fig_index_hor}).228 (northward $v$-point) have the same index (see the dashed area in \autoref{fig:index_hor}). 229 229 A $t$-point and its nearest northeast $f$-point have the same $i$-and $j$-indices. 230 230 … … 233 233 % ----------------------------------- 234 234 \subsubsection{Vertical indexing} 235 \label{ DOM_Num_Index_vertical}235 \label{subsec:DOM_Num_Index_vertical} 236 236 237 237 In the vertical, the chosen indexing requires special attention since the 238 238 $k$-axis is re-orientated downward in the \textsc{Fortran} code compared 239 to the indexing used in the semi-discrete equations and given in \ S\ref{DOM_cell}.239 to the indexing used in the semi-discrete equations and given in \autoref{subsec:DOM_cell}. 240 240 The sea surface corresponds to the $w$-level $k=1$ which is the same index 241 as $t$-level just below ( Fig.\ref{Fig_index_vert}). The last $w$-level ($k=jpk$)241 as $t$-level just below (\autoref{fig:index_vert}). The last $w$-level ($k=jpk$) 242 242 either corresponds to the ocean floor or is inside the bathymetry while the last 243 $t$-level is always inside the bathymetry ( Fig.\ref{Fig_index_vert}). Note that243 $t$-level is always inside the bathymetry (\autoref{fig:index_vert}). Note that 244 244 for an increasing $k$ index, a $w$-point and the $t$-point just below have the 245 245 same $k$ index, in opposition to what is done in the horizontal plane where 246 246 it is the $t$-point and the nearest velocity points in the direction of the horizontal 247 247 axis that have the same $i$ or $j$ index (compare the dashed area in 248 Fig.\ref{Fig_index_hor} and \ref{Fig_index_vert}). Since the scale factors are248 \autoref{fig:index_hor} and \autoref{fig:index_vert}). Since the scale factors are 249 249 chosen to be strictly positive, a \emph{minus sign} appears in the \textsc{Fortran} 250 250 code \emph{before all the vertical derivatives} of the discrete equations given in … … 254 254 \begin{figure}[!pt] \begin{center} 255 255 \includegraphics[width=.90\textwidth]{Fig_index_vert} 256 \caption{ \protect\label{ Fig_index_vert}256 \caption{ \protect\label{fig:index_vert} 257 257 Vertical integer indexing used in the \textsc{Fortran } code. Note that 258 258 the $k$-axis is orientated downward. The dashed area indicates the cell in … … 265 265 % ----------------------------------- 266 266 \subsubsection{Domain size} 267 \label{ DOM_size}267 \label{subsec:DOM_size} 268 268 269 269 The total size of the computational domain is set by the parameters \np{jpiglo}, … … 273 273 %%% 274 274 Parameters $jpi$ and $jpj$ refer to the size of each processor subdomain when the code is 275 run in parallel using domain decomposition (\key{mpp\_mpi} defined, see \ S\ref{LBC_mpp}).275 run in parallel using domain decomposition (\key{mpp\_mpi} defined, see \autoref{sec:LBC_mpp}). 276 276 277 277 … … 282 282 % ================================================================ 283 283 \section{Needed fields} 284 \label{ DOM_fields}284 \label{sec:DOM_fields} 285 285 The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined 286 286 by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$. 287 287 The grid-points are located at integer or integer and a half values of as indicated 288 in Table~\ref{Tab_cell}. The associated scale factors are defined using the289 analytical first derivative of the transformation \ eqref{Eq_scale_factors}.288 in \autoref{tab:cell}. The associated scale factors are defined using the 289 analytical first derivative of the transformation \autoref{eq:scale_factors}. 290 290 Necessary fields for configuration definition are: \\ 291 291 Geographic position : … … 316 316 % ------------------------------------------------------------------------------------------------------------- 317 317 %\subsection{List of needed fields to build DOMAIN} 318 %\label{ DOM_fields_list}318 %\label{subsec:DOM_fields_list} 319 319 320 320 … … 323 323 % ================================================================ 324 324 \section{Horizontal grid mesh (\protect\mdl{domhgr})} 325 \label{ DOM_hgr}325 \label{sec:DOM_hgr} 326 326 327 327 % ------------------------------------------------------------------------------------------------------------- … … 329 329 % ------------------------------------------------------------------------------------------------------------- 330 330 \subsection{Coordinates and scale factors} 331 \label{ DOM_hgr_coord_e}331 \label{subsec:DOM_hgr_coord_e} 332 332 333 333 The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined 334 334 by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$. 335 335 The grid-points are located at integer or integer and a half values of as indicated 336 in Table~\ref{Tab_cell}. The associated scale factors are defined using the337 analytical first derivative of the transformation \ eqref{Eq_scale_factors}. These336 in \autoref{tab:cell}. The associated scale factors are defined using the 337 analytical first derivative of the transformation \autoref{eq:scale_factors}. These 338 338 definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr}, which 339 339 provide the horizontal and vertical meshes, respectively. This section deals with … … 343 343 analytical expressions of the longitude $\lambda$ and latitude $\varphi$ as a 344 344 function of $(i,j)$. The horizontal scale factors are calculated using 345 \ eqref{Eq_scale_factors}. For example, when the longitude and latitude are345 \autoref{eq:scale_factors}. For example, when the longitude and latitude are 346 346 function of a single value ($i$ and $j$, respectively) (geographical configuration 347 347 of the mesh), the horizontal mesh definition reduces to define the wanted … … 382 382 allowing the user to set arbitrary jumps in thickness between adjacent layers) 383 383 \citep{Treguier1996}. An example of the effect of such a choice is shown in 384 Fig.~\ref{Fig_zgr_e3}.384 \autoref{fig:zgr_e3}. 385 385 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 386 386 \begin{figure}[!t] \begin{center} 387 387 \includegraphics[width=0.90\textwidth]{Fig_zgr_e3} 388 \caption{ \protect\label{ Fig_zgr_e3}388 \caption{ \protect\label{fig:zgr_e3} 389 389 Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical, 390 390 and (b) analytically derived grid-point position and scale factors. … … 401 401 % ------------------------------------------------------------------------------------------------------------- 402 402 \subsection{Choice of horizontal grid} 403 \label{ DOM_hgr_msh_choice}403 \label{subsec:DOM_hgr_msh_choice} 404 404 405 405 … … 408 408 % ------------------------------------------------------------------------------------------------------------- 409 409 \subsection{Output grid files} 410 \label{ DOM_hgr_files}410 \label{subsec:DOM_hgr_files} 411 411 412 412 All the arrays relating to a particular ocean model configuration (grid-point … … 426 426 % ================================================================ 427 427 \section{Vertical grid (\protect\mdl{domzgr})} 428 \label{ DOM_zgr}428 \label{sec:DOM_zgr} 429 429 %-----------------------------------------nam_zgr & namdom------------------------------------------- 430 430 %\forfile{../namelists/namzgr} … … 444 444 \begin{figure}[!tb] \begin{center} 445 445 \includegraphics[width=1.0\textwidth]{Fig_z_zps_s_sps} 446 \caption{ \protect\label{ Fig_z_zps_s_sps}446 \caption{ \protect\label{fig:z_zps_s_sps} 447 447 The ocean bottom as seen by the model: 448 448 (a) $z$-coordinate with full step, … … 451 451 (d) hybrid $s-z$ coordinate, 452 452 (e) hybrid $s-z$ coordinate with partial step, and 453 (f) same as (e) but in the non-linear free surface (\ np{ln\_linssh}\forcode{ = .false.}).453 (f) same as (e) but in the non-linear free surface (\protect\np{ln\_linssh}\forcode{ = .false.}). 454 454 Note that the non-linear free surface can be used with any of the 455 455 5 coordinates (a) to (e).} … … 460 460 must be done once of all at the beginning of an experiment. It is not intended as an 461 461 option which can be enabled or disabled in the middle of an experiment. Three main 462 choices are offered ( Fig.~\ref{Fig_z_zps_s_sps}a to c): $z$-coordinate with full step462 choices are offered (\autoref{fig:z_zps_s_sps}a to c): $z$-coordinate with full step 463 463 bathymetry (\np{ln\_zco}\forcode{ = .true.}), $z$-coordinate with partial step bathymetry 464 464 (\np{ln\_zps}\forcode{ = .true.}), or generalized, $s$-coordinate (\np{ln\_sco}\forcode{ = .true.}). 465 465 Hybridation of the three main coordinates are available: $s-z$ or $s-zps$ coordinate 466 ( Fig.~\ref{Fig_z_zps_s_sps}d and \ref{Fig_z_zps_s_sps}e). By default a non-linear free surface is used:466 (\autoref{fig:z_zps_s_sps}d and \autoref{fig:z_zps_s_sps}e). By default a non-linear free surface is used: 467 467 the coordinate follow the time-variation of the free surface so that the transformation is time dependent: 468 $z(i,j,k,t)$ ( Fig.~\ref{Fig_z_zps_s_sps}f). When a linear free surface is assumed (\np{ln\_linssh}\forcode{ = .true.}),468 $z(i,j,k,t)$ (\autoref{fig:z_zps_s_sps}f). When a linear free surface is assumed (\np{ln\_linssh}\forcode{ = .true.}), 469 469 the vertical coordinate are fixed in time, but the seawater can move up and down across the z=0 surface 470 470 (in other words, the top of the ocean in not a rigid-lid). … … 513 513 % ------------------------------------------------------------------------------------------------------------- 514 514 \subsection{Meter bathymetry} 515 \label{ DOM_bathy}515 \label{subsec:DOM_bathy} 516 516 517 517 Three options are possible for defining the bathymetry, according to the … … 541 541 This is unnecessary when the ocean is forced by fixed atmospheric conditions, 542 542 so these seas can be removed from the ocean domain. The user has the option 543 to set the bathymetry in closed seas to zero (see \ S\ref{MISC_closea}), but the543 to set the bathymetry in closed seas to zero (see \autoref{sec:MISC_closea}), but the 544 544 code has to be adapted to the user's configuration. 545 545 … … 549 549 \subsection[$Z$-coordinate (\protect\np{ln\_zco}\forcode{ = .true.}) and ref. coordinate] 550 550 {$Z$-coordinate (\protect\np{ln\_zco}\forcode{ = .true.}) and reference coordinate} 551 \label{ DOM_zco}551 \label{subsec:DOM_zco} 552 552 553 553 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 554 554 \begin{figure}[!tb] \begin{center} 555 555 \includegraphics[width=0.90\textwidth]{Fig_zgr} 556 \caption{ \protect\label{ Fig_zgr}556 \caption{ \protect\label{fig:zgr} 557 557 Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level functions for 558 558 (a) T-point depth and (b) the associated scale factor as computed 559 from \ eqref{DOM_zgr_ana} using \eqref{DOM_zgr_coef} in $z$-coordinate.}559 from \autoref{eq:DOM_zgr_ana} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.} 560 560 \end{center} \end{figure} 561 561 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 563 563 The reference coordinate transformation $z_0 (k)$ defines the arrays $gdept_0$ 564 564 and $gdepw_0$ for $t$- and $w$-points, respectively. As indicated on 565 Fig.\ref{Fig_index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the565 \autoref{fig:index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the 566 566 ocean surface. There are at most \jp{jpk}-1 $t$-points inside the ocean, the 567 567 additional $t$-point at $jk=jpk$ is below the sea floor and is not used. … … 579 579 near the ocean surface. The following function is proposed as a standard for a 580 580 $z$-coordinate (with either full or partial steps): 581 \begin{equation} \label{ DOM_zgr_ana}581 \begin{equation} \label{eq:DOM_zgr_ana} 582 582 \begin{split} 583 583 z_0 (k) &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\ … … 588 588 expression allows us to define a nearly uniform vertical location of levels at the 589 589 ocean top and bottom with a smooth hyperbolic tangent transition in between 590 ( Fig.~\ref{Fig_zgr}).590 (\autoref{fig:zgr}). 591 591 592 592 If the ice shelf cavities are opened (\np{ln\_isfcav}\forcode{ = .true.}), the definition of $z_0$ is the same. 593 593 However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to: 594 \begin{equation} \label{ DOM_zgr_ana}594 \begin{equation} \label{eq:DOM_zgr_ana} 595 595 \begin{split} 596 596 e_3^T(k) &= z_W (k+1) - z_W (k) \\ … … 605 605 surface (bottom) layers and a depth which varies from 0 at the sea surface to a 606 606 minimum of $-5000~m$. This leads to the following conditions: 607 \begin{equation} \label{ DOM_zgr_coef}607 \begin{equation} \label{eq:DOM_zgr_coef} 608 608 \begin{split} 609 609 e_3 (1+1/2) &=10. \\ … … 616 616 With the choice of the stretching $h_{cr} =3$ and the number of levels 617 617 \jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in 618 \ eqref{DOM_zgr_ana} have been determined such that \eqref{DOM_zgr_coef} is618 \autoref{eq:DOM_zgr_ana} have been determined such that \autoref{eq:DOM_zgr_coef} is 619 619 satisfied, through an optimisation procedure using a bisection method. For the first 620 620 standard ORCA2 vertical grid this led to the following values: $h_{sur} =4762.96$, 621 621 $h_0 =255.58, h_1 =245.5813$, and $h_{th} =21.43336$. The resulting depths and 622 scale factors as a function of the model levels are shown in Fig.~\ref{Fig_zgr} and623 given in Table \ref{Tab_orca_zgr}. Those values correspond to the parameters622 scale factors as a function of the model levels are shown in \autoref{fig:zgr} and 623 given in \autoref{tab:orca_zgr}. Those values correspond to the parameters 624 624 \np{ppsur}, \np{ppa0}, \np{ppa1}, \np{ppkth} in \ngn{namcfg} namelist. 625 625 … … 675 675 31 & \textbf{5250.23}& 5000.00 & \textbf{500.56} & 500.33 \\ \hline 676 676 \end{tabular} \end{center} 677 \caption{ \protect\label{ Tab_orca_zgr}677 \caption{ \protect\label{tab:orca_zgr} 678 678 Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed 679 from \ eqref{DOM_zgr_ana} using the coefficients given in \eqref{DOM_zgr_coef}}679 from \autoref{eq:DOM_zgr_ana} using the coefficients given in \autoref{eq:DOM_zgr_coef}} 680 680 \end{table} 681 681 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 685 685 % ------------------------------------------------------------------------------------------------------------- 686 686 \subsection{$Z$-coordinate with partial step (\protect\np{ln\_zps}\forcode{ = .true.})} 687 \label{ DOM_zps}687 \label{subsec:DOM_zps} 688 688 %--------------------------------------------namdom------------------------------------------------------- 689 689 \forfile{../namelists/namdom} … … 717 717 % ------------------------------------------------------------------------------------------------------------- 718 718 \subsection{$S$-coordinate (\protect\np{ln\_sco}\forcode{ = .true.})} 719 \label{ DOM_sco}719 \label{subsec:DOM_sco} 720 720 %------------------------------------------nam_zgr_sco--------------------------------------------------- 721 721 %\forfile{../namelists/namzgr_sco} … … 726 726 function or its derivative, respectively: 727 727 728 \begin{equation} \label{ DOM_sco_ana}728 \begin{equation} \label{eq:DOM_sco_ana} 729 729 \begin{split} 730 730 z(k) &= h(i,j) \; z_0(k) \\ … … 737 737 surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean 738 738 depth, since a mixed step-like and bottom-following representation of the 739 topography can be used ( Fig.~\ref{Fig_z_zps_s_sps}d-e) or an envelop bathymetry can be defined (Fig.~\ref{Fig_z_zps_s_sps}f).739 topography can be used (\autoref{fig:z_zps_s_sps}d-e) or an envelop bathymetry can be defined (\autoref{fig:z_zps_s_sps}f). 740 740 The namelist parameter \np{rn\_rmax} determines the slope at which the terrain-following coordinate intersects 741 741 the sea bed and becomes a pseudo z-coordinate. … … 764 764 \end{equation} 765 765 766 \begin{equation} \label{ DOM_sco_function}766 \begin{equation} \label{eq:DOM_sco_function} 767 767 \begin{split} 768 768 C(s) &= \frac{ \left[ \tanh{ \left( \theta \, (s+b) \right)} … … 784 784 \begin{figure}[!ht] \begin{center} 785 785 \includegraphics[width=1.0\textwidth]{Fig_sco_function} 786 \caption{ \protect\label{ Fig_sco_function}786 \caption{ \protect\label{fig:sco_function} 787 787 Examples of the stretching function applied to a seamount; from left to right: 788 788 surface, surface and bottom, and bottom intensified resolutions} … … 794 794 are the surface and bottom control parameters such that $0\leqslant \theta \leqslant 20$, and 795 795 $0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom 796 increase of the vertical resolution ( Fig.~\ref{Fig_sco_function}).796 increase of the vertical resolution (\autoref{fig:sco_function}). 797 797 798 798 Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows … … 807 807 The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate: 808 808 809 \begin{equation} \label{ DOM_gamma_deriv}809 \begin{equation} \label{eq:DOM_gamma_deriv} 810 810 \gamma= A\left(\sigma-\frac{1}{2}\left(\sigma^{2}+f\left(\sigma\right)\right)\right)+B\left(\sigma^{3}-f\left(\sigma\right)\right)+f\left(\sigma\right) 811 811 \end{equation} 812 812 813 813 Where: 814 \begin{equation} \label{ DOM_gamma}814 \begin{equation} \label{eq:DOM_gamma} 815 815 f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} \quad \text{ and } \quad \sigma = \frac{k}{n-1} 816 816 \end{equation} … … 821 821 and bottom depths. The bottom cell depth in this example is given as a function of water depth: 822 822 823 \begin{equation} \label{ DOM_zb}823 \begin{equation} \label{eq:DOM_zb} 824 824 Z_b= h a + b 825 825 \end{equation} … … 831 831 \includegraphics[width=1.0\textwidth]{FIG_DOM_compare_coordinates_surface} 832 832 \caption{A comparison of the \citet{Song_Haidvogel_JCP94} $S$-coordinate (solid lines), a 50 level $Z$-coordinate (contoured surfaces) and the \citet{Siddorn_Furner_OM12} $S$-coordinate (dashed lines) in the surface 100m for a idealised bathymetry that goes from 50m to 5500m depth. For clarity every third coordinate surface is shown.} 833 \label{fig _compare_coordinates_surface}833 \label{fig:fig_compare_coordinates_surface} 834 834 \end{figure} 835 835 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 845 845 % ------------------------------------------------------------------------------------------------------------- 846 846 \subsection{$Z^*$- or $S^*$-coordinate (\protect\np{ln\_linssh}\forcode{ = .false.}) } 847 \label{ DOM_zgr_star}847 \label{subsec:DOM_zgr_star} 848 848 849 849 This option is described in the Report by Levier \textit{et al.} (2007), available on the \NEMO web site. … … 855 855 % ------------------------------------------------------------------------------------------------------------- 856 856 \subsection{Level bathymetry and mask} 857 \label{ DOM_msk}857 \label{subsec:DOM_msk} 858 858 859 859 Whatever the vertical coordinate used, the model offers the possibility of … … 892 892 893 893 Note that, without ice shelves cavities, masks at $t-$ and $w-$points are identical with 894 the numerical indexing used (\ S~\ref{DOM_Num_Index}). Nevertheless, $wmask$ are required894 the numerical indexing used (\autoref{subsec:DOM_Num_Index}). Nevertheless, $wmask$ are required 895 895 with oceean cavities to deal with the top boundary (ice shelf/ocean interface) 896 896 exactly in the same way as for the bottom boundary. … … 900 900 case of an east-west cyclical boundary condition, \textit{mbathy} has its last 901 901 column equal to the second one and its first column equal to the last but one 902 (and so too the mask arrays) (see \ S~\ref{LBC_jperio}).902 (and so too the mask arrays) (see \autoref{fig:LBC_jperio}). 903 903 904 904 … … 907 907 % ================================================================ 908 908 \section{Initial state (\protect\mdl{istate} and \protect\mdl{dtatsd})} 909 \label{ DTA_tsd}909 \label{sec:DTA_tsd} 910 910 %-----------------------------------------namtsd------------------------------------------- 911 911 \forfile{../namelists/namtsd} … … 918 918 \item[\np{ln\_tsd\_init}\forcode{ = .true.}] use a T and S input files that can be given on the model grid itself or 919 919 on their native input data grid. In the latter case, the data will be interpolated on-the-fly both in the 920 horizontal and the vertical to the model grid (see \ S~\ref{SBC_iof}). The information relative to the920 horizontal and the vertical to the model grid (see \autoref{subsec:SBC_iof}). The information relative to the 921 921 input files are given in the \np{sn\_tem} and \np{sn\_sal} structures. 922 922 The computation is done in the \mdl{dtatsd} module. -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_DYN.tex
r9394 r9407 5 5 % ================================================================ 6 6 \chapter{Ocean Dynamics (DYN)} 7 \label{ DYN}7 \label{chap:DYN} 8 8 \minitoc 9 9 … … 11 11 $\ $\newline %force an empty line 12 12 13 Using the representation described in Chapter \ref{DOM}, several semi-discrete13 Using the representation described in \autoref{chap:DOM}, several semi-discrete 14 14 space forms of the dynamical equations are available depending on the vertical 15 15 coordinate used and on the conservation properties of the vorticity term. In all … … 36 36 inputs (surface wind stress calculation using bulk formulae, estimation of mixing 37 37 coefficients) that are carried out in modules SBC, LDF and ZDF and are described 38 in Chapters \ref{SBC}, \ref{LDF} and \ref{ZDF}, respectively.38 in \autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively. 39 39 40 40 In the present chapter we also describe the diagnostic equations used to compute … … 51 51 The user has the option of extracting and outputting each tendency term from the 52 52 3D momentum equations (\key{trddyn} defined), as described in 53 Chap.\ref{MISC}. Furthermore, the tendency terms associated with the 2D53 \autoref{chap:MISC}. Furthermore, the tendency terms associated with the 2D 54 54 barotropic vorticity balance (when \key{trdvor} is defined) can be derived from the 55 55 3D terms. … … 64 64 % ================================================================ 65 65 \section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)} 66 \label{ DYN_divcur_wzv}66 \label{sec:DYN_divcur_wzv} 67 67 68 68 %-------------------------------------------------------------------------------------------------------------- … … 70 70 %-------------------------------------------------------------------------------------------------------------- 71 71 \subsection{Horizontal divergence and relative vorticity (\protect\mdl{divcur})} 72 \label{ DYN_divcur}72 \label{subsec:DYN_divcur} 73 73 74 74 The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows: 75 \begin{equation} \label{ Eq_divcur_cur}75 \begin{equation} \label{eq:divcur_cur} 76 76 \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta _{i+1/2} \left[ {e_{2v}\;v} \right] 77 77 -\delta _{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right) … … 79 79 80 80 The horizontal divergence is defined at a $T$-point. It is given by: 81 \begin{equation} \label{ Eq_divcur_div}81 \begin{equation} \label{eq:divcur_div} 82 82 \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 83 83 \left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u} \right] … … 102 102 %-------------------------------------------------------------------------------------------------------------- 103 103 \subsection{Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})} 104 \label{ DYN_sshwzv}104 \label{subsec:DYN_sshwzv} 105 105 106 106 The sea surface height is given by : 107 \begin{equation} \label{ Eq_dynspg_ssh}107 \begin{equation} \label{eq:dynspg_ssh} 108 108 \begin{aligned} 109 109 \frac{\partial \eta }{\partial t} … … 117 117 expressed in Kg/m$^2$/s (which is equal to mm/s), and $\rho _w$=1,035~Kg/m$^3$ 118 118 is the reference density of sea water (Boussinesq approximation). If river runoff is 119 expressed as a surface freshwater flux (see \ S\ref{SBC}) then \textit{emp} can be119 expressed as a surface freshwater flux (see \autoref{chap:SBC}) then \textit{emp} can be 120 120 written as the evaporation minus precipitation, minus the river runoff. 121 121 The sea-surface height is evaluated using exactly the same time stepping scheme 122 as the tracer equation \ eqref{Eq_tra_nxt}:122 as the tracer equation \autoref{eq:tra_nxt}: 123 123 a leapfrog scheme in combination with an Asselin time filter, $i.e.$ the velocity appearing 124 in \ eqref{Eq_dynspg_ssh} is centred in time (\textit{now} velocity).124 in \autoref{eq:dynspg_ssh} is centred in time (\textit{now} velocity). 125 125 This is of paramount importance. Replacing $T$ by the number $1$ in the tracer equation and summing 126 126 over the water column must lead to the sea surface height equation otherwise tracer content … … 129 129 The vertical velocity is computed by an upward integration of the horizontal 130 130 divergence starting at the bottom, taking into account the change of the thickness of the levels : 131 \begin{equation} \label{ Eq_wzv}131 \begin{equation} \label{eq:wzv} 132 132 \left\{ \begin{aligned} 133 133 &\left. w \right|_{k_b-1/2} \quad= 0 \qquad \text{where } k_b \text{ is the level just above the sea floor } \\ … … 141 141 of the level thicknesses, re-orientated downward. 142 142 \gmcomment{not sure of this... to be modified with the change in emp setting} 143 In the case of a linear free surface, the time derivative in \ eqref{Eq_wzv} disappears.143 In the case of a linear free surface, the time derivative in \autoref{eq:wzv} disappears. 144 144 The upper boundary condition applies at a fixed level $z=0$. The top vertical velocity 145 145 is thus equal to the divergence of the barotropic transport ($i.e.$ the first term in the 146 right-hand-side of \ eqref{Eq_dynspg_ssh}).146 right-hand-side of \autoref{eq:dynspg_ssh}). 147 147 148 148 Note also that whereas the vertical velocity has the same discrete … … 150 150 in the second case, $w$ is the velocity normal to the $s$-surfaces. 151 151 Note also that the $k$-axis is re-orientated downwards in the \textsc{fortran} code compared 152 to the indexing used in the semi-discrete equations such as \ eqref{Eq_wzv}153 (see \S\ref{DOM_Num_Index_vertical}).152 to the indexing used in the semi-discrete equations such as \autoref{eq:wzv} 153 (see \autoref{subsec:DOM_Num_Index_vertical}). 154 154 155 155 … … 158 158 % ================================================================ 159 159 \section{Coriolis and advection: vector invariant form} 160 \label{ DYN_adv_cor_vect}160 \label{sec:DYN_adv_cor_vect} 161 161 %-----------------------------------------nam_dynadv---------------------------------------------------- 162 162 \forfile{../namelists/namdyn_adv} … … 171 171 time (\textit{now} velocity). 172 172 At the lateral boundaries either free slip, no slip or partial slip boundary 173 conditions are applied following Chap.\ref{LBC}.173 conditions are applied following \autoref{chap:LBC}. 174 174 175 175 % ------------------------------------------------------------------------------------------------------------- … … 177 177 % ------------------------------------------------------------------------------------------------------------- 178 178 \subsection{Vorticity term (\protect\mdl{dynvor})} 179 \label{ DYN_vor}179 \label{subsec:DYN_vor} 180 180 %------------------------------------------nam_dynvor---------------------------------------------------- 181 181 \forfile{../namelists/namdyn_vor} … … 188 188 the relative vorticity term and horizontal kinetic energy for the planetary vorticity 189 189 term (MIX scheme) ; or conserving both the potential enstrophy of horizontally non-divergent 190 flow and horizontal kinetic energy (EEN scheme) (see Appendix~\ref{Apdx_C_vorEEN}). In the190 flow and horizontal kinetic energy (EEN scheme) (see \autoref{subsec:C_vorEEN}). In the 191 191 case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the 192 192 consistency of vorticity term with analytical equations (\np{ln\_dynvor\_con}\forcode{ = .true.}). … … 198 198 %------------------------------------------------------------- 199 199 \subsubsection{Enstrophy conserving scheme (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})} 200 \label{ DYN_vor_ens}200 \label{subsec:DYN_vor_ens} 201 201 202 202 In the enstrophy conserving case (ENS scheme), the discrete formulation of the … … 204 204 ($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent 205 205 flow ($i.e.$ $\chi$=$0$), but does not conserve the total kinetic energy. It is given by: 206 \begin{equation} \label{ Eq_dynvor_ens}206 \begin{equation} \label{eq:dynvor_ens} 207 207 \left\{ 208 208 \begin{aligned} … … 219 219 %------------------------------------------------------------- 220 220 \subsubsection{Energy conserving scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})} 221 \label{ DYN_vor_ene}221 \label{subsec:DYN_vor_ene} 222 222 223 223 The kinetic energy conserving scheme (ENE scheme) conserves the global 224 224 kinetic energy but not the global enstrophy. It is given by: 225 \begin{equation} \label{ Eq_dynvor_ene}225 \begin{equation} \label{eq:dynvor_ene} 226 226 \left\{ \begin{aligned} 227 227 {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) … … 236 236 %------------------------------------------------------------- 237 237 \subsubsection{Mixed energy/enstrophy conserving scheme (\protect\np{ln\_dynvor\_mix}\forcode{ = .true.}) } 238 \label{ DYN_vor_mix}238 \label{subsec:DYN_vor_mix} 239 239 240 240 For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the 241 two previous schemes is used. It consists of the ENS scheme (\ ref{Eq_dynvor_ens})242 for the relative vorticity term, and of the ENE scheme (\ ref{Eq_dynvor_ene}) applied241 two previous schemes is used. It consists of the ENS scheme (\autoref{eq:dynvor_ens}) 242 for the relative vorticity term, and of the ENE scheme (\autoref{eq:dynvor_ene}) applied 243 243 to the planetary vorticity term. 244 \begin{equation} \label{ Eq_dynvor_mix}244 \begin{equation} \label{eq:dynvor_mix} 245 245 \left\{ { \begin{aligned} 246 246 {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} … … 259 259 %------------------------------------------------------------- 260 260 \subsubsection{Energy and enstrophy conserving scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.}) } 261 \label{ DYN_vor_een}261 \label{subsec:DYN_vor_een} 262 262 263 263 In both the ENS and ENE schemes, it is apparent that the combination of $i$ and $j$ … … 277 277 The idea is to get rid of the double averaging by considering triad combinations of vorticity. 278 278 It is noteworthy that this solution is conceptually quite similar to the one proposed by 279 \citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see App.\ref{Apdx_C}).279 \citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:C}). 280 280 281 281 The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified 282 282 for spherical coordinates as described by \citet{Arakawa_Lamb_MWR81} to obtain the EEN scheme. 283 283 First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point: 284 \begin{equation} \label{ Eq_pot_vor}284 \begin{equation} \label{eq:pot_vor} 285 285 q = \frac{\zeta +f} {e_{3f} } 286 286 \end{equation} 287 where the relative vorticity is defined by (\ ref{Eq_divcur_cur}), the Coriolis parameter287 where the relative vorticity is defined by (\autoref{eq:divcur_cur}), the Coriolis parameter 288 288 is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is: 289 \begin{equation} \label{ Eq_een_e3f}289 \begin{equation} \label{eq:een_e3f} 290 290 e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2} 291 291 \end{equation} … … 294 294 \begin{figure}[!ht] \begin{center} 295 295 \includegraphics[width=0.70\textwidth]{Fig_DYN_een_triad} 296 \caption{ \protect\label{ Fig_DYN_een_triad}296 \caption{ \protect\label{fig:DYN_een_triad} 297 297 Triads used in the energy and enstrophy conserving scheme (een) for 298 298 $u$-component (upper panel) and $v$-component (lower panel).} … … 300 300 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 301 301 302 A key point in \ eqref{Eq_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made.302 A key point in \autoref{eq:een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 303 303 It uses the sum of masked t-point vertical scale factor divided either 304 304 by the sum of the four t-point masks (\np{nn\_een\_e3f}\forcode{ = 1}), … … 312 312 Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as 313 313 the following triad combinations of the neighbouring potential vorticities defined at f-points 314 ( Fig.~\ref{Fig_DYN_een_triad}):315 \begin{equation} \label{ Q_triads}314 (\autoref{fig:DYN_een_triad}): 315 \begin{equation} \label{eq:Q_triads} 316 316 _i^j \mathbb{Q}^{i_p}_{j_p} 317 317 = \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right) … … 320 320 321 321 Finally, the vorticity terms are represented as: 322 \begin{equation} \label{ Eq_dynvor_een}322 \begin{equation} \label{eq:dynvor_een} 323 323 \left\{ { 324 324 \begin{aligned} … … 333 333 This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes. 334 334 It conserves both total energy and potential enstrophy in the limit of horizontally 335 nondivergent flow ($i.e.$ $\chi$=$0$) (see Appendix~\ref{Apdx_C_vorEEN}).335 nondivergent flow ($i.e.$ $\chi$=$0$) (see \autoref{subsec:C_vorEEN}). 336 336 Applied to a realistic ocean configuration, it has been shown that it leads to a significant 337 337 reduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}. … … 344 344 %-------------------------------------------------------------------------------------------------------------- 345 345 \subsection{Kinetic energy gradient term (\protect\mdl{dynkeg})} 346 \label{ DYN_keg}347 348 As demonstrated in Appendix~\ref{Apdx_C}, there is a single discrete formulation346 \label{subsec:DYN_keg} 347 348 As demonstrated in \autoref{apdx:C}, there is a single discrete formulation 349 349 of the kinetic energy gradient term that, together with the formulation chosen for 350 350 the vertical advection (see below), conserves the total kinetic energy: 351 \begin{equation} \label{ Eq_dynkeg}351 \begin{equation} \label{eq:dynkeg} 352 352 \left\{ \begin{aligned} 353 353 -\frac{1}{2 \; e_{1u} } & \ \delta _{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right] \\ … … 360 360 %-------------------------------------------------------------------------------------------------------------- 361 361 \subsection{Vertical advection term (\protect\mdl{dynzad}) } 362 \label{ DYN_zad}362 \label{subsec:DYN_zad} 363 363 364 364 The discrete formulation of the vertical advection, together with the formulation 365 365 chosen for the gradient of kinetic energy (KE) term, conserves the total kinetic 366 366 energy. Indeed, the change of KE due to the vertical advection is exactly 367 balanced by the change of KE due to the gradient of KE (see Appendix~\ref{Apdx_C}).368 \begin{equation} \label{ Eq_dynzad}367 balanced by the change of KE due to the gradient of KE (see \autoref{apdx:C}). 368 \begin{equation} \label{eq:dynzad} 369 369 \left\{ \begin{aligned} 370 370 -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2} \;\delta _{k+1/2} \left[ u \right]\ }^{\,k} \\ … … 377 377 Note that in this case, a similar split-explicit time stepping should be used on 378 378 vertical advection of tracer to ensure a better stability, 379 an option which is only available with a TVD scheme (see \np{ln\_traadv\_tvd\_zts} in \ S\ref{TRA_adv_tvd}).379 an option which is only available with a TVD scheme (see \np{ln\_traadv\_tvd\_zts} in \autoref{subsec:TRA_adv_tvd}). 380 380 381 381 … … 384 384 % ================================================================ 385 385 \section{Coriolis and advection: flux form} 386 \label{ DYN_adv_cor_flux}386 \label{sec:DYN_adv_cor_flux} 387 387 %------------------------------------------nam_dynadv---------------------------------------------------- 388 388 \forfile{../namelists/namdyn_adv} … … 394 394 appearing in their expressions is centred in time (\textit{now} velocity). At the 395 395 lateral boundaries either free slip, no slip or partial slip boundary conditions 396 are applied following Chap.\ref{LBC}.396 are applied following \autoref{chap:LBC}. 397 397 398 398 … … 401 401 %-------------------------------------------------------------------------------------------------------------- 402 402 \subsection{Coriolis plus curvature metric terms (\protect\mdl{dynvor}) } 403 \label{ DYN_cor_flux}403 \label{subsec:DYN_cor_flux} 404 404 405 405 In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis 406 406 parameter has been modified to account for the "metric" term. This altered 407 407 Coriolis parameter is thus discretised at $f$-points. It is given by: 408 \begin{multline} \label{ Eq_dyncor_metric}408 \begin{multline} \label{eq:dyncor_metric} 409 409 f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i} - u\frac{\partial e_1 }{\partial j}} \right) \\ 410 410 \equiv f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta _{i+1/2} \left[ {e_{2u} } \right] … … 412 412 \end{multline} 413 413 414 Any of the (\ ref{Eq_dynvor_ens}), (\ref{Eq_dynvor_ene}) and (\ref{Eq_dynvor_een})414 Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een}) 415 415 schemes can be used to compute the product of the Coriolis parameter and the 416 vorticity. However, the energy-conserving scheme (\ ref{Eq_dynvor_een}) has416 vorticity. However, the energy-conserving scheme (\autoref{eq:dynvor_een}) has 417 417 exclusively been used to date. This term is evaluated using a leapfrog scheme, 418 418 $i.e.$ the velocity is centred in time (\textit{now} velocity). … … 422 422 %-------------------------------------------------------------------------------------------------------------- 423 423 \subsection{Flux form advection term (\protect\mdl{dynadv}) } 424 \label{ DYN_adv_flux}424 \label{subsec:DYN_adv_flux} 425 425 426 426 The discrete expression of the advection term is given by : 427 \begin{equation} \label{ Eq_dynadv}427 \begin{equation} \label{eq:dynadv} 428 428 \left\{ 429 429 \begin{aligned} … … 454 454 %------------------------------------------------------------- 455 455 \subsubsection{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln\_dynadv\_cen2}\forcode{ = .true.})} 456 \label{ DYN_adv_cen2}456 \label{subsec:DYN_adv_cen2} 457 457 458 458 In the centered $2^{nd}$ order formulation, the velocity is evaluated as the 459 459 mean of the two neighbouring points : 460 \begin{equation} \label{ Eq_dynadv_cen2}460 \begin{equation} \label{eq:dynadv_cen2} 461 461 \left\{ \begin{aligned} 462 462 u_T^{cen2} &=\overline u^{i } \quad & u_F^{cen2} &=\overline u^{j+1/2} \quad & u_{uw}^{cen2} &=\overline u^{k+1/2} \\ … … 475 475 %------------------------------------------------------------- 476 476 \subsubsection{UBS: Upstream Biased Scheme (\protect\np{ln\_dynadv\_ubs}\forcode{ = .true.})} 477 \label{ DYN_adv_ubs}477 \label{subsec:DYN_adv_ubs} 478 478 479 479 The UBS advection scheme is an upstream biased third order scheme based on 480 480 an upstream-biased parabolic interpolation. For example, the evaluation of 481 481 $u_T^{ubs} $ is done as follows: 482 \begin{equation} \label{ Eq_dynadv_ubs}482 \begin{equation} \label{eq:dynadv_ubs} 483 483 u_T^{ubs} =\overline u ^i-\;\frac{1}{6} \begin{cases} 484 484 u"_{i-1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i \geqslant 0$ } \\ … … 498 498 The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ 499 499 order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and 500 $u_{vw}^{ubs}$ in \ eqref{Eq_dynadv_cen2} are used. UBS is diffusive and is500 $u_{vw}^{ubs}$ in \autoref{eq:dynadv_cen2} are used. UBS is diffusive and is 501 501 associated with vertical mixing of momentum. \gmcomment{ gm pursue the 502 502 sentence:Since vertical mixing of momentum is a source term of the TKE equation... } 503 503 504 For stability reasons, the first term in (\ref{Eq_dynadv_ubs}), which corresponds504 For stability reasons, the first term in (\autoref{eq:dynadv_ubs}), which corresponds 505 505 to a second order centred scheme, is evaluated using the \textit{now} velocity 506 506 (centred in time), while the second term, which is the diffusion part of the scheme, … … 510 510 Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) 511 511 schemes only differ by one coefficient. Replacing $1/6$ by $1/8$ in 512 (\ ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.512 (\autoref{eq:dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. 513 513 This option is not available through a namelist parameter, since the $1/6$ coefficient 514 514 is hard coded. Nevertheless it is quite easy to make the substitution in the … … 526 526 % ================================================================ 527 527 \section{Hydrostatic pressure gradient (\protect\mdl{dynhpg})} 528 \label{ DYN_hpg}528 \label{sec:DYN_hpg} 529 529 %------------------------------------------nam_dynhpg--------------------------------------------------- 530 530 \forfile{../namelists/namdyn_hpg} … … 547 547 %-------------------------------------------------------------------------------------------------------------- 548 548 \subsection{Full step $Z$-coordinate (\protect\np{ln\_dynhpg\_zco}\forcode{ = .true.})} 549 \label{ DYN_hpg_zco}549 \label{subsec:DYN_hpg_zco} 550 550 551 551 The hydrostatic pressure can be obtained by integrating the hydrostatic equation … … 556 556 557 557 for $k=km$ (surface layer, $jk=1$ in the code) 558 \begin{equation} \label{ Eq_dynhpg_zco_surf}558 \begin{equation} \label{eq:dynhpg_zco_surf} 559 559 \left\{ \begin{aligned} 560 560 \left. \delta _{i+1/2} \left[ p^h \right] \right|_{k=km} … … 566 566 567 567 for $1<k<km$ (interior layer) 568 \begin{equation} \label{ Eq_dynhpg_zco}568 \begin{equation} \label{eq:dynhpg_zco} 569 569 \left\{ \begin{aligned} 570 570 \left. \delta _{i+1/2} \left[ p^h \right] \right|_{k} … … 577 577 \end{equation} 578 578 579 Note that the $1/2$ factor in (\ ref{Eq_dynhpg_zco_surf}) is adequate because of579 Note that the $1/2$ factor in (\autoref{eq:dynhpg_zco_surf}) is adequate because of 580 580 the definition of $e_{3w}$ as the vertical derivative of the scale factor at the surface 581 581 level ($z=0$). Note also that in case of variable volume level (\key{vvl} defined), the 582 surface pressure gradient is included in \ eqref{Eq_dynhpg_zco_surf} and \eqref{Eq_dynhpg_zco}582 surface pressure gradient is included in \autoref{eq:dynhpg_zco_surf} and \autoref{eq:dynhpg_zco} 583 583 through the space and time variations of the vertical scale factor $e_{3w}$. 584 584 … … 587 587 %-------------------------------------------------------------------------------------------------------------- 588 588 \subsection{Partial step $Z$-coordinate (\protect\np{ln\_dynhpg\_zps}\forcode{ = .true.})} 589 \label{ DYN_hpg_zps}589 \label{subsec:DYN_hpg_zps} 590 590 591 591 With partial bottom cells, tracers in horizontally adjacent cells generally live at … … 596 596 Apart from this modification, the horizontal hydrostatic pressure gradient evaluated 597 597 in the $z$-coordinate with partial step is exactly as in the pure $z$-coordinate case. 598 As explained in detail in section \ S\ref{TRA_zpshde}, the nonlinearity of pressure598 As explained in detail in section \autoref{sec:TRA_zpshde}, the nonlinearity of pressure 599 599 effects in the equation of state is such that it is better to interpolate temperature and 600 600 salinity vertically before computing the density. Horizontal gradients of temperature 601 601 and salinity are needed for the TRA modules, which is the reason why the horizontal 602 602 gradients of density at the deepest model level are computed in module \mdl{zpsdhe} 603 located in the TRA directory and described in \ S\ref{TRA_zpshde}.603 located in the TRA directory and described in \autoref{sec:TRA_zpshde}. 604 604 605 605 %-------------------------------------------------------------------------------------------------------------- … … 607 607 %-------------------------------------------------------------------------------------------------------------- 608 608 \subsection{$S$- and $Z$-$S$-coordinates} 609 \label{ DYN_hpg_sco}609 \label{subsec:DYN_hpg_sco} 610 610 611 611 Pressure gradient formulations in an $s$-coordinate have been the subject of a vast … … 615 615 616 616 $\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{ = .true.}) 617 \begin{equation} \label{ Eq_dynhpg_sco}617 \begin{equation} \label{eq:dynhpg_sco} 618 618 \left\{ \begin{aligned} 619 619 - \frac{1} {\rho_o \, e_{1u}} \; \delta _{i+1/2} \left[ p^h \right] … … 625 625 626 626 Where the first term is the pressure gradient along coordinates, computed as in 627 \ eqref{Eq_dynhpg_zco_surf} - \eqref{Eq_dynhpg_zco}, and $z_T$ is the depth of627 \autoref{eq:dynhpg_zco_surf} - \autoref{eq:dynhpg_zco}, and $z_T$ is the depth of 628 628 the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point 629 629 ($e_{3w}$). … … 637 637 (\np{ln\_dynhpg\_djc}\forcode{ = .true.}) (currently disabled; under development) 638 638 639 Note that expression \ eqref{Eq_dynhpg_sco} is commonly used when the variable volume formulation is639 Note that expression \autoref{eq:dynhpg_sco} is commonly used when the variable volume formulation is 640 640 activated (\key{vvl}) because in that case, even with a flat bottom, the coordinate surfaces are not 641 641 horizontal but follow the free surface \citep{Levier2007}. The pressure jacobian scheme … … 648 648 649 649 \subsection{Ice shelf cavity} 650 \label{ DYN_hpg_isf}650 \label{subsec:DYN_hpg_isf} 651 651 Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and 652 652 the pressure gradient due to the ocean load. If cavity opened (\np{ln\_isfcav}\forcode{ = .true.}) these 2 terms can be … … 658 658 This top pressure is constant over time. A detailed description of this method is described in \citet{Losch2008}.\\ 659 659 660 $\bullet$ The ocean load is computed using the expression \ eqref{Eq_dynhpg_sco} described in \ref{DYN_hpg_sco}.660 $\bullet$ The ocean load is computed using the expression \autoref{eq:dynhpg_sco} described in \autoref{subsec:DYN_hpg_sco}. 661 661 662 662 %-------------------------------------------------------------------------------------------------------------- … … 664 664 %-------------------------------------------------------------------------------------------------------------- 665 665 \subsection{Time-scheme (\protect\np{ln\_dynhpg\_imp}\forcode{ = .true./.false.})} 666 \label{ DYN_hpg_imp}666 \label{subsec:DYN_hpg_imp} 667 667 668 668 The default time differencing scheme used for the horizontal pressure gradient is … … 680 680 $\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}): 681 681 682 \begin{equation} \label{ Eq_dynhpg_lf}682 \begin{equation} \label{eq:dynhpg_lf} 683 683 \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; 684 684 -\frac{1}{\rho _o \,e_{1u} }\delta _{i+1/2} \left[ {p_h^t } \right] … … 686 686 687 687 $\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}): 688 \begin{equation} \label{ Eq_dynhpg_imp}688 \begin{equation} \label{eq:dynhpg_imp} 689 689 \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; 690 690 -\frac{1}{4\,\rho _o \,e_{1u} } \delta_{i+1/2} \left[ p_h^{t+\rdt} +2\,p_h^t +p_h^{t-\rdt} \right] 691 691 \end{equation} 692 692 693 The semi-implicit time scheme \ eqref{Eq_dynhpg_imp} is made possible without693 The semi-implicit time scheme \autoref{eq:dynhpg_imp} is made possible without 694 694 significant additional computation since the density can be updated to time level 695 695 $t+\rdt$ before computing the horizontal hydrostatic pressure gradient. It can 696 696 be easily shown that the stability limit associated with the hydrostatic pressure 697 gradient doubles using \ eqref{Eq_dynhpg_imp} compared to that using the698 standard leapfrog scheme \ eqref{Eq_dynhpg_lf}. Note that \eqref{Eq_dynhpg_imp}697 gradient doubles using \autoref{eq:dynhpg_imp} compared to that using the 698 standard leapfrog scheme \autoref{eq:dynhpg_lf}. Note that \autoref{eq:dynhpg_imp} 699 699 is equivalent to applying a time filter to the pressure gradient to eliminate high 700 frequency IGWs. Obviously, when using \ eqref{Eq_dynhpg_imp}, the doubling of700 frequency IGWs. Obviously, when using \autoref{eq:dynhpg_imp}, the doubling of 701 701 the time-step is achievable only if no other factors control the time-step, such as 702 702 the stability limits associated with advection or diffusion. … … 708 708 compute the hydrostatic pressure gradient (whatever the formulation) is evaluated 709 709 as follows: 710 \begin{equation} \label{ Eq_rho_flt}710 \begin{equation} \label{eq:rho_flt} 711 711 \rho^t = \rho( \widetilde{T},\widetilde {S},z_t) 712 712 \quad \text{with} \quad … … 722 722 % ================================================================ 723 723 \section{Surface pressure gradient (\protect\mdl{dynspg})} 724 \label{ DYN_spg}724 \label{sec:DYN_spg} 725 725 %-----------------------------------------nam_dynspg---------------------------------------------------- 726 726 \forfile{../namelists/namdyn_spg} … … 730 730 731 731 Options are defined through the \ngn{namdyn\_spg} namelist variables. 732 The surface pressure gradient term is related to the representation of the free surface (\ S\ref{PE_hor_pg}).732 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). 733 733 The main distinction is between the fixed volume case (linear free surface) and the variable volume case 734 (nonlinear free surface, \key{vvl} is defined). In the linear free surface case (\ S\ref{PE_free_surface})734 (nonlinear free surface, \key{vvl} is defined). In the linear free surface case (\autoref{subsec:PE_free_surface}) 735 735 the vertical scale factors $e_{3}$ are fixed in time, while they are time-dependent in the nonlinear case 736 (\ S\ref{PE_free_surface}).736 (\autoref{subsec:PE_free_surface}). 737 737 With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 738 738 which imposes a very small time step when an explicit time stepping is used. 739 739 Two methods are proposed to allow a longer time step for the three-dimensional equations: 740 the filtered free surface, which is a modification of the continuous equations (see \ eqref{Eq_PE_flt}),740 the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt}), 741 741 and the split-explicit free surface described below. 742 742 The extra term introduced in the filtered method is calculated implicitly, … … 745 745 746 746 The form of the surface pressure gradient term depends on how the user wants to handle 747 the fast external gravity waves that are a solution of the analytical equation (\ S\ref{PE_hor_pg}).747 the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:PE_hor_pg}). 748 748 Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx): 749 749 an explicit formulation which requires a small time step ; … … 761 761 %-------------------------------------------------------------------------------------------------------------- 762 762 \subsection{Explicit free surface (\protect\key{dynspg\_exp})} 763 \label{ DYN_spg_exp}763 \label{subsec:DYN_spg_exp} 764 764 765 765 In the explicit free surface formulation (\key{dynspg\_exp} defined), the model time step … … 767 767 The surface pressure gradient, evaluated using a leap-frog scheme ($i.e.$ centered in time), 768 768 is thus simply given by : 769 \begin{equation} \label{ Eq_dynspg_exp}769 \begin{equation} \label{eq:dynspg_exp} 770 770 \left\{ \begin{aligned} 771 771 - \frac{1}{e_{1u}\,\rho_o} \; \delta _{i+1/2} \left[ \,\rho \,\eta\, \right] \\ … … 782 782 %-------------------------------------------------------------------------------------------------------------- 783 783 \subsection{Split-explicit free surface (\protect\key{dynspg\_ts})} 784 \label{ DYN_spg_ts}784 \label{subsec:DYN_spg_ts} 785 785 %------------------------------------------namsplit----------------------------------------------------------- 786 786 %\forfile{../namelists/namsplit} … … 792 792 equation and the associated barotropic velocity equations with a smaller time 793 793 step than $\rdt$, the time step used for the three dimensional prognostic 794 variables ( Fig.~\ref{Fig_DYN_dynspg_ts}).794 variables (\autoref{fig:DYN_dynspg_ts}). 795 795 The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) 796 796 is provided through the \np{nn\_baro} namelist parameter as: … … 802 802 %%% 803 803 The barotropic mode solves the following equations: 804 \begin{subequations} \label{ Eq_BT}805 \begin{equation} \label{ Eq_BT_dyn}804 \begin{subequations} \label{eq:BT} 805 \begin{equation} \label{eq:BT_dyn} 806 806 \frac{\partial {\rm \overline{{\bf U}}_h} }{\partial t}= 807 807 -f\;{\rm {\bf k}}\times {\rm \overline{{\bf U}}_h} … … 809 809 \end{equation} 810 810 811 \begin{equation} \label{ Eq_BT_ssh}811 \begin{equation} \label{eq:BT_ssh} 812 812 \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]+P-E 813 813 \end{equation} 814 814 \end{subequations} 815 where $\rm {\overline{\bf G}}$ is a forcing term held constant, containing coupling term between modes, surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency. The third term on the right hand side of \ eqref{Eq_BT_dyn} represents the bottom stress (see section \S\ref{ZDF_bfr}), explicitly accounted for at each barotropic iteration. Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm detailed in \citet{Shchepetkin_McWilliams_OM05}. AB3-AM4 coefficients used in \NEMO follow the second-order accurate, "multi-purpose" stability compromise as defined in \citet{Shchepetkin_McWilliams_Bk08} (see their figure 12, lower left).815 where $\rm {\overline{\bf G}}$ is a forcing term held constant, containing coupling term between modes, surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency. The third term on the right hand side of \autoref{eq:BT_dyn} represents the bottom stress (see section \autoref{sec:ZDF_bfr}), explicitly accounted for at each barotropic iteration. Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm detailed in \citet{Shchepetkin_McWilliams_OM05}. AB3-AM4 coefficients used in \NEMO follow the second-order accurate, "multi-purpose" stability compromise as defined in \citet{Shchepetkin_McWilliams_Bk08} (see their figure 12, lower left). 816 816 817 817 %> > > > > > > > > > > > > > > > > > > > > > > > > > > > 818 818 \begin{figure}[!t] \begin{center} 819 819 \includegraphics[width=0.7\textwidth]{Fig_DYN_dynspg_ts} 820 \caption{ \protect\label{ Fig_DYN_dynspg_ts}820 \caption{ \protect\label{fig:DYN_dynspg_ts} 821 821 Schematic of the split-explicit time stepping scheme for the external 822 822 and internal modes. Time increases to the right. In this particular exemple, … … 827 827 The former are used to obtain time filtered quantities at $t+\rdt$ while the latter are used to obtain time averaged 828 828 transports to advect tracers. 829 a) Forward time integration: \ np{ln\_bt\_fw}\forcode{ = .true.},\np{ln\_bt\_av}\forcode{ = .true.}.830 b) Centred time integration: \ np{ln\_bt\_fw}\forcode{ = .false.},\np{ln\_bt\_av}\forcode{ = .true.}.831 c) Forward time integration with no time filtering (POM-like scheme): \ np{ln\_bt\_fw}\forcode{ = .true.},\np{ln\_bt\_av}\forcode{ = .false.}. }829 a) Forward time integration: \protect\np{ln\_bt\_fw}\forcode{ = .true.}, \protect\np{ln\_bt\_av}\forcode{ = .true.}. 830 b) Centred time integration: \protect\np{ln\_bt\_fw}\forcode{ = .false.}, \protect\np{ln\_bt\_av}\forcode{ = .true.}. 831 c) Forward time integration with no time filtering (POM-like scheme): \protect\np{ln\_bt\_fw}\forcode{ = .true.}, \protect\np{ln\_bt\_av}\forcode{ = .false.}. } 832 832 \end{center} \end{figure} 833 833 %> > > > > > > > > > > > > > > > > > > > > > > > > > > > 834 834 835 835 In the default case (\np{ln\_bt\_fw}\forcode{ = .true.}), the external mode is integrated 836 between \textit{now} and \textit{after} baroclinic time-steps ( Fig.~\ref{Fig_DYN_dynspg_ts}a). To avoid aliasing of fast barotropic motions into three dimensional equations, time filtering is eventually applied on barotropic836 between \textit{now} and \textit{after} baroclinic time-steps (\autoref{fig:DYN_dynspg_ts}a). To avoid aliasing of fast barotropic motions into three dimensional equations, time filtering is eventually applied on barotropic 837 837 quantities (\np{ln\_bt\_av}\forcode{ = .true.}). In that case, the integration is extended slightly beyond \textit{after} time step to provide time filtered quantities. 838 838 These are used for the subsequent initialization of the barotropic mode in the following baroclinic step. … … 850 850 at \textit{now} time step. This implies to change the traditional order of computations in \NEMO: most of momentum 851 851 trends (including the barotropic mode calculation) updated first, tracers' after. This \textit{de facto} makes semi-implicit hydrostatic 852 pressure gradient (see section \ S\ref{DYN_hpg_imp}) and time splitting not compatible.852 pressure gradient (see section \autoref{subsec:DYN_hpg_imp}) and time splitting not compatible. 853 853 Advective barotropic velocities are obtained by using a secondary set of filtering weights, uniquely defined from the filter 854 854 coefficients used for the time averaging (\citet{Shchepetkin_McWilliams_OM05}). Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to obtain exact conservation. … … 872 872 scheme using the small barotropic time step $\rdt$. We have 873 873 874 \begin{equation} \label{ DYN_spg_ts_eta}874 \begin{equation} \label{eq:DYN_spg_ts_eta} 875 875 \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) 876 876 = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 877 877 \end{equation} 878 \begin{multline} \label{ DYN_spg_ts_u}878 \begin{multline} \label{eq:DYN_spg_ts_u} 879 879 \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\ 880 880 = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) … … 886 886 and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time 887 887 that sets the barotropic time steps via 888 \begin{equation} \label{ DYN_spg_ts_t}888 \begin{equation} \label{eq:DYN_spg_ts_t} 889 889 t_n=\tau+n\rdt 890 890 \end{equation} 891 891 with $n$ an integer. The density scaled surface pressure is evaluated via 892 \begin{equation} \label{ DYN_spg_ts_ps}892 \begin{equation} \label{eq:DYN_spg_ts_ps} 893 893 p_s^{(b)}(\tau,t_{n}) = \begin{cases} 894 894 g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ … … 897 897 \end{equation} 898 898 To get started, we assume the following initial conditions 899 \begin{equation} \label{ DYN_spg_ts_eta}899 \begin{equation} \label{eq:DYN_spg_ts_eta} 900 900 \begin{split} 901 901 \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} … … 905 905 \end{equation} 906 906 with 907 \begin{equation} \label{ DYN_spg_ts_etaF}907 \begin{equation} \label{eq:DYN_spg_ts_etaF} 908 908 \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n}) 909 909 \end{equation} 910 910 the time averaged surface height taken from the previous barotropic cycle. Likewise, 911 \begin{equation} \label{ DYN_spg_ts_u}911 \begin{equation} \label{eq:DYN_spg_ts_u} 912 912 \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ 913 913 \\ … … 915 915 \end{equation} 916 916 with 917 \begin{equation} \label{ DYN_spg_ts_u}917 \begin{equation} \label{eq:DYN_spg_ts_u} 918 918 \overline{\textbf{U}^{(b)}(\tau)} 919 919 = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n}) … … 922 922 923 923 Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \rdt \tau$ 924 \begin{equation} \label{ DYN_spg_ts_u}924 \begin{equation} \label{eq:DYN_spg_ts_u} 925 925 \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} 926 926 = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) … … 928 928 The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form 929 929 930 \begin{equation} \label{ DYN_spg_ts_ssh}930 \begin{equation} \label{eq:DYN_spg_ts_ssh} 931 931 \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 932 932 \end{equation} … … 935 935 936 936 In general, some form of time filter is needed to maintain integrity of the surface 937 height field due to the leap-frog splitting mode in equation \ ref{DYN_spg_ts_ssh}. We937 height field due to the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. We 938 938 have tried various forms of such filtering, with the following method discussed in 939 939 \cite{Griffies_al_MWR01} chosen due to its stability and reasonably good maintenance of 940 tracer conservation properties (see Section??)941 942 \begin{equation} \label{ DYN_spg_ts_sshf}940 tracer conservation properties (see ??) 941 942 \begin{equation} \label{eq:DYN_spg_ts_sshf} 943 943 \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} 944 944 \end{equation} 945 945 Another approach tried was 946 946 947 \begin{equation} \label{ DYN_spg_ts_sshf2}947 \begin{equation} \label{eq:DYN_spg_ts_sshf2} 948 948 \eta^{F}(\tau-\Delta) = \eta(\tau) 949 949 + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt) … … 953 953 which is useful since it isolates all the time filtering aspects into the term multiplied 954 954 by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when 955 eliminating tracer and surface height time filtering (see Section ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \ref{DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.955 eliminating tracer and surface height time filtering (see ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended. 956 956 957 957 } %%end gm comment (copy of griffies book) … … 964 964 %-------------------------------------------------------------------------------------------------------------- 965 965 \subsection{Filtered free surface (\protect\key{dynspg\_flt})} 966 \label{ DYN_spg_fltp}966 \label{subsec:DYN_spg_fltp} 967 967 968 968 The filtered formulation follows the \citet{Roullet_Madec_JGR00} implementation. 969 The extra term introduced in the equations (see \ S\ref{PE_free_surface}) is solved implicitly.970 The elliptic solvers available in the code are documented in \ S\ref{MISC}.969 The extra term introduced in the equations (see \autoref{subsec:PE_free_surface}) is solved implicitly. 970 The elliptic solvers available in the code are documented in \autoref{chap:MISC}. 971 971 972 972 %% gm %%======>>>> given here the discrete eqs provided to the solver 973 973 \gmcomment{ %%% copy from chap-model basics 974 \begin{equation} \label{ Eq_spg_flt}974 \begin{equation} \label{eq:spg_flt} 975 975 \frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}} 976 976 - g \nabla \left( \tilde{\rho} \ \eta \right) … … 980 980 $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ 981 981 represents the collected contributions of the Coriolis, hydrostatic pressure gradient, 982 non-linear and viscous terms in \ eqref{Eq_PE_dyn}.982 non-linear and viscous terms in \autoref{eq:PE_dyn}. 983 983 } %end gmcomment 984 984 … … 990 990 % ================================================================ 991 991 \section{Lateral diffusion term and operators (\protect\mdl{dynldf})} 992 \label{ DYN_ldf}992 \label{sec:DYN_ldf} 993 993 %------------------------------------------nam_dynldf---------------------------------------------------- 994 994 \forfile{../namelists/namdyn_ldf} … … 999 999 (rotated or not) or biharmonic operators. The coefficients may be constant 1000 1000 or spatially variable; the description of the coefficients is found in the chapter 1001 on lateral physics ( Chap.\ref{LDF}). The lateral diffusion of momentum is1001 on lateral physics (\autoref{chap:LDF}). The lateral diffusion of momentum is 1002 1002 evaluated using a forward scheme, $i.e.$ the velocity appearing in its expression 1003 1003 is the \textit{before} velocity in time, except for the pure vertical component 1004 1004 that appears when a tensor of rotation is used. This latter term is solved 1005 implicitly together with the vertical diffusion term (see \ S\ref{STP})1005 implicitly together with the vertical diffusion term (see \autoref{chap:STP}) 1006 1006 1007 1007 At the lateral boundaries either free slip, no slip or partial slip boundary 1008 conditions are applied according to the user's choice (see Chap.\ref{LBC}).1008 conditions are applied according to the user's choice (see \autoref{chap:LBC}). 1009 1009 1010 1010 \gmcomment{ … … 1025 1025 \subsection[Iso-level laplacian (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})] 1026 1026 {Iso-level laplacian operator (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})} 1027 \label{ DYN_ldf_lap}1027 \label{subsec:DYN_ldf_lap} 1028 1028 1029 1029 For lateral iso-level diffusion, the discrete operator is: 1030 \begin{equation} \label{ Eq_dynldf_lap}1030 \begin{equation} \label{eq:dynldf_lap} 1031 1031 \left\{ \begin{aligned} 1032 1032 D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta _{i+1/2} \left[ {A_T^{lm} … … 1040 1040 \end{equation} 1041 1041 1042 As explained in \ S\ref{PE_ldf}, this formulation (as the gradient of a divergence1042 As explained in \autoref{subsec:PE_ldf}, this formulation (as the gradient of a divergence 1043 1043 and curl of the vorticity) preserves symmetry and ensures a complete 1044 1044 separation between the vorticity and divergence parts of the momentum diffusion. … … 1049 1049 \subsection[Rotated laplacian (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})] 1050 1050 {Rotated laplacian operator (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})} 1051 \label{ DYN_ldf_iso}1051 \label{subsec:DYN_ldf_iso} 1052 1052 1053 1053 A rotation of the lateral momentum diffusion operator is needed in several cases: … … 1061 1061 constraints on the stress tensor such as symmetry. The resulting discrete 1062 1062 representation is: 1063 \begin{equation} \label{ Eq_dyn_ldf_iso}1063 \begin{equation} \label{eq:dyn_ldf_iso} 1064 1064 \begin{split} 1065 1065 D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\ … … 1111 1111 diffusion operator acts and the surface of computation ($z$- or $s$-surfaces). 1112 1112 The way these slopes are evaluated is given in the lateral physics chapter 1113 ( Chap.\ref{LDF}).1113 (\autoref{chap:LDF}). 1114 1114 1115 1115 %-------------------------------------------------------------------------------------------------------------- … … 1118 1118 \subsection[Iso-level bilaplacian (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})] 1119 1119 {Iso-level bilaplacian operator (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})} 1120 \label{ DYN_ldf_bilap}1120 \label{subsec:DYN_ldf_bilap} 1121 1121 1122 1122 The lateral fourth order operator formulation on momentum is obtained by 1123 applying \ eqref{Eq_dynldf_lap} twice. It requires an additional assumption on1123 applying \autoref{eq:dynldf_lap} twice. It requires an additional assumption on 1124 1124 boundary conditions: the first derivative term normal to the coast depends on 1125 1125 the free or no-slip lateral boundary conditions chosen, while the third 1126 derivative terms normal to the coast are set to zero (see Chap.\ref{LBC}).1126 derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}). 1127 1127 %%% 1128 1128 \gmcomment{add a remark on the the change in the position of the coefficient} … … 1133 1133 % ================================================================ 1134 1134 \section{Vertical diffusion term (\protect\mdl{dynzdf})} 1135 \label{ DYN_zdf}1135 \label{sec:DYN_zdf} 1136 1136 %----------------------------------------------namzdf------------------------------------------------------ 1137 1137 \forfile{../namelists/namzdf} … … 1145 1145 scheme (\np{ln\_zdfexp}\forcode{ = .true.}) using a time splitting technique 1146 1146 (\np{nn\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme 1147 (\np{ln\_zdfexp}\forcode{ = .false.}) (see \ S\ref{STP}). Note that namelist variables1147 (\np{ln\_zdfexp}\forcode{ = .false.}) (see \autoref{chap:STP}). Note that namelist variables 1148 1148 \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics. 1149 1149 1150 1150 The formulation of the vertical subgrid scale physics is the same whatever 1151 1151 the vertical coordinate is. The vertical diffusion operators given by 1152 \ eqref{Eq_PE_zdf} take the following semi-discrete space form:1153 \begin{equation} \label{ Eq_dynzdf}1152 \autoref{eq:PE_zdf} take the following semi-discrete space form: 1153 \begin{equation} \label{eq:dynzdf} 1154 1154 \left\{ \begin{aligned} 1155 1155 D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta _k \left[ \frac{A_{uw}^{vm} }{e_{3uw} } … … 1162 1162 where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and 1163 1163 diffusivity coefficients. The way these coefficients are evaluated 1164 depends on the vertical physics used (see \ S\ref{ZDF}).1164 depends on the vertical physics used (see \autoref{chap:ZDF}). 1165 1165 1166 1166 The surface boundary condition on momentum is the stress exerted by 1167 1167 the wind. At the surface, the momentum fluxes are prescribed as the boundary 1168 1168 condition on the vertical turbulent momentum fluxes, 1169 \begin{equation} \label{ Eq_dynzdf_sbc}1169 \begin{equation} \label{eq:dynzdf_sbc} 1170 1170 \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1} 1171 1171 = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v } … … 1177 1177 is small (when no mixed layer scheme is used) the surface stress enters only 1178 1178 the top model level, as a body force. The surface wind stress is calculated 1179 in the surface module routines (SBC, see Chap.\ref{SBC})1179 in the surface module routines (SBC, see \autoref{chap:SBC}) 1180 1180 1181 1181 The turbulent flux of momentum at the bottom of the ocean is specified through 1182 a bottom friction parameterisation (see \ S\ref{ZDF_bfr})1182 a bottom friction parameterisation (see \autoref{sec:ZDF_bfr}) 1183 1183 1184 1184 % ================================================================ … … 1186 1186 % ================================================================ 1187 1187 \section{External forcings} 1188 \label{ DYN_forcing}1188 \label{sec:DYN_forcing} 1189 1189 1190 1190 Besides the surface and bottom stresses (see the above section) which are … … 1192 1192 may enter the dynamical equations by affecting the surface pressure gradient. 1193 1193 1194 (1) When \np{ln\_apr\_dyn}\forcode{ = .true.} (see \ S\ref{SBC_apr}), the atmospheric pressure is taken1194 (1) When \np{ln\_apr\_dyn}\forcode{ = .true.} (see \autoref{sec:SBC_apr}), the atmospheric pressure is taken 1195 1195 into account when computing the surface pressure gradient. 1196 1196 1197 (2) When \np{ln\_tide\_pot}\forcode{ = .true.} and \np{ln\_tide}\forcode{ = .true.} (see \ S\ref{SBC_tide}),1197 (2) When \np{ln\_tide\_pot}\forcode{ = .true.} and \np{ln\_tide}\forcode{ = .true.} (see \autoref{sec:SBC_tide}), 1198 1198 the tidal potential is taken into account when computing the surface pressure gradient. 1199 1199 … … 1209 1209 % ================================================================ 1210 1210 \section{Time evolution term (\protect\mdl{dynnxt})} 1211 \label{ DYN_nxt}1211 \label{sec:DYN_nxt} 1212 1212 1213 1213 %----------------------------------------------namdom---------------------------------------------------- … … 1218 1218 The general framework for dynamics time stepping is a leap-frog scheme, 1219 1219 $i.e.$ a three level centred time scheme associated with an Asselin time filter 1220 (cf. Chap.\ref{STP}). The scheme is applied to the velocity, except when using1221 the flux form of momentum advection (cf. \ S\ref{DYN_adv_cor_flux}) in the variable1220 (cf. \autoref{chap:STP}). The scheme is applied to the velocity, except when using 1221 the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux}) in the variable 1222 1222 volume case (\key{vvl} defined), where it has to be applied to the thickness 1223 weighted velocity (see \ S\ref{Apdx_A_momentum})1223 weighted velocity (see \autoref{sec:A_momentum}) 1224 1224 1225 1225 $\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}\forcode{ = .true.} ; \key{vvl} not defined): 1226 \begin{equation} \label{ Eq_dynnxt_vec}1226 \begin{equation} \label{eq:dynnxt_vec} 1227 1227 \left\{ \begin{aligned} 1228 1228 &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt \ \text{RHS}_u^t \\ … … 1232 1232 1233 1233 $\bullet$ flux form and nonlinear free surface (\np{ln\_dynhpg\_vec}\forcode{ = .false.} ; \key{vvl} defined): 1234 \begin{equation} \label{ Eq_dynnxt_flux}1234 \begin{equation} \label{eq:dynnxt_flux} 1235 1235 \left\{ \begin{aligned} 1236 1236 &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t \\ -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_LBC.tex
r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Lateral Boundary Condition (LBC)} 7 \label{ LBC}7 \label{chap:LBC} 8 8 \minitoc 9 9 … … 18 18 % ================================================================ 19 19 \section{Boundary condition at the coast (\protect\np{rn\_shlat})} 20 \label{ LBC_coast}20 \label{sec:LBC_coast} 21 21 %--------------------------------------------nam_lbc------------------------------------------------------- 22 22 \forfile{../namelists/namlbc} 23 23 %-------------------------------------------------------------------------------------------------------------- 24 24 25 %The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt (no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip). They are handled automatically by the mask system (see \ S\ref{DOM_msk}).26 27 %OPA allows land and topography grid points in the computational domain due to the presence of continents or islands, and includes the use of a full or partial step representation of bottom topography. The computation is performed over the whole domain, i.e. we do not try to restrict the computation to ocean-only points. This choice has two motivations. Firstly, working on ocean only grid points overloads the code and harms the code readability. Secondly, and more importantly, it drastically reduces the vector portion of the computation, leading to a dramatic increase of CPU time requirement on vector computers. The current section describes how the masking affects the computation of the various terms of the equations with respect to the boundary condition at solid walls. The process of defining which areas are to be masked is described in \ S\ref{DOM_msk}.25 %The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt (no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip). They are handled automatically by the mask system (see \autoref{subsec:DOM_msk}). 26 27 %OPA allows land and topography grid points in the computational domain due to the presence of continents or islands, and includes the use of a full or partial step representation of bottom topography. The computation is performed over the whole domain, i.e. we do not try to restrict the computation to ocean-only points. This choice has two motivations. Firstly, working on ocean only grid points overloads the code and harms the code readability. Secondly, and more importantly, it drastically reduces the vector portion of the computation, leading to a dramatic increase of CPU time requirement on vector computers. The current section describes how the masking affects the computation of the various terms of the equations with respect to the boundary condition at solid walls. The process of defining which areas are to be masked is described in \autoref{subsec:DOM_msk}. 28 28 29 29 Options are defined through the \ngn{namlbc} namelist variables. … … 44 44 at $u$-points. Evaluating this quantity as, 45 45 46 \begin{equation} \label{ Eq_lbc_aaaa}46 \begin{equation} \label{eq:lbc_aaaa} 47 47 \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT} 48 48 }{e_{1u} } \; \delta _{i+1 / 2} \left[ T \right]\;\;mask_u … … 51 51 zero inside land and at the boundaries, since mask$_{u}$ is zero at solid boundaries 52 52 which in this case are defined at $u$-points (normal velocity $u$ remains zero at 53 the coast) ( Fig.~\ref{Fig_LBC_uv}).53 the coast) (\autoref{fig:LBC_uv}). 54 54 55 55 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 56 56 \begin{figure}[!t] \begin{center} 57 57 \includegraphics[width=0.90\textwidth]{Fig_LBC_uv} 58 \caption{ \protect\label{ Fig_LBC_uv}58 \caption{ \protect\label{fig:LBC_uv} 59 59 Lateral boundary (thick line) at T-level. The velocity normal to the boundary is set to zero.} 60 60 \end{center} \end{figure} … … 66 66 For example, at a given $T$-level, the lateral boundary (a coastline or an intersection 67 67 with the bottom topography) is made of segments joining $f$-points, and normal 68 velocity points are located between two $f-$points ( Fig.~\ref{Fig_LBC_uv}).68 velocity points are located between two $f-$points (\autoref{fig:LBC_uv}). 69 69 The boundary condition on the normal velocity (no flux through solid boundaries) 70 70 can thus be easily implemented using the mask system. The boundary condition … … 79 79 \begin{figure}[!p] \begin{center} 80 80 \includegraphics[width=0.90\textwidth]{Fig_LBC_shlat} 81 \caption{ \protect\label{ Fig_LBC_shlat}81 \caption{ \protect\label{fig:LBC_shlat} 82 82 lateral boundary condition (a) free-slip ($rn\_shlat=0$) ; (b) no-slip ($rn\_shlat=2$) 83 83 ; (c) "partial" free-slip ($0<rn\_shlat<2$) and (d) "strong" no-slip ($2<rn\_shlat$). … … 91 91 coastline is equal to the offshore velocity, $i.e.$ the normal derivative of the 92 92 tangential velocity is zero at the coast, so the vorticity: mask$_{f}$ array is set 93 to zero inside the land and just at the coast ( Fig.~\ref{Fig_LBC_shlat}-a).93 to zero inside the land and just at the coast (\autoref{fig:LBC_shlat}-a). 94 94 95 95 \item[no-slip boundary condition (\np{rn\_shlat}\forcode{ = 2}): ] the tangential velocity vanishes … … 98 98 evaluated as if the velocities at the closest land velocity gridpoint and the closest 99 99 ocean velocity gridpoint were of the same magnitude but in the opposite direction 100 ( Fig.~\ref{Fig_LBC_shlat}-b). Therefore, the vorticity along the coastlines is given by:100 (\autoref{fig:LBC_shlat}-b). Therefore, the vorticity along the coastlines is given by: 101 101 102 102 \begin{equation*} … … 106 106 the coastline provides a vorticity field computed with the no-slip boundary condition, 107 107 simply by multiplying it by the mask$_{f}$ : 108 \begin{equation} \label{ Eq_lbc_bbbb}108 \begin{equation} \label{eq:lbc_bbbb} 109 109 \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta _{i+1/2} 110 110 \left[ {e_{2v} \,v} \right]-\delta _{j+1/2} \left[ {e_{1u} \,u} \right]} … … 115 115 velocity at the coastline is smaller than the offshore velocity, $i.e.$ there is a lateral 116 116 friction but not strong enough to make the tangential velocity at the coast vanish 117 ( Fig.~\ref{Fig_LBC_shlat}-c). This can be selected by providing a value of mask$_{f}$117 (\autoref{fig:LBC_shlat}-c). This can be selected by providing a value of mask$_{f}$ 118 118 strictly inbetween $0$ and $2$. 119 119 120 120 \item["strong" no-slip boundary condition (2$<$\np{rn\_shlat}): ] the viscous boundary 121 layer is assumed to be smaller than half the grid size ( Fig.~\ref{Fig_LBC_shlat}-d).121 layer is assumed to be smaller than half the grid size (\autoref{fig:LBC_shlat}-d). 122 122 The friction is thus larger than in the no-slip case. 123 123 … … 134 134 % ================================================================ 135 135 \section{Model domain boundary condition (\protect\np{jperio})} 136 \label{ LBC_jperio}136 \label{sec:LBC_jperio} 137 137 138 138 At the model domain boundaries several choices are offered: closed, cyclic east-west, … … 144 144 % ------------------------------------------------------------------------------------------------------------- 145 145 \subsection{Closed, cyclic, south symmetric (\protect\np{jperio}\forcode{= 0..2})} 146 \label{ LBC_jperio012}146 \label{subsec:LBC_jperio012} 147 147 148 148 The choice of closed, cyclic or symmetric model domain boundary condition is made … … 160 160 \item[For cyclic east-west boundary (\np{jperio}\forcode{ = 1})], first and last rows are set 161 161 to zero (closed) whilst the first column is set to the value of the last-but-one column 162 and the last column to the value of the second one ( Fig.~\ref{Fig_LBC_jperio}-a).162 and the last column to the value of the second one (\autoref{fig:LBC_jperio}-a). 163 163 Whatever flows out of the eastern (western) end of the basin enters the western 164 164 (eastern) end. Note that there is no option for north-south cyclic or for doubly … … 171 171 to the value of the third row while for most of $v$- and $f$-point arrays ($v$, $\zeta$, 172 172 $j\psi$, but \gmcomment{not sure why this is "but"} scalar arrays such as eddy coefficients) 173 the first row is set to minus the value of the second row ( Fig.~\ref{Fig_LBC_jperio}-b).173 the first row is set to minus the value of the second row (\autoref{fig:LBC_jperio}-b). 174 174 Note that this boundary condition is not yet available for the case of a massively 175 175 parallel computer (\textbf{key{\_}mpp} defined). … … 180 180 \begin{figure}[!t] \begin{center} 181 181 \includegraphics[width=1.0\textwidth]{Fig_LBC_jperio} 182 \caption{ \protect\label{ Fig_LBC_jperio}182 \caption{ \protect\label{fig:LBC_jperio} 183 183 setting of (a) east-west cyclic (b) symmetric across the equator boundary conditions.} 184 184 \end{center} \end{figure} … … 189 189 % ------------------------------------------------------------------------------------------------------------- 190 190 \subsection{North-fold (\protect\np{jperio}\forcode{ = 3..6})} 191 \label{ LBC_north_fold}191 \label{subsec:LBC_north_fold} 192 192 193 193 The north fold boundary condition has been introduced in order to handle the north 194 194 boundary of a three-polar ORCA grid. Such a grid has two poles in the northern hemisphere 195 ( Fig.\ref{Fig_MISC_ORCA_msh}, and thus requires a specific treatment illustrated in Fig.\ref{Fig_North_Fold_T}.195 (\autoref{fig:MISC_ORCA_msh}, and thus requires a specific treatment illustrated in \autoref{fig:North_Fold_T}. 196 196 Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition. 197 197 … … 199 199 \begin{figure}[!t] \begin{center} 200 200 \includegraphics[width=0.90\textwidth]{Fig_North_Fold_T} 201 \caption{ \protect\label{ Fig_North_Fold_T}201 \caption{ \protect\label{fig:North_Fold_T} 202 202 North fold boundary with a $T$-point pivot and cyclic east-west boundary condition 203 203 ($jperio=4$), as used in ORCA 2, 1/4, and 1/12. Pink shaded area corresponds … … 210 210 % ==================================================================== 211 211 \section{Exchange with neighbouring processors (\protect\mdl{lbclnk}, \protect\mdl{lib\_mpp})} 212 \label{ LBC_mpp}212 \label{sec:LBC_mpp} 213 213 214 214 For massively parallel processing (mpp), a domain decomposition method is used. … … 261 261 \begin{figure}[!t] \begin{center} 262 262 \includegraphics[width=0.90\textwidth]{Fig_mpp} 263 \caption{ \protect\label{ Fig_mpp}263 \caption{ \protect\label{fig:mpp} 264 264 Positioning of a sub-domain when massively parallel processing is used. } 265 265 \end{center} \end{figure} … … 279 279 \begin{eqnarray} 280 280 jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci \nonumber \\ 281 jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj \label{ Eq_lbc_jpi}281 jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj \label{eq:lbc_jpi} 282 282 \end{eqnarray} 283 283 where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis. … … 287 287 An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$, 288 288 a global array (whole domain) by the relationship: 289 \begin{equation} \label{ Eq_lbc_nimpp}289 \begin{equation} \label{eq:lbc_nimpp} 290 290 T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k), 291 291 \end{equation} … … 315 315 global ocean where more than 50 \% of points are land points. For this reason, a 316 316 pre-processing tool can be used to choose the mpp domain decomposition with a 317 maximum number of only land points processors, which can then be eliminated ( Fig. \ref{Fig_mppini2})317 maximum number of only land points processors, which can then be eliminated (\autoref{fig:mppini2}) 318 318 (For example, the mpp\_optimiz tools, available from the DRAKKAR web site). 319 319 This optimisation is dependent on the specific bathymetry employed. The user … … 335 335 \begin{figure}[!ht] \begin{center} 336 336 \includegraphics[width=0.90\textwidth]{Fig_mppini2} 337 \caption { \protect\label{ Fig_mppini2}337 \caption { \protect\label{fig:mppini2} 338 338 Example of Atlantic domain defined for the CLIPPER projet. Initial grid is 339 339 composed of 773 x 1236 horizontal points. … … 350 350 % ==================================================================== 351 351 \section{Unstructured open boundary conditions (BDY)} 352 \label{ LBC_bdy}352 \label{sec:LBC_bdy} 353 353 354 354 %-----------------------------------------nambdy-------------------------------------------- … … 384 384 %---------------------------------------------- 385 385 \subsection{Namelists} 386 \label{ BDY_namelist}386 \label{subsec:BDY_namelist} 387 387 388 388 The BDY module is activated by setting \np{ln\_bdy} to true. … … 400 400 a file and the second is defined in a namelist. For more details of 401 401 the definition of the boundary geometry see section 402 \ ref{BDY_geometry}.402 \autoref{subsec:BDY_geometry}. 403 403 404 404 For each boundary set a boundary … … 457 457 %---------------------------------------------- 458 458 \subsection{Flow relaxation scheme} 459 \label{ BDY_FRS_scheme}459 \label{subsec:BDY_FRS_scheme} 460 460 461 461 The Flow Relaxation Scheme (FRS) \citep{Davies_QJRMS76,Engerdahl_Tel95}, … … 463 463 externally-specified values over a zone next to the edge of the model 464 464 domain. Given a model prognostic variable $\Phi$ 465 \begin{equation} \label{ Eq_bdy_frs1}465 \begin{equation} \label{eq:bdy_frs1} 466 466 \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N 467 467 \end{equation} … … 472 472 to adding a relaxation term to the prognostic equation for $\Phi$ of 473 473 the form: 474 \begin{equation} \label{ Eq_bdy_frs2}474 \begin{equation} \label{eq:bdy_frs2} 475 475 -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right) 476 476 \end{equation} 477 477 where the relaxation time scale $\tau$ is given by a function of 478 478 $\alpha$ and the model time step $\Delta t$: 479 \begin{equation} \label{ Eq_bdy_frs3}479 \begin{equation} \label{eq:bdy_frs3} 480 480 \tau = \frac{1-\alpha}{\alpha} \,\rdt 481 481 \end{equation} … … 487 487 488 488 The function $\alpha$ is specified as a $tanh$ function: 489 \begin{equation} \label{ Eq_bdy_frs4}489 \begin{equation} \label{eq:bdy_frs4} 490 490 \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right), \quad d=1,N 491 491 \end{equation} … … 495 495 %---------------------------------------------- 496 496 \subsection{Flather radiation scheme} 497 \label{ BDY_flather_scheme}497 \label{subsec:BDY_flather_scheme} 498 498 499 499 The \citet{Flather_JPO94} scheme is a radiation condition on the normal, depth-mean 500 500 transport across the open boundary. It takes the form 501 \begin{equation} \label{ Eq_bdy_fla1}501 \begin{equation} \label{eq:bdy_fla1} 502 502 U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right), 503 503 \end{equation} … … 510 510 external depth-mean normal velocity, plus a correction term that 511 511 allows gravity waves generated internally to exit the model boundary. 512 Note that the sea-surface height gradient in \ eqref{Eq_bdy_fla1}512 Note that the sea-surface height gradient in \autoref{eq:bdy_fla1} 513 513 is a spatial gradient across the model boundary, so that $\eta_{e}$ is 514 514 defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the … … 518 518 %---------------------------------------------- 519 519 \subsection{Boundary geometry} 520 \label{ BDY_geometry}520 \label{subsec:BDY_geometry} 521 521 522 522 Each open boundary set is defined as a list of points. The information … … 529 529 further away from the edge of the model domain. A set of $nbi$, $nbj$, 530 530 and $nbr$ arrays is defined for each of the $T$, $U$ and $V$ 531 grids. Figure \ ref{Fig_LBC_bdy_geom} shows an example of an irregular531 grids. Figure \autoref{fig:LBC_bdy_geom} shows an example of an irregular 532 532 boundary. 533 533 … … 545 545 546 546 The boundary geometry may also be defined from a 547 ``\ifile{coordinates.bdy}'' file. Figure \ ref{Fig_LBC_nc_header}547 ``\ifile{coordinates.bdy}'' file. Figure \autoref{fig:LBC_nc_header} 548 548 gives an example of the header information from such a file. The file 549 549 should contain the index arrays for each of the $T$, $U$ and $V$ … … 566 566 \begin{figure}[!t] \begin{center} 567 567 \includegraphics[width=1.0\textwidth]{Fig_LBC_bdy_geom} 568 \caption { \protect\label{ Fig_LBC_bdy_geom}568 \caption { \protect\label{fig:LBC_bdy_geom} 569 569 Example of geometry of unstructured open boundary} 570 570 \end{center} \end{figure} … … 573 573 %---------------------------------------------- 574 574 \subsection{Input boundary data files} 575 \label{ BDY_data}575 \label{subsec:BDY_data} 576 576 577 577 The data files contain the data arrays … … 607 607 \begin{figure}[!t] \begin{center} 608 608 \includegraphics[width=1.0\textwidth]{Fig_LBC_nc_header} 609 \caption { \protect\label{ Fig_LBC_nc_header}610 Example of the header for a \ ifile{coordinates.bdy} file}609 \caption { \protect\label{fig:LBC_nc_header} 610 Example of the header for a \protect\ifile{coordinates.bdy} file} 611 611 \end{center} \end{figure} 612 612 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 614 614 %---------------------------------------------- 615 615 \subsection{Volume correction} 616 \label{ BDY_vol_corr}616 \label{subsec:BDY_vol_corr} 617 617 618 618 There is an option to force the total volume in the regional model to be constant, … … 631 631 %---------------------------------------------- 632 632 \subsection{Tidal harmonic forcing} 633 \label{ BDY_tides}633 \label{subsec:BDY_tides} 634 634 635 635 %-----------------------------------------nambdy_tide-------------------------------------------- -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_LDF.tex
r9394 r9407 6 6 % ================================================================ 7 7 \chapter{Lateral Ocean Physics (LDF)} 8 \label{ LDF}8 \label{chap:LDF} 9 9 \minitoc 10 10 … … 15 15 16 16 The lateral physics terms in the momentum and tracer equations have been 17 described in \ S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf}18 and \ S\ref{DYN_ldf}). In this section we further discuss each lateral physics option.17 described in \autoref{eq:PE_zdf} and their discrete formulation in \autoref{sec:TRA_ldf} 18 and \autoref{sec:DYN_ldf}). In this section we further discuss each lateral physics option. 19 19 Choosing one lateral physics scheme means for the user defining, 20 20 (1) the type of operator used (laplacian or bilaplacian operators, or no lateral mixing term) ; … … 25 25 Note that this chapter describes the standard implementation of iso-neutral 26 26 tracer mixing, and Griffies's implementation, which is used if 27 \np{traldf\_grif}\forcode{ = .true.}, is described in Appdx\ ref{sec:triad}27 \np{traldf\_grif}\forcode{ = .true.}, is described in Appdx\autoref{apdx:triad} 28 28 29 29 %-----------------------------------nam_traldf - nam_dynldf-------------------------------------------- … … 37 37 % ================================================================ 38 38 \section{Direction of lateral mixing (\protect\mdl{ldfslp})} 39 \label{ LDF_slp}39 \label{sec:LDF_slp} 40 40 41 41 %%% … … 50 50 slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the 51 51 quantity to be diffused. For a tracer, this leads to the following four slopes : 52 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \ eqref{Eq_tra_ldf_iso}), while52 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:tra_ldf_iso}), while 53 53 for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for 54 54 $u$ and $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. … … 60 60 In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and 61 61 $r_2$ are the slopes between the geopotential and computational surfaces. 62 Their discrete formulation is found by locally solving \ eqref{Eq_tra_ldf_iso}62 Their discrete formulation is found by locally solving \autoref{eq:tra_ldf_iso} 63 63 when the diffusive fluxes in the three directions are set to zero and $T$ is 64 64 assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the … … 66 66 %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} 67 67 68 \begin{equation} \label{ Eq_ldfslp_geo}68 \begin{equation} \label{eq:ldfslp_geo} 69 69 \begin{aligned} 70 70 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} … … 91 91 92 92 \subsection{Slopes for tracer iso-neutral mixing} 93 \label{ LDF_slp_iso}93 \label{subsec:LDF_slp_iso} 94 94 In iso-neutral mixing $r_1$ and $r_2$ are the slopes between the iso-neutral 95 95 and computational surfaces. Their formulation does not depend on the vertical 96 96 coordinate used. Their discrete formulation is found using the fact that the 97 97 diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density) 98 vanish. So, substituting $T$ by $\rho$ in \ eqref{Eq_tra_ldf_iso} and setting the98 vanish. So, substituting $T$ by $\rho$ in \autoref{eq:tra_ldf_iso} and setting the 99 99 diffusive fluxes in the three directions to zero leads to the following definition for 100 100 the neutral slopes: 101 101 102 \begin{equation} \label{ Eq_ldfslp_iso}102 \begin{equation} \label{eq:ldfslp_iso} 103 103 \begin{split} 104 104 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} … … 120 120 121 121 %gm% rewrite this as the explanation is not very clear !!! 122 %In practice, \ eqref{Eq_ldfslp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \eqref{Eq_ldfslp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth.123 124 %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{McDougall1987}, therefore in \ eqref{Eq_ldfslp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters).125 126 %In the $z$-coordinate, the derivative of the \ eqref{Eq_ldfslp_iso} numerator is evaluated at the same depth \nocite{as what?} ($T$-level, which is the same as the $u$- and $v$-levels), so the $in situ$ density can be used for its evaluation.122 %In practice, \autoref{eq:ldfslp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \autoref{eq:ldfslp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth. 123 124 %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{McDougall1987}, therefore in \autoref{eq:ldfslp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters). 125 126 %In the $z$-coordinate, the derivative of the \autoref{eq:ldfslp_iso} numerator is evaluated at the same depth \nocite{as what?} ($T$-level, which is the same as the $u$- and $v$-levels), so the $in situ$ density can be used for its evaluation. 127 127 128 128 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in 129 \ eqref{Eq_ldfslp_iso} has to be evaluated at the same local pressure (which,129 \autoref{eq:ldfslp_iso} has to be evaluated at the same local pressure (which, 130 130 in decibars, is approximated by the depth in meters in the model). Therefore 131 \ eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is131 \autoref{eq:ldfslp_iso} cannot be used as such, but further transformation is 132 132 needed depending on the vertical coordinate used: 133 133 134 134 \begin{description} 135 135 136 \item[$z$-coordinate with full step : ] in \ eqref{Eq_ldfslp_iso} the densities136 \item[$z$-coordinate with full step : ] in \autoref{eq:ldfslp_iso} the densities 137 137 appearing in the $i$ and $j$ derivatives are taken at the same depth, thus 138 138 the $in situ$ density can be used. This is not the case for the vertical 139 139 derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$ 140 140 is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following 141 \citet{McDougall1987} (see \ S\ref{TRA_bn2}).141 \citet{McDougall1987} (see \autoref{subsec:TRA_bn2}). 142 142 143 143 \item[$z$-coordinate with partial step : ] this case is identical to the full step 144 144 case except that at partial step level, the \emph{horizontal} density gradient 145 is evaluated as described in \ S\ref{TRA_zpshde}.145 is evaluated as described in \autoref{sec:TRA_zpshde}. 146 146 147 147 \item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO, 148 148 iso-neutral mixing is only employed for $s$-coordinates if the 149 Griffies scheme is used (\np{traldf\_grif}\forcode{ = .true.}; see Appdx \ ref{sec:triad}).149 Griffies scheme is used (\np{traldf\_grif}\forcode{ = .true.}; see Appdx \autoref{apdx:triad}). 150 150 In other words, iso-neutral mixing will only be accurately represented with a 151 151 linear equation of state (\np{nn\_eos}\forcode{ = 1..2}). In the case of a "true" equation 152 of state, the evaluation of $i$ and $j$ derivatives in \ eqref{Eq_ldfslp_iso}152 of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:ldfslp_iso} 153 153 will include a pressure dependent part, leading to the wrong evaluation of 154 154 the neutral slopes. … … 168 168 This constraint leads to the following definition for the slopes: 169 169 170 \begin{equation} \label{ Eq_ldfslp_iso2}170 \begin{equation} \label{eq:ldfslp_iso2} 171 171 \begin{split} 172 172 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac … … 193 193 \end{equation} 194 194 where $\alpha$ and $\beta$, the thermal expansion and saline contraction 195 coefficients introduced in \ S\ref{TRA_bn2}, have to be evaluated at the three195 coefficients introduced in \autoref{subsec:TRA_bn2}, have to be evaluated at the three 196 196 velocity points. In order to save computation time, they should be approximated 197 197 by the mean of their values at $T$-points (for example in the case of $\alpha$: … … 212 212 ocean model are modified \citep{Weaver_Eby_JPO97, 213 213 Griffies_al_JPO98}. Griffies's scheme is now available in \NEMO if 214 \np{traldf\_grif\_iso} is set true; see Appdx \ ref{sec:triad}. Here,214 \np{traldf\_grif\_iso} is set true; see Appdx \autoref{apdx:triad}. Here, 215 215 another strategy is presented \citep{Lazar_PhD97}: a local 216 216 filtering of the iso-neutral slopes (made on 9 grid-points) prevents 217 217 the development of grid point noise generated by the iso-neutral 218 diffusion operator ( Fig.~\ref{Fig_LDF_ZDF1}). This allows an218 diffusion operator (\autoref{fig:LDF_ZDF1}). This allows an 219 219 iso-neutral diffusion scheme without additional background horizontal 220 220 mixing. This technique can be viewed as a diffusion operator that acts … … 231 231 \begin{figure}[!ht] \begin{center} 232 232 \includegraphics[width=0.70\textwidth]{Fig_LDF_ZDF1} 233 \caption { \protect\label{ Fig_LDF_ZDF1}233 \caption { \protect\label{fig:LDF_ZDF1} 234 234 averaging procedure for isopycnal slope computation.} 235 235 \end{center} \end{figure} … … 259 259 \begin{figure}[!ht] \begin{center} 260 260 \includegraphics[width=0.70\textwidth]{Fig_eiv_slp} 261 \caption { \protect\label{ Fig_eiv_slp}261 \caption { \protect\label{fig:eiv_slp} 262 262 Vertical profile of the slope used for lateral mixing in the mixed layer : 263 263 \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, … … 280 280 The iso-neutral diffusion operator on momentum is the same as the one used on 281 281 tracers but applied to each component of the velocity separately (see 282 \ eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the282 \autoref{eq:dyn_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). The slopes between the 283 283 surface along which the diffusion operator acts and the surface of computation 284 284 ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the 285 285 $u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component. 286 286 They are computed from the slopes used for tracer diffusion, $i.e.$ 287 \ eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} :288 289 \begin{equation} \label{ Eq_ldfslp_dyn}287 \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso} : 288 289 \begin{equation} \label{eq:ldfslp_dyn} 290 290 \begin{aligned} 291 291 &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ … … 300 300 diffusion along model level surfaces, i.e. using the shear computed along 301 301 the model levels and with no additional friction at the ocean bottom (see 302 \ S\ref{LBC_coast}).302 \autoref{sec:LBC_coast}). 303 303 304 304 … … 307 307 % ================================================================ 308 308 \section{Lateral mixing operators (\protect\mdl{traldf}, \protect\mdl{traldf}) } 309 \label{ LDF_op}309 \label{sec:LDF_op} 310 310 311 311 … … 315 315 % ================================================================ 316 316 \section{Lateral mixing coefficient (\protect\mdl{ldftra}, \protect\mdl{ldfdyn}) } 317 \label{ LDF_coef}317 \label{sec:LDF_coef} 318 318 319 319 Introducing a space variation in the lateral eddy mixing coefficients changes … … 362 362 By default the horizontal variation of the eddy coefficient depends on the local mesh 363 363 size and the type of operator used: 364 \begin{equation} \label{ Eq_title}364 \begin{equation} \label{eq:title} 365 365 A_l = \left\{ 366 366 \begin{aligned} … … 378 378 such as global ocean models. Indeed, in such a case, a constant mixing coefficient 379 379 can lead to a blow up of the model due to large coefficient compare to the smallest 380 grid size (see \ S\ref{STP_forward_imp}), especially when using a bilaplacian operator.380 grid size (see \autoref{sec:STP_forward_imp}), especially when using a bilaplacian operator. 381 381 382 382 Other formulations can be introduced by the user for a given configuration. … … 411 411 (1) the momentum diffusion operator acting along model level surfaces is 412 412 written in terms of curl and divergent components of the horizontal current 413 (see \ S\ref{PE_ldf}). Although the eddy coefficient could be set to different values413 (see \autoref{subsec:PE_ldf}). Although the eddy coefficient could be set to different values 414 414 in these two terms, this option is not currently available. 415 415 … … 417 417 on enstrophy and on the square of the horizontal divergence for operators 418 418 acting along model-surfaces are no longer satisfied 419 ( Appendix~\ref{Apdx_dynldf_properties}).419 (\autoref{sec:dynldf_properties}). 420 420 421 421 (3) for isopycnal diffusion on momentum or tracers, an additional purely … … 425 425 values are $0$). However, the technique used to compute the isopycnal 426 426 slopes is intended to get rid of such a background diffusion, since it introduces 427 spurious diapycnal diffusion (see \ S\ref{LDF_slp}).427 spurious diapycnal diffusion (see \autoref{sec:LDF_slp}). 428 428 429 429 (4) when an eddy induced advection term is used (\key{traldf\_eiv}), $A^{eiv}$, … … 438 438 (7) it is possible to run without explicit lateral diffusion on momentum (\np{ln\_dynldf\_lap}\forcode{ = 439 439 }\np{ln\_dynldf\_bilap}\forcode{ = .false.}). This is recommended when using the UBS advection 440 scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{ = .true.}, see \ ref{DYN_adv_ubs})440 scheme on momentum (\np{ln\_dynadv\_ubs}\forcode{ = .true.}, see \autoref{subsec:DYN_adv_ubs}) 441 441 and can be useful for testing purposes. 442 442 … … 445 445 % ================================================================ 446 446 \section{Eddy induced velocity (\protect\mdl{traadv\_eiv}, \protect\mdl{ldfeiv})} 447 \label{ LDF_eiv}447 \label{sec:LDF_eiv} 448 448 449 449 %%gm from Triad appendix : to be incorporated.... 450 450 \gmcomment{ 451 451 Values of iso-neutral diffusivity and GM coefficient are set as 452 described in \ S\ref{LDF_coef}. If none of the keys \key{traldf\_cNd},452 described in \autoref{sec:LDF_coef}. If none of the keys \key{traldf\_cNd}, 453 453 N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and 454 454 GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and 455 455 \np{rn\_aeiv\_0}. If 2D-varying coefficients are set with 456 456 \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal 457 scale factor according to \ eqref{Eq_title} \footnote{Except in global ORCA457 scale factor according to \autoref{eq:title} \footnote{Except in global ORCA 458 458 $0.5^{\circ}$ runs with \key{traldf\_eiv}, where 459 459 $A_l$ is set like $A_e$ but with a minimum vale of … … 472 472 depends on the slopes of iso-neutral surfaces. Contrary to the case of iso-neutral 473 473 mixing, the slopes used here are referenced to the geopotential surfaces, $i.e.$ 474 \ eqref{Eq_ldfslp_geo} is used in $z$-coordinates, and the sum \eqref{Eq_ldfslp_geo}475 + \ eqref{Eq_ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by:476 \begin{equation} \label{ Eq_ldfeiv}474 \autoref{eq:ldfslp_geo} is used in $z$-coordinates, and the sum \autoref{eq:ldfslp_geo} 475 + \autoref{eq:ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by: 476 \begin{equation} \label{eq:ldfeiv} 477 477 \begin{split} 478 478 u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ … … 487 487 separate computation of the advective trends associated with the eiv velocity, 488 488 since it allows us to take advantage of all the advection schemes offered for 489 the tracers (see \ S\ref{TRA_adv}) and not just the $2^{nd}$ order advection489 the tracers (see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection 490 490 scheme as in previous releases of OPA \citep{Madec1998}. This is particularly 491 491 useful for passive tracers where \emph{positivity} of the advection scheme is -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_OBS.tex
r9394 r9407 5 5 % ================================================================ 6 6 \chapter{Observation and Model Comparison (OBS)} 7 \label{ OBS}7 \label{chap:OBS} 8 8 9 9 Authors: D. Lea, M. Martin, K. Mogensen, A. Vidard, A. Weaver, A. Ryan, ... % do we keep that ? … … 42 42 details on setting up the namelist. 43 43 44 Section~\ref{OBS_example} introduces a test example of the observation operator code including45 where to obtain data and how to setup the namelist. Section~\ref{OBS_details} introduces some44 \autoref{sec:OBS_example} introduces a test example of the observation operator code including 45 where to obtain data and how to setup the namelist. \autoref{sec:OBS_details} introduces some 46 46 more technical details of the different observation types used and also shows a more complete 47 namelist. Section~\ref{OBS_theory} introduces some of the theoretical aspects of the observation47 namelist. \autoref{sec:OBS_theory} introduces some of the theoretical aspects of the observation 48 48 operator including interpolation methods and running on multiple processors. 49 Section~\ref{OBS_ooo} describes the offline observation operator code.50 Section~\ref{OBS_obsutils} introduces some utilities to help working with the files49 \autoref{sec:OBS_ooo} describes the offline observation operator code. 50 \autoref{sec:OBS_obsutils} introduces some utilities to help working with the files 51 51 produced by the OBS code. 52 52 … … 55 55 % ================================================================ 56 56 \section{Running the observation operator code example} 57 \label{ OBS_example}57 \label{sec:OBS_example} 58 58 59 59 This section describes an example of running the observation operator code using … … 99 99 Setting \np{ln\_grid\_global} means that the code distributes the observations 100 100 evenly between processors. Alternatively each processor will work with 101 observations located within the model subdomain (see section~\ ref{OBS_parallel}).101 observations located within the model subdomain (see section~\autoref{subsec:OBS_parallel}). 102 102 103 103 A number of utilities are now provided to plot the feedback files, convert and 104 recombine the files. These are explained in more detail in section~\ ref{OBS_obsutils}.104 recombine the files. These are explained in more detail in section~\autoref{sec:OBS_obsutils}. 105 105 Utilites to convert other input data formats into the feedback format are also 106 described in section~\ ref{OBS_obsutils}.106 described in section~\autoref{sec:OBS_obsutils}. 107 107 108 108 \section{Technical details (feedback type observation file headers)} 109 \label{ OBS_details}109 \label{sec:OBS_details} 110 110 111 111 Here we show a more complete example namelist \ngn{namobs} and also show the NetCDF headers … … 545 545 546 546 \section{Theoretical details} 547 \label{ OBS_theory}547 \label{sec:OBS_theory} 548 548 549 549 \subsection{Horizontal interpolation and averaging methods} … … 683 683 \end{itemize} 684 684 685 Examples of the weights calculated for an observation with rectangular and radial footprints are shown in Figs.~\ ref{fig:obsavgrec} and~\ref{fig:obsavgrad}.685 Examples of the weights calculated for an observation with rectangular and radial footprints are shown in Figs.~\autoref{fig:obsavgrec} and~\autoref{fig:obsavgrad}. 686 686 687 687 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 768 768 769 769 \subsection{Parallel aspects of horizontal interpolation} 770 \label{ OBS_parallel}770 \label{subsec:OBS_parallel} 771 771 772 772 For horizontal interpolation, there is the basic problem that the … … 794 794 795 795 This is the simplest option in which the observations are distributed according 796 to the domain of the grid-point parallelization. Figure~\ref{fig:obslocal}796 to the domain of the grid-point parallelization. \autoref{fig:obslocal} 797 797 shows an example of the distribution of the {\em in situ} data on processors 798 798 with a different colour for each observation … … 823 823 among processors and use message passing in order to retrieve 824 824 the stencil for interpolation. The simplest distribution of the observations 825 is to distribute them using a round-robin scheme. Figure~\ref{fig:obsglobal}825 is to distribute them using a round-robin scheme. \autoref{fig:obsglobal} 826 826 shows the distribution of the {\em in situ} data on processors for the 827 827 round-robin distribution of observations with a different colour for 828 828 each observation on a given processor for a 4 $\times$ 2 decomposition 829 with ORCA2 for the same input data as in Fig.~\ref{fig:obslocal}.829 with ORCA2 for the same input data as in \autoref{fig:obslocal}. 830 830 The observations are now clearly randomly distributed on the globe. 831 831 In order to be able to perform horizontal interpolation in this case, … … 850 850 851 851 \section{Offline observation operator} 852 \label{ OBS_ooo}852 \label{sec:OBS_ooo} 853 853 854 854 \subsection{Concept} … … 1183 1183 1184 1184 \section{Observation utilities} 1185 \label{ OBS_obsutils}1185 \label{sec:OBS_obsutils} 1186 1186 1187 1187 Some tools for viewing and processing of observation and feedback files are provided in the … … 1354 1354 \end{minted} 1355 1355 1356 Fig~\ref{fig:obsdataplotmain} shows the main window which is launched when dataplot starts.1356 \autoref{fig:obsdataplotmain} shows the main window which is launched when dataplot starts. 1357 1357 This is split into three parts. At the top there is a menu bar which contains a variety of 1358 1358 drop down menus. Areas - zooms into prespecified regions; plot - plots the data as a … … 1389 1389 1390 1390 If a profile point is clicked with the mouse button a plot of the observation and background 1391 values as a function of depth ( Fig~\ref{fig:obsdataplotprofile}).1391 values as a function of depth (\autoref{fig:obsdataplotprofile}). 1392 1392 1393 1393 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_SBC.tex
r9394 r9407 5 5 % ================================================================ 6 6 \chapter{Surface Boundary Condition (SBC, ISF, ICB) } 7 \label{ SBC}7 \label{chap:SBC} 8 8 \minitoc 9 9 … … 40 40 need not be supplied on the model grid. Instead a file of coordinates and weights can 41 41 be supplied which maps the data from the supplied grid to the model points 42 (so called "Interpolation on the Fly", see \ S\ref{SBC_iof}).42 (so called "Interpolation on the Fly", see \autoref{subsec:SBC_iof}). 43 43 If the Interpolation on the Fly option is used, input data belonging to land points (in the native grid), 44 44 can be masked to avoid spurious results in proximity of the coasts as large sea-land gradients characterize … … 65 65 Next the scheme for interpolation on the fly is described. 66 66 Finally, the different options that further modify the fluxes applied to the ocean are discussed. 67 One of these is modification by icebergs (see \ S\ref{ICB_icebergs}), which act as drifting sources of fresh water.68 Another example of modification is that due to the ice shelf melting/freezing (see \ S\ref{SBC_isf}),67 One of these is modification by icebergs (see \autoref{sec:ICB_icebergs}), which act as drifting sources of fresh water. 68 Another example of modification is that due to the ice shelf melting/freezing (see \autoref{sec:SBC_isf}), 69 69 which provides additional sources of fresh water. 70 70 … … 74 74 % ================================================================ 75 75 \section{Surface boundary condition for the ocean} 76 \label{ SBC_general}76 \label{sec:SBC_general} 77 77 78 78 The surface ocean stress is the stress exerted by the wind and the sea-ice 79 79 on the ocean. It is applied in \mdl{dynzdf} module as a surface boundary condition of the 80 computation of the momentum vertical mixing trend (see \ eqref{Eq_dynzdf_sbc} in \S\ref{DYN_zdf}).80 computation of the momentum vertical mixing trend (see \autoref{eq:dynzdf_sbc} in \autoref{sec:DYN_zdf}). 81 81 As such, it has to be provided as a 2D vector interpolated 82 82 onto the horizontal velocity ocean mesh, $i.e.$ resolved onto the model … … 88 88 plus the heat content of the mass exchange with the atmosphere and sea-ice). 89 89 It is applied in \mdl{trasbc} module as a surface boundary condition trend of 90 the first level temperature time evolution equation (see \ eqref{Eq_tra_sbc}91 and \ eqref{Eq_tra_sbc_lin} in \S\ref{TRA_sbc}).90 the first level temperature time evolution equation (see \autoref{eq:tra_sbc} 91 and \autoref{eq:tra_sbc_lin} in \autoref{subsec:TRA_sbc}). 92 92 The latter is the penetrative part of the heat flux. It is applied as a 3D 93 93 trends of the temperature equation (\mdl{traqsr} module) when \np{ln\_traqsr}\forcode{ = .true.}. 94 94 The way the light penetrates inside the water column is generally a sum of decreasing 95 exponentials (see \ S\ref{TRA_qsr}).95 exponentials (see \autoref{subsec:TRA_qsr}). 96 96 97 97 The surface freshwater budget is provided by the \textit{emp} field. … … 130 130 The ocean model provides, at each time step, to the surface module (\mdl{sbcmod}) 131 131 the surface currents, temperature and salinity. 132 These variables are averaged over \np{nn\_fsbc} time-step (\ ref{Tab_ssm}),132 These variables are averaged over \np{nn\_fsbc} time-step (\autoref{tab:ssm}), 133 133 and it is these averaged fields which are used to computes the surface fluxes 134 134 at a frequency of \np{nn\_fsbc} time-step. … … 144 144 Sea surface salinty & sss\_m & $psu$ & T \\ \hline 145 145 \end{tabular} 146 \caption{ \protect\label{ Tab_ssm}146 \caption{ \protect\label{tab:ssm} 147 147 Ocean variables provided by the ocean to the surface module (SBC). 148 148 The variable are averaged over nn{\_}fsbc time step, … … 158 158 % ================================================================ 159 159 \section{Input data generic interface} 160 \label{ SBC_input}160 \label{sec:SBC_input} 161 161 162 162 A generic interface has been introduced to manage the way input data (2D or 3D fields, … … 181 181 182 182 The only constraints are that the input file is a NetCDF file, the file name follows a nomenclature 183 (see \ S\ref{SBC_fldread}), the period it cover is one year, month, week or day, and, if on-the-fly184 interpolation is used, a file of weights must be supplied (see \ S\ref{SBC_iof}).183 (see \autoref{subsec:SBC_fldread}), the period it cover is one year, month, week or day, and, if on-the-fly 184 interpolation is used, a file of weights must be supplied (see \autoref{subsec:SBC_iof}). 185 185 186 186 Note that when an input data is archived on a disc which is accessible directly … … 193 193 % ------------------------------------------------------------------------------------------------------------- 194 194 \subsection{Input data specification (\protect\mdl{fldread})} 195 \label{ SBC_fldread}195 \label{subsec:SBC_fldread} 196 196 197 197 The structure associated with an input variable contains the following information: … … 205 205 This stem will be completed automatically by the model, with the addition of a '.nc' at its end 206 206 and by date information and possibly a prefix (when using AGRIF). 207 Tab.\ ref{Tab_fldread} provides the resulting file name in all possible cases according to whether207 Tab.\autoref{tab:fldread} provides the resulting file name in all possible cases according to whether 208 208 it is a climatological file or not, and to the open/close frequency (see below for definition). 209 209 … … 218 218 \end{tabular} 219 219 \end{center} 220 \caption{ \protect\label{ Tab_fldread} naming nomenclature for climatological or interannual input file,220 \caption{ \protect\label{tab:fldread} naming nomenclature for climatological or interannual input file, 221 221 as a function of the Open/close frequency. The stem name is assumed to be 'fn'. 222 222 For weekly files, the 'LLL' corresponds to the first three letters of the first day of the week ($i.e.$ 'sun','sat','fri','thu','wed','tue','mon'). The 'YYYY', 'MM' and 'DD' should be replaced by the … … 259 259 260 260 \item[Others]: 'weights filename', 'pairing rotation' and 'land/sea mask' are associted with on-the-fly interpolation 261 which is described in \ S\ref{SBC_iof}.261 which is described in \autoref{subsec:SBC_iof}. 262 262 263 263 \end{description} … … 301 301 % ------------------------------------------------------------------------------------------------------------- 302 302 \subsection{Interpolation on-the-fly} 303 \label{ SBC_iof}303 \label{subsec:SBC_iof} 304 304 305 305 Interpolation on the Fly allows the user to supply input files required … … 325 325 326 326 \subsubsection{Bilinear interpolation} 327 \label{ SBC_iof_bilinear}327 \label{subsec:SBC_iof_bilinear} 328 328 329 329 The input weights file in this case has two sets of variables: src01, src02, … … 347 347 348 348 \subsubsection{Bicubic interpolation} 349 \label{ SBC_iof_bicubic}349 \label{subsec:SBC_iof_bicubic} 350 350 351 351 Again there are two sets of variables: "src" and "wgt". … … 363 363 364 364 \subsubsection{Implementation} 365 \label{ SBC_iof_imp}365 \label{subsec:SBC_iof_imp} 366 366 367 367 To activate this option, a non-empty string should be supplied in the weights filename column … … 398 398 399 399 \subsubsection{Limitations} 400 \label{ SBC_iof_lim}400 \label{subsec:SBC_iof_lim} 401 401 402 402 \begin{enumerate} … … 412 412 413 413 \subsubsection{Utilities} 414 \label{ SBC_iof_util}414 \label{subsec:SBC_iof_util} 415 415 416 416 % to be completed … … 422 422 % ------------------------------------------------------------------------------------------------------------- 423 423 \subsection{Standalone surface boundary condition scheme} 424 \label{ SAS_iof}424 \label{subsec:SAS_iof} 425 425 426 426 %---------------------------------------namsbc_ana-------------------------------------------------- … … 482 482 % ================================================================ 483 483 \section{Analytical formulation (\protect\mdl{sbcana})} 484 \label{ SBC_ana}484 \label{sec:SBC_ana} 485 485 486 486 %---------------------------------------namsbc_ana-------------------------------------------------- … … 506 506 % ================================================================ 507 507 \section{Flux formulation (\protect\mdl{sbcflx})} 508 \label{ SBC_flx}508 \label{sec:SBC_flx} 509 509 %------------------------------------------namsbc_flx---------------------------------------------------- 510 510 \forfile{../namelists/namsbc_flx} … … 516 516 read in the file, the time frequency at which it is given (in hours), and a logical 517 517 setting whether a time interpolation to the model time step is required 518 for this field. See \ S\ref{SBC_fldread} for a more detailed description of the parameters.518 for this field. See \autoref{subsec:SBC_fldread} for a more detailed description of the parameters. 519 519 520 520 Note that in general, a flux formulation is used in associated with a 521 restoring term to observed SST and/or SSS. See \ S\ref{SBC_ssr} for its521 restoring term to observed SST and/or SSS. See \autoref{subsec:SBC_ssr} for its 522 522 specification. 523 523 … … 528 528 \section[Bulk formulation {(\textit{sbcblk\{\_core,\_clio,\_mfs\}.F90})}] 529 529 {Bulk formulation {(\protect\mdl{sbcblk\_core}, \protect\mdl{sbcblk\_clio}, \protect\mdl{sbcblk\_mfs})}} 530 \label{ SBC_blk}530 \label{sec:SBC_blk} 531 531 532 532 In the bulk formulation, the surface boundary condition fields are computed … … 545 545 % ------------------------------------------------------------------------------------------------------------- 546 546 \subsection{CORE formulea (\protect\mdl{sbcblk\_core}, \protect\np{ln\_core}\forcode{ = .true.})} 547 \label{ SBC_blk_core}547 \label{subsec:SBC_blk_core} 548 548 %------------------------------------------namsbc_core---------------------------------------------------- 549 549 %\forfile{../namelists/namsbc_core} … … 566 566 567 567 %--------------------------------------------------TABLE-------------------------------------------------- 568 \begin{table}[htbp] \label{ Tab_CORE}568 \begin{table}[htbp] \label{tab:CORE} 569 569 \begin{center} 570 570 \begin{tabular}{|l|c|c|c|} … … 609 609 % ------------------------------------------------------------------------------------------------------------- 610 610 \subsection{CLIO formulea (\protect\mdl{sbcblk\_clio}, \protect\np{ln\_clio}\forcode{ = .true.})} 611 \label{ SBC_blk_clio}611 \label{subsec:SBC_blk_clio} 612 612 %------------------------------------------namsbc_clio---------------------------------------------------- 613 613 %\forfile{../namelists/namsbc_clio} … … 623 623 624 624 %--------------------------------------------------TABLE-------------------------------------------------- 625 \begin{table}[htbp] \label{ Tab_CLIO}625 \begin{table}[htbp] \label{tab:CLIO} 626 626 \begin{center} 627 627 \begin{tabular}{|l|l|l|l|} … … 643 643 As for the flux formulation, information about the input data required by the 644 644 model is provided in the namsbc\_blk\_core or namsbc\_blk\_clio 645 namelist (see \ S\ref{SBC_fldread}).645 namelist (see \autoref{subsec:SBC_fldread}). 646 646 647 647 % ------------------------------------------------------------------------------------------------------------- … … 649 649 % ------------------------------------------------------------------------------------------------------------- 650 650 \subsection{MFS formulea (\protect\mdl{sbcblk\_mfs}, \protect\np{ln\_mfs}\forcode{ = .true.})} 651 \label{ SBC_blk_mfs}651 \label{subsec:SBC_blk_mfs} 652 652 %------------------------------------------namsbc_mfs---------------------------------------------------- 653 653 %\forfile{../namelists/namsbc_mfs} … … 687 687 % ================================================================ 688 688 \section{Coupled formulation (\protect\mdl{sbccpl})} 689 \label{ SBC_cpl}689 \label{sec:SBC_cpl} 690 690 %------------------------------------------namsbc_cpl---------------------------------------------------- 691 691 \forfile{../namelists/namsbc_cpl} … … 725 725 % ================================================================ 726 726 \section{Atmospheric pressure (\protect\mdl{sbcapr})} 727 \label{ SBC_apr}727 \label{sec:SBC_apr} 728 728 %------------------------------------------namsbc_apr---------------------------------------------------- 729 729 \forfile{../namelists/namsbc_apr} … … 737 737 pressure is further transformed into an equivalent inverse barometer sea surface height, 738 738 $\eta_{ib}$, using: 739 \begin{equation} \label{ SBC_ssh_ib}739 \begin{equation} \label{eq:SBC_ssh_ib} 740 740 \eta_{ib} = - \frac{1}{g\,\rho_o} \left( P_{atm} - P_o \right) 741 741 \end{equation} … … 759 759 % ================================================================ 760 760 \section{Tidal potential (\protect\mdl{sbctide})} 761 \label{ SBC_tide}761 \label{sec:SBC_tide} 762 762 763 763 %------------------------------------------nam_tide--------------------------------------- … … 814 814 % ================================================================ 815 815 \section{River runoffs (\protect\mdl{sbcrnf})} 816 \label{ SBC_rnf}816 \label{sec:SBC_rnf} 817 817 %------------------------------------------namsbc_rnf---------------------------------------------------- 818 818 \forfile{../namelists/namsbc_rnf} … … 826 826 %coastal modelling and becomes more and more often open ocean and climate modelling 827 827 %\footnote{At least a top cells thickness of 1~meter and a 3 hours forcing frequency are 828 %required to properly represent the diurnal cycle \citep{Bernie_al_JC05}. see also \ S\ref{SBC_dcy}.}.828 %required to properly represent the diurnal cycle \citep{Bernie_al_JC05}. see also \autoref{fig:SBC_dcy}.}. 829 829 830 830 … … 847 847 more common in open ocean and climate modelling 848 848 \footnote{At least a top cells thickness of 1~meter and a 3 hours forcing frequency are 849 required to properly represent the diurnal cycle \citep{Bernie_al_JC05}. see also \ S\ref{SBC_dcy}.}.849 required to properly represent the diurnal cycle \citep{Bernie_al_JC05}. see also \autoref{fig:SBC_dcy}.}. 850 850 851 851 As such from V~3.3 onwards it is possible to add river runoff through a non-zero depth, and for the … … 929 929 % ================================================================ 930 930 \section{Ice shelf melting (\protect\mdl{sbcisf})} 931 \label{ SBC_isf}931 \label{sec:SBC_isf} 932 932 %------------------------------------------namsbc_isf---------------------------------------------------- 933 933 \forfile{../namelists/namsbc_isf} … … 1006 1006 The fw addition due to the ice shelf melting is, at each relevant depth level, added to the horizontal divergence 1007 1007 (\textit{hdivn}) in the subroutine \rou{sbc\_isf\_div}, called from \mdl{divcur}. 1008 See the runoff section \ ref{SBC_rnf} for all the details about the divergence correction.1008 See the runoff section \autoref{sec:SBC_rnf} for all the details about the divergence correction. 1009 1009 1010 1010 1011 1011 \section{Ice sheet coupling} 1012 \label{ SBC_iscpl}1012 \label{sec:SBC_iscpl} 1013 1013 %------------------------------------------namsbc_iscpl---------------------------------------------------- 1014 1014 \forfile{../namelists/namsbc_iscpl} … … 1048 1048 % ================================================================ 1049 1049 \section{Handling of icebergs (ICB)} 1050 \label{ ICB_icebergs}1050 \label{sec:ICB_icebergs} 1051 1051 %------------------------------------------namberg---------------------------------------------------- 1052 1052 \forfile{../namelists/namberg} … … 1113 1113 % ================================================================ 1114 1114 \section{Miscellaneous options} 1115 \label{ SBC_misc}1115 \label{sec:SBC_misc} 1116 1116 1117 1117 % ------------------------------------------------------------------------------------------------------------- … … 1119 1119 % ------------------------------------------------------------------------------------------------------------- 1120 1120 \subsection{Diurnal cycle (\protect\mdl{sbcdcy})} 1121 \label{ SBC_dcy}1121 \label{subsec:SBC_dcy} 1122 1122 %------------------------------------------namsbc_rnf---------------------------------------------------- 1123 1123 %\forfile{../namelists/namsbc} … … 1127 1127 \begin{figure}[!t] \begin{center} 1128 1128 \includegraphics[width=0.8\textwidth]{Fig_SBC_diurnal} 1129 \caption{ \protect\label{ Fig_SBC_diurnal}1129 \caption{ \protect\label{fig:SBC_diurnal} 1130 1130 Example of recontruction of the diurnal cycle variation of short wave flux 1131 1131 from daily mean values. The reconstructed diurnal cycle (black line) is chosen … … 1149 1149 can be found in the appendix~A of \cite{Bernie_al_CD07}. The algorithm preserve the daily 1150 1150 mean incomming SWF as the reconstructed SWF at a given time step is the mean value 1151 of the analytical cycle over this time step ( Fig.\ref{Fig_SBC_diurnal}).1151 of the analytical cycle over this time step (\autoref{fig:SBC_diurnal}). 1152 1152 The use of diurnal cycle reconstruction requires the input SWF to be daily 1153 1153 ($i.e.$ a frequency of 24 and a time interpolation set to true in \np{sn\_qsr} namelist parameter). 1154 1154 Furthermore, it is recommended to have a least 8 surface module time step per day, 1155 1155 that is $\rdt \ nn\_fsbc < 10,800~s = 3~h$. An example of recontructed SWF 1156 is given in Fig.\ref{Fig_SBC_dcy} for a 12 reconstructed diurnal cycle, one every 2~hours1156 is given in \autoref{fig:SBC_dcy} for a 12 reconstructed diurnal cycle, one every 2~hours 1157 1157 (from 1am to 11pm). 1158 1158 … … 1160 1160 \begin{figure}[!t] \begin{center} 1161 1161 \includegraphics[width=0.7\textwidth]{Fig_SBC_dcy} 1162 \caption{ \protect\label{ Fig_SBC_dcy}1162 \caption{ \protect\label{fig:SBC_dcy} 1163 1163 Example of recontruction of the diurnal cycle variation of short wave flux 1164 1164 from daily mean values on an ORCA2 grid with a time sampling of 2~hours (from 1am to 11pm). … … 1176 1176 % ------------------------------------------------------------------------------------------------------------- 1177 1177 \subsection{Rotation of vector pairs onto the model grid directions} 1178 \label{ SBC_rotation}1178 \label{subsec:SBC_rotation} 1179 1179 1180 1180 When using a flux (\np{ln\_flx}\forcode{ = .true.}) or bulk (\np{ln\_clio}\forcode{ = .true.} or \np{ln\_core}\forcode{ = .true.}) formulation, … … 1195 1195 % ------------------------------------------------------------------------------------------------------------- 1196 1196 \subsection{Surface restoring to observed SST and/or SSS (\protect\mdl{sbcssr})} 1197 \label{ SBC_ssr}1197 \label{subsec:SBC_ssr} 1198 1198 %------------------------------------------namsbc_ssr---------------------------------------------------- 1199 1199 \forfile{../namelists/namsbc_ssr} … … 1203 1203 n forced mode using a flux formulation (\np{ln\_flx}\forcode{ = .true.}), a 1204 1204 feedback term \emph{must} be added to the surface heat flux $Q_{ns}^o$: 1205 \begin{equation} \label{ Eq_sbc_dmp_q}1205 \begin{equation} \label{eq:sbc_dmp_q} 1206 1206 Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right) 1207 1207 \end{equation} … … 1216 1216 equivalent freshwater flux, it takes the following expression : 1217 1217 1218 \begin{equation} \label{ Eq_sbc_dmp_emp}1218 \begin{equation} \label{eq:sbc_dmp_emp} 1219 1219 \textit{emp} = \textit{emp}_o + \gamma_s^{-1} e_{3t} \frac{ \left(\left.S\right|_{k=1}-SSS_{Obs}\right)} 1220 1220 {\left.S\right|_{k=1}} … … 1226 1226 $\left.S\right|_{k=1}$ is the model surface layer salinity and $\gamma_s$ is a negative 1227 1227 feedback coefficient which is provided as a namelist parameter. Unlike heat flux, there is no 1228 physical justification for the feedback term in \ ref{Eq_sbc_dmp_emp} as the atmosphere1228 physical justification for the feedback term in \autoref{eq:sbc_dmp_emp} as the atmosphere 1229 1229 does not care about ocean surface salinity \citep{Madec1997}. The SSS restoring 1230 1230 term should be viewed as a flux correction on freshwater fluxes to reduce the … … 1235 1235 % ------------------------------------------------------------------------------------------------------------- 1236 1236 \subsection{Handling of ice-covered area (\textit{sbcice\_...})} 1237 \label{ SBC_ice-cover}1237 \label{subsec:SBC_ice-cover} 1238 1238 1239 1239 The presence at the sea surface of an ice covered area modifies all the fluxes … … 1264 1264 1265 1265 \subsection{Interface to CICE (\protect\mdl{sbcice\_cice})} 1266 \label{ SBC_cice}1266 \label{subsec:SBC_cice} 1267 1267 1268 1268 It is now possible to couple a regional or global NEMO configuration (without AGRIF) to the CICE sea-ice … … 1291 1291 % ------------------------------------------------------------------------------------------------------------- 1292 1292 \subsection{Freshwater budget control (\protect\mdl{sbcfwb})} 1293 \label{ SBC_fwb}1293 \label{subsec:SBC_fwb} 1294 1294 1295 1295 For global ocean simulation it can be useful to introduce a control of the mean sea … … 1313 1313 \subsection[Neutral drag coeff. from external wave model (\protect\mdl{sbcwave})] 1314 1314 {Neutral drag coefficient from external wave model (\protect\mdl{sbcwave})} 1315 \label{ SBC_wave}1315 \label{subsec:SBC_wave} 1316 1316 %------------------------------------------namwave---------------------------------------------------- 1317 1317 \forfile{../namelists/namsbc_wave} … … 1322 1322 The \mdl{sbcwave} module containing the routine \np{sbc\_wave} reads the 1323 1323 namelist \ngn{namsbc\_wave} (for external data names, locations, frequency, interpolation and all 1324 the miscellanous options allowed by Input Data generic Interface see \ S\ref{SBC_input})1324 the miscellanous options allowed by Input Data generic Interface see \autoref{sec:SBC_input}) 1325 1325 and a 2D field of neutral drag coefficient. 1326 1326 Then using the routine TURB\_CORE\_1Z or TURB\_CORE\_2Z, and starting from the neutral drag coefficent provided, -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_STO.tex
r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Stochastic Parametrization of EOS (STO)} 7 \label{ STO}7 \label{chap:STO} 8 8 9 9 Authors: P.-A. Bouttier … … 39 39 40 40 \section{Stochastic processes} 41 \label{ STO_the_details}41 \label{sec:STO_the_details} 42 42 43 43 The starting point of our implementation of stochastic parameterizations … … 104 104 \noindent 105 105 In this way, higher order processes can be easily generated recursively using 106 the same piece of code implementing Eq.~(\ref{eq:autoreg}),106 the same piece of code implementing (\autoref{eq:autoreg}), 107 107 and using succesively processes from order $0$ to~$n-1$ as~$w^{(i)}$. 108 The parameters in Eq.~(\ref{eq:ord2}) are computed so that this recursive application109 of Eq.~(\ref{eq:autoreg}) leads to processes with the required standard deviation108 The parameters in (\autoref{eq:ord2}) are computed so that this recursive application 109 of (\autoref{eq:autoreg}) leads to processes with the required standard deviation 110 110 and correlation timescale, with the additional condition that 111 111 the $n-1$ first derivatives of the autocorrelation function … … 121 121 122 122 \section{Implementation details} 123 \label{ STO_thech_details}123 \label{sec:STO_thech_details} 124 124 125 125 %---------------------------------------namsbc-------------------------------------------------- … … 135 135 (see \href{https://groups.google.com/forum/#!searchin/comp.lang.fortran/64-bit$20KISS$20RNGs}{here}) 136 136 \item[\mdl{stopts}] : stochastic parametrisation associated with the non-linearity of the equation of seawater, 137 implementing Eq~\ref{eq:sto_pert} and specific piece of code in the equation of state implementing Eq~\ref{eq:eos_sto}.137 implementing \autoref{eq:sto_pert} and specific piece of code in the equation of state implementing \autoref{eq:eos_sto}. 138 138 \end{description} 139 139 140 140 The \mdl{stopar} module has 3 public routines to be called by the model (in our case, NEMO): 141 141 142 The first routine (\rou{sto\_par}) is a direct implementation of Eq.~(\ref{eq:autoreg}),142 The first routine (\rou{sto\_par}) is a direct implementation of (\autoref{eq:autoreg}), 143 143 applied at each model grid point (in 2D or 3D), 144 144 and called at each model time step ($k$) to update -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_TRA.tex
r9394 r9407 5 5 % ================================================================ 6 6 \chapter{Ocean Tracers (TRA)} 7 \label{ TRA}7 \label{chap:TRA} 8 8 \minitoc 9 9 … … 17 17 %$\ $\newline % force a new ligne 18 18 19 Using the representation described in Chap.~\ref{DOM}, several semi-discrete19 Using the representation described in \autoref{chap:DOM}, several semi-discrete 20 20 space forms of the tracer equations are available depending on the vertical 21 21 coordinate used and on the physics used. In all the equations presented … … 40 40 require complex inputs and complex calculations ($e.g.$ bulk formulae, estimation 41 41 of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and 42 described in chapters \S\ref{SBC}, \S\ref{LDF} and \S\ref{ZDF}, respectively.42 described in \autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively. 43 43 Note that \mdl{tranpc}, the non-penetrative convection module, although 44 44 located in the NEMO/OPA/TRA directory as it directly modifies the tracer fields, … … 57 57 58 58 The user has the option of extracting each tendency term on the RHS of the tracer 59 equation for output (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}\forcode{ = .true.}), as described in Chap.~\ref{DIA}.59 equation for output (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}\forcode{ = .true.}), as described in \autoref{chap:DIA}. 60 60 61 61 $\ $\newline % force a new ligne … … 64 64 % ================================================================ 65 65 \section{Tracer advection (\protect\mdl{traadv})} 66 \label{ TRA_adv}66 \label{sec:TRA_adv} 67 67 %------------------------------------------namtra_adv----------------------------------------------------- 68 68 \forfile{../namelists/namtra_adv} … … 72 72 the advection tendency of a tracer is expressed in flux form, 73 73 $i.e.$ as the divergence of the advective fluxes. Its discrete expression is given by : 74 \begin{equation} \label{ Eq_tra_adv}74 \begin{equation} \label{eq:tra_adv} 75 75 ADV_\tau =-\frac{1}{b_t} \left( 76 76 \;\delta _i \left[ e_{2u}\,e_{3u} \; u\; \tau _u \right] … … 79 79 \end{equation} 80 80 where $\tau$ is either T or S, and $b_t= e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells. 81 The flux form in \ eqref{Eq_tra_adv}81 The flux form in \autoref{eq:tra_adv} 82 82 implicitly requires the use of the continuity equation. Indeed, it is obtained 83 83 by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ … … 87 87 advection tendency so that it is consistent with the continuity equation in order to 88 88 enforce the conservation properties of the continuous equations. In other words, 89 by setting $\tau = 1$ in (\ ref{Eq_tra_adv}) we recover the discrete form of89 by setting $\tau = 1$ in (\autoref{eq:tra_adv}) we recover the discrete form of 90 90 the continuity equation which is used to calculate the vertical velocity. 91 91 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 92 92 \begin{figure}[!t] \begin{center} 93 93 \includegraphics[width=0.9\textwidth]{Fig_adv_scheme} 94 \caption{ \protect\label{ Fig_adv_scheme}94 \caption{ \protect\label{fig:adv_scheme} 95 95 Schematic representation of some ways used to evaluate the tracer value 96 96 at $u$-point and the amount of tracer exchanged between two neighbouring grid … … 107 107 The key difference between the advection schemes available in \NEMO is the choice 108 108 made in space and time interpolation to define the value of the tracer at the 109 velocity points ( Fig.~\ref{Fig_adv_scheme}).109 velocity points (\autoref{fig:adv_scheme}). 110 110 111 111 Along solid lateral and bottom boundaries a zero tracer flux is automatically … … 131 131 height, two quantities that are not correlated \citep{Roullet_Madec_JGR00,Griffies_al_MWR01,Campin2004}. 132 132 133 The velocity field that appears in (\ ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_zco})133 The velocity field that appears in (\autoref{eq:tra_adv}) and (\autoref{eq:tra_adv_zco}) 134 134 is the centred (\textit{now}) \textit{effective} ocean velocity, $i.e.$ the \textit{eulerian} velocity 135 (see Chap.~\ref{DYN}) plus the eddy induced velocity (\textit{eiv})135 (see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) 136 136 and/or the mixed layer eddy induced velocity (\textit{eiv}) 137 when those parameterisations are used (see Chap.~\ref{LDF}).137 when those parameterisations are used (see \autoref{chap:LDF}). 138 138 139 139 Several tracer advection scheme are proposed, namely … … 174 174 % ------------------------------------------------------------------------------------------------------------- 175 175 \subsection{CEN: Centred scheme (\protect\np{ln\_traadv\_cen}\forcode{ = .true.})} 176 \label{ TRA_adv_cen}176 \label{subsec:TRA_adv_cen} 177 177 178 178 % 2nd order centred scheme … … 186 186 is evaluated as the mean of the two neighbouring $T$-point values. 187 187 For example, in the $i$-direction : 188 \begin{equation} \label{ Eq_tra_adv_cen2}188 \begin{equation} \label{eq:tra_adv_cen2} 189 189 \tau _u^{cen2} =\overline T ^{i+1/2} 190 190 \end{equation} … … 195 195 produce a sensible solution. The associated time-stepping is performed using 196 196 a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in 197 (\ ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value.197 (\autoref{eq:tra_adv_cen2}) is the \textit{now} tracer value. 198 198 199 199 Note that using the CEN2, the overall tracer advection is of second 200 order accuracy since both (\ ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2})200 order accuracy since both (\autoref{eq:tra_adv}) and (\autoref{eq:tra_adv_cen2}) 201 201 have this order of accuracy. 202 202 … … 206 206 a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. 207 207 For example, in the $i$-direction: 208 \begin{equation} \label{ Eq_tra_adv_cen4}208 \begin{equation} \label{eq:tra_adv_cen4} 209 209 \tau _u^{cen4} 210 210 =\overline{ T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} … … 219 219 Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme 220 220 but a $4^{th}$ order evaluation of advective fluxes, since the divergence of 221 advective fluxes \ eqref{Eq_tra_adv} is kept at $2^{nd}$ order.221 advective fluxes \autoref{eq:tra_adv} is kept at $2^{nd}$ order. 222 222 The expression \textit{$4^{th}$ order scheme} used in oceanographic literature 223 223 is usually associated with the scheme presented here. … … 232 232 to produce a sensible solution. 233 233 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction 234 with an Asselin time-filter, so $T$ in (\ ref{Eq_tra_adv_cen4}) is the \textit{now} tracer.234 with an Asselin time-filter, so $T$ in (\autoref{eq:tra_adv_cen4}) is the \textit{now} tracer. 235 235 236 236 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), … … 245 245 % ------------------------------------------------------------------------------------------------------------- 246 246 \subsection{FCT: Flux Corrected Transport scheme (\protect\np{ln\_traadv\_fct}\forcode{ = .true.})} 247 \label{ TRA_adv_tvd}247 \label{subsec:TRA_adv_tvd} 248 248 249 249 The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}\forcode{ = .true.}. … … 254 254 In FCT formulation, the tracer at velocity points is evaluated using a combination of 255 255 an upstream and a centred scheme. For example, in the $i$-direction : 256 \begin{equation} \label{ Eq_tra_adv_fct}256 \begin{equation} \label{eq:tra_adv_fct} 257 257 \begin{split} 258 258 \tau _u^{ups}&= \begin{cases} … … 280 280 by vertical advection \citep{Lemarie_OM2015}. Note that in this case, a similar split-explicit 281 281 time stepping should be used on vertical advection of momentum to insure a better stability 282 (see \ S\ref{DYN_zad}).283 284 For stability reasons (see \ S\ref{STP}), $\tau _u^{cen}$ is evaluated in (\ref{Eq_tra_adv_fct})282 (see \autoref{subsec:DYN_zad}). 283 284 For stability reasons (see \autoref{chap:STP}), $\tau _u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) 285 285 using the \textit{now} tracer while $\tau _u^{ups}$ is evaluated using the \textit{before} tracer. In other words, 286 286 the advective part of the scheme is time stepped with a leap-frog scheme … … 291 291 % ------------------------------------------------------------------------------------------------------------- 292 292 \subsection{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln\_traadv\_mus}\forcode{ = .true.})} 293 \label{ TRA_adv_mus}293 \label{subsec:TRA_adv_mus} 294 294 295 295 The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}\forcode{ = .true.}. … … 298 298 MUSCL has been first implemented in \NEMO by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points 299 299 is evaluated assuming a linear tracer variation between two $T$-points 300 ( Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction :301 \begin{equation} \label{ Eq_tra_adv_mus}300 (\autoref{fig:adv_scheme}). For example, in the $i$-direction : 301 \begin{equation} \label{eq:tra_adv_mus} 302 302 \tau _u^{mus} = \left\{ \begin{aligned} 303 303 &\tau _i &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) … … 323 323 % ------------------------------------------------------------------------------------------------------------- 324 324 \subsection{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})} 325 \label{ TRA_adv_ubs}325 \label{subsec:TRA_adv_ubs} 326 326 327 327 The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}\forcode{ = .true.}. … … 332 332 third order scheme based on an upstream-biased parabolic interpolation. 333 333 For example, in the $i$-direction : 334 \begin{equation} \label{ Eq_tra_adv_ubs}334 \begin{equation} \label{eq:tra_adv_ubs} 335 335 \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{ 336 336 \begin{aligned} … … 355 355 or a $4^th$ order COMPACT scheme (\np{nn\_cen\_v}\forcode{ = 2 or 4}). 356 356 357 For stability reasons (see \ S\ref{STP}),358 the first term in \ eqref{Eq_tra_adv_ubs} (which corresponds to a second order357 For stability reasons (see \autoref{chap:STP}), 358 the first term in \autoref{eq:tra_adv_ubs} (which corresponds to a second order 359 359 centred scheme) is evaluated using the \textit{now} tracer (centred in time) 360 360 while the second term (which is the diffusive part of the scheme), is … … 362 362 This choice is discussed by \citet{Webb_al_JAOT98} in the context of the 363 363 QUICK advection scheme. UBS and QUICK schemes only differ 364 by one coefficient. Replacing 1/6 with 1/8 in \ eqref{Eq_tra_adv_ubs}364 by one coefficient. Replacing 1/6 with 1/8 in \autoref{eq:tra_adv_ubs} 365 365 leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. 366 366 This option is not available through a namelist parameter, since the … … 368 368 substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. 369 369 370 Note that it is straightforward to rewrite \ eqref{Eq_tra_adv_ubs} as follows:371 \begin{equation} \label{ Eq_traadv_ubs2}370 Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows: 371 \begin{equation} \label{eq:traadv_ubs2} 372 372 \tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{ 373 373 \begin{aligned} … … 377 377 \end{equation} 378 378 or equivalently 379 \begin{equation} \label{ Eq_traadv_ubs2b}379 \begin{equation} \label{eq:traadv_ubs2b} 380 380 u_{i+1/2} \ \tau _u^{ubs} 381 381 =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta _i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} … … 383 383 \end{equation} 384 384 385 \ eqref{Eq_traadv_ubs2} has several advantages. Firstly, it clearly reveals385 \autoref{eq:traadv_ubs2} has several advantages. Firstly, it clearly reveals 386 386 that the UBS scheme is based on the fourth order scheme to which an 387 387 upstream-biased diffusion term is added. Secondly, this emphasises that the 388 388 $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has 389 to be evaluated at the \emph{now} time step using \ eqref{Eq_tra_adv_ubs}.389 to be evaluated at the \emph{now} time step using \autoref{eq:tra_adv_ubs}. 390 390 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy 391 391 coefficient which is simply proportional to the velocity: 392 392 $A_u^{lm}= \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note the current version of NEMO uses 393 the computationally more efficient formulation \ eqref{Eq_tra_adv_ubs}.393 the computationally more efficient formulation \autoref{eq:tra_adv_ubs}. 394 394 395 395 % ------------------------------------------------------------------------------------------------------------- … … 397 397 % ------------------------------------------------------------------------------------------------------------- 398 398 \subsection{QCK: QuiCKest scheme (\protect\np{ln\_traadv\_qck}\forcode{ = .true.})} 399 \label{ TRA_adv_qck}399 \label{subsec:TRA_adv_qck} 400 400 401 401 The Quadratic Upstream Interpolation for Convective Kinematics with … … 423 423 % ================================================================ 424 424 \section{Tracer lateral diffusion (\protect\mdl{traldf})} 425 \label{ TRA_ldf}425 \label{sec:TRA_ldf} 426 426 %-----------------------------------------nam_traldf------------------------------------------------------ 427 427 \forfile{../namelists/namtra_ldf} … … 434 434 $(iii)$ some specific options related to the rotated operators ($i.e.$ non-iso-level operator), and 435 435 $(iv)$ the specification of eddy diffusivity coefficient (either constant or variable in space and time). 436 Item $(iv)$ will be described in Chap.\ref{LDF} .436 Item $(iv)$ will be described in \autoref{chap:LDF} . 437 437 The direction along which the operators act is defined through the slope between this direction and the iso-level surfaces. 438 The slope is computed in the \mdl{ldfslp} module and will also be described in Chap.~\ref{LDF}.438 The slope is computed in the \mdl{ldfslp} module and will also be described in \autoref{chap:LDF}. 439 439 440 440 The lateral diffusion of tracers is evaluated using a forward scheme, 441 441 $i.e.$ the tracers appearing in its expression are the \textit{before} tracers in time, 442 442 except for the pure vertical component that appears when a rotation tensor is used. 443 This latter component is solved implicitly together with the vertical diffusion term (see \ S\ref{STP}).443 This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}). 444 444 When \np{ln\_traldf\_msc}\forcode{ = .true.}, a Method of Stabilizing Correction is used in which 445 445 the pure vertical component is split into an explicit and an implicit part \citep{Lemarie_OM2012}. … … 450 450 \subsection[Type of operator (\protect\np{ln\_traldf}\{\_NONE,\_lap,\_blp\}\})] 451 451 {Type of operator (\protect\np{ln\_traldf\_NONE}, \protect\np{ln\_traldf\_lap}, or \protect\np{ln\_traldf\_blp}) } 452 \label{ TRA_ldf_op}452 \label{subsec:TRA_ldf_op} 453 453 454 454 Three operator options are proposed and, one and only one of them must be selected: … … 459 459 \item [\np{ln\_traldf\_lap}\forcode{ = .true.}]: a laplacian operator is selected. This harmonic operator 460 460 takes the following expression: $\mathpzc{L}(T)=\nabla \cdot A_{ht}\;\nabla T $, 461 where the gradient operates along the selected direction (see \ S\ref{TRA_ldf_dir}),462 and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see Chap.~\ref{LDF}).461 where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}), 462 and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}). 463 463 \item [\np{ln\_traldf\_blp}\forcode{ = .true.}]: a bilaplacian operator is selected. This biharmonic operator 464 464 takes the following expression: 465 465 $\mathpzc{B}=- \mathpzc{L}\left(\mathpzc{L}(T) \right) = -\nabla \cdot b\nabla \left( {\nabla \cdot b\nabla T} \right)$ 466 466 where the gradient operats along the selected direction, 467 and $b^2=B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see Chap.~\ref{LDF}).467 and $b^2=B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see \autoref{chap:LDF}). 468 468 In the code, the bilaplacian operator is obtained by calling the laplacian twice. 469 469 \end{description} … … 483 483 \subsection[Action direction (\protect\np{ln\_traldf}\{\_lev,\_hor,\_iso,\_triad\})] 484 484 {Direction of action (\protect\np{ln\_traldf\_lev}, \protect\np{ln\_traldf\_hor}, \protect\np{ln\_traldf\_iso}, or \protect\np{ln\_traldf\_triad}) } 485 \label{ TRA_ldf_dir}485 \label{subsec:TRA_ldf_dir} 486 486 487 487 The choice of a direction of action determines the form of operator used. … … 509 509 % ------------------------------------------------------------------------------------------------------------- 510 510 \subsection{Iso-level (bi-)laplacian operator ( \protect\np{ln\_traldf\_iso}) } 511 \label{ TRA_ldf_lev}511 \label{subsec:TRA_ldf_lev} 512 512 513 513 The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by: 514 \begin{equation} \label{ Eq_tra_ldf_lap}514 \begin{equation} \label{eq:tra_ldf_lap} 515 515 D_t^{lT} =\frac{1}{b_t} \left( \; 516 516 \delta _{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta _{i+1/2} [T] \right] … … 533 533 Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}), tracers in horizontally 534 534 adjacent cells are located at different depths in the vicinity of the bottom. 535 In this case, horizontal derivatives in (\ ref{Eq_tra_ldf_lap}) at the bottom level535 In this case, horizontal derivatives in (\autoref{eq:tra_ldf_lap}) at the bottom level 536 536 require a specific treatment. They are calculated in the \mdl{zpshde} module, 537 described in \ S\ref{TRA_zpshde}.537 described in \autoref{sec:TRA_zpshde}. 538 538 539 539 … … 542 542 % ------------------------------------------------------------------------------------------------------------- 543 543 \subsection{Standard and triad (bi-)laplacian operator} 544 \label{ TRA_ldf_iso_triad}544 \label{subsec:TRA_ldf_iso_triad} 545 545 546 546 %&& Standard rotated (bi-)laplacian operator 547 547 %&& ---------------------------------------------- 548 548 \subsubsection{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})} 549 \label{ TRA_ldf_iso}549 \label{subsec:TRA_ldf_iso} 550 550 The general form of the second order lateral tracer subgrid scale physics 551 (\ ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and $s$-coordinates:552 \begin{equation} \label{ Eq_tra_ldf_iso}551 (\autoref{eq:PE_zdf}) takes the following semi-discrete space form in $z$- and $s$-coordinates: 552 \begin{equation} \label{eq:tra_ldf_iso} 553 553 \begin{split} 554 554 D_T^{lT} = \frac{1}{b_t} & \left\{ \,\;\delta_i \left[ A_u^{lT} \left( … … 576 576 in addition to \np{ln\_traldf\_lap}\forcode{ = .true.}, we have \np{ln\_traldf\_iso}\forcode{ = .true.}, 577 577 or both \np{ln\_traldf\_hor}\forcode{ = .true.} and \np{ln\_zco}\forcode{ = .true.}. The way these 578 slopes are evaluated is given in \ S\ref{LDF_slp}. At the surface, bottom578 slopes are evaluated is given in \autoref{sec:LDF_slp}. At the surface, bottom 579 579 and lateral boundaries, the turbulent fluxes of heat and salt are set to zero 580 using the mask technique (see \ S\ref{LBC_coast}).581 582 The operator in \ eqref{Eq_tra_ldf_iso} involves both lateral and vertical580 using the mask technique (see \autoref{sec:LBC_coast}). 581 582 The operator in \autoref{eq:tra_ldf_iso} involves both lateral and vertical 583 583 derivatives. For numerical stability, the vertical second derivative must 584 584 be solved using the same implicit time scheme as that used in the vertical 585 physics (see \ S\ref{TRA_zdf}). For computer efficiency reasons, this term585 physics (see \autoref{sec:TRA_zdf}). For computer efficiency reasons, this term 586 586 is not computed in the \mdl{traldf\_iso} module, but in the \mdl{trazdf} module 587 587 where, if iso-neutral mixing is used, the vertical mixing coefficient is simply … … 590 590 This formulation conserves the tracer but does not ensure the decrease 591 591 of the tracer variance. Nevertheless the treatment performed on the slopes 592 (see \ S\ref{LDF}) allows the model to run safely without any additional592 (see \autoref{chap:LDF}) allows the model to run safely without any additional 593 593 background horizontal diffusion \citep{Guilyardi_al_CD01}. 594 594 595 595 Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}), the horizontal derivatives 596 at the bottom level in \ eqref{Eq_tra_ldf_iso} require a specific treatment.597 They are calculated in module zpshde, described in \ S\ref{TRA_zpshde}.596 at the bottom level in \autoref{eq:tra_ldf_iso} require a specific treatment. 597 They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}. 598 598 599 599 %&& Triad rotated (bi-)laplacian operator 600 600 %&& ------------------------------------------- 601 601 \subsubsection{Triad rotated (bi-)laplacian operator (\protect\np{ln\_traldf\_triad})} 602 \label{ TRA_ldf_triad}603 604 If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}\forcode{ = .true.} ; see App.\ref{sec:triad})602 \label{subsec:TRA_ldf_triad} 603 604 If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}\forcode{ = .true.} ; see \autoref{apdx:triad}) 605 605 606 606 An alternative scheme developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases 607 607 is also available in \NEMO (\np{ln\_traldf\_grif}\forcode{ = .true.}). A complete description of 608 the algorithm is given in App.\ref{sec:triad}.608 the algorithm is given in \autoref{apdx:triad}. 609 609 610 610 The lateral fourth order bilaplacian operator on tracers is obtained by 611 applying (\ ref{Eq_tra_ldf_lap}) twice. The operator requires an additional assumption611 applying (\autoref{eq:tra_ldf_lap}) twice. The operator requires an additional assumption 612 612 on boundary conditions: both first and third derivative terms normal to the 613 613 coast are set to zero. 614 614 615 615 The lateral fourth order operator formulation on tracers is obtained by 616 applying (\ ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption616 applying (\autoref{eq:tra_ldf_iso}) twice. It requires an additional assumption 617 617 on boundary conditions: first and third derivative terms normal to the 618 618 coast, normal to the bottom and normal to the surface are set to zero. … … 621 621 %&& ---------------------------------------------- 622 622 \subsubsection{Option for the rotated operators} 623 \label{ TRA_ldf_options}623 \label{subsec:TRA_ldf_options} 624 624 625 625 \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators) … … 637 637 % ================================================================ 638 638 \section{Tracer vertical diffusion (\protect\mdl{trazdf})} 639 \label{ TRA_zdf}639 \label{sec:TRA_zdf} 640 640 %--------------------------------------------namzdf--------------------------------------------------------- 641 641 \forfile{../namelists/namzdf} … … 645 645 The formulation of the vertical subgrid scale tracer physics is the same 646 646 for all the vertical coordinates, and is based on a laplacian operator. 647 The vertical diffusion operator given by (\ ref{Eq_PE_zdf}) takes the647 The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the 648 648 following semi-discrete space form: 649 \begin{equation} \label{ Eq_tra_zdf}649 \begin{equation} \label{eq:tra_zdf} 650 650 \begin{split} 651 651 D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}} \delta_{k+1/2}[T] \;\right] … … 658 658 $A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is 659 659 parameterised ($i.e.$ \key{zdfddm} is defined). The way these coefficients 660 are evaluated is given in \ S\ref{ZDF} (ZDF). Furthermore, when660 are evaluated is given in \autoref{chap:ZDF} (ZDF). Furthermore, when 661 661 iso-neutral mixing is used, both mixing coefficients are increased 662 662 by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ 663 to account for the vertical second derivative of \ eqref{Eq_tra_ldf_iso}.663 to account for the vertical second derivative of \autoref{eq:tra_ldf_iso}. 664 664 665 665 At the surface and bottom boundaries, the turbulent fluxes of 666 666 heat and salt must be specified. At the surface they are prescribed 667 from the surface forcing and added in a dedicated routine (see \ S\ref{TRA_sbc}),667 from the surface forcing and added in a dedicated routine (see \autoref{subsec:TRA_sbc}), 668 668 whilst at the bottom they are set to zero for heat and salt unless 669 669 a geothermal flux forcing is prescribed as a bottom boundary 670 condition (see \ S\ref{TRA_bbc}).670 condition (see \autoref{subsec:TRA_bbc}). 671 671 672 672 The large eddy coefficient found in the mixed layer together with high … … 684 684 % ================================================================ 685 685 \section{External forcing} 686 \label{ TRA_sbc_qsr_bbc}686 \label{sec:TRA_sbc_qsr_bbc} 687 687 688 688 % ------------------------------------------------------------------------------------------------------------- … … 690 690 % ------------------------------------------------------------------------------------------------------------- 691 691 \subsection{Surface boundary condition (\protect\mdl{trasbc})} 692 \label{ TRA_sbc}692 \label{subsec:TRA_sbc} 693 693 694 694 The surface boundary condition for tracers is implemented in a separate … … 703 703 of the ocean is due both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) 704 704 and to the heat and salt content of the mass exchange. They are both included directly in $Q_{ns}$, 705 the surface heat flux, and $F_{salt}$, the surface salt flux (see \ S\ref{SBC} for further details).705 the surface heat flux, and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details). 706 706 By doing this, the forcing formulation is the same for any tracer (including temperature and salinity). 707 707 708 The surface module (\mdl{sbcmod}, see \ S\ref{SBC}) provides the following708 The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following 709 709 forcing fields (used on tracers): 710 710 711 711 $\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 712 712 (i.e. the difference between the total surface heat flux and the fraction of the short wave flux that 713 penetrates into the water column, see \ S\ref{TRA_qsr}) plus the heat content associated with713 penetrates into the water column, see \autoref{subsec:TRA_qsr}) plus the heat content associated with 714 714 of the mass exchange with the atmosphere and lands. 715 715 … … 720 720 721 721 $\bullet$ \textit{rnf}, the mass flux associated with runoff 722 (see \ S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies)722 (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 723 723 724 724 $\bullet$ \textit{fwfisf}, the mass flux associated with ice shelf melt, 725 (see \ S\ref{SBC_isf} for further details on how the ice shelf melt is computed and applied).725 (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied). 726 726 727 727 The surface boundary condition on temperature and salinity is applied as follows: 728 \begin{equation} \label{ Eq_tra_sbc}728 \begin{equation} \label{eq:tra_sbc} 729 729 \begin{aligned} 730 730 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ … … 734 734 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps 735 735 ($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the 736 divergence of odd and even time step (see \ S\ref{STP}).736 divergence of odd and even time step (see \autoref{chap:STP}). 737 737 738 738 In the linear free surface case (\np{ln\_linssh}\forcode{ = .true.}), … … 742 742 would have resulted from a change in the volume of the first level. 743 743 The resulting surface boundary condition is applied as follows: 744 \begin{equation} \label{ Eq_tra_sbc_lin}744 \begin{equation} \label{eq:tra_sbc_lin} 745 745 \begin{aligned} 746 746 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } … … 754 754 In the linear free surface case, there is a small imbalance. The imbalance is larger 755 755 than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}. 756 This is the reason why the modified filter is not applied in the linear free surface case (see \ S\ref{STP}).756 This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:STP}). 757 757 758 758 % ------------------------------------------------------------------------------------------------------------- … … 760 760 % ------------------------------------------------------------------------------------------------------------- 761 761 \subsection{Solar radiation penetration (\protect\mdl{traqsr})} 762 \label{ TRA_qsr}762 \label{subsec:TRA_qsr} 763 763 %--------------------------------------------namqsr-------------------------------------------------------- 764 764 \forfile{../namelists/namtra_qsr} 765 765 %-------------------------------------------------------------------------------------------------------------- 766 766 767 Options are defined through the 767 Options are defined through the \ngn{namtra\_qsr} namelist variables. 768 768 When the penetrative solar radiation option is used (\np{ln\_flxqsr}\forcode{ = .true.}), 769 769 the solar radiation penetrates the top few tens of meters of the ocean. If it is not used 770 770 (\np{ln\_flxqsr}\forcode{ = .false.}) all the heat flux is absorbed in the first ocean level. 771 771 Thus, in the former case a term is added to the time evolution equation of 772 temperature \ eqref{Eq_PE_tra_T} and the surface boundary condition is772 temperature \autoref{eq:PE_tra_T} and the surface boundary condition is 773 773 modified to take into account only the non-penetrative part of the surface 774 774 heat flux: 775 \begin{equation} \label{ Eq_PE_qsr}775 \begin{equation} \label{eq:PE_qsr} 776 776 \begin{split} 777 777 \frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \\ … … 781 781 where $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation) 782 782 and $I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$). 783 The additional term in \ eqref{Eq_PE_qsr} is discretized as follows:784 \begin{equation} \label{ Eq_tra_qsr}783 The additional term in \autoref{eq:PE_qsr} is discretized as follows: 784 \begin{equation} \label{eq:tra_qsr} 785 785 \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3t}} \delta_k \left[ I_w \right] 786 786 \end{equation} 787 787 788 The shortwave radiation, 788 The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range. 789 789 The ocean is strongly absorbing for wavelengths longer than 700~nm and these 790 790 wavelengths contribute to heating the upper few tens of centimetres. The fraction of $Q_{sr}$ 791 791 that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ (specified 792 through namelist parameter \np{rn\_abs}). 792 through namelist parameter \np{rn\_abs}). It is assumed to penetrate the ocean 793 793 with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 794 of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the namtra\_qsrnamelist).794 of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the \ngn{namtra\_qsr} namelist). 795 795 For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy 796 796 propagates to larger depths where it contributes to 797 797 local heating. 798 798 The way this second part of the solar energy penetrates into the ocean depends on 799 which formulation is chosen. In the simple 2-waveband light penetration scheme 799 which formulation is chosen. In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}\forcode{ = .true.}) 800 800 a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, 801 leading to the following expression 802 \begin{equation} \label{ Eq_traqsr_iradiance}801 leading to the following expression \citep{Paulson1977}: 802 \begin{equation} \label{eq:traqsr_iradiance} 803 803 I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right] 804 804 \end{equation} … … 810 810 Such assumptions have been shown to provide a very crude and simplistic 811 811 representation of observed light penetration profiles (\cite{Morel_JGR88}, see also 812 Fig.\ref{Fig_traqsr_irradiance}). Light absorption in the ocean depends on812 \autoref{fig:traqsr_irradiance}). Light absorption in the ocean depends on 813 813 particle concentration and is spectrally selective. \cite{Morel_JGR88} has shown 814 814 that an accurate representation of light penetration can be provided by a 61 waveband … … 819 819 attenuation coefficient is fitted to the coefficients computed from the full spectral model 820 820 of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}), assuming 821 the same power-law relationship. As shown in Fig.\ref{Fig_traqsr_irradiance},821 the same power-law relationship. As shown in \autoref{fig:traqsr_irradiance}, 822 822 this formulation, called RGB (Red-Green-Blue), reproduces quite closely 823 823 the light penetration profiles predicted by the full spectal model, but with much greater … … 842 842 light limitation in PISCES or LOBSTER and the oceanic heating rate. 843 843 \end{description} 844 The trend in \ eqref{Eq_tra_qsr} associated with the penetration of the solar radiation844 The trend in \autoref{eq:tra_qsr} associated with the penetration of the solar radiation 845 845 is added to the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. 846 846 … … 857 857 \begin{figure}[!t] \begin{center} 858 858 \includegraphics[width=1.0\textwidth]{Fig_TRA_Irradiance} 859 \caption{ \protect\label{ Fig_traqsr_irradiance}859 \caption{ \protect\label{fig:traqsr_irradiance} 860 860 Penetration profile of the downward solar irradiance calculated by four models. 861 861 Two waveband chlorophyll-independent formulation (blue), a chlorophyll-dependent … … 870 870 % ------------------------------------------------------------------------------------------------------------- 871 871 \subsection{Bottom boundary condition (\protect\mdl{trabbc})} 872 \label{ TRA_bbc}872 \label{subsec:TRA_bbc} 873 873 %--------------------------------------------nambbc-------------------------------------------------------- 874 874 \forfile{../namelists/nambbc} … … 877 877 \begin{figure}[!t] \begin{center} 878 878 \includegraphics[width=1.0\textwidth]{Fig_TRA_geoth} 879 \caption{ \protect\label{ Fig_geothermal}879 \caption{ \protect\label{fig:geothermal} 880 880 Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}. 881 881 It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}.} … … 902 902 When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is 903 903 introduced which is provided in the \ifile{geothermal\_heating} NetCDF file 904 ( Fig.\ref{Fig_geothermal}) \citep{Emile-Geay_Madec_OS09}.904 (\autoref{fig:geothermal}) \citep{Emile-Geay_Madec_OS09}. 905 905 906 906 % ================================================================ … … 908 908 % ================================================================ 909 909 \section{Bottom boundary layer (\protect\mdl{trabbl} - \protect\key{trabbl})} 910 \label{ TRA_bbl}910 \label{sec:TRA_bbl} 911 911 %--------------------------------------------nambbl--------------------------------------------------------- 912 912 \forfile{../namelists/nambbl} … … 943 943 % ------------------------------------------------------------------------------------------------------------- 944 944 \subsection{Diffusive bottom boundary layer (\protect\np{nn\_bbl\_ldf}\forcode{ = 1})} 945 \label{ TRA_bbl_diff}945 \label{subsec:TRA_bbl_diff} 946 946 947 947 When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1), 948 948 the diffusive flux between two adjacent cells at the ocean floor is given by 949 \begin{equation} \label{ Eq_tra_bbl_diff}949 \begin{equation} \label{eq:tra_bbl_diff} 950 950 {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T 951 951 \end{equation} … … 953 953 and $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{Beckmann_Doscher1997}, 954 954 the latter is prescribed with a spatial dependence, $i.e.$ in the conditional form 955 \begin{equation} \label{ Eq_tra_bbl_coef}955 \begin{equation} \label{eq:tra_bbl_coef} 956 956 A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l} 957 957 A_{bbl} \quad \quad \mbox{if} \quad \nabla_\sigma \rho \cdot \nabla H<0 \\ … … 962 962 where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist 963 963 parameter \np{rn\_ahtbbl} and usually set to a value much larger 964 than the one used for lateral mixing in the open ocean. The constraint in \ eqref{Eq_tra_bbl_coef}964 than the one used for lateral mixing in the open ocean. The constraint in \autoref{eq:tra_bbl_coef} 965 965 implies that sigma-like diffusion only occurs when the density above the sea floor, at the top of 966 the slope, is larger than in the deeper ocean (see green arrow in Fig.\ref{Fig_bbl}).966 the slope, is larger than in the deeper ocean (see green arrow in \autoref{fig:bbl}). 967 967 In practice, this constraint is applied separately in the two horizontal directions, 968 and the density gradient in \ eqref{Eq_tra_bbl_coef} is evaluated with the log gradient formulation:969 \begin{equation} \label{ Eq_tra_bbl_Drho}968 and the density gradient in \autoref{eq:tra_bbl_coef} is evaluated with the log gradient formulation: 969 \begin{equation} \label{eq:tra_bbl_Drho} 970 970 \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta \,\nabla_\sigma S 971 971 \end{equation} … … 978 978 % ------------------------------------------------------------------------------------------------------------- 979 979 \subsection{Advective bottom boundary layer (\protect\np{nn\_bbl\_adv}\forcode{ = 1..2})} 980 \label{ TRA_bbl_adv}980 \label{subsec:TRA_bbl_adv} 981 981 982 982 \sgacomment{"downsloping flow" has been replaced by "downslope flow" in the following … … 986 986 \begin{figure}[!t] \begin{center} 987 987 \includegraphics[width=0.7\textwidth]{Fig_BBL_adv} 988 \caption{ \protect\label{ Fig_bbl}988 \caption{ \protect\label{fig:bbl} 989 989 Advective/diffusive Bottom Boundary Layer. The BBL parameterisation is 990 990 activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$. … … 1011 1011 1012 1012 \np{nn\_bbl\_adv}\forcode{ = 1} : the downslope velocity is chosen to be the Eulerian 1013 ocean velocity just above the topographic step (see black arrow in Fig.\ref{Fig_bbl})1013 ocean velocity just above the topographic step (see black arrow in \autoref{fig:bbl}) 1014 1014 \citep{Beckmann_Doscher1997}. It is a \textit{conditional advection}, that is, advection 1015 1015 is allowed only if dense water overlies less dense water on the slope ($i.e.$ … … 1021 1021 The advection is allowed only if dense water overlies less dense water on the slope ($i.e.$ 1022 1022 $\nabla_\sigma \rho \cdot \nabla H<0$). For example, the resulting transport of the 1023 downslope flow, here in the $i$-direction ( Fig.\ref{Fig_bbl}), is simply given by the1023 downslope flow, here in the $i$-direction (\autoref{fig:bbl}), is simply given by the 1024 1024 following expression: 1025 \begin{equation} \label{ Eq_bbl_Utr}1025 \begin{equation} \label{eq:bbl_Utr} 1026 1026 u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o} e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right) 1027 1027 \end{equation} … … 1039 1039 water at intermediate depths. The entrainment is replaced by the vertical mixing 1040 1040 implicit in the advection scheme. Let us consider as an example the 1041 case displayed in Fig.\ref{Fig_bbl} where the density at level $(i,kup)$ is1041 case displayed in \autoref{fig:bbl} where the density at level $(i,kup)$ is 1042 1042 larger than the one at level $(i,kdwn)$. The advective BBL scheme 1043 1043 modifies the tracer time tendency of the ocean cells near the 1044 topographic step by the downslope flow \ eqref{Eq_bbl_dw},1045 the horizontal \ eqref{Eq_bbl_hor} and the upward \eqref{Eq_bbl_up}1044 topographic step by the downslope flow \autoref{eq:bbl_dw}, 1045 the horizontal \autoref{eq:bbl_hor} and the upward \autoref{eq:bbl_up} 1046 1046 return flows as follows: 1047 1047 \begin{align} 1048 1048 \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 1049 + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} \left( T^{sh}_{kup} - T^{do}_{kdw} \right) \label{ Eq_bbl_dw} \\1049 + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} \left( T^{sh}_{kup} - T^{do}_{kdw} \right) \label{eq:bbl_dw} \\ 1050 1050 % 1051 1051 \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 1052 + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} \left( T^{do}_{kup} - T^{sh}_{kup} \right) \label{ Eq_bbl_hor} \\1052 + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} \left( T^{do}_{kup} - T^{sh}_{kup} \right) \label{eq:bbl_hor} \\ 1053 1053 % 1054 1054 \intertext{and for $k =kdw-1,\;..., \; kup$ :} 1055 1055 % 1056 1056 \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 1057 + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} \left( T^{do}_{k+1} - T^{sh}_{k} \right) \label{ Eq_bbl_up}1057 + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} \left( T^{do}_{k+1} - T^{sh}_{k} \right) \label{eq:bbl_up} 1058 1058 \end{align} 1059 1059 where $b_t$ is the $T$-cell volume. … … 1067 1067 % ================================================================ 1068 1068 \section{Tracer damping (\protect\mdl{tradmp})} 1069 \label{ TRA_dmp}1069 \label{sec:TRA_dmp} 1070 1070 %--------------------------------------------namtra_dmp------------------------------------------------- 1071 1071 \forfile{../namelists/namtra_dmp} … … 1074 1074 In some applications it can be useful to add a Newtonian damping term 1075 1075 into the temperature and salinity equations: 1076 \begin{equation} \label{ Eq_tra_dmp}1076 \begin{equation} \label{eq:tra_dmp} 1077 1077 \begin{split} 1078 1078 \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right) \\ … … 1087 1087 in \textit{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are 1088 1088 correctly set ($i.e.$ that $T_o$ and $S_o$ are provided in input files and read 1089 using \mdl{fldread}, see \ S\ref{SBC_fldread}).1089 using \mdl{fldread}, see \autoref{subsec:SBC_fldread}). 1090 1090 The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine. The file name is specified by the namelist variable \np{cn\_resto}. The DMP\_TOOLS tool is provided to allow users to generate the netcdf file. 1091 1091 1092 The two main cases in which \ eqref{Eq_tra_dmp} is used are \textit{(a)}1092 The two main cases in which \autoref{eq:tra_dmp} is used are \textit{(a)} 1093 1093 the specification of the boundary conditions along artificial walls of a 1094 1094 limited domain basin and \textit{(b)} the computation of the velocity … … 1151 1151 % ================================================================ 1152 1152 \section{Tracer time evolution (\protect\mdl{tranxt})} 1153 \label{ TRA_nxt}1153 \label{sec:TRA_nxt} 1154 1154 %--------------------------------------------namdom----------------------------------------------------- 1155 1155 \forfile{../namelists/namdom} … … 1159 1159 The general framework for tracer time stepping is a modified leap-frog scheme 1160 1160 \citep{Leclair_Madec_OM09}, $i.e.$ a three level centred time scheme associated 1161 with a Asselin time filter (cf. \ S\ref{STP_mLF}):1162 \begin{equation} \label{ Eq_tra_nxt}1161 with a Asselin time filter (cf. \autoref{sec:STP_mLF}): 1162 \begin{equation} \label{eq:tra_nxt} 1163 1163 \begin{aligned} 1164 1164 (e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt \,e_{3t}^t\ \text{RHS}^t & \\ … … 1174 1174 $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). 1175 1175 Its default value is \np{rn\_atfp}\forcode{ = 10.e-3}. Note that the forcing correction term in the filter 1176 is not applied in linear free surface (\jp{lk\_vvl}\forcode{ = .false.}) (see \ S\ref{TRA_sbc}.1176 is not applied in linear free surface (\jp{lk\_vvl}\forcode{ = .false.}) (see \autoref{subsec:TRA_sbc}. 1177 1177 Not also that in constant volume case, the time stepping is performed on $T$, 1178 1178 not on its content, $e_{3t}T$. … … 1189 1189 % ================================================================ 1190 1190 \section{Equation of state (\protect\mdl{eosbn2}) } 1191 \label{ TRA_eosbn2}1191 \label{sec:TRA_eosbn2} 1192 1192 %--------------------------------------------nameos----------------------------------------------------- 1193 1193 \forfile{../namelists/nameos} … … 1198 1198 % ------------------------------------------------------------------------------------------------------------- 1199 1199 \subsection{Equation of seawater (\protect\np{nn\_eos}\forcode{ = -1..1})} 1200 \label{ TRA_eos}1200 \label{subsec:TRA_eos} 1201 1201 1202 1202 The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship … … 1272 1272 and \textit{practical} salinity. 1273 1273 S-EOS takes the following expression: 1274 \begin{equation} \label{ Eq_tra_S-EOS}1274 \begin{equation} \label{eq:tra_S-EOS} 1275 1275 \begin{split} 1276 1276 d_a(T,S,z) = ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a \\ … … 1280 1280 \end{split} 1281 1281 \end{equation} 1282 where the computer name of the coefficients as well as their standard value are given in \ ref{Tab_SEOS}.1282 where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}. 1283 1283 In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing 1284 1284 the associated coefficients. … … 1303 1303 $\mu_2$ & \np{rn\_mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline 1304 1304 \end{tabular} 1305 \caption{ \protect\label{ Tab_SEOS}1305 \caption{ \protect\label{tab:SEOS} 1306 1306 Standard value of S-EOS coefficients. } 1307 1307 \end{center} … … 1314 1314 % ------------------------------------------------------------------------------------------------------------- 1315 1315 \subsection{Brunt-V\"{a}is\"{a}l\"{a} frequency (\protect\np{nn\_eos}\forcode{ = 0..2})} 1316 \label{ TRA_bn2}1316 \label{subsec:TRA_bn2} 1317 1317 1318 1318 An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} … … 1323 1323 (pressure in decibar being approximated by the depth in meters). The expression for $N^2$ 1324 1324 is given by: 1325 \begin{equation} \label{ Eq_tra_bn2}1325 \begin{equation} \label{eq:tra_bn2} 1326 1326 N^2 = \frac{g}{e_{3w}} \left( \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T] \right) 1327 1327 \end{equation} … … 1336 1336 % ------------------------------------------------------------------------------------------------------------- 1337 1337 \subsection{Freezing point of seawater} 1338 \label{ TRA_fzp}1338 \label{subsec:TRA_fzp} 1339 1339 1340 1340 The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}: 1341 \begin{equation} \label{ Eq_tra_eos_fzp}1341 \begin{equation} \label{eq:tra_eos_fzp} 1342 1342 \begin{split} 1343 1343 T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} … … 1347 1347 \end{equation} 1348 1348 1349 \ eqref{Eq_tra_eos_fzp} is only used to compute the potential freezing point of1349 \autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of 1350 1350 sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent 1351 terms in \ eqref{Eq_tra_eos_fzp} (last term) have been dropped. The freezing1351 terms in \autoref{eq:tra_eos_fzp} (last term) have been dropped. The freezing 1352 1352 point is computed through \textit{eos\_fzp}, a \textsc{Fortran} function that can be found 1353 1353 in \mdl{eosbn2}. … … 1358 1358 % ------------------------------------------------------------------------------------------------------------- 1359 1359 %\subsection{Potential Energy anomalies} 1360 %\label{ TRA_bn2}1360 %\label{subsec:TRA_bn2} 1361 1361 1362 1362 % =====>>>>> TO BE written … … 1368 1368 % ================================================================ 1369 1369 \section{Horizontal derivative in \textit{zps}-coordinate (\protect\mdl{zpshde})} 1370 \label{ TRA_zpshde}1370 \label{sec:TRA_zpshde} 1371 1371 1372 1372 \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, … … 1382 1382 Before taking horizontal gradients between the tracers next to the bottom, a linear 1383 1383 interpolation in the vertical is used to approximate the deeper tracer as if it actually 1384 lived at the depth of the shallower tracer point ( Fig.~\ref{Fig_Partial_step_scheme}).1384 lived at the depth of the shallower tracer point (\autoref{fig:Partial_step_scheme}). 1385 1385 For example, for temperature in the $i$-direction the needed interpolated 1386 1386 temperature, $\widetilde{T}$, is: … … 1389 1389 \begin{figure}[!p] \begin{center} 1390 1390 \includegraphics[width=0.9\textwidth]{Partial_step_scheme} 1391 \caption{ \protect\label{ Fig_Partial_step_scheme}1391 \caption{ \protect\label{fig:Partial_step_scheme} 1392 1392 Discretisation of the horizontal difference and average of tracers in the $z$-partial 1393 step coordinate (\ np{ln\_zps}\forcode{ = .true.}) in the case $( e3w_k^{i+1} - e3w_k^i )>0$.1393 step coordinate (\protect\np{ln\_zps}\forcode{ = .true.}) in the case $( e3w_k^{i+1} - e3w_k^i )>0$. 1394 1394 A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value 1395 1395 at the depth of the shallower tracer point of the two adjacent bottom $T$-points. … … 1409 1409 and the resulting forms for the horizontal difference and the horizontal average 1410 1410 value of $T$ at a $U$-point are: 1411 \begin{equation} \label{ Eq_zps_hde}1411 \begin{equation} \label{eq:zps_hde} 1412 1412 \begin{aligned} 1413 1413 \delta _{i+1/2} T= \begin{cases} … … 1432 1432 of density, we compute $\widetilde{\rho }$ from the interpolated values of $T$ 1433 1433 and $S$, and the pressure at a $u$-point (in the equation of state pressure is 1434 approximated by depth, see \ S\ref{TRA_eos} ) :1435 \begin{equation} \label{ Eq_zps_hde_rho}1434 approximated by depth, see \autoref{subsec:TRA_eos} ) : 1435 \begin{equation} \label{eq:zps_hde_rho} 1436 1436 \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) 1437 1437 \quad \text{where }\ z_u = \min \left( {z_T^{i+1} ,z_T^i } \right) … … 1441 1441 thus pressure) is highly non-linear with a true equation of state and thus is badly 1442 1442 approximated with a linear interpolation. This approximation is used to compute 1443 both the horizontal pressure gradient (\ S\ref{DYN_hpg}) and the slopes of neutral1444 surfaces (\ S\ref{LDF_slp})1443 both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and the slopes of neutral 1444 surfaces (\autoref{sec:LDF_slp}) 1445 1445 1446 1446 Note that in almost all the advection schemes presented in this Chapter, both 1447 averaging and differencing operators appear. Yet \ eqref{Eq_zps_hde} has not1447 averaging and differencing operators appear. Yet \autoref{eq:zps_hde} has not 1448 1448 been used in these schemes: in contrast to diffusion and pressure gradient 1449 1449 computations, no correction for partial steps is applied for advection. The main -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_ZDF.tex
r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Vertical Ocean Physics (ZDF)} 7 \label{ ZDF}7 \label{chap:ZDF} 8 8 \minitoc 9 9 … … 19 19 % ================================================================ 20 20 \section{Vertical mixing} 21 \label{ ZDF_zdf}21 \label{sec:ZDF_zdf} 22 22 23 23 The discrete form of the ocean subgrid scale physics has been presented in 24 \ S\ref{TRA_zdf} and \S\ref{DYN_zdf}. At the surface and bottom boundaries,24 \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}. At the surface and bottom boundaries, 25 25 the turbulent fluxes of momentum, heat and salt have to be defined. At the 26 surface they are prescribed from the surface forcing (see Chap.~\ref{SBC}),26 surface they are prescribed from the surface forcing (see \autoref{chap:SBC}), 27 27 while at the bottom they are set to zero for heat and salt, unless a geothermal 28 28 flux forcing is prescribed as a bottom boundary condition ($i.e.$ \key{trabbl} 29 defined, see \ S\ref{TRA_bbc}), and specified through a bottom friction30 parameterisation for momentum (see \ S\ref{ZDF_bfr}).29 defined, see \autoref{subsec:TRA_bbc}), and specified through a bottom friction 30 parameterisation for momentum (see \autoref{sec:ZDF_bfr}). 31 31 32 32 In this section we briefly discuss the various choices offered to compute 33 33 the vertical eddy viscosity and diffusivity coefficients, $A_u^{vm}$ , 34 34 $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- 35 points, respectively (see \ S\ref{TRA_zdf} and \S\ref{DYN_zdf}). These35 points, respectively (see \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}). These 36 36 coefficients can be assumed to be either constant, or a function of the local 37 37 Richardson number, or computed from a turbulent closure model (either … … 44 44 (namelist parameter \np{ln\_zdfexp}\forcode{ = .true.}) or a backward time stepping 45 45 scheme (\np{ln\_zdfexp}\forcode{ = .false.}) depending on the magnitude of the mixing 46 coefficients, and thus of the formulation used (see \ S\ref{STP}).46 coefficients, and thus of the formulation used (see \autoref{chap:STP}). 47 47 48 48 % ------------------------------------------------------------------------------------------------------------- … … 50 50 % ------------------------------------------------------------------------------------------------------------- 51 51 \subsection{Constant (\protect\key{zdfcst})} 52 \label{ ZDF_cst}52 \label{subsec:ZDF_cst} 53 53 %--------------------------------------------namzdf--------------------------------------------------------- 54 54 \forfile{../namelists/namzdf} … … 75 75 % ------------------------------------------------------------------------------------------------------------- 76 76 \subsection{Richardson number dependent (\protect\key{zdfric})} 77 \label{ ZDF_ric}77 \label{subsec:ZDF_ric} 78 78 79 79 %--------------------------------------------namric--------------------------------------------------------- … … 91 91 ratio of stratification to vertical shear). Following \citet{Pacanowski_Philander_JPO81}, the following 92 92 formulation has been implemented: 93 \begin{equation} \label{ Eq_zdfric}93 \begin{equation} \label{eq:zdfric} 94 94 \left\{ \begin{aligned} 95 95 A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT} \\ … … 98 98 \end{equation} 99 99 where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson 100 number, $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \ S\ref{TRA_bn2}),100 number, $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), 101 101 $A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the 102 constant case (see \ S\ref{ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$102 constant case (see \autoref{subsec:ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ 103 103 is the maximum value that can be reached by the coefficient when $Ri\leq 0$, 104 104 $a=5$ and $n=2$. The last three values can be modified by setting the … … 133 133 % ------------------------------------------------------------------------------------------------------------- 134 134 \subsection{TKE turbulent closure scheme (\protect\key{zdftke})} 135 \label{ ZDF_tke}135 \label{subsec:ZDF_tke} 136 136 137 137 %--------------------------------------------namzdf_tke-------------------------------------------------- … … 150 150 $\bar{e}$ through vertical shear, its destruction through stratification, its vertical 151 151 diffusion, and its dissipation of \citet{Kolmogorov1942} type: 152 \begin{equation} \label{ Eq_zdftke_e}152 \begin{equation} \label{eq:zdftke_e} 153 153 \frac{\partial \bar{e}}{\partial t} = 154 154 \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2 … … 159 159 - c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon } 160 160 \end{equation} 161 \begin{equation} \label{ Eq_zdftke_kz}161 \begin{equation} \label{eq:zdftke_kz} 162 162 \begin{split} 163 163 K_m &= C_k\ l_k\ \sqrt {\bar{e}\; } \\ … … 165 165 \end{split} 166 166 \end{equation} 167 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \ S\ref{TRA_bn2}),167 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), 168 168 $l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales, 169 169 $P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity … … 173 173 $P_{rt}$ can be set to unity or, following \citet{Blanke1993}, be a function 174 174 of the local Richardson number, $R_i$: 175 \begin{align*} \label{ Eq_prt}175 \begin{align*} \label{eq:prt} 176 176 P_{rt} = \begin{cases} 177 177 \ \ \ 1 & \text{if $\ R_i \leq 0.2$} \\ … … 187 187 namelist parameter. The default value of $e_{bb}$ is 3.75. \citep{Gaspar1990}), 188 188 however a much larger value can be used when taking into account the 189 surface wave breaking (see below Eq. \ eqref{ZDF_Esbc}).189 surface wave breaking (see below Eq. \autoref{eq:ZDF_Esbc}). 190 190 The bottom value of TKE is assumed to be equal to the value of the level just above. 191 191 The time integration of the $\bar{e}$ equation may formally lead to negative values … … 199 199 instabilities associated with too weak vertical diffusion. They must be 200 200 specified at least larger than the molecular values, and are set through 201 \np{rn\_avm0} and \np{rn\_avt0} (namzdf namelist, see \ S\ref{ZDF_cst}).201 \np{rn\_avm0} and \np{rn\_avt0} (namzdf namelist, see \autoref{subsec:ZDF_cst}). 202 202 203 203 \subsubsection{Turbulent length scale} … … 207 207 parameter. The first two are based on the following first order approximation 208 208 \citep{Blanke1993}: 209 \begin{equation} \label{ Eq_tke_mxl0_1}209 \begin{equation} \label{eq:tke_mxl0_1} 210 210 l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N 211 211 \end{equation} … … 219 219 \np{nn\_mxl}\forcode{ = 2..3} cases, which add an extra assumption concerning the vertical 220 220 gradient of the computed length scale. So, the length scales are first evaluated 221 as in \ eqref{Eq_tke_mxl0_1} and then bounded such that:222 \begin{equation} \label{ Eq_tke_mxl_constraint}221 as in \autoref{eq:tke_mxl0_1} and then bounded such that: 222 \begin{equation} \label{eq:tke_mxl_constraint} 223 223 \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1 224 224 \qquad \text{with }\ l = l_k = l_\epsilon 225 225 \end{equation} 226 \ eqref{Eq_tke_mxl_constraint} means that the vertical variations of the length226 \autoref{eq:tke_mxl_constraint} means that the vertical variations of the length 227 227 scale cannot be larger than the variations of depth. It provides a better 228 228 approximation of the \citet{Gaspar1990} formulation while being much less … … 230 230 by the distance to the surface or to the ocean bottom but also by the distance 231 231 to a strongly stratified portion of the water column such as the thermocline 232 ( Fig.~\ref{Fig_mixing_length}). In order to impose the \eqref{Eq_tke_mxl_constraint}232 (\autoref{fig:mixing_length}). In order to impose the \autoref{eq:tke_mxl_constraint} 233 233 constraint, we introduce two additional length scales: $l_{up}$ and $l_{dwn}$, 234 234 the upward and downward length scales, and evaluate the dissipation and … … 237 237 \begin{figure}[!t] \begin{center} 238 238 \includegraphics[width=1.00\textwidth]{Fig_mixing_length} 239 \caption{ \protect\label{ Fig_mixing_length}239 \caption{ \protect\label{fig:mixing_length} 240 240 Illustration of the mixing length computation. } 241 241 \end{center} 242 242 \end{figure} 243 243 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 244 \begin{equation} \label{ Eq_tke_mxl2}244 \begin{equation} \label{eq:tke_mxl2} 245 245 \begin{aligned} 246 246 l_{up\ \ }^{(k)} &= \min \left( l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \; \right) … … 250 250 \end{aligned} 251 251 \end{equation} 252 where $l^{(k)}$ is computed using \ eqref{Eq_tke_mxl0_1},252 where $l^{(k)}$ is computed using \autoref{eq:tke_mxl0_1}, 253 253 $i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. 254 254 … … 257 257 \np{nn\_mxl}\forcode{ = 3} case, the dissipation and mixing turbulent length scales are give 258 258 as in \citet{Gaspar1990}: 259 \begin{equation} \label{ Eq_tke_mxl_gaspar}259 \begin{equation} \label{eq:tke_mxl_gaspar} 260 260 \begin{aligned} 261 261 & l_k = \sqrt{\ l_{up} \ \ l_{dwn}\ } \\ … … 282 282 283 283 Following \citet{Craig_Banner_JPO94}, the boundary condition on surface TKE value is : 284 \begin{equation} \label{ ZDF_Esbc}284 \begin{equation} \label{eq:ZDF_Esbc} 285 285 \bar{e}_o = \frac{1}{2}\,\left( 15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o} 286 286 \end{equation} … … 289 289 younger waves \citep{Mellor_Blumberg_JPO04}. 290 290 The boundary condition on the turbulent length scale follows the Charnock's relation: 291 \begin{equation} \label{ ZDF_Lsbc}291 \begin{equation} \label{eq:ZDF_Lsbc} 292 292 l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o} 293 293 \end{equation} … … 297 297 As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$, 298 298 with $e_{bb}$ the \np{rn\_ebb} namelist parameter, setting \np{rn\_ebb}\forcode{ = 67.83} corresponds 299 to $\alpha_{CB} = 100$. Further setting \np{ln\_mxl0} to true applies \ eqref{ZDF_Lsbc}299 to $\alpha_{CB} = 100$. Further setting \np{ln\_mxl0} to true applies \autoref{eq:ZDF_Lsbc} 300 300 as surface boundary condition on length scale, with $\beta$ hard coded to the Stacey's value. 301 301 Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) … … 316 316 The parameterization, tuned against large-eddy simulation, includes the whole effect 317 317 of LC in an extra source terms of TKE, $P_{LC}$. 318 The presence of $P_{LC}$ in \ eqref{Eq_zdftke_e}, the TKE equation, is controlled318 The presence of $P_{LC}$ in \autoref{eq:zdftke_e}, the TKE equation, is controlled 319 319 by setting \np{ln\_lc} to \forcode{.true.} in the namtke namelist. 320 320 … … 368 368 When using this parameterization ($i.e.$ when \np{nn\_etau}\forcode{ = 1}), the TKE input to the ocean ($S$) 369 369 imposed by the winds in the form of near-inertial oscillations, swell and waves is parameterized 370 by \ eqref{ZDF_Esbc} the standard TKE surface boundary condition, plus a depth depend one given by:371 \begin{equation} \label{ ZDF_Ehtau}370 by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition, plus a depth depend one given by: 371 \begin{equation} \label{eq:ZDF_Ehtau} 372 372 S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau} 373 373 \end{equation} … … 385 385 386 386 Note that two other option existe, \np{nn\_etau}\forcode{ = 2..3}. They correspond to applying 387 \ eqref{ZDF_Ehtau} only at the base of the mixed layer, or to using the high frequency part387 \autoref{eq:ZDF_Ehtau} only at the base of the mixed layer, or to using the high frequency part 388 388 of the stress to evaluate the fraction of TKE that penetrate the ocean. 389 389 Those two options are obsolescent features introduced for test purposes. … … 406 406 % ------------------------------------------------------------------------------------------------------------- 407 407 \subsection{TKE discretization considerations (\protect\key{zdftke})} 408 \label{ ZDF_tke_ene}408 \label{subsec:ZDF_tke_ene} 409 409 410 410 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 411 411 \begin{figure}[!t] \begin{center} 412 412 \includegraphics[width=1.00\textwidth]{Fig_ZDF_TKE_time_scheme} 413 \caption{ \protect\label{ Fig_TKE_time_scheme}413 \caption{ \protect\label{fig:TKE_time_scheme} 414 414 Illustration of the TKE time integration and its links to the momentum and tracer time integration. } 415 415 \end{center} … … 418 418 419 419 The production of turbulence by vertical shear (the first term of the right hand side 420 of \ eqref{Eq_zdftke_e}) should balance the loss of kinetic energy associated with421 the vertical momentum diffusion (first line in \ eqref{Eq_PE_zdf}). To do so a special care420 of \autoref{eq:zdftke_e}) should balance the loss of kinetic energy associated with 421 the vertical momentum diffusion (first line in \autoref{eq:PE_zdf}). To do so a special care 422 422 have to be taken for both the time and space discretization of the TKE equation 423 423 \citep{Burchard_OM02,Marsaleix_al_OM08}. 424 424 425 Let us first address the time stepping issue. Fig.~\ref{Fig_TKE_time_scheme} shows425 Let us first address the time stepping issue. \autoref{fig:TKE_time_scheme} shows 426 426 how the two-level Leap-Frog time stepping of the momentum and tracer equations interplays 427 427 with the one-level forward time stepping of TKE equation. With this framework, the total loss 428 428 of kinetic energy (in 1D for the demonstration) due to the vertical momentum diffusion is 429 429 obtained by multiplying this quantity by $u^t$ and summing the result vertically: 430 \begin{equation} \label{ Eq_energ1}430 \begin{equation} \label{eq:energ1} 431 431 \begin{split} 432 432 \int_{-H}^{\eta} u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt} \right) \,dz \\ … … 436 436 \end{equation} 437 437 Here, the vertical diffusion of momentum is discretized backward in time 438 with a coefficient, $K_m$, known at time $t$ ( Fig.~\ref{Fig_TKE_time_scheme}),439 as it is required when using the TKE scheme (see \ S\ref{STP_forward_imp}).440 The first term of the right hand side of \ eqref{Eq_energ1} represents the kinetic energy438 with a coefficient, $K_m$, known at time $t$ (\autoref{fig:TKE_time_scheme}), 439 as it is required when using the TKE scheme (see \autoref{sec:STP_forward_imp}). 440 The first term of the right hand side of \autoref{eq:energ1} represents the kinetic energy 441 441 transfer at the surface (atmospheric forcing) and at the bottom (friction effect). 442 442 The second term is always negative. It is the dissipation rate of kinetic energy, 443 and thus minus the shear production rate of $\bar{e}$. \ eqref{Eq_energ1}443 and thus minus the shear production rate of $\bar{e}$. \autoref{eq:energ1} 444 444 implies that, to be energetically consistent, the production rate of $\bar{e}$ 445 445 used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as … … 448 448 449 449 A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification 450 (second term of the right hand side of \ eqref{Eq_zdftke_e}). This term450 (second term of the right hand side of \autoref{eq:zdftke_e}). This term 451 451 must balance the input of potential energy resulting from vertical mixing. 452 452 The rate of change of potential energy (in 1D for the demonstration) due vertical 453 453 mixing is obtained by multiplying vertical density diffusion 454 454 tendency by $g\,z$ and and summing the result vertically: 455 \begin{equation} \label{ Eq_energ2}455 \begin{equation} \label{eq:energ2} 456 456 \begin{split} 457 457 \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt} \right) \,dz \\ … … 463 463 \end{equation} 464 464 where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$. 465 The first term of the right hand side of \ eqref{Eq_energ2} is always zero465 The first term of the right hand side of \autoref{eq:energ2} is always zero 466 466 because there is no diffusive flux through the ocean surface and bottom). 467 467 The second term is minus the destruction rate of $\bar{e}$ due to stratification. 468 Therefore \ eqref{Eq_energ1} implies that, to be energetically consistent, the product469 ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \ eqref{Eq_zdftke_e}, the TKE equation.468 Therefore \autoref{eq:energ1} implies that, to be energetically consistent, the product 469 ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:zdftke_e}, the TKE equation. 470 470 471 471 Let us now address the space discretization issue. 472 472 The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity 473 473 components are in the centre of the side faces of a $t$-box in staggered C-grid 474 ( Fig.\ref{Fig_cell}). A space averaging is thus required to obtain the shear TKE production term.475 By redoing the \ eqref{Eq_energ1} in the 3D case, it can be shown that the product of474 (\autoref{fig:cell}). A space averaging is thus required to obtain the shear TKE production term. 475 By redoing the \autoref{eq:energ1} in the 3D case, it can be shown that the product of 476 476 eddy coefficient by the shear at $t$ and $t-\rdt$ must be performed prior to the averaging. 477 477 Furthermore, the possible time variation of $e_3$ (\key{vvl} case) have to be taken into … … 480 480 The above energetic considerations leads to 481 481 the following final discrete form for the TKE equation: 482 \begin{equation} \label{ Eq_zdftke_ene}482 \begin{equation} \label{eq:zdftke_ene} 483 483 \begin{split} 484 484 \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt} \equiv … … 497 497 \end{split} 498 498 \end{equation} 499 where the last two terms in \ eqref{Eq_zdftke_ene} (vertical diffusion and Kolmogorov dissipation)500 are time stepped using a backward scheme (see\ S\ref{STP_forward_imp}).499 where the last two terms in \autoref{eq:zdftke_ene} (vertical diffusion and Kolmogorov dissipation) 500 are time stepped using a backward scheme (see\autoref{sec:STP_forward_imp}). 501 501 Note that the Kolmogorov term has been linearized in time in order to render 502 502 the implicit computation possible. The restart of the TKE scheme 503 503 requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as they all appear in 504 the right hand side of \ eqref{Eq_zdftke_ene}. For the latter, it is in fact504 the right hand side of \autoref{eq:zdftke_ene}. For the latter, it is in fact 505 505 the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored. 506 506 … … 509 509 % ------------------------------------------------------------------------------------------------------------- 510 510 \subsection{GLS: Generic Length Scale (\protect\key{zdfgls})} 511 \label{ ZDF_gls}511 \label{subsec:ZDF_gls} 512 512 513 513 %--------------------------------------------namzdf_gls--------------------------------------------------------- … … 519 519 for the generic length scale, $\psi$ \citep{Umlauf_Burchard_JMS03, Umlauf_Burchard_CSR05}. 520 520 This later variable is defined as : $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$, 521 where the triplet $(p, m, n)$ value given in Tab.\ ref{Tab_GLS} allows to recover521 where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:GLS} allows to recover 522 522 a number of well-known turbulent closures ($k$-$kl$ \citep{Mellor_Yamada_1982}, 523 523 $k$-$\epsilon$ \citep{Rodi_1987}, $k$-$\omega$ \citep{Wilcox_1988} 524 524 among others \citep{Umlauf_Burchard_JMS03,Kantha_Carniel_CSR05}). 525 525 The GLS scheme is given by the following set of equations: 526 \begin{equation} \label{ Eq_zdfgls_e}526 \begin{equation} \label{eq:zdfgls_e} 527 527 \frac{\partial \bar{e}}{\partial t} = 528 528 \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 … … 533 533 \end{equation} 534 534 535 \begin{equation} \label{ Eq_zdfgls_psi}535 \begin{equation} \label{eq:zdfgls_psi} 536 536 \begin{split} 537 537 \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{ … … 544 544 \end{equation} 545 545 546 \begin{equation} \label{ Eq_zdfgls_kz}546 \begin{equation} \label{eq:zdfgls_kz} 547 547 \begin{split} 548 548 K_m &= C_{\mu} \ \sqrt {\bar{e}} \ l \\ … … 551 551 \end{equation} 552 552 553 \begin{equation} \label{ Eq_zdfgls_eps}553 \begin{equation} \label{eq:zdfgls_eps} 554 554 {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \; 555 555 \end{equation} 556 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \ S\ref{TRA_bn2})556 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) 557 557 and $\epsilon$ the dissipation rate. 558 558 The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) 559 559 depends of the choice of the turbulence model. Four different turbulent models are pre-defined 560 (Tab.\ ref{Tab_GLS}). They are made available through the \np{nn\_clo} namelist parameter.560 (Tab.\autoref{tab:GLS}). They are made available through the \np{nn\_clo} namelist parameter. 561 561 562 562 %--------------------------------------------------TABLE-------------------------------------------------- … … 579 579 \hline 580 580 \end{tabular} 581 \caption{ \protect\label{ Tab_GLS}581 \caption{ \protect\label{tab:GLS} 582 582 Set of predefined GLS parameters, or equivalently predefined turbulence models available 583 583 with \protect\key{zdfgls} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\ngn{namzdf\_gls} .} … … 596 596 As for TKE closure , the wave effect on the mixing is considered when \np{ln\_crban}\forcode{ = .true.} 597 597 \citep{Craig_Banner_JPO94, Mellor_Blumberg_JPO04}. The \np{rn\_crban} namelist parameter 598 is $\alpha_{CB}$ in \ eqref{ZDF_Esbc} and \np{rn\_charn} provides the value of $\beta$ in \eqref{ZDF_Lsbc}.598 is $\alpha_{CB}$ in \autoref{eq:ZDF_Esbc} and \np{rn\_charn} provides the value of $\beta$ in \autoref{eq:ZDF_Lsbc}. 599 599 600 600 The $\psi$ equation is known to fail in stably stratified flows, and for this reason … … 609 609 610 610 The time and space discretization of the GLS equations follows the same energetic 611 consideration as for the TKE case described in \ S\ref{ZDF_tke_ene} \citep{Burchard_OM02}.611 consideration as for the TKE case described in \autoref{subsec:ZDF_tke_ene} \citep{Burchard_OM02}. 612 612 Examples of performance of the 4 turbulent closure scheme can be found in \citet{Warner_al_OM05}. 613 613 … … 616 616 % ------------------------------------------------------------------------------------------------------------- 617 617 \subsection{OSM: OSMOSIS boundary layer scheme (\protect\key{zdfosm})} 618 \label{ ZDF_osm}618 \label{subsec:ZDF_osm} 619 619 620 620 %--------------------------------------------namzdf_osm--------------------------------------------------------- … … 628 628 % ================================================================ 629 629 \section{Convection} 630 \label{ ZDF_conv}630 \label{sec:ZDF_conv} 631 631 632 632 %--------------------------------------------namzdf-------------------------------------------------------- … … 648 648 \subsection[Non-penetrative convective adjmt (\protect\np{ln\_tranpc}\forcode{ = .true.})] 649 649 {Non-penetrative convective adjustment (\protect\np{ln\_tranpc}\forcode{ = .true.})} 650 \label{ ZDF_npc}650 \label{subsec:ZDF_npc} 651 651 652 652 %--------------------------------------------namzdf-------------------------------------------------------- … … 657 657 \begin{figure}[!htb] \begin{center} 658 658 \includegraphics[width=0.90\textwidth]{Fig_npc} 659 \caption{ \protect\label{ Fig_npc}659 \caption{ \protect\label{fig:npc} 660 660 Example of an unstable density profile treated by the non penetrative 661 661 convective adjustment algorithm. $1^{st}$ step: the initial profile is checked from … … 677 677 column has \textit{exactly} the density of the water just below) \citep{Madec_al_JPO91}. 678 678 The associated algorithm is an iterative process used in the following way 679 ( Fig. \ref{Fig_npc}): starting from the top of the ocean, the first instability is679 (\autoref{fig:npc}): starting from the top of the ocean, the first instability is 680 680 found. Assume in the following that the instability is located between levels 681 681 $k$ and $k+1$. The temperature and salinity in the two levels are … … 714 714 % ------------------------------------------------------------------------------------------------------------- 715 715 \subsection{Enhanced vertical diffusion (\protect\np{ln\_zdfevd}\forcode{ = .true.})} 716 \label{ ZDF_evd}716 \label{subsec:ZDF_evd} 717 717 718 718 %--------------------------------------------namzdf-------------------------------------------------------- … … 739 739 Note that the stability test is performed on both \textit{before} and \textit{now} 740 740 values of $N^2$. This removes a potential source of divergence of odd and 741 even time step in a leapfrog environment \citep{Leclair_PhD2010} (see \ S\ref{STP_mLF}).741 even time step in a leapfrog environment \citep{Leclair_PhD2010} (see \autoref{sec:STP_mLF}). 742 742 743 743 % ------------------------------------------------------------------------------------------------------------- … … 745 745 % ------------------------------------------------------------------------------------------------------------- 746 746 \subsection[Turbulent closure scheme (\protect\key{zdf}\{tke,gls,osm\})]{Turbulent Closure Scheme (\protect\key{zdftke}, \protect\key{zdfgls} or \protect\key{zdfosm})} 747 \label{ ZDF_tcs}748 749 The turbulent closure scheme presented in \ S\ref{ZDF_tke} and \S\ref{ZDF_gls}747 \label{subsec:ZDF_tcs} 748 749 The turbulent closure scheme presented in \autoref{subsec:ZDF_tke} and \autoref{subsec:ZDF_gls} 750 750 (\key{zdftke} or \key{zdftke} is defined) in theory solves the problem of statically 751 751 unstable density profiles. In such a case, the term corresponding to the 752 destruction of turbulent kinetic energy through stratification in \ eqref{Eq_zdftke_e}753 or \ eqref{Eq_zdfgls_e} becomes a source term, since $N^2$ is negative.752 destruction of turbulent kinetic energy through stratification in \autoref{eq:zdftke_e} 753 or \autoref{eq:zdfgls_e} becomes a source term, since $N^2$ is negative. 754 754 It results in large values of $A_T^{vT}$ and $A_T^{vT}$, and also the four neighbouring 755 755 $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1})$. These large values 756 756 restore the static stability of the water column in a way similar to that of the 757 enhanced vertical diffusion parameterisation (\ S\ref{ZDF_evd}). However,757 enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}). However, 758 758 in the vicinity of the sea surface (first ocean layer), the eddy coefficients 759 759 computed by the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$, … … 772 772 % ================================================================ 773 773 \section{Double diffusion mixing (\protect\key{zdfddm})} 774 \label{ ZDF_ddm}774 \label{sec:ZDF_ddm} 775 775 776 776 %-------------------------------------------namzdf_ddm------------------------------------------------- … … 789 789 790 790 Diapycnal mixing of S and T are described by diapycnal diffusion coefficients 791 \begin{align*} % \label{ Eq_zdfddm_Kz}791 \begin{align*} % \label{eq:zdfddm_Kz} 792 792 &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\ 793 793 &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS} … … 797 797 mixing depend on the buoyancy ratio $R_\rho = \alpha \partial_z T / \beta \partial_z S$, 798 798 where $\alpha$ and $\beta$ are coefficients of thermal expansion and saline 799 contraction (see \ S\ref{TRA_eos}). To represent mixing of $S$ and $T$ by salt799 contraction (see \autoref{subsec:TRA_eos}). To represent mixing of $S$ and $T$ by salt 800 800 fingering, we adopt the diapycnal diffusivities suggested by Schmitt (1981): 801 \begin{align} \label{ Eq_zdfddm_f}801 \begin{align} \label{eq:zdfddm_f} 802 802 A_f^{vS} &= \begin{cases} 803 803 \frac{A^{\ast v}}{1+(R_\rho / R_c)^n } &\text{if $R_\rho > 1$ and $N^2>0$ } \\ 804 804 0 &\text{otherwise} 805 805 \end{cases} 806 \\ \label{ Eq_zdfddm_f_T}806 \\ \label{eq:zdfddm_f_T} 807 807 A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho 808 808 \end{align} … … 811 811 \begin{figure}[!t] \begin{center} 812 812 \includegraphics[width=0.99\textwidth]{Fig_zdfddm} 813 \caption{ \protect\label{ Fig_zdfddm}813 \caption{ \protect\label{fig:zdfddm} 814 814 From \citet{Merryfield1999} : (a) Diapycnal diffusivities $A_f^{vT}$ 815 815 and $A_f^{vS}$ for temperature and salt in regions of salt fingering. Heavy … … 822 822 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 823 823 824 The factor 0.7 in \ eqref{Eq_zdfddm_f_T} reflects the measured ratio824 The factor 0.7 in \autoref{eq:zdfddm_f_T} reflects the measured ratio 825 825 $\alpha F_T /\beta F_S \approx 0.7$ of buoyancy flux of heat to buoyancy 826 826 flux of salt ($e.g.$, \citet{McDougall_Taylor_JMR84}). Following \citet{Merryfield1999}, … … 828 828 829 829 To represent mixing of S and T by diffusive layering, the diapycnal diffusivities suggested by Federov (1988) is used: 830 \begin{align} \label{ Eq_zdfddm_d}830 \begin{align} \label{eq:zdfddm_d} 831 831 A_d^{vT} &= \begin{cases} 832 832 1.3635 \, \exp{\left( 4.6\, \exp{ \left[ -0.54\,( R_{\rho}^{-1} - 1 ) \right] } \right)} … … 834 834 0 &\text{otherwise} 835 835 \end{cases} 836 \\ \label{ Eq_zdfddm_d_S}836 \\ \label{eq:zdfddm_d_S} 837 837 A_d^{vS} &= \begin{cases} 838 838 A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) … … 844 844 \end{align} 845 845 846 The dependencies of \ eqref{Eq_zdfddm_f} to \eqref{Eq_zdfddm_d_S} on $R_\rho$847 are illustrated in Fig.~\ref{Fig_zdfddm}. Implementing this requires computing846 The dependencies of \autoref{eq:zdfddm_f} to \autoref{eq:zdfddm_d_S} on $R_\rho$ 847 are illustrated in \autoref{fig:zdfddm}. Implementing this requires computing 848 848 $R_\rho$ at each grid point on every time step. This is done in \mdl{eosbn2} at the 849 849 same time as $N^2$ is computed. This avoids duplication in the computation of … … 854 854 % ================================================================ 855 855 \section{Bottom and top friction (\protect\mdl{zdfbfr})} 856 \label{ ZDF_bfr}856 \label{sec:ZDF_bfr} 857 857 858 858 %--------------------------------------------nambfr-------------------------------------------------------- … … 870 870 flux (bottom friction) enter the equations as a condition on the vertical 871 871 diffusive flux. For the bottom boundary layer, one has: 872 \begin{equation} \label{ Eq_zdfbfr_flux}872 \begin{equation} \label{eq:zdfbfr_flux} 873 873 A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U} 874 874 \end{equation} … … 886 886 as a body force over the depth of the top or bottom model layer. To illustrate 887 887 this, consider the equation for $u$ at $k$, the last ocean level: 888 \begin{equation} \label{ Eq_zdfbfr_flux2}888 \begin{equation} \label{eq:zdfbfr_flux2} 889 889 \frac{\partial u_k}{\partial t} = \frac{1}{e_{3u}} \left[ \frac{A_{uw}^{vm}}{e_{3uw}} \delta_{k+1/2}\;[u] - {\cal F}^u_h \right] \approx - \frac{{\cal F}^u_{h}}{e_{3u}} 890 890 \end{equation} … … 907 907 These coefficients are computed in \mdl{zdfbfr} and generally take the form 908 908 $c_b^{\textbf U}$ where: 909 \begin{equation} \label{ Eq_zdfbfr_bdef}909 \begin{equation} \label{eq:zdfbfr_bdef} 910 910 \frac{\partial {\textbf U_h}}{\partial t} = 911 911 - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b … … 917 917 % ------------------------------------------------------------------------------------------------------------- 918 918 \subsection{Linear bottom friction (\protect\np{nn\_botfr}\forcode{ = 0..1})} 919 \label{ ZDF_bfr_linear}919 \label{subsec:ZDF_bfr_linear} 920 920 921 921 The linear bottom friction parameterisation (including the special case … … 923 923 is proportional to the interior velocity (i.e. the velocity of the last 924 924 model level): 925 \begin{equation} \label{ Eq_zdfbfr_linear}925 \begin{equation} \label{eq:zdfbfr_linear} 926 926 {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b 927 927 \end{equation} … … 941 941 942 942 For the linear friction case the coefficients defined in the general 943 expression \ eqref{Eq_zdfbfr_bdef} are:944 \begin{equation} \label{ Eq_zdfbfr_linbfr_b}943 expression \autoref{eq:zdfbfr_bdef} are: 944 \begin{equation} \label{eq:zdfbfr_linbfr_b} 945 945 \begin{split} 946 946 c_b^u &= - r\\ … … 961 961 % ------------------------------------------------------------------------------------------------------------- 962 962 \subsection{Non-linear bottom friction (\protect\np{nn\_botfr}\forcode{ = 2})} 963 \label{ ZDF_bfr_nonlinear}963 \label{subsec:ZDF_bfr_nonlinear} 964 964 965 965 The non-linear bottom friction parameterisation assumes that the bottom 966 966 friction is quadratic: 967 \begin{equation} \label{ Eq_zdfbfr_nonlinear}967 \begin{equation} \label{eq:zdfbfr_nonlinear} 968 968 {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h 969 969 }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b … … 983 983 For the non-linear friction case the terms 984 984 computed in \mdl{zdfbfr} are: 985 \begin{equation} \label{ Eq_zdfbfr_nonlinbfr}985 \begin{equation} \label{eq:zdfbfr_nonlinbfr} 986 986 \begin{split} 987 987 c_b^u &= - \; C_D\;\left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^{1/2}\\ … … 1003 1003 \subsection[Log-layer btm frict enhncmnt (\protect\np{nn\_botfr}\forcode{ = 2}, \protect\np{ln\_loglayer}\forcode{ = .true.})] 1004 1004 {Log-layer bottom friction enhancement (\protect\np{nn\_botfr}\forcode{ = 2}, \protect\np{ln\_loglayer}\forcode{ = .true.})} 1005 \label{ ZDF_bfr_loglayer}1005 \label{subsec:ZDF_bfr_loglayer} 1006 1006 1007 1007 In the non-linear bottom friction case, the drag coefficient, $C_D$, can be optionally … … 1033 1033 % ------------------------------------------------------------------------------------------------------------- 1034 1034 \subsection{Bottom friction stability considerations} 1035 \label{ ZDF_bfr_stability}1035 \label{subsec:ZDF_bfr_stability} 1036 1036 1037 1037 Some care needs to exercised over the choice of parameters to ensure that the 1038 1038 implementation of bottom friction does not induce numerical instability. For 1039 the purposes of stability analysis, an approximation to \ eqref{Eq_zdfbfr_flux2}1039 the purposes of stability analysis, an approximation to \autoref{eq:zdfbfr_flux2} 1040 1040 is: 1041 \begin{equation} \label{ Eqn_bfrstab}1041 \begin{equation} \label{eq:Eqn_bfrstab} 1042 1042 \begin{split} 1043 1043 \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt \\ … … 1050 1050 |\Delta u| < \;|u| 1051 1051 \end{equation} 1052 \noindent which, using \ eqref{Eqn_bfrstab}, gives:1052 \noindent which, using \autoref{eq:Eqn_bfrstab}, gives: 1053 1053 \begin{equation} 1054 1054 r\frac{2\rdt}{e_{3u}} < 1 \qquad \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\ … … 1075 1075 1076 1076 Limits on the bottom friction coefficient are not imposed if the user has elected to 1077 handle the bottom friction implicitly (see \ S\ref{ZDF_bfr_imp}). The number of potential1077 handle the bottom friction implicitly (see \autoref{subsec:ZDF_bfr_imp}). The number of potential 1078 1078 breaches of the explicit stability criterion are still reported for information purposes. 1079 1079 … … 1082 1082 % ------------------------------------------------------------------------------------------------------------- 1083 1083 \subsection{Implicit bottom friction (\protect\np{ln\_bfrimp}\forcode{ = .true.})} 1084 \label{ ZDF_bfr_imp}1084 \label{subsec:ZDF_bfr_imp} 1085 1085 1086 1086 An optional implicit form of bottom friction has been implemented to improve … … 1093 1093 bottom boundary condition is implemented implicitly. 1094 1094 1095 \begin{equation} \label{ Eq_dynzdf_bfr}1095 \begin{equation} \label{eq:dynzdf_bfr} 1096 1096 \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{mbk} 1097 1097 = \binom{c_{b}^{u}u^{n+1}_{mbk}}{c_{b}^{v}v^{n+1}_{mbk}} … … 1112 1112 following: 1113 1113 1114 \begin{equation} \label{ Eq_dynspg_ts_bfr1}1114 \begin{equation} \label{eq:dynspg_ts_bfr1} 1115 1115 \frac{\textbf{U}_{med}-\textbf{U}^{m-1}}{2\Delta t}=-g\nabla\eta-f\textbf{k}\times\textbf{U}^{m}+c_{b} 1116 1116 \left(\textbf{U}_{med}-\textbf{U}^{m-1}\right) 1117 1117 \end{equation} 1118 \begin{equation} \label{ Eq_dynspg_ts_bfr2}1118 \begin{equation} \label{eq:dynspg_ts_bfr2} 1119 1119 \frac{\textbf{U}^{m+1}-\textbf{U}_{med}}{2\Delta t}=\textbf{T}+ 1120 1120 \left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{U}^{'}\right)- … … 1136 1136 \subsection[Bottom friction w/ split-explicit time splitting (\protect\np{ln\_bfrimp})] 1137 1137 {Bottom friction with split-explicit time splitting (\protect\np{ln\_bfrimp})} 1138 \label{ ZDF_bfr_ts}1138 \label{subsec:ZDF_bfr_ts} 1139 1139 1140 1140 When calculating the momentum trend due to bottom friction in \mdl{dynbfr}, the … … 1175 1175 the barotropic component which uses the unrestricted value of the coefficient. However, if the 1176 1176 limiting is thought to be having a major effect (a more likely prospect in coastal and shelf seas 1177 applications) then the fully implicit form of the bottom friction should be used (see \ S\ref{ZDF_bfr_imp} )1177 applications) then the fully implicit form of the bottom friction should be used (see \autoref{subsec:ZDF_bfr_imp} ) 1178 1178 which can be selected by setting \np{ln\_bfrimp} $=$ \forcode{.true.}. 1179 1179 1180 1180 Otherwise, the implicit formulation takes the form: 1181 \begin{equation} \label{ Eq_zdfbfr_implicitts}1181 \begin{equation} \label{eq:zdfbfr_implicitts} 1182 1182 \bar{U}^{t+ \rdt} = \; \left [ \bar{U}^{t-\rdt}\; + 2 \rdt\;RHS \right ] / \left [ 1 - 2 \rdt \;c_b^{u} / H_e \right ] 1183 1183 \end{equation} … … 1193 1193 % ================================================================ 1194 1194 \section{Tidal mixing (\protect\key{zdftmx})} 1195 \label{ ZDF_tmx}1195 \label{sec:ZDF_tmx} 1196 1196 1197 1197 %--------------------------------------------namzdf_tmx-------------------------------------------------- … … 1204 1204 % ------------------------------------------------------------------------------------------------------------- 1205 1205 \subsection{Bottom intensified tidal mixing} 1206 \label{ ZDF_tmx_bottom}1206 \label{subsec:ZDF_tmx_bottom} 1207 1207 1208 1208 Options are defined through the \ngn{namzdf\_tmx} namelist variables. … … 1213 1213 $A^{vT}_{tides}$ is expressed as a function of $E(x,y)$, the energy transfer from barotropic 1214 1214 tides to baroclinic tides : 1215 \begin{equation} \label{ Eq_Ktides}1215 \begin{equation} \label{eq:Ktides} 1216 1216 A^{vT}_{tides} = q \,\Gamma \,\frac{ E(x,y) \, F(z) }{ \rho \, N^2 } 1217 1217 \end{equation} 1218 1218 where $\Gamma$ is the mixing efficiency, $N$ the Brunt-Vais\"{a}l\"{a} frequency 1219 (see \ S\ref{TRA_bn2}), $\rho$ the density, $q$ the tidal dissipation efficiency,1219 (see \autoref{subsec:TRA_bn2}), $\rho$ the density, $q$ the tidal dissipation efficiency, 1220 1220 and $F(z)$ the vertical structure function. 1221 1221 … … 1230 1230 It is implemented as a simple exponential decaying upward away from the bottom, 1231 1231 with a vertical scale of $h_o$ (\np{rn\_htmx} namelist parameter, with a typical value of $500\,m$) \citep{St_Laurent_Nash_DSR04}, 1232 \begin{equation} \label{ Eq_Fz}1232 \begin{equation} \label{eq:Fz} 1233 1233 F(i,j,k) = \frac{ e^{ -\frac{H+z}{h_o} } }{ h_o \left( 1- e^{ -\frac{H}{h_o} } \right) } 1234 1234 \end{equation} … … 1241 1241 usually set to $10^{-8} s^{-2}$. These bounds are usually rarely encountered. 1242 1242 1243 The internal wave energy map, $E(x, y)$ in \ eqref{Eq_Ktides}, is derived1243 The internal wave energy map, $E(x, y)$ in \autoref{eq:Ktides}, is derived 1244 1244 from a barotropic model of the tides utilizing a parameterization of the 1245 1245 conversion of barotropic tidal energy into internal waves. … … 1250 1250 the barotropic global ocean tide model MOG2D-G \citep{Carrere_Lyard_GRL03}. 1251 1251 This model provides the dissipation associated with internal wave energy for the M2 and K1 1252 tides component ( Fig.~\ref{Fig_ZDF_M2_K1_tmx}). The S2 dissipation is simply approximated1252 tides component (\autoref{fig:ZDF_M2_K1_tmx}). The S2 dissipation is simply approximated 1253 1253 as being $1/4$ of the M2 one. The internal wave energy is thus : $E(x, y) = 1.25 E_{M2} + E_{K1}$. 1254 1254 Its global mean value is $1.1$ TW, in agreement with independent estimates … … 1258 1258 \begin{figure}[!t] \begin{center} 1259 1259 \includegraphics[width=0.90\textwidth]{Fig_ZDF_M2_K1_tmx} 1260 \caption{ \protect\label{ Fig_ZDF_M2_K1_tmx}1260 \caption{ \protect\label{fig:ZDF_M2_K1_tmx} 1261 1261 (a) M2 and (b) K1 internal wave drag energy from \citet{Carrere_Lyard_GRL03} ($W/m^2$). } 1262 1262 \end{center} \end{figure} … … 1267 1267 % ------------------------------------------------------------------------------------------------------------- 1268 1268 \subsection{Indonesian area specific treatment (\protect\np{ln\_zdftmx\_itf})} 1269 \label{ ZDF_tmx_itf}1269 \label{subsec:ZDF_tmx_itf} 1270 1270 1271 1271 When the Indonesian Through Flow (ITF) area is included in the model domain, … … 1294 1294 proportional to $N^2$ below the core of the thermocline and to $N$ above. 1295 1295 The resulting $F(z)$ is: 1296 \begin{equation} \label{ Eq_Fz_itf}1296 \begin{equation} \label{eq:Fz_itf} 1297 1297 F(i,j,k) \sim \left\{ \begin{aligned} 1298 1298 \frac{q\,\Gamma E(i,j) } {\rho N \, \int N dz} \qquad \text{when $\partial_z N < 0$} \\ … … 1315 1315 % ================================================================ 1316 1316 \section{Internal wave-driven mixing (\protect\key{zdftmx\_new})} 1317 \label{ ZDF_tmx_new}1317 \label{sec:ZDF_tmx_new} 1318 1318 1319 1319 %--------------------------------------------namzdf_tmx_new------------------------------------------ … … 1325 1325 A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed, 1326 1326 and the resulting diffusivity is obtained as 1327 \begin{equation} \label{ Eq_Kwave}1327 \begin{equation} \label{eq:Kwave} 1328 1328 A^{vT}_{wave} = R_f \,\frac{ \epsilon }{ \rho \, N^2 } 1329 1329 \end{equation} -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_conservation.tex
r9393 r9407 6 6 % ================================================================ 7 7 \chapter{Invariants of the Primitive Equations} 8 \label{ Invariant}8 \label{chap:Invariant} 9 9 \minitoc 10 10 … … 48 48 % ------------------------------------------------------------------------------------------------------------- 49 49 \section{Conservation properties on ocean dynamics} 50 \label{ Invariant_dyn}50 \label{sec:Invariant_dyn} 51 51 52 52 The non linear term of the momentum equations has been split into a … … 68 68 vorticity, i.e. , , and , respectively. The continuous formulation of the 69 69 vorticity term satisfies following integral constraints: 70 \begin{equation} \label{ Eq_vor_vorticity}70 \begin{equation} \label{eq:vor_vorticity} 71 71 \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma 72 72 \;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0 73 73 \end{equation} 74 74 75 \begin{equation} \label{ Eq_vor_enstrophy}75 \begin{equation} \label{eq:vor_enstrophy} 76 76 if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot 77 77 \frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv} … … 79 79 \end{equation} 80 80 81 \begin{equation} \label{ Eq_vor_energy}81 \begin{equation} \label{eq:vor_energy} 82 82 \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0 83 83 \end{equation} … … 107 107 energy due to the horizontal gradient of horizontal kinetic energy: 108 108 109 \begin{equation} \label{ Eq_keg_zad}109 \begin{equation} \label{eq:keg_zad} 110 110 \int_D {{\textbf{U}}_h \cdot \nabla _h \left( {1/2\;{\textbf{U}}_h ^2} \right)\;dv} =-\int_D {{\textbf{U}}_h \cdot \frac{w}{e_3 }\;\frac{\partial 111 111 {\textbf{U}}_h }{\partial k}\;dv} … … 113 113 114 114 Using the discrete form given in {\S}II.2-a and the symmetry or 115 anti-symmetry properties of the mean and difference operators, \ eqref{Eq_keg_zad} is115 anti-symmetry properties of the mean and difference operators, \autoref{eq:keg_zad} is 116 116 demonstrated in the Appendix C. The main point here is that satisfying 117 \ eqref{Eq_keg_zad} links the choice of the discrete forms of the vertical advection117 \autoref{eq:keg_zad} links the choice of the discrete forms of the vertical advection 118 118 and of the horizontal gradient of horizontal kinetic energy. Choosing one 119 119 imposes the other. The discrete form of the vertical advection given in … … 132 132 energy due to buoyancy forces: 133 133 134 \begin{equation} \label{ Eq_hpg_pe}134 \begin{equation} \label{eq:hpg_pe} 135 135 \int_D {-\frac{1}{\rho _o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv} 136 136 \end{equation} … … 155 155 approximation, the change of horizontal kinetic energy due to the work of 156 156 surface pressure forces is exactly zero: 157 \begin{equation} \label{ Eq_spg}157 \begin{equation} \label{eq:spg} 158 158 \int_D {-\frac{1}{\rho _o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 159 159 \end{equation} … … 169 169 % ------------------------------------------------------------------------------------------------------------- 170 170 \section{Conservation properties on ocean thermodynamics} 171 \label{ Invariant_tra}171 \label{sec:Invariant_tra} 172 172 173 173 In continuous formulation, the advective terms of the tracer equations 174 174 conserve the tracer content and the quadratic form of the tracer, i.e. 175 \begin{equation} \label{ Eq_tra_tra2}175 \begin{equation} \label{eq:tra_tra2} 176 176 \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 177 177 \;\text{and} … … 189 189 % ------------------------------------------------------------------------------------------------------------- 190 190 \subsection{Conservation properties on momentum physics} 191 \label{ Invariant_dyn_physics}191 \label{subsec:Invariant_dyn_physics} 192 192 193 193 \textbf{* lateral momentum diffusion term} … … 195 195 The continuous formulation of the horizontal diffusion of momentum satisfies 196 196 the following integral constraints~: 197 \begin{equation} \label{ Eq_dynldf_dyn}197 \begin{equation} \label{eq:dynldf_dyn} 198 198 \int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla 199 199 _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta … … 201 201 \end{equation} 202 202 203 \begin{equation} \label{ Eq_dynldf_div}203 \begin{equation} \label{eq:dynldf_div} 204 204 \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 205 205 \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} … … 207 207 \end{equation} 208 208 209 \begin{equation} \label{ Eq_dynldf_curl}209 \begin{equation} \label{eq:dynldf_curl} 210 210 \int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 211 211 \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} … … 213 213 \end{equation} 214 214 215 \begin{equation} \label{ Eq_dynldf_curl2}215 \begin{equation} \label{eq:dynldf_curl2} 216 216 \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot 217 217 \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h … … 220 220 \end{equation} 221 221 222 \begin{equation} \label{ Eq_dynldf_div2}222 \begin{equation} \label{eq:dynldf_div2} 223 223 \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ 224 224 {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( … … 256 256 conservation of momentum, dissipation of horizontal kinetic energy 257 257 258 \begin{equation} \label{ Eq_dynzdf_dyn}258 \begin{equation} \label{eq:dynzdf_dyn} 259 259 \begin{aligned} 260 260 & \int_D {\frac{1}{e_3 }} \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\ … … 263 263 \end{equation} 264 264 conservation of vorticity, dissipation of enstrophy 265 \begin{equation} \label{ Eq_dynzdf_vor}265 \begin{equation} \label{eq:dynzdf_vor} 266 266 \begin{aligned} 267 267 & \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 … … 275 275 conservation of horizontal divergence, dissipation of square of the 276 276 horizontal divergence 277 \begin{equation} \label{ Eq_dynzdf_div}277 \begin{equation} \label{eq:dynzdf_div} 278 278 \begin{aligned} 279 279 &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial … … 295 295 % ------------------------------------------------------------------------------------------------------------- 296 296 \subsection{Conservation properties on tracer physics} 297 \label{ Invariant_tra_physics}297 \label{subsec:Invariant_tra_physics} 298 298 299 299 The numerical schemes used for tracer subgridscale physics are written in … … 308 308 variance, i.e. 309 309 310 \begin{equation} \label{ Eq_traldf_t_t2}310 \begin{equation} \label{eq:traldf_t_t2} 311 311 \begin{aligned} 312 312 &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ … … 318 318 variance, i.e. 319 319 320 \begin{equation} \label{ Eq_trazdf_t_t2}320 \begin{equation} \label{eq:trazdf_t_t2} 321 321 \begin{aligned} 322 322 & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 } \frac{\partial T}{\partial k} \right)\;dv = 0 \\ -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_misc.tex
r9394 r9407 5 5 % ================================================================ 6 6 \chapter{Miscellaneous Topics} 7 \label{ MISC}7 \label{chap:MISC} 8 8 \minitoc 9 9 … … 15 15 % ================================================================ 16 16 \section{Representation of unresolved straits} 17 \label{ MISC_strait}17 \label{sec:MISC_strait} 18 18 19 19 In climate modeling, it often occurs that a crucial connections between water masses … … 43 43 % ------------------------------------------------------------------------------------------------------------- 44 44 \subsection{Hand made geometry changes} 45 \label{ MISC_strait_hand}45 \label{subsec:MISC_strait_hand} 46 46 47 47 $\bullet$ reduced scale factor in the cross-strait direction to a value in better agreement 48 with the true mean width of the strait. ( Fig.~\ref{Fig_MISC_strait_hand}).48 with the true mean width of the strait. (\autoref{fig:MISC_strait_hand}). 49 49 This technique is sometime called "partially open face" or "partially closed cells". 50 50 The key issue here is only to reduce the faces of $T$-cell ($i.e.$ change the value … … 56 56 57 57 $\bullet$ increase of the viscous boundary layer thickness by local increase of the 58 fmask value at the coast ( Fig.~\ref{Fig_MISC_strait_hand}). This is done in58 fmask value at the coast (\autoref{fig:MISC_strait_hand}). This is done in 59 59 \mdl{dommsk} together with the setting of the coastal value of fmask 60 (see Section \ref{LBC_coast})60 (see \autoref{sec:LBC_coast}) 61 61 62 62 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 64 64 \includegraphics[width=0.80\textwidth]{Fig_Gibraltar} 65 65 \includegraphics[width=0.80\textwidth]{Fig_Gibraltar2} 66 \caption{ \protect\label{ Fig_MISC_strait_hand}66 \caption{ \protect\label{fig:MISC_strait_hand} 67 67 Example of the Gibraltar strait defined in a $1^{\circ} \times 1^{\circ}$ mesh. 68 68 \textit{Top}: using partially open cells. The meridional scale factor at $v$-point … … 71 71 \textit{Bottom}: using viscous boundary layers. The four fmask parameters 72 72 along the strait coastlines are set to a value larger than 4, $i.e.$ "strong" no-slip 73 case (see Fig.\ref{Fig_LBC_shlat}) creating a large viscous boundary layer73 case (see \autoref{fig:LBC_shlat}) creating a large viscous boundary layer 74 74 that allows a reduced transport through the strait.} 75 75 \end{center} \end{figure} … … 81 81 % ================================================================ 82 82 \section{Closed seas (\protect\mdl{closea})} 83 \label{ MISC_closea}83 \label{sec:MISC_closea} 84 84 85 85 \colorbox{yellow}{Add here a short description of the way closed seas are managed} … … 90 90 % ================================================================ 91 91 \section{Sub-domain functionality} 92 \label{ MISC_zoom}92 \label{sec:MISC_zoom} 93 93 94 94 \subsection{Simple subsetting of input files via NetCDF attributes} … … 140 140 \begin{figure}[!ht] \begin{center} 141 141 \includegraphics[width=0.90\textwidth]{Fig_LBC_zoom} 142 \caption{ \protect\label{ Fig_LBC_zoom}142 \caption{ \protect\label{fig:LBC_zoom} 143 143 Position of a model domain compared to the data input domain when the zoom functionality is used.} 144 144 \end{center} \end{figure} … … 150 150 % ================================================================ 151 151 \section{Accuracy and reproducibility (\protect\mdl{lib\_fortran})} 152 \label{ MISC_fortran}152 \label{sec:MISC_fortran} 153 153 154 154 \subsection{Issues with intrinsinc SIGN function (\protect\key{nosignedzero})} 155 \label{ MISC_sign}155 \label{subsec:MISC_sign} 156 156 157 157 The SIGN(A, B) is the \textsc {Fortran} intrinsic function delivers the magnitude … … 179 179 180 180 \subsection{MPP reproducibility} 181 \label{ MISC_glosum}181 \label{subsec:MISC_glosum} 182 182 183 183 The numerical reproducibility of simulations on distributed memory parallel computers … … 207 207 208 208 \subsection{MPP scalability} 209 \label{ MISC_mppsca}209 \label{subsec:MISC_mppsca} 210 210 211 211 The default method of communicating values across the north-fold in distributed memory applications … … 231 231 % ================================================================ 232 232 \section{Model optimisation, control print and benchmark} 233 \label{ MISC_opt}233 \label{sec:MISC_opt} 234 234 %--------------------------------------------namctl------------------------------------------------------- 235 235 \forfile{../namelists/namctl} … … 270 270 271 271 $\bullet$ Benchmark (\np{nn\_bench}). This option defines a benchmark run based on 272 a GYRE configuration (see \ S\ref{CFG_gyre}) in which the resolution remains the same272 a GYRE configuration (see \autoref{sec:CFG_gyre}) in which the resolution remains the same 273 273 whatever the domain size. This allows a very large model domain to be used, just by 274 274 changing the domain size (\jp{jpiglo}, \jp{jpjglo}) and without adjusting either the time-step -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_model_basics.tex
r9393 r9407 6 6 7 7 \chapter{Model Basics} 8 \label{ PE}8 \label{chap:PE} 9 9 \minitoc 10 10 … … 16 16 % ================================================================ 17 17 \section{Primitive equations} 18 \label{ PE_PE}18 \label{sec:PE_PE} 19 19 20 20 % ------------------------------------------------------------------------------------------------------------- … … 23 23 24 24 \subsection{Vector invariant formulation} 25 \label{ PE_Vector}25 \label{subsec:PE_Vector} 26 26 27 27 … … 61 61 hydrostatic equilibrium, the incompressibility equation, the heat and salt conservation 62 62 equations and an equation of state): 63 \begin{subequations} \label{ Eq_PE}64 \begin{equation} \label{ Eq_PE_dyn}63 \begin{subequations} \label{eq:PE} 64 \begin{equation} \label{eq:PE_dyn} 65 65 \frac{\partial {\rm {\bf U}}_h }{\partial t}= 66 66 -\left[ {\left( {\nabla \times {\rm {\bf U}}} \right)\times {\rm {\bf U}} … … 69 69 -\frac{1}{\rho _o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}} 70 70 \end{equation} 71 \begin{equation} \label{ Eq_PE_hydrostatic}71 \begin{equation} \label{eq:PE_hydrostatic} 72 72 \frac{\partial p }{\partial z} = - \rho \ g 73 73 \end{equation} 74 \begin{equation} \label{ Eq_PE_continuity}74 \begin{equation} \label{eq:PE_continuity} 75 75 \nabla \cdot {\bf U}= 0 76 76 \end{equation} 77 \begin{equation} \label{ Eq_PE_tra_T}77 \begin{equation} \label{eq:PE_tra_T} 78 78 \frac{\partial T}{\partial t} = - \nabla \cdot \left( T \ \rm{\bf U} \right) + D^T + F^T 79 79 \end{equation} 80 \begin{equation} \label{ Eq_PE_tra_S}80 \begin{equation} \label{eq:PE_tra_S} 81 81 \frac{\partial S}{\partial t} = - \nabla \cdot \left( S \ \rm{\bf U} \right) + D^S + F^S 82 82 \end{equation} 83 \begin{equation} \label{ Eq_PE_eos}83 \begin{equation} \label{eq:PE_eos} 84 84 \rho = \rho \left( T,S,p \right) 85 85 \end{equation} … … 87 87 where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions, 88 88 $t$ is the time, $z$ is the vertical coordinate, $\rho $ is the \textit{in situ} density given by 89 the equation of state (\ ref{Eq_PE_eos}), $\rho_o$ is a reference density, $p$ the pressure,89 the equation of state (\autoref{eq:PE_eos}), $\rho_o$ is a reference density, $p$ the pressure, 90 90 $f=2 \bf \Omega \cdot \bf k$ is the Coriolis acceleration (where $\bf \Omega$ is the Earth's 91 91 angular velocity vector), and $g$ is the gravitational acceleration. … … 93 93 physics for momentum, temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ 94 94 and $F^S$ surface forcing terms. Their nature and formulation are discussed in 95 \ S\ref{PE_zdf_ldf} and page \S\ref{PE_boundary_condition}.95 \autoref{sec:PE_zdf_ldf} and \autoref{subsec:PE_boundary_condition}. 96 96 97 97 . … … 101 101 % ------------------------------------------------------------------------------------------------------------- 102 102 \subsection{Boundary conditions} 103 \label{ PE_boundary_condition}103 \label{subsec:PE_boundary_condition} 104 104 105 105 An ocean is bounded by complex coastlines, bottom topography at its base and an air-sea … … 107 107 and $z=\eta(i,j,k,t)$, where $H$ is the depth of the ocean bottom and $\eta$ is the height 108 108 of the sea surface. Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$, 109 chosen as a mean sea surface ( Fig.~\ref{Fig_ocean_bc}). Through these two boundaries,109 chosen as a mean sea surface (\autoref{fig:ocean_bc}). Through these two boundaries, 110 110 the ocean can exchange fluxes of heat, fresh water, salt, and momentum with the solid earth, 111 111 the continental margins, the sea ice and the atmosphere. However, some of these fluxes are … … 117 117 \begin{figure}[!ht] \begin{center} 118 118 \includegraphics[width=0.90\textwidth]{Fig_I_ocean_bc} 119 \caption{ \protect\label{ Fig_ocean_bc}119 \caption{ \protect\label{fig:ocean_bc} 120 120 The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,t)$, where $H$ 121 121 is the depth of the sea floor and $\eta$ the height of the sea surface. … … 137 137 \footnote{In fact, it has been shown that the heat flux associated with the solid Earth cooling 138 138 ($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world 139 ocean (see \ ref{TRA_bbc}).}.139 ocean (see \autoref{subsec:TRA_bbc}).}. 140 140 The boundary condition is thus set to no flux of heat and salt across solid boundaries. 141 141 For momentum, the situation is different. There is no flow across solid boundaries, … … 143 143 the bottom velocity is parallel to solid boundaries). This kinematic boundary condition 144 144 can be expressed as: 145 \begin{equation} \label{ Eq_PE_w_bbc}145 \begin{equation} \label{eq:PE_w_bbc} 146 146 w = -{\rm {\bf U}}_h \cdot \nabla _h \left( H \right) 147 147 \end{equation} … … 150 150 in terms of turbulent fluxes using bottom and/or lateral boundary conditions. Its specification 151 151 depends on the nature of the physical parameterisation used for ${\rm {\bf D}}^{\rm {\bf U}}$ 152 in \ eqref{Eq_PE_dyn}. It is discussed in \S\ref{PE_zdf}, page~\pageref{PE_zdf}.% and Chap. III.6 to 9.152 in \autoref{eq:PE_dyn}. It is discussed in \autoref{eq:PE_zdf}.% and Chap. III.6 to 9. 153 153 \item[Atmosphere - ocean interface:] the kinematic surface condition plus the mass flux 154 154 of fresh water PE (the precipitation minus evaporation budget) leads to: 155 \begin{equation} \label{ Eq_PE_w_sbc}155 \begin{equation} \label{eq:PE_w_sbc} 156 156 w = \frac{\partial \eta }{\partial t} 157 157 + \left. {{\rm {\bf U}}_h } \right|_{z=\eta } \cdot \nabla _h \left( \eta \right) … … 176 176 % ================================================================ 177 177 \section{Horizontal pressure gradient } 178 \label{ PE_hor_pg}178 \label{sec:PE_hor_pg} 179 179 180 180 % ------------------------------------------------------------------------------------------------------------- … … 182 182 % ------------------------------------------------------------------------------------------------------------- 183 183 \subsection{Pressure formulation} 184 \label{ PE_p_formulation}184 \label{subsec:PE_p_formulation} 185 185 186 186 The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at a 187 187 reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that: 188 $p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\ ref{Eq_PE_hydrostatic}),189 assuming that pressure in decibars can be approximated by depth in meters in (\ ref{Eq_PE_eos}).188 $p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\autoref{eq:PE_hydrostatic}), 189 assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:PE_eos}). 190 190 The hydrostatic pressure is then given by: 191 \begin{equation} \label{ Eq_PE_pressure}191 \begin{equation} \label{eq:PE_pressure} 192 192 p_h \left( {i,j,z,t} \right) 193 193 = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma } … … 213 213 % ------------------------------------------------------------------------------------------------------------- 214 214 \subsection{Free surface formulation} 215 \label{ PE_free_surface}215 \label{subsec:PE_free_surface} 216 216 217 217 In the free surface formulation, a variable $\eta$, the sea-surface height, is introduced 218 218 which describes the shape of the air-sea interface. This variable is solution of a 219 219 prognostic equation which is established by forming the vertical average of the kinematic 220 surface condition (\ ref{Eq_PE_w_bbc}):221 \begin{equation} \label{ Eq_PE_ssh}220 surface condition (\autoref{eq:PE_w_bbc}): 221 \begin{equation} \label{eq:PE_ssh} 222 222 \frac{\partial \eta }{\partial t}=-D+P-E 223 223 \quad \text{where} \ 224 224 D=\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right] 225 225 \end{equation} 226 and using (\ ref{Eq_PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$.226 and using (\autoref{eq:PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$. 227 227 228 228 Allowing the air-sea interface to move introduces the external gravity waves (EGWs) … … 237 237 with the baroclinic structure of the ocean (internal waves) possibly in shallow seas, 238 238 then a non linear free surface is the most appropriate. This means that no 239 approximation is made in (\ ref{Eq_PE_ssh}) and that the variation of the ocean239 approximation is made in (\autoref{eq:PE_ssh}) and that the variation of the ocean 240 240 volume is fully taken into account. Note that in order to study the fast time scales 241 241 associated with EGWs it is necessary to minimize time filtering effects (use an 242 242 explicit time scheme with very small time step, or a split-explicit scheme with 243 reasonably small time step, see \ S\ref{DYN_spg_exp} or \S\ref{DYN_spg_ts}.243 reasonably small time step, see \autoref{subsec:DYN_spg_exp} or \autoref{subsec:DYN_spg_ts}. 244 244 245 245 $\bullet$ If one is not interested in EGW but rather sees them as high frequency … … 247 247 not altering the slow barotropic Rossby waves. If further, an approximative conservation 248 248 of heat and salt contents is sufficient for the problem solved, then it is 249 sufficient to solve a linearized version of (\ ref{Eq_PE_ssh}), which still allows249 sufficient to solve a linearized version of (\autoref{eq:PE_ssh}), which still allows 250 250 to take into account freshwater fluxes applied at the ocean surface \citep{Roullet_Madec_JGR00}. 251 251 Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost. … … 255 255 or the implicit scheme \citep{Dukowicz1994} or the addition of a filtering force in the momentum equation 256 256 \citep{Roullet_Madec_JGR00}. With the present release, \NEMO offers the choice between 257 an explicit free surface (see \ S\ref{DYN_spg_exp}) or a split-explicit scheme strongly258 inspired the one proposed by \citet{Shchepetkin_McWilliams_OM05} (see \ S\ref{DYN_spg_ts}).257 an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or a split-explicit scheme strongly 258 inspired the one proposed by \citet{Shchepetkin_McWilliams_OM05} (see \autoref{subsec:DYN_spg_ts}). 259 259 260 260 %\newpage … … 265 265 % ================================================================ 266 266 \section{Curvilinear \textit{z-}coordinate system} 267 \label{ PE_zco}267 \label{sec:PE_zco} 268 268 269 269 … … 272 272 % ------------------------------------------------------------------------------------------------------------- 273 273 \subsection{Tensorial formalism} 274 \label{ PE_tensorial}274 \label{subsec:PE_tensorial} 275 275 276 276 In many ocean circulation problems, the flow field has regions of enhanced dynamics … … 294 294 associated with the positively oriented orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) 295 295 linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are 296 two vectors orthogonal to \textbf{k}, $i.e.$ along geopotential surfaces ( Fig.\ref{Fig_referential}).296 two vectors orthogonal to \textbf{k}, $i.e.$ along geopotential surfaces (\autoref{fig:referential}). 297 297 Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined 298 298 by the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and the distance from the centre of 299 299 the earth $a+z(k)$ where $a$ is the earth's radius and $z$ the altitude above a reference sea 300 level ( Fig.\ref{Fig_referential}). The local deformation of the curvilinear coordinate system is300 level (\autoref{fig:referential}). The local deformation of the curvilinear coordinate system is 301 301 given by $e_1$, $e_2$ and $e_3$, the three scale factors: 302 \begin{equation} \label{ Eq_scale_factors}302 \begin{equation} \label{eq:scale_factors} 303 303 \begin{aligned} 304 304 e_1 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda … … 315 315 \begin{figure}[!tb] \begin{center} 316 316 \includegraphics[width=0.60\textwidth]{Fig_I_earth_referential} 317 \caption{ \protect\label{ Fig_referential}317 \caption{ \protect\label{fig:referential} 318 318 the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear 319 319 coordinate system (\textbf{i},\textbf{j},\textbf{k}). } … … 322 322 323 323 Since the ocean depth is far smaller than the earth's radius, $a+z$, can be replaced by 324 $a$ in (\ ref{Eq_scale_factors}) (thin-shell approximation). The resulting horizontal scale324 $a$ in (\autoref{eq:scale_factors}) (thin-shell approximation). The resulting horizontal scale 325 325 factors $e_1$, $e_2$ are independent of $k$ while the vertical scale factor is a single 326 326 function of $k$ as \textbf{k} is parallel to \textbf{z}. The scalar and vector operators that 327 appear in the primitive equations ( Eqs. \eqref{Eq_PE_dyn} to \eqref{Eq_PE_eos}) can327 appear in the primitive equations (\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can 328 328 be written in the tensorial form, invariant in any orthogonal horizontal curvilinear coordinate 329 329 system transformation: 330 \begin{subequations} \label{ Eq_PE_discrete_operators}331 \begin{equation} \label{ Eq_PE_grad}330 \begin{subequations} \label{eq:PE_discrete_operators} 331 \begin{equation} \label{eq:PE_grad} 332 332 \nabla q=\frac{1}{e_1 }\frac{\partial q}{\partial i}\;{\rm {\bf 333 333 i}}+\frac{1}{e_2 }\frac{\partial q}{\partial j}\;{\rm {\bf j}}+\frac{1}{e_3 334 334 }\frac{\partial q}{\partial k}\;{\rm {\bf k}} \\ 335 335 \end{equation} 336 \begin{equation} \label{ Eq_PE_div}336 \begin{equation} \label{eq:PE_div} 337 337 \nabla \cdot {\rm {\bf A}} 338 338 = \frac{1}{e_1 \; e_2} \left[ … … 341 341 + \frac{1}{e_3} \left[ \frac{\partial a_3}{\partial k } \right] 342 342 \end{equation} 343 \begin{equation} \label{ Eq_PE_curl}343 \begin{equation} \label{eq:PE_curl} 344 344 \begin{split} 345 345 \nabla \times \vect{A} = … … 352 352 \end{split} 353 353 \end{equation} 354 \begin{equation} \label{ Eq_PE_lap}354 \begin{equation} \label{eq:PE_lap} 355 355 \Delta q = \nabla \cdot \left( \nabla q \right) 356 356 \end{equation} 357 \begin{equation} \label{ Eq_PE_lap_vector}357 \begin{equation} \label{eq:PE_lap_vector} 358 358 \Delta {\rm {\bf A}} = 359 359 \nabla \left( \nabla \cdot {\rm {\bf A}} \right) … … 367 367 % ------------------------------------------------------------------------------------------------------------- 368 368 \subsection{Continuous model equations} 369 \label{ PE_zco_Eq}369 \label{subsec:PE_zco_Eq} 370 370 371 371 In order to express the Primitive Equations in tensorial formalism, it is necessary to compute 372 372 the horizontal component of the non-linear and viscous terms of the equation using 373 \ eqref{Eq_PE_grad}) to \eqref{Eq_PE_lap_vector}.373 \autoref{eq:PE_grad}) to \autoref{eq:PE_lap_vector}. 374 374 Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate 375 375 system and define the relative vorticity $\zeta$ and the divergence of the horizontal velocity 376 376 field $\chi$, by: 377 \begin{equation} \label{ Eq_PE_curl_Uh}377 \begin{equation} \label{eq:PE_curl_Uh} 378 378 \zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v} 379 379 \right)}{\partial i}-\frac{\partial \left( {e_1 \,u} \right)}{\partial j}} 380 380 \right] 381 381 \end{equation} 382 \begin{equation} \label{ Eq_PE_div_Uh}382 \begin{equation} \label{eq:PE_div_Uh} 383 383 \chi =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,u} 384 384 \right)}{\partial i}+\frac{\partial \left( {e_1 \,v} \right)}{\partial j}} … … 388 388 Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ 389 389 and that $e_3$ is a function of the single variable $k$, the nonlinear term of 390 \ eqref{Eq_PE_dyn} can be transformed as follows:390 \autoref{eq:PE_dyn} can be transformed as follows: 391 391 \begin{flalign*} 392 392 &\left[ {\left( { \nabla \times {\rm {\bf U}} } \right) \times {\rm {\bf U}} … … 427 427 428 428 The last term of the right hand side is obviously zero, and thus the nonlinear term of 429 \ eqref{Eq_PE_dyn} is written in the $(i,j,k)$ coordinate system:430 \begin{equation} \label{ Eq_PE_vector_form}429 \autoref{eq:PE_dyn} is written in the $(i,j,k)$ coordinate system: 430 \begin{equation} \label{eq:PE_vector_form} 431 431 \left[ {\left( { \nabla \times {\rm {\bf U}} } \right) \times {\rm {\bf U}} 432 432 +\frac{1}{2} \nabla \left( {{\rm {\bf U}}^2} \right)} \right]_h … … 440 440 For some purposes, it can be advantageous to write this term in the so-called flux form, 441 441 $i.e.$ to write it as the divergence of fluxes. For example, the first component of 442 \ eqref{Eq_PE_vector_form} (the $i$-component) is transformed as follows:442 \autoref{eq:PE_vector_form} (the $i$-component) is transformed as follows: 443 443 \begin{flalign*} 444 444 &{ \begin{array}{*{20}l} … … 509 509 510 510 The flux form of the momentum advection term is therefore given by: 511 \begin{multline} \label{ Eq_PE_flux_form}511 \begin{multline} \label{eq:PE_flux_form} 512 512 \left[ 513 513 \left( {\nabla \times {\rm {\bf U}}} \right) \times {\rm {\bf U}} … … 529 529 the curvilinear nature of the coordinate system used. The latter is called the \emph{metric} 530 530 term and can be viewed as a modification of the Coriolis parameter: 531 \begin{equation} \label{ Eq_PE_cor+metric}531 \begin{equation} \label{eq:PE_cor+metric} 532 532 f \to f + \frac{1}{e_1\;e_2} \left( v \frac{\partial e_2}{\partial i} 533 533 -u \frac{\partial e_1}{\partial j} \right) … … 547 547 $\bullet$ \textbf{Vector invariant form of the momentum equations} : 548 548 549 \begin{subequations} \label{ Eq_PE_dyn_vect}550 \begin{equation} \label{ Eq_PE_dyn_vect_u} \begin{split}549 \begin{subequations} \label{eq:PE_dyn_vect} 550 \begin{equation} \label{eq:PE_dyn_vect_u} \begin{split} 551 551 \frac{\partial u}{\partial t} 552 552 = + \left( {\zeta +f} \right)\,v … … 568 568 \vspace{+10pt} 569 569 $\bullet$ \textbf{flux form of the momentum equations} : 570 \begin{subequations} \label{ Eq_PE_dyn_flux}571 \begin{multline} \label{ Eq_PE_dyn_flux_u}570 \begin{subequations} \label{eq:PE_dyn_flux} 571 \begin{multline} \label{eq:PE_dyn_flux_u} 572 572 \frac{\partial u}{\partial t}= 573 573 + \left( { f + \frac{1}{e_1 \; e_2} … … 581 581 + D_u^{\vect{U}} + F_u^{\vect{U}} 582 582 \end{multline} 583 \begin{multline} \label{ Eq_PE_dyn_flux_v}583 \begin{multline} \label{eq:PE_dyn_flux_v} 584 584 \frac{\partial v}{\partial t}= 585 585 - \left( { f + \frac{1}{e_1 \; e_2} … … 594 594 \end{multline} 595 595 \end{subequations} 596 where $\zeta$, the relative vorticity, is given by \ eqref{Eq_PE_curl_Uh} and $p_s $,596 where $\zeta$, the relative vorticity, is given by \autoref{eq:PE_curl_Uh} and $p_s $, 597 597 the surface pressure, is given by: 598 \begin{equation} \label{ Eq_PE_spg}598 \begin{equation} \label{eq:PE_spg} 599 599 p_s = \rho \,g \,\eta 600 600 \end{equation} 601 with $\eta$ is solution of \ eqref{Eq_PE_ssh}601 with $\eta$ is solution of \autoref{eq:PE_ssh} 602 602 603 603 The vertical velocity and the hydrostatic pressure are diagnosed from the following equations: 604 \begin{equation} \label{ Eq_w_diag}604 \begin{equation} \label{eq:w_diag} 605 605 \frac{\partial w}{\partial k}=-\chi \;e_3 606 606 \end{equation} 607 \begin{equation} \label{ Eq_hp_diag}607 \begin{equation} \label{eq:hp_diag} 608 608 \frac{\partial p_h }{\partial k}=-\rho \;g\;e_3 609 609 \end{equation} 610 where the divergence of the horizontal velocity, $\chi$ is given by \ eqref{Eq_PE_div_Uh}.610 where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:PE_div_Uh}. 611 611 612 612 \vspace{+10pt} 613 613 $\bullet$ \textit{tracer equations} : 614 \begin{equation} \label{ Eq_S}614 \begin{equation} \label{eq:S} 615 615 \frac{\partial T}{\partial t} = 616 616 -\frac{1}{e_1 e_2 }\left[ { \frac{\partial \left( {e_2 T\,u} \right)}{\partial i} … … 618 618 -\frac{1}{e_3 }\frac{\partial \left( {T\,w} \right)}{\partial k} + D^T + F^T 619 619 \end{equation} 620 \begin{equation} \label{ Eq_T}620 \begin{equation} \label{eq:T} 621 621 \frac{\partial S}{\partial t} = 622 622 -\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 S\,u} \right)}{\partial i} … … 624 624 -\frac{1}{e_3 }\frac{\partial \left( {S\,w} \right)}{\partial k} + D^S + F^S 625 625 \end{equation} 626 \begin{equation} \label{ Eq_rho}626 \begin{equation} \label{eq:rho} 627 627 \rho =\rho \left( {T,S,z(k)} \right) 628 628 \end{equation} 629 629 630 630 The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale 631 parameterisation used. It will be defined in \ S\ref{PE_zdf}. The nature and formulation of631 parameterisation used. It will be defined in \autoref{eq:PE_zdf}. The nature and formulation of 632 632 ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms, are discussed 633 in Chapter~\ref{SBC}.633 in \autoref{chap:SBC}. 634 634 635 635 … … 640 640 % ================================================================ 641 641 \section{Curvilinear generalised vertical coordinate system} 642 \label{ PE_gco}642 \label{sec:PE_gco} 643 643 644 644 The ocean domain presents a huge diversity of situation in the vertical. First the ocean surface is a time dependent surface (moving surface). Second the ocean floor depends on the geographical position, varying from more than 6,000 meters in abyssal trenches to zero at the coast. Last but not least, the ocean stratification exerts a strong barrier to vertical motions and mixing. … … 648 648 649 649 In fact one is totally free to choose any space and time vertical coordinate by introducing an arbitrary vertical coordinate : 650 \begin{equation} \label{ Eq_s}650 \begin{equation} \label{eq:s} 651 651 s=s(i,j,k,t) 652 652 \end{equation} 653 with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \ eqref{Eq_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \eqref{Eq_s}.653 with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \autoref{eq:s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \autoref{eq:s}. 654 654 This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact an Arbitrary Lagrangian--Eulerian (ALE) coordinate. Indeed, choosing an expression for $s$ is an arbitrary choice that determines which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and which part will be used to move them (Lagrangian part). 655 655 The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09}, since the coordinate system is adapted in the course of the simulation. Its most often used implementation is via an ALE algorithm, in which a pure lagrangian step is followed by regridding and remapping steps, the later step implicitly embedding the vertical advection \citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}. Here we follow the \citep{Kasahara_MWR74} strategy : a regridding step (an update of the vertical coordinate) followed by an eulerian step with an explicit computation of vertical advection relative to the moving s-surfaces. … … 693 693 \subsection{\textit{S-}coordinate formulation} 694 694 695 Starting from the set of equations established in \ S\ref{PE_zco} for the special case $k=z$695 Starting from the set of equations established in \autoref{sec:PE_zco} for the special case $k=z$ 696 696 and thus $e_3=1$, we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$, which includes 697 697 $z$-, \textit{z*}- and $\sigma-$coordinates as special cases ($s=z$, $s=\textit{z*}$, and 698 698 $s=\sigma=z/H$ or $=z/\left(H+\eta \right)$, resp.). A formal derivation of the transformed 699 equations is given in Appendix~\ref{Apdx_A}. Let us define the vertical scale factor by699 equations is given in \autoref{apdx:A}. Let us define the vertical scale factor by 700 700 $e_3=\partial_s z$ ($e_3$ is now a function of $(i,j,k,t)$ ), and the slopes in the 701 701 (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by : 702 \begin{equation} \label{ Eq_PE_sco_slope}702 \begin{equation} \label{eq:PE_sco_slope} 703 703 \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s 704 704 \quad \text{, and } \quad … … 707 707 We also introduce $\omega $, a dia-surface velocity component, defined as the velocity 708 708 relative to the moving $s$-surfaces and normal to them: 709 \begin{equation} \label{ Eq_PE_sco_w}709 \begin{equation} \label{eq:PE_sco_w} 710 710 \omega = w - e_3 \, \frac{\partial s}{\partial t} - \sigma _1 \,u - \sigma _2 \,v \\ 711 711 \end{equation} 712 712 713 The equations solved by the ocean model \ eqref{Eq_PE} in $s-$coordinate can be written as follows (see Appendix~\ref{Apdx_A_momentum}):713 The equations solved by the ocean model \autoref{eq:PE} in $s-$coordinate can be written as follows (see \autoref{sec:A_momentum}): 714 714 715 715 \vspace{0.5cm} 716 716 $\bullet$ Vector invariant form of the momentum equation : 717 \begin{multline} \label{ Eq_PE_sco_u}717 \begin{multline} \label{eq:PE_sco_u} 718 718 \frac{\partial u }{\partial t}= 719 719 + \left( {\zeta +f} \right)\,v … … 724 724 + D_u^{\vect{U}} + F_u^{\vect{U}} \quad 725 725 \end{multline} 726 \begin{multline} \label{ Eq_PE_sco_v}726 \begin{multline} \label{eq:PE_sco_v} 727 727 \frac{\partial v }{\partial t}= 728 728 - \left( {\zeta +f} \right)\,u … … 736 736 \vspace{0.5cm} 737 737 $\bullet$ Vector invariant form of the momentum equation : 738 \begin{multline} \label{ Eq_PE_sco_u}738 \begin{multline} \label{eq:PE_sco_u} 739 739 \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t}= 740 740 + \left( { f + \frac{1}{e_1 \; e_2 } … … 749 749 + D_u^{\vect{U}} + F_u^{\vect{U}} \quad 750 750 \end{multline} 751 \begin{multline} \label{ Eq_PE_sco_v}751 \begin{multline} \label{eq:PE_sco_v} 752 752 \frac{1}{e_3} \frac{\partial \left( e_3\,v \right) }{\partial t}= 753 753 - \left( { f + \frac{1}{e_1 \; e_2} … … 766 766 pressure have the same expressions as in $z$-coordinates although they do not represent 767 767 exactly the same quantities. $\omega$ is provided by the continuity equation 768 (see Appendix~\ref{Apdx_A}):769 \begin{equation} \label{ Eq_PE_sco_continuity}768 (see \autoref{apdx:A}): 769 \begin{equation} \label{eq:PE_sco_continuity} 770 770 \frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0 771 771 \qquad \text{with }\;\; … … 777 777 \vspace{0.5cm} 778 778 $\bullet$ tracer equations: 779 \begin{multline} \label{ Eq_PE_sco_t}779 \begin{multline} \label{eq:PE_sco_t} 780 780 \frac{1}{e_3} \frac{\partial \left( e_3\,T \right) }{\partial t}= 781 781 -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,T} \right)}{\partial i} … … 784 784 \end{multline} 785 785 786 \begin{multline} \label{ Eq_PE_sco_s}786 \begin{multline} \label{eq:PE_sco_s} 787 787 \frac{1}{e_3} \frac{\partial \left( e_3\,S \right) }{\partial t}= 788 788 -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,S} \right)}{\partial i} … … 805 805 % ------------------------------------------------------------------------------------------------------------- 806 806 \subsection{Curvilinear \textit{z*}--coordinate system} 807 \label{ PE_zco_star}807 \label{subsec:PE_zco_star} 808 808 809 809 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 810 810 \begin{figure}[!b] \begin{center} 811 811 \includegraphics[width=1.0\textwidth]{Fig_z_zstar} 812 \caption{ \protect\label{ Fig_z_zstar}812 \caption{ \protect\label{fig:z_zstar} 813 813 (a) $z$-coordinate in linear free-surface case ; 814 814 (b) $z-$coordinate in non-linear free surface case ; … … 837 837 detailed in Adcroft and Campin (2004). The major points are summarized 838 838 here. The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as: 839 \begin{equation} \label{ Eq_z-star}839 \begin{equation} \label{eq:z-star} 840 840 H + \textit{z*} = (H + z) / r \quad \text{and} \ \delta \textit{z*} = \delta z / r \quad \text{with} \ r = \frac{H+\eta} {H} 841 841 \end{equation} … … 855 855 To overcome problems with vanishing surface and/or bottom cells, we consider the 856 856 zstar coordinate 857 \begin{equation} \label{ PE_}857 \begin{equation} \label{eq:PE_} 858 858 z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 859 859 \end{equation} … … 867 867 The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between 868 868 undulations of the bottom topography versus undulations in the surface height, it 869 is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \ S\ref{PE_sco}.869 is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \autoref{subsec:PE_sco}. 870 870 Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an 871 871 unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in … … 873 873 gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using 874 874 the same techniques as in $z$-models (see Chapters 13-16 of \cite{Griffies_Bk04}) for a 875 discussion of neutral physics in $z$-models, as well as Section \S\ref{LDF_slp}875 discussion of neutral physics in $z$-models, as well as \autoref{sec:LDF_slp} 876 876 in this document for treatment in \NEMO). 877 877 … … 902 902 % ------------------------------------------------------------------------------------------------------------- 903 903 \subsection{Curvilinear terrain-following \textit{s}--coordinate} 904 \label{ PE_sco}904 \label{subsec:PE_sco} 905 905 906 906 % ------------------------------------------------------------------------------------------------------------- … … 915 915 one along continental slopes. Topographic Rossby waves can be excited and can interact 916 916 with the mean current. In the $z-$coordinate system presented in the previous section 917 (\ S\ref{PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is917 (\autoref{sec:PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is 918 918 discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom 919 919 and to large localized depth gradients associated with large localized vertical velocities. … … 937 937 The main two problems come from the truncation error in the horizontal pressure 938 938 gradient and a possibly increased diapycnal diffusion. The horizontal pressure force 939 in $s$-coordinate consists of two terms (see Appendix~\ref{Apdx_A}),940 941 \begin{equation} \label{ Eq_PE_p_sco}939 in $s$-coordinate consists of two terms (see \autoref{apdx:A}), 940 941 \begin{equation} \label{eq:PE_p_sco} 942 942 \left. {\nabla p} \right|_z =\left. {\nabla p} \right|_s -\frac{\partial 943 943 p}{\partial s}\left. {\nabla z} \right|_s 944 944 \end{equation} 945 945 946 The second term in \ eqref{Eq_PE_p_sco} depends on the tilt of the coordinate surface946 The second term in \autoref{eq:PE_p_sco} depends on the tilt of the coordinate surface 947 947 and introduces a truncation error that is not present in a $z$-model. In the special case 948 948 of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), … … 958 958 topography: a envelope topography is defined in $s$-coordinate on which a full or 959 959 partial step bottom topography is then applied in order to adjust the model depth to 960 the observed one (see \ S\ref{DOM_zgr}.960 the observed one (see \autoref{sec:DOM_zgr}. 961 961 962 962 For numerical reasons a minimum of diffusion is required along the coordinate surfaces … … 973 973 the ocean will stay at rest in a $z$-model. As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below the strongly stratified portion of the water column 974 974 ($i.e.$ the main thermocline) \citep{Madec_al_JPO96}. An alternate solution consists of rotating 975 the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \ S\ref{PE_ldf}.975 the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \autoref{subsec:PE_ldf}. 976 976 Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large, 977 strongly exceeding the stability limit of such a operator when it is discretized (see Chapter~\ref{LDF}).977 strongly exceeding the stability limit of such a operator when it is discretized (see \autoref{chap:LDF}). 978 978 979 979 The $s-$coordinates introduced here \citep{Lott_al_OM90,Madec_al_JPO96} differ mainly in two … … 988 988 % ------------------------------------------------------------------------------------------------------------- 989 989 \subsection{\texorpdfstring{Curvilinear $\tilde{z}$--coordinate}{}} 990 \label{ PE_zco_tilde}990 \label{subsec:PE_zco_tilde} 991 991 992 992 The $\tilde{z}$-coordinate has been developed by \citet{Leclair_Madec_OM11}. … … 1000 1000 % ================================================================ 1001 1001 \section{Subgrid scale physics} 1002 \label{ PE_zdf_ldf}1002 \label{sec:PE_zdf_ldf} 1003 1003 1004 1004 The primitive equations describe the behaviour of a geophysical fluid at … … 1019 1019 The control exerted by gravity on the flow induces a strong anisotropy 1020 1020 between the lateral and vertical motions. Therefore subgrid-scale physics 1021 \textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$ in \ eqref{Eq_PE_dyn},1022 \ eqref{Eq_PE_tra_T} and \eqref{Eq_PE_tra_S} are divided into a lateral part1021 \textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$ in \autoref{eq:PE_dyn}, 1022 \autoref{eq:PE_tra_T} and \autoref{eq:PE_tra_S} are divided into a lateral part 1023 1023 \textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and a vertical part 1024 1024 \textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$. The formulation of these terms … … 1029 1029 % ------------------------------------------------------------------------------------------------------------- 1030 1030 \subsection{Vertical subgrid scale physics} 1031 \label{ PE_zdf}1031 \label{subsec:PE_zdf} 1032 1032 1033 1033 The model resolution is always larger than the scale at which the major … … 1044 1044 turbulent motions is simply impractical. The resulting vertical momentum and 1045 1045 tracer diffusive operators are of second order: 1046 \begin{equation} \label{ Eq_PE_zdf}1046 \begin{equation} \label{eq:PE_zdf} 1047 1047 \begin{split} 1048 1048 {\vect{D}}^{v \vect{U}} &=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\ … … 1054 1054 where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, 1055 1055 respectively. At the sea surface and at the bottom, turbulent fluxes of momentum, heat 1056 and salt must be specified (see Chap.~\ref{SBC} and \ref{ZDF} and \S\ref{TRA_bbl}).1056 and salt must be specified (see \autoref{chap:SBC} and \autoref{chap:ZDF} and \autoref{sec:TRA_bbl}). 1057 1057 All the vertical physics is embedded in the specification of the eddy coefficients. 1058 1058 They can be assumed to be either constant, or function of the local fluid properties 1059 1059 ($e.g.$ Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), or computed from a 1060 turbulent closure model. The choices available in \NEMO are discussed in \ S\ref{ZDF}).1060 turbulent closure model. The choices available in \NEMO are discussed in \autoref{chap:ZDF}). 1061 1061 1062 1062 % ------------------------------------------------------------------------------------------------------------- … … 1064 1064 % ------------------------------------------------------------------------------------------------------------- 1065 1065 \subsection{Formulation of the lateral diffusive and viscous operators} 1066 \label{ PE_ldf}1066 \label{subsec:PE_ldf} 1067 1067 1068 1068 Lateral turbulence can be roughly divided into a mesoscale turbulence … … 1124 1124 \subsubsection{Lateral laplacian tracer diffusive operator} 1125 1125 1126 The lateral Laplacian tracer diffusive operator is defined by (see Appendix~\ref{Apdx_B}):1127 \begin{equation} \label{ Eq_PE_iso_tensor}1126 The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx:B}): 1127 \begin{equation} \label{eq:PE_iso_tensor} 1128 1128 D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 1129 1129 \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} … … 1135 1135 where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along 1136 1136 which the diffusive operator acts and the model level ($e. g.$ $z$- or 1137 $s$-surfaces). Note that the formulation \ eqref{Eq_PE_iso_tensor} is exact for the1137 $s$-surfaces). Note that the formulation \autoref{eq:PE_iso_tensor} is exact for the 1138 1138 rotation between geopotential and $s$-surfaces, while it is only an approximation 1139 1139 for the rotation between isoneutral and $z$- or $s$-surfaces. Indeed, in the latter 1140 case, two assumptions are made to simplify \ eqref{Eq_PE_iso_tensor} \citep{Cox1987}.1140 case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{Cox1987}. 1141 1141 First, the horizontal contribution of the dianeutral mixing is neglected since the ratio 1142 1142 between iso and dia-neutral diffusive coefficients is known to be several orders of 1143 1143 magnitude smaller than unity. Second, the two isoneutral directions of diffusion are 1144 1144 assumed to be independent since the slopes are generally less than $10^{-2}$ in the 1145 ocean (see Appendix~\ref{Apdx_B}).1145 ocean (see \autoref{apdx:B}). 1146 1146 1147 1147 For \textit{iso-level} diffusion, $r_1$ and $r_2 $ are zero. $\Re $ reduces to the identity … … 1150 1150 For \textit{geopotential} diffusion, $r_1$ and $r_2 $ are the slopes between the 1151 1151 geopotential and computational surfaces: they are equal to $\sigma _1$ and $\sigma _2$, 1152 respectively (see \ eqref{Eq_PE_sco_slope}).1152 respectively (see \autoref{eq:PE_sco_slope}). 1153 1153 1154 1154 For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral 1155 1155 and computational surfaces. Therefore, they are different quantities, 1156 1156 but have similar expressions in $z$- and $s$-coordinates. In $z$-coordinates: 1157 \begin{equation} \label{ Eq_PE_iso_slopes}1157 \begin{equation} \label{eq:PE_iso_slopes} 1158 1158 r_1 =\frac{e_3 }{e_1 } \left( \pd[\rho]{i} \right) \left( \pd[\rho]{k} \right)^{-1} \, \quad 1159 1159 r_2 =\frac{e_3 }{e_2 } \left( \pd[\rho]{j} \right) \left( \pd[\rho]{k} \right)^{-1} \, … … 1164 1164 When the \textit{eddy induced velocity} parametrisation (eiv) \citep{Gent1990} is used, 1165 1165 an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: 1166 \begin{equation} \label{ Eq_PE_iso+eiv}1166 \begin{equation} \label{eq:PE_iso+eiv} 1167 1167 D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right) 1168 1168 +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right) … … 1170 1170 where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent, 1171 1171 eddy-induced transport velocity. This velocity field is defined by: 1172 \begin{equation} \label{ Eq_PE_eiv}1172 \begin{equation} \label{eq:PE_eiv} 1173 1173 \begin{split} 1174 1174 u^\ast &= +\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_1 } \right] \\ … … 1183 1183 between isoneutral and \emph{geopotential} surfaces. Their values are 1184 1184 thus independent of the vertical coordinate, but their expression depends on the coordinate: 1185 \begin{align} \label{ Eq_PE_slopes_eiv}1185 \begin{align} \label{eq:PE_slopes_eiv} 1186 1186 \tilde{r}_n = \begin{cases} 1187 1187 r_n & \text{in $z$-coordinate} \\ … … 1193 1193 The normal component of the eddy induced velocity is zero at all the boundaries. 1194 1194 This can be achieved in a model by tapering either the eddy coefficient or the slopes 1195 to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. Chap.~\ref{LDF}).1195 to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. \autoref{chap:LDF}). 1196 1196 1197 1197 \subsubsection{Lateral bilaplacian tracer diffusive operator} 1198 1198 1199 1199 The lateral bilaplacian tracer diffusive operator is defined by: 1200 \begin{equation} \label{ Eq_PE_bilapT}1200 \begin{equation} \label{eq:PE_bilapT} 1201 1201 D^{lT}= - \Delta \left( \;\Delta T \right) 1202 1202 \qquad \text{where} \;\; \Delta \bullet = \nabla \left( {\sqrt{B^{lT}\,}\;\Re \;\nabla \bullet} \right) 1203 1203 \end{equation} 1204 It is the Laplacian operator given by \ eqref{Eq_PE_iso_tensor} applied twice with1204 It is the Laplacian operator given by \autoref{eq:PE_iso_tensor} applied twice with 1205 1205 the harmonic eddy diffusion coefficient set to the square root of the biharmonic one. 1206 1206 … … 1209 1209 1210 1210 The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by 1211 applying \ eqref{Eq_PE_lap_vector} to the horizontal velocity vector (see Appendix~\ref{Apdx_B}):1212 \begin{equation} \label{ Eq_PE_lapU}1211 applying \autoref{eq:PE_lap_vector} to the horizontal velocity vector (see \autoref{apdx:B}): 1212 \begin{equation} \label{eq:PE_lapU} 1213 1213 \begin{split} 1214 1214 {\rm {\bf D}}^{l{\rm {\bf U}}} … … 1225 1225 1226 1226 Such a formulation ensures a complete separation between the vorticity and 1227 horizontal divergence fields (see Appendix~\ref{Apdx_C}).1227 horizontal divergence fields (see \autoref{apdx:C}). 1228 1228 Unfortunately, it is only available in \textit{iso-level} direction. 1229 1229 When a rotation is required ($i.e.$ geopotential diffusion in $s-$coordinates 1230 1230 or isoneutral diffusion in both $z$- and $s$-coordinates), the $u$ and $v-$fields 1231 1231 are considered as independent scalar fields, so that the diffusive operator is given by: 1232 \begin{equation} \label{ Eq_PE_lapU_iso}1232 \begin{equation} \label{eq:PE_lapU_iso} 1233 1233 \begin{split} 1234 1234 D_u^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla u} \right) \\ … … 1236 1236 \end{split} 1237 1237 \end{equation} 1238 where $\Re$ is given by \eqref{Eq_PE_iso_tensor}. It is the same expression as1238 where $\Re$ is given by \autoref{eq:PE_iso_tensor}. It is the same expression as 1239 1239 those used for diffusive operator on tracers. It must be emphasised that such a 1240 1240 formulation is only exact in a Cartesian coordinate system, $i.e.$ on a $f-$ or -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_model_basics_zstar.tex
r9393 r9407 27 27 To overcome problems with vanishing surface and/or bottom cells, we consider the 28 28 zstar coordinate 29 \begin{equation} \label{ PE_}29 \begin{equation} \label{eq:PE_} 30 30 z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 31 31 \end{equation} … … 39 39 The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between 40 40 undulations of the bottom topography versus undulations in the surface height, it 41 is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \ S\ref{PE_sco}.41 is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \autoref{subsec:PE_sco}. 42 42 Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an 43 43 unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in … … 45 45 gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using 46 46 the same techniques as in $z$-models (see Chapters 13-16 of Griffies (2004) for a 47 discussion of neutral physics in $z$-models, as well as Section \S\ref{LDF_slp}47 discussion of neutral physics in $z$-models, as well as \autoref{sec:LDF_slp} 48 48 in this document for treatment in \NEMO). 49 49 … … 76 76 % ================================================================ 77 77 \section{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})} 78 \label{ DYN_hpg_spg}78 \label{sec:DYN_hpg_spg} 79 79 %-----------------------------------------nam_dynspg---------------------------------------------------- 80 80 \forfile{../namelists/nam_dynspg} 81 81 %------------------------------------------------------------------------------------------------------------ 82 82 Options are defined through the \ngn{nam\_dynspg} namelist variables. 83 The surface pressure gradient term is related to the representation of the free surface (\ S\ref{PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface or rigid lid) and the variable volume case (nonlinear free surface, \key{vvl} is active). In the linear free surface case (\S\ref{PE_free_surface}) and rigid lid (\S\ref{PE_rigid_lid}), the vertical scale factors $e_{3}$ are fixed in time, while in the nonlinear case (\S\ref{PE_free_surface}) they are time-dependent. With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.83 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface or rigid lid) and the variable volume case (nonlinear free surface, \key{vvl} is active). In the linear free surface case (\autoref{subsec:PE_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), the vertical scale factors $e_{3}$ are fixed in time, while in the nonlinear case (\autoref{subsec:PE_free_surface}) they are time-dependent. With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 84 84 85 85 %------------------------------------------------------------- … … 87 87 %------------------------------------------------------------- 88 88 \subsubsection{Explicit (\protect\key{dynspg\_exp})} 89 \label{ DYN_spg_exp}89 \label{subsec:DYN_spg_exp} 90 90 91 91 In the explicit free surface formulation, the model time step is chosen small enough to describe the external gravity waves (typically a few ten seconds). The sea surface height is given by : 92 \begin{equation} \label{ Eq_dynspg_ssh}92 \begin{equation} \label{eq:dynspg_ssh} 93 93 \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T} 94 94 e_{2T} }\sum\limits_k {\left( {\delta _i \left[ {e_{2u} e_{3u} u} … … 96 96 \end{equation} 97 97 98 where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs (if the later are introduced as a surface freshwater flux, see \ S\ref{SBC}) expressed in $Kg.m^{-2}.s^{-1}$, and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, i.e. the velocity appearing in (\ref{Eq_dynspg_ssh}) is centred in time (\textit{now} velocity).98 where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs (if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$, and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, i.e. the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity). 99 99 100 100 The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by : 101 \begin{equation} \label{ Eq_dynspg_exp}101 \begin{equation} \label{eq:dynspg_exp} 102 102 \left\{ \begin{aligned} 103 103 - \frac{1} {e_{1u}} \; \delta _{i+1/2} \left[ \,\eta\, \right] \\ … … 107 107 \end{equation} 108 108 109 Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in (\ ref{Eq_dynspg_exp}).109 Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in (\autoref{eq:dynspg_exp}). 110 110 111 111 %------------------------------------------------------------- … … 113 113 %------------------------------------------------------------- 114 114 \subsubsection{Split-explicit time-stepping (\protect\key{dynspg\_ts})} 115 \label{ DYN_spg_ts}115 \label{subsec:DYN_spg_ts} 116 116 %--------------------------------------------namdom---------------------------------------------------- 117 117 \forfile{../namelists/namdom} … … 124 124 \begin{figure}[!t] \begin{center} 125 125 \includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts} 126 \caption{ \protect\label{ Fig_DYN_dynspg_ts}126 \caption{ \protect\label{fig:DYN_dynspg_ts} 127 127 Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, 128 128 after \citet{Griffies2004}. Time increases to the right. Baroclinic time steps are denoted by … … 151 151 scheme using the small barotropic time step $\Delta t$. We have 152 152 153 \begin{equation} \label{ DYN_spg_ts_eta}153 \begin{equation} \label{eq:DYN_spg_ts_eta} 154 154 \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) 155 155 = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 156 156 \end{equation} 157 \begin{multline} \label{ DYN_spg_ts_u}157 \begin{multline} \label{eq:DYN_spg_ts_u} 158 158 \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\ 159 159 = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) … … 165 165 and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time 166 166 that sets the barotropic time steps via 167 \begin{equation} \label{ DYN_spg_ts_t}167 \begin{equation} \label{eq:DYN_spg_ts_t} 168 168 t_n=\tau+n\Delta t 169 169 \end{equation} 170 170 with $n$ an integer. The density scaled surface pressure is evaluated via 171 \begin{equation} \label{ DYN_spg_ts_ps}171 \begin{equation} \label{eq:DYN_spg_ts_ps} 172 172 p_s^{(b)}(\tau,t_{n}) = \begin{cases} 173 173 g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ … … 176 176 \end{equation} 177 177 To get started, we assume the following initial conditions 178 \begin{equation} \label{ DYN_spg_ts_eta}178 \begin{equation} \label{eq:DYN_spg_ts_eta} 179 179 \begin{split} 180 180 \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} … … 184 184 \end{equation} 185 185 with 186 \begin{equation} \label{ DYN_spg_ts_etaF}186 \begin{equation} \label{eq:DYN_spg_ts_etaF} 187 187 \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) 188 188 \end{equation} 189 189 the time averaged surface height taken from the previous barotropic cycle. Likewise, 190 \begin{equation} \label{ DYN_spg_ts_u}190 \begin{equation} \label{eq:DYN_spg_ts_u} 191 191 \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ 192 192 \\ … … 194 194 \end{equation} 195 195 with 196 \begin{equation} \label{ DYN_spg_ts_u}196 \begin{equation} \label{eq:DYN_spg_ts_u} 197 197 \overline{\textbf{U}^{(b)}(\tau)} 198 198 = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) … … 201 201 202 202 Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 203 \begin{equation} \label{ DYN_spg_ts_u}203 \begin{equation} \label{eq:DYN_spg_ts_u} 204 204 \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} 205 205 = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) … … 207 207 The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form 208 208 209 \begin{equation} \label{ DYN_spg_ts_ssh}209 \begin{equation} \label{eq:DYN_spg_ts_ssh} 210 210 \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 211 211 \end{equation} … … 214 214 215 215 In general, some form of time filter is needed to maintain integrity of the surface 216 height field due to the leap-frog splitting mode in equation \ ref{DYN_spg_ts_ssh}. We216 height field due to the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. We 217 217 have tried various forms of such filtering, with the following method discussed in 218 218 Griffies et al. (2001) chosen due to its stability and reasonably good maintenance of 219 tracer conservation properties (see Section??)220 221 \begin{equation} \label{ DYN_spg_ts_sshf}219 tracer conservation properties (see ??) 220 221 \begin{equation} \label{eq:DYN_spg_ts_sshf} 222 222 \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} 223 223 \end{equation} 224 224 Another approach tried was 225 225 226 \begin{equation} \label{ DYN_spg_ts_sshf2}226 \begin{equation} \label{eq:DYN_spg_ts_sshf2} 227 227 \eta^{F}(\tau-\Delta) = \eta(\tau) 228 228 + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) … … 232 232 which is useful since it isolates all the time filtering aspects into the term multiplied 233 233 by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when 234 eliminating tracer and surface height time filtering (see Section ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \ref{DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.234 eliminating tracer and surface height time filtering (see ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended. 235 235 236 236 … … 242 242 %------------------------------------------------------------- 243 243 \subsubsection{Filtered formulation (\protect\key{dynspg\_flt})} 244 \label{ DYN_spg_flt}244 \label{subsec:DYN_spg_flt} 245 245 246 246 The filtered formulation follows the \citet{Roullet2000} implementation. The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic solvers available in the code are 247 documented in \ S\ref{MISC}. The amplitude of the extra term is given by the namelist variable \np{rnu}. The default value is 1, as recommended by \citet{Roullet2000}247 documented in \autoref{chap:MISC}. The amplitude of the extra term is given by the namelist variable \np{rnu}. The default value is 1, as recommended by \citet{Roullet2000} 248 248 249 249 \colorbox{red}{\np{rnu}\forcode{ = 1} to be suppressed from namelist !} … … 253 253 %------------------------------------------------------------- 254 254 \subsection{Non-linear free surface formulation (\protect\key{vvl})} 255 \label{ DYN_spg_vvl}256 257 In the non-linear free surface formulation, the variations of volume are fully taken into account. This option is presented in a report \citep{Levier2007} available on the NEMO web site. The three time-stepping methods (explicit, split-explicit and filtered) are the same as in \ S\ref{DYN_spg_linear} except that the ocean depth is now time-dependent. In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.255 \label{subsec:DYN_spg_vvl} 256 257 In the non-linear free surface formulation, the variations of volume are fully taken into account. This option is presented in a report \citep{Levier2007} available on the NEMO web site. The three time-stepping methods (explicit, split-explicit and filtered) are the same as in \autoref{DYN_spg_linear} except that the ocean depth is now time-dependent. In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step. 258 258 259 259 -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_time_domain.tex
r9394 r9407 6 6 % ================================================================ 7 7 \chapter{Time Domain (STP) } 8 \label{ STP}8 \label{chap:STP} 9 9 \minitoc 10 10 … … 22 22 23 23 24 Having defined the continuous equations in Chap.~\ref{PE}, we need now to choose24 Having defined the continuous equations in \autoref{chap:PE}, we need now to choose 25 25 a time discretization, a key feature of an ocean model as it exerts a strong influence 26 26 on the structure of the computer code ($i.e.$ on its flowchart). … … 34 34 % ================================================================ 35 35 \section{Time stepping environment} 36 \label{ STP_environment}36 \label{sec:STP_environment} 37 37 38 38 The time stepping used in \NEMO is a three level scheme that can be 39 39 represented as follows: 40 \begin{equation} \label{ Eq_STP}40 \begin{equation} \label{eq:STP} 41 41 x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ \text{RHS}_x^{t-\rdt,\,t,\,t+\rdt} 42 42 \end{equation} … … 57 57 although referred to as $x_a$ (after) in the code, is usually not the variable at 58 58 the after time step; but rather it is used to store the time derivative (RHS in 59 \ eqref{Eq_STP}) prior to time-stepping the equation. Generally, the time59 \autoref{eq:STP}) prior to time-stepping the equation. Generally, the time 60 60 stepping is performed once at each time step in the \mdl{tranxt} and \mdl{dynnxt} 61 61 modules, except when using implicit vertical diffusion or calculating sea surface height … … 66 66 % ------------------------------------------------------------------------------------------------------------- 67 67 \section{Non-diffusive part --- Leapfrog scheme} 68 \label{ STP_leap_frog}68 \label{sec:STP_leap_frog} 69 69 70 70 The time stepping used for processes other than diffusion is the well-known leapfrog 71 71 scheme \citep{Mesinger_Arakawa_Bk76}. This scheme is widely used for advection 72 72 processes in low-viscosity fluids. It is a time centred scheme, $i.e.$ 73 the RHS in \ eqref{Eq_STP} is evaluated at time step $t$, the now time step.73 the RHS in \autoref{eq:STP} is evaluated at time step $t$, the now time step. 74 74 It may be used for momentum and tracer advection, 75 75 pressure gradient, and Coriolis terms, but not for diffusion terms. … … 87 87 by \citet{Asselin_MWR72}, is a kind of laplacian diffusion in time that mixes odd and 88 88 even time steps: 89 \begin{equation} \label{ Eq_STP_asselin}89 \begin{equation} \label{eq:STP_asselin} 90 90 x_F^t = x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right] 91 91 \end{equation} 92 92 where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin 93 93 coefficient. $\gamma$ is initialized as \np{rn\_atfp} (namelist parameter). 94 Its default value is \np{rn\_atfp}\forcode{ = 10.e-3} (see \ S~\ref{STP_mLF}),94 Its default value is \np{rn\_atfp}\forcode{ = 10.e-3} (see \autoref{sec:STP_mLF}), 95 95 causing only a weak dissipation of high frequency motions (\citep{Farge1987}). 96 96 The addition of a time filter degrades the accuracy of the … … 110 110 % ------------------------------------------------------------------------------------------------------------- 111 111 \section{Diffusive part --- Forward or backward scheme} 112 \label{ STP_forward_imp}112 \label{sec:STP_forward_imp} 113 113 114 114 The leapfrog differencing scheme is unsuitable for the representation of 115 115 diffusion and damping processes. For a tendancy $D_x$, representing a 116 116 diffusion term or a restoring term to a tracer climatology 117 (when present, see \ S~\ref{TRA_dmp}), a forward time differencing scheme117 (when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme 118 118 is used : 119 \begin{equation} \label{ Eq_STP_euler}119 \begin{equation} \label{eq:STP_euler} 120 120 x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ {D_x}^{t-\rdt} 121 121 \end{equation} … … 123 123 This is diffusive in time and conditionally stable. The 124 124 conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}: 125 \begin{equation} \label{ Eq_STP_euler_stability}125 \begin{equation} \label{eq:STP_euler_stability} 126 126 A^h < \left\{ 127 127 \begin{aligned} … … 132 132 \end{equation} 133 133 where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is 134 the mixing coefficient. The linear constraint \ eqref{Eq_STP_euler_stability}134 the mixing coefficient. The linear constraint \autoref{eq:STP_euler_stability} 135 135 is a necessary condition, but not sufficient. If it is not satisfied, even mildly, 136 136 then the model soon becomes wildly unstable. The instability can be removed … … 146 146 stability criterion is reduced by a factor of $N$. The computation is performed as 147 147 follows: 148 \begin{equation} \label{ Eq_STP_ts}148 \begin{equation} \label{eq:STP_ts} 149 149 \begin{split} 150 150 & x_\ast ^{t-\rdt} = x^{t-\rdt} \\ … … 158 158 by setting \np{nn\_zdfexp}, (namelist parameter). The scheme $(b)$ is unconditionally 159 159 stable but diffusive. It can be written as follows: 160 \begin{equation} \label{ Eq_STP_imp}160 \begin{equation} \label{eq:STP_imp} 161 161 x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ \text{RHS}_x^{t+\rdt} 162 162 \end{equation} … … 170 170 the forward time differencing scheme. For example, the finite difference 171 171 approximation of the temperature equation is: 172 \begin{equation} \label{ Eq_STP_imp_zdf}172 \begin{equation} \label{eq:STP_imp_zdf} 173 173 \frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\rdt}\equiv \text{RHS}+\frac{1}{e_{3t} }\delta 174 174 _k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} … … 176 176 \end{equation} 177 177 where RHS is the right hand side of the equation except for the vertical diffusion term. 178 We rewrite \ eqref{Eq_STP_imp} as:179 \begin{equation} \label{ Eq_STP_imp_mat}178 We rewrite \autoref{eq:STP_imp} as: 179 \begin{equation} \label{eq:STP_imp_mat} 180 180 -c(k+1)\;T^{t+1}(k+1) + d(k)\;T^{t+1}(k) - \;c(k)\;T^{t+1}(k-1) \equiv b(k) 181 181 \end{equation} … … 187 187 \end{align*} 188 188 189 \ eqref{Eq_STP_imp_mat} is a linear system of equations with an associated189 \autoref{eq:STP_imp_mat} is a linear system of equations with an associated 190 190 matrix which is tridiagonal. Moreover, $c(k)$ and $d(k)$ are positive and the diagonal 191 191 term is greater than the sum of the two extra-diagonal terms, therefore a special … … 199 199 % ------------------------------------------------------------------------------------------------------------- 200 200 \section{Surface pressure gradient} 201 \label{ STP_spg_ts}201 \label{sec:STP_spg_ts} 202 202 203 203 ===>>>> TO BE written.... :-) … … 207 207 \begin{figure}[!t] \begin{center} 208 208 \includegraphics[width=0.7\textwidth]{Fig_TimeStepping_flowchart} 209 \caption{ \protect\label{ Fig_TimeStep_flowchart}209 \caption{ \protect\label{fig:TimeStep_flowchart} 210 210 Sketch of the leapfrog time stepping sequence in \NEMO from \citet{Leclair_Madec_OM09}. 211 211 The use of a semi-implicit computation of the hydrostatic pressure gradient requires … … 215 215 prior to the computation of the tracer equation. 216 216 Note that the method for the evaluation of the surface pressure gradient $\nabla p_s$ is not presented here 217 (see \ S~\ref{DYN_spg}). }217 (see \autoref{sec:DYN_spg}). } 218 218 \end{center} \end{figure} 219 219 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 224 224 % ------------------------------------------------------------------------------------------------------------- 225 225 \section{Modified Leapfrog -- Asselin filter scheme} 226 \label{ STP_mLF}226 \label{sec:STP_mLF} 227 227 228 228 Significant changes have been introduced by \cite{Leclair_Madec_OM09} in the … … 233 233 In a classical LF-RA environment, the forcing term is centred in time, $i.e.$ 234 234 it is time-stepped over a $2\rdt$ period: $x^t = x^t + 2\rdt Q^t $ where $Q$ 235 is the forcing applied to $x$, and the time filter is given by \ eqref{Eq_STP_asselin}235 is the forcing applied to $x$, and the time filter is given by \autoref{eq:STP_asselin} 236 236 so that $Q$ is redistributed over several time step. 237 237 In the modified LF-RA environment, these two formulations have been replaced by: 238 238 \begin{align} 239 x^{t+\rdt} &= x^{t-\rdt} + \rdt \left( Q^{t-\rdt/2} + Q^{t+\rdt/2} \right) \label{ Eq_STP_forcing} \\239 x^{t+\rdt} &= x^{t-\rdt} + \rdt \left( Q^{t-\rdt/2} + Q^{t+\rdt/2} \right) \label{eq:STP_forcing} \\ 240 240 % 241 241 x_F^t &= x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right] 242 - \gamma\,\rdt \, \left[ Q^{t+\rdt/2} - Q^{t-\rdt/2} \right] \label{ Eq_STP_RA}242 - \gamma\,\rdt \, \left[ Q^{t+\rdt/2} - Q^{t-\rdt/2} \right] \label{eq:STP_RA} 243 243 \end{align} 244 The change in the forcing formulation given by \ eqref{Eq_STP_forcing}245 (see Fig.\ref{Fig_MLF_forcing}) has a significant effect: the forcing term no244 The change in the forcing formulation given by \autoref{eq:STP_forcing} 245 (see \autoref{fig:MLF_forcing}) has a significant effect: the forcing term no 246 246 longer excites the divergence of odd and even time steps \citep{Leclair_Madec_OM09}. 247 247 % forcing seen by the model.... … … 250 250 Indeed, time filtering is no longer required on the forcing part. The influence of 251 251 the Asselin filter on the forcing is be removed by adding a new term in the filter 252 (last term in \ eqref{Eq_STP_RA} compared to \eqref{Eq_STP_asselin}). Since252 (last term in \autoref{eq:STP_RA} compared to \autoref{eq:STP_asselin}). Since 253 253 the filtering of the forcing was the source of non-conservation in the classical 254 254 LF-RA scheme, the modified formulation becomes conservative \citep{Leclair_Madec_OM09}. 255 255 Second, the LF-RA becomes a truly quasi-second order scheme. Indeed, 256 \ eqref{Eq_STP_forcing} used in combination with a careful treatment of static257 instability (\ S\ref{ZDF_evd}) and of the TKE physics (\S\ref{ZDF_tke_ene}),256 \autoref{eq:STP_forcing} used in combination with a careful treatment of static 257 instability (\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}), 258 258 the two other main sources of time step divergence, allows a reduction by 259 259 two orders of magnitude of the Asselin filter parameter. … … 261 261 Note that the forcing is now provided at the middle of a time step: $Q^{t+\rdt/2}$ 262 262 is the forcing applied over the $[t,t+\rdt]$ time interval. This and the change 263 in the time filter, \ eqref{Eq_STP_RA}, allows an exact evaluation of the263 in the time filter, \autoref{eq:STP_RA}, allows an exact evaluation of the 264 264 contribution due to the forcing term between any two time steps, 265 265 even if separated by only $\rdt$ since the time filter is no longer applied to the … … 269 269 \begin{figure}[!t] \begin{center} 270 270 \includegraphics[width=0.90\textwidth]{Fig_MLF_forcing} 271 \caption{ \protect\label{ Fig_MLF_forcing}271 \caption{ \protect\label{fig:MLF_forcing} 272 272 Illustration of forcing integration methods. 273 273 (top) ''Traditional'' formulation : the forcing is defined at the same time as the variable … … 283 283 % ------------------------------------------------------------------------------------------------------------- 284 284 \section{Start/Restart strategy} 285 \label{ STP_rst}285 \label{sec:STP_rst} 286 286 287 287 %--------------------------------------------namrun------------------------------------------- … … 291 291 The first time step of this three level scheme when starting from initial conditions 292 292 is a forward step (Euler time integration): 293 \begin{equation} \label{ Eq_DOM_euler}293 \begin{equation} \label{eq:DOM_euler} 294 294 x^1 = x^0 + \rdt \ \text{RHS}^0 295 295 \end{equation} 296 This is done simply by keeping the leapfrog environment ($i.e.$ the \ eqref{Eq_STP}296 This is done simply by keeping the leapfrog environment ($i.e.$ the \autoref{eq:STP} 297 297 three level time stepping) but setting all $x^0$ (\textit{before}) and $x^{1}$ (\textit{now}) fields 298 298 equal at the first time step and using half the value of $\rdt$. … … 307 307 308 308 Note that when a semi-implicit scheme is used to evaluate the hydrostatic pressure 309 gradient (see \ S\ref{DYN_hpg_imp}), an extra three-dimensional field has to be309 gradient (see \autoref{subsec:DYN_hpg_imp}), an extra three-dimensional field has to be 310 310 added to the restart file to ensure an exact restartability. This is done optionally 311 311 via the \np{nn\_dynhpg\_rst} namelist parameter, so that the size of the … … 335 335 % ------------------------------------------------------------------------------------------------------------- 336 336 \subsection{Time domain} 337 \label{ STP_time}337 \label{subsec:STP_time} 338 338 %--------------------------------------------namrun------------------------------------------- 339 339 \forfile{../namelists/namdom} -
branches/2017/dev_merge_2017/DOC/tex_sub/introduction.tex
r9393 r9407 29 29 model coupled with the sea-ice and/or the atmosphere. 30 30 31 This manual is organised in as follows. Chapter~\ref{PE} presents the model basics,31 This manual is organised in as follows. \autoref{chap:PE} presents the model basics, 32 32 $i.e.$ the equations and their assumptions, the vertical coordinates used, and the 33 33 subgrid scale physics. This part deals with the continuous equations of the model … … 39 39 are used throughout. 40 40 41 The following chapters deal with the discrete equations. Chapter~\ref{STP} presents the41 The following chapters deal with the discrete equations. \autoref{chap:STP} presents the 42 42 time domain. The model time stepping environment is a three level scheme in which 43 43 the tendency terms of the equations are evaluated either centered in time, or forward, 44 44 or backward depending of the nature of the term. 45 Chapter~\ref{DOM} presents the space domain. The model is discretised on a staggered45 \autoref{chap:DOM} presents the space domain. The model is discretised on a staggered 46 46 grid (Arakawa C grid) with masking of land areas. Vertical discretisation used depends 47 47 on both how the bottom topography is represented and whether the free surface is linear or not. … … 50 50 the corresponding rescaled height coordinate formulation (\textit{z*} or \textit{s*}) is used 51 51 (the level position then vary in time as a function of the sea surface heigh). 52 The following two chapters (\ ref{TRA} and \ref{DYN}) describe the discretisation of the52 The following two chapters (\autoref{chap:TRA} and \autoref{chap:DYN}) describe the discretisation of the 53 53 prognostic equations for the active tracers and the momentum. Explicit, split-explicit 54 54 and filtered free surface formulations are implemented. … … 57 57 order advection schemes, including positive ones). 58 58 59 Surface boundary conditions ( chapter~\ref{SBC}) can be implemented as prescribed59 Surface boundary conditions (\autoref{chap:SBC}) can be implemented as prescribed 60 60 fluxes, or bulk formulations for the surface fluxes (wind stress, heat, freshwater). The 61 61 model allows penetration of solar radiation There is an optional geothermal heating at … … 69 69 is still not available. 70 70 71 Other model characteristics are the lateral boundary conditions ( chapter~\ref{LBC}).71 Other model characteristics are the lateral boundary conditions (\autoref{chap:LBC}). 72 72 Global configurations of the model make use of the ORCA tripolar grid, with special north 73 73 fold boundary condition. Free-slip or no-slip boundary conditions are allowed at land … … 75 75 conditions are possible. 76 76 77 Physical parameterisations are described in chapters~\ref{LDF} and \ref{ZDF}. The77 Physical parameterisations are described in \autoref{chap:LDF} and \autoref{chap:ZDF}. The 78 78 model includes an implicit treatment of vertical viscosity and diffusivity. The lateral 79 79 Laplacian and biharmonic viscosity and diffusion can be rotated following a geopotential … … 112 112 %%gm end 113 113 114 Model outputs management and specific online diagnostics are described in chapters~\ref{DIA}.114 Model outputs management and specific online diagnostics are described in \autoref{chap:DIA}. 115 115 The diagnostics includes the output of all the tendencies of the momentum and tracers equations, 116 116 the output of tracers tendencies averaged over the time evolving mixed layer, the output of 117 117 the tendencies of the barotropic vorticity equation, the computation of on-line floats trajectories... 118 Chapter~\ref{OBS} describes a tool which reads in observation files (profile temperature118 \autoref{chap:OBS} describes a tool which reads in observation files (profile temperature 119 119 and salinity, sea surface temperature, sea level anomaly and sea ice concentration) 120 120 and calculates an interpolated model equivalent value at the observation location 121 121 and nearest model timestep. Originally developed of data assimilation, it is a fantastic 122 tool for model and data comparison. Chapter~\ref{ASM} describes how increments122 tool for model and data comparison. \autoref{chap:ASM} describes how increments 123 123 produced by data assimilation may be applied to the model equations. 124 Finally, Chapter~\ref{CFG} provides a brief introduction to the pre-defined model124 Finally, \autoref{chap:CFG} provides a brief introduction to the pre-defined model 125 125 configurations (water column model, ORCA and GYRE families of configurations). 126 126 … … 132 132 include conventions for naming variables, with different starting letters for different types 133 133 of variables (real, integer, parameter\ldots). Those rules are briefly presented in 134 Appendix~\ref{Apdx_D} and a more complete document is available on the \NEMO web site.134 \autoref{apdx:D} and a more complete document is available on the \NEMO web site. 135 135 136 136 The model is organized with a high internal modularity based on physics. For example, … … 139 139 around the code, the module names follow a three-letter rule. For example, \mdl{traldf} 140 140 is a module related to the TRAcers equation, computing the Lateral DiFfussion. 141 %The complete list of module names is presented in Appendix~\ref{Apdx_D}. %====>>>> to be done !141 %The complete list of module names is presented in \autoref{apdx:D}. %====>>>> to be done ! 142 142 Furthermore, modules are organized in a few directories that correspond to their category, 143 as indicated by the first three letters of their name ( Tab.~\ref{Tab_chap}).143 as indicated by the first three letters of their name (\autoref{tab:chap}). 144 144 145 145 The manual mirrors the organization of the model. 146 After the presentation of the continuous equations ( Chapter \ref{PE}), the following chapters147 refer to specific terms of the equations each associated with a group of modules ( Tab.~\ref{Tab_chap}).146 After the presentation of the continuous equations (\autoref{chap:PE}), the following chapters 147 refer to specific terms of the equations each associated with a group of modules (\autoref{tab:chap}). 148 148 149 149 … … 151 151 \begin{table}[!t] 152 152 %\begin{center} \begin{tabular}{|p{143pt}|l|l|} \hline 153 \caption{ \protect\label{ Tab_chap} Organization of Chapters mimicking the one of the model directories. }153 \caption{ \protect\label{tab:chap} Organization of Chapters mimicking the one of the model directories. } 154 154 \begin{center} \begin{tabular}{|l|l|l|} \hline 155 Chapter \ref{STP} & - & model time STePping environment \\ \hline156 Chapter \ref{DOM} & DOM & model DOMain \\ \hline157 Chapter \ref{TRA} & TRA & TRAcer equations (potential temperature and salinity) \\ \hline158 Chapter \ref{DYN} & DYN & DYNamic equations (momentum) \\ \hline159 Chapter \ref{SBC} & SBC & Surface Boundary Conditions \\ \hline160 Chapter \ref{LBC} & LBC & Lateral Boundary Conditions (also OBC and BDY) \\ \hline161 Chapter \ref{LDF} & LDF & Lateral DiFfusion (parameterisations) \\ \hline162 Chapter \ref{ZDF} & ZDF & vertical (Z) DiFfusion (parameterisations) \\ \hline163 Chapter \ref{DIA} & DIA & I/O and DIAgnostics (also IOM, FLO and TRD) \\ \hline164 Chapter \ref{OBS} & OBS & OBServation and model comparison \\ \hline165 Chapter \ref{ASM} & ASM & ASsiMilation increment \\ \hline166 Chapter \ref{MISC} & SOL & Miscellaneous topics (including solvers) \\ \hline167 Chapter \ref{CFG} & - & predefined configurations (including C1D) \\ \hline155 \autoref{chap:STP} & - & model time STePping environment \\ \hline 156 \autoref{chap:DOM} & DOM & model DOMain \\ \hline 157 \autoref{chap:TRA} & TRA & TRAcer equations (potential temperature and salinity) \\ \hline 158 \autoref{chap:DYN} & DYN & DYNamic equations (momentum) \\ \hline 159 \autoref{chap:SBC} & SBC & Surface Boundary Conditions \\ \hline 160 \autoref{chap:LBC} & LBC & Lateral Boundary Conditions (also OBC and BDY) \\ \hline 161 \autoref{chap:LDF} & LDF & Lateral DiFfusion (parameterisations) \\ \hline 162 \autoref{chap:ZDF} & ZDF & vertical (Z) DiFfusion (parameterisations) \\ \hline 163 \autoref{chap:DIA} & DIA & I/O and DIAgnostics (also IOM, FLO and TRD) \\ \hline 164 \autoref{chap:OBS} & OBS & OBServation and model comparison \\ \hline 165 \autoref{chap:ASM} & ASM & ASsiMilation increment \\ \hline 166 \autoref{chap:MISC} & SOL & Miscellaneous topics (including solvers) \\ \hline 167 \autoref{chap:CFG} & - & predefined configurations (including C1D) \\ \hline 168 168 \end{tabular} 169 169 \end{center} \end{table}
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