Changeset 11543 for NEMO/trunk
- Timestamp:
- 2019-09-13T15:57:52+02:00 (5 years ago)
- Location:
- NEMO/trunk/doc/latex
- Files:
-
- 19 edited
- 7 moved
Legend:
- Unmodified
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NEMO/trunk/doc/latex/NEMO/main/appendices.tex
r11330 r11543 1 1 2 \subfile{../subfiles/a nnex_A}%% Generalised vertical coordinate3 \subfile{../subfiles/a nnex_B} %% Diffusive operator4 \subfile{../subfiles/a nnex_C}%% Discrete invariants of the eqs.5 \subfile{../subfiles/a nnex_iso}%% Isoneutral diffusion using triads6 \subfile{../subfiles/a nnex_DOMAINcfg}%% Brief notes on DOMAINcfg2 \subfile{../subfiles/apdx_s_coord} %% Generalised vertical coordinate 3 \subfile{../subfiles/apdx_diff_opers} %% Diffusive operators 4 \subfile{../subfiles/apdx_invariants} %% Discrete invariants of the eqs. 5 \subfile{../subfiles/apdx_triads} %% Isoneutral diffusion using triads 6 \subfile{../subfiles/apdx_DOMAINcfg} %% Brief notes on DOMAINcfg 7 7 8 8 %% Not included … … 10 10 %\subfile{../subfiles/chap_DIU} 11 11 %\subfile{../subfiles/chap_conservation} 12 %\subfile{../subfiles/annex_E} %% Notes on some on going staff 13 12 %\subfile{../subfiles/apdx_algos} %% Notes on some on going staff -
NEMO/trunk/doc/latex/NEMO/main/chapters.tex
r11522 r11543 13 13 \subfile{../subfiles/chap_STO} %% Stochastic param. 14 14 \subfile{../subfiles/chap_misc} %% Miscellaneous topics 15 \subfile{../subfiles/chap_ CONFIG}%% Predefined configurations15 \subfile{../subfiles/chap_cfgs} %% Predefined configurations 16 16 17 17 %% Not included -
NEMO/trunk/doc/latex/NEMO/main/introduction.tex
r11522 r11543 56 56 57 57 \begin{description} 58 \item [\nameref{chap: PE}] presents the equations and their assumptions, the vertical coordinates used,58 \item [\nameref{chap:MB}] presents the equations and their assumptions, the vertical coordinates used, 59 59 and the subgrid scale physics. 60 60 The equations are written in a curvilinear coordinate system, with a choice of vertical coordinates … … 63 63 Dimensional units in the meter, kilogram, second (MKS) international system are used throughout. 64 64 The following chapters deal with the discrete equations. 65 \item [\nameref{chap: STP}] presents the model time stepping environment.65 \item [\nameref{chap:TD}] presents the model time stepping environment. 66 66 it is a three level scheme in which the tendency terms of the equations are evaluated either 67 67 centered in time, or forward, or backward depending of the nature of the term. … … 123 123 \item [\nameref{chap:ASM}] describes how increments produced by 124 124 data \textbf{A}s\textbf{S}i\textbf{M}ilation may be applied to the model equations. 125 \item [\nameref{chap:STO}] 125 126 \item [\nameref{chap:MISC}] (including solvers) 126 \item [\nameref{chap:CFG }] provides finally a brief introduction to127 \item [\nameref{chap:CFGS}] provides finally a brief introduction to 127 128 the pre-defined model configurations 128 129 (water column model \texttt{C1D}, ORCA and GYRE families of configurations). … … 133 134 134 135 \begin{description} 135 \item [\nameref{apdx: s_coord}]136 \item [\nameref{apdx: diff_oper}]137 \item [\nameref{apdx: invariants}]138 \item [\nameref{apdx: triads}]139 \item [\nameref{apdx:DOM AINcfg}]140 \item [\nameref{apdx: coding}]136 \item [\nameref{apdx:SCOORD}] 137 \item [\nameref{apdx:DIFFOPERS}] 138 \item [\nameref{apdx:INVARIANTS}] 139 \item [\nameref{apdx:TRIADS}] 140 \item [\nameref{apdx:DOMCFG}] 141 \item [\nameref{apdx:CODING}] 141 142 \end{description} -
NEMO/trunk/doc/latex/NEMO/subfiles/apdx_DOMAINcfg.tex
r11529 r11543 6 6 % ================================================================ 7 7 \chapter{A brief guide to the DOMAINcfg tool} 8 \label{apdx:DOM AINcfg}8 \label{apdx:DOMCFG} 9 9 10 10 \chaptertoc … … 121 121 The reference coordinate transformation $z_0(k)$ defines the arrays $gdept_0$ and 122 122 $gdepw_0$ for $t$- and $w$-points, respectively. See \autoref{sec:DOMCFG_sco} for the 123 S-coordinate options. As indicated on \autoref{fig: index_vert} \jp{jpk} is the number of123 S-coordinate options. As indicated on \autoref{fig:DOM_index_vert} \jp{jpk} is the number of 124 124 $w$-levels. $gdepw_0(1)$ is the ocean surface. There are at most \jp{jpk}-1 $t$-points 125 125 inside the ocean, the additional $t$-point at $jk = jpk$ is below the sea floor and is not … … 421 421 The depth field $h$ is not necessary the ocean depth, 422 422 since a mixed step-like and bottom-following representation of the topography can be used 423 (\autoref{fig: z_zps_s_sps}) or an envelop bathymetry can be defined (\autoref{fig:z_zps_s_sps}).423 (\autoref{fig:DOM_z_zps_s_sps}) or an envelop bathymetry can be defined (\autoref{fig:DOM_z_zps_s_sps}). 424 424 The namelist parameter \np{rn\_rmax} determines the slope at which 425 425 the terrain-following coordinate intersects the sea bed and becomes a pseudo z-coordinate. … … 436 436 \[ 437 437 z = s_{min} + C (s) (H - s_{min}) 438 % \label{eq: SH94_1}438 % \label{eq:DOMCFG_SH94_1} 439 439 \] 440 440 … … 458 458 + b \frac{\tanh \lt[ \theta \lt(s + \frac{1}{2} \rt) \rt] - \tanh \lt( \frac{\theta}{2} \rt)} 459 459 { 2 \tanh \lt( \frac{\theta}{2} \rt)} 460 \label{eq: SH94_2}460 \label{eq:DOMCFG_SH94_2} 461 461 \] 462 462 … … 466 466 \includegraphics[width=\textwidth]{Fig_sco_function} 467 467 \caption{ 468 \protect\label{fig: sco_function}468 \protect\label{fig:DOMCFG_sco_function} 469 469 Examples of the stretching function applied to a seamount; 470 470 from left to right: surface, surface and bottom, and bottom intensified resolutions … … 478 478 bottom control parameters such that $0 \leqslant \theta \leqslant 20$, and $0 \leqslant b \leqslant 1$. 479 479 $b$ has been designed to allow surface and/or bottom increase of the vertical resolution 480 (\autoref{fig: sco_function}).480 (\autoref{fig:DOMCFG_sco_function}). 481 481 482 482 Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows a fixed surface resolution in … … 486 486 \begin{equation} 487 487 z = - \gamma h \quad \text{with} \quad 0 \leq \gamma \leq 1 488 % \label{eq: z}488 % \label{eq:DOMCFG_z} 489 489 \end{equation} 490 490 … … 524 524 For clarity every third coordinate surface is shown. 525 525 } 526 \label{fig: fig_compare_coordinates_surface}526 \label{fig:DOMCFG_fig_compare_coordinates_surface} 527 527 \end{figure} 528 528 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> -
NEMO/trunk/doc/latex/NEMO/subfiles/apdx_algos.tex
r11529 r11543 6 6 % ================================================================ 7 7 \chapter{Note on some algorithms} 8 \label{apdx: E}8 \label{apdx:ALGOS} 9 9 10 10 \chaptertoc … … 12 12 \newpage 13 13 14 This appendix some on going consideration on algorithms used or planned to be used in \NEMO. 14 This appendix some on going consideration on algorithms used or planned to be used in \NEMO. 15 15 16 16 % ------------------------------------------------------------------------------------------------------------- 17 % UBS scheme 17 % UBS scheme 18 18 % ------------------------------------------------------------------------------------------------------------- 19 19 \section{Upstream Biased Scheme (UBS) (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})} … … 45 45 a constant i-grid spacing ($\Delta i=1$). 46 46 47 Alternative choice: introduce the scale factors: 47 Alternative choice: introduce the scale factors: 48 48 $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta_{i+1/2}[\tau] \right]$. 49 49 … … 54 54 It is not a \emph{positive} scheme meaning false extrema are permitted but 55 55 the amplitude of such are significantly reduced over the centred second order method. 56 Nevertheless it is not recommended to apply it to a passive tracer that requires positivity. 56 Nevertheless it is not recommended to apply it to a passive tracer that requires positivity. 57 57 58 58 The intrinsic diffusion of UBS makes its use risky in the vertical direction where … … 61 61 \np{ln\_traadv\_ubs}\forcode{ = .true.}. 62 62 63 For stability reasons, in \autoref{eq: tra_adv_ubs}, the first term which corresponds to63 For stability reasons, in \autoref{eq:TRA_adv_ubs}, the first term which corresponds to 64 64 a second order centred scheme is evaluated using the \textit{now} velocity (centred in time) while 65 65 the second term which is the diffusive part of the scheme, is evaluated using the \textit{before} velocity … … 67 67 This is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the Quick advection scheme. 68 68 UBS and QUICK schemes only differ by one coefficient. 69 Substituting 1/6 with 1/8 in (\autoref{eq: tra_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}.69 Substituting 1/6 with 1/8 in (\autoref{eq:TRA_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 70 70 This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded. 71 71 Nevertheless it is quite easy to make the substitution in \mdl{traadv\_ubs} module and obtain a QUICK scheme. … … 75 75 Computer time can be saved by using a time-splitting technique on vertical advection. 76 76 This possibility have been implemented and validated in ORCA05-L301. 77 It is not currently offered in the current reference version. 77 It is not currently offered in the current reference version. 78 78 79 79 NB 2: In a forthcoming release four options will be proposed for the vertical component used in the UBS scheme. … … 83 83 The $3^{rd}$ case has dispersion properties similar to an eight-order accurate conventional scheme. 84 84 85 NB 3: It is straight forward to rewrite \autoref{eq: tra_adv_ubs} as follows:85 NB 3: It is straight forward to rewrite \autoref{eq:TRA_adv_ubs} as follows: 86 86 \begin{equation} 87 87 \label{eq:tra_adv_ubs2} … … 93 93 \right. 94 94 \end{equation} 95 or equivalently 95 or equivalently 96 96 \begin{equation} 97 97 \label{eq:tra_adv_ubs2} … … 102 102 \end{split} 103 103 \end{equation} 104 \autoref{eq: tra_adv_ubs2} has several advantages.104 \autoref{eq:TRA_adv_ubs2} has several advantages. 105 105 First it clearly evidences that the UBS scheme is based on the fourth order scheme to which 106 106 is added an upstream biased diffusive term. 107 107 Second, this emphasises that the $4^{th}$ order part have to be evaluated at \emph{now} time step, 108 not only the $2^{th}$ order part as stated above using \autoref{eq: tra_adv_ubs}.108 not only the $2^{th}$ order part as stated above using \autoref{eq:TRA_adv_ubs}. 109 109 Third, the diffusive term is in fact a biharmonic operator with a eddy coefficient which 110 110 is simply proportional to the velocity. … … 134 134 \end{split} 135 135 \end{equation} 136 with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$, 136 with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$, 137 137 \ie\ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$ 138 138 it comes: … … 189 189 190 190 % ------------------------------------------------------------------------------------------------------------- 191 % Leap-Frog energetic 191 % Leap-Frog energetic 192 192 % ------------------------------------------------------------------------------------------------------------- 193 193 \section{Leapfrog energetic} … … 214 214 \equiv \frac{1}{\rdt} \overline{ \delta_{t+\rdt/2}[q]}^{\,t} 215 215 = \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} 216 \] 216 \] 217 217 Note that \autoref{chap:LF} shows that the leapfrog time step is $\rdt$, 218 218 not $2\rdt$ as it can be found sometimes in literature. … … 226 226 \] 227 227 is satisfied in discrete form. 228 Indeed, 228 Indeed, 229 229 \[ 230 230 \begin{split} … … 240 240 \] 241 241 NB here pb of boundary condition when applying the adjoint! 242 In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition 242 In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition 243 243 (equivalently of the boundary value of the integration by part). 244 244 In time this boundary condition is not physical and \textbf{add something here!!!} 245 245 246 246 % ================================================================ 247 % Iso-neutral diffusion : 247 % Iso-neutral diffusion : 248 248 % ================================================================ 249 249 … … 251 251 252 252 % ================================================================ 253 % Griffies' iso-neutral diffusion operator : 253 % Griffies' iso-neutral diffusion operator : 254 254 % ================================================================ 255 255 \subsection{Griffies iso-neutral diffusion operator} … … 258 258 but is formulated within the \NEMO\ framework 259 259 (\ie\ using scale factors rather than grid-size and having a position of $T$-points that 260 is not necessary in the middle of vertical velocity points, see \autoref{fig: zgr_e3}).261 262 In the formulation \autoref{eq: tra_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO,260 is not necessary in the middle of vertical velocity points, see \autoref{fig:DOM_zgr_e3}). 261 262 In the formulation \autoref{eq:TRA_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO, 263 263 the off-diagonal terms of the small angle diffusion tensor contain several double spatial averages of a gradient, 264 264 for example $\overline{\overline{\delta_k \cdot}}^{\,i,k}$. … … 318 318 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 319 319 320 The four iso-neutral fluxes associated with the triads are defined at $T$-point. 320 The four iso-neutral fluxes associated with the triads are defined at $T$-point. 321 321 They take the following expression: 322 322 \begin{flalign*} … … 332 332 333 333 The resulting iso-neutral fluxes at $u$- and $w$-points are then given by 334 the sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig: ISO_triad}):334 the sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:TRIADS_ISO_triad}): 335 335 \begin{flalign} 336 336 \label{eq:iso_flux} … … 369 369 + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} 370 370 \end{equation} 371 where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. 371 where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. 372 372 373 373 This expression of the iso-neutral diffusion has been chosen in order to satisfy the following six properties: … … 448 448 449 449 % ================================================================ 450 % Skew flux formulation for Eddy Induced Velocity : 450 % Skew flux formulation for Eddy Induced Velocity : 451 451 % ================================================================ 452 452 \subsection{Eddy induced velocity and skew flux formulation} … … 457 457 the formulation of which depends on the slopes of iso-neutral surfaces. 458 458 Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, 459 \ie\ \autoref{eq: ldfslp_geo} is used in $z$-coordinate,460 and the sum \autoref{eq: ldfslp_geo} + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates.461 462 The eddy induced velocity is given by: 459 \ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinate, 460 and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $z^*$ or $s$-coordinates. 461 462 The eddy induced velocity is given by: 463 463 \begin{equation} 464 464 \label{eq:eiv_v} … … 484 484 (see \autoref{sec:TRA_adv}) and not just a $2^{nd}$ order advection scheme. 485 485 This is particularly useful for passive tracers where 486 \emph{positivity} of the advection scheme is of paramount importance. 487 % give here the expression using the triads. It is different from the one given in \autoref{eq: ldfeiv}486 \emph{positivity} of the advection scheme is of paramount importance. 487 % give here the expression using the triads. It is different from the one given in \autoref{eq:LDF_eiv} 488 488 % see just below a copy of this equation: 489 489 %\begin{equation} \label{eq:ldfeiv} … … 593 593 \right) 594 594 \end{equation} 595 Note that \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells. 595 Note that \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells. 596 596 In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces must be added to 597 $\mathbb{R}$ for the discret form to be exact. 597 $\mathbb{R}$ for the discret form to be exact. 598 598 599 599 Such a choice of discretisation is consistent with the iso-neutral operator as … … 604 604 $\ $\newpage %force an empty line 605 605 % ================================================================ 606 % Discrete Invariants of the iso-neutral diffrusion 606 % Discrete Invariants of the iso-neutral diffrusion 607 607 % ================================================================ 608 608 \subsection{Discrete invariants of the iso-neutral diffrusion} 609 609 \label{subsec:Gf_operator} 610 610 611 Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane. 611 Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane. 612 612 613 613 This part will be moved in an Appendix. … … 617 617 \int_D D_l^T \; T \;dv \leq 0 618 618 \] 619 The discrete form of its left hand side is obtained using \autoref{eq: iso_flux}619 The discrete form of its left hand side is obtained using \autoref{eq:TRIADS_iso_flux} 620 620 621 621 \begin{align*} … … 740 740 \right\} 741 741 \quad \leq 0 742 \end{align*} 742 \end{align*} 743 743 The last inequality is obviously obtained as we succeed in obtaining a negative summation of square quantities. 744 744 … … 764 764 % 765 765 &\equiv \sum_{i,k} \left\{ D_l^S \ T \ b_T \right\} 766 \end{align*} 766 \end{align*} 767 767 This means that the iso-neutral operator is self-adjoint. 768 768 There is no need to develop a specific to obtain it. … … 776 776 \label{subsec:eiv_skew} 777 777 778 Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. 778 Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. 779 779 780 780 This have to be moved in an Appendix. … … 830 830 &{\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{i+1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] 831 831 &\Bigr\} \\ 832 \end{matrix} 832 \end{matrix} 833 833 \end{align*} 834 834 The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the same but of opposite signs, 835 they cancel out. 835 they cancel out. 836 836 Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{-1/2}}$. 837 837 The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{-1/2}}$ are the same but both of opposite signs and -
NEMO/trunk/doc/latex/NEMO/subfiles/apdx_diff_opers.tex
r11529 r11543 5 5 % Chapter Appendix B : Diffusive Operators 6 6 % ================================================================ 7 \chapter{ Appendix B :Diffusive Operators}8 \label{apdx: B}7 \chapter{Diffusive Operators} 8 \label{apdx:DIFFOPERS} 9 9 10 10 \chaptertoc … … 16 16 % ================================================================ 17 17 \section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators} 18 \label{sec: B_1}18 \label{sec:DIFFOPERS_1} 19 19 20 20 \subsubsection*{In z-coordinates} … … 22 22 In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by: 23 23 \begin{align} 24 \label{ apdx:B1}24 \label{eq:DIFFOPERS_1} 25 25 &D^T = \frac{1}{e_1 \, e_2} \left[ 26 26 \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. … … 32 32 \subsubsection*{In generalized vertical coordinates} 33 33 34 In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{ apdx:A_s_slope} and34 In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{eq:SCOORD_s_slope} and 35 35 the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$. 36 36 The diffusion operator is given by: 37 37 38 38 \begin{equation} 39 \label{ apdx:B2}39 \label{eq:DIFFOPERS_2} 40 40 D^T = \left. \nabla \right|_s \cdot 41 41 \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ … … 54 54 \begin{array}{*{20}l} 55 55 D^T= \frac{1}{e_1\,e_2\,e_3 } & \left\{ \quad \quad \frac{\partial }{\partial i} \left. \left[ e_2\,e_3 \, A^{lT} 56 \left( \ \frac{1}{e_1}\; \left. \frac{\partial T}{\partial i} \right|_s 56 \left( \ \frac{1}{e_1}\; \left. \frac{\partial T}{\partial i} \right|_s 57 57 -\frac{\sigma_1 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right] \right|_s \right. \\ 58 58 & \quad \ + \ \left. \frac{\partial }{\partial j} \left. \left[ e_1\,e_3 \, A^{lT} 59 \left( \ \frac{1}{e_2 }\; \left. \frac{\partial T}{\partial j} \right|_s 59 \left( \ \frac{1}{e_2 }\; \left. \frac{\partial T}{\partial j} \right|_s 60 60 -\frac{\sigma_2 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right] \right|_s \right. \\ 61 & \quad \ + \ \left. e_1\,e_2\, \frac{\partial }{\partial s} \left[ A^{lT} \; \left( 62 -\frac{\sigma_1 }{e_1 } \; \left. \frac{\partial T}{\partial i} \right|_s 63 -\frac{\sigma_2 }{e_2 } \; \left. \frac{\partial T}{\partial j} \right|_s 61 & \quad \ + \ \left. e_1\,e_2\, \frac{\partial }{\partial s} \left[ A^{lT} \; \left( 62 -\frac{\sigma_1 }{e_1 } \; \left. \frac{\partial T}{\partial i} \right|_s 63 -\frac{\sigma_2 }{e_2 } \; \left. \frac{\partial T}{\partial j} \right|_s 64 64 +\left( \varepsilon +\sigma_1^2+\sigma_2 ^2 \right) \; \frac{1}{e_3 } \; \frac{\partial T}{\partial s} \right) \; \right] \; \right\} . 65 65 \end{array} … … 67 67 \end{align*} 68 68 69 \autoref{ apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption.69 \autoref{eq:DIFFOPERS_2} is obtained from \autoref{eq:DIFFOPERS_1} without any additional assumption. 70 70 Indeed, for the special case $k=z$ and thus $e_3 =1$, 71 we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx: A} and72 use \autoref{ apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}.73 Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{ apdx:B1},71 we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:SCOORD} and 72 use \autoref{eq:SCOORD_s_slope} and \autoref{eq:SCOORD_s_chain_rule}. 73 Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{eq:DIFFOPERS_1}, 74 74 the ($i$,$z$) and ($j$,$z$) planes are independent. 75 75 The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without … … 160 160 % ================================================================ 161 161 \section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators} 162 \label{sec: B_2}162 \label{sec:DIFFOPERS_2} 163 163 164 164 \subsubsection*{In z-coordinates} … … 170 170 171 171 \begin{equation} 172 \label{ apdx:B3}172 \label{eq:DIFFOPERS_3} 173 173 \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} 174 174 \left[ {{ … … 193 193 194 194 In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean, 195 so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0) 195 so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0) 196 196 and (0,1) elements which are set to zero. See \citet{griffies_bk04}, section 14.1.4.1 for a discussion of this point.}: 197 197 \begin{subequations} 198 \label{ apdx:B4}198 \label{eq:DIFFOPERS_4} 199 199 \begin{equation} 200 \label{ apdx:B4a}200 \label{eq:DIFFOPERS_4a} 201 201 {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = 202 202 \left[ {{ … … 210 210 and the iso/dianeutral diffusive operator in $z$-coordinates is then 211 211 \begin{equation} 212 \label{ apdx:B4b}212 \label{eq:DIFFOPERS_4b} 213 213 D^T = \left. \nabla \right|_z \cdot 214 214 \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T \right]. \\ … … 216 216 \end{subequations} 217 217 218 Physically, the full tensor \autoref{ apdx:B3} represents strong isoneutral diffusion on a plane parallel to218 Physically, the full tensor \autoref{eq:DIFFOPERS_3} represents strong isoneutral diffusion on a plane parallel to 219 219 the isoneutral surface and weak dianeutral diffusion perpendicular to this plane. 220 220 However, 221 the approximate `weak-slope' tensor \autoref{ apdx:B4a} represents strong diffusion along the isoneutral surface,221 the approximate `weak-slope' tensor \autoref{eq:DIFFOPERS_4a} represents strong diffusion along the isoneutral surface, 222 222 with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal. 223 223 This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor. 224 The weak-slope operator therefore takes the same form, \autoref{ apdx:B4}, as \autoref{apdx:B2},224 The weak-slope operator therefore takes the same form, \autoref{eq:DIFFOPERS_4}, as \autoref{eq:DIFFOPERS_2}, 225 225 the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates. 226 226 Written out explicitly, 227 227 228 228 \begin{multline} 229 \label{ apdx:B_ldfiso}229 \label{eq:DIFFOPERS_ldfiso} 230 230 D^T=\frac{1}{e_1 e_2 }\left\{ 231 231 {\;\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]} … … 234 234 \end{multline} 235 235 236 The isopycnal diffusion operator \autoref{ apdx:B4},237 \autoref{ apdx:B_ldfiso} conserves tracer quantity and dissipates its square.238 As \autoref{ apdx:B4} is the divergence of a flux, the demonstration of the first property is trivial, providing that the flux normal to the boundary is zero236 The isopycnal diffusion operator \autoref{eq:DIFFOPERS_4}, 237 \autoref{eq:DIFFOPERS_ldfiso} conserves tracer quantity and dissipates its square. 238 As \autoref{eq:DIFFOPERS_4} is the divergence of a flux, the demonstration of the first property is trivial, providing that the flux normal to the boundary is zero 239 239 (as it is when $A_h$ is zero at the boundary). Let us demonstrate the second one: 240 240 \[ … … 256 256 j}-a_2 \frac{\partial T}{\partial k}} \right)^2} 257 257 +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right] \\ 258 & \geq 0 . 258 & \geq 0 . 259 259 \end{array} 260 260 } … … 265 265 \subsubsection*{In generalized vertical coordinates} 266 266 267 Because the weak-slope operator \autoref{ apdx:B4},268 \autoref{ apdx:B_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes,267 Because the weak-slope operator \autoref{eq:DIFFOPERS_4}, 268 \autoref{eq:DIFFOPERS_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes, 269 269 it may be transformed into generalized $s$-coordinates in the same way as 270 \autoref{sec: B_1} was transformed into \autoref{sec:B_2}.270 \autoref{sec:DIFFOPERS_1} was transformed into \autoref{sec:DIFFOPERS_2}. 271 271 The resulting operator then takes the simple form 272 272 273 273 \begin{equation} 274 \label{ apdx:B_ldfiso_s}274 \label{eq:DIFFOPERS_ldfiso_s} 275 275 D^T = \left. \nabla \right|_s \cdot 276 276 \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ … … 295 295 \] 296 296 297 To prove \autoref{ apdx:B_ldfiso_s} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious.298 An easier way is first to note (by reversing the derivation of \autoref{sec: B_2} from \autoref{sec:B_1} ) that297 To prove \autoref{eq:DIFFOPERS_ldfiso_s} by direct re-expression of \autoref{eq:DIFFOPERS_ldfiso} is straightforward, but laborious. 298 An easier way is first to note (by reversing the derivation of \autoref{sec:DIFFOPERS_2} from \autoref{sec:DIFFOPERS_1} ) that 299 299 the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as 300 300 301 301 \begin{equation} 302 \label{ apdx:B5}302 \label{eq:DIFFOPERS_5} 303 303 D^T = \left. \nabla \right|_\rho \cdot 304 304 \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T \right] \\ … … 312 312 \end{equation} 313 313 Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives 314 \autoref{ apdx:B_ldfiso_s} immediately.314 \autoref{eq:DIFFOPERS_ldfiso_s} immediately. 315 315 316 316 Note that the weak-slope approximation is only made in transforming from … … 318 318 The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces, 319 319 in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in 320 \autoref{sec: B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.320 \autoref{sec:DIFFOPERS_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. 321 321 322 322 … … 325 325 % ================================================================ 326 326 \section{Lateral/Vertical momentum diffusive operators} 327 \label{sec: B_3}327 \label{sec:DIFFOPERS_3} 328 328 329 329 The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by 330 applying \autoref{eq: PE_lap_vector}, the expression for the Laplacian of a vector,330 applying \autoref{eq:MB_lap_vector}, the expression for the Laplacian of a vector, 331 331 to the horizontal velocity vector: 332 332 \begin{align*} … … 371 371 }} \right) 372 372 \end{align*} 373 Using \autoref{eq: PE_div}, the definition of the horizontal divergence,373 Using \autoref{eq:MB_div}, the definition of the horizontal divergence, 374 374 the third component of the second vector is obviously zero and thus : 375 375 \[ 376 \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) . 376 \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) . 377 377 \] 378 378 379 379 Note that this operator ensures a full separation between 380 the vorticity and horizontal divergence fields (see \autoref{apdx: C}).380 the vorticity and horizontal divergence fields (see \autoref{apdx:INVARIANTS}). 381 381 It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere. 382 382 … … 384 384 the $z$-coordinate therefore takes the following form: 385 385 \begin{equation} 386 \label{ apdx:B_Lap_U}386 \label{eq:DIFFOPERS_Lap_U} 387 387 { 388 388 \textbf{D}}^{\textbf{U}} = … … 404 404 \end{align*} 405 405 406 Note Bene: introducing a rotation in \autoref{ apdx:B_Lap_U} does not lead to406 Note Bene: introducing a rotation in \autoref{eq:DIFFOPERS_Lap_U} does not lead to 407 407 a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. 408 408 Similarly, we did not found an expression of practical use for 409 409 the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate. 410 Generally, \autoref{ apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems,410 Generally, \autoref{eq:DIFFOPERS_Lap_U} is used in both $z$- and $s$-coordinate systems, 411 411 that is a Laplacian diffusion is applied on momentum along the coordinate directions. 412 412 -
NEMO/trunk/doc/latex/NEMO/subfiles/apdx_invariants.tex
r11529 r11543 6 6 % ================================================================ 7 7 \chapter{Discrete Invariants of the Equations} 8 \label{apdx: C}8 \label{apdx:INVARIANTS} 9 9 10 10 \chaptertoc … … 21 21 % ================================================================ 22 22 \section{Introduction / Notations} 23 \label{sec: C.0}23 \label{sec:INVARIANTS_0} 24 24 25 25 Notation used in this appendix in the demonstations: … … 72 72 that is in a more compact form : 73 73 \begin{flalign} 74 \label{eq: Q2_flux}74 \label{eq:INVARIANTS_Q2_flux} 75 75 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 76 76 =& \int_D { \frac{Q}{e_3} \partial_t \left( e_3 \, Q \right) dv } … … 87 87 that is in a more compact form: 88 88 \begin{flalign} 89 \label{eq: Q2_vect}89 \label{eq:INVARIANTS_Q2_vect} 90 90 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 91 91 =& \int_D { Q \,\partial_t Q \;dv } … … 97 97 % ================================================================ 98 98 \section{Continuous conservation} 99 \label{sec: C.1}99 \label{sec:INVARIANTS_1} 100 100 101 101 The discretization of pimitive equation in $s$-coordinate (\ie\ time and space varying vertical coordinate) … … 105 105 The total energy (\ie\ kinetic plus potential energies) is conserved: 106 106 \begin{flalign} 107 \label{eq: Tot_Energy}107 \label{eq:INVARIANTS_Tot_Energy} 108 108 \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 + \rho \, g \, z \right) \;dv \right) = & 0 109 109 \end{flalign} … … 114 114 The transformation for the advection term depends on whether the vector invariant form or 115 115 the flux form is used for the momentum equation. 116 Using \autoref{eq: Q2_vect} and introducing \autoref{apdx:A_dyn_vect} in117 \autoref{eq: Tot_Energy} for the former form and118 using \autoref{eq: Q2_flux} and introducing \autoref{apdx:A_dyn_flux} in119 \autoref{eq: Tot_Energy} for the latter form leads to:120 121 % \label{eq: E_tot}116 Using \autoref{eq:INVARIANTS_Q2_vect} and introducing \autoref{eq:SCOORD_dyn_vect} in 117 \autoref{eq:INVARIANTS_Tot_Energy} for the former form and 118 using \autoref{eq:INVARIANTS_Q2_flux} and introducing \autoref{eq:SCOORD_dyn_flux} in 119 \autoref{eq:INVARIANTS_Tot_Energy} for the latter form leads to: 120 121 % \label{eq:INVARIANTS_E_tot} 122 122 advection term (vector invariant form): 123 123 \[ 124 % \label{eq: E_tot_vect_vor_1}124 % \label{eq:INVARIANTS_E_tot_vect_vor_1} 125 125 \int\limits_D \zeta \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ 126 126 \] 127 127 % 128 128 \[ 129 % \label{eq: E_tot_vect_adv_1}129 % \label{eq:INVARIANTS_E_tot_vect_adv_1} 130 130 \int\limits_D \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right) dv 131 131 + \int\limits_D \textbf{U}_h \cdot \nabla_z \textbf{U}_h \;dv … … 134 134 advection term (flux form): 135 135 \[ 136 % \label{eq: E_tot_flux_metric}136 % \label{eq:INVARIANTS_E_tot_flux_metric} 137 137 \int\limits_D \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1 \right)\; 138 138 \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 139 139 \] 140 140 \[ 141 % \label{eq: E_tot_flux_adv}141 % \label{eq:INVARIANTS_E_tot_flux_adv} 142 142 \int\limits_D \textbf{U}_h \cdot \left( {{ 143 143 \begin{array} {*{20}c} … … 150 150 coriolis term 151 151 \[ 152 % \label{eq: E_tot_cor}152 % \label{eq:INVARIANTS_E_tot_cor} 153 153 \int\limits_D f \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 154 154 \] 155 155 pressure gradient: 156 156 \[ 157 % \label{eq: E_tot_pg_1}157 % \label{eq:INVARIANTS_E_tot_pg_1} 158 158 - \int\limits_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 159 159 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv … … 173 173 174 174 Vector invariant form: 175 % \label{eq: E_tot_vect}176 \[ 177 % \label{eq: E_tot_vect_vor_2}175 % \label{eq:INVARIANTS_E_tot_vect} 176 \[ 177 % \label{eq:INVARIANTS_E_tot_vect_vor_2} 178 178 \int\limits_D \textbf{U}_h \cdot \text{VOR} \;dv = 0 179 179 \] 180 180 \[ 181 % \label{eq: E_tot_vect_adv_2}181 % \label{eq:INVARIANTS_E_tot_vect_adv_2} 182 182 \int\limits_D \textbf{U}_h \cdot \text{KEG} \;dv 183 183 + \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv … … 185 185 \] 186 186 \[ 187 % \label{eq: E_tot_pg_2}187 % \label{eq:INVARIANTS_E_tot_pg_2} 188 188 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 189 189 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv … … 193 193 Flux form: 194 194 \begin{subequations} 195 \label{eq: E_tot_flux}195 \label{eq:INVARIANTS_E_tot_flux} 196 196 \[ 197 % \label{eq: E_tot_flux_metric_2}197 % \label{eq:INVARIANTS_E_tot_flux_metric_2} 198 198 \int\limits_D \textbf{U}_h \cdot \text {COR} \; dv = 0 199 199 \] 200 200 \[ 201 % \label{eq: E_tot_flux_adv_2}201 % \label{eq:INVARIANTS_E_tot_flux_adv_2} 202 202 \int\limits_D \textbf{U}_h \cdot \text{ADV} \;dv 203 203 + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 204 204 \] 205 205 \begin{equation} 206 \label{eq: E_tot_pg_3}206 \label{eq:INVARIANTS_E_tot_pg_3} 207 207 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 208 208 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv … … 211 211 \end{subequations} 212 212 213 \autoref{eq: E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE.214 Indeed the left hand side of \autoref{eq: E_tot_pg_3} can be transformed as follows:213 \autoref{eq:INVARIANTS_E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. 214 Indeed the left hand side of \autoref{eq:INVARIANTS_E_tot_pg_3} can be transformed as follows: 215 215 \begin{flalign*} 216 216 \partial_t \left( \int\limits_D { \rho \, g \, z \;dv} \right) … … 225 225 \end{flalign*} 226 226 where the last equality is obtained by noting that the brackets is exactly the expression of $w$, 227 the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{ apdx:A_w_s}).228 229 The left hand side of \autoref{eq: E_tot_pg_3} can be transformed as follows:227 the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{eq:SCOORD_w_s}). 228 229 The left hand side of \autoref{eq:INVARIANTS_E_tot_pg_3} can be transformed as follows: 230 230 \begin{flalign*} 231 231 - \int\limits_D \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv … … 326 326 % ================================================================ 327 327 \section{Discrete total energy conservation: vector invariant form} 328 \label{sec: C.2}328 \label{sec:INVARIANTS_2} 329 329 330 330 % ------------------------------------------------------------------------------------------------------------- … … 332 332 % ------------------------------------------------------------------------------------------------------------- 333 333 \subsection{Total energy conservation} 334 \label{subsec: C_KE+PE_vect}335 336 The discrete form of the total energy conservation, \autoref{eq: Tot_Energy}, is given by:334 \label{subsec:INVARIANTS_KE+PE_vect} 335 336 The discrete form of the total energy conservation, \autoref{eq:INVARIANTS_Tot_Energy}, is given by: 337 337 \begin{flalign*} 338 338 \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 … … 340 340 which in vector invariant forms, it leads to: 341 341 \begin{equation} 342 \label{eq: KE+PE_vect_discrete}342 \label{eq:INVARIANTS_KE+PE_vect_discrete} 343 343 \begin{split} 344 344 \sum\limits_{i,j,k} \biggl\{ u\, \partial_t u \;b_u … … 352 352 353 353 Substituting the discrete expression of the time derivative of the velocity either in vector invariant, 354 leads to the discrete equivalent of the four equations \autoref{eq: E_tot_flux}.354 leads to the discrete equivalent of the four equations \autoref{eq:INVARIANTS_E_tot_flux}. 355 355 356 356 % ------------------------------------------------------------------------------------------------------------- … … 358 358 % ------------------------------------------------------------------------------------------------------------- 359 359 \subsection{Vorticity term (coriolis + vorticity part of the advection)} 360 \label{subsec: C_vor}360 \label{subsec:INVARIANTS_vor} 361 361 362 362 Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$), … … 367 367 % ------------------------------------------------------------------------------------------------------------- 368 368 \subsubsection{Vorticity term with ENE scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})} 369 \label{subsec: C_vorENE}369 \label{subsec:INVARIANTS_vorENE} 370 370 371 371 For the ENE scheme, the two components of the vorticity term are given by: … … 407 407 % ------------------------------------------------------------------------------------------------------------- 408 408 \subsubsection{Vorticity term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 409 \label{subsec: C_vorEEN_vect}409 \label{subsec:INVARIANTS_vorEEN_vect} 410 410 411 411 With the EEN scheme, the vorticity terms are represented as: 412 412 \begin{equation} 413 \ tag{\ref{eq:dynvor_een}}413 \label{eq:INVARIANTS_dynvor_een} 414 414 \left\{ { 415 415 \begin{aligned} … … 424 424 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 425 425 \begin{equation} 426 \ tag{\ref{eq:Q_triads}}426 \label{eq:INVARIANTS_Q_triads} 427 427 _i^j \mathbb{Q}^{i_p}_{j_p} 428 428 = \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) … … 479 479 % ------------------------------------------------------------------------------------------------------------- 480 480 \subsubsection{Gradient of kinetic energy / Vertical advection} 481 \label{subsec: C_zad}481 \label{subsec:INVARIANTS_zad} 482 482 483 483 The change of Kinetic Energy (KE) due to the vertical advection is exactly balanced by the change of KE due to the horizontal gradient of KE~: … … 542 542 % 543 543 \intertext{The first term provides the discrete expression for the vertical advection of momentum (ZAD), 544 while the second term corresponds exactly to \autoref{eq: KE+PE_vect_discrete}, therefore:}544 while the second term corresponds exactly to \autoref{eq:INVARIANTS_KE+PE_vect_discrete}, therefore:} 545 545 \equiv& \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 546 546 + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t (e_3) \;dv } &&&\\ … … 578 578 which is (over-)satified by defining the vertical scale factor as follows: 579 579 \begin{flalign*} 580 % \label{eq: e3u-e3v}580 % \label{eq:INVARIANTS_e3u-e3v} 581 581 e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2} \\ 582 582 e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2} … … 590 590 % ------------------------------------------------------------------------------------------------------------- 591 591 \subsection{Pressure gradient term} 592 \label{subsec: C.2.6}592 \label{subsec:INVARIANTS_2.6} 593 593 594 594 \gmcomment{ … … 622 622 \allowdisplaybreaks 623 623 \intertext{Using successively \autoref{eq:DOM_di_adj}, \ie\ the skew symmetry property of 624 the $\delta$ operator, \autoref{eq: wzv}, the continuity equation, \autoref{eq:dynhpg_sco},624 the $\delta$ operator, \autoref{eq:DYN_wzv}, the continuity equation, \autoref{eq:DYN_hpg_sco}, 625 625 the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $, 626 626 which comes from the definition of $z_t$, it becomes: } … … 667 667 % 668 668 \end{flalign*} 669 The first term is exactly the first term of the right-hand-side of \autoref{eq: KE+PE_vect_discrete}.669 The first term is exactly the first term of the right-hand-side of \autoref{eq:INVARIANTS_KE+PE_vect_discrete}. 670 670 It remains to demonstrate that the last term, 671 671 which is obviously a discrete analogue of $\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to 672 the last term of \autoref{eq: KE+PE_vect_discrete}.672 the last term of \autoref{eq:INVARIANTS_KE+PE_vect_discrete}. 673 673 In other words, the following property must be satisfied: 674 674 \begin{flalign*} … … 735 735 % ================================================================ 736 736 \section{Discrete total energy conservation: flux form} 737 \label{sec: C.3}737 \label{sec:INVARIANTS_3} 738 738 739 739 % ------------------------------------------------------------------------------------------------------------- … … 741 741 % ------------------------------------------------------------------------------------------------------------- 742 742 \subsection{Total energy conservation} 743 \label{subsec: C_KE+PE_flux}744 745 The discrete form of the total energy conservation, \autoref{eq: Tot_Energy}, is given by:743 \label{subsec:INVARIANTS_KE+PE_flux} 744 745 The discrete form of the total energy conservation, \autoref{eq:INVARIANTS_Tot_Energy}, is given by: 746 746 \begin{flalign*} 747 747 \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 \\ … … 765 765 % ------------------------------------------------------------------------------------------------------------- 766 766 \subsection{Coriolis and advection terms: flux form} 767 \label{subsec: C.3.2}767 \label{subsec:INVARIANTS_3.2} 768 768 769 769 % ------------------------------------------------------------------------------------------------------------- … … 771 771 % ------------------------------------------------------------------------------------------------------------- 772 772 \subsubsection{Coriolis plus ``metric'' term} 773 \label{subsec: C.3.3}773 \label{subsec:INVARIANTS_3.3} 774 774 775 775 In flux from the vorticity term reduces to a Coriolis term in which … … 786 786 Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. 787 787 It therefore conserves the total KE. 788 The derivation is the same as for the vorticity term in the vector invariant form (\autoref{subsec: C_vor}).788 The derivation is the same as for the vorticity term in the vector invariant form (\autoref{subsec:INVARIANTS_vor}). 789 789 790 790 % ------------------------------------------------------------------------------------------------------------- … … 792 792 % ------------------------------------------------------------------------------------------------------------- 793 793 \subsubsection{Flux form advection} 794 \label{subsec: C.3.4}794 \label{subsec:INVARIANTS_3.4} 795 795 796 796 The flux form operator of the momentum advection is evaluated using … … 800 800 801 801 \begin{equation} 802 \label{eq: C_ADV_KE_flux}802 \label{eq:INVARIANTS_ADV_KE_flux} 803 803 - \int_D \textbf{U}_h \cdot \left( {{ 804 804 \begin{array} {*{20}c} … … 863 863 \] 864 864 which is the discrete form of $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv $. 865 \autoref{eq: C_ADV_KE_flux} is thus satisfied.865 \autoref{eq:INVARIANTS_ADV_KE_flux} is thus satisfied. 866 866 867 867 When the UBS scheme is used to evaluate the flux form momentum advection, … … 873 873 % ================================================================ 874 874 \section{Discrete enstrophy conservation} 875 \label{sec: C.4}875 \label{sec:INVARIANTS_4} 876 876 877 877 % ------------------------------------------------------------------------------------------------------------- … … 879 879 % ------------------------------------------------------------------------------------------------------------- 880 880 \subsubsection{Vorticity term with ENS scheme (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})} 881 \label{subsec: C_vorENS}881 \label{subsec:INVARIANTS_vorENS} 882 882 883 883 In the ENS scheme, the vorticity term is descretized as follows: 884 884 \begin{equation} 885 \ tag{\ref{eq:dynvor_ens}}885 \label{eq:INVARIANTS_dynvor_ens} 886 886 \left\{ 887 887 \begin{aligned} … … 898 898 it can be shown that: 899 899 \begin{equation} 900 \label{eq: C_1.1}900 \label{eq:INVARIANTS_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. 904 Indeed, using \autoref{eq: dynvor_ens},905 the discrete form of the right hand side of \autoref{eq: C_1.1} can be transformed as follow:904 Indeed, using \autoref{eq:DYN_vor_ens}, 905 the discrete form of the right hand side of \autoref{eq:INVARIANTS_1.1} can be transformed as follow: 906 906 \begin{flalign*} 907 907 &\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times … … 948 948 % ------------------------------------------------------------------------------------------------------------- 949 949 \subsubsection{Vorticity Term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 950 \label{subsec: C_vorEEN}950 \label{subsec:INVARIANTS_vorEEN} 951 951 952 952 With the EEN scheme, the vorticity terms are represented as: 953 953 \begin{equation} 954 \ tag{\ref{eq:dynvor_een}}954 \label{eq:INVARIANTS_dynvor_een} 955 955 \left\{ { 956 956 \begin{aligned} … … 966 966 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 967 967 \begin{equation} 968 \tag{\ref{eq: Q_triads}}968 \tag{\ref{eq:INVARIANTS_Q_triads}} 969 969 _i^j \mathbb{Q}^{i_p}_{j_p} 970 970 = \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) … … 975 975 Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $, 976 976 similar manipulation can be done for the 3 others. 977 The discrete form of the right hand side of \autoref{eq: C_1.1} applied to977 The discrete form of the right hand side of \autoref{eq:INVARIANTS_1.1} applied to 978 978 this triad only can be transformed as follow: 979 979 … … 1021 1021 % ================================================================ 1022 1022 \section{Conservation properties on tracers} 1023 \label{sec: C.5}1023 \label{sec:INVARIANTS_5} 1024 1024 1025 1025 All the numerical schemes used in \NEMO\ are written such that the tracer content is conserved by … … 1037 1037 % ------------------------------------------------------------------------------------------------------------- 1038 1038 \subsection{Advection term} 1039 \label{subsec: C.5.1}1039 \label{subsec:INVARIANTS_5.1} 1040 1040 1041 1041 conservation of a tracer, $T$: … … 1103 1103 % ================================================================ 1104 1104 \section{Conservation properties on lateral momentum physics} 1105 \label{sec: dynldf_properties}1105 \label{sec:INVARIANTS_dynldf_properties} 1106 1106 1107 1107 The discrete formulation of the horizontal diffusion of momentum ensures … … 1124 1124 % ------------------------------------------------------------------------------------------------------------- 1125 1125 \subsection{Conservation of potential vorticity} 1126 \label{subsec: C.6.1}1126 \label{subsec:INVARIANTS_6.1} 1127 1127 1128 1128 The lateral momentum diffusion term conserves the potential vorticity: … … 1158 1158 % ------------------------------------------------------------------------------------------------------------- 1159 1159 \subsection{Dissipation of horizontal kinetic energy} 1160 \label{subsec: C.6.2}1160 \label{subsec:INVARIANTS_6.2} 1161 1161 1162 1162 The lateral momentum diffusion term dissipates the horizontal kinetic energy: … … 1210 1210 % ------------------------------------------------------------------------------------------------------------- 1211 1211 \subsection{Dissipation of enstrophy} 1212 \label{subsec: C.6.3}1212 \label{subsec:INVARIANTS_6.3} 1213 1213 1214 1214 The lateral momentum diffusion term dissipates the enstrophy when the eddy coefficients are horizontally uniform: … … 1234 1234 % ------------------------------------------------------------------------------------------------------------- 1235 1235 \subsection{Conservation of horizontal divergence} 1236 \label{subsec: C.6.4}1236 \label{subsec:INVARIANTS_6.4} 1237 1237 1238 1238 When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken, … … 1261 1261 % ------------------------------------------------------------------------------------------------------------- 1262 1262 \subsection{Dissipation of horizontal divergence variance} 1263 \label{subsec: C.6.5}1263 \label{subsec:INVARIANTS_6.5} 1264 1264 1265 1265 \begin{flalign*} … … 1287 1287 % ================================================================ 1288 1288 \section{Conservation properties on vertical momentum physics} 1289 \label{sec: C.7}1289 \label{sec:INVARIANTS_7} 1290 1290 1291 1291 As for the lateral momentum physics, … … 1458 1458 % ================================================================ 1459 1459 \section{Conservation properties on tracer physics} 1460 \label{sec: C.8}1460 \label{sec:INVARIANTS_8} 1461 1461 1462 1462 The numerical schemes used for tracer subgridscale physics are written such that … … 1470 1470 % ------------------------------------------------------------------------------------------------------------- 1471 1471 \subsection{Conservation of tracers} 1472 \label{subsec: C.8.1}1472 \label{subsec:INVARIANTS_8.1} 1473 1473 1474 1474 constraint of conservation of tracers: … … 1503 1503 % ------------------------------------------------------------------------------------------------------------- 1504 1504 \subsection{Dissipation of tracer variance} 1505 \label{subsec: C.8.2}1505 \label{subsec:INVARIANTS_8.2} 1506 1506 1507 1507 constraint on the dissipation of tracer variance: -
NEMO/trunk/doc/latex/NEMO/subfiles/apdx_s_coord.tex
r11529 r11543 7 7 % ================================================================ 8 8 \chapter{Curvilinear $s-$Coordinate Equations} 9 \label{apdx: A}9 \label{apdx:SCOORD} 10 10 11 11 \chaptertoc … … 28 28 % ================================================================ 29 29 \section{Chain rule for $s-$coordinates} 30 \label{sec: A_chain}30 \label{sec:SCOORD_chain} 31 31 32 32 In order to establish the set of Primitive Equation in curvilinear $s$-coordinates 33 33 (\ie\ an orthogonal curvilinear coordinate in the horizontal and 34 34 an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical), 35 we start from the set of equations established in \autoref{subsec: PE_zco_Eq} for35 we start from the set of equations established in \autoref{subsec:MB_zco_Eq} for 36 36 the special case $k = z$ and thus $e_3 = 1$, 37 37 and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$. … … 39 39 the horizontal slope of $s-$surfaces by: 40 40 \begin{equation} 41 \label{ apdx:A_s_slope}41 \label{eq:SCOORD_s_slope} 42 42 \sigma_1 =\frac{1}{e_1 } \; \left. {\frac{\partial z}{\partial i}} \right|_s 43 43 \quad \text{and} \quad … … 46 46 47 47 The model fields (e.g. pressure $p$) can be viewed as functions of $(i,j,z,t)$ (e.g. $p(i,j,z,t)$) or as 48 functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of 49 these fields. Any ``infinitesimal'' change in $\bullet$ can be written in two forms: 50 \begin{equation} 51 \label{ apdx:A_s_infin_changes}48 functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of 49 these fields. Any ``infinitesimal'' change in $\bullet$ can be written in two forms: 50 \begin{equation} 51 \label{eq:SCOORD_s_infin_changes} 52 52 \begin{aligned} 53 & \delta \bullet = \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t} 54 + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t} 55 + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t} 53 & \delta \bullet = \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t} 54 + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t} 55 + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t} 56 56 + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,s} , \\ 57 & \delta \bullet = \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t} 58 + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t} 59 + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t} 57 & \delta \bullet = \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t} 58 + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t} 59 + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t} 60 60 + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,z} . 61 61 \end{aligned} … … 63 63 Using the first form and considering a change $\delta i$ with $j, z$ and $t$ held constant, shows that 64 64 \begin{equation} 65 \label{ apdx:A_s_chain_rule}65 \label{eq:SCOORD_s_chain_rule} 66 66 \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,z,t} = 67 67 \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,s,t} 68 + \left. {\frac{\partial s }{\partial i}} \right|_{j,z,t} \; 69 \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} . 70 \end{equation} 71 The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces, 72 (\autoref{ apdx:A_s_slope}), by applying the second of (\autoref{apdx:A_s_infin_changes}) with $\bullet$ set to68 + \left. {\frac{\partial s }{\partial i}} \right|_{j,z,t} \; 69 \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} . 70 \end{equation} 71 The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces, 72 (\autoref{eq:SCOORD_s_slope}), by applying the second of (\autoref{eq:SCOORD_s_infin_changes}) with $\bullet$ set to 73 73 $s$ and $j, t$ held constant 74 74 \begin{equation} 75 \label{ apdx:a_delta_s}76 \delta s|_{j,t} = 77 \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} 75 \label{eq:SCOORD_delta_s} 76 \delta s|_{j,t} = 77 \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} 78 78 + \delta z \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} . 79 79 \end{equation} 80 80 Choosing to look at a direction in the $(i,z)$ plane in which $\delta s = 0$ and using 81 (\autoref{ apdx:A_s_slope}) we obtain82 \begin{equation} 83 \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} = 81 (\autoref{eq:SCOORD_s_slope}) we obtain 82 \begin{equation} 83 \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} = 84 84 - \left. \frac{ \partial z }{\partial i} \right|_{j,s,t} \; 85 85 \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} 86 86 = - \frac{e_1 }{e_3 }\sigma_1 . 87 \label{ apdx:a_ds_di_z}88 \end{equation} 89 Another identity, similar in form to (\autoref{ apdx:a_ds_di_z}), can be derived90 by choosing $\bullet$ to be $s$ and using the second form of (\autoref{ apdx:A_s_infin_changes}) to consider87 \label{eq:SCOORD_ds_di_z} 88 \end{equation} 89 Another identity, similar in form to (\autoref{eq:SCOORD_ds_di_z}), can be derived 90 by choosing $\bullet$ to be $s$ and using the second form of (\autoref{eq:SCOORD_s_infin_changes}) to consider 91 91 changes in which $i , j$ and $s$ are constant. This shows that 92 92 \begin{equation} 93 \label{ apdx:A_w_in_s}94 w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} = 93 \label{eq:SCOORD_w_in_s} 94 w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} = 95 95 - \left. \frac{ \partial z }{\partial s} \right|_{i,j,t} 96 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} 97 = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} . 98 \end{equation} 99 100 In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is 101 usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish 96 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} 97 = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} . 98 \end{equation} 99 100 In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is 101 usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish 102 102 the model equations in the curvilinear $s-$coordinate system are: 103 103 \begin{equation} 104 \label{ apdx:A_s_chain_rule}104 \label{eq:SCOORD_s_chain_rule} 105 105 \begin{aligned} 106 106 &\left. {\frac{\partial \bullet }{\partial t}} \right|_z = 107 \left. {\frac{\partial \bullet }{\partial t}} \right|_s 107 \left. {\frac{\partial \bullet }{\partial t}} \right|_s 108 108 + \frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial t} , \\ 109 109 &\left. {\frac{\partial \bullet }{\partial i}} \right|_z = 110 110 \left. {\frac{\partial \bullet }{\partial i}} \right|_s 111 111 +\frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial i}= 112 \left. {\frac{\partial \bullet }{\partial i}} \right|_s 112 \left. {\frac{\partial \bullet }{\partial i}} \right|_s 113 113 -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} , \\ 114 114 &\left. {\frac{\partial \bullet }{\partial j}} \right|_z = 115 \left. {\frac{\partial \bullet }{\partial j}} \right|_s 115 \left. {\frac{\partial \bullet }{\partial j}} \right|_s 116 116 + \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= 117 \left. {\frac{\partial \bullet }{\partial j}} \right|_s 117 \left. {\frac{\partial \bullet }{\partial j}} \right|_s 118 118 - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} , \\ 119 119 &\;\frac{\partial \bullet }{\partial z} \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} . … … 126 126 % ================================================================ 127 127 \section{Continuity equation in $s-$coordinates} 128 \label{sec: A_continuity}129 130 Using (\autoref{ apdx:A_s_chain_rule}) and128 \label{sec:SCOORD_continuity} 129 130 Using (\autoref{eq:SCOORD_s_chain_rule}) and 131 131 the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, 132 132 the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to … … 189 189 \end{subequations} 190 190 191 Here, $w$ is the vertical velocity relative to the $z-$coordinate system. 192 Using the first form of (\autoref{ apdx:A_s_infin_changes})193 and the definitions (\autoref{ apdx:A_s_slope}) and (\autoref{apdx:A_w_in_s}) for $\sigma_1$, $\sigma_2$ and $w_s$,191 Here, $w$ is the vertical velocity relative to the $z-$coordinate system. 192 Using the first form of (\autoref{eq:SCOORD_s_infin_changes}) 193 and the definitions (\autoref{eq:SCOORD_s_slope}) and (\autoref{eq:SCOORD_w_in_s}) for $\sigma_1$, $\sigma_2$ and $w_s$, 194 194 one can show that the vertical velocity, $w_p$ of a point 195 moving with the horizontal velocity of the fluid along an $s$ surface is given by 196 \begin{equation} 197 \label{ apdx:A_w_p}195 moving with the horizontal velocity of the fluid along an $s$ surface is given by 196 \begin{equation} 197 \label{eq:SCOORD_w_p} 198 198 \begin{split} 199 199 w_p = & \left. \frac{ \partial z }{\partial t} \right|_s 200 + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s 200 + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s 201 201 + \frac{v}{e_2} \left. \frac{ \partial z }{\partial j} \right|_s \\ 202 202 = & w_s + u \sigma_1 + v \sigma_2 . 203 \end{split} 203 \end{split} 204 204 \end{equation} 205 205 The vertical velocity across this surface is denoted by 206 206 \begin{equation} 207 \label{ apdx:A_w_s}208 \omega = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v ) . 209 \end{equation} 210 Hence 211 \begin{equation} 212 \frac{1}{e_3 } \frac{\partial}{\partial s} \left[ w - u\;\sigma_1 - v\;\sigma_2 \right] = 213 \frac{1}{e_3 } \frac{\partial}{\partial s} \left[ \omega + w_s \right] = 214 \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s} 215 + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] = 216 \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s 217 \end{equation} 218 219 Using (\autoref{ apdx:A_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain207 \label{eq:SCOORD_w_s} 208 \omega = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v ) . 209 \end{equation} 210 Hence 211 \begin{equation} 212 \frac{1}{e_3 } \frac{\partial}{\partial s} \left[ w - u\;\sigma_1 - v\;\sigma_2 \right] = 213 \frac{1}{e_3 } \frac{\partial}{\partial s} \left[ \omega + w_s \right] = 214 \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s} 215 + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] = 216 \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s 217 \end{equation} 218 219 Using (\autoref{eq:SCOORD_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain 220 220 our final expression for the divergence of the velocity in the curvilinear $s-$coordinate system: 221 221 \begin{equation} … … 228 228 \end{equation} 229 229 230 As a result, the continuity equation \autoref{eq: PE_continuity} in the $s-$coordinates is:231 \begin{equation} 232 \label{ apdx:A_sco_Continuity}230 As a result, the continuity equation \autoref{eq:MB_PE_continuity} in the $s-$coordinates is: 231 \begin{equation} 232 \label{eq:SCOORD_sco_Continuity} 233 233 \frac{1}{e_3 } \frac{\partial e_3}{\partial t} 234 234 + \frac{1}{e_1 \,e_2 \,e_3 }\left[ … … 245 245 % ================================================================ 246 246 \section{Momentum equation in $s-$coordinate} 247 \label{sec: A_momentum}247 \label{sec:SCOORD_momentum} 248 248 249 249 Here we only consider the first component of the momentum equation, … … 252 252 $\bullet$ \textbf{Total derivative in vector invariant form} 253 253 254 Let us consider \autoref{eq: PE_dyn_vect}, the first component of the momentum equation in the vector invariant form.254 Let us consider \autoref{eq:MB_dyn_vect}, the first component of the momentum equation in the vector invariant form. 255 255 Its total $z-$coordinate time derivative, 256 256 $\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain … … 272 272 + w \;\frac{\partial u}{\partial z} \\ 273 273 % 274 \intertext{introducing the chain rule (\autoref{ apdx:A_s_chain_rule}) }274 \intertext{introducing the chain rule (\autoref{eq:SCOORD_s_chain_rule}) } 275 275 % 276 276 &= \left. {\frac{\partial u }{\partial t}} \right|_z … … 306 306 \; \frac{\partial u}{\partial s} . \\ 307 307 % 308 \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{ apdx:A_w_s}) }308 \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{eq:SCOORD_w_s}) } 309 309 % 310 310 &= \left. {\frac{\partial u }{\partial t}} \right|_z … … 317 317 \end{subequations} 318 318 % 319 Applying the time derivative chain rule (first equation of (\autoref{ apdx:A_s_chain_rule})) to $u$ and320 using (\autoref{ apdx:A_w_in_s}) provides the expression of the last term of the right hand side,319 Applying the time derivative chain rule (first equation of (\autoref{eq:SCOORD_s_chain_rule})) to $u$ and 320 using (\autoref{eq:SCOORD_w_in_s}) provides the expression of the last term of the right hand side, 321 321 \[ 322 322 { … … 331 331 \ie\ the total $s-$coordinate time derivative : 332 332 \begin{align} 333 \label{ apdx:A_sco_Dt_vect}333 \label{eq:SCOORD_sco_Dt_vect} 334 334 \left. \frac{D u}{D t} \right|_s 335 335 = \left. {\frac{\partial u }{\partial t}} \right|_s 336 336 - \left. \zeta \right|_s \;v 337 337 + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 338 + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} . 338 + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} . 339 339 \end{align} 340 340 Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in … … 345 345 346 346 Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish. 347 Following the procedure used to establish (\autoref{eq: PE_flux_form}), it can be transformed into :347 Following the procedure used to establish (\autoref{eq:MB_flux_form}), it can be transformed into : 348 348 % \begin{subequations} 349 349 \begin{align*} … … 367 367 \end{align*} 368 368 % 369 Introducing the vertical scale factor inside the horizontal derivative of the first two terms 369 Introducing the vertical scale factor inside the horizontal derivative of the first two terms 370 370 (\ie\ the horizontal divergence), it becomes : 371 371 \begin{align*} … … 373 373 \begin{array}{*{20}l} 374 374 % \begin{align*} {\begin{array}{*{20}l} 375 % {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s 375 % {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s 376 376 &= \left. {\frac{\partial u }{\partial t}} \right|_s 377 377 &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u^2 )}{\partial i} … … 398 398 % 399 399 \intertext {Introducing a more compact form for the divergence of the momentum fluxes, 400 and using (\autoref{ apdx:A_sco_Continuity}), the $s-$coordinate continuity equation,400 and using (\autoref{eq:SCOORD_sco_Continuity}), the $s-$coordinate continuity equation, 401 401 it becomes : } 402 402 % … … 410 410 } 411 411 \end{align*} 412 which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, 412 which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, 413 413 \ie\ the total $s-$coordinate time derivative in flux form: 414 414 \begin{flalign} 415 \label{ apdx:A_sco_Dt_flux}415 \label{eq:SCOORD_sco_Dt_flux} 416 416 \left. \frac{D u}{D t} \right|_s = \frac{1}{e_3} \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s 417 417 + \left. \nabla \cdot \left( {{\mathrm {\mathbf U}}\,u} \right) \right|_s … … 422 422 It has the same form as in the $z-$coordinate but for 423 423 the vertical scale factor that has appeared inside the time derivative which 424 comes from the modification of (\autoref{ apdx:A_sco_Continuity}),424 comes from the modification of (\autoref{eq:SCOORD_sco_Continuity}), 425 425 the continuity equation. 426 426 … … 437 437 \] 438 438 Applying similar manipulation to the second component and 439 replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{ apdx:A_s_slope}, it becomes:440 \begin{equation} 441 \label{ apdx:A_grad_p_1}439 replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{eq:SCOORD_s_slope}, it becomes: 440 \begin{equation} 441 \label{eq:SCOORD_grad_p_1} 442 442 \begin{split} 443 443 -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z … … 451 451 \end{equation} 452 452 453 An additional term appears in (\autoref{ apdx:A_grad_p_1}) which accounts for453 An additional term appears in (\autoref{eq:SCOORD_grad_p_1}) which accounts for 454 454 the tilt of $s-$surfaces with respect to geopotential $z-$surfaces. 455 455 … … 467 467 Therefore, $p$ and $p_h'$ are linked through: 468 468 \begin{equation} 469 \label{ apdx:A_pressure}469 \label{eq:SCOORD_pressure} 470 470 p = \rho_o \; p_h' + \rho_o \, g \, ( \eta - z ) 471 471 \end{equation} … … 475 475 \] 476 476 477 Substituing \autoref{ apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and477 Substituing \autoref{eq:SCOORD_pressure} in \autoref{eq:SCOORD_grad_p_1} and 478 478 using the definition of the density anomaly it becomes an expression in two parts: 479 479 \begin{equation} 480 \label{ apdx:A_grad_p_2}480 \label{eq:SCOORD_grad_p_2} 481 481 \begin{split} 482 482 -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z … … 491 491 This formulation of the pressure gradient is characterised by the appearance of 492 492 a term depending on the sea surface height only 493 (last term on the right hand side of expression \autoref{ apdx:A_grad_p_2}).493 (last term on the right hand side of expression \autoref{eq:SCOORD_grad_p_2}). 494 494 This term will be loosely termed \textit{surface pressure gradient} whereas 495 495 the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to … … 502 502 The coriolis and forcing terms as well as the the vertical physics remain unchanged as 503 503 they involve neither time nor space derivatives. 504 The form of the lateral physics is discussed in \autoref{apdx: B}.504 The form of the lateral physics is discussed in \autoref{apdx:DIFFOPERS}. 505 505 506 506 $\bullet$ \textbf{Full momentum equation} … … 510 510 the one in a curvilinear $z-$coordinate, except for the pressure gradient term: 511 511 \begin{subequations} 512 \label{ apdx:A_dyn_vect}512 \label{eq:SCOORD_dyn_vect} 513 513 \begin{multline} 514 \label{ apdx:A_PE_dyn_vect_u}514 \label{eq:SCOORD_PE_dyn_vect_u} 515 515 \frac{\partial u}{\partial t}= 516 516 + \left( {\zeta +f} \right)\,v … … 522 522 \end{multline} 523 523 \begin{multline} 524 \label{ apdx:A_dyn_vect_v}524 \label{eq:SCOORD_dyn_vect_v} 525 525 \frac{\partial v}{\partial t}= 526 526 - \left( {\zeta +f} \right)\,u … … 535 535 the formulation of both the time derivative and the pressure gradient term: 536 536 \begin{subequations} 537 \label{ apdx:A_dyn_flux}537 \label{eq:SCOORD_dyn_flux} 538 538 \begin{multline} 539 \label{ apdx:A_PE_dyn_flux_u}539 \label{eq:SCOORD_PE_dyn_flux_u} 540 540 \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t} = 541 541 - \nabla \cdot \left( {{\mathrm {\mathbf U}}\,u} \right) … … 547 547 \end{multline} 548 548 \begin{multline} 549 \label{ apdx:A_dyn_flux_v}549 \label{eq:SCOORD_dyn_flux_v} 550 550 \frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}= 551 551 - \nabla \cdot \left( {{\mathrm {\mathbf U}}\,v} \right) … … 554 554 - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) 555 555 - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 556 + D_v^{\vect{U}} + F_v^{\vect{U}} . 556 + D_v^{\vect{U}} + F_v^{\vect{U}} . 557 557 \end{multline} 558 558 \end{subequations} … … 560 560 hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: 561 561 \begin{equation} 562 \label{ apdx:A_dyn_zph}562 \label{eq:SCOORD_dyn_zph} 563 563 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 . 564 564 \end{equation} … … 569 569 in particular the pressure gradient. 570 570 By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component, 571 \ie\ the volume flux across the moving $s$-surfaces per unit horizontal area. 571 \ie\ the volume flux across the moving $s$-surfaces per unit horizontal area. 572 572 573 573 … … 576 576 % ================================================================ 577 577 \section{Tracer equation} 578 \label{sec: A_tracer}578 \label{sec:SCOORD_tracer} 579 579 580 580 The tracer equation is obtained using the same calculation as for the continuity equation and then … … 582 582 583 583 \begin{multline} 584 \label{ apdx:A_tracer}584 \label{eq:SCOORD_tracer} 585 585 \frac{1}{e_3} \frac{\partial \left( e_3 T \right)}{\partial t} 586 586 = -\frac{1}{e_1 \,e_2 \,e_3} … … 591 591 \end{multline} 592 592 593 The expression for the advection term is a straight consequence of (\autoref{ apdx:A_sco_Continuity}),594 the expression of the 3D divergence in the $s-$coordinates established above. 593 The expression for the advection term is a straight consequence of (\autoref{eq:SCOORD_sco_Continuity}), 594 the expression of the 3D divergence in the $s-$coordinates established above. 595 595 596 596 \biblio -
NEMO/trunk/doc/latex/NEMO/subfiles/apdx_triads.tex
r11529 r11543 16 16 % ================================================================ 17 17 \chapter{Iso-Neutral Diffusion and Eddy Advection using Triads} 18 \label{apdx: triad}18 \label{apdx:TRIADS} 19 19 20 20 \chaptertoc … … 45 45 \begin{description} 46 46 \item[\np{ln\_triad\_iso}] 47 See \autoref{sec: taper}.47 See \autoref{sec:TRIADS_taper}. 48 48 If this is set false (the default), 49 49 then `iso-neutral' mixing is accomplished within the surface mixed-layer along slopes linearly decreasing with 50 depth from the value immediately below the mixed-layer to zero (flat) at the surface (\autoref{sec: lintaper}).50 depth from the value immediately below the mixed-layer to zero (flat) at the surface (\autoref{sec:TRIADS_lintaper}). 51 51 This is the same treatment as used in the default implementation 52 \autoref{subsec:LDF_slp_iso}; \autoref{fig: eiv_slp}.52 \autoref{subsec:LDF_slp_iso}; \autoref{fig:LDF_eiv_slp}. 53 53 Where \np{ln\_triad\_iso} is set true, 54 54 the vertical skew flux is further reduced to ensure no vertical buoyancy flux, 55 55 giving an almost pure horizontal diffusive tracer flux within the mixed layer. 56 This is similar to the tapering suggested by \citet{gerdes.koberle.ea_CD91}. See \autoref{subsec: Gerdes-taper}56 This is similar to the tapering suggested by \citet{gerdes.koberle.ea_CD91}. See \autoref{subsec:TRIADS_Gerdes-taper} 57 57 \item[\np{ln\_botmix\_triad}] 58 See \autoref{sec: iso_bdry}.58 See \autoref{sec:TRIADS_iso_bdry}. 59 59 If this is set false (the default) then the lateral diffusive fluxes 60 associated with triads partly masked by topography are neglected. 61 If it is set true, however, then these lateral diffusive fluxes are applied, 60 associated with triads partly masked by topography are neglected. 61 If it is set true, however, then these lateral diffusive fluxes are applied, 62 62 giving smoother bottom tracer fields at the cost of introducing diapycnal mixing. 63 63 \item[\np{rn\_sw\_triad}] … … 71 71 72 72 \section{Triad formulation of iso-neutral diffusion} 73 \label{sec: iso}73 \label{sec:TRIADS_iso} 74 74 75 75 We have implemented into \NEMO\ a scheme inspired by \citet{griffies.gnanadesikan.ea_JPO98}, … … 79 79 80 80 The iso-neutral second order tracer diffusive operator for small angles between 81 iso-neutral surfaces and geopotentials is given by \autoref{eq: iso_tensor_1}:81 iso-neutral surfaces and geopotentials is given by \autoref{eq:TRIADS_iso_tensor_1}: 82 82 \begin{subequations} 83 \label{eq: iso_tensor_1}83 \label{eq:TRIADS_iso_tensor_1} 84 84 \begin{equation} 85 85 D^{lT}=-\nabla \cdot\vect{f}^{lT}\equiv … … 92 92 \end{equation} 93 93 \begin{equation} 94 \label{eq: iso_tensor_2}94 \label{eq:TRIADS_iso_tensor_2} 95 95 \mbox{with}\quad \;\;\Re = 96 96 \begin{pmatrix} … … 113 113 % {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ 114 114 % \end{array} }} \right) 115 Here \autoref{eq: PE_iso_slopes}115 Here \autoref{eq:MB_iso_slopes} 116 116 \begin{align*} 117 117 r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i} … … 128 128 We will find it useful to consider the fluxes per unit area in $i,j,k$ space; we write 129 129 \[ 130 % \label{eq: Fijk}130 % \label{eq:TRIADS_Fijk} 131 131 \vect{F}_{\mathrm{iso}}=\left(f_1^{lT}e_2e_3, f_2^{lT}e_1e_3, f_3^{lT}e_1e_2\right). 132 132 \] … … 136 136 137 137 The off-diagonal terms of the small angle diffusion tensor 138 \autoref{eq: iso_tensor_1}, \autoref{eq:iso_tensor_2} produce skew-fluxes along138 \autoref{eq:TRIADS_iso_tensor_1}, \autoref{eq:TRIADS_iso_tensor_2} produce skew-fluxes along 139 139 the $i$- and $j$-directions resulting from the vertical tracer gradient: 140 140 \begin{align} 141 \label{eq: i13c}141 \label{eq:TRIADS_i13c} 142 142 f_{13}=&+{A^{lT}} r_1\frac{1}{e_3}\frac{\partial T}{\partial k},\qquad f_{23}=+{A^{lT}} r_2\frac{1}{e_3}\frac{\partial T}{\partial k}\\ 143 143 \intertext{and in the k-direction resulting from the lateral tracer gradients} 144 \label{eq: i31c}144 \label{eq:TRIADS_i31c} 145 145 f_{31}+f_{32}=& {A^{lT}} r_1\frac{1}{e_1}\frac{\partial T}{\partial i}+{A^{lT}} r_2\frac{1}{e_1}\frac{\partial T}{\partial i} 146 146 \end{align} … … 148 148 The vertical diffusive flux associated with the $_{33}$ component of the small angle diffusion tensor is 149 149 \begin{equation} 150 \label{eq: i33c}150 \label{eq:TRIADS_i33c} 151 151 f_{33}=-{A^{lT}}(r_1^2 +r_2^2) \frac{1}{e_3}\frac{\partial T}{\partial k}. 152 152 \end{equation} … … 157 157 The following description will describe the fluxes on the $i$-$k$ plane. 158 158 159 There is no natural discretization for the $i$-component of the skew-flux, \autoref{eq: i13c},159 There is no natural discretization for the $i$-component of the skew-flux, \autoref{eq:TRIADS_i13c}, 160 160 as although it must be evaluated at $u$-points, 161 161 it involves vertical gradients (both for the tracer and the slope $r_1$), defined at $w$-points. 162 Similarly, the vertical skew flux, \autoref{eq: i31c},162 Similarly, the vertical skew flux, \autoref{eq:TRIADS_i31c}, 163 163 is evaluated at $w$-points but involves horizontal gradients defined at $u$-points. 164 164 … … 166 166 167 167 The straightforward approach to discretize the lateral skew flux 168 \autoref{eq: i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 into OPA,169 \autoref{eq: tra_ldf_iso}, is to calculate a mean vertical gradient at the $u$-point from168 \autoref{eq:TRIADS_i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 into OPA, 169 \autoref{eq:TRA_ldf_iso}, is to calculate a mean vertical gradient at the $u$-point from 170 170 the average of the four surrounding vertical tracer gradients, and multiply this by a mean slope at the $u$-point, 171 171 calculated from the averaged surrounding vertical density gradients. 172 172 The total area-integrated skew-flux (flux per unit area in $ijk$ space) from tracer cell $i,k$ to $i+1,k$, 173 173 noting that the $e_{{3}_{i+1/2}^k}$ in the area $e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with 174 the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer gradient, is then \autoref{eq: tra_ldf_iso}174 the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer gradient, is then \autoref{eq:TRA_ldf_iso} 175 175 \[ 176 176 \left(F_u^{13} \right)_{i+\frac{1}{2}}^k = {A}_{i+\frac{1}{2}}^k … … 205 205 \includegraphics[width=\textwidth]{Fig_GRIFF_triad_fluxes} 206 206 \caption{ 207 \protect\label{fig: ISO_triad}207 \protect\label{fig:TRIADS_ISO_triad} 208 208 (a) Arrangement of triads $S_i$ and tracer gradients to 209 209 give lateral tracer flux from box $i,k$ to $i+1,k$ … … 217 217 the corresponding `triad' slope calculated from the lateral density gradient across the $u$-point divided by 218 218 the vertical density gradient at the same $w$-point as the tracer gradient. 219 See \autoref{fig: ISO_triad}a, where the thick lines denote the tracer gradients,219 See \autoref{fig:TRIADS_ISO_triad}a, where the thick lines denote the tracer gradients, 220 220 and the thin lines the corresponding triads, with slopes $s_1, \dotsc s_4$. 221 221 The total area-integrated skew-flux from tracer cell $i,k$ to $i+1,k$ 222 222 \begin{multline} 223 \label{eq: i13}223 \label{eq:TRIADS_i13} 224 224 \left( F_u^{13} \right)_{i+\frac{1}{2}}^k = {A}_{i+1}^k a_1 s_1 225 225 \delta_{k+\frac{1}{2}} \left[ T^{i+1} … … 235 235 This discretization gives a much closer stencil, and disallows the two-point computational modes. 236 236 237 The vertical skew flux \autoref{eq: i31c} from tracer cell $i,k$ to $i,k+1$ at238 the $w$-point $i,k+\frac{1}{2}$ is constructed similarly (\autoref{fig: ISO_triad}b) by237 The vertical skew flux \autoref{eq:TRIADS_i31c} from tracer cell $i,k$ to $i,k+1$ at 238 the $w$-point $i,k+\frac{1}{2}$ is constructed similarly (\autoref{fig:TRIADS_ISO_triad}b) by 239 239 multiplying lateral tracer gradients from each of the four surrounding $u$-points by the appropriate triad slope: 240 240 \begin{multline} 241 \label{eq: i31}241 \label{eq:TRIADS_i31} 242 242 \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} = {A}_i^{k+1} a_{1}' 243 243 s_{1}' \delta_{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1} … … 250 250 (appearing in both the vertical and lateral gradient), 251 251 and the $u$- and $w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the triad as follows 252 (see also \autoref{fig: ISO_triad}):253 \begin{equation} 254 \label{eq: R}252 (see also \autoref{fig:TRIADS_ISO_triad}): 253 \begin{equation} 254 \label{eq:TRIADS_R} 255 255 _i^k \mathbb{R}_{i_p}^{k_p} 256 256 =-\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} … … 269 269 \includegraphics[width=\textwidth]{Fig_GRIFF_qcells} 270 270 \caption{ 271 \protect\label{fig: qcells}271 \protect\label{fig:TRIADS_qcells} 272 272 Triad notation for quarter cells. $T$-cells are inside boxes, 273 273 while the $i+\fractext{1}{2},k$ $u$-cell is shaded in green and … … 278 278 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 279 279 280 Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (\autoref{fig: qcells}) with the quarter cell that is280 Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (\autoref{fig:TRIADS_qcells}) with the quarter cell that is 281 281 the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ $u$-cell and the $i,k+k_p$ $w$-cell. 282 Expressing the slopes $s_i$ and $s'_i$ in \autoref{eq: i13} and \autoref{eq:i31} in this notation,282 Expressing the slopes $s_i$ and $s'_i$ in \autoref{eq:TRIADS_i13} and \autoref{eq:TRIADS_i31} in this notation, 283 283 we have \eg\ \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. 284 284 Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to … … 289 289 and we notate these areas, similarly to the triad slopes, 290 290 as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, 291 where \eg\ in \autoref{eq: i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$,292 and in \autoref{eq: i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$.291 where \eg\ in \autoref{eq:TRIADS_i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, 292 and in \autoref{eq:TRIADS_i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. 293 293 294 294 \subsection{Full triad fluxes} … … 299 299 tracer cell $i,k$ to $i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the form 300 300 \begin{equation} 301 \label{eq: i11}301 \label{eq:TRIADS_i11} 302 302 \left( F_u^{11} \right) _{i+\frac{1}{2}} ^{k} = 303 303 - \left( {A}_i^{k+1} a_{1} + {A}_i^{k+1} a_{2} + {A}_i^k … … 305 305 \frac{\delta_{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}}, 306 306 \end{equation} 307 where the areas $a_i$ are as in \autoref{eq: i13}.308 In this case, separating the total lateral flux, the sum of \autoref{eq: i13} and \autoref{eq:i11},307 where the areas $a_i$ are as in \autoref{eq:TRIADS_i13}. 308 In this case, separating the total lateral flux, the sum of \autoref{eq:TRIADS_i13} and \autoref{eq:TRIADS_i11}, 309 309 into triad components, a lateral tracer flux 310 310 \begin{equation} 311 \label{eq: latflux-triad}311 \label{eq:TRIADS_latflux-triad} 312 312 _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) = - {A}_i^k{ \:}_i^k{\mathbb{A}_u}_{i_p}^{k_p} 313 313 \left( … … 322 322 the lateral density flux associated with each triad separately disappears. 323 323 \begin{equation} 324 \label{eq: latflux-rho}324 \label{eq:TRIADS_latflux-rho} 325 325 {\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)=0 326 326 \end{equation} … … 328 328 tracer cell $i,k$ to $i+1,k$ must also vanish since it is a sum of four such triad fluxes. 329 329 330 The squared slope $r_1^2$ in the expression \autoref{eq: i33c} for the $_{33}$ component is also expressed in330 The squared slope $r_1^2$ in the expression \autoref{eq:TRIADS_i33c} for the $_{33}$ component is also expressed in 331 331 terms of area-weighted squared triad slopes, 332 332 so the area-integrated vertical flux from tracer cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is 333 333 \begin{equation} 334 \label{eq: i33}334 \label{eq:TRIADS_i33} 335 335 \left( F_w^{33} \right) _i^{k+\frac{1}{2}} = 336 336 - \left( {A}_i^{k+1} a_{1}' s_{1}'^2 … … 339 339 + {A}_i^k a_{4}' s_{4}'^2 \right)\delta_{k+\frac{1}{2}} \left[ T^{i+1} \right], 340 340 \end{equation} 341 where the areas $a'$ and slopes $s'$ are the same as in \autoref{eq: i31}.342 Then, separating the total vertical flux, the sum of \autoref{eq: i31} and \autoref{eq:i33},341 where the areas $a'$ and slopes $s'$ are the same as in \autoref{eq:TRIADS_i31}. 342 Then, separating the total vertical flux, the sum of \autoref{eq:TRIADS_i31} and \autoref{eq:TRIADS_i33}, 343 343 into triad components, a vertical flux 344 344 \begin{align} 345 \label{eq: vertflux-triad}345 \label{eq:TRIADS_vertflux-triad} 346 346 _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) 347 347 &= {A}_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p} … … 352 352 \right) \\ 353 353 &= - \left(\left.{ }_i^k{\mathbb{A}_w}_{i_p}^{k_p}\right/{ }_i^k{\mathbb{A}_u}_{i_p}^{k_p}\right) 354 {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq: vertflux-triad2}354 {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:TRIADS_vertflux-triad2} 355 355 \end{align} 356 356 may be associated with each triad. … … 361 361 tracer cell $i,k$ to $i,k+1$ must also vanish since it is a sum of four such triad fluxes. 362 362 363 We can explicitly identify (\autoref{fig: qcells}) the triads associated with the $s_i$, $a_i$,364 and $s'_i$, $a'_i$ used in the definition of the $u$-fluxes and $w$-fluxes in \autoref{eq: i31},365 \autoref{eq: i13}, \autoref{eq:i11} \autoref{eq:i33} and \autoref{fig:ISO_triad} to write out363 We can explicitly identify (\autoref{fig:TRIADS_qcells}) the triads associated with the $s_i$, $a_i$, 364 and $s'_i$, $a'_i$ used in the definition of the $u$-fluxes and $w$-fluxes in \autoref{eq:TRIADS_i31}, 365 \autoref{eq:TRIADS_i13}, \autoref{eq:TRIADS_i11} \autoref{eq:TRIADS_i33} and \autoref{fig:TRIADS_ISO_triad} to write out 366 366 the iso-neutral fluxes at $u$- and $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces: 367 %(\autoref{fig: ISO_triad}):367 %(\autoref{fig:TRIADS_ISO_triad}): 368 368 \begin{flalign} 369 \label{eq: iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv369 \label{eq:TRIADS_iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv 370 370 \sum_{\substack{i_p,\,k_p}} 371 371 \begin{pmatrix} … … 376 376 377 377 \subsection{Ensuring the scheme does not increase tracer variance} 378 \label{subsec: variance}378 \label{subsec:TRIADS_variance} 379 379 380 380 We now require that this operator should not increase the globally-integrated tracer variance. … … 400 400 &= -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 401 401 {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\ 402 &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq: dvar_iso_i}402 &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:TRIADS_dvar_iso_i} 403 403 \end{aligned} 404 404 \end{multline} … … 406 406 the $T$-points $i,k+k_p-\fractext{1}{2}$ (above) and $i,k+k_p+\fractext{1}{2}$ (below) of 407 407 \begin{equation} 408 \label{eq: dvar_iso_k}408 \label{eq:TRIADS_dvar_iso_k} 409 409 _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. 410 410 \end{equation} 411 411 The total variance tendency driven by the triad is the sum of these two. 412 412 Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with 413 \autoref{eq: latflux-triad} and \autoref{eq:vertflux-triad}, it is413 \autoref{eq:TRIADS_latflux-triad} and \autoref{eq:TRIADS_vertflux-triad}, it is 414 414 \begin{multline*} 415 415 -{A}_i^k\left \{ … … 430 430 be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by 431 431 \begin{equation} 432 \label{eq: V-A}432 \label{eq:TRIADS_V-A} 433 433 _i^k\mathbb{V}_{i_p}^{k_p} 434 434 ={\;}_i^k{\mathbb{A}_u}_{i_p}^{k_p}\,{e_{1u}}_{\,i + i_p}^{\,k} … … 437 437 the variance tendency reduces to the perfect square 438 438 \begin{equation} 439 \label{eq: perfect-square}439 \label{eq:TRIADS_perfect-square} 440 440 -{A}_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} 441 441 \left( … … 445 445 \right)^2\leq 0. 446 446 \end{equation} 447 Thus, the constraint \autoref{eq: V-A} ensures that the fluxes448 (\autoref{eq: latflux-triad}, \autoref{eq:vertflux-triad}) associated with447 Thus, the constraint \autoref{eq:TRIADS_V-A} ensures that the fluxes 448 (\autoref{eq:TRIADS_latflux-triad}, \autoref{eq:TRIADS_vertflux-triad}) associated with 449 449 a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase the net variance. 450 450 Since the total fluxes are sums of such fluxes from the various triads, this constraint, applied to all triads, 451 451 is sufficient to ensure that the globally integrated variance does not increase. 452 452 453 The expression \autoref{eq: V-A} can be interpreted as a discretization of the global integral454 \begin{equation} 455 \label{eq: cts-var}453 The expression \autoref{eq:TRIADS_V-A} can be interpreted as a discretization of the global integral 454 \begin{equation} 455 \label{eq:TRIADS_cts-var} 456 456 \frac{\partial}{\partial t}\int\!\fractext{1}{2} T^2\, dV = 457 457 \int\!\mathbf{F}\cdot\nabla T\, dV, … … 477 477 \citet{griffies.gnanadesikan.ea_JPO98} identifies these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter cells, 478 478 defined in terms of the distances between $T$, $u$,$f$ and $w$-points. 479 This is the natural discretization of \autoref{eq: cts-var}.479 This is the natural discretization of \autoref{eq:TRIADS_cts-var}. 480 480 The \NEMO\ model, however, operates with scale factors instead of grid sizes, 481 481 and scale factors for the quarter cells are not defined. 482 482 Instead, therefore we simply choose 483 483 \begin{equation} 484 \label{eq: V-NEMO}484 \label{eq:TRIADS_V-NEMO} 485 485 _i^k\mathbb{V}_{i_p}^{k_p}=\fractext{1}{4} {b_u}_{i+i_p}^k, 486 486 \end{equation} … … 489 489 the lateral flux from tracer cell $i,k$ to $i+1,k$ reduces to the classical form 490 490 \begin{equation} 491 \label{eq: lat-normal}491 \label{eq:TRIADS_lat-normal} 492 492 -\overline{A}_{\,i+1/2}^k\; 493 493 \frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} … … 497 497 In fact if the diffusive coefficient is defined at $u$-points, 498 498 so that we employ ${A}_{i+i_p}^k$ instead of ${A}_i^k$ in the definitions of the triad fluxes 499 \autoref{eq: latflux-triad} and \autoref{eq:vertflux-triad},499 \autoref{eq:TRIADS_latflux-triad} and \autoref{eq:TRIADS_vertflux-triad}, 500 500 we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above. 501 501 … … 503 503 504 504 The iso-neutral fluxes at $u$- and $w$-points are the sums of the triad fluxes that 505 cross the $u$- and $w$-faces \autoref{eq: iso_flux}:505 cross the $u$- and $w$-faces \autoref{eq:TRIADS_iso_flux}: 506 506 \begin{subequations} 507 % \label{eq: alltriadflux}507 % \label{eq:TRIADS_alltriadflux} 508 508 \begin{flalign*} 509 % \label{eq: vect_isoflux}509 % \label{eq:TRIADS_vect_isoflux} 510 510 \vect{F}_{\mathrm{iso}}(T) &\equiv 511 511 \sum_{\substack{i_p,\,k_p}} … … 515 515 \end{pmatrix}, 516 516 \end{flalign*} 517 where \autoref{eq: latflux-triad}:517 where \autoref{eq:TRIADS_latflux-triad}: 518 518 \begin{align} 519 \label{eq: triadfluxu}519 \label{eq:TRIADS_triadfluxu} 520 520 _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - {A}_i^k{ 521 521 \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}} … … 532 532 -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ 533 533 \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 534 \right),\label{eq: triadfluxw}534 \right),\label{eq:TRIADS_triadfluxw} 535 535 \end{align} 536 with \autoref{eq: V-NEMO}536 with \autoref{eq:TRIADS_V-NEMO} 537 537 \[ 538 % \label{eq: V-NEMO2}538 % \label{eq:TRIADS_V-NEMO2} 539 539 _i^k{\mathbb{V}}_{i_p}^{k_p}=\fractext{1}{4} {b_u}_{i+i_p}^k. 540 540 \] 541 541 \end{subequations} 542 542 543 The divergence of the expression \autoref{eq: iso_flux} for the fluxes gives the iso-neutral diffusion tendency at543 The divergence of the expression \autoref{eq:TRIADS_iso_flux} for the fluxes gives the iso-neutral diffusion tendency at 544 544 each tracer point: 545 545 \[ 546 % \label{eq: iso_operator}546 % \label{eq:TRIADS_iso_operator} 547 547 D_l^T = \frac{1}{b_T} 548 548 \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k … … 555 555 \item[$\bullet$ horizontal diffusion] 556 556 The discretization of the diffusion operator recovers the traditional five-point Laplacian 557 \autoref{eq: lat-normal} in the limit of flat iso-neutral direction:557 \autoref{eq:TRIADS_lat-normal} in the limit of flat iso-neutral direction: 558 558 \[ 559 % \label{eq: iso_property0}559 % \label{eq:TRIADS_iso_property0} 560 560 D_l^T = \frac{1}{b_T} \ 561 561 \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; … … 565 565 566 566 \item[$\bullet$ implicit treatment in the vertical] 567 Only tracer values associated with a single water column appear in the expression \autoref{eq: i33} for567 Only tracer values associated with a single water column appear in the expression \autoref{eq:TRIADS_i33} for 568 568 the $_{33}$ fluxes, vertical fluxes driven by vertical gradients. 569 569 This is of paramount importance since it means that a time-implicit algorithm can be used to … … 582 582 \item[$\bullet$ pure iso-neutral operator] 583 583 The iso-neutral flux of locally referenced potential density is zero. 584 See \autoref{eq: latflux-rho} and \autoref{eq:vertflux-triad2}.584 See \autoref{eq:TRIADS_latflux-rho} and \autoref{eq:TRIADS_vertflux-triad2}. 585 585 586 586 \item[$\bullet$ conservation of tracer] 587 587 The iso-neutral diffusion conserves tracer content, \ie 588 588 \[ 589 % \label{eq: iso_property1}589 % \label{eq:TRIADS_iso_property1} 590 590 \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 591 591 \] … … 595 595 The iso-neutral diffusion does not increase the tracer variance, \ie 596 596 \[ 597 % \label{eq: iso_property2}597 % \label{eq:TRIADS_iso_property2} 598 598 \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 599 599 \] 600 The property is demonstrated in \autoref{subsec: variance} above.600 The property is demonstrated in \autoref{subsec:TRIADS_variance} above. 601 601 It is a key property for a diffusion term. 602 602 It means that it is also a dissipation term, … … 608 608 The iso-neutral diffusion operator is self-adjoint, \ie 609 609 \begin{equation} 610 \label{eq: iso_property3}610 \label{eq:TRIADS_iso_property3} 611 611 \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 612 612 \end{equation} … … 614 614 We just have to apply the same routine. 615 615 This property can be demonstrated similarly to the proof of the `no increase of tracer variance' property. 616 The contribution by a single triad towards the left hand side of \autoref{eq: iso_property3},617 can be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq: dvar_iso_i} and \autoref{eq:dvar_iso_k}.618 This results in a term similar to \autoref{eq: perfect-square},616 The contribution by a single triad towards the left hand side of \autoref{eq:TRIADS_iso_property3}, 617 can be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq:TRIADS_dvar_iso_i} and \autoref{eq:TRIADS_dvar_iso_k}. 618 This results in a term similar to \autoref{eq:TRIADS_perfect-square}, 619 619 \[ 620 % \label{eq:T Scovar}620 % \label{eq:TRIADS_TScovar} 621 621 - {A}_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} 622 622 \left( … … 632 632 \] 633 633 This is symmetrical in $T $ and $S$, so exactly the same term arises from 634 the discretization of this triad's contribution towards the RHS of \autoref{eq: iso_property3}.634 the discretization of this triad's contribution towards the RHS of \autoref{eq:TRIADS_iso_property3}. 635 635 \end{description} 636 636 637 637 \subsection{Treatment of the triads at the boundaries} 638 \label{sec: iso_bdry}638 \label{sec:TRIADS_iso_bdry} 639 639 640 640 The triad slope can only be defined where both the grid boxes centred at the end of the arms exist. 641 641 Triads that would poke up through the upper ocean surface into the atmosphere, 642 642 or down into the ocean floor, must be masked out. 643 See \autoref{fig: bdry_triads}.643 See \autoref{fig:TRIADS_bdry_triads}. 644 644 Surface layer triads \triad{i}{1}{R}{1/2}{-1/2} (magenta) and \triad{i+1}{1}{R}{-1/2}{-1/2} (blue) that 645 require density to be specified above the ocean surface are masked (\autoref{fig: bdry_triads}a):645 require density to be specified above the ocean surface are masked (\autoref{fig:TRIADS_bdry_triads}a): 646 646 this ensures that lateral tracer gradients produce no flux through the ocean surface. 647 647 However, to prevent surface noise, it is customary to retain the $_{11}$ contributions towards 648 648 the lateral triad fluxes \triad[u]{i}{1}{F}{1/2}{-1/2} and \triad[u]{i+1}{1}{F}{-1/2}{-1/2}; 649 649 this drives diapycnal tracer fluxes. 650 Similar comments apply to triads that would intersect the ocean floor (\autoref{fig: bdry_triads}b).650 Similar comments apply to triads that would intersect the ocean floor (\autoref{fig:TRIADS_bdry_triads}b). 651 651 Note that both near bottom triad slopes \triad{i}{k}{R}{1/2}{1/2} and \triad{i+1}{k}{R}{-1/2}{1/2} are masked when 652 652 either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie\ the $i,k+1$ $u$-point is masked. … … 662 662 \includegraphics[width=\textwidth]{Fig_GRIFF_bdry_triads} 663 663 \caption{ 664 \protect\label{fig: bdry_triads}664 \protect\label{fig:TRIADS_bdry_triads} 665 665 (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer points (black dots), 666 666 and $i+1/2,1$ $u$-point (blue square). … … 683 683 684 684 \subsection{ Limiting of the slopes within the interior} 685 \label{sec: limit}685 \label{sec:TRIADS_limit} 686 686 687 687 As discussed in \autoref{subsec:LDF_slp_iso}, … … 693 693 It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to geopotentials 694 694 (here the $\sigma_i$ are the slopes of the coordinate surfaces relative to geopotentials) 695 \autoref{eq: PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate surfaces, so we require695 \autoref{eq:MB_slopes_eiv} rather than the slope $r_i$ relative to coordinate surfaces, so we require 696 696 \[ 697 697 |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01. … … 700 700 Each individual triad slope 701 701 \begin{equation} 702 \label{eq: Rtilde}702 \label{eq:TRIADS_Rtilde} 703 703 _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}} 704 704 \end{equation} … … 711 711 712 712 \subsection{Tapering within the surface mixed layer} 713 \label{sec: taper}713 \label{sec:TRIADS_taper} 714 714 715 715 Additional tapering of the iso-neutral fluxes is necessary within the surface mixed layer. … … 717 717 718 718 \subsubsection{Linear slope tapering within the surface mixed layer} 719 \label{sec: lintaper}719 \label{sec:TRIADS_lintaper} 720 720 721 721 This is the option activated by the default choice \np{ln\_triad\_iso}\forcode{ = .false.}. 722 722 Slopes $\tilde{r}_i$ relative to geopotentials are tapered linearly from their value immediately below 723 the mixed layer to zero at the surface, as described in option (c) of \autoref{fig: eiv_slp}, to values724 \begin{equation} 725 \label{eq: rmtilde}723 the mixed layer to zero at the surface, as described in option (c) of \autoref{fig:LDF_eiv_slp}, to values 724 \begin{equation} 725 \label{eq:TRIADS_rmtilde} 726 726 \rMLt = -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for } z>-h, 727 727 \end{equation} 728 728 and then the $r_i$ relative to vertical coordinate surfaces are appropriately adjusted to 729 729 \[ 730 % \label{eq: rm}730 % \label{eq:TRIADS_rm} 731 731 \rML =\rMLt -\sigma_i \quad \text{ for } z>-h. 732 732 \] 733 733 Thus the diffusion operator within the mixed layer is given by: 734 734 \[ 735 % \label{eq: iso_tensor_ML}735 % \label{eq:TRIADS_iso_tensor_ML} 736 736 D^{lT}=\nabla {\mathrm {\mathbf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 737 737 \mbox{with}\quad \;\;\Re =\left( {{ … … 747 747 in isopycnal layers immediately below, in the thermocline. 748 748 It is consistent with the way the $\tilde{r}_i$ are tapered within the mixed layer 749 (see \autoref{sec: taperskew} below) so as to ensure a uniform GM eddy-induced velocity throughout the mixed layer.749 (see \autoref{sec:TRIADS_taperskew} below) so as to ensure a uniform GM eddy-induced velocity throughout the mixed layer. 750 750 However, it gives a downwards density flux and so acts so as to reduce potential energy in the same way as 751 does the slope limiting discussed above in \autoref{sec: limit}.752 753 As in \autoref{sec: limit} above, the tapering \autoref{eq:rmtilde} is applied separately to751 does the slope limiting discussed above in \autoref{sec:TRIADS_limit}. 752 753 As in \autoref{sec:TRIADS_limit} above, the tapering \autoref{eq:TRIADS_rmtilde} is applied separately to 754 754 each triad $_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the $_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted. 755 755 For clarity, we assume $z$-coordinates in the following; 756 756 the conversion from $\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as 757 described above by \autoref{eq: Rtilde}.757 described above by \autoref{eq:TRIADS_Rtilde}. 758 758 \begin{enumerate} 759 759 \item 760 760 Mixed-layer depth is defined so as to avoid including regions of weak vertical stratification in 761 761 the slope definition. 762 At each $i,j$ (simplified to $i$ in \autoref{fig: MLB_triad}),762 At each $i,j$ (simplified to $i$ in \autoref{fig:TRIADS_MLB_triad}), 763 763 we define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer, 764 764 $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that 765 765 the potential density ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, 766 766 where $i,k_{10}$ is the tracer gridbox within which the depth reaches 10~m. 767 See the left side of \autoref{fig: MLB_triad}.767 See the left side of \autoref{fig:TRIADS_MLB_triad}. 768 768 We use the $k_{10}$-gridbox instead of the surface gridbox to avoid problems \eg\ with thin daytime mixed-layers. 769 769 Currently we use the same $\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is used to … … 776 776 This is to ensure that the vertical density gradients associated with 777 777 these basal triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are representative of the thermocline. 778 The four basal triads defined in the bottom part of \autoref{fig: MLB_triad} are then778 The four basal triads defined in the bottom part of \autoref{fig:TRIADS_MLB_triad} are then 779 779 \begin{align*} 780 780 {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &= 781 781 {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}, 782 % \label{eq: Rbase}782 % \label{eq:TRIADS_Rbase} 783 783 \\ 784 784 \intertext{with \eg\ the green triad} … … 789 789 the $w$-point $i,k_{\mathrm{ML}}-1/2$ lying \emph{below} the $i,k_{\mathrm{ML}}$ tracer point, so it is this depth 790 790 \[ 791 % \label{eq: zbase}791 % \label{eq:TRIADS_zbase} 792 792 {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2} 793 793 \] 794 794 one gridbox deeper than the diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper the slopes in 795 \autoref{eq: rmtilde}.795 \autoref{eq:TRIADS_rmtilde}. 796 796 \item 797 797 Finally, we calculate the adjusted triads ${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within … … 805 805 {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p} &= 806 806 \frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}. 807 % \label{eq: RML}807 % \label{eq:TRIADS_RML} 808 808 \end{align*} 809 809 \end{enumerate} … … 813 813 % \fcapside { 814 814 \caption{ 815 \protect\label{fig: MLB_triad}815 \protect\label{fig:TRIADS_MLB_triad} 816 816 Definition of mixed-layer depth and calculation of linearly tapered triads. 817 817 The figure shows a water column at a given $i,j$ (simplified to $i$), with the ocean surface at the top. … … 836 836 837 837 \subsubsection{Additional truncation of skew iso-neutral flux components} 838 \label{subsec: Gerdes-taper}838 \label{subsec:TRIADS_Gerdes-taper} 839 839 840 840 The alternative option is activated by setting \np{ln\_triad\_iso} = true. … … 843 843 but replaces the $\rML$ in the skew term by 844 844 \begin{equation} 845 \label{eq: rm*}845 \label{eq:TRIADS_rm*} 846 846 \rML^*=\left.\rMLt^2\right/\tilde{r}_i-\sigma_i, 847 847 \end{equation} 848 848 giving a ML diffusive operator 849 849 \[ 850 % \label{eq: iso_tensor_ML2}850 % \label{eq:TRIADS_iso_tensor_ML2} 851 851 D^{lT}=\nabla {\mathrm {\mathbf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 852 852 \mbox{with}\quad \;\;\Re =\left( {{ … … 881 881 % ================================================================ 882 882 \section{Eddy induced advection formulated as a skew flux} 883 \label{sec: skew-flux}883 \label{sec:TRIADS_skew-flux} 884 884 885 885 \subsection{Continuous skew flux formulation} 886 \label{sec: continuous-skew-flux}886 \label{sec:TRIADS_continuous-skew-flux} 887 887 888 888 When Gent and McWilliams's [1990] diffusion is used, an additional advection term is added. … … 890 890 the formulation of which depends on the slopes of iso-neutral surfaces. 891 891 Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, 892 \ie\ \autoref{eq: ldfslp_geo} is used in $z$-coordinate,893 and the sum \autoref{eq: ldfslp_geo} + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates.892 \ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinate, 893 and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $z^*$ or $s$-coordinates. 894 894 895 895 The eddy induced velocity is given by: 896 896 \begin{subequations} 897 % \label{eq: eiv}897 % \label{eq:TRIADS_eiv} 898 898 \begin{equation} 899 \label{eq: eiv_v}899 \label{eq:TRIADS_eiv_v} 900 900 \begin{split} 901 901 u^* & = - \frac{1}{e_{3}}\; \partial_i\psi_1, \\ … … 907 907 where the streamfunctions $\psi_i$ are given by 908 908 \begin{equation} 909 \label{eq: eiv_psi}909 \label{eq:TRIADS_eiv_psi} 910 910 \begin{split} 911 911 \psi_1 & = A_{e} \; \tilde{r}_1, \\ … … 958 958 we end up with the skew form of the eddy induced advective fluxes per unit area in $ijk$ space: 959 959 \begin{equation} 960 \label{eq: eiv_skew_ijk}960 \label{eq:TRIADS_eiv_skew_ijk} 961 961 \textbf{F}_\mathrm{eiv}^T = 962 962 \begin{pmatrix} … … 967 967 The total fluxes per unit physical area are then 968 968 \begin{equation} 969 \label{eq: eiv_skew_physical}969 \label{eq:TRIADS_eiv_skew_physical} 970 970 \begin{split} 971 971 f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T \\ … … 974 974 \end{split} 975 975 \end{equation} 976 Note that \autoref{eq: eiv_skew_physical} takes the same form whatever the vertical coordinate,977 though of course the slopes $\tilde{r}_i$ which define the $\psi_i$ in \autoref{eq: eiv_psi} are relative to976 Note that \autoref{eq:TRIADS_eiv_skew_physical} takes the same form whatever the vertical coordinate, 977 though of course the slopes $\tilde{r}_i$ which define the $\psi_i$ in \autoref{eq:TRIADS_eiv_psi} are relative to 978 978 geopotentials. 979 979 The tendency associated with eddy induced velocity is then simply the convergence of the fluxes 980 (\autoref{eq: eiv_skew_ijk}, \autoref{eq:eiv_skew_physical}), so981 \[ 982 % \label{eq: skew_eiv_conv}980 (\autoref{eq:TRIADS_eiv_skew_ijk}, \autoref{eq:TRIADS_eiv_skew_physical}), so 981 \[ 982 % \label{eq:TRIADS_skew_eiv_conv} 983 983 \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 } \left[ 984 984 \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right) … … 993 993 \subsection{Discrete skew flux formulation} 994 994 995 The skew fluxes in (\autoref{eq: eiv_skew_physical}, \autoref{eq:eiv_skew_ijk}),996 like the off-diagonal terms (\autoref{eq: i13c}, \autoref{eq:i31c}) of the small angle diffusion tensor,997 are best expressed in terms of the triad slopes, as in \autoref{fig: ISO_triad} and998 (\autoref{eq: i13}, \autoref{eq:i31});995 The skew fluxes in (\autoref{eq:TRIADS_eiv_skew_physical}, \autoref{eq:TRIADS_eiv_skew_ijk}), 996 like the off-diagonal terms (\autoref{eq:TRIADS_i13c}, \autoref{eq:TRIADS_i31c}) of the small angle diffusion tensor, 997 are best expressed in terms of the triad slopes, as in \autoref{fig:TRIADS_ISO_triad} and 998 (\autoref{eq:TRIADS_i13}, \autoref{eq:TRIADS_i31}); 999 999 but now in terms of the triad slopes $\tilde{\mathbb{R}}$ relative to geopotentials instead of 1000 1000 the $\mathbb{R}$ relative to coordinate surfaces. 1001 The discrete form of \autoref{eq: eiv_skew_ijk} using the slopes \autoref{eq:R} and1001 The discrete form of \autoref{eq:TRIADS_eiv_skew_ijk} using the slopes \autoref{eq:TRIADS_R} and 1002 1002 defining $A_e$ at $T$-points is then given by: 1003 1003 1004 1004 \begin{subequations} 1005 % \label{eq: allskewflux}1005 % \label{eq:TRIADS_allskewflux} 1006 1006 \begin{flalign*} 1007 % \label{eq: vect_skew_flux}1007 % \label{eq:TRIADS_vect_skew_flux} 1008 1008 \vect{F}_{\mathrm{eiv}}(T) &\equiv \sum_{\substack{i_p,\,k_p}} 1009 1009 \begin{pmatrix} … … 1012 1012 \end{pmatrix}, 1013 1013 \end{flalign*} 1014 where the skew flux in the $i$-direction associated with a given triad is (\autoref{eq: latflux-triad},1015 \autoref{eq: triadfluxu}):1014 where the skew flux in the $i$-direction associated with a given triad is (\autoref{eq:TRIADS_latflux-triad}, 1015 \autoref{eq:TRIADS_triadfluxu}): 1016 1016 \begin{align} 1017 \label{eq: skewfluxu}1017 \label{eq:TRIADS_skewfluxu} 1018 1018 _i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) &= + \fractext{1}{4} {A_e}_i^k{ 1019 1019 \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} … … 1021 1021 \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }, \\ 1022 1022 \intertext{ 1023 and \autoref{eq: triadfluxw} in the $k$-direction, changing the sign1024 to be consistent with \autoref{eq: eiv_skew_ijk}:1023 and \autoref{eq:TRIADS_triadfluxw} in the $k$-direction, changing the sign 1024 to be consistent with \autoref{eq:TRIADS_eiv_skew_ijk}: 1025 1025 } 1026 1026 _i^k {\mathbb{S}_w}_{i_p}^{k_p} (T) 1027 1027 &= -\fractext{1}{4} {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}} 1028 {_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}1028 {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:TRIADS_skewfluxw} 1029 1029 \end{align} 1030 1030 \end{subequations} … … 1038 1038 This can be seen %either from Appendix \autoref{apdx:eiv_skew} or 1039 1039 by considering the fluxes associated with a given triad slope $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. 1040 For, following \autoref{subsec: variance} and \autoref{eq:dvar_iso_i},1040 For, following \autoref{subsec:TRIADS_variance} and \autoref{eq:TRIADS_dvar_iso_i}, 1041 1041 the associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$ drives a net rate of change of variance, 1042 1042 summed over the two $T$-points $i+i_p-\fractext{1}{2},k$ and $i+i_p+\fractext{1}{2},k$, of 1043 1043 \begin{equation} 1044 \label{eq: dvar_eiv_i}1044 \label{eq:TRIADS_dvar_eiv_i} 1045 1045 _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], 1046 1046 \end{equation} … … 1048 1048 the $T$-points $i,k+k_p-\fractext{1}{2}$ (above) and $i,k+k_p+\fractext{1}{2}$ (below) of 1049 1049 \begin{equation} 1050 \label{eq: dvar_eiv_k}1050 \label{eq:TRIADS_dvar_eiv_k} 1051 1051 _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. 1052 1052 \end{equation} 1053 Inspection of the definitions (\autoref{eq: skewfluxu}, \autoref{eq:skewfluxw}) shows that1054 these two variance changes (\autoref{eq: dvar_eiv_i}, \autoref{eq:dvar_eiv_k}) sum to zero.1053 Inspection of the definitions (\autoref{eq:TRIADS_skewfluxu}, \autoref{eq:TRIADS_skewfluxw}) shows that 1054 these two variance changes (\autoref{eq:TRIADS_dvar_eiv_i}, \autoref{eq:TRIADS_dvar_eiv_k}) sum to zero. 1055 1055 Hence the two fluxes associated with each triad make no net contribution to the variance budget. 1056 1056 … … 1064 1064 For the change in gravitational PE driven by the $k$-flux is 1065 1065 \begin{align} 1066 \label{eq: vert_densityPE}1066 \label{eq:TRIADS_vert_densityPE} 1067 1067 g {e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) 1068 1068 &=g {e_{3w}}_{\,i}^{\,k+k_p}\left[-\alpha _i^k {\:}_i^k 1069 1069 {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k 1070 1070 {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\ 1071 \intertext{Substituting ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from \autoref{eq: skewfluxw}, gives}1071 \intertext{Substituting ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from \autoref{eq:TRIADS_skewfluxw}, gives} 1072 1072 % and separating out 1073 1073 % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$, … … 1080 1080 \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}}, 1081 1081 \end{align} 1082 using the definition of the triad slope $\rtriad{R}$, \autoref{eq: R} to1082 using the definition of the triad slope $\rtriad{R}$, \autoref{eq:TRIADS_R} to 1083 1083 express $-\alpha _i^k\delta_{i+ i_p}[T^k]+\beta_i^k\delta_{i+ i_p}[S^k]$ in terms of 1084 1084 $-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$. … … 1086 1086 Where the coordinates slope, the $i$-flux gives a PE change 1087 1087 \begin{multline} 1088 \label{eq: lat_densityPE}1088 \label{eq:TRIADS_lat_densityPE} 1089 1089 g \delta_{i+i_p}[z_T^k] 1090 1090 \left[ … … 1096 1096 \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}}, 1097 1097 \end{multline} 1098 (using \autoref{eq: skewfluxu}) and so the total PE change \autoref{eq:vert_densityPE} +1099 \autoref{eq: lat_densityPE} associated with the triad fluxes is1098 (using \autoref{eq:TRIADS_skewfluxu}) and so the total PE change \autoref{eq:TRIADS_vert_densityPE} + 1099 \autoref{eq:TRIADS_lat_densityPE} associated with the triad fluxes is 1100 1100 \begin{multline*} 1101 % \label{eq: tot_densityPE}1101 % \label{eq:TRIADS_tot_densityPE} 1102 1102 g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) + 1103 1103 g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\ … … 1110 1110 1111 1111 \subsection{Treatment of the triads at the boundaries} 1112 \label{sec: skew_bdry}1113 1114 Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes are masked at the boundaries 1115 in exactly the same way as are the triad slopes \rtriad{R} used for the iso-neutral diffusive fluxes, 1116 as described in \autoref{sec: iso_bdry} and \autoref{fig:bdry_triads}.1117 Thus surface layer triads $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are masked, 1118 and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when 1119 either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie\ the $i,k+1$ $u$-point is masked. 1112 \label{sec:TRIADS_skew_bdry} 1113 1114 Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes are masked at the boundaries 1115 in exactly the same way as are the triad slopes \rtriad{R} used for the iso-neutral diffusive fluxes, 1116 as described in \autoref{sec:TRIADS_iso_bdry} and \autoref{fig:TRIADS_bdry_triads}. 1117 Thus surface layer triads $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are masked, 1118 and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when 1119 either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie\ the $i,k+1$ $u$-point is masked. 1120 1120 The namelist parameter \np{ln\_botmix\_triad} has no effect on the eddy-induced skew-fluxes. 1121 1121 1122 1122 \subsection{Limiting of the slopes within the interior} 1123 \label{sec: limitskew}1124 1125 Presently, the iso-neutral slopes $\tilde{r}_i$ relative to geopotentials are limited to be less than $1/100$, 1126 exactly as in calculating the iso-neutral diffusion, \S \autoref{sec: limit}.1123 \label{sec:TRIADS_limitskew} 1124 1125 Presently, the iso-neutral slopes $\tilde{r}_i$ relative to geopotentials are limited to be less than $1/100$, 1126 exactly as in calculating the iso-neutral diffusion, \S \autoref{sec:TRIADS_limit}. 1127 1127 Each individual triad \rtriadt{R} is so limited. 1128 1128 1129 1129 \subsection{Tapering within the surface mixed layer} 1130 \label{sec: taperskew}1131 1132 The slopes $\tilde{r}_i$ relative to geopotentials (and thus the individual triads \rtriadt{R}) 1133 are always tapered linearly from their value immediately below the mixed layer to zero at the surface 1134 \autoref{eq: rmtilde}, as described in \autoref{sec:lintaper}.1135 This is option (c) of \autoref{fig: eiv_slp}.1136 This linear tapering for the slopes used to calculate the eddy-induced fluxes is unaffected by 1130 \label{sec:TRIADS_taperskew} 1131 1132 The slopes $\tilde{r}_i$ relative to geopotentials (and thus the individual triads \rtriadt{R}) 1133 are always tapered linearly from their value immediately below the mixed layer to zero at the surface 1134 \autoref{eq:TRIADS_rmtilde}, as described in \autoref{sec:TRIADS_lintaper}. 1135 This is option (c) of \autoref{fig:LDF_eiv_slp}. 1136 This linear tapering for the slopes used to calculate the eddy-induced fluxes is unaffected by 1137 1137 the value of \np{ln\_triad\_iso}. 1138 1138 … … 1140 1140 the horizontal (the most commonly used options in \NEMO: see \autoref{sec:LDF_coef}), 1141 1141 it is equivalent to a horizontal eiv (eddy-induced velocity) that is uniform within the mixed layer 1142 \autoref{eq: eiv_v}.1142 \autoref{eq:TRIADS_eiv_v}. 1143 1143 This ensures that the eiv velocities do not restratify the mixed layer \citep{treguier.held.ea_JPO97,danabasoglu.ferrari.ea_JC08}. 1144 1144 Equivantly, in terms of the skew-flux formulation we use here, … … 1148 1148 1149 1149 \subsection{Streamfunction diagnostics} 1150 \label{sec: sfdiag}1150 \label{sec:TRIADS_sfdiag} 1151 1151 1152 1152 Where the namelist parameter \np{ln\_traldf\_gdia}\forcode{ = .true.}, … … 1154 1154 Each time step, streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at 1155 1155 $uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$ (integer $i$, integer +1/2 $j$, integer +1/2 $k$) 1156 points (see Table \autoref{tab: cell}) respectively.1156 points (see Table \autoref{tab:DOM_cell}) respectively. 1157 1157 We follow \citep{griffies_bk04} and calculate the streamfunction at a given $uw$-point from 1158 1158 the surrounding four triads according to: 1159 1159 \[ 1160 % \label{eq: sfdiagi}1160 % \label{eq:TRIADS_sfdiagi} 1161 1161 {\psi_1}_{i+1/2}^{k+1/2}={\fractext{1}{4}}\sum_{\substack{i_p,\,k_p}} 1162 1162 {A_e}_{i+1/2-i_p}^{k+1/2-k_p}\:\triadd{i+1/2-i_p}{k+1/2-k_p}{R}{i_p}{k_p}. 1163 1163 \] 1164 1164 The streamfunction $\psi_1$ is calculated similarly at $vw$ points. 1165 The eddy-induced velocities are then calculated from the straightforward discretisation of \autoref{eq: eiv_v}:1166 \[ 1167 % \label{eq: eiv_v_discrete}1165 The eddy-induced velocities are then calculated from the straightforward discretisation of \autoref{eq:TRIADS_eiv_v}: 1166 \[ 1167 % \label{eq:TRIADS_eiv_v_discrete} 1168 1168 \begin{split} 1169 1169 {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), \\ -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_ASM.tex
r11537 r11543 53 53 additional tendency terms to the prognostic equations: 54 54 \begin{align*} 55 % \label{eq: wa_traj_iau}55 % \label{eq:ASM_wa_traj_iau} 56 56 {\mathbf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\mathbf x}^{b}(t_{0})] \; + \; F_{i} \delta \tilde{\mathbf x}^{a} 57 57 \end{align*} … … 66 66 The first function (namelist option \np{niaufn}=0) employs constant weights, 67 67 \begin{align} 68 \label{eq: F1_i}68 \label{eq:ASM_F1_i} 69 69 F^{(1)}_{i} 70 70 =\left\{ … … 80 80 with the weighting reduced linearly to a small value at the window end-points: 81 81 \begin{align} 82 \label{eq: F2_i}82 \label{eq:ASM_F2_i} 83 83 F^{(2)}_{i} 84 84 =\left\{ … … 92 92 \end{align} 93 93 where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even. 94 The weights described by \autoref{eq: F2_i} provide a smoother transition of the analysis trajectory from95 one assimilation cycle to the next than that described by \autoref{eq: F1_i}.94 The weights described by \autoref{eq:ASM_F2_i} provide a smoother transition of the analysis trajectory from 95 one assimilation cycle to the next than that described by \autoref{eq:ASM_F1_i}. 96 96 97 97 %========================================================================== … … 106 106 107 107 \begin{equation} 108 \label{eq: asm_dmp}108 \label{eq:ASM_dmp} 109 109 \left\{ 110 110 \begin{aligned} … … 120 120 121 121 \[ 122 % \label{eq: asm_div}122 % \label{eq:ASM_div} 123 123 \chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 124 124 \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right] … … 126 126 \] 127 127 128 By the application of \autoref{eq: asm_dmp} the divergence is filtered in each iteration,128 By the application of \autoref{eq:ASM_dmp} the divergence is filtered in each iteration, 129 129 and the vorticity is left unchanged. 130 130 In the presence of coastal boundaries with zero velocity increments perpendicular to the coast -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_DIA.tex
r11537 r11543 696 696 \begin{table} 697 697 \scriptsize 698 \begin{tabular x}{\textwidth}{|X|c|c|c|}698 \begin{tabular}{|l|c|c|} 699 699 \hline 700 700 tag ids affected by automatic definition of some of their attributes & 701 701 name attribute & 702 attribute value \\702 attribute value \\ 703 703 \hline 704 704 \hline 705 705 field\_definition & 706 706 freq\_op & 707 \np{rn\_rdt} \\707 \np{rn\_rdt} \\ 708 708 \hline 709 709 SBC & 710 710 freq\_op & 711 \np{rn\_rdt} $\times$ \np{nn\_fsbc} \\711 \np{rn\_rdt} $\times$ \np{nn\_fsbc} \\ 712 712 \hline 713 713 ptrc\_T & 714 714 freq\_op & 715 \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\715 \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\ 716 716 \hline 717 717 diad\_T & 718 718 freq\_op & 719 \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\719 \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\ 720 720 \hline 721 721 EqT, EqU, EqW & 722 722 jbegin, ni, & 723 according to the grid \\724 &723 according to the grid \\ 724 & 725 725 name\_suffix & 726 \\726 \\ 727 727 \hline 728 728 TAO, RAMA and PIRATA moorings & 729 729 zoom\_ibegin, zoom\_jbegin, & 730 according to the grid \\731 &730 according to the grid \\ 731 & 732 732 name\_suffix & 733 \\734 \hline 735 \end{tabular x}733 \\ 734 \hline 735 \end{tabular} 736 736 \end{table} 737 737 … … 739 739 740 740 \subsection{XML reference tables} 741 \label{subsec: IOM_xmlref}741 \label{subsec:DIA_IOM_xmlref} 742 742 743 743 \begin{enumerate} … … 1336 1336 the CF metadata standard. 1337 1337 Therefore while a user may wish to add their own metadata to the output files (as demonstrated in example 4 of 1338 section \autoref{subsec: IOM_xmlref}) the metadata should, for the most part, comply with the CF-1.5 standard.1338 section \autoref{subsec:DIA_IOM_xmlref}) the metadata should, for the most part, comply with the CF-1.5 standard. 1339 1339 1340 1340 Some metadata that may significantly increase the file size (horizontal cell areas and vertices) are controlled by … … 1407 1407 the mono-processor case (\ie\ global domain of {\small\ttfamily 182x149x31}). 1408 1408 An illustration of the potential space savings that NetCDF4 chunking and compression provides is given in 1409 table \autoref{tab: NC4} which compares the results of two short runs of the ORCA2\_LIM reference configuration with1409 table \autoref{tab:DIA_NC4} which compares the results of two short runs of the ORCA2\_LIM reference configuration with 1410 1410 a 4x2 mpi partitioning. 1411 1411 Note the variation in the compression ratio achieved which reflects chiefly the dry to wet volume ratio of … … 1447 1447 \end{tabular} 1448 1448 \caption{ 1449 \protect\label{tab: NC4}1449 \protect\label{tab:DIA_NC4} 1450 1450 Filesize comparison between NetCDF3 and NetCDF4 with chunking and compression 1451 1451 } … … 1515 1515 \section[FLO: On-Line Floats trajectories (\texttt{\textbf{key\_floats}})] 1516 1516 {FLO: On-Line Floats trajectories (\protect\key{floats})} 1517 \label{sec: FLO}1517 \label{sec:DIA_FLO} 1518 1518 %--------------------------------------------namflo------------------------------------------------------- 1519 1519 … … 1847 1847 \mathcal{V} &= \mathcal{A} \;\bar{\eta} 1848 1848 \end{split} 1849 \label{eq: MV_nBq}1849 \label{eq:DIA_MV_nBq} 1850 1850 \end{equation} 1851 1851 … … 1855 1855 \frac{1}{e_3} \partial_t ( e_3\,\rho) + \nabla( \rho \, \textbf{U} ) 1856 1856 = \left. \frac{\textit{emp}}{e_3}\right|_\textit{surface} 1857 \label{eq: Co_nBq}1857 \label{eq:DIA_Co_nBq} 1858 1858 \end{equation} 1859 1859 … … 1864 1864 \begin{equation} 1865 1865 \partial_t \mathcal{M} = \mathcal{A} \;\overline{\textit{emp}} 1866 \label{eq: Mass_nBq}1866 \label{eq:DIA_Mass_nBq} 1867 1867 \end{equation} 1868 1868 1869 1869 where $\overline{\textit{emp}} = \int_S \textit{emp}\,ds$ is the net mass flux through the ocean surface. 1870 Bringing \autoref{eq: Mass_nBq} and the time derivative of \autoref{eq:MV_nBq} together leads to1870 Bringing \autoref{eq:DIA_Mass_nBq} and the time derivative of \autoref{eq:DIA_MV_nBq} together leads to 1871 1871 the evolution equation of the mean sea level 1872 1872 … … 1874 1874 \partial_t \bar{\eta} = \frac{\overline{\textit{emp}}}{ \bar{\rho}} 1875 1875 - \frac{\mathcal{V}}{\mathcal{A}} \;\frac{\partial_t \bar{\rho} }{\bar{\rho}} 1876 \label{eq: ssh_nBq}1876 \label{eq:DIA_ssh_nBq} 1877 1877 \end{equation} 1878 1878 1879 The first term in equation \autoref{eq: ssh_nBq} alters sea level by adding or subtracting mass from the ocean.1879 The first term in equation \autoref{eq:DIA_ssh_nBq} alters sea level by adding or subtracting mass from the ocean. 1880 1880 The second term arises from temporal changes in the global mean density; \ie\ from steric effects. 1881 1881 1882 1882 In a Boussinesq fluid, $\rho$ is replaced by $\rho_o$ in all the equation except when $\rho$ appears multiplied by 1883 1883 the gravity (\ie\ in the hydrostatic balance of the primitive Equations). 1884 In particular, the mass conservation equation, \autoref{eq: Co_nBq}, degenerates into the incompressibility equation:1884 In particular, the mass conservation equation, \autoref{eq:DIA_Co_nBq}, degenerates into the incompressibility equation: 1885 1885 1886 1886 \[ 1887 1887 \frac{1}{e_3} \partial_t ( e_3 ) + \nabla( \textbf{U} ) = \left. \frac{\textit{emp}}{\rho_o \,e_3}\right|_ \textit{surface} 1888 % \label{eq: Co_Bq}1888 % \label{eq:DIA_Co_Bq} 1889 1889 \] 1890 1890 … … 1893 1893 \[ 1894 1894 \partial_t \mathcal{V} = \mathcal{A} \;\frac{\overline{\textit{emp}}}{\rho_o} 1895 % \label{eq: V_Bq}1895 % \label{eq:DIA_V_Bq} 1896 1896 \] 1897 1897 … … 1912 1912 \begin{equation} 1913 1913 \mathcal{M}_o = \mathcal{M} + \rho_o \,\eta_s \,\mathcal{A} 1914 \label{eq: M_Bq}1914 \label{eq:DIA_M_Bq} 1915 1915 \end{equation} 1916 1916 … … 1919 1919 Introducing the total density anomaly, $\mathcal{D}= \int_D d_a \,dv$, 1920 1920 where $d_a = (\rho -\rho_o ) / \rho_o$ is the density anomaly used in \NEMO\ (cf. \autoref{subsec:TRA_eos}) 1921 in \autoref{eq: M_Bq} leads to a very simple form for the steric height:1921 in \autoref{eq:DIA_M_Bq} leads to a very simple form for the steric height: 1922 1922 1923 1923 \begin{equation} 1924 1924 \eta_s = - \frac{1}{\mathcal{A}} \mathcal{D} 1925 \label{eq: steric_Bq}1925 \label{eq:DIA_steric_Bq} 1926 1926 \end{equation} 1927 1927 … … 1943 1943 (wetting and drying of grid point is not allowed). 1944 1944 1945 Third, the discretisation of \autoref{eq: steric_Bq} depends on the type of free surface which is considered.1945 Third, the discretisation of \autoref{eq:DIA_steric_Bq} depends on the type of free surface which is considered. 1946 1946 In the non linear free surface case, \ie\ \np{ln\_linssh}\forcode{=.true.}, it is given by 1947 1947 1948 1948 \[ 1949 1949 \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t} e_{2t} e_{3t} }{ \sum_{i,\,j,\,k} e_{1t} e_{2t} e_{3t} } 1950 % \label{eq: discrete_steric_Bq_nfs}1950 % \label{eq:DIA_discrete_steric_Bq_nfs} 1951 1951 \] 1952 1952 … … 1958 1958 \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 } 1959 1959 { \sum_{i,\,j,\,k} e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} e_{1t}e_{2t} \eta } 1960 % \label{eq: discrete_steric_Bq_fs}1960 % \label{eq:DIA_discrete_steric_Bq_fs} 1961 1961 \] 1962 1962 … … 1978 1978 \[ 1979 1979 \eta_s = - \frac{1}{\mathcal{A}} \int_D d_a(T,S_o,p_o) \,dv 1980 % \label{eq: thermosteric_Bq}1980 % \label{eq:DIA_thermosteric_Bq} 1981 1981 \] 1982 1982 … … 2014 2014 \includegraphics[width=\textwidth]{Fig_mask_subasins} 2015 2015 \caption{ 2016 \protect\label{fig: mask_subasins}2016 \protect\label{fig:DIA_mask_subasins} 2017 2017 Decomposition of the World Ocean (here ORCA2) into sub-basin used in to 2018 2018 compute the heat and salt transports as well as the meridional stream-function: … … 2045 2045 Pacific and Indo-Pacific Oceans (defined north of 30\deg{S}) as well as for the World Ocean. 2046 2046 The sub-basin decomposition requires an input file (\ifile{subbasins}) which contains three 2D mask arrays, 2047 the Indo-Pacific mask been deduced from the sum of the Indian and Pacific mask (\autoref{fig: mask_subasins}).2047 the Indo-Pacific mask been deduced from the sum of the Indian and Pacific mask (\autoref{fig:DIA_mask_subasins}). 2048 2048 2049 2049 %------------------------------------------namptr----------------------------------------- … … 2093 2093 \[ 2094 2094 C_u = |u|\frac{\rdt}{e_{1u}}, \quad C_v = |v|\frac{\rdt}{e_{2v}}, \quad C_w = |w|\frac{\rdt}{e_{3w}} 2095 % \label{eq: CFL}2095 % \label{eq:DIA_CFL} 2096 2096 \] 2097 2097 -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_DOM.tex
r11537 r11543 29 29 {\em 30 30 Compatibility changes Major simplification has moved many of the options to external domain configuration tools. 31 (see \autoref{apdx:DOM AINcfg})31 (see \autoref{apdx:DOMCFG}) 32 32 } \\ 33 33 {\em 3.x} & {\em Rachid Benshila, Gurvan Madec \& S\'{e}bastien Masson} & … … 38 38 \newpage 39 39 40 Having defined the continuous equations in \autoref{chap: PE} and chosen a time discretisation \autoref{chap:STP},40 Having defined the continuous equations in \autoref{chap:MB} and chosen a time discretisation \autoref{chap:TD}, 41 41 we need to choose a grid for spatial discretisation and related numerical algorithms. 42 42 In the present chapter, we provide a general description of the staggered grid used in \NEMO, … … 60 60 \includegraphics[width=\textwidth]{Fig_cell} 61 61 \caption{ 62 \protect\label{fig: cell}62 \protect\label{fig:DOM_cell} 63 63 Arrangement of variables. 64 64 $t$ indicates scalar points where temperature, salinity, density, pressure and … … 76 76 The arrangement of variables is the same in all directions. 77 77 It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector points $(u, v, w)$ defined in 78 the centre of each face of the cells (\autoref{fig: cell}).78 the centre of each face of the cells (\autoref{fig:DOM_cell}). 79 79 This is the generalisation to three dimensions of the well-known ``C'' grid in Arakawa's classification 80 80 \citep{mesinger.arakawa_bk76}. … … 84 84 The ocean mesh (\ie\ the position of all the scalar and vector points) is defined by the transformation that 85 85 gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$. 86 The grid-points are located at integer or integer and a half value of $(i,j,k)$ as indicated on \autoref{tab: cell}.86 The grid-points are located at integer or integer and a half value of $(i,j,k)$ as indicated on \autoref{tab:DOM_cell}. 87 87 In all the following, subscripts $u$, $v$, $w$, $f$, $uw$, $vw$ or $fw$ indicate the position of 88 88 the grid-point where the scale factors are defined. 89 Each scale factor is defined as the local analytical value provided by \autoref{eq: scale_factors}.89 Each scale factor is defined as the local analytical value provided by \autoref{eq:MB_scale_factors}. 90 90 As a result, the mesh on which partial derivatives $\pd[]{\lambda}$, $\pd[]{\varphi}$ and 91 91 $\pd[]{z}$ are evaluated is a uniform mesh with a grid size of unity. … … 95 95 centred finite difference approximation, not from their analytical expression. 96 96 This preserves the symmetry of the discrete set of equations and therefore satisfies many of 97 the continuous properties (see \autoref{apdx: C}).97 the continuous properties (see \autoref{apdx:INVARIANTS}). 98 98 A similar, related remark can be made about the domain size: 99 99 when needed, an area, volume, or the total ocean depth must be evaluated as the product or sum of the relevant scale factors … … 123 123 \end{tabular} 124 124 \caption{ 125 \protect\label{tab: cell}125 \protect\label{tab:DOM_cell} 126 126 Location of grid-points as a function of integer or integer and a half value of the column, line or level. 127 127 This indexing is only used for the writing of the semi -discrete equations. … … 145 145 secondly, analytical transformations encourage good practice by the definition of smoothly varying grids 146 146 (rather than allowing the user to set arbitrary jumps in thickness between adjacent layers) \citep{treguier.dukowicz.ea_JGR96}. 147 An example of the effect of such a choice is shown in \autoref{fig: zgr_e3}.147 An example of the effect of such a choice is shown in \autoref{fig:DOM_zgr_e3}. 148 148 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 149 149 \begin{figure}[!t] … … 151 151 \includegraphics[width=\textwidth]{Fig_zgr_e3} 152 152 \caption{ 153 \protect\label{fig: zgr_e3}153 \protect\label{fig:DOM_zgr_e3} 154 154 Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical, 155 155 and (b) analytically derived grid-point position and scale factors. … … 174 174 the midpoint between them are: 175 175 \begin{alignat*}{2} 176 % \label{eq: di_mi}176 % \label{eq:DOM_di_mi} 177 177 \delta_i [q] &= & &q (i + 1/2) - q (i - 1/2) \\ 178 178 \overline q^{\, i} &= &\big\{ &q (i + 1/2) + q (i - 1/2) \big\} / 2 … … 180 180 181 181 Similar operators are defined with respect to $i + 1/2$, $j$, $j + 1/2$, $k$, and $k + 1/2$. 182 Following \autoref{eq: PE_grad} and \autoref{eq:PE_lap}, the gradient of a variable $q$ defined at a $t$-point has182 Following \autoref{eq:MB_grad} and \autoref{eq:MB_lap}, the gradient of a variable $q$ defined at a $t$-point has 183 183 its three components defined at $u$-, $v$- and $w$-points while its Laplacian is defined at the $t$-point. 184 184 These operators have the following discrete forms in the curvilinear $s$-coordinates system: … … 198 198 \end{multline*} 199 199 200 Following \autoref{eq: PE_curl} and \autoref{eq:PE_div}, a vector $\vect A = (a_1,a_2,a_3)$ defined at200 Following \autoref{eq:MB_curl} and \autoref{eq:MB_div}, a vector $\vect A = (a_1,a_2,a_3)$ defined at 201 201 vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$, and $f$-points, and 202 202 its divergence defined at $t$-points: … … 255 255 In other words, the adjoint of the differencing and averaging operators are $\delta_i^* = \delta_{i + 1/2}$ and 256 256 $(\overline{\cdots}^{\, i})^* = \overline{\cdots}^{\, i + 1/2}$, respectively. 257 These two properties will be used extensively in the \autoref{apdx: C} to257 These two properties will be used extensively in the \autoref{apdx:INVARIANTS} to 258 258 demonstrate integral conservative properties of the discrete formulation chosen. 259 259 … … 269 269 \includegraphics[width=\textwidth]{Fig_index_hor} 270 270 \caption{ 271 \protect\label{fig: index_hor}271 \protect\label{fig:DOM_index_hor} 272 272 Horizontal integer indexing used in the \fortran code. 273 273 The dashed area indicates the cell in which variables contained in arrays have the same $i$- and $j$-indices … … 290 290 \label{subsec:DOM_Num_Index_hor} 291 291 292 The indexing in the horizontal plane has been chosen as shown in \autoref{fig: index_hor}.292 The indexing in the horizontal plane has been chosen as shown in \autoref{fig:DOM_index_hor}. 293 293 For an increasing $i$ index ($j$ index), 294 294 the $t$-point and the eastward $u$-point (northward $v$-point) have the same index 295 (see the dashed area in \autoref{fig: index_hor}).295 (see the dashed area in \autoref{fig:DOM_index_hor}). 296 296 A $t$-point and its nearest north-east $f$-point have the same $i$-and $j$-indices. 297 297 … … 306 306 given in \autoref{subsec:DOM_cell}. 307 307 The sea surface corresponds to the $w$-level $k = 1$, which is the same index as the $t$-level just below 308 (\autoref{fig: index_vert}).308 (\autoref{fig:DOM_index_vert}). 309 309 The last $w$-level ($k = jpk$) either corresponds to or is below the ocean floor while 310 the last $t$-level is always outside the ocean domain (\autoref{fig: index_vert}).310 the last $t$-level is always outside the ocean domain (\autoref{fig:DOM_index_vert}). 311 311 Note that a $w$-point and the directly underlaying $t$-point have a common $k$ index 312 312 (\ie\ $t$-points and their nearest $w$-point neighbour in negative index direction), 313 313 in contrast to the indexing on the horizontal plane where the $t$-point has the same index as 314 314 the nearest velocity points in the positive direction of the respective horizontal axis index 315 (compare the dashed area in \autoref{fig: index_hor} and \autoref{fig:index_vert}).315 (compare the dashed area in \autoref{fig:DOM_index_hor} and \autoref{fig:DOM_index_vert}). 316 316 Since the scale factors are chosen to be strictly positive, 317 317 a \textit{minus sign} is included in the \fortran implementations of … … 324 324 \includegraphics[width=\textwidth]{Fig_index_vert} 325 325 \caption{ 326 \protect\label{fig: index_vert}326 \protect\label{fig:DOM_index_vert} 327 327 Vertical integer indexing used in the \fortran code. 328 328 Note that the $k$-axis is oriented downward. … … 363 363 the model domain itself can be altered by runtime selections. 364 364 The code previously used to perform vertical discretisation has been incorporated into an external tool 365 (\path{./tools/DOMAINcfg}) which is briefly described in \autoref{apdx:DOM AINcfg}.365 (\path{./tools/DOMAINcfg}) which is briefly described in \autoref{apdx:DOMCFG}. 366 366 367 367 The next subsections summarise the parameter and fields related to the configuration of the whole model domain. … … 418 418 The values of the geographic longitude and latitude arrays at indices $i,j$ correspond to 419 419 the analytical expressions of the longitude $\lambda$ and latitude $\varphi$ as a function of $(i,j)$, 420 evaluated at the values as specified in \autoref{tab: cell} for the respective grid-point position.420 evaluated at the values as specified in \autoref{tab:DOM_cell} for the respective grid-point position. 421 421 The calculation of the values of the horizontal scale factor arrays in general additionally involves 422 422 partial derivatives of $\lambda$ and $\varphi$ with respect to $i$ and $j$, … … 485 485 \includegraphics[width=\textwidth]{Fig_z_zps_s_sps} 486 486 \caption{ 487 \protect\label{fig: z_zps_s_sps}487 \protect\label{fig:DOM_z_zps_s_sps} 488 488 The ocean bottom as seen by the model: 489 489 (a) $z$-coordinate with full step, … … 510 510 By default a non-linear free surface is used (\np{ln\_linssh} set to \forcode{=.false.} in \nam{dom}): 511 511 the coordinate follow the time-variation of the free surface so that the transformation is time dependent: 512 $z(i,j,k,t)$ (\eg\ \autoref{fig: z_zps_s_sps}f).512 $z(i,j,k,t)$ (\eg\ \autoref{fig:DOM_z_zps_s_sps}f). 513 513 When a linear free surface is assumed (\np{ln\_linssh} set to \forcode{=.true.} in \nam{dom}), 514 514 the vertical coordinates are fixed in time, but the seawater can move up and down across the $z_0$ surface … … 527 527 \medskip 528 528 The decision on these choices must be made when the \np{cn\_domcfg} file is constructed. 529 Three main choices are offered (\autoref{fig: z_zps_s_sps}a-c):529 Three main choices are offered (\autoref{fig:DOM_z_zps_s_sps}a-c): 530 530 531 531 \begin{itemize} … … 536 536 537 537 Additionally, hybrid combinations of the three main coordinates are available: 538 $s-z$ or $s-zps$ coordinate (\autoref{fig: z_zps_s_sps}d and \autoref{fig:z_zps_s_sps}e).538 $s-z$ or $s-zps$ coordinate (\autoref{fig:DOM_z_zps_s_sps}d and \autoref{fig:DOM_z_zps_s_sps}e). 539 539 540 540 A further choice related to vertical coordinate concerns … … 678 678 \section[Initial state (\textit{istate.F90} and \textit{dtatsd.F90})] 679 679 {Initial state (\protect\mdl{istate} and \protect\mdl{dtatsd})} 680 \label{sec:D TA_tsd}680 \label{sec:DOM_DTA_tsd} 681 681 %-----------------------------------------namtsd------------------------------------------- 682 682 \nlst{namtsd} … … 697 697 Initial values for T and S are set via a user supplied \rou{usr\_def\_istate} routine contained in \mdl{userdef\_istate}. 698 698 The default version sets horizontally uniform T and profiles as used in the GYRE configuration 699 (see \autoref{sec:CFG _gyre}).699 (see \autoref{sec:CFGS_gyre}). 700 700 \end{description} 701 701 -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_DYN.tex
r11537 r11543 65 65 % Horizontal divergence and relative vorticity 66 66 %-------------------------------------------------------------------------------------------------------------- 67 \subsection[Horizontal divergence and relative vorticity (\textit{divcur.F90})] 68 {Horizontal divergence and relative vorticity (\protect\mdl{divcur})} 67 \subsection[Horizontal divergence and relative vorticity (\textit{divcur.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{divcur})} 69 68 \label{subsec:DYN_divcur} 70 69 71 70 The vorticity is defined at an $f$-point (\ie\ corner point) as follows: 72 71 \begin{equation} 73 \label{eq: divcur_cur}72 \label{eq:DYN_divcur_cur} 74 73 \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right] 75 74 -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right) … … 79 78 It is given by: 80 79 \[ 81 % \label{eq: divcur_div}80 % \label{eq:DYN_divcur_div} 82 81 \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 83 82 \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right] … … 102 101 % Sea Surface Height evolution 103 102 %-------------------------------------------------------------------------------------------------------------- 104 \subsection[Horizontal divergence and relative vorticity (\textit{sshwzv.F90})] 105 {Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})} 103 \subsection[Horizontal divergence and relative vorticity (\textit{sshwzv.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})} 106 104 \label{subsec:DYN_sshwzv} 107 105 108 106 The sea surface height is given by: 109 107 \begin{equation} 110 \label{eq: dynspg_ssh}108 \label{eq:DYN_spg_ssh} 111 109 \begin{aligned} 112 110 \frac{\partial \eta }{\partial t} … … 123 121 \textit{emp} can be written as the evaporation minus precipitation, minus the river runoff. 124 122 The sea-surface height is evaluated using exactly the same time stepping scheme as 125 the tracer equation \autoref{eq: tra_nxt}:123 the tracer equation \autoref{eq:TRA_nxt}: 126 124 a leapfrog scheme in combination with an Asselin time filter, 127 \ie\ the velocity appearing in \autoref{eq: dynspg_ssh} is centred in time (\textit{now} velocity).125 \ie\ the velocity appearing in \autoref{eq:DYN_spg_ssh} is centred in time (\textit{now} velocity). 128 126 This is of paramount importance. 129 127 Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to … … 134 132 taking into account the change of the thickness of the levels: 135 133 \begin{equation} 136 \label{eq: wzv}134 \label{eq:DYN_wzv} 137 135 \left\{ 138 136 \begin{aligned} … … 148 146 re-orientated downward. 149 147 \gmcomment{not sure of this... to be modified with the change in emp setting} 150 In the case of a linear free surface, the time derivative in \autoref{eq: wzv} disappears.148 In the case of a linear free surface, the time derivative in \autoref{eq:DYN_wzv} disappears. 151 149 The upper boundary condition applies at a fixed level $z=0$. 152 150 The top vertical velocity is thus equal to the divergence of the barotropic transport 153 (\ie\ the first term in the right-hand-side of \autoref{eq: dynspg_ssh}).151 (\ie\ the first term in the right-hand-side of \autoref{eq:DYN_spg_ssh}). 154 152 155 153 Note also that whereas the vertical velocity has the same discrete expression in $z$- and $s$-coordinates, … … 157 155 in the second case, $w$ is the velocity normal to the $s$-surfaces. 158 156 Note also that the $k$-axis is re-orientated downwards in the \fortran code compared to 159 the indexing used in the semi-discrete equations such as \autoref{eq: wzv}157 the indexing used in the semi-discrete equations such as \autoref{eq:DYN_wzv} 160 158 (see \autoref{subsec:DOM_Num_Index_vertical}). 161 159 … … 183 181 % Vorticity term 184 182 % ------------------------------------------------------------------------------------------------------------- 185 \subsection[Vorticity term (\textit{dynvor.F90})] 186 {Vorticity term (\protect\mdl{dynvor})} 183 \subsection[Vorticity term (\textit{dynvor.F90})]{Vorticity term (\protect\mdl{dynvor})} 187 184 \label{subsec:DYN_vor} 188 185 %------------------------------------------nam_dynvor---------------------------------------------------- … … 198 195 horizontal kinetic energy for the planetary vorticity term (MIX scheme); 199 196 or conserving both the potential enstrophy of horizontally non-divergent flow and horizontal kinetic energy 200 (EEN scheme) (see \autoref{subsec: C_vorEEN}).197 (EEN scheme) (see \autoref{subsec:INVARIANTS_vorEEN}). 201 198 In the case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the consistency of 202 199 vorticity term with analytical equations (\np{ln\_dynvor\_con}\forcode{=.true.}). … … 206 203 % enstrophy conserving scheme 207 204 %------------------------------------------------------------- 208 \subsubsection[Enstrophy conserving scheme (\forcode{ln_dynvor_ens=.true.})] 209 {Enstrophy conserving scheme (\protect\np{ln\_dynvor\_ens}\forcode{=.true.})} 205 \subsubsection[Enstrophy conserving scheme (\forcode{ln_dynvor_ens = .true.})]{Enstrophy conserving scheme (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})} 210 206 \label{subsec:DYN_vor_ens} 211 207 … … 216 212 It is given by: 217 213 \begin{equation} 218 \label{eq: dynvor_ens}214 \label{eq:DYN_vor_ens} 219 215 \left\{ 220 216 \begin{aligned} … … 230 226 % energy conserving scheme 231 227 %------------------------------------------------------------- 232 \subsubsection[Energy conserving scheme (\forcode{ln_dynvor_ene=.true.})] 233 {Energy conserving scheme (\protect\np{ln\_dynvor\_ene}\forcode{=.true.})} 228 \subsubsection[Energy conserving scheme (\forcode{ln_dynvor_ene = .true.})]{Energy conserving scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})} 234 229 \label{subsec:DYN_vor_ene} 235 230 … … 237 232 It is given by: 238 233 \begin{equation} 239 \label{eq: dynvor_ene}234 \label{eq:DYN_vor_ene} 240 235 \left\{ 241 236 \begin{aligned} … … 251 246 % mix energy/enstrophy conserving scheme 252 247 %------------------------------------------------------------- 253 \subsubsection[Mixed energy/enstrophy conserving scheme (\forcode{ln_dynvor_mix=.true.})] 254 {Mixed energy/enstrophy conserving scheme (\protect\np{ln\_dynvor\_mix}\forcode{=.true.})} 248 \subsubsection[Mixed energy/enstrophy conserving scheme (\forcode{ln_dynvor_mix = .true.})]{Mixed energy/enstrophy conserving scheme (\protect\np{ln\_dynvor\_mix}\forcode{ = .true.})} 255 249 \label{subsec:DYN_vor_mix} 256 250 257 251 For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the two previous schemes is used. 258 It consists of the ENS scheme (\autoref{eq: dynvor_ens}) for the relative vorticity term,259 and of the ENE scheme (\autoref{eq: dynvor_ene}) applied to the planetary vorticity term.260 \[ 261 % \label{eq: dynvor_mix}252 It consists of the ENS scheme (\autoref{eq:DYN_vor_ens}) for the relative vorticity term, 253 and of the ENE scheme (\autoref{eq:DYN_vor_ene}) applied to the planetary vorticity term. 254 \[ 255 % \label{eq:DYN_vor_mix} 262 256 \left\{ { 263 257 \begin{aligned} … … 277 271 % energy and enstrophy conserving scheme 278 272 %------------------------------------------------------------- 279 \subsubsection[Energy and enstrophy conserving scheme (\forcode{ln_dynvor_een=.true.})] 280 {Energy and enstrophy conserving scheme (\protect\np{ln\_dynvor\_een}\forcode{=.true.})} 273 \subsubsection[Energy and enstrophy conserving scheme (\forcode{ln_dynvor_een = .true.})]{Energy and enstrophy conserving scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 281 274 \label{subsec:DYN_vor_een} 282 275 … … 297 290 The idea is to get rid of the double averaging by considering triad combinations of vorticity. 298 291 It is noteworthy that this solution is conceptually quite similar to the one proposed by 299 \citep{griffies.gnanadesikan.ea_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx: C}).292 \citep{griffies.gnanadesikan.ea_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:INVARIANTS}). 300 293 301 294 The \citet{arakawa.hsu_MWR90} vorticity advection scheme for a single layer is modified … … 303 296 First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point: 304 297 \[ 305 % \label{eq: pot_vor}298 % \label{eq:DYN_pot_vor} 306 299 q = \frac{\zeta +f} {e_{3f} } 307 300 \] 308 where the relative vorticity is defined by (\autoref{eq: divcur_cur}),301 where the relative vorticity is defined by (\autoref{eq:DYN_divcur_cur}), 309 302 the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is: 310 303 \begin{equation} 311 \label{eq: een_e3f}304 \label{eq:DYN_een_e3f} 312 305 e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2} 313 306 \end{equation} … … 326 319 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 327 320 328 A key point in \autoref{eq: een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made.321 A key point in \autoref{eq:DYN_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 329 322 It uses the sum of masked t-point vertical scale factor divided either by the sum of the four t-point masks 330 323 (\np{nn\_een\_e3f}\forcode{=1}), or just by $4$ (\np{nn\_een\_e3f}\forcode{=.true.}). … … 340 333 (\autoref{fig:DYN_een_triad}): 341 334 \begin{equation} 342 \label{eq: Q_triads}335 \label{eq:DYN_Q_triads} 343 336 _i^j \mathbb{Q}^{i_p}_{j_p} 344 337 = \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) … … 348 341 Finally, the vorticity terms are represented as: 349 342 \begin{equation} 350 \label{eq: dynvor_een}343 \label{eq:DYN_vor_een} 351 344 \left\{ { 352 345 \begin{aligned} … … 361 354 This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes. 362 355 It conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow 363 (\ie\ $\chi$=$0$) (see \autoref{subsec: C_vorEEN}).356 (\ie\ $\chi$=$0$) (see \autoref{subsec:INVARIANTS_vorEEN}). 364 357 Applied to a realistic ocean configuration, it has been shown that it leads to a significant reduction of 365 358 the noise in the vertical velocity field \citep{le-sommer.penduff.ea_OM09}. … … 371 364 % Kinetic Energy Gradient term 372 365 %-------------------------------------------------------------------------------------------------------------- 373 \subsection[Kinetic energy gradient term (\textit{dynkeg.F90})] 374 {Kinetic energy gradient term (\protect\mdl{dynkeg})} 366 \subsection[Kinetic energy gradient term (\textit{dynkeg.F90})]{Kinetic energy gradient term (\protect\mdl{dynkeg})} 375 367 \label{subsec:DYN_keg} 376 368 377 As demonstrated in \autoref{apdx: C},369 As demonstrated in \autoref{apdx:INVARIANTS}, 378 370 there is a single discrete formulation of the kinetic energy gradient term that, 379 371 together with the formulation chosen for the vertical advection (see below), 380 372 conserves the total kinetic energy: 381 373 \[ 382 % \label{eq: dynkeg}374 % \label{eq:DYN_keg} 383 375 \left\{ 384 376 \begin{aligned} … … 392 384 % Vertical advection term 393 385 %-------------------------------------------------------------------------------------------------------------- 394 \subsection[Vertical advection term (\textit{dynzad.F90})] 395 {Vertical advection term (\protect\mdl{dynzad})} 386 \subsection[Vertical advection term (\textit{dynzad.F90})]{Vertical advection term (\protect\mdl{dynzad})} 396 387 \label{subsec:DYN_zad} 397 388 … … 400 391 conserves the total kinetic energy. 401 392 Indeed, the change of KE due to the vertical advection is exactly balanced by 402 the change of KE due to the gradient of KE (see \autoref{apdx: C}).403 \[ 404 % \label{eq: dynzad}393 the change of KE due to the gradient of KE (see \autoref{apdx:INVARIANTS}). 394 \[ 395 % \label{eq:DYN_zad} 405 396 \left\{ 406 397 \begin{aligned} … … 439 430 % Coriolis plus curvature metric terms 440 431 %-------------------------------------------------------------------------------------------------------------- 441 \subsection[Coriolis plus curvature metric terms (\textit{dynvor.F90})] 442 {Coriolis plus curvature metric terms (\protect\mdl{dynvor})} 432 \subsection[Coriolis plus curvature metric terms (\textit{dynvor.F90})]{Coriolis plus curvature metric terms (\protect\mdl{dynvor})} 443 433 \label{subsec:DYN_cor_flux} 444 434 … … 447 437 It is given by: 448 438 \begin{multline*} 449 % \label{eq: dyncor_metric}439 % \label{eq:DYN_cor_metric} 450 440 f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i} - u\frac{\partial e_1 }{\partial j}} \right) \\ 451 441 \equiv f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right] … … 453 443 \end{multline*} 454 444 455 Any of the (\autoref{eq: dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een}) schemes can be used to445 Any of the (\autoref{eq:DYN_vor_ens}), (\autoref{eq:DYN_vor_ene}) and (\autoref{eq:DYN_vor_een}) schemes can be used to 456 446 compute the product of the Coriolis parameter and the vorticity. 457 However, the energy-conserving scheme (\autoref{eq: dynvor_een}) has exclusively been used to date.447 However, the energy-conserving scheme (\autoref{eq:DYN_vor_een}) has exclusively been used to date. 458 448 This term is evaluated using a leapfrog scheme, \ie\ the velocity is centred in time (\textit{now} velocity). 459 449 … … 461 451 % Flux form Advection term 462 452 %-------------------------------------------------------------------------------------------------------------- 463 \subsection[Flux form advection term (\textit{dynadv.F90})] 464 {Flux form advection term (\protect\mdl{dynadv})} 453 \subsection[Flux form advection term (\textit{dynadv.F90})]{Flux form advection term (\protect\mdl{dynadv})} 465 454 \label{subsec:DYN_adv_flux} 466 455 467 456 The discrete expression of the advection term is given by: 468 457 \[ 469 % \label{eq: dynadv}458 % \label{eq:DYN_adv} 470 459 \left\{ 471 460 \begin{aligned} … … 495 484 % 2nd order centred scheme 496 485 %------------------------------------------------------------- 497 \subsubsection[CEN2: $2^{nd}$ order centred scheme (\forcode{ln_dynadv_cen2=.true.})] 498 {CEN2: $2^{nd}$ order centred scheme (\protect\np{ln\_dynadv\_cen2}\forcode{=.true.})} 486 \subsubsection[CEN2: $2^{nd}$ order centred scheme (\forcode{ln_dynadv_cen2 = .true.})]{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln\_dynadv\_cen2}\forcode{ = .true.})} 499 487 \label{subsec:DYN_adv_cen2} 500 488 501 489 In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points: 502 490 \begin{equation} 503 \label{eq: dynadv_cen2}491 \label{eq:DYN_adv_cen2} 504 492 \left\{ 505 493 \begin{aligned} … … 519 507 % UBS scheme 520 508 %------------------------------------------------------------- 521 \subsubsection[UBS: Upstream Biased Scheme (\forcode{ln_dynadv_ubs=.true.})] 522 {UBS: Upstream Biased Scheme (\protect\np{ln\_dynadv\_ubs}\forcode{=.true.})} 509 \subsubsection[UBS: Upstream Biased Scheme (\forcode{ln_dynadv_ubs = .true.})]{UBS: Upstream Biased Scheme (\protect\np{ln\_dynadv\_ubs}\forcode{ = .true.})} 523 510 \label{subsec:DYN_adv_ubs} 524 511 … … 527 514 For example, the evaluation of $u_T^{ubs} $ is done as follows: 528 515 \begin{equation} 529 \label{eq: dynadv_ubs}516 \label{eq:DYN_adv_ubs} 530 517 u_T^{ubs} =\overline u ^i-\;\frac{1}{6} 531 518 \begin{cases} … … 547 534 The UBS scheme is not used in all directions. 548 535 In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, \ie\ $u_{uw}^{ubs}$ and 549 $u_{vw}^{ubs}$ in \autoref{eq: dynadv_cen2} are used.536 $u_{vw}^{ubs}$ in \autoref{eq:DYN_adv_cen2} are used. 550 537 UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm pursue the 551 538 sentence:Since vertical mixing of momentum is a source term of the TKE equation... } 552 539 553 For stability reasons, the first term in (\autoref{eq: dynadv_ubs}),540 For stability reasons, the first term in (\autoref{eq:DYN_adv_ubs}), 554 541 which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time), 555 542 while the second term, which is the diffusion part of the scheme, … … 559 546 Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by 560 547 one coefficient. 561 Replacing $1/6$ by $1/8$ in (\autoref{eq: dynadv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}.548 Replacing $1/6$ by $1/8$ in (\autoref{eq:DYN_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 562 549 This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded. 563 550 Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_ubs} module and obtain a QUICK scheme. … … 573 560 % Hydrostatic pressure gradient term 574 561 % ================================================================ 575 \section[Hydrostatic pressure gradient (\textit{dynhpg.F90})] 576 {Hydrostatic pressure gradient (\protect\mdl{dynhpg})} 562 \section[Hydrostatic pressure gradient (\textit{dynhpg.F90})]{Hydrostatic pressure gradient (\protect\mdl{dynhpg})} 577 563 \label{sec:DYN_hpg} 578 564 %------------------------------------------nam_dynhpg--------------------------------------------------- … … 596 582 % z-coordinate with full step 597 583 %-------------------------------------------------------------------------------------------------------------- 598 \subsection[Full step $Z$-coordinate (\forcode{ln_dynhpg_zco=.true.})] 599 {Full step $Z$-coordinate (\protect\np{ln\_dynhpg\_zco}\forcode{=.true.})} 584 \subsection[Full step $Z$-coordinate (\forcode{ln_dynhpg_zco = .true.})]{Full step $Z$-coordinate (\protect\np{ln\_dynhpg\_zco}\forcode{ = .true.})} 600 585 \label{subsec:DYN_hpg_zco} 601 586 … … 607 592 for $k=km$ (surface layer, $jk=1$ in the code) 608 593 \begin{equation} 609 \label{eq: dynhpg_zco_surf}594 \label{eq:DYN_hpg_zco_surf} 610 595 \left\{ 611 596 \begin{aligned} … … 620 605 for $1<k<km$ (interior layer) 621 606 \begin{equation} 622 \label{eq: dynhpg_zco}607 \label{eq:DYN_hpg_zco} 623 608 \left\{ 624 609 \begin{aligned} … … 633 618 \end{equation} 634 619 635 Note that the $1/2$ factor in (\autoref{eq: dynhpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as620 Note that the $1/2$ factor in (\autoref{eq:DYN_hpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as 636 621 the vertical derivative of the scale factor at the surface level ($z=0$). 637 622 Note also that in case of variable volume level (\texttt{vvl?} defined), 638 the surface pressure gradient is included in \autoref{eq: dynhpg_zco_surf} and639 \autoref{eq: dynhpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$.623 the surface pressure gradient is included in \autoref{eq:DYN_hpg_zco_surf} and 624 \autoref{eq:DYN_hpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$. 640 625 641 626 %-------------------------------------------------------------------------------------------------------------- 642 627 % z-coordinate with partial step 643 628 %-------------------------------------------------------------------------------------------------------------- 644 \subsection[Partial step $Z$-coordinate (\forcode{ln_dynhpg_zps=.true.})] 645 {Partial step $Z$-coordinate (\protect\np{ln\_dynhpg\_zps}\forcode{=.true.})} 629 \subsection[Partial step $Z$-coordinate (\forcode{ln_dynhpg_zps = .true.})]{Partial step $Z$-coordinate (\protect\np{ln\_dynhpg\_zps}\forcode{ = .true.})} 646 630 \label{subsec:DYN_hpg_zps} 647 631 … … 674 658 $\bullet$ Traditional coding (see for example \citet{madec.delecluse.ea_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{=.true.}) 675 659 \begin{equation} 676 \label{eq: dynhpg_sco}660 \label{eq:DYN_hpg_sco} 677 661 \left\{ 678 662 \begin{aligned} … … 686 670 687 671 Where the first term is the pressure gradient along coordinates, 688 computed as in \autoref{eq: dynhpg_zco_surf} - \autoref{eq:dynhpg_zco},672 computed as in \autoref{eq:DYN_hpg_zco_surf} - \autoref{eq:DYN_hpg_zco}, 689 673 and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point 690 674 ($e_{3w}$). … … 698 682 (\np{ln\_dynhpg\_djc}\forcode{=.true.}) (currently disabled; under development) 699 683 700 Note that expression \autoref{eq: dynhpg_sco} is commonly used when the variable volume formulation is activated684 Note that expression \autoref{eq:DYN_hpg_sco} is commonly used when the variable volume formulation is activated 701 685 (\texttt{vvl?}) because in that case, even with a flat bottom, 702 686 the coordinate surfaces are not horizontal but follow the free surface \citep{levier.treguier.ea_rpt07}. … … 712 696 \subsection{Ice shelf cavity} 713 697 \label{subsec:DYN_hpg_isf} 698 714 699 Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and 715 700 the pressure gradient due to the ocean load (\np{ln\_dynhpg\_isf}\forcode{=.true.}).\\ … … 722 707 A detailed description of this method is described in \citet{losch_JGR08}.\\ 723 708 724 The pressure gradient due to ocean load is computed using the expression \autoref{eq: dynhpg_sco} described in709 The pressure gradient due to ocean load is computed using the expression \autoref{eq:DYN_hpg_sco} described in 725 710 \autoref{subsec:DYN_hpg_sco}. 726 711 … … 728 713 % Time-scheme 729 714 %-------------------------------------------------------------------------------------------------------------- 730 \subsection[Time-scheme (\forcode{ln_dynhpg_imp={.true.,.false.}})] 731 {Time-scheme (\protect\np{ln\_dynhpg\_imp}\forcode{=.true.,.false.})} 715 \subsection[Time-scheme (\forcode{ln_dynhpg_imp = .{true,false}.})]{Time-scheme (\protect\np{ln\_dynhpg\_imp}\forcode{ = .\{true,false\}}.)} 732 716 \label{subsec:DYN_hpg_imp} 733 717 … … 748 732 749 733 \begin{equation} 750 \label{eq: dynhpg_lf}734 \label{eq:DYN_hpg_lf} 751 735 \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; 752 736 -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right] … … 755 739 $\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}\forcode{=.true.}): 756 740 \begin{equation} 757 \label{eq: dynhpg_imp}741 \label{eq:DYN_hpg_imp} 758 742 \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; 759 743 -\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] 760 744 \end{equation} 761 745 762 The semi-implicit time scheme \autoref{eq: dynhpg_imp} is made possible without746 The semi-implicit time scheme \autoref{eq:DYN_hpg_imp} is made possible without 763 747 significant additional computation since the density can be updated to time level $t+\rdt$ before 764 748 computing the horizontal hydrostatic pressure gradient. 765 749 It can be easily shown that the stability limit associated with the hydrostatic pressure gradient doubles using 766 \autoref{eq: dynhpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:dynhpg_lf}.767 Note that \autoref{eq: dynhpg_imp} is equivalent to applying a time filter to the pressure gradient to750 \autoref{eq:DYN_hpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:DYN_hpg_lf}. 751 Note that \autoref{eq:DYN_hpg_imp} is equivalent to applying a time filter to the pressure gradient to 768 752 eliminate high frequency IGWs. 769 Obviously, when using \autoref{eq: dynhpg_imp},753 Obviously, when using \autoref{eq:DYN_hpg_imp}, 770 754 the doubling of the time-step is achievable only if no other factors control the time-step, 771 755 such as the stability limits associated with advection or diffusion. … … 777 761 The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows: 778 762 \[ 779 % \label{eq: rho_flt}763 % \label{eq:DYN_rho_flt} 780 764 \rho^t = \rho( \widetilde{T},\widetilde {S},z_t) 781 765 \quad \text{with} \quad … … 790 774 % Surface Pressure Gradient 791 775 % ================================================================ 792 \section[Surface pressure gradient (\textit{dynspg.F90})] 793 {Surface pressure gradient (\protect\mdl{dynspg})} 776 \section[Surface pressure gradient (\textit{dynspg.F90})]{Surface pressure gradient (\protect\mdl{dynspg})} 794 777 \label{sec:DYN_spg} 795 778 %-----------------------------------------nam_dynspg---------------------------------------------------- … … 799 782 800 783 Options are defined through the \nam{dyn\_spg} namelist variables. 801 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec: PE_hor_pg}).784 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}). 802 785 The main distinction is between the fixed volume case (linear free surface) and 803 786 the variable volume case (nonlinear free surface, \texttt{vvl?} is defined). 804 In the linear free surface case (\autoref{subsec: PE_free_surface})787 In the linear free surface case (\autoref{subsec:MB_free_surface}) 805 788 the vertical scale factors $e_{3}$ are fixed in time, 806 while they are time-dependent in the nonlinear case (\autoref{subsec: PE_free_surface}).789 while they are time-dependent in the nonlinear case (\autoref{subsec:MB_free_surface}). 807 790 With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 808 791 which imposes a very small time step when an explicit time stepping is used. 809 792 Two methods are proposed to allow a longer time step for the three-dimensional equations: 810 the filtered free surface, which is a modification of the continuous equations (see \autoref{eq: PE_flt?}),793 the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:MB_flt?}), 811 794 and the split-explicit free surface described below. 812 795 The extra term introduced in the filtered method is calculated implicitly, … … 815 798 816 799 The form of the surface pressure gradient term depends on how the user wants to 817 handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec: PE_hor_pg}).800 handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:MB_hor_pg}). 818 801 Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx): 819 802 an explicit formulation which requires a small time step; … … 829 812 % Explicit free surface formulation 830 813 %-------------------------------------------------------------------------------------------------------------- 831 \subsection[Explicit free surface (\texttt{ln\_dynspg\_exp}\forcode{=.true.})] 832 {Explicit free surface (\protect\np{ln\_dynspg\_exp}\forcode{=.true.})} 814 \subsection[Explicit free surface (\texttt{ln\_dynspg\_exp}\forcode{ = .true.})]{Explicit free surface (\protect\np{ln\_dynspg\_exp}\forcode{ = .true.})} 833 815 \label{subsec:DYN_spg_exp} 834 816 … … 839 821 is thus simply given by : 840 822 \begin{equation} 841 \label{eq: dynspg_exp}823 \label{eq:DYN_spg_exp} 842 824 \left\{ 843 825 \begin{aligned} … … 856 838 % Split-explict free surface formulation 857 839 %-------------------------------------------------------------------------------------------------------------- 858 \subsection[Split-explicit free surface (\texttt{ln\_dynspg\_ts}\forcode{=.true.})] 859 {Split-explicit free surface (\protect\np{ln\_dynspg\_ts}\forcode{=.true.})} 840 \subsection[Split-explicit free surface (\texttt{ln\_dynspg\_ts}\forcode{ = .true.})]{Split-explicit free surface (\protect\np{ln\_dynspg\_ts}\forcode{ = .true.})} 860 841 \label{subsec:DYN_spg_ts} 861 842 %------------------------------------------namsplit----------------------------------------------------------- … … 868 849 The general idea is to solve the free surface equation and the associated barotropic velocity equations with 869 850 a smaller time step than $\rdt$, the time step used for the three dimensional prognostic variables 870 (\autoref{fig:DYN_ dynspg_ts}).851 (\autoref{fig:DYN_spg_ts}). 871 852 The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) is provided through 872 853 the \np{nn\_baro} namelist parameter as: $\rdt_e = \rdt / nn\_baro$. … … 879 860 The barotropic mode solves the following equations: 880 861 % \begin{subequations} 881 % \label{eq: BT}882 \begin{equation} 883 \label{eq: BT_dyn}862 % \label{eq:DYN_BT} 863 \begin{equation} 864 \label{eq:DYN_BT_dyn} 884 865 \frac{\partial {\mathrm \overline{{\mathbf U}}_h} }{\partial t}= 885 866 -f\;{\mathrm {\mathbf k}}\times {\mathrm \overline{{\mathbf U}}_h} … … 887 868 \end{equation} 888 869 \[ 889 % \label{eq: BT_ssh}870 % \label{eq:DYN_BT_ssh} 890 871 \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\mathrm{\mathbf \overline{U}}}_h \,} \right]+P-E 891 872 \] … … 893 874 where $\mathrm {\overline{\mathbf G}}$ is a forcing term held constant, containing coupling term between modes, 894 875 surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency. 895 The third term on the right hand side of \autoref{eq: BT_dyn} represents the bottom stress896 (see section \autoref{sec:ZDF_ bfr}), explicitly accounted for at each barotropic iteration.876 The third term on the right hand side of \autoref{eq:DYN_BT_dyn} represents the bottom stress 877 (see section \autoref{sec:ZDF_drg}), explicitly accounted for at each barotropic iteration. 897 878 Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm 898 879 detailed in \citet{shchepetkin.mcwilliams_OM05}. … … 906 887 \includegraphics[width=\textwidth]{Fig_DYN_dynspg_ts} 907 888 \caption{ 908 \protect\label{fig:DYN_ dynspg_ts}889 \protect\label{fig:DYN_spg_ts} 909 890 Schematic of the split-explicit time stepping scheme for the external and internal modes. 910 891 Time increases to the right. In this particular exemple, … … 929 910 In the default case (\np{ln\_bt\_fw}\forcode{=.true.}), 930 911 the external mode is integrated between \textit{now} and \textit{after} baroclinic time-steps 931 (\autoref{fig:DYN_ dynspg_ts}a).912 (\autoref{fig:DYN_spg_ts}a). 932 913 To avoid aliasing of fast barotropic motions into three dimensional equations, 933 914 time filtering is eventually applied on barotropic quantities (\np{ln\_bt\_av}\forcode{=.true.}). … … 1101 1082 % Filtered free surface formulation 1102 1083 %-------------------------------------------------------------------------------------------------------------- 1103 \subsection[Filtered free surface (\texttt{dynspg\_flt?})] 1104 {Filtered free surface (\protect\texttt{dynspg\_flt?})} 1084 \subsection[Filtered free surface (\texttt{dynspg\_flt?})]{Filtered free surface (\protect\texttt{dynspg\_flt?})} 1105 1085 \label{subsec:DYN_spg_fltp} 1106 1086 1107 1087 The filtered formulation follows the \citet{roullet.madec_JGR00} implementation. 1108 The extra term introduced in the equations (see \autoref{subsec: PE_free_surface}) is solved implicitly.1088 The extra term introduced in the equations (see \autoref{subsec:MB_free_surface}) is solved implicitly. 1109 1089 The elliptic solvers available in the code are documented in \autoref{chap:MISC}. 1110 1090 … … 1112 1092 \gmcomment{ %%% copy from chap-model basics 1113 1093 \[ 1114 % \label{eq: spg_flt}1094 % \label{eq:DYN_spg_flt} 1115 1095 \frac{\partial {\mathrm {\mathbf U}}_h }{\partial t}= {\mathrm {\mathbf M}} 1116 1096 - g \nabla \left( \tilde{\rho} \ \eta \right) … … 1120 1100 $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, 1121 1101 and $\mathrm {\mathbf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient, 1122 non-linear and viscous terms in \autoref{eq: PE_dyn}.1102 non-linear and viscous terms in \autoref{eq:MB_dyn}. 1123 1103 } %end gmcomment 1124 1104 … … 1130 1110 % Lateral diffusion term 1131 1111 % ================================================================ 1132 \section[Lateral diffusion term and operators (\textit{dynldf.F90})] 1133 {Lateral diffusion term and operators (\protect\mdl{dynldf})} 1112 \section[Lateral diffusion term and operators (\textit{dynldf.F90})]{Lateral diffusion term and operators (\protect\mdl{dynldf})} 1134 1113 \label{sec:DYN_ldf} 1135 1114 %------------------------------------------nam_dynldf---------------------------------------------------- … … 1145 1124 \ie\ the velocity appearing in its expression is the \textit{before} velocity in time, 1146 1125 except for the pure vertical component that appears when a tensor of rotation is used. 1147 This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap: STP}).1126 This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}). 1148 1127 1149 1128 At the lateral boundaries either free slip, … … 1165 1144 1166 1145 % ================================================================ 1167 \subsection[Iso-level laplacian (\forcode{ln_dynldf_lap=.true.})] 1168 {Iso-level laplacian operator (\protect\np{ln\_dynldf\_lap}\forcode{=.true.})} 1146 \subsection[Iso-level laplacian (\forcode{ln_dynldf_lap = .true.})]{Iso-level laplacian operator (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})} 1169 1147 \label{subsec:DYN_ldf_lap} 1170 1148 1171 1149 For lateral iso-level diffusion, the discrete operator is: 1172 1150 \begin{equation} 1173 \label{eq: dynldf_lap}1151 \label{eq:DYN_ldf_lap} 1174 1152 \left\{ 1175 1153 \begin{aligned} … … 1184 1162 \end{equation} 1185 1163 1186 As explained in \autoref{subsec: PE_ldf},1164 As explained in \autoref{subsec:MB_ldf}, 1187 1165 this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and 1188 1166 ensures a complete separation between the vorticity and divergence parts of the momentum diffusion. … … 1191 1169 % Rotated laplacian operator 1192 1170 %-------------------------------------------------------------------------------------------------------------- 1193 \subsection[Rotated laplacian (\forcode{ln_dynldf_iso=.true.})] 1194 {Rotated laplacian operator (\protect\np{ln\_dynldf\_iso}\forcode{=.true.})} 1171 \subsection[Rotated laplacian (\forcode{ln_dynldf_iso = .true.})]{Rotated laplacian operator (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})} 1195 1172 \label{subsec:DYN_ldf_iso} 1196 1173 … … 1206 1183 The resulting discrete representation is: 1207 1184 \begin{equation} 1208 \label{eq: dyn_ldf_iso}1185 \label{eq:DYN_ldf_iso} 1209 1186 \begin{split} 1210 1187 D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\ … … 1250 1227 % Iso-level bilaplacian operator 1251 1228 %-------------------------------------------------------------------------------------------------------------- 1252 \subsection[Iso-level bilaplacian (\forcode{ln_dynldf_bilap=.true.})] 1253 {Iso-level bilaplacian operator (\protect\np{ln\_dynldf\_bilap}\forcode{=.true.})} 1229 \subsection[Iso-level bilaplacian (\forcode{ln_dynldf_bilap = .true.})]{Iso-level bilaplacian operator (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})} 1254 1230 \label{subsec:DYN_ldf_bilap} 1255 1231 1256 The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq: dynldf_lap} twice.1232 The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:DYN_ldf_lap} twice. 1257 1233 It requires an additional assumption on boundary conditions: 1258 1234 the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen, … … 1265 1241 % Vertical diffusion term 1266 1242 % ================================================================ 1267 \section[Vertical diffusion term (\textit{dynzdf.F90})] 1268 {Vertical diffusion term (\protect\mdl{dynzdf})} 1243 \section[Vertical diffusion term (\textit{dynzdf.F90})]{Vertical diffusion term (\protect\mdl{dynzdf})} 1269 1244 \label{sec:DYN_zdf} 1270 1245 %----------------------------------------------namzdf------------------------------------------------------ … … 1279 1254 (\np{ln\_zdfexp}\forcode{=.true.}) using a time splitting technique (\np{nn\_zdfexp} $>$ 1) or 1280 1255 $(b)$ a backward (or implicit) time differencing scheme (\np{ln\_zdfexp}\forcode{=.false.}) 1281 (see \autoref{chap: STP}).1256 (see \autoref{chap:TD}). 1282 1257 Note that namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics. 1283 1258 1284 1259 The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is. 1285 The vertical diffusion operators given by \autoref{eq: PE_zdf} take the following semi-discrete space form:1286 \[ 1287 % \label{eq: dynzdf}1260 The vertical diffusion operators given by \autoref{eq:MB_zdf} take the following semi-discrete space form: 1261 \[ 1262 % \label{eq:DYN_zdf} 1288 1263 \left\{ 1289 1264 \begin{aligned} … … 1303 1278 the vertical turbulent momentum fluxes, 1304 1279 \begin{equation} 1305 \label{eq: dynzdf_sbc}1280 \label{eq:DYN_zdf_sbc} 1306 1281 \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1} 1307 1282 = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v } … … 1316 1291 1317 1292 The turbulent flux of momentum at the bottom of the ocean is specified through a bottom friction parameterisation 1318 (see \autoref{sec:ZDF_ bfr})1293 (see \autoref{sec:ZDF_drg}) 1319 1294 1320 1295 % ================================================================ … … 1347 1322 \section{Wetting and drying } 1348 1323 \label{sec:DYN_wetdry} 1324 1349 1325 There are two main options for wetting and drying code (wd): 1350 1326 (a) an iterative limiter (il) and (b) a directional limiter (dl). … … 1395 1371 % Iterative limiters 1396 1372 %----------------------------------------------------------------------------------------- 1397 \subsection[Directional limiter (\textit{wet\_dry.F90})] 1398 {Directional limiter (\mdl{wet\_dry})} 1373 \subsection[Directional limiter (\textit{wet\_dry.F90})]{Directional limiter (\mdl{wet\_dry})} 1399 1374 \label{subsec:DYN_wd_directional_limiter} 1375 1400 1376 The principal idea of the directional limiter is that 1401 1377 water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than \np{rn\_wdmin1}). … … 1435 1411 %----------------------------------------------------------------------------------------- 1436 1412 1437 \subsection[Iterative limiter (\textit{wet\_dry.F90})] 1438 {Iterative limiter (\mdl{wet\_dry})} 1413 \subsection[Iterative limiter (\textit{wet\_dry.F90})]{Iterative limiter (\mdl{wet\_dry})} 1439 1414 \label{subsec:DYN_wd_iterative_limiter} 1440 1415 1441 \subsubsection[Iterative flux limiter (\textit{wet\_dry.F90})] 1442 {Iterative flux limiter (\mdl{wet\_dry})} 1443 \label{subsubsec:DYN_wd_il_spg_limiter} 1416 \subsubsection[Iterative flux limiter (\textit{wet\_dry.F90})]{Iterative flux limiter (\mdl{wet\_dry})} 1417 \label{subsec:DYN_wd_il_spg_limiter} 1444 1418 1445 1419 The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry'' … … 1449 1423 1450 1424 The continuity equation for the total water depth in a column 1451 \begin{equation} \label{dyn_wd_continuity} 1452 \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 . 1425 \begin{equation} 1426 \label{eq:DYN_wd_continuity} 1427 \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 . 1453 1428 \end{equation} 1454 1429 can be written in discrete form as 1455 1430 1456 \begin{align} \label{dyn_wd_continuity_2} 1457 \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) ) 1458 &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j} + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\ 1459 &= \mathrm{zzflx}_{i,j} . 1431 \begin{align} 1432 \label{eq:DYN_wd_continuity_2} 1433 \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) ) 1434 &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j} + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\ 1435 &= \mathrm{zzflx}_{i,j} . 1460 1436 \end{align} 1461 1437 … … 1470 1446 (zzflxp) and fluxes that are into the cell (zzflxn). Clearly 1471 1447 1472 \begin{equation} \label{dyn_wd_zzflx_p_n_1} 1473 \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} . 1448 \begin{equation} 1449 \label{eq:DYN_wd_zzflx_p_n_1} 1450 \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} . 1474 1451 \end{equation} 1475 1452 … … 1482 1459 $\mathrm{zcoef}_{i,j}^{(m)}$ such that: 1483 1460 1484 \begin{equation} \label{dyn_wd_continuity_coef} 1485 \begin{split} 1486 \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\ 1487 \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j} 1488 \end{split} 1461 \begin{equation} 1462 \label{eq:DYN_wd_continuity_coef} 1463 \begin{split} 1464 \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\ 1465 \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j} 1466 \end{split} 1489 1467 \end{equation} 1490 1468 … … 1494 1472 The iteration is initialised by setting 1495 1473 1496 \begin{equation} \label{dyn_wd_zzflx_initial} 1497 \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} . 1474 \begin{equation} 1475 \label{eq:DYN_wd_zzflx_initial} 1476 \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} . 1498 1477 \end{equation} 1499 1478 1500 1479 The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the 1501 1480 cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell 1502 times the timestep divided by the cell area. Using (\ ref{dyn_wd_continuity_2}) this1481 times the timestep divided by the cell area. Using (\autoref{eq:DYN_wd_continuity_2}) this 1503 1482 condition is 1504 1483 1505 \begin{equation} \label{dyn_wd_continuity_if} 1506 h_{i,j}(t_e) - \mathrm{rn\_wdmin1} < \frac{\Delta t}{e_1 e_2} ( \mathrm{zzflxp}^{(m)}_{i,j} + \mathrm{zzflxn}^{(m)}_{i,j} ) . 1507 \end{equation} 1508 1509 Rearranging (\ref{dyn_wd_continuity_if}) we can obtain an expression for the maximum 1484 \begin{equation} 1485 \label{eq:DYN_wd_continuity_if} 1486 h_{i,j}(t_e) - \mathrm{rn\_wdmin1} < \frac{\Delta t}{e_1 e_2} ( \mathrm{zzflxp}^{(m)}_{i,j} + \mathrm{zzflxn}^{(m)}_{i,j} ) . 1487 \end{equation} 1488 1489 Rearranging (\autoref{eq:DYN_wd_continuity_if}) we can obtain an expression for the maximum 1510 1490 outward flux that can be allowed and still maintain the minimum wet depth: 1511 1491 1512 \begin{equation} \label{dyn_wd_max_flux} 1513 \begin{split} 1514 \mathrm{zzflxp}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2}) \frac{e_1 e_2}{\Delta t} \phantom{]} \\ 1515 \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] 1516 \end{split} 1492 \begin{equation} 1493 \label{eq:DYN_wd_max_flux} 1494 \begin{split} 1495 \mathrm{zzflxp}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2}) \frac{e_1 e_2}{\Delta t} \phantom{]} \\ 1496 \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] 1497 \end{split} 1517 1498 \end{equation} 1518 1499 1519 1500 Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\itshape [Q: Why is 1520 this necessary/desirable?]}. Substituting from (\ ref{dyn_wd_continuity_coef}) gives an1501 this necessary/desirable?]}. Substituting from (\autoref{eq:DYN_wd_continuity_coef}) gives an 1521 1502 expression for the coefficient needed to multiply the outward flux at this cell in order 1522 1503 to avoid drying. 1523 1504 1524 \begin{equation} \label{dyn_wd_continuity_nxtcoef} 1525 \begin{split} 1526 \mathrm{zcoef}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2}) \frac{e_1 e_2}{\Delta t} \phantom{]} \\ 1527 \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} } 1528 \end{split} 1505 \begin{equation} 1506 \label{eq:DYN_wd_continuity_nxtcoef} 1507 \begin{split} 1508 \mathrm{zcoef}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2}) \frac{e_1 e_2}{\Delta t} \phantom{]} \\ 1509 \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} } 1510 \end{split} 1529 1511 \end{equation} 1530 1512 … … 1545 1527 % Surface pressure gradients 1546 1528 %---------------------------------------------------------------------------------------- 1547 \subsubsection[Modification of surface pressure gradients (\textit{dynhpg.F90})] 1548 {Modification of surface pressure gradients (\mdl{dynhpg})} 1549 \label{subsubsec:DYN_wd_il_spg} 1529 \subsubsection[Modification of surface pressure gradients (\textit{dynhpg.F90})]{Modification of surface pressure gradients (\mdl{dynhpg})} 1530 \label{subsec:DYN_wd_il_spg} 1550 1531 1551 1532 At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the … … 1560 1541 neighbouring $(i+1,j)$ and $(i,j)$ tracer points. zcpx is calculated using two logicals 1561 1542 variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid 1562 column. The three possible combinations are illustrated in figure \ ref{Fig_WAD_dynhpg}.1543 column. The three possible combinations are illustrated in figure \autoref{fig:DYN_WAD_dynhpg}. 1563 1544 1564 1545 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1565 1546 \begin{figure}[!ht] \begin{center} 1566 1547 \includegraphics[width=\textwidth]{Fig_WAD_dynhpg} 1567 \caption{ \label{Fig_WAD_dynhpg} 1568 Illustrations of the three possible combinations of the logical variables controlling the 1569 limiting of the horizontal pressure gradient in wetting and drying regimes} 1570 \end{center}\end{figure} 1548 \caption{ 1549 \label{fig:DYN_WAD_dynhpg} 1550 Illustrations of the three possible combinations of the logical variables controlling the 1551 limiting of the horizontal pressure gradient in wetting and drying regimes} 1552 \end{center} 1553 \end{figure} 1571 1554 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1572 1555 … … 1576 1559 of the topography at the two points: 1577 1560 1578 \begin{equation} \label{dyn_ll_tmp1} 1579 \begin{split} 1580 \mathrm{ll\_tmp1} = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\ 1561 \begin{equation} 1562 \label{eq:DYN_ll_tmp1} 1563 \begin{split} 1564 \mathrm{ll\_tmp1} = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\ 1581 1565 & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\ .and.} \\ 1582 & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\1583 & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\1584 & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 }1585 \end{split}1566 & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\ 1567 & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\ 1568 & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 } 1569 \end{split} 1586 1570 \end{equation} 1587 1571 … … 1590 1574 at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ 1591 1575 1592 \begin{equation} \label{dyn_ll_tmp2} 1593 \begin{split} 1594 \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\ 1595 & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\ 1596 & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) . 1597 \end{split} 1576 \begin{equation} 1577 \label{eq:DYN_ll_tmp2} 1578 \begin{split} 1579 \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\ 1580 & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\ 1581 & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) . 1582 \end{split} 1598 1583 \end{equation} 1599 1584 … … 1611 1596 conditions. 1612 1597 1613 \subsubsection[Additional considerations (\textit{usrdef\_zgr.F90})] 1614 {Additional considerations (\mdl{usrdef\_zgr})} 1615 \label{subsubsec:WAD_additional} 1598 \subsubsection[Additional considerations (\textit{usrdef\_zgr.F90})]{Additional considerations (\mdl{usrdef\_zgr})} 1599 \label{subsec:DYN_WAD_additional} 1616 1600 1617 1601 In the very shallow water where wetting and drying occurs the parametrisation of … … 1626 1610 % The WAD test cases 1627 1611 %---------------------------------------------------------------------------------------- 1628 \subsection[The WAD test cases (\textit{usrdef\_zgr.F90})] 1629 {The WAD test cases (\mdl{usrdef\_zgr})} 1630 \label{WAD_test_cases} 1612 \subsection[The WAD test cases (\textit{usrdef\_zgr.F90})]{The WAD test cases (\mdl{usrdef\_zgr})} 1613 \label{subsec:DYN_WAD_test_cases} 1631 1614 1632 1615 See the WAD tests MY\_DOC documention for details of the WAD test cases. … … 1637 1620 % Time evolution term 1638 1621 % ================================================================ 1639 \section[Time evolution term (\textit{dynnxt.F90})] 1640 {Time evolution term (\protect\mdl{dynnxt})} 1622 \section[Time evolution term (\textit{dynnxt.F90})]{Time evolution term (\protect\mdl{dynnxt})} 1641 1623 \label{sec:DYN_nxt} 1642 1624 … … 1648 1630 Options are defined through the \nam{dom} namelist variables. 1649 1631 The general framework for dynamics time stepping is a leap-frog scheme, 1650 \ie\ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap: STP}).1632 \ie\ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:TD}). 1651 1633 The scheme is applied to the velocity, except when 1652 1634 using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux}) 1653 1635 in the variable volume case (\texttt{vvl?} defined), 1654 where it has to be applied to the thickness weighted velocity (see \autoref{sec: A_momentum})1636 where it has to be applied to the thickness weighted velocity (see \autoref{sec:SCOORD_momentum}) 1655 1637 1656 1638 $\bullet$ vector invariant form or linear free surface 1657 1639 (\np{ln\_dynhpg\_vec}\forcode{=.true.} ; \texttt{vvl?} not defined): 1658 1640 \[ 1659 % \label{eq: dynnxt_vec}1641 % \label{eq:DYN_nxt_vec} 1660 1642 \left\{ 1661 1643 \begin{aligned} … … 1669 1651 (\np{ln\_dynhpg\_vec}\forcode{=.false.} ; \texttt{vvl?} defined): 1670 1652 \[ 1671 % \label{eq: dynnxt_flux}1653 % \label{eq:DYN_nxt_flux} 1672 1654 \left\{ 1673 1655 \begin{aligned} -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_LBC.tex
r11537 r11543 57 57 58 58 \[ 59 % \label{eq: lbc_aaaa}59 % \label{eq:LBC_aaaa} 60 60 \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT} 61 61 }{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u … … 134 134 the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ : 135 135 \[ 136 % \label{eq: lbc_bbbb}136 % \label{eq:LBC_bbbb} 137 137 \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta_{i+1/2} 138 138 \left[ {e_{2v} \,v} \right]-\delta_{j+1/2} \left[ {e_{1u} \,u} \right]} … … 226 226 The north fold boundary condition has been introduced in order to handle the north boundary of 227 227 a three-polar ORCA grid. 228 Such a grid has two poles in the northern hemisphere (\autoref{fig: MISC_ORCA_msh},229 and thus requires a specific treatment illustrated in \autoref{fig: North_Fold_T}.228 Such a grid has two poles in the northern hemisphere (\autoref{fig:CFGS_ORCA_msh}, 229 and thus requires a specific treatment illustrated in \autoref{fig:LBC_North_Fold_T}. 230 230 Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition. 231 231 … … 235 235 \includegraphics[width=\textwidth]{Fig_North_Fold_T} 236 236 \caption{ 237 \protect\label{fig: North_Fold_T}237 \protect\label{fig:LBC_North_Fold_T} 238 238 North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$), 239 239 as used in ORCA 2, 1/4, and 1/12. … … 256 256 %----------------------------------------------------------------------------------------------- 257 257 258 For massively parallel processing (mpp), a domain decomposition method is used. The basic idea of the method is to split the large computation domain of a numerical experiment into several smaller domains and solve the set of equations by addressing independent local problems. Each processor has its own local memory and computes the model equation over a subdomain of the whole model domain. The subdomain boundary conditions are specified through communications between processors which are organized by explicit statements (message passing method). The present implementation is largely inspired by Guyon's work [Guyon 1995]. 258 For massively parallel processing (mpp), a domain decomposition method is used. 259 The basic idea of the method is to split the large computation domain of a numerical experiment into several smaller domains and 260 solve the set of equations by addressing independent local problems. 261 Each processor has its own local memory and computes the model equation over a subdomain of the whole model domain. 262 The subdomain boundary conditions are specified through communications between processors which are organized by 263 explicit statements (message passing method). 264 The present implementation is largely inspired by Guyon's work [Guyon 1995]. 259 265 260 266 The parallelization strategy is defined by the physical characteristics of the ocean model. … … 272 278 each processor sends to its neighbouring processors the update values of the points corresponding to 273 279 the interior overlapping area to its neighbouring sub-domain (\ie\ the innermost of the two overlapping rows). 274 Communications are first done according to the east-west direction and next according to the north-south direction. There is no specific communications for the corners. The communication is done through the Message Passing Interface (MPI) and requires \key{mpp\_mpi}. Use also \key{mpi2} if MPI3 is not available on your computer. 280 Communications are first done according to the east-west direction and next according to the north-south direction. 281 There is no specific communications for the corners. 282 The communication is done through the Message Passing Interface (MPI) and requires \key{mpp\_mpi}. 283 Use also \key{mpi2} if MPI3 is not available on your computer. 275 284 The data exchanges between processors are required at the very place where 276 285 lateral domain boundary conditions are set in the mono-domain computation: … … 285 294 \includegraphics[width=\textwidth]{Fig_mpp} 286 295 \caption{ 287 \protect\label{fig: mpp}296 \protect\label{fig:LBC_mpp} 288 297 Positioning of a sub-domain when massively parallel processing is used. 289 298 } … … 292 301 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 293 302 294 In \NEMO, the splitting is regular and arithmetic. The total number of subdomains corresponds to the number of MPI processes allocated to \NEMO\ when the model is launched (\ie\ mpirun -np x ./nemo will automatically give x subdomains). The i-axis is divided by \np{jpni} and the j-axis by \np{jpnj}. These parameters are defined in \nam{mpp} namelist. If \np{jpni} and \np{jpnj} are < 1, they will be automatically redefined in the code to give the best domain decomposition (see bellow). 295 296 Each processor is independent and without message passing or synchronous process, programs run alone and access just its own local memory. For this reason, the main model dimensions are now the local dimensions of the subdomain (pencil) that are named \jp{jpi}, \jp{jpj}, \jp{jpk}. 297 These dimensions include the internal domain and the overlapping rows. The number of rows to exchange (known as the halo) is usually set to one (nn\_hls=1, in \mdl{par\_oce}, and must be kept to one until further notice). The whole domain dimensions are named \jp{jpiglo}, \jp{jpjglo} and \jp{jpk}. The relationship between the whole domain and a sub-domain is: 298 \[ 299 jpi = ( jpiglo-2\times nn\_hls + (jpni-1) ) / jpni + 2\times nn\_hls 300 \] 301 \[ 303 In \NEMO, the splitting is regular and arithmetic. 304 The total number of subdomains corresponds to the number of MPI processes allocated to \NEMO\ when the model is launched 305 (\ie\ mpirun -np x ./nemo will automatically give x subdomains). 306 The i-axis is divided by \np{jpni} and the j-axis by \np{jpnj}. 307 These parameters are defined in \nam{mpp} namelist. 308 If \np{jpni} and \np{jpnj} are < 1, they will be automatically redefined in the code to give the best domain decomposition 309 (see bellow). 310 311 Each processor is independent and without message passing or synchronous process, programs run alone and access just its own local memory. 312 For this reason, 313 the main model dimensions are now the local dimensions of the subdomain (pencil) that are named \jp{jpi}, \jp{jpj}, \jp{jpk}. 314 These dimensions include the internal domain and the overlapping rows. 315 The number of rows to exchange (known as the halo) is usually set to one (nn\_hls=1, in \mdl{par\_oce}, 316 and must be kept to one until further notice). 317 The whole domain dimensions are named \jp{jpiglo}, \jp{jpjglo} and \jp{jpk}. 318 The relationship between the whole domain and a sub-domain is: 319 \begin{gather*} 320 jpi = ( jpiglo-2\times nn\_hls + (jpni-1) ) / jpni + 2\times nn\_hls \\ 302 321 jpj = ( jpjglo-2\times nn\_hls + (jpnj-1) ) / jpnj + 2\times nn\_hls 303 \ ]304 305 One also defines variables nldi and nlei which correspond to the internal domain bounds, and the variables nimpp and njmpp which are the position of the (1,1) grid-point in the global domain (\autoref{fig: mpp}). Note that since the version 4, there is no more extra-halo area as defined in \autoref{fig:mpp} so \jp{jpi} is now always equal to nlci and \jp{jpj} equal to nlcj.322 \end{gather*} 323 324 One also defines variables nldi and nlei which correspond to the internal domain bounds, and the variables nimpp and njmpp which are the position of the (1,1) grid-point in the global domain (\autoref{fig:LBC_mpp}). Note that since the version 4, there is no more extra-halo area as defined in \autoref{fig:LBC_mpp} so \jp{jpi} is now always equal to nlci and \jp{jpj} equal to nlcj. 306 325 307 326 An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$, 308 327 a global array (whole domain) by the relationship: 309 328 \[ 310 % \label{eq: lbc_nimpp}329 % \label{eq:LBC_nimpp} 311 330 T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k), 312 331 \] … … 322 341 323 342 If the domain decomposition is automatically defined (when \np{jpni} and \np{jpnj} are < 1), the decomposition chosen by the model will minimise the sub-domain size (defined as $max_{all domains}(jpi \times jpj)$) and maximize the number of eliminated land subdomains. This means that no other domain decomposition (a set of \np{jpni} and \np{jpnj} values) will use less processes than $(jpni \times jpnj - N_{land})$ and get a smaller subdomain size. 324 In order to specify $N_{mpi}$ properly (minimize $N_{useless}$), you must run the model once with \np{ln\_list} activated. In this case, the model will start the initialisation phase, print the list of optimum decompositions ($N_{mpi}$, \np{jpni} and \np{jpnj}) in \texttt{ocean.output} and directly abort. The maximum value of $N_{mpi}$ tested in this list is given by $max(N_{MPI\_tasks}, \np{jpni} \times \np{jpnj})$. For example, run the model on 40 nodes with ln\_list activated and $\np{jpni} = 10000$ and $\np{jpnj} = 1$, will print the list of optimum domains decomposition from 1 to about 10000. 325 326 Processors are numbered from 0 to $N_{mpi} - 1$. Subdomains containning some ocean points are numbered first from 0 to $jpni * jpnj - N_{land} -1$. The remaining $N_{useless}$ land subdomains are numbered next, which means that, for a given (\np{jpni}, \np{jpnj}), the numbers attributed to he ocean subdomains do not vary with $N_{useless}$. 343 In order to specify $N_{mpi}$ properly (minimize $N_{useless}$), you must run the model once with \np{ln\_list} activated. In this case, the model will start the initialisation phase, print the list of optimum decompositions ($N_{mpi}$, \np{jpni} and \np{jpnj}) in \texttt{ocean.output} and directly abort. The maximum value of $N_{mpi}$ tested in this list is given by $max(N_{MPI\_tasks}, \np{jpni} \times \np{jpnj})$. For example, run the model on 40 nodes with ln\_list activated and $\np{jpni} = 10000$ and $\np{jpnj} = 1$, will print the list of optimum domains decomposition from 1 to about 10000. 344 345 Processors are numbered from 0 to $N_{mpi} - 1$. Subdomains containning some ocean points are numbered first from 0 to $jpni * jpnj - N_{land} -1$. The remaining $N_{useless}$ land subdomains are numbered next, which means that, for a given (\np{jpni}, \np{jpnj}), the numbers attributed to he ocean subdomains do not vary with $N_{useless}$. 327 346 328 347 When land processors are eliminated, the value corresponding to these locations in the model output files is undefined. \np{ln\_mskland} must be activated in order avoid Not a Number values in output files. Note that it is better to not eliminate land processors when creating a meshmask file (\ie\ when setting a non-zero value to \np{nn\_msh}). … … 332 351 \begin{center} 333 352 \includegraphics[width=\textwidth]{Fig_mppini2} 334 \caption 335 \protect\label{fig: mppini2}353 \caption[Atlantic domain]{ 354 \protect\label{fig:LBC_mppini2} 336 355 Example of Atlantic domain defined for the CLIPPER projet. 337 356 Initial grid is composed of 773 x 1236 horizontal points. … … 374 393 %---------------------------------------------- 375 394 \subsection{Namelists} 376 \label{subsec: BDY_namelist}395 \label{subsec:LBC_bdy_namelist} 377 396 378 397 The BDY module is activated by setting \np{ln\_bdy}\forcode{=.true.} . … … 384 403 In the example above, there are two boundary sets, the first of which is defined via a file and 385 404 the second is defined in the namelist. 386 For more details of the definition of the boundary geometry see section \autoref{subsec: BDY_geometry}.405 For more details of the definition of the boundary geometry see section \autoref{subsec:LBC_bdy_geometry}. 387 406 388 407 For each boundary set a boundary condition has to be chosen for the barotropic solution … … 441 460 %---------------------------------------------- 442 461 \subsection{Flow relaxation scheme} 443 \label{subsec: BDY_FRS_scheme}462 \label{subsec:LBC_bdy_FRS_scheme} 444 463 445 464 The Flow Relaxation Scheme (FRS) \citep{davies_QJRMS76,engedahl_T95}, … … 448 467 Given a model prognostic variable $\Phi$ 449 468 \[ 450 % \label{eq: bdy_frs1}469 % \label{eq:LBC_bdy_frs1} 451 470 \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N 452 471 \] … … 457 476 the prognostic equation for $\Phi$ of the form: 458 477 \[ 459 % \label{eq: bdy_frs2}478 % \label{eq:LBC_bdy_frs2} 460 479 -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right) 461 480 \] 462 481 where the relaxation time scale $\tau$ is given by a function of $\alpha$ and the model time step $\Delta t$: 463 482 \[ 464 % \label{eq: bdy_frs3}483 % \label{eq:LBC_bdy_frs3} 465 484 \tau = \frac{1-\alpha}{\alpha} \,\rdt 466 485 \] … … 472 491 The function $\alpha$ is specified as a $tanh$ function: 473 492 \[ 474 % \label{eq: bdy_frs4}493 % \label{eq:LBC_bdy_frs4} 475 494 \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right), \quad d=1,N 476 495 \] … … 480 499 %---------------------------------------------- 481 500 \subsection{Flather radiation scheme} 482 \label{subsec: BDY_flather_scheme}501 \label{subsec:LBC_bdy_flather_scheme} 483 502 484 503 The \citet{flather_JPO94} scheme is a radiation condition on the normal, 485 504 depth-mean transport across the open boundary. 486 505 It takes the form 487 \begin{equation} \label{eq:bdy_fla1} 488 U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right), 506 \begin{equation} 507 \label{eq:LBC_bdy_fla1} 508 U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right), 489 509 \end{equation} 490 510 where $U$ is the depth-mean velocity normal to the boundary and $\eta$ is the sea surface height, … … 495 515 the external depth-mean normal velocity, 496 516 plus a correction term that allows gravity waves generated internally to exit the model boundary. 497 Note that the sea-surface height gradient in \autoref{eq: bdy_fla1} is a spatial gradient across the model boundary,517 Note that the sea-surface height gradient in \autoref{eq:LBC_bdy_fla1} is a spatial gradient across the model boundary, 498 518 so that $\eta_{e}$ is defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the $T$ points with $nbr=2$. 499 519 $U$ and $U_{e}$ are defined on the $U$ or $V$ points with $nbr=1$, \ie\ between the two $T$ grid points. … … 501 521 %---------------------------------------------- 502 522 \subsection{Orlanski radiation scheme} 503 \label{subsec: BDY_orlanski_scheme}523 \label{subsec:LBC_bdy_orlanski_scheme} 504 524 505 525 The Orlanski scheme is based on the algorithm described by \citep{marchesiello.mcwilliams.ea_OM01}, hereafter MMS. … … 507 527 The adaptive Orlanski condition solves a wave plus relaxation equation at the boundary: 508 528 \begin{equation} 509 \frac{\partial\phi}{\partial t} + c_x \frac{\partial\phi}{\partial x} + c_y \frac{\partial\phi}{\partial y} = 510 -\frac{1}{\tau}(\phi - \phi^{ext})511 \label{eq:wave_continuous} 529 \label{eq:LBC_wave_continuous} 530 \frac{\partial\phi}{\partial t} + c_x \frac{\partial\phi}{\partial x} + c_y \frac{\partial\phi}{\partial y} = 531 -\frac{1}{\tau}(\phi - \phi^{ext}) 512 532 \end{equation} 513 533 … … 515 535 velocities are diagnosed from the model fields as: 516 536 517 \begin{equation} \label{eq:cx} 518 c_x = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial x}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2} 537 \begin{equation} 538 \label{eq:LBC_cx} 539 c_x = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial x}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2} 519 540 \end{equation} 520 541 \begin{equation} 521 \label{eq:cy}522 c_y = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial y}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2}542 \label{eq:LBC_cy} 543 c_y = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial y}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2} 523 544 \end{equation} 524 545 525 546 (As noted by MMS, this is a circular diagnosis of the phase speeds which only makes sense on a discrete grid). 526 Equation (\autoref{eq: wave_continuous}) is defined adaptively depending on the sign of the phase velocity normal to the boundary $c_x$.547 Equation (\autoref{eq:LBC_wave_continuous}) is defined adaptively depending on the sign of the phase velocity normal to the boundary $c_x$. 527 548 For $c_x$ outward, we have 528 549 … … 534 555 535 556 \begin{equation} 536 \tau = \tau_{in}\,\,\,;\,\,\, c_x = c_y = 0 537 \label{eq:tau_in} 557 \label{eq:LBC_tau_in} 558 \tau = \tau_{in}\,\,\,;\,\,\, c_x = c_y = 0 538 559 \end{equation} 539 560 540 561 Generally the relaxation time scale at inward propagation points (\np{rn\_time\_dmp}) is set much shorter than the time scale at outward propagation 541 562 points (\np{rn\_time\_dmp\_out}) so that the solution is constrained more strongly by the external data at inward propagation points. 542 See \autoref{subsec: BDY_relaxation} for detailed on the spatial shape of the scaling.\\563 See \autoref{subsec:LBC_bdy_relaxation} for detailed on the spatial shape of the scaling.\\ 543 564 The ``normal propagation of oblique radiation'' or NPO approximation (called \forcode{'orlanski_npo'}) involves assuming 544 that $c_y$ is zero in equation (\autoref{eq: wave_continuous}), but including545 this term in the denominator of equation (\autoref{eq: cx}). Both versions of the scheme are options in BDY. Equations546 (\autoref{eq: wave_continuous}) - (\autoref{eq:tau_in}) correspond to equations (13) - (15) and (2) - (3) in MMS.\\565 that $c_y$ is zero in equation (\autoref{eq:LBC_wave_continuous}), but including 566 this term in the denominator of equation (\autoref{eq:LBC_cx}). Both versions of the scheme are options in BDY. Equations 567 (\autoref{eq:LBC_wave_continuous}) - (\autoref{eq:LBC_tau_in}) correspond to equations (13) - (15) and (2) - (3) in MMS.\\ 547 568 548 569 %---------------------------------------------- 549 570 \subsection{Relaxation at the boundary} 550 \label{subsec: BDY_relaxation}571 \label{subsec:LBC_bdy_relaxation} 551 572 552 573 In addition to a specific boundary condition specified as \np{cn\_tra} and \np{cn\_dyn3d}, relaxation on baroclinic velocities and tracers variables are available. … … 564 585 %---------------------------------------------- 565 586 \subsection{Boundary geometry} 566 \label{subsec: BDY_geometry}587 \label{subsec:LBC_bdy_geometry} 567 588 568 589 Each open boundary set is defined as a list of points. … … 615 636 %---------------------------------------------- 616 637 \subsection{Input boundary data files} 617 \label{subsec: BDY_data}638 \label{subsec:LBC_bdy_data} 618 639 619 640 The data files contain the data arrays in the order in which the points are defined in the $nbi$ and $nbj$ arrays. … … 655 676 %---------------------------------------------- 656 677 \subsection{Volume correction} 657 \label{subsec: BDY_vol_corr}678 \label{subsec:LBC_bdy_vol_corr} 658 679 659 680 There is an option to force the total volume in the regional model to be constant. … … 672 693 %---------------------------------------------- 673 694 \subsection{Tidal harmonic forcing} 674 \label{subsec: BDY_tides}695 \label{subsec:LBC_bdy_tides} 675 696 676 697 %-----------------------------------------nambdy_tide-------------------------------------------- -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_LDF.tex
r11537 r11543 13 13 \newpage 14 14 15 The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq: PE_zdf} and15 The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:MB_zdf} and 16 16 their discrete formulation in \autoref{sec:TRA_ldf} and \autoref{sec:DYN_ldf}). 17 17 In this section we further discuss each lateral physics option. … … 25 25 Note that this chapter describes the standard implementation of iso-neutral tracer mixing. 26 26 Griffies's implementation, which is used if \np{ln\_traldf\_triad}\forcode{=.true.}, 27 is described in \autoref{apdx: triad}27 is described in \autoref{apdx:TRIADS} 28 28 29 29 %-----------------------------------namtra_ldf - namdyn_ldf-------------------------------------------- … … 82 82 the cell of the quantity to be diffused. 83 83 For a tracer, this leads to the following four slopes: 84 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq: tra_ldf_iso}),84 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:TRA_ldf_iso}), 85 85 while for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and 86 86 $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. … … 92 92 In $s$-coordinates, geopotential mixing (\ie\ horizontal mixing) $r_1$ and $r_2$ are the slopes between 93 93 the geopotential and computational surfaces. 94 Their discrete formulation is found by locally solving \autoref{eq: tra_ldf_iso} when94 Their discrete formulation is found by locally solving \autoref{eq:TRA_ldf_iso} when 95 95 the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform, 96 96 \ie\ a linear function of $z_T$, the depth of a $T$-point. … … 98 98 99 99 \begin{equation} 100 \label{eq: ldfslp_geo}100 \label{eq:LDF_slp_geo} 101 101 \begin{aligned} 102 102 r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} … … 125 125 Their discrete formulation is found using the fact that the diffusive fluxes of 126 126 locally referenced potential density (\ie\ $in situ$ density) vanish. 127 So, substituting $T$ by $\rho$ in \autoref{eq: tra_ldf_iso} and setting the diffusive fluxes in127 So, substituting $T$ by $\rho$ in \autoref{eq:TRA_ldf_iso} and setting the diffusive fluxes in 128 128 the three directions to zero leads to the following definition for the neutral slopes: 129 129 130 130 \begin{equation} 131 \label{eq: ldfslp_iso}131 \label{eq:LDF_slp_iso} 132 132 \begin{split} 133 133 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} … … 145 145 146 146 %gm% rewrite this as the explanation is not very clear !!! 147 %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.148 149 %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{mcdougall_JPO87}, 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).150 151 %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.152 153 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq: ldfslp_iso} has to147 %In practice, \autoref{eq:LDF_slp_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:LDF_slp_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. 148 149 %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{mcdougall_JPO87}, therefore in \autoref{eq:LDF_slp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters). 150 151 %In the $z$-coordinate, the derivative of the \autoref{eq:LDF_slp_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. 152 153 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:LDF_slp_iso} has to 154 154 be evaluated at the same local pressure 155 155 (which, in decibars, is approximated by the depth in meters in the model). 156 Therefore \autoref{eq: ldfslp_iso} cannot be used as such,156 Therefore \autoref{eq:LDF_slp_iso} cannot be used as such, 157 157 but further transformation is needed depending on the vertical coordinate used: 158 158 … … 160 160 161 161 \item[$z$-coordinate with full step: ] 162 in \autoref{eq: ldfslp_iso} the densities appearing in the $i$ and $j$ derivatives are taken at the same depth,162 in \autoref{eq:LDF_slp_iso} the densities appearing in the $i$ and $j$ derivatives are taken at the same depth, 163 163 thus the $in situ$ density can be used. 164 164 This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, … … 173 173 in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if 174 174 the Griffies scheme is used (\np{ln\_traldf\_triad}\forcode{=.true.}; 175 see \autoref{apdx: triad}).175 see \autoref{apdx:TRIADS}). 176 176 In other words, iso-neutral mixing will only be accurately represented with a linear equation of state 177 177 (\np{ln\_seos}\forcode{=.true.}). 178 In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq: ldfslp_iso}178 In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:LDF_slp_iso} 179 179 will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. 180 180 … … 193 193 194 194 \[ 195 % \label{eq: ldfslp_iso2}195 % \label{eq:LDF_slp_iso2} 196 196 \begin{split} 197 197 r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac … … 230 230 To overcome this problem, several techniques have been proposed in which the numerical schemes of 231 231 the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}. 232 Griffies's scheme is now available in \NEMO\ if \np{ln\_traldf\_triad}\forcode{ =.true.}; see \autoref{apdx:triad}.232 Griffies's scheme is now available in \NEMO\ if \np{ln\_traldf\_triad}\forcode{ = .true.}; see \autoref{apdx:TRIADS}. 233 233 Here, another strategy is presented \citep{lazar_phd97}: 234 234 a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of … … 280 280 \includegraphics[width=\textwidth]{Fig_eiv_slp} 281 281 \caption{ 282 \protect\label{fig: eiv_slp}282 \protect\label{fig:LDF_eiv_slp} 283 283 Vertical profile of the slope used for lateral mixing in the mixed layer: 284 284 \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, … … 304 304 The iso-neutral diffusion operator on momentum is the same as the one used on tracers but 305 305 applied to each component of the velocity separately 306 (see \autoref{eq: dyn_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}).306 (see \autoref{eq:DYN_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). 307 307 The slopes between the surface along which the diffusion operator acts and the surface of computation 308 308 ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and 309 309 \textit{vw}- points for the $v$-component. 310 310 They are computed from the slopes used for tracer diffusion, 311 \ie\ \autoref{eq: ldfslp_geo} and \autoref{eq:ldfslp_iso}:311 \ie\ \autoref{eq:LDF_slp_geo} and \autoref{eq:LDF_slp_iso}: 312 312 313 313 \[ 314 % \label{eq: ldfslp_dyn}314 % \label{eq:LDF_slp_dyn} 315 315 \begin{aligned} 316 316 &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ … … 371 371 372 372 \begin{equation} 373 \label{eq: constantah}373 \label{eq:LDF_constantah} 374 374 A_o^l = \left\{ 375 375 \begin{aligned} … … 386 386 387 387 In the vertically varying case, a hyperbolic variation of the lateral mixing coefficient is introduced in which 388 the surface value is given by \autoref{eq: constantah}, the bottom value is 1/4 of the surface value,388 the surface value is given by \autoref{eq:LDF_constantah}, the bottom value is 1/4 of the surface value, 389 389 and the transition takes place around z=500~m with a width of 200~m. 390 390 This profile is hard coded in module \mdl{ldfc1d\_c2d}, but can be easily modified by users. … … 396 396 the type of operator used: 397 397 \begin{equation} 398 \label{eq: title}398 \label{eq:LDF_title} 399 399 A_l = \left\{ 400 400 \begin{aligned} … … 411 411 model configurations presenting large changes in grid spacing such as global ocean models. 412 412 Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to 413 large coefficient compare to the smallest grid size (see \autoref{sec: STP_forward_imp}),413 large coefficient compare to the smallest grid size (see \autoref{sec:TD_forward_imp}), 414 414 especially when using a bilaplacian operator. 415 415 … … 429 429 430 430 \begin{equation} 431 \label{eq: flowah}431 \label{eq:LDF_flowah} 432 432 A_l = \left\{ 433 433 \begin{aligned} … … 445 445 446 446 \begin{equation} 447 \label{eq: smag1}447 \label{eq:LDF_smag1} 448 448 \begin{split} 449 449 T_{smag}^{-1} & = \sqrt{\left( \partial_x u - \partial_y v\right)^2 + \left( \partial_y u + \partial_x v\right)^2 } \\ … … 455 455 456 456 \begin{equation} 457 \label{eq: smag2}457 \label{eq:LDF_smag2} 458 458 A_{smag} = \left\{ 459 459 \begin{aligned} … … 464 464 \end{equation} 465 465 466 For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec: STP_forward_imp}) so that:467 \begin{equation} 468 \label{eq: smag3}466 For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:TD_forward_imp}) so that: 467 \begin{equation} 468 \label{eq:LDF_smag3} 469 469 \begin{aligned} 470 470 & C_{min} \frac{1}{2} \lvert U \rvert e < A_{smag} < C_{max} \frac{e^2}{ 8\rdt} & \text{for laplacian operator } \\ … … 480 480 481 481 (1) the momentum diffusion operator acting along model level surfaces is written in terms of curl and 482 divergent components of the horizontal current (see \autoref{subsec: PE_ldf}).482 divergent components of the horizontal current (see \autoref{subsec:MB_ldf}). 483 483 Although the eddy coefficient could be set to different values in these two terms, 484 484 this option is not currently available. … … 486 486 (2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of 487 487 the horizontal divergence for operators acting along model-surfaces are no longer satisfied 488 (\autoref{sec: dynldf_properties}).488 (\autoref{sec:INVARIANTS_dynldf_properties}). 489 489 490 490 % ================================================================ … … 527 527 the formulation of which depends on the slopes of iso-neutral surfaces. 528 528 Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, 529 \ie\ \autoref{eq: ldfslp_geo} is used in $z$-coordinates,530 and the sum \autoref{eq: ldfslp_geo} + \autoref{eq:ldfslp_iso} in $s$-coordinates.529 \ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinates, 530 and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $s$-coordinates. 531 531 532 532 If isopycnal mixing is used in the standard way, \ie\ \np{ln\_traldf\_triad}\forcode{=.false.}, the eddy induced velocity is given by: 533 533 \begin{equation} 534 \label{eq: ldfeiv}534 \label{eq:LDF_eiv} 535 535 \begin{split} 536 536 u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ … … 554 554 \colorbox{yellow}{CASE \np{nn\_aei\_ijk\_t} = 21 to be added} 555 555 556 In case of setting \np{ln\_traldf\_triad}\forcode{ =.true.}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:triad}.556 In case of setting \np{ln\_traldf\_triad}\forcode{ = .true.}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:TRIADS}. 557 557 558 558 % ================================================================ -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_OBS.tex
r11435 r11543 691 691 692 692 Examples of the weights calculated for an observation with rectangular and radial footprints are shown in 693 \autoref{fig: obsavgrec} and~\autoref{fig:obsavgrad}.693 \autoref{fig:OBS_avgrec} and~\autoref{fig:OBS_avgrad}. 694 694 695 695 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 698 698 \includegraphics[width=\textwidth]{Fig_OBS_avg_rec} 699 699 \caption{ 700 \protect\label{fig: obsavgrec}700 \protect\label{fig:OBS_avgrec} 701 701 Weights associated with each model grid box (blue lines and numbers) 702 702 for an observation at -170.5\deg{E}, 56.0\deg{N} with a rectangular footprint of 1\deg x 1\deg. … … 711 711 \includegraphics[width=\textwidth]{Fig_OBS_avg_rad} 712 712 \caption{ 713 \protect\label{fig: obsavgrad}713 \protect\label{fig:OBS_avgrad} 714 714 Weights associated with each model grid box (blue lines and numbers) 715 715 for an observation at -170.5\deg{E}, 56.0\deg{N} with a radial footprint with diameter 1\deg. … … 756 756 ({\phi_{}}_{\mathrm D} \; - \; {\phi_{}}_{\mathrm P} )] \; \widehat{\mathbf k} \\ 757 757 \end{array} 758 % \label{eq: cross}758 % \label{eq:OBS_cross} 759 759 \end{align*} 760 760 point in the opposite direction to the unit normal $\widehat{\mathbf k}$ … … 791 791 \includegraphics[width=\textwidth]{Fig_ASM_obsdist_local} 792 792 \caption{ 793 \protect\label{fig: obslocal}793 \protect\label{fig:OBS_local} 794 794 Example of the distribution of observations with the geographical distribution of observational data. 795 795 } … … 800 800 This is the simplest option in which the observations are distributed according to 801 801 the domain of the grid-point parallelization. 802 \autoref{fig: obslocal} shows an example of the distribution of the {\em in situ} data on processors with802 \autoref{fig:OBS_local} shows an example of the distribution of the {\em in situ} data on processors with 803 803 a different colour for each observation on a given processor for a 4 $\times$ 2 decomposition with ORCA2. 804 804 The grid-point domain decomposition is clearly visible on the plot. … … 820 820 \includegraphics[width=\textwidth]{Fig_ASM_obsdist_global} 821 821 \caption{ 822 \protect\label{fig: obsglobal}822 \protect\label{fig:OBS_global} 823 823 Example of the distribution of observations with the round-robin distribution of observational data. 824 824 } … … 830 830 use message passing in order to retrieve the stencil for interpolation. 831 831 The simplest distribution of the observations is to distribute them using a round-robin scheme. 832 \autoref{fig: obsglobal} shows the distribution of the {\em in situ} data on processors for832 \autoref{fig:OBS_global} shows the distribution of the {\em in situ} data on processors for 833 833 the round-robin distribution of observations with a different colour for each observation on a given processor for 834 a 4 $\times$ 2 decomposition with ORCA2 for the same input data as in \autoref{fig: obslocal}.834 a 4 $\times$ 2 decomposition with ORCA2 for the same input data as in \autoref{fig:OBS_local}. 835 835 The observations are now clearly randomly distributed on the globe. 836 836 In order to be able to perform horizontal interpolation in this case, … … 1118 1118 \end{minted} 1119 1119 1120 \autoref{fig: obsdataplotmain} shows the main window which is launched when dataplot starts.1120 \autoref{fig:OBS_dataplotmain} shows the main window which is launched when dataplot starts. 1121 1121 This is split into three parts. 1122 1122 At the top there is a menu bar which contains a variety of drop down menus. … … 1154 1154 \includegraphics[width=\textwidth]{Fig_OBS_dataplot_main} 1155 1155 \caption{ 1156 \protect\label{fig: obsdataplotmain}1156 \protect\label{fig:OBS_dataplotmain} 1157 1157 Main window of dataplot. 1158 1158 } … … 1162 1162 1163 1163 If a profile point is clicked with the mouse button a plot of the observation and background values as 1164 a function of depth (\autoref{fig: obsdataplotprofile}).1164 a function of depth (\autoref{fig:OBS_dataplotprofile}). 1165 1165 1166 1166 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 1170 1170 \includegraphics[width=\textwidth]{Fig_OBS_dataplot_prof} 1171 1171 \caption{ 1172 \protect\label{fig: obsdataplotprofile}1172 \protect\label{fig:OBS_dataplotprofile} 1173 1173 Profile plot from dataplot produced by right clicking on a point in the main window. 1174 1174 } -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_SBC.tex
r11537 r11543 109 109 Next, the scheme for interpolation on the fly is described. 110 110 Finally, the different options that further modify the fluxes applied to the ocean are discussed. 111 One of these is modification by icebergs (see \autoref{sec: ICB_icebergs}),111 One of these is modification by icebergs (see \autoref{sec:SBC_ICB_icebergs}), 112 112 which act as drifting sources of fresh water. 113 113 Another example of modification is that due to the ice shelf melting/freezing (see \autoref{sec:SBC_isf}), … … 124 124 The surface ocean stress is the stress exerted by the wind and the sea-ice on the ocean. 125 125 It is applied in \mdl{dynzdf} module as a surface boundary condition of the computation of 126 the momentum vertical mixing trend (see \autoref{eq: dynzdf_sbc} in \autoref{sec:DYN_zdf}).126 the momentum vertical mixing trend (see \autoref{eq:DYN_zdf_sbc} in \autoref{sec:DYN_zdf}). 127 127 As such, it has to be provided as a 2D vector interpolated onto the horizontal velocity ocean mesh, 128 128 \ie\ resolved onto the model (\textbf{i},\textbf{j}) direction at $u$- and $v$-points. … … 135 135 It is applied in \mdl{trasbc} module as a surface boundary condition trend of 136 136 the first level temperature time evolution equation 137 (see \autoref{eq: tra_sbc} and \autoref{eq:tra_sbc_lin} in \autoref{subsec:TRA_sbc}).137 (see \autoref{eq:TRA_sbc} and \autoref{eq:TRA_sbc_lin} in \autoref{subsec:TRA_sbc}). 138 138 The latter is the penetrative part of the heat flux. 139 139 It is applied as a 3D trend of the temperature equation (\mdl{traqsr} module) when … … 177 177 The ocean model provides, at each time step, to the surface module (\mdl{sbcmod}) 178 178 the surface currents, temperature and salinity. 179 These variables are averaged over \np{nn\_fsbc} time-step (\autoref{tab: ssm}), and179 These variables are averaged over \np{nn\_fsbc} time-step (\autoref{tab:SBC_ssm}), and 180 180 these averaged fields are used to compute the surface fluxes at the frequency of \np{nn\_fsbc} time-steps. 181 181 … … 193 193 \end{tabular} 194 194 \caption{ 195 \protect\label{tab: ssm}195 \protect\label{tab:SBC_ssm} 196 196 Ocean variables provided by the ocean to the surface module (SBC). 197 197 The variable are averaged over \np{nn\_fsbc} time-step, … … 264 264 This stem will be completed automatically by the model, with the addition of a '.nc' at its end and 265 265 by date information and possibly a prefix (when using AGRIF). 266 \autoref{tab: fldread} provides the resulting file name in all possible cases according to266 \autoref{tab:SBC_fldread} provides the resulting file name in all possible cases according to 267 267 whether it is a climatological file or not, and to the open/close frequency (see below for definition). 268 268 … … 278 278 \end{center} 279 279 \caption{ 280 \protect\label{tab: fldread}280 \protect\label{tab:SBC_fldread} 281 281 naming nomenclature for climatological or interannual input file(s), as a function of the open/close frequency. 282 282 The stem name is assumed to be 'fn'. … … 515 515 % ------------------------------------------------------------------------------------------------------------- 516 516 \subsection{Standalone surface boundary condition scheme (SAS)} 517 \label{subsec:S AS}517 \label{subsec:SBC_SAS} 518 518 519 519 %---------------------------------------namsbc_sas-------------------------------------------------- … … 649 649 %--------------------------------------------------TABLE-------------------------------------------------- 650 650 \begin{table}[htbp] 651 \label{tab: BULK}651 \label{tab:SBC_BULK} 652 652 \begin{center} 653 653 \begin{tabular}{|l|c|c|c|} … … 852 852 The tidal forcing, generated by the gravity forces of the Earth-Moon and Earth-Sun sytems, 853 853 is activated if \np{ln\_tide} and \np{ln\_tide\_pot} are both set to \forcode{.true.} in \nam{\_tide}. 854 This translates as an additional barotropic force in the momentum equations \ref{eq:PE_dyn} such that:855 \[ 856 % \label{eq: PE_dyn_tides}854 This translates as an additional barotropic force in the momentum \autoref{eq:MB_PE_dyn} such that: 855 \[ 856 % \label{eq:SBC_PE_dyn_tides} 857 857 \frac{\partial {\mathrm {\mathbf U}}_h }{\partial t}= ... 858 858 +g\nabla (\Pi_{eq} + \Pi_{sal}) … … 895 895 % River runoffs 896 896 % ================================================================ 897 \section[River runoffs (\textit{sbcrnf.F90})] 898 {River runoffs (\protect\mdl{sbcrnf})} 897 \section[River runoffs (\textit{sbcrnf.F90})]{River runoffs (\protect\mdl{sbcrnf})} 899 898 \label{sec:SBC_rnf} 900 899 %------------------------------------------namsbc_rnf---------------------------------------------------- … … 1022 1021 % Ice shelf melting 1023 1022 % ================================================================ 1024 \section[Ice shelf melting (\textit{sbcisf.F90})] 1025 {Ice shelf melting (\protect\mdl{sbcisf})} 1023 \section[Ice shelf melting (\textit{sbcisf.F90})]{Ice shelf melting (\protect\mdl{sbcisf})} 1026 1024 \label{sec:SBC_isf} 1027 1025 %------------------------------------------namsbc_isf---------------------------------------------------- … … 1066 1064 The salt and heat exchange coefficients are constant and defined by \np{rn\_gammas0} and \np{rn\_gammat0}. 1067 1065 \[ 1068 % \label{eq: sbc_isf_gamma_iso}1066 % \label{eq:SBC_isf_gamma_iso} 1069 1067 \gamma^{T} = \np{rn\_gammat0} 1070 1068 \] … … 1213 1211 % ================================================================ 1214 1212 \section{Handling of icebergs (ICB)} 1215 \label{sec: ICB_icebergs}1213 \label{sec:SBC_ICB_icebergs} 1216 1214 %------------------------------------------namberg---------------------------------------------------- 1217 1215 … … 1282 1280 % Interactions with waves (sbcwave.F90, ln_wave) 1283 1281 % ============================================================================================================= 1284 \section[Interactions with waves (\textit{sbcwave.F90}, \texttt{ln\_wave})] 1285 {Interactions with waves (\protect\mdl{sbcwave}, \protect\np{ln\_wave})} 1282 \section[Interactions with waves (\textit{sbcwave.F90}, \texttt{ln\_wave})]{Interactions with waves (\protect\mdl{sbcwave}, \protect\np{ln\_wave})} 1286 1283 \label{sec:SBC_wave} 1287 1284 %------------------------------------------namsbc_wave-------------------------------------------------------- … … 1314 1311 1315 1312 % ---------------------------------------------------------------- 1316 \subsection[Neutral drag coefficient from wave model (\texttt{ln\_cdgw})] 1317 {Neutral drag coefficient from wave model (\protect\np{ln\_cdgw})} 1313 \subsection[Neutral drag coefficient from wave model (\texttt{ln\_cdgw})]{Neutral drag coefficient from wave model (\protect\np{ln\_cdgw})} 1318 1314 \label{subsec:SBC_wave_cdgw} 1319 1315 … … 1328 1324 % 3D Stokes Drift (ln_sdw, nn_sdrift) 1329 1325 % ---------------------------------------------------------------- 1330 \subsection[3D Stokes Drift (\texttt{ln\_sdw}, \texttt{nn\_sdrift})] 1331 {3D Stokes Drift (\protect\np{ln\_sdw, nn\_sdrift})} 1326 \subsection[3D Stokes Drift (\texttt{ln\_sdw}, \texttt{nn\_sdrift})]{3D Stokes Drift (\protect\np{ln\_sdw, nn\_sdrift})} 1332 1327 \label{subsec:SBC_wave_sdw} 1333 1328 … … 1343 1338 1344 1339 \[ 1345 % \label{eq: sbc_wave_sdw}1340 % \label{eq:SBC_wave_sdw} 1346 1341 \mathbf{U}_{st} = \frac{16{\pi^3}} {g} 1347 1342 \int_0^\infty \int_{-\pi}^{\pi} (cos{\theta},sin{\theta}) {f^3} … … 1368 1363 1369 1364 \[ 1370 % \label{eq: sbc_wave_sdw_0a}1365 % \label{eq:SBC_wave_sdw_0a} 1371 1366 \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \frac{\mathrm{e}^{-2k_ez}} {1-8k_ez} 1372 1367 \] … … 1375 1370 1376 1371 \[ 1377 % \label{eq: sbc_wave_sdw_0b}1372 % \label{eq:SBC_wave_sdw_0b} 1378 1373 k_e = \frac{|\mathbf{U}_{\left.st\right|_{z=0}}|} {|T_{st}|} 1379 1374 \quad \text{and }\ … … 1388 1383 1389 1384 \[ 1390 % \label{eq: sbc_wave_sdw_1}1385 % \label{eq:SBC_wave_sdw_1} 1391 1386 \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \Big[exp(2k_pz)-\beta \sqrt{-2 \pi k_pz} 1392 1387 \textit{ erf } \Big(\sqrt{-2 k_pz}\Big)\Big] … … 1404 1399 1405 1400 \[ 1406 % \label{eq: sbc_wave_eta_sdw}1401 % \label{eq:SBC_wave_eta_sdw} 1407 1402 \frac{\partial{\eta}}{\partial{t}} = 1408 1403 -\nabla_h \int_{-H}^{\eta} (\mathbf{U} + \mathbf{U}_{st}) dz … … 1416 1411 1417 1412 \[ 1418 % \label{eq: sbc_wave_tra_sdw}1413 % \label{eq:SBC_wave_tra_sdw} 1419 1414 \frac{\partial{c}}{\partial{t}} = 1420 1415 - (\mathbf{U} + \mathbf{U}_{st}) \cdot \nabla{c} … … 1425 1420 % Stokes-Coriolis term (ln_stcor) 1426 1421 % ---------------------------------------------------------------- 1427 \subsection[Stokes-Coriolis term (\texttt{ln\_stcor})] 1428 {Stokes-Coriolis term (\protect\np{ln\_stcor})} 1422 \subsection[Stokes-Coriolis term (\texttt{ln\_stcor})]{Stokes-Coriolis term (\protect\np{ln\_stcor})} 1429 1423 \label{subsec:SBC_wave_stcor} 1430 1424 … … 1440 1434 % Waves modified stress (ln_tauwoc, ln_tauw) 1441 1435 % ---------------------------------------------------------------- 1442 \subsection[Wave modified stress (\texttt{ln\_tauwoc}, \texttt{ln\_tauw})] 1443 {Wave modified sress (\protect\np{ln\_tauwoc, ln\_tauw})} 1436 \subsection[Wave modified stress (\texttt{ln\_tauwoc}, \texttt{ln\_tauw})]{Wave modified sress (\protect\np{ln\_tauwoc, ln\_tauw})} 1444 1437 \label{subsec:SBC_wave_tauw} 1445 1438 … … 1453 1446 1454 1447 \[ 1455 % \label{eq: sbc_wave_tauoc}1448 % \label{eq:SBC_wave_tauoc} 1456 1449 \tau_{oc,a} = \tau_a - \tau_w 1457 1450 \] … … 1461 1454 1462 1455 \[ 1463 % \label{eq: sbc_wave_tauw}1456 % \label{eq:SBC_wave_tauw} 1464 1457 \tau_w = \rho g \int {\frac{dk}{c_p} (S_{in}+S_{nl}+S_{diss})} 1465 1458 \] … … 1490 1483 % Diurnal cycle 1491 1484 % ------------------------------------------------------------------------------------------------------------- 1492 \subsection[Diurnal cycle (\textit{sbcdcy.F90})] 1493 {Diurnal cycle (\protect\mdl{sbcdcy})} 1485 \subsection[Diurnal cycle (\textit{sbcdcy.F90})]{Diurnal cycle (\protect\mdl{sbcdcy})} 1494 1486 \label{subsec:SBC_dcy} 1495 1487 %------------------------------------------namsbc------------------------------------------------------------- … … 1577 1569 % Surface restoring to observed SST and/or SSS 1578 1570 % ------------------------------------------------------------------------------------------------------------- 1579 \subsection[Surface restoring to observed SST and/or SSS (\textit{sbcssr.F90})] 1580 {Surface restoring to observed SST and/or SSS (\protect\mdl{sbcssr})} 1571 \subsection[Surface restoring to observed SST and/or SSS (\textit{sbcssr.F90})]{Surface restoring to observed SST and/or SSS (\protect\mdl{sbcssr})} 1581 1572 \label{subsec:SBC_ssr} 1582 1573 %------------------------------------------namsbc_ssr---------------------------------------------------- … … 1589 1580 a feedback term \emph{must} be added to the surface heat flux $Q_{ns}^o$: 1590 1581 \[ 1591 % \label{eq: sbc_dmp_q}1582 % \label{eq:SBC_dmp_q} 1592 1583 Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right) 1593 1584 \] … … 1602 1593 1603 1594 \begin{equation} 1604 \label{eq: sbc_dmp_emp}1595 \label{eq:SBC_dmp_emp} 1605 1596 \textit{emp} = \textit{emp}_o + \gamma_s^{-1} e_{3t} \frac{ \left(\left.S\right|_{k=1}-SSS_{Obs}\right)} 1606 1597 {\left.S\right|_{k=1}} … … 1613 1604 $\left.S\right|_{k=1}$ is the model surface layer salinity and 1614 1605 $\gamma_s$ is a negative feedback coefficient which is provided as a namelist parameter. 1615 Unlike heat flux, there is no physical justification for the feedback term in \autoref{eq: sbc_dmp_emp} as1606 Unlike heat flux, there is no physical justification for the feedback term in \autoref{eq:SBC_dmp_emp} as 1616 1607 the atmosphere does not care about ocean surface salinity \citep{madec.delecluse_IWN97}. 1617 1608 The SSS restoring term should be viewed as a flux correction on freshwater fluxes to … … 1663 1654 % CICE-ocean Interface 1664 1655 % ------------------------------------------------------------------------------------------------------------- 1665 \subsection[Interface to CICE (\textit{sbcice\_cice.F90})] 1666 {Interface to CICE (\protect\mdl{sbcice\_cice})} 1656 \subsection[Interface to CICE (\textit{sbcice\_cice.F90})]{Interface to CICE (\protect\mdl{sbcice\_cice})} 1667 1657 \label{subsec:SBC_cice} 1668 1658 … … 1698 1688 % Freshwater budget control 1699 1689 % ------------------------------------------------------------------------------------------------------------- 1700 \subsection[Freshwater budget control (\textit{sbcfwb.F90})] 1701 {Freshwater budget control (\protect\mdl{sbcfwb})} 1690 \subsection[Freshwater budget control (\textit{sbcfwb.F90})]{Freshwater budget control (\protect\mdl{sbcfwb})} 1702 1691 \label{subsec:SBC_fwb} 1703 1692 -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex
r11537 r11543 73 73 Its discrete expression is given by : 74 74 \begin{equation} 75 \label{eq: tra_adv}75 \label{eq:TRA_adv} 76 76 ADV_\tau = - \frac{1}{b_t} \Big( \delta_i [ e_{2u} \, e_{3u} \; u \; \tau_u] 77 77 + \delta_j [ e_{1v} \, e_{3v} \; v \; \tau_v] \Big) … … 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 \autoref{eq: tra_adv} implicitly requires the use of the continuity equation.81 The flux form in \autoref{eq:TRA_adv} implicitly requires the use of the continuity equation. 82 82 Indeed, it is obtained by using the following equality: $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which 83 83 results from the use of the continuity equation, $\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$ … … 86 86 it is consistent with the continuity equation in order to enforce the conservation properties of 87 87 the continuous equations. 88 In other words, by setting $\tau =1$ in (\autoref{eq:tra_adv}) we recover the discrete form of88 In other words, by setting $\tau = 1$ in (\autoref{eq:TRA_adv}) we recover the discrete form of 89 89 the continuity equation which is used to calculate the vertical velocity. 90 90 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 93 93 \includegraphics[width=\textwidth]{Fig_adv_scheme} 94 94 \caption{ 95 \protect\label{fig: adv_scheme}95 \protect\label{fig:TRA_adv_scheme} 96 96 Schematic representation of some ways used to evaluate the tracer value at $u$-point and 97 97 the amount of tracer exchanged between two neighbouring grid points. … … 112 112 The key difference between the advection schemes available in \NEMO\ is the choice made in space and 113 113 time interpolation to define the value of the tracer at the velocity points 114 (\autoref{fig: adv_scheme}).114 (\autoref{fig:TRA_adv_scheme}). 115 115 116 116 Along solid lateral and bottom boundaries a zero tracer flux is automatically specified, … … 139 139 two quantities that are not correlated \citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}. 140 140 141 The velocity field that appears in (\autoref{eq: tra_adv} is141 The velocity field that appears in (\autoref{eq:TRA_adv} is 142 142 the centred (\textit{now}) \textit{effective} ocean velocity, \ie\ the \textit{eulerian} velocity 143 143 (see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) and/or … … 199 199 For example, in the $i$-direction : 200 200 \begin{equation} 201 \label{eq: tra_adv_cen2}201 \label{eq:TRA_adv_cen2} 202 202 \tau_u^{cen2} = \overline T ^{i + 1/2} 203 203 \end{equation} … … 208 208 produce a sensible solution. 209 209 The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 210 so $T$ in (\autoref{eq: tra_adv_cen2}) is the \textit{now} tracer value.210 so $T$ in (\autoref{eq:TRA_adv_cen2}) is the \textit{now} tracer value. 211 211 212 212 Note that using the CEN2, the overall tracer advection is of second order accuracy since 213 both (\autoref{eq: tra_adv}) and (\autoref{eq:tra_adv_cen2}) have this order of accuracy.213 both (\autoref{eq:TRA_adv}) and (\autoref{eq:TRA_adv_cen2}) have this order of accuracy. 214 214 215 215 % 4nd order centred scheme … … 219 219 For example, in the $i$-direction: 220 220 \begin{equation} 221 \label{eq: tra_adv_cen4}221 \label{eq:TRA_adv_cen4} 222 222 \tau_u^{cen4} = \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \, \Big]}^{\,i + 1/2} 223 223 \end{equation} … … 229 229 Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but 230 230 a $4^{th}$ order evaluation of advective fluxes, 231 since the divergence of advective fluxes \autoref{eq: tra_adv} is kept at $2^{nd}$ order.231 since the divergence of advective fluxes \autoref{eq:TRA_adv} is kept at $2^{nd}$ order. 232 232 The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is usually associated with 233 233 the scheme presented here. … … 240 240 Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution. 241 241 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 242 so $T$ in (\autoref{eq: tra_adv_cen4}) is the \textit{now} tracer.242 so $T$ in (\autoref{eq:TRA_adv_cen4}) is the \textit{now} tracer. 243 243 244 244 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), … … 265 265 For example, in the $i$-direction : 266 266 \begin{equation} 267 \label{eq: tra_adv_fct}267 \label{eq:TRA_adv_fct} 268 268 \begin{split} 269 269 \tau_u^{ups} &= … … 287 287 288 288 289 For stability reasons (see \autoref{chap: STP}),290 $\tau_u^{cen}$ is evaluated in (\autoref{eq: tra_adv_fct}) using the \textit{now} tracer while289 For stability reasons (see \autoref{chap:TD}), 290 $\tau_u^{cen}$ is evaluated in (\autoref{eq:TRA_adv_fct}) using the \textit{now} tracer while 291 291 $\tau_u^{ups}$ is evaluated using the \textit{before} tracer. 292 292 In other words, the advective part of the scheme is time stepped with a leap-frog scheme … … 305 305 MUSCL has been first implemented in \NEMO\ by \citet{levy.estublier.ea_GRL01}. 306 306 In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between 307 two $T$-points (\autoref{fig: adv_scheme}).307 two $T$-points (\autoref{fig:TRA_adv_scheme}). 308 308 For example, in the $i$-direction : 309 309 \begin{equation} 310 % \label{eq: tra_adv_mus}310 % \label{eq:TRA_adv_mus} 311 311 \tau_u^{mus} = \lt\{ 312 312 \begin{split} … … 345 345 For example, in the $i$-direction: 346 346 \begin{equation} 347 \label{eq: tra_adv_ubs}347 \label{eq:TRA_adv_ubs} 348 348 \tau_u^{ubs} = \overline T ^{i + 1/2} - \frac{1}{6} 349 349 \begin{cases} … … 369 369 (\np{nn\_ubs\_v}\forcode{=2 or 4}). 370 370 371 For stability reasons (see \autoref{chap: STP}), the first term in \autoref{eq:tra_adv_ubs}371 For stability reasons (see \autoref{chap:TD}), the first term in \autoref{eq:TRA_adv_ubs} 372 372 (which corresponds to a second order centred scheme) 373 373 is evaluated using the \textit{now} tracer (centred in time) while the second term … … 376 376 This choice is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the QUICK advection scheme. 377 377 UBS and QUICK schemes only differ by one coefficient. 378 Replacing 1/6 with 1/8 in \autoref{eq: tra_adv_ubs} leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}.378 Replacing 1/6 with 1/8 in \autoref{eq:TRA_adv_ubs} leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 379 379 This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded. 380 380 Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. 381 381 382 Note that it is straightforward to rewrite \autoref{eq: tra_adv_ubs} as follows:382 Note that it is straightforward to rewrite \autoref{eq:TRA_adv_ubs} as follows: 383 383 \begin{gather} 384 \label{eq: traadv_ubs2}384 \label{eq:TRA_adv_ubs2} 385 385 \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12} 386 386 \begin{cases} … … 389 389 \end{cases} 390 390 \intertext{or equivalently} 391 % \label{eq: traadv_ubs2b}391 % \label{eq:TRA_adv_ubs2b} 392 392 u_{i + 1/2} \ \tau_u^{ubs} = u_{i + 1/2} \, \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \Big]}^{\,i + 1/2} 393 393 - \frac{1}{2} |u|_{i + 1/2} \, \frac{1}{6} \, \delta_{i + 1/2} [\tau"_i] \nonumber 394 394 \end{gather} 395 395 396 \autoref{eq: traadv_ubs2} has several advantages.396 \autoref{eq:TRA_adv_ubs2} has several advantages. 397 397 Firstly, it clearly reveals that the UBS scheme is based on the fourth order scheme to which 398 398 an upstream-biased diffusion term is added. 399 399 Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to 400 be evaluated at the \textit{now} time step using \autoref{eq: tra_adv_ubs}.400 be evaluated at the \textit{now} time step using \autoref{eq:TRA_adv_ubs}. 401 401 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which 402 402 is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$. 403 Note the current version of \NEMO\ uses the computationally more efficient formulation \autoref{eq: tra_adv_ubs}.403 Note the current version of \NEMO\ uses the computationally more efficient formulation \autoref{eq:TRA_adv_ubs}. 404 404 405 405 % ------------------------------------------------------------------------------------------------------------- … … 452 452 \ie\ the tracers appearing in its expression are the \textit{before} tracers in time, 453 453 except for the pure vertical component that appears when a rotation tensor is used. 454 This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap: STP}).454 This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}). 455 455 When \np{ln\_traldf\_msc}\forcode{=.true.}, a Method of Stabilizing Correction is used in which 456 456 the pure vertical component is split into an explicit and an implicit part \citep{lemarie.debreu.ea_OM12}. … … 527 527 The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by: 528 528 \begin{equation} 529 \label{eq: tra_ldf_lap}529 \label{eq:TRA_ldf_lap} 530 530 D_t^{lT} = \frac{1}{b_t} \Bigg( \delta_{i} \lt[ A_u^{lT} \; \frac{e_{2u} \, e_{3u}}{e_{1u}} \; \delta_{i + 1/2} [T] \rt] 531 531 + \delta_{j} \lt[ A_v^{lT} \; \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [T] \rt] \Bigg) … … 547 547 Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{=.true.}), 548 548 tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom. 549 In this case, horizontal derivatives in (\autoref{eq: tra_ldf_lap}) at the bottom level require a specific treatment.549 In this case, horizontal derivatives in (\autoref{eq:TRA_ldf_lap}) at the bottom level require a specific treatment. 550 550 They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}. 551 551 … … 561 561 {Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})} 562 562 \label{subsec:TRA_ldf_iso} 563 The general form of the second order lateral tracer subgrid scale physics (\autoref{eq: PE_zdf})563 The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:MB_zdf}) 564 564 takes the following semi -discrete space form in $z$- and $s$-coordinates: 565 565 \begin{equation} 566 \label{eq: tra_ldf_iso}566 \label{eq:TRA_ldf_iso} 567 567 \begin{split} 568 568 D_T^{lT} = \frac{1}{b_t} \Bigg[ \quad &\delta_i A_u^{lT} \lt( \frac{e_{2u} e_{3u}}{e_{1u}} \, \delta_{i + 1/2} [T] … … 585 585 the mask technique (see \autoref{sec:LBC_coast}). 586 586 587 The operator in \autoref{eq: tra_ldf_iso} involves both lateral and vertical derivatives.587 The operator in \autoref{eq:TRA_ldf_iso} involves both lateral and vertical derivatives. 588 588 For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as that 589 589 used in the vertical physics (see \autoref{sec:TRA_zdf}). … … 597 597 598 598 Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{=.true.}), 599 the horizontal derivatives at the bottom level in \autoref{eq: tra_ldf_iso} require a specific treatment.599 the horizontal derivatives at the bottom level in \autoref{eq:TRA_ldf_iso} require a specific treatment. 600 600 They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}. 601 601 … … 608 608 An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which ensures tracer variance decreases 609 609 is also available in \NEMO\ (\np{ln\_traldf\_triad}\forcode{=.true.}). 610 A complete description of the algorithm is given in \autoref{apdx: triad}.611 612 The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq: tra_ldf_lap}) twice.610 A complete description of the algorithm is given in \autoref{apdx:TRIADS}. 611 612 The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq:TRA_ldf_lap}) twice. 613 613 The operator requires an additional assumption on boundary conditions: 614 614 both first and third derivative terms normal to the coast are set to zero. 615 615 616 The lateral fourth order operator formulation on tracers is obtained by applying (\autoref{eq: tra_ldf_iso}) twice.616 The lateral fourth order operator formulation on tracers is obtained by applying (\autoref{eq:TRA_ldf_iso}) twice. 617 617 It requires an additional assumption on boundary conditions: 618 618 first and third derivative terms normal to the coast, … … 646 646 The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates, 647 647 and is based on a laplacian operator. 648 The vertical diffusion operator given by (\autoref{eq: PE_zdf}) takes the following semi -discrete space form:648 The vertical diffusion operator given by (\autoref{eq:MB_zdf}) takes the following semi -discrete space form: 649 649 \begin{gather*} 650 % \label{eq: tra_zdf}650 % \label{eq:TRA_zdf} 651 651 D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \\ 652 652 D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt] … … 659 659 Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by 660 660 $\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to account for the vertical second derivative of 661 \autoref{eq: tra_ldf_iso}.661 \autoref{eq:TRA_ldf_iso}. 662 662 663 663 At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified. … … 721 721 The surface boundary condition on temperature and salinity is applied as follows: 722 722 \begin{equation} 723 \label{eq: tra_sbc}723 \label{eq:TRA_sbc} 724 724 \begin{alignedat}{2} 725 725 F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{Q_{ns} }^t \\ … … 729 729 where $\overline x^t$ means that $x$ is averaged over two consecutive time steps 730 730 ($t - \rdt / 2$ and $t + \rdt / 2$). 731 Such time averaging prevents the divergence of odd and even time step (see \autoref{chap: STP}).731 Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:TD}). 732 732 733 733 In the linear free surface case (\np{ln\_linssh}\forcode{=.true.}), an additional term has to be added on … … 738 738 The resulting surface boundary condition is applied as follows: 739 739 \begin{equation} 740 \label{eq: tra_sbc_lin}740 \label{eq:TRA_sbc_lin} 741 741 \begin{alignedat}{2} 742 742 F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} … … 749 749 In the linear free surface case, there is a small imbalance. 750 750 The imbalance is larger than the imbalance associated with the Asselin time filter \citep{leclair.madec_OM09}. 751 This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap: STP}).751 This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:TD}). 752 752 753 753 % ------------------------------------------------------------------------------------------------------------- … … 766 766 the solar radiation penetrates the top few tens of meters of the ocean. 767 767 If it is not used (\np{ln\_traqsr}\forcode{=.false.}) all the heat flux is absorbed in the first ocean level. 768 Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq: PE_tra_T} and768 Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:MB_PE_tra_T} and 769 769 the surface boundary condition is modified to take into account only the non-penetrative part of the surface 770 770 heat flux: 771 771 \begin{equation} 772 \label{eq: PE_qsr}772 \label{eq:TRA_PE_qsr} 773 773 \begin{gathered} 774 774 \pd[T]{t} = \ldots + \frac{1}{\rho_o \, C_p \, e_3} \; \pd[I]{k} \\ … … 778 778 where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie\ the shortwave radiation) and 779 779 $I$ is the downward irradiance ($\lt. I \rt|_{z = \eta} = Q_{sr}$). 780 The additional term in \autoref{eq: PE_qsr} is discretized as follows:781 \begin{equation} 782 \label{eq: tra_qsr}780 The additional term in \autoref{eq:TRA_PE_qsr} is discretized as follows: 781 \begin{equation} 782 \label{eq:TRA_qsr} 783 783 \frac{1}{\rho_o \, C_p \, e_3} \, \pd[I]{k} \equiv \frac{1}{\rho_o \, C_p \, e_{3t}} \delta_k [I_w] 784 784 \end{equation} … … 798 798 leading to the following expression \citep{paulson.simpson_JPO77}: 799 799 \[ 800 % \label{eq: traqsr_iradiance}800 % \label{eq:TRA_qsr_iradiance} 801 801 I(z) = Q_{sr} \lt[ Re^{- z / \xi_0} + (1 - R) e^{- z / \xi_1} \rt] 802 802 \] … … 807 807 808 808 Such assumptions have been shown to provide a very crude and simplistic representation of 809 observed light penetration profiles (\cite{morel_JGR88}, see also \autoref{fig: traqsr_irradiance}).809 observed light penetration profiles (\cite{morel_JGR88}, see also \autoref{fig:TRA_qsr_irradiance}). 810 810 Light absorption in the ocean depends on particle concentration and is spectrally selective. 811 811 \cite{morel_JGR88} has shown that an accurate representation of light penetration can be provided by … … 817 817 the full spectral model of \cite{morel_JGR88} (as modified by \cite{morel.maritorena_JGR01}), 818 818 assuming the same power-law relationship. 819 As shown in \autoref{fig: traqsr_irradiance}, this formulation, called RGB (Red-Green-Blue),819 As shown in \autoref{fig:TRA_qsr_irradiance}, this formulation, called RGB (Red-Green-Blue), 820 820 reproduces quite closely the light penetration profiles predicted by the full spectal model, 821 821 but with much greater computational efficiency. … … 843 843 \end{description} 844 844 845 The trend in \autoref{eq: tra_qsr} associated with the penetration of the solar radiation is added to845 The trend in \autoref{eq:TRA_qsr} associated with the penetration of the solar radiation is added to 846 846 the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. 847 847 … … 860 860 \includegraphics[width=\textwidth]{Fig_TRA_Irradiance} 861 861 \caption{ 862 \protect\label{fig: traqsr_irradiance}862 \protect\label{fig:TRA_qsr_irradiance} 863 863 Penetration profile of the downward solar irradiance calculated by four models. 864 864 Two waveband chlorophyll-independent formulation (blue), … … 888 888 \includegraphics[width=\textwidth]{Fig_TRA_geoth} 889 889 \caption{ 890 \protect\label{fig: geothermal}890 \protect\label{fig:TRA_geothermal} 891 891 Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{emile-geay.madec_OS09}. 892 892 It is inferred from the age of the sea floor and the formulae of \citet{stein.stein_N92}. … … 910 910 the \np{rn\_geoflx\_cst}, which is also a namelist parameter. 911 911 When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in 912 the \ifile{geothermal\_heating} NetCDF file (\autoref{fig: geothermal}) \citep{emile-geay.madec_OS09}.912 the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:TRA_geothermal}) \citep{emile-geay.madec_OS09}. 913 913 914 914 % ================================================================ … … 955 955 the diffusive flux between two adjacent cells at the ocean floor is given by 956 956 \[ 957 % \label{eq: tra_bbl_diff}957 % \label{eq:TRA_bbl_diff} 958 958 \vect F_\sigma = A_l^\sigma \, \nabla_\sigma T 959 959 \] … … 963 963 \ie\ in the conditional form 964 964 \begin{equation} 965 \label{eq: tra_bbl_coef}965 \label{eq:TRA_bbl_coef} 966 966 A_l^\sigma (i,j,t) = 967 967 \begin{cases} … … 973 973 where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn\_ahtbbl} and 974 974 usually set to a value much larger than the one used for lateral mixing in the open ocean. 975 The constraint in \autoref{eq: tra_bbl_coef} implies that sigma-like diffusion only occurs when975 The constraint in \autoref{eq:TRA_bbl_coef} implies that sigma-like diffusion only occurs when 976 976 the density above the sea floor, at the top of the slope, is larger than in the deeper ocean 977 (see green arrow in \autoref{fig: bbl}).977 (see green arrow in \autoref{fig:TRA_bbl}). 978 978 In practice, this constraint is applied separately in the two horizontal directions, 979 and the density gradient in \autoref{eq: tra_bbl_coef} is evaluated with the log gradient formulation:979 and the density gradient in \autoref{eq:TRA_bbl_coef} is evaluated with the log gradient formulation: 980 980 \[ 981 % \label{eq: tra_bbl_Drho}981 % \label{eq:TRA_bbl_Drho} 982 982 \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \, \nabla_\sigma S 983 983 \] … … 1002 1002 \includegraphics[width=\textwidth]{Fig_BBL_adv} 1003 1003 \caption{ 1004 \protect\label{fig: bbl}1004 \protect\label{fig:TRA_bbl} 1005 1005 Advective/diffusive Bottom Boundary Layer. 1006 1006 The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i + 1}_{kdnw}$. … … 1026 1026 \np{nn\_bbl\_adv}\forcode{=1}: 1027 1027 the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step 1028 (see black arrow in \autoref{fig: bbl}) \citep{beckmann.doscher_JPO97}.1028 (see black arrow in \autoref{fig:TRA_bbl}) \citep{beckmann.doscher_JPO97}. 1029 1029 It is a \textit{conditional advection}, that is, advection is allowed only 1030 1030 if dense water overlies less dense water on the slope (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$) and … … 1036 1036 The advection is allowed only if dense water overlies less dense water on the slope 1037 1037 (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$). 1038 For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig: bbl}),1038 For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:TRA_bbl}), 1039 1039 is simply given by the following expression: 1040 1040 \[ 1041 % \label{eq: bbl_Utr}1041 % \label{eq:TRA_bbl_Utr} 1042 1042 u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn}) 1043 1043 \] … … 1053 1053 the surrounding water at intermediate depths. 1054 1054 The entrainment is replaced by the vertical mixing implicit in the advection scheme. 1055 Let us consider as an example the case displayed in \autoref{fig: bbl} where1055 Let us consider as an example the case displayed in \autoref{fig:TRA_bbl} where 1056 1056 the density at level $(i,kup)$ is larger than the one at level $(i,kdwn)$. 1057 1057 The advective BBL scheme modifies the tracer time tendency of the ocean cells near the topographic step by 1058 the downslope flow \autoref{eq: bbl_dw}, the horizontal \autoref{eq:bbl_hor} and1059 the upward \autoref{eq: bbl_up} return flows as follows:1058 the downslope flow \autoref{eq:TRA_bbl_dw}, the horizontal \autoref{eq:TRA_bbl_hor} and 1059 the upward \autoref{eq:TRA_bbl_up} return flows as follows: 1060 1060 \begin{alignat}{3} 1061 \label{eq: bbl_dw}1061 \label{eq:TRA_bbl_dw} 1062 1062 \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 1063 1063 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\ 1064 \label{eq: bbl_hor}1064 \label{eq:TRA_bbl_hor} 1065 1065 \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 1066 1066 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\ … … 1068 1068 \intertext{and for $k =kdw-1,\;..., \; kup$ :} 1069 1069 % 1070 \label{eq: bbl_up}1070 \label{eq:TRA_bbl_up} 1071 1071 \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 1072 1072 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} &&\lt( T^{do}_{k +1} - T^{sh}_{k} \rt) … … 1090 1090 In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations: 1091 1091 \begin{equation} 1092 \label{eq: tra_dmp}1092 \label{eq:TRA_dmp} 1093 1093 \begin{gathered} 1094 1094 \pd[T]{t} = \cdots - \gamma (T - T_o) \\ … … 1108 1108 The DMP\_TOOLS tool is provided to allow users to generate the netcdf file. 1109 1109 1110 The two main cases in which \autoref{eq: tra_dmp} is used are1110 The two main cases in which \autoref{eq:TRA_dmp} is used are 1111 1111 \textit{(a)} the specification of the boundary conditions along artificial walls of a limited domain basin and 1112 1112 \textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field … … 1146 1146 Options are defined through the \nam{dom} namelist variables. 1147 1147 The general framework for tracer time stepping is a modified leap-frog scheme \citep{leclair.madec_OM09}, 1148 \ie\ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec: STP_mLF}):1149 \begin{equation} 1150 \label{eq: tra_nxt}1148 \ie\ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:TD_mLF}): 1149 \begin{equation} 1150 \label{eq:TRA_nxt} 1151 1151 \begin{alignedat}{3} 1152 1152 &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\ … … 1263 1263 1264 1264 \begin{gather*} 1265 % \label{eq: tra_S-EOS}1265 % \label{eq:TRA_S-EOS} 1266 1266 \begin{alignedat}{2} 1267 1267 &d_a(T,S,z) = \frac{1}{\rho_o} \big[ &- a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * &T_a \big. \\ … … 1272 1272 \text{with~} T_a = T - 10 \, ; \, S_a = S - 35 \, ; \, \rho_o = 1026~Kg/m^3 1273 1273 \end{gather*} 1274 where the computer name of the coefficients as well as their standard value are given in \autoref{tab: SEOS}.1274 where the computer name of the coefficients as well as their standard value are given in \autoref{tab:TRA_SEOS}. 1275 1275 In fact, when choosing S-EOS, various approximation of EOS can be specified simply by 1276 1276 changing the associated coefficients. … … 1304 1304 \end{tabular} 1305 1305 \caption{ 1306 \protect\label{tab: SEOS}1306 \protect\label{tab:TRA_SEOS} 1307 1307 Standard value of S-EOS coefficients. 1308 1308 } … … 1326 1326 The expression for $N^2$ is given by: 1327 1327 \[ 1328 % \label{eq: tra_bn2}1328 % \label{eq:TRA_bn2} 1329 1329 N^2 = \frac{g}{e_{3w}} \lt( \beta \; \delta_{k + 1/2}[S] - \alpha \; \delta_{k + 1/2}[T] \rt) 1330 1330 \] … … 1343 1343 The freezing point of seawater is a function of salinity and pressure \citep{fofonoff.millard_bk83}: 1344 1344 \begin{equation} 1345 \label{eq: tra_eos_fzp}1345 \label{eq:TRA_eos_fzp} 1346 1346 \begin{split} 1347 1347 &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\ … … 1351 1351 \end{equation} 1352 1352 1353 \autoref{eq: tra_eos_fzp} is only used to compute the potential freezing point of sea water1353 \autoref{eq:TRA_eos_fzp} is only used to compute the potential freezing point of sea water 1354 1354 (\ie\ referenced to the surface $p = 0$), 1355 thus the pressure dependent terms in \autoref{eq: tra_eos_fzp} (last term) have been dropped.1355 thus the pressure dependent terms in \autoref{eq:TRA_eos_fzp} (last term) have been dropped. 1356 1356 The freezing point is computed through \textit{eos\_fzp}, 1357 1357 a \fortran function that can be found in \mdl{eosbn2}. … … 1386 1386 Before taking horizontal gradients between the tracers next to the bottom, 1387 1387 a linear interpolation in the vertical is used to approximate the deeper tracer as if 1388 it actually lived at the depth of the shallower tracer point (\autoref{fig: Partial_step_scheme}).1388 it actually lived at the depth of the shallower tracer point (\autoref{fig:TRA_Partial_step_scheme}). 1389 1389 For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is: 1390 1390 … … 1394 1394 \includegraphics[width=\textwidth]{Fig_partial_step_scheme} 1395 1395 \caption{ 1396 \protect\label{fig: Partial_step_scheme}1396 \protect\label{fig:TRA_Partial_step_scheme} 1397 1397 Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate 1398 1398 (\protect\np{ln\_zps}\forcode{=.true.}) in the case $(e3w_k^{i + 1} - e3w_k^i) > 0$. … … 1417 1417 and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are: 1418 1418 \begin{equation} 1419 \label{eq: zps_hde}1419 \label{eq:TRA_zps_hde} 1420 1420 \begin{split} 1421 1421 \delta_{i + 1/2} T &= … … 1443 1443 (in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}): 1444 1444 \[ 1445 % \label{eq: zps_hde_rho}1445 % \label{eq:TRA_zps_hde_rho} 1446 1446 \widetilde \rho = \rho (\widetilde T,\widetilde S,z_u) \quad \text{where~} z_u = \min \lt( z_T^{i + 1},z_T^i \rt) 1447 1447 \] … … 1454 1454 Note that in almost all the advection schemes presented in this Chapter, 1455 1455 both averaging and differencing operators appear. 1456 Yet \autoref{eq: zps_hde} has not been used in these schemes:1456 Yet \autoref{eq:TRA_zps_hde} has not been used in these schemes: 1457 1457 in contrast to diffusion and pressure gradient computations, 1458 1458 no correction for partial steps is applied for advection. -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex
r11537 r11543 18 18 % ================================================================ 19 19 \section{Vertical mixing} 20 \label{sec:ZDF _zdf}20 \label{sec:ZDF} 21 21 22 22 The discrete form of the ocean subgrid scale physics has been presented in … … 41 41 %(namelist parameter \np{ln\_zdfexp}\forcode{=.true.}) or a backward time stepping scheme 42 42 %(\np{ln\_zdfexp}\forcode{=.false.}) depending on the magnitude of the mixing coefficients, 43 %and thus of the formulation used (see \autoref{chap: STP}).43 %and thus of the formulation used (see \autoref{chap:TD}). 44 44 45 45 %--------------------------------------------namzdf-------------------------------------------------------- … … 92 92 Following \citet{pacanowski.philander_JPO81}, the following formulation has been implemented: 93 93 \[ 94 % \label{eq: zdfric}94 % \label{eq:ZDF_ric} 95 95 \left\{ 96 96 \begin{aligned} … … 151 151 its destruction through stratification, its vertical diffusion, and its dissipation of \citet{kolmogorov_IANS42} type: 152 152 \begin{equation} 153 \label{eq: zdftke_e}153 \label{eq:ZDF_tke_e} 154 154 \frac{\partial \bar{e}}{\partial t} = 155 155 \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2 … … 161 161 \end{equation} 162 162 \[ 163 % \label{eq: zdftke_kz}163 % \label{eq:ZDF_tke_kz} 164 164 \begin{split} 165 165 K_m &= C_k\ l_k\ \sqrt {\bar{e}\; } \\ … … 175 175 $P_{rt}$ can be set to unity or, following \citet{blanke.delecluse_JPO93}, be a function of the local Richardson number, $R_i$: 176 176 \begin{align*} 177 % \label{eq: prt}177 % \label{eq:ZDF_prt} 178 178 P_{rt} = 179 179 \begin{cases} … … 208 208 The first two are based on the following first order approximation \citep{blanke.delecluse_JPO93}: 209 209 \begin{equation} 210 \label{eq: tke_mxl0_1}210 \label{eq:ZDF_tke_mxl0_1} 211 211 l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N 212 212 \end{equation} … … 219 219 To overcome these drawbacks, \citet{madec.delecluse.ea_NPM98} introduces the \np{nn\_mxl}\forcode{=2, 3} cases, 220 220 which add an extra assumption concerning the vertical gradient of the computed length scale. 221 So, the length scales are first evaluated as in \autoref{eq: tke_mxl0_1} and then bounded such that:221 So, the length scales are first evaluated as in \autoref{eq:ZDF_tke_mxl0_1} and then bounded such that: 222 222 \begin{equation} 223 \label{eq: tke_mxl_constraint}223 \label{eq:ZDF_tke_mxl_constraint} 224 224 \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1 225 225 \qquad \text{with }\ l = l_k = l_\epsilon 226 226 \end{equation} 227 \autoref{eq: tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than227 \autoref{eq:ZDF_tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than 228 228 the variations of depth. 229 229 It provides a better approximation of the \citet{gaspar.gregoris.ea_JGR90} formulation while being much less … … 231 231 In particular, it allows the length scale to be limited not only by the distance to the surface or 232 232 to the ocean bottom but also by the distance to a strongly stratified portion of the water column such as 233 the thermocline (\autoref{fig: mixing_length}).234 In order to impose the \autoref{eq: tke_mxl_constraint} constraint, we introduce two additional length scales:233 the thermocline (\autoref{fig:ZDF_mixing_length}). 234 In order to impose the \autoref{eq:ZDF_tke_mxl_constraint} constraint, we introduce two additional length scales: 235 235 $l_{up}$ and $l_{dwn}$, the upward and downward length scales, and 236 236 evaluate the dissipation and mixing length scales as … … 241 241 \includegraphics[width=\textwidth]{Fig_mixing_length} 242 242 \caption{ 243 \protect\label{fig: mixing_length}243 \protect\label{fig:ZDF_mixing_length} 244 244 Illustration of the mixing length computation. 245 245 } … … 248 248 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 249 249 \[ 250 % \label{eq: tke_mxl2}250 % \label{eq:ZDF_tke_mxl2} 251 251 \begin{aligned} 252 252 l_{up\ \ }^{(k)} &= \min \left( l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \; \right) … … 256 256 \end{aligned} 257 257 \] 258 where $l^{(k)}$ is computed using \autoref{eq: tke_mxl0_1}, \ie\ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$.258 where $l^{(k)}$ is computed using \autoref{eq:ZDF_tke_mxl0_1}, \ie\ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. 259 259 260 260 In the \np{nn\_mxl}\forcode{=2} case, the dissipation and mixing length scales take the same value: … … 262 262 the dissipation and mixing turbulent length scales are give as in \citet{gaspar.gregoris.ea_JGR90}: 263 263 \[ 264 % \label{eq: tke_mxl_gaspar}264 % \label{eq:ZDF_tke_mxl_gaspar} 265 265 \begin{aligned} 266 266 & l_k = \sqrt{\ l_{up} \ \ l_{dwn}\ } \\ … … 325 325 The parameterization, tuned against large-eddy simulation, includes the whole effect of LC in 326 326 an extra source term of TKE, $P_{LC}$. 327 The presence of $P_{LC}$ in \autoref{eq: zdftke_e}, the TKE equation, is controlled by setting \np{ln\_lc} to327 The presence of $P_{LC}$ in \autoref{eq:ZDF_tke_e}, the TKE equation, is controlled by setting \np{ln\_lc} to 328 328 \forcode{.true.} in the \nam{zdf\_tke} namelist. 329 329 … … 428 428 $\psi$ \citep{umlauf.burchard_JMR03, umlauf.burchard_CSR05}. 429 429 This later variable is defined as: $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$, 430 where the triplet $(p, m, n)$ value given in Tab.\autoref{tab: GLS} allows to recover a number of430 where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:ZDF_GLS} allows to recover a number of 431 431 well-known turbulent closures ($k$-$kl$ \citep{mellor.yamada_RG82}, $k$-$\epsilon$ \citep{rodi_JGR87}, 432 432 $k$-$\omega$ \citep{wilcox_AJ88} among others \citep{umlauf.burchard_JMR03,kantha.carniel_JMR03}). 433 433 The GLS scheme is given by the following set of equations: 434 434 \begin{equation} 435 \label{eq: zdfgls_e}435 \label{eq:ZDF_gls_e} 436 436 \frac{\partial \bar{e}}{\partial t} = 437 437 \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 … … 443 443 444 444 \[ 445 % \label{eq: zdfgls_psi}445 % \label{eq:ZDF_gls_psi} 446 446 \begin{split} 447 447 \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{ … … 455 455 456 456 \[ 457 % \label{eq: zdfgls_kz}457 % \label{eq:ZDF_gls_kz} 458 458 \begin{split} 459 459 K_m &= C_{\mu} \ \sqrt {\bar{e}} \ l \\ … … 463 463 464 464 \[ 465 % \label{eq: zdfgls_eps}465 % \label{eq:ZDF_gls_eps} 466 466 {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \; 467 467 \] … … 470 470 The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) depends of 471 471 the choice of the turbulence model. 472 Four different turbulent models are pre-defined (\autoref{tab: GLS}).472 Four different turbulent models are pre-defined (\autoref{tab:ZDF_GLS}). 473 473 They are made available through the \np{nn\_clo} namelist parameter. 474 474 … … 495 495 \end{tabular} 496 496 \caption{ 497 \protect\label{tab: GLS}497 \protect\label{tab:ZDF_GLS} 498 498 Set of predefined GLS parameters, or equivalently predefined turbulence models available with 499 499 \protect\np{ln\_zdfgls}\forcode{=.true.} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\nam{zdf\_gls}. … … 559 559 \includegraphics[width=\textwidth]{Fig_ZDF_TKE_time_scheme} 560 560 \caption{ 561 \protect\label{fig: TKE_time_scheme}561 \protect\label{fig:ZDF_TKE_time_scheme} 562 562 Illustration of the subgrid kinetic energy integration in GLS and TKE schemes and its links to the momentum and tracer time integration. 563 563 } … … 567 567 568 568 The production of turbulence by vertical shear (the first term of the right hand side of 569 \autoref{eq: zdftke_e}) and \autoref{eq:zdfgls_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion570 (first line in \autoref{eq: PE_zdf}).569 \autoref{eq:ZDF_tke_e}) and \autoref{eq:ZDF_gls_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion 570 (first line in \autoref{eq:MB_zdf}). 571 571 To do so a special care has to be taken for both the time and space discretization of 572 572 the kinetic energy equation \citep{burchard_OM02,marsaleix.auclair.ea_OM08}. 573 573 574 Let us first address the time stepping issue. \autoref{fig: TKE_time_scheme} shows how574 Let us first address the time stepping issue. \autoref{fig:ZDF_TKE_time_scheme} shows how 575 575 the two-level Leap-Frog time stepping of the momentum and tracer equations interplays with 576 576 the one-level forward time stepping of the equation for $\bar{e}$. … … 579 579 summing the result vertically: 580 580 \begin{equation} 581 \label{eq: energ1}581 \label{eq:ZDF_energ1} 582 582 \begin{split} 583 583 \int_{-H}^{\eta} u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt} \right) \,dz \\ … … 587 587 \end{equation} 588 588 Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$, 589 known at time $t$ (\autoref{fig: TKE_time_scheme}), as it is required when using the TKE scheme590 (see \autoref{sec: STP_forward_imp}).591 The first term of the right hand side of \autoref{eq: energ1} represents the kinetic energy transfer at589 known at time $t$ (\autoref{fig:ZDF_TKE_time_scheme}), as it is required when using the TKE scheme 590 (see \autoref{sec:TD_forward_imp}). 591 The first term of the right hand side of \autoref{eq:ZDF_energ1} represents the kinetic energy transfer at 592 592 the surface (atmospheric forcing) and at the bottom (friction effect). 593 593 The second term is always negative. 594 594 It is the dissipation rate of kinetic energy, and thus minus the shear production rate of $\bar{e}$. 595 \autoref{eq: energ1} implies that, to be energetically consistent,595 \autoref{eq:ZDF_energ1} implies that, to be energetically consistent, 596 596 the production rate of $\bar{e}$ used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as 597 597 ${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$ … … 599 599 600 600 A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification 601 (second term of the right hand side of \autoref{eq: zdftke_e} and \autoref{eq:zdfgls_e}).601 (second term of the right hand side of \autoref{eq:ZDF_tke_e} and \autoref{eq:ZDF_gls_e}). 602 602 This term must balance the input of potential energy resulting from vertical mixing. 603 603 The rate of change of potential energy (in 1D for the demonstration) due to vertical mixing is obtained by 604 604 multiplying the vertical density diffusion tendency by $g\,z$ and and summing the result vertically: 605 605 \begin{equation} 606 \label{eq: energ2}606 \label{eq:ZDF_energ2} 607 607 \begin{split} 608 608 \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt} \right) \,dz \\ … … 614 614 \end{equation} 615 615 where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$. 616 The first term of the right hand side of \autoref{eq: energ2} is always zero because616 The first term of the right hand side of \autoref{eq:ZDF_energ2} is always zero because 617 617 there is no diffusive flux through the ocean surface and bottom). 618 618 The second term is minus the destruction rate of $\bar{e}$ due to stratification. 619 Therefore \autoref{eq: energ1} implies that, to be energetically consistent,620 the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq: zdftke_e} and \autoref{eq:zdfgls_e}.619 Therefore \autoref{eq:ZDF_energ1} implies that, to be energetically consistent, 620 the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:ZDF_tke_e} and \autoref{eq:ZDF_gls_e}. 621 621 622 622 Let us now address the space discretization issue. 623 623 The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity components are in 624 the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig: cell}).624 the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:DOM_cell}). 625 625 A space averaging is thus required to obtain the shear TKE production term. 626 By redoing the \autoref{eq: energ1} in the 3D case, it can be shown that the product of eddy coefficient by626 By redoing the \autoref{eq:ZDF_energ1} in the 3D case, it can be shown that the product of eddy coefficient by 627 627 the shear at $t$ and $t-\rdt$ must be performed prior to the averaging. 628 628 Furthermore, the time variation of $e_3$ has be taken into account. … … 630 630 The above energetic considerations leads to the following final discrete form for the TKE equation: 631 631 \begin{equation} 632 \label{eq: zdftke_ene}632 \label{eq:ZDF_tke_ene} 633 633 \begin{split} 634 634 \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt} \equiv … … 647 647 \end{split} 648 648 \end{equation} 649 where the last two terms in \autoref{eq: zdftke_ene} (vertical diffusion and Kolmogorov dissipation)650 are time stepped using a backward scheme (see\autoref{sec: STP_forward_imp}).649 where the last two terms in \autoref{eq:ZDF_tke_ene} (vertical diffusion and Kolmogorov dissipation) 650 are time stepped using a backward scheme (see\autoref{sec:TD_forward_imp}). 651 651 Note that the Kolmogorov term has been linearized in time in order to render the implicit computation possible. 652 652 %The restart of the TKE scheme requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as 653 %they all appear in the right hand side of \autoref{eq: zdftke_ene}.653 %they all appear in the right hand side of \autoref{eq:ZDF_tke_ene}. 654 654 %For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored. 655 655 … … 679 679 \includegraphics[width=\textwidth]{Fig_npc} 680 680 \caption{ 681 \protect\label{fig: npc}681 \protect\label{fig:ZDF_npc} 682 682 Example of an unstable density profile treated by the non penetrative convective adjustment algorithm. 683 683 $1^{st}$ step: the initial profile is checked from the surface to the bottom. … … 702 702 (\ie\ until the mixed portion of the water column has \textit{exactly} the density of the water just below) 703 703 \citep{madec.delecluse.ea_JPO91}. 704 The associated algorithm is an iterative process used in the following way (\autoref{fig: npc}):704 The associated algorithm is an iterative process used in the following way (\autoref{fig:ZDF_npc}): 705 705 starting from the top of the ocean, the first instability is found. 706 706 Assume in the following that the instability is located between levels $k$ and $k+1$. … … 759 759 Note that the stability test is performed on both \textit{before} and \textit{now} values of $N^2$. 760 760 This removes a potential source of divergence of odd and even time step in 761 a leapfrog environment \citep{leclair_phd10} (see \autoref{sec: STP_mLF}).761 a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:TD_mLF}). 762 762 763 763 % ------------------------------------------------------------------------------------------------------------- … … 772 772 with statically unstable density profiles. 773 773 In such a case, the term corresponding to the destruction of turbulent kinetic energy through stratification in 774 \autoref{eq: zdftke_e} or \autoref{eq:zdfgls_e} becomes a source term, since $N^2$ is negative.774 \autoref{eq:ZDF_tke_e} or \autoref{eq:ZDF_gls_e} becomes a source term, since $N^2$ is negative. 775 775 It results in large values of $A_T^{vT}$ and $A_T^{vT}$, and also of the four neighboring values at 776 776 velocity points $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1}$). … … 814 814 Diapycnal mixing of S and T are described by diapycnal diffusion coefficients 815 815 \begin{align*} 816 % \label{eq: zdfddm_Kz}816 % \label{eq:ZDF_ddm_Kz} 817 817 &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\ 818 818 &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS} … … 826 826 (1981): 827 827 \begin{align} 828 \label{eq: zdfddm_f}828 \label{eq:ZDF_ddm_f} 829 829 A_f^{vS} &= 830 830 \begin{cases} … … 832 832 0 &\text{otherwise} 833 833 \end{cases} 834 \\ \label{eq: zdfddm_f_T}834 \\ \label{eq:ZDF_ddm_f_T} 835 835 A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho 836 836 \end{align} … … 841 841 \includegraphics[width=\textwidth]{Fig_zdfddm} 842 842 \caption{ 843 \protect\label{fig: zdfddm}843 \protect\label{fig:ZDF_ddm} 844 844 From \citet{merryfield.holloway.ea_JPO99} : 845 845 (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in regions of salt fingering. … … 854 854 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 855 855 856 The factor 0.7 in \autoref{eq: zdfddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx 0.7$ of856 The factor 0.7 in \autoref{eq:ZDF_ddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx 0.7$ of 857 857 buoyancy flux of heat to buoyancy flux of salt (\eg, \citet{mcdougall.taylor_JMR84}). 858 858 Following \citet{merryfield.holloway.ea_JPO99}, we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$. … … 861 861 Federov (1988) is used: 862 862 \begin{align} 863 % \label{eq: zdfddm_d}863 % \label{eq:ZDF_ddm_d} 864 864 A_d^{vT} &= 865 865 \begin{cases} … … 869 869 \end{cases} 870 870 \nonumber \\ 871 \label{eq: zdfddm_d_S}871 \label{eq:ZDF_ddm_d_S} 872 872 A_d^{vS} &= 873 873 \begin{cases} … … 878 878 \end{align} 879 879 880 The dependencies of \autoref{eq: zdfddm_f} to \autoref{eq:zdfddm_d_S} on $R_\rho$ are illustrated in881 \autoref{fig: zdfddm}.880 The dependencies of \autoref{eq:ZDF_ddm_f} to \autoref{eq:ZDF_ddm_d_S} on $R_\rho$ are illustrated in 881 \autoref{fig:ZDF_ddm}. 882 882 Implementing this requires computing $R_\rho$ at each grid point on every time step. 883 883 This is done in \mdl{eosbn2} at the same time as $N^2$ is computed. … … 891 891 \label{sec:ZDF_drg} 892 892 893 %--------------------------------------------nam bfr--------------------------------------------------------893 %--------------------------------------------namdrg-------------------------------------------------------- 894 894 % 895 895 \nlst{namdrg} … … 910 910 For the bottom boundary layer, one has: 911 911 \[ 912 % \label{eq: zdfbfr_flux}912 % \label{eq:ZDF_bfr_flux} 913 913 A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U} 914 914 \] … … 926 926 To illustrate this, consider the equation for $u$ at $k$, the last ocean level: 927 927 \begin{equation} 928 \label{eq: zdfdrg_flux2}928 \label{eq:ZDF_drg_flux2} 929 929 \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}} 930 930 \end{equation} … … 946 946 These coefficients are computed in \mdl{zdfdrg} and generally take the form $c_b^{\textbf U}$ where: 947 947 \begin{equation} 948 \label{eq: zdfbfr_bdef}948 \label{eq:ZDF_bfr_bdef} 949 949 \frac{\partial {\textbf U_h}}{\partial t} = 950 950 - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b … … 963 963 the friction is proportional to the interior velocity (\ie\ the velocity of the first/last model level): 964 964 \[ 965 % \label{eq: zdfbfr_linear}965 % \label{eq:ZDF_bfr_linear} 966 966 {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b 967 967 \] … … 978 978 It can be changed by specifying \np{rn\_Uc0} (namelist parameter). 979 979 980 For the linear friction case the drag coefficient used in the general expression \autoref{eq: zdfbfr_bdef} is:981 \[ 982 % \label{eq: zdfbfr_linbfr_b}980 For the linear friction case the drag coefficient used in the general expression \autoref{eq:ZDF_bfr_bdef} is: 981 \[ 982 % \label{eq:ZDF_bfr_linbfr_b} 983 983 c_b^T = - r 984 984 \] … … 1002 1002 The non-linear bottom friction parameterisation assumes that the top/bottom friction is quadratic: 1003 1003 \[ 1004 % \label{eq: zdfdrg_nonlinear}1004 % \label{eq:ZDF_drg_nonlinear} 1005 1005 {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h 1006 1006 }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b … … 1018 1018 For the non-linear friction case the term computed in \mdl{zdfdrg} is: 1019 1019 \[ 1020 % \label{eq: zdfdrg_nonlinbfr}1020 % \label{eq:ZDF_drg_nonlinbfr} 1021 1021 c_b^T = - \; C_D\;\left[ \left(\bar{u_b}^{i}\right)^2 + \left(\bar{v_b}^{j}\right)^2 + e_b \right]^{1/2} 1022 1022 \] … … 1077 1077 1078 1078 Since this is conditionally stable, some care needs to exercised over the choice of parameters to ensure that the implementation of explicit top/bottom friction does not induce numerical instability. 1079 For the purposes of stability analysis, an approximation to \autoref{eq: zdfdrg_flux2} is:1079 For the purposes of stability analysis, an approximation to \autoref{eq:ZDF_drg_flux2} is: 1080 1080 \begin{equation} 1081 \label{eq: Eqn_drgstab}1081 \label{eq:ZDF_Eqn_drgstab} 1082 1082 \begin{split} 1083 1083 \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt \\ … … 1090 1090 |\Delta u| < \;|u| 1091 1091 \] 1092 \noindent which, using \autoref{eq: Eqn_drgstab}, gives:1092 \noindent which, using \autoref{eq:ZDF_Eqn_drgstab}, gives: 1093 1093 \[ 1094 1094 r\frac{2\rdt}{e_{3u}} < 1 \qquad \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\ … … 1133 1133 At the top (below an ice shelf cavity): 1134 1134 \[ 1135 % \label{eq: dynzdf_drg_top}1135 % \label{eq:ZDF_dynZDF__drg_top} 1136 1136 \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t} 1137 1137 = c_{t}^{\textbf{U}}\textbf{u}^{n+1}_{t} … … 1140 1140 At the bottom (above the sea floor): 1141 1141 \[ 1142 % \label{eq: dynzdf_drg_bot}1142 % \label{eq:ZDF_dynZDF__drg_bot} 1143 1143 \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b} 1144 1144 = c_{b}^{\textbf{U}}\textbf{u}^{n+1}_{b} … … 1183 1183 and the resulting diffusivity is obtained as 1184 1184 \[ 1185 % \label{eq: Kwave}1185 % \label{eq:ZDF_Kwave} 1186 1186 A^{vT}_{wave} = R_f \,\frac{ \epsilon }{ \rho \, N^2 } 1187 1187 \] … … 1242 1242 1243 1243 \begin{equation} 1244 \label{eq: Bv}1244 \label{eq:ZDF_Bv} 1245 1245 B_{v} = \alpha {A} {U}_{st} {exp(3kz)} 1246 1246 \end{equation} … … 1276 1276 criteria for a range of advection schemes. The values for the Leap-Frog with Robert 1277 1277 asselin filter time-stepping (as used in NEMO) are reproduced in 1278 \autoref{tab: zad_Aimp_CFLcrit}. Treating the vertical advection implicitly can avoid these1278 \autoref{tab:ZDF_zad_Aimp_CFLcrit}. Treating the vertical advection implicitly can avoid these 1279 1279 restrictions but at the cost of large dispersive errors and, possibly, large numerical 1280 1280 viscosity. The adaptive-implicit vertical advection option provides a targetted use of the … … 1296 1296 \end{tabular} 1297 1297 \caption{ 1298 \protect\label{tab: zad_Aimp_CFLcrit}1298 \protect\label{tab:ZDF_zad_Aimp_CFLcrit} 1299 1299 The advective CFL criteria for a range of spatial discretizations for the Leap-Frog with Robert Asselin filter time-stepping 1300 1300 ($\nu=0.1$) as given in \citep{lemarie.debreu.ea_OM15}. … … 1313 1313 1314 1314 \begin{equation} 1315 \label{eq: Eqn_zad_Aimp_Courant}1315 \label{eq:ZDF_Eqn_zad_Aimp_Courant} 1316 1316 \begin{split} 1317 1317 Cu &= {2 \rdt \over e^n_{3t_{ijk}}} \bigg (\big [ \texttt{Max}(w^n_{ijk},0.0) - \texttt{Min}(w^n_{ijk+1},0.0) \big ] \\ … … 1326 1326 1327 1327 \begin{align} 1328 \label{eq: Eqn_zad_Aimp_partition}1328 \label{eq:ZDF_Eqn_zad_Aimp_partition} 1329 1329 Cu_{min} &= 0.15 \nonumber \\ 1330 1330 Cu_{max} &= 0.3 \nonumber \\ … … 1343 1343 \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_coeff} 1344 1344 \caption{ 1345 \protect\label{fig: zad_Aimp_coeff}1345 \protect\label{fig:ZDF_zad_Aimp_coeff} 1346 1346 The value of the partitioning coefficient ($C\kern-0.14em f$) used to partition vertical velocities into parts to 1347 1347 be treated implicitly and explicitly for a range of typical Courant numbers (\forcode{ln_zad_Aimp=.true.}) … … 1355 1355 1356 1356 \begin{align} 1357 \label{eq: Eqn_zad_Aimp_partition2}1357 \label{eq:ZDF_Eqn_zad_Aimp_partition2} 1358 1358 w_{i_{ijk}} &= C\kern-0.14em f_{ijk} w_{n_{ijk}} \nonumber \\ 1359 w_{n_{ijk}} &= (1-C\kern-0.14em f_{ijk}) w_{n_{ijk}} 1359 w_{n_{ijk}} &= (1-C\kern-0.14em f_{ijk}) w_{n_{ijk}} 1360 1360 \end{align} 1361 1361 1362 1362 \noindent Note that the coefficient is such that the treatment is never fully implicit; 1363 the three cases from \autoref{eq: Eqn_zad_Aimp_partition} can be considered as:1363 the three cases from \autoref{eq:ZDF_Eqn_zad_Aimp_partition} can be considered as: 1364 1364 fully-explicit; mixed explicit/implicit and mostly-implicit. With the settings shown the 1365 coefficient ($C\kern-0.14em f$) varies as shown in \autoref{fig: zad_Aimp_coeff}. Note with these values1365 coefficient ($C\kern-0.14em f$) varies as shown in \autoref{fig:ZDF_zad_Aimp_coeff}. Note with these values 1366 1366 the $Cu_{cut}$ boundary between the mixed implicit-explicit treatment and 'mostly 1367 1367 implicit' is 0.45 which is just below the stability limited given in 1368 \autoref{tab: zad_Aimp_CFLcrit} for a 3rd order scheme.1368 \autoref{tab:ZDF_zad_Aimp_CFLcrit} for a 3rd order scheme. 1369 1369 1370 1370 The $w_i$ component is added to the implicit solvers for the vertical mixing in … … 1376 1376 vertical fluxes are then removed since they are added by the implicit solver later on. 1377 1377 1378 The adaptive-implicit vertical advection option is new to NEMO at v4.0 and has yet to be 1378 The adaptive-implicit vertical advection option is new to NEMO at v4.0 and has yet to be 1379 1379 used in a wide range of simulations. The following test simulation, however, does illustrate 1380 1380 the potential benefits and will hopefully encourage further testing and feedback from users: … … 1384 1384 \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_overflow_frames} 1385 1385 \caption{ 1386 \protect\label{fig: zad_Aimp_overflow_frames}1386 \protect\label{fig:ZDF_zad_Aimp_overflow_frames} 1387 1387 A time-series of temperature vertical cross-sections for the OVERFLOW test case. These results are for the default 1388 1388 settings with \forcode{nn_rdt=10.0} and without adaptive implicit vertical advection (\forcode{ln_zad_Aimp=.false.}). … … 1408 1408 \noindent which were chosen to provide a slightly more stable and less noisy solution. The 1409 1409 result when using the default value of \forcode{nn_rdt=10.} without adaptive-implicit 1410 vertical velocity is illustrated in \autoref{fig: zad_Aimp_overflow_frames}. The mass of1410 vertical velocity is illustrated in \autoref{fig:ZDF_zad_Aimp_overflow_frames}. The mass of 1411 1411 cold water, initially sitting on the shelf, moves down the slope and forms a 1412 1412 bottom-trapped, dense plume. Even with these extra physics choices the model is close to … … 1418 1418 1419 1419 The results with \forcode{ln_zad_Aimp=.true.} and a variety of model timesteps 1420 are shown in \autoref{fig: zad_Aimp_overflow_all_rdt} (together with the equivalent1420 are shown in \autoref{fig:ZDF_zad_Aimp_overflow_all_rdt} (together with the equivalent 1421 1421 frames from the base run). In this simple example the use of the adaptive-implicit 1422 1422 vertcal advection scheme has enabled a 12x increase in the model timestep without … … 1434 1434 \autoref{sec:MISC_opt} for activation details). 1435 1435 1436 \autoref{fig: zad_Aimp_maxCf} shows examples of the maximum partitioning coefficient for1436 \autoref{fig:ZDF_zad_Aimp_maxCf} shows examples of the maximum partitioning coefficient for 1437 1437 the various overflow tests. Note that the adaptive-implicit vertical advection scheme is 1438 1438 active even in the base run with \forcode{nn_rdt=10.0s} adding to the evidence that the … … 1441 1441 oscillatory nature of this measure appears to be linked to the progress of the plume front 1442 1442 as each cusp is associated with the location of the maximum shifting to the adjacent cell. 1443 This is illustrated in \autoref{fig: zad_Aimp_maxCf_loc} where the i- and k- locations of the1443 This is illustrated in \autoref{fig:ZDF_zad_Aimp_maxCf_loc} where the i- and k- locations of the 1444 1444 maximum have been overlaid for the base run case. 1445 1445 … … 1463 1463 \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_overflow_all_rdt} 1464 1464 \caption{ 1465 \protect\label{fig: zad_Aimp_overflow_all_rdt}1466 Sample temperature vertical cross-sections from mid- and end-run using different values for \forcode{nn_rdt} 1465 \protect\label{fig:ZDF_zad_Aimp_overflow_all_rdt} 1466 Sample temperature vertical cross-sections from mid- and end-run using different values for \forcode{nn_rdt} 1467 1467 and with or without adaptive implicit vertical advection. Without the adaptive implicit vertical advection only 1468 1468 the run with the shortest timestep is able to run to completion. Note also that the colour-scale has been … … 1476 1476 \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_maxCf} 1477 1477 \caption{ 1478 \protect\label{fig: zad_Aimp_maxCf}1478 \protect\label{fig:ZDF_zad_Aimp_maxCf} 1479 1479 The maximum partitioning coefficient during a series of test runs with increasing model timestep length. 1480 At the larger timesteps, the vertical velocity is treated mostly implicitly at some location throughout 1480 At the larger timesteps, the vertical velocity is treated mostly implicitly at some location throughout 1481 1481 the run. 1482 1482 } … … 1488 1488 \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_maxCf_loc} 1489 1489 \caption{ 1490 \protect\label{fig: zad_Aimp_maxCf_loc}1490 \protect\label{fig:ZDF_zad_Aimp_maxCf_loc} 1491 1491 The maximum partitioning coefficient for the \forcode{nn_rdt=10.0s} case overlaid with information on the gridcell i- and k- 1492 locations of the maximum value. 1492 locations of the maximum value. 1493 1493 } 1494 1494 \end{center} -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_cfgs.tex
r11541 r11543 6 6 % ================================================================ 7 7 \chapter{Configurations} 8 \label{chap:CFG }8 \label{chap:CFGS} 9 9 10 10 \chaptertoc … … 16 16 % ================================================================ 17 17 \section{Introduction} 18 \label{sec:CFG _intro}18 \label{sec:CFGS_intro} 19 19 20 20 The purpose of this part of the manual is to introduce the \NEMO\ reference configurations. … … 36 36 \section[C1D: 1D Water column model (\texttt{\textbf{key\_c1d}})] 37 37 {C1D: 1D Water column model (\protect\key{c1d})} 38 \label{sec:CFG _c1d}38 \label{sec:CFGS_c1d} 39 39 40 40 The 1D model option simulates a stand alone water column within the 3D \NEMO\ system. … … 76 76 % ================================================================ 77 77 \section{ORCA family: global ocean with tripolar grid} 78 \label{sec:CFG _orca}78 \label{sec:CFGS_orca} 79 79 80 80 The ORCA family is a series of global ocean configurations that are run together with … … 92 92 \includegraphics[width=\textwidth]{Fig_ORCA_NH_mesh} 93 93 \caption{ 94 \protect\label{fig: MISC_ORCA_msh}94 \protect\label{fig:CFGS_ORCA_msh} 95 95 ORCA mesh conception. 96 96 The departure from an isotropic Mercator grid start poleward of 20\deg{N}. … … 108 108 % ------------------------------------------------------------------------------------------------------------- 109 109 \subsection{ORCA tripolar grid} 110 \label{subsec:CFG _orca_grid}110 \label{subsec:CFGS_orca_grid} 111 111 112 112 The ORCA grid is a tripolar grid 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 poles 118 (\autoref{fig: MISC_ORCA_msh}).118 (\autoref{fig:CFGS_ORCA_msh}). 119 119 The resulting mesh presents no loss of continuity in either the mesh lines or the scale factors, 120 120 or even the scale factor derivatives over the whole ocean domain, as the mesh is not a composite mesh. … … 125 125 \includegraphics[width=\textwidth]{Fig_ORCA_aniso} 126 126 \caption { 127 \protect\label{fig: MISC_ORCA_e1e2}127 \protect\label{fig:CFGS_ORCA_e1e2} 128 128 \textit{Top}: Horizontal scale factors ($e_1$, $e_2$) and 129 129 \textit{Bottom}: ratio of anisotropy ($e_1 / e_2$) … … 143 143 (especially in area of strong eddy activities such as the Gulf Stream) and keeping the smallest scale factor in 144 144 the northern hemisphere larger than the smallest one in the southern hemisphere. 145 The resulting mesh is shown in \autoref{fig: MISC_ORCA_msh} and \autoref{fig:MISC_ORCA_e1e2} for145 The resulting mesh is shown in \autoref{fig:CFGS_ORCA_msh} and \autoref{fig:CFGS_ORCA_e1e2} for 146 146 a half a degree grid (ORCA\_R05). 147 147 The smallest ocean scale factor is found in along Antarctica, … … 152 152 % ------------------------------------------------------------------------------------------------------------- 153 153 \subsection{ORCA pre-defined resolution} 154 \label{subsec:CFG _orca_resolution}154 \label{subsec:CFGS_orca_resolution} 155 155 156 156 The \NEMO\ system is provided with five built-in ORCA configurations which differ in the horizontal resolution. … … 159 159 which sets the grid size and configuration name parameters. 160 160 The \NEMO\ System Team provides only ORCA2 domain input file "\ifile{ORCA\_R2\_zps\_domcfg}" file 161 (\autoref{tab: ORCA}).161 (\autoref{tab:CFGS_ORCA}). 162 162 163 163 %--------------------------------------------------TABLE-------------------------------------------------- … … 175 175 \end{tabular} 176 176 \caption{ 177 \protect\label{tab: ORCA}177 \protect\label{tab:CFGS_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. … … 184 184 185 185 186 The ORCA\_R2 configuration has the following specificity: starting from a 2\deg ~ORCA mesh,186 The ORCA\_R2 configuration has the following specificity: starting from a 2\deg\ ORCA mesh, 187 187 local mesh refinements were applied to the Mediterranean, Red, Black and Caspian Seas, 188 so that the resolution is 1\deg ~there.188 so that the resolution is 1\deg\ there. 189 189 A local transformation were also applied with in the Tropics in order to refine the meridional resolution up to 190 0.5\deg ~at the Equator.190 0.5\deg\ at the Equator. 191 191 192 192 The ORCA\_R1 configuration has only a local tropical transformation to refine the meridional resolution up to 193 1/3\deg ~at the Equator.193 1/3\deg\ at the Equator. 194 194 Note that the tropical mesh refinements in ORCA\_R2 and R1 strongly increases the mesh anisotropy there. 195 195 … … 198 198 For ORCA\_R1 and R025, setting the configuration key to 75 allows to use 75 vertical levels, otherwise 46 are used. 199 199 In the other ORCA configurations, 31 levels are used 200 (see \autoref{tab: orca_zgr}). %\sfcomment{HERE I need to put new table for ORCA2 values} and \autoref{fig:zgr}).200 (see \autoref{tab:CFGS_ORCA}). %\sfcomment{HERE I need to put new table for ORCA2 values} and \autoref{fig:DOM_zgr_e3}). 201 201 202 202 Only the ORCA\_R2 is provided with all its input files in the \NEMO\ distribution. … … 207 207 208 208 This version of ORCA\_R2 has 31 levels in the vertical, with the highest resolution (10m) in the upper 150m 209 (see \autoref{tab: orca_zgr} and \autoref{fig:zgr}).209 (see \autoref{tab:CFGS_ORCA} and \autoref{fig:DOM_zgr_e3}). 210 210 The bottom topography and the coastlines are derived from the global atlas of Smith and Sandwell (1997). 211 The default forcing uses the boundary forcing from \citet{large.yeager_rpt04} (see \autoref{subsec:SBC_blk_ core}),211 The default forcing uses the boundary forcing from \citet{large.yeager_rpt04} (see \autoref{subsec:SBC_blk_ocean}), 212 212 which was developed for the purpose of running global coupled ocean-ice simulations without 213 213 an interactive atmosphere. … … 226 226 % ------------------------------------------------------------------------------------------------------------- 227 227 \section{GYRE family: double gyre basin} 228 \label{sec:CFG _gyre}228 \label{sec:CFGS_gyre} 229 229 230 230 The GYRE configuration \citep{levy.klein.ea_OM10} has been built to … … 254 254 255 255 The GYRE configuration is set like an analytical configuration. 256 Through \np{ln\_read\_cfg}\forcode{ =.false.} in \nam{cfg} namelist defined in256 Through \np{ln\_read\_cfg}\forcode{ = .false.} in \nam{cfg} namelist defined in 257 257 the reference configuration \path{./cfgs/GYRE_PISCES/EXPREF/namelist_cfg} 258 258 analytical definition of grid in GYRE is done in usrdef\_hrg, usrdef\_zgr routines. … … 266 266 Obviously, the namelist parameters have to be adjusted to the chosen resolution, 267 267 see the Configurations pages on the \NEMO\ web site (\NEMO\ Configurations). 268 In the vertical, GYRE uses the default 30 ocean levels (\jp{jpk}\forcode{ =31}) (\autoref{fig:zgr}).268 In the vertical, GYRE uses the default 30 ocean levels (\jp{jpk}\forcode{ = 31}) (\autoref{fig:DOM_zgr_e3}). 269 269 270 270 The GYRE configuration is also used in benchmark test as it is very simple to increase its resolution and … … 272 272 For example, keeping a same model size on each processor while increasing the number of processor used is very easy, 273 273 even though the physical integrity of the solution can be compromised. 274 Benchmark is activate via \np{ln\_bench}\forcode{ =.true.} in \nam{usr\_def} in274 Benchmark is activate via \np{ln\_bench}\forcode{ = .true.} in \nam{usr\_def} in 275 275 namelist \path{./cfgs/GYRE_PISCES/EXPREF/namelist_cfg}. 276 276 … … 280 280 \includegraphics[width=\textwidth]{Fig_GYRE} 281 281 \caption{ 282 \protect\label{fig: GYRE}282 \protect\label{fig:CFGS_GYRE} 283 283 Snapshot of relative vorticity at the surface of the model domain in GYRE R9, R27 and R54. 284 284 From \citet{levy.klein.ea_OM10}. … … 292 292 % ------------------------------------------------------------------------------------------------------------- 293 293 \section{AMM: atlantic margin configuration} 294 \label{sec: MISC_config_AMM}294 \label{sec:CFGS_config_AMM} 295 295 296 296 The AMM, Atlantic Margins Model, is a regional model covering the Northwest European Shelf domain on -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_misc.tex
r11435 r11543 110 110 \includegraphics[width=\textwidth]{Fig_closea_mask_example} 111 111 \caption{ 112 \protect\label{fig: closea_mask_example}112 \protect\label{fig:MISC_closea_mask_example} 113 113 Example of mask fields for the closea module. \textit{Left}: a 114 114 closea\_mask field; \textit{Right}: a closea\_mask\_rnf … … 159 159 configuration file and ln\_closea=.true. in namelist namcfg.} Each 160 160 inland sea or group of inland seas is set to a positive integer value 161 in the closea\_mask field (see Figure \ref{fig:closea_mask_example}161 in the closea\_mask field (see \autoref{fig:MISC_closea_mask_example} 162 162 for an example). The net surface flux over each inland sea or group of 163 163 inland seas is set to zero each timestep and the residual flux is … … 174 174 by the closea\_mask\_rnf field. Each mapping is defined by a positive 175 175 integer value for the inland sea(s) and the corresponding runoff 176 points. An example is given in Figure177 \ ref{fig:closea_mask_example}. If no mapping is provided for a176 points. An example is given in 177 \autoref{fig:MISC_closea_mask_example}. If no mapping is provided for a 178 178 particular inland sea then the residual is spread over the global 179 179 ocean.} -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex
r11435 r11543 8 8 % ================================================================ 9 9 \chapter{Model Basics} 10 \label{chap: PE}10 \label{chap:MB} 11 11 12 12 \chaptertoc … … 18 18 % ================================================================ 19 19 \section{Primitive equations} 20 \label{sec: PE_PE}20 \label{sec:MB_PE} 21 21 22 22 % ------------------------------------------------------------------------------------------------------------- … … 25 25 26 26 \subsection{Vector invariant formulation} 27 \label{subsec: PE_Vector}27 \label{subsec:MB_PE_vector} 28 28 29 29 The ocean is a fluid that can be described to a good approximation by the primitive equations, … … 48 48 the buoyancy force 49 49 \begin{equation} 50 \label{eq: PE_eos}50 \label{eq:MB_PE_eos} 51 51 \rho = \rho \ (T,S,p) 52 52 \end{equation} … … 57 57 convective processes must be parameterized instead) 58 58 \begin{equation} 59 \label{eq: PE_hydrostatic}59 \label{eq:MB_PE_hydrostatic} 60 60 \pd[p]{z} = - \rho \ g 61 61 \end{equation} … … 64 64 is assumed to be zero. 65 65 \begin{equation} 66 \label{eq: PE_continuity}66 \label{eq:MB_PE_continuity} 67 67 \nabla \cdot \vect U = 0 68 68 \end{equation} … … 85 85 the following equations: 86 86 \begin{subequations} 87 \label{eq: PE}87 \label{eq:MB_PE} 88 88 \begin{gather} 89 89 \intertext{$-$ the momentum balance} 90 \label{eq: PE_dyn}90 \label{eq:MB_PE_dyn} 91 91 \pd[\vect U_h]{t} = - \lt[ (\nabla \times \vect U) \times \vect U + \frac{1}{2} \nabla \lt( \vect U^2 \rt) \rt]_h 92 92 - f \; k \times \vect U_h - \frac{1}{\rho_o} \nabla_h p 93 93 + \vect D^{\vect U} + \vect F^{\vect U} \\ 94 94 \intertext{$-$ the heat and salt conservation equations} 95 \label{eq: PE_tra_T}95 \label{eq:MB_PE_tra_T} 96 96 \pd[T]{t} = - \nabla \cdot (T \ \vect U) + D^T + F^T \\ 97 \label{eq: PE_tra_S}97 \label{eq:MB_PE_tra_S} 98 98 \pd[S]{t} = - \nabla \cdot (S \ \vect U) + D^S + F^S 99 99 \end{gather} … … 101 101 where $\nabla$ is the generalised derivative vector operator in $(i,j,k)$ directions, $t$ is the time, 102 102 $z$ is the vertical coordinate, $\rho$ is the \textit{in situ} density given by the equation of state 103 (\autoref{eq: PE_eos}), $\rho_o$ is a reference density, $p$ the pressure,103 (\autoref{eq:MB_PE_eos}), $\rho_o$ is a reference density, $p$ the pressure, 104 104 $f = 2 \vect \Omega \cdot k$ is the Coriolis acceleration 105 105 (where $\vect \Omega$ is the Earth's angular velocity vector), and $g$ is the gravitational acceleration. 106 106 $\vect D^{\vect U}$, $D^T$ and $D^S$ are the parameterisations of small-scale physics for momentum, 107 107 temperature and salinity, and $\vect F^{\vect U}$, $F^T$ and $F^S$ surface forcing terms. 108 Their nature and formulation are discussed in \autoref{sec: PE_zdf_ldf} and \autoref{subsec:PE_boundary_condition}.108 Their nature and formulation are discussed in \autoref{sec:MB_zdf_ldf} and \autoref{subsec:MB_boundary_condition}. 109 109 110 110 % ------------------------------------------------------------------------------------------------------------- … … 112 112 % ------------------------------------------------------------------------------------------------------------- 113 113 \subsection{Boundary conditions} 114 \label{subsec: PE_boundary_condition}114 \label{subsec:MB_boundary_condition} 115 115 116 116 An ocean is bounded by complex coastlines, bottom topography at its base and … … 120 120 (discretisation can introduce additional artificial ``side-wall'' boundaries). 121 121 Both $H$ and $\eta$ are referenced to a surface of constant geopotential (\ie\ a mean sea surface height) on which $z = 0$. 122 (\autoref{fig: ocean_bc}).122 (\autoref{fig:MB_ocean_bc}). 123 123 Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with 124 124 the solid earth, the continental margins, the sea ice and the atmosphere. … … 133 133 \includegraphics[width=\textwidth]{Fig_I_ocean_bc} 134 134 \caption{ 135 \protect\label{fig: ocean_bc}135 \protect\label{fig:MB_ocean_bc} 136 136 The ocean is bounded by two surfaces, $z = - H(i,j)$ and $z = \eta(i,j,t)$, 137 137 where $H$ is the depth of the sea floor and $\eta$ the height of the sea surface. … … 164 164 can be expressed as: 165 165 \begin{equation} 166 \label{eq: PE_w_bbc}166 \label{eq:MB_w_bbc} 167 167 w = - \vect U_h \cdot \nabla_h (H) 168 168 \end{equation} … … 171 171 It must be parameterized in terms of turbulent fluxes using bottom and/or lateral boundary conditions. 172 172 Its specification depends on the nature of the physical parameterisation used for 173 $\vect D^{\vect U}$ in \autoref{eq: PE_dyn}.174 It is discussed in \autoref{eq: PE_zdf}.% and Chap. III.6 to 9.173 $\vect D^{\vect U}$ in \autoref{eq:MB_PE_dyn}. 174 It is discussed in \autoref{eq:MB_zdf}.% and Chap. III.6 to 9. 175 175 \item[Atmosphere - ocean interface:] 176 176 the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget) 177 177 leads to: 178 178 \[ 179 % \label{eq: PE_w_sbc}179 % \label{eq:MB_w_sbc} 180 180 w = \pd[\eta]{t} + \lt. \vect U_h \rt|_{z = \eta} \cdot \nabla_h (\eta) + P - E 181 181 \] … … 194 194 % ================================================================ 195 195 \section{Horizontal pressure gradient} 196 \label{sec: PE_hor_pg}196 \label{sec:MB_hor_pg} 197 197 198 198 % ------------------------------------------------------------------------------------------------------------- … … 200 200 % ------------------------------------------------------------------------------------------------------------- 201 201 \subsection{Pressure formulation} 202 \label{subsec: PE_p_formulation}202 \label{subsec:MB_p_formulation} 203 203 204 204 The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at 205 205 a reference geopotential surface ($z = 0$) and a hydrostatic pressure $p_h$ such that: 206 206 $p(i,j,k,t) = p_s(i,j,t) + p_h(i,j,k,t)$. 207 The latter is computed by integrating (\autoref{eq: PE_hydrostatic}),208 assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq: PE_eos}).207 The latter is computed by integrating (\autoref{eq:MB_PE_hydrostatic}), 208 assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:MB_PE_eos}). 209 209 The hydrostatic pressure is then given by: 210 210 \[ 211 % \label{eq: PE_pressure}211 % \label{eq:MB_pressure} 212 212 p_h (i,j,z,t) = \int_{\varsigma = z}^{\varsigma = 0} g \; \rho (T,S,\varsigma) \; d \varsigma 213 213 \] … … 234 234 % ------------------------------------------------------------------------------------------------------------- 235 235 \subsection{Free surface formulation} 236 \label{subsec: PE_free_surface}236 \label{subsec:MB_free_surface} 237 237 238 238 In the free surface formulation, a variable $\eta$, the sea-surface height, 239 239 is introduced which describes the shape of the air-sea interface. 240 240 This variable is solution of a prognostic equation which is established by forming the vertical average of 241 the kinematic surface condition (\autoref{eq: PE_w_bbc}):241 the kinematic surface condition (\autoref{eq:MB_w_bbc}): 242 242 \begin{equation} 243 \label{eq: PE_ssh}243 \label{eq:MB_ssh} 244 244 \pd[\eta]{t} = - D + P - E \quad \text{where} \quad D = \nabla \cdot \lt[ (H + \eta) \; \overline{U}_h \, \rt] 245 245 \end{equation} 246 and using (\autoref{eq: PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$.246 and using (\autoref{eq:MB_PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$. 247 247 248 248 Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as … … 257 257 the baroclinic structure of the ocean (internal waves) possibly in shallow seas, 258 258 then a non linear free surface is the most appropriate. 259 This means that no approximation is made in \autoref{eq: PE_ssh} and that259 This means that no approximation is made in \autoref{eq:MB_ssh} and that 260 260 the variation of the ocean volume is fully taken into account. 261 261 Note that in order to study the fast time scales associated with EGWs it is necessary to … … 268 268 not altering the slow barotropic Rossby waves. 269 269 If further, an approximative conservation of heat and salt contents is sufficient for the problem solved, 270 then it is sufficient to solve a linearized version of \autoref{eq: PE_ssh},270 then it is sufficient to solve a linearized version of \autoref{eq:MB_ssh}, 271 271 which still allows to take into account freshwater fluxes applied at the ocean surface \citep{roullet.madec_JGR00}. 272 272 Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost. … … 286 286 % ================================================================ 287 287 \section{Curvilinear \textit{z-}coordinate system} 288 \label{sec: PE_zco}288 \label{sec:MB_zco} 289 289 290 290 % ------------------------------------------------------------------------------------------------------------- … … 292 292 % ------------------------------------------------------------------------------------------------------------- 293 293 \subsection{Tensorial formalism} 294 \label{subsec: PE_tensorial}294 \label{subsec:MB_tensorial} 295 295 296 296 In many ocean circulation problems, the flow field has regions of enhanced dynamics … … 315 315 $(i,j,k)$ linked to the earth such that 316 316 $k$ is the local upward vector and $(i,j)$ are two vectors orthogonal to $k$, 317 \ie\ along geopotential surfaces (\autoref{fig: referential}).317 \ie\ along geopotential surfaces (\autoref{fig:MB_referential}). 318 318 Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined by 319 319 the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and 320 320 the distance from the centre of the earth $a + z(k)$ where $a$ is the earth's radius and 321 $z$ the altitude above a reference sea level (\autoref{fig: referential}).321 $z$ the altitude above a reference sea level (\autoref{fig:MB_referential}). 322 322 The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$, 323 323 the three scale factors: 324 324 \begin{equation} 325 \label{eq: scale_factors}325 \label{eq:MB_scale_factors} 326 326 \begin{aligned} 327 327 e_1 &= (a + z) \lt[ \lt( \pd[\lambda]{i} \cos \varphi \rt)^2 + \lt( \pd[\varphi]{i} \rt)^2 \rt]^{1/2} \\ … … 336 336 \includegraphics[width=\textwidth]{Fig_I_earth_referential} 337 337 \caption{ 338 \protect\label{fig: referential}338 \protect\label{fig:MB_referential} 339 339 the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear 340 340 coordinate system $(i,j,k)$. … … 345 345 346 346 Since the ocean depth is far smaller than the earth's radius, $a + z$, can be replaced by $a$ in 347 (\autoref{eq: scale_factors}) (thin-shell approximation).347 (\autoref{eq:MB_scale_factors}) (thin-shell approximation). 348 348 The resulting horizontal scale factors $e_1$, $e_2$ are independent of $k$ while 349 349 the vertical scale factor is a single function of $k$ as $k$ is parallel to $z$. 350 350 The scalar and vector operators that appear in the primitive equations 351 (\autoref{eq: PE_dyn} to \autoref{eq:PE_eos}) can then be written in the tensorial form,351 (\autoref{eq:MB_PE_dyn} to \autoref{eq:MB_PE_eos}) can then be written in the tensorial form, 352 352 invariant in any orthogonal horizontal curvilinear coordinate system transformation: 353 353 \begin{subequations} 354 % \label{eq: PE_discrete_operators}354 % \label{eq:MB_discrete_operators} 355 355 \begin{gather} 356 \label{eq: PE_grad}356 \label{eq:MB_grad} 357 357 \nabla q = \frac{1}{e_1} \pd[q]{i} \; \vect i 358 358 + \frac{1}{e_2} \pd[q]{j} \; \vect j 359 359 + \frac{1}{e_3} \pd[q]{k} \; \vect k \\ 360 \label{eq: PE_div}360 \label{eq:MB_div} 361 361 \nabla \cdot \vect A = \frac{1}{e_1 \; e_2} \lt[ \pd[(e_2 \; a_1)]{\partial i} + \pd[(e_1 \; a_2)]{j} \rt] 362 362 + \frac{1}{e_3} \lt[ \pd[a_3]{k} \rt] 363 363 \end{gather} 364 364 \begin{multline} 365 \label{eq: PE_curl}365 \label{eq:MB_curl} 366 366 \nabla \times \vect{A} = \lt[ \frac{1}{e_2} \pd[a_3]{j} - \frac{1}{e_3} \pd[a_2]{k} \rt] \vect i \\ 367 367 + \lt[ \frac{1}{e_3} \pd[a_1]{k} - \frac{1}{e_1} \pd[a_3]{i} \rt] \vect j \\ … … 369 369 \end{multline} 370 370 \begin{gather} 371 \label{eq: PE_lap}371 \label{eq:MB_lap} 372 372 \Delta q = \nabla \cdot (\nabla q) \\ 373 \label{eq: PE_lap_vector}373 \label{eq:MB_lap_vector} 374 374 \Delta \vect A = \nabla (\nabla \cdot \vect A) - \nabla \times (\nabla \times \vect A) 375 375 \end{gather} … … 381 381 % ------------------------------------------------------------------------------------------------------------- 382 382 \subsection{Continuous model equations} 383 \label{subsec: PE_zco_Eq}383 \label{subsec:MB_zco_Eq} 384 384 385 385 In order to express the Primitive Equations in tensorial formalism, 386 386 it is necessary to compute the horizontal component of the non-linear and viscous terms of the equation using 387 \autoref{eq: PE_grad}) to \autoref{eq:PE_lap_vector}.387 \autoref{eq:MB_grad}) to \autoref{eq:MB_lap_vector}. 388 388 Let us set $\vect U = (u,v,w) = \vect U_h + w \; \vect k $, the velocity in the $(i,j,k)$ coordinates system and 389 389 define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by: 390 390 \begin{gather} 391 \label{eq: PE_curl_Uh}391 \label{eq:MB_curl_Uh} 392 392 \zeta = \frac{1}{e_1 e_2} \lt[ \pd[(e_2 \, v)]{i} - \pd[(e_1 \, u)]{j} \rt] \\ 393 \label{eq: PE_div_Uh}393 \label{eq:MB_div_Uh} 394 394 \chi = \frac{1}{e_1 e_2} \lt[ \pd[(e_2 \, u)]{i} + \pd[(e_1 \, v)]{j} \rt] 395 395 \end{gather} … … 397 397 Using again the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that 398 398 $e_3$ is a function of the single variable $k$, 399 $NLT$ the nonlinear term of \autoref{eq: PE_dyn} can be transformed as follows:399 $NLT$ the nonlinear term of \autoref{eq:MB_PE_dyn} can be transformed as follows: 400 400 \begin{alignat*}{2} 401 401 &NLT &= &\lt[ (\nabla \times {\vect U}) \times {\vect U} + \frac{1}{2} \nabla \lt( {\vect U}^2 \rt) \rt]_h \\ … … 438 438 \end{alignat*} 439 439 The last term of the right hand side is obviously zero, and thus the nonlinear term of 440 \autoref{eq: PE_dyn} is written in the $(i,j,k)$ coordinate system:440 \autoref{eq:MB_PE_dyn} is written in the $(i,j,k)$ coordinate system: 441 441 \begin{equation} 442 \label{eq: PE_vector_form}442 \label{eq:MB_vector_form} 443 443 NLT = \zeta \; \vect k \times \vect U_h + \frac{1}{2} \nabla_h \lt( \vect U_h^2 \rt) 444 444 + \frac{1}{e_3} w \pd[\vect U_h]{k} … … 448 448 For some purposes, it can be advantageous to write this term in the so-called flux form, 449 449 \ie\ to write it as the divergence of fluxes. 450 For example, the first component of \autoref{eq: PE_vector_form} (the $i$-component) is transformed as follows:450 For example, the first component of \autoref{eq:MB_vector_form} (the $i$-component) is transformed as follows: 451 451 \begin{alignat*}{2} 452 452 &NLT_i &= &- \zeta \; v + \frac{1}{2 \; e_1} \pd[ (u^2 + v^2) ]{i} + \frac{1}{e_3} w \ \pd[u]{k} \\ … … 473 473 The flux form of the momentum advection term is therefore given by: 474 474 \begin{equation} 475 \label{eq: PE_flux_form}475 \label{eq:MB_flux_form} 476 476 NLT = \nabla \cdot \lt( 477 477 \begin{array}{*{20}c} … … 488 488 The latter is called the \textit{metric} term and can be viewed as a modification of the Coriolis parameter: 489 489 \[ 490 % \label{eq: PE_cor+metric}490 % \label{eq:MB_cor+metric} 491 491 f \to f + \frac{1}{e_1 e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) 492 492 \] … … 503 503 \textbf{Vector invariant form of the momentum equations}: 504 504 \begin{equation} 505 \label{eq: PE_dyn_vect}505 \label{eq:MB_dyn_vect} 506 506 \begin{split} 507 % \label{eq: PE_dyn_vect_u}507 % \label{eq:MB_dyn_vect_u} 508 508 \pd[u]{t} = &+ (\zeta + f) \, v - \frac{1}{2 e_1} \pd[]{i} (u^2 + v^2) 509 509 - \frac{1}{e_3} w \pd[u]{k} - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) \\ … … 516 516 \item 517 517 \textbf{flux form of the momentum equations}: 518 % \label{eq: PE_dyn_flux}518 % \label{eq:MB_dyn_flux} 519 519 \begin{multline*} 520 % \label{eq: PE_dyn_flux_u}520 % \label{eq:MB_dyn_flux_u} 521 521 \pd[u]{t} = + \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v \\ 522 522 - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, u)]{i} + \pd[(e_1 \, v \, u)]{j} \rt) \\ … … 525 525 \end{multline*} 526 526 \begin{multline*} 527 % \label{eq: PE_dyn_flux_v}527 % \label{eq:MB_dyn_flux_v} 528 528 \pd[v]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u \\ 529 529 - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) \\ … … 531 531 + D_v^{\vect U} + F_v^{\vect U} 532 532 \end{multline*} 533 where $\zeta$, the relative vorticity, is given by \autoref{eq: PE_curl_Uh} and $p_s$, the surface pressure,533 where $\zeta$, the relative vorticity, is given by \autoref{eq:MB_curl_Uh} and $p_s$, the surface pressure, 534 534 is given by: 535 535 \[ 536 % \label{eq: PE_spg}536 % \label{eq:MB_spg} 537 537 p_s = \rho \,g \, \eta 538 538 \] 539 and $\eta$ is the solution of \autoref{eq: PE_ssh}.539 and $\eta$ is the solution of \autoref{eq:MB_ssh}. 540 540 541 541 The vertical velocity and the hydrostatic pressure are diagnosed from the following equations: 542 542 \[ 543 % \label{eq: w_diag}543 % \label{eq:MB_w_diag} 544 544 \pd[w]{k} = - \chi \; e_3 \qquad 545 % \label{eq: hp_diag}545 % \label{eq:MB_hp_diag} 546 546 \pd[p_h]{k} = - \rho \; g \; e_3 547 547 \] 548 where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq: PE_div_Uh}.548 where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:MB_div_Uh}. 549 549 550 550 \item … … 562 562 563 563 The expression of $\vect D^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale parameterisation used. 564 It will be defined in \autoref{eq: PE_zdf}.564 It will be defined in \autoref{eq:MB_zdf}. 565 565 The nature and formulation of $\vect F^{\vect U}$, $F^T$ and $F^S$, the surface forcing terms, 566 566 are discussed in \autoref{chap:SBC}. … … 572 572 % ================================================================ 573 573 \section{Curvilinear generalised vertical coordinate system} 574 \label{sec: PE_gco}574 \label{sec:MB_gco} 575 575 576 576 The ocean domain presents a huge diversity of situation in the vertical. … … 596 596 introducing an arbitrary vertical coordinate : 597 597 \begin{equation} 598 \label{eq: PE_s}598 \label{eq:MB_s} 599 599 s = s(i,j,k,t) 600 600 \end{equation} 601 601 with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, 602 602 when $i$, $j$ and $t$ are held fixed. 603 \autoref{eq: PE_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into603 \autoref{eq:MB_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into 604 604 the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through 605 \autoref{eq: PE_s}.605 \autoref{eq:MB_s}. 606 606 This so-called \textit{generalised vertical coordinate} \citep{kasahara_MWR74} is in fact 607 607 an Arbitrary Lagrangian--Eulerian (ALE) coordinate. … … 656 656 \subsection{\textit{S}-coordinate formulation} 657 657 658 Starting from the set of equations established in \autoref{sec: PE_zco} for the special case $k = z$ and658 Starting from the set of equations established in \autoref{sec:MB_zco} for the special case $k = z$ and 659 659 thus $e_3 = 1$, we introduce an arbitrary vertical coordinate $s = s(i,j,k,t)$, 660 660 which includes $z$-, \zstar- and $\sigma$-coordinates as special cases 661 661 ($s = z$, $s = \zstar$, and $s = \sigma = z / H$ or $ = z / \lt( H + \eta \rt)$, resp.). 662 A formal derivation of the transformed equations is given in \autoref{apdx: A}.662 A formal derivation of the transformed equations is given in \autoref{apdx:SCOORD}. 663 663 Let us define the vertical scale factor by $e_3 = \partial_s z$ ($e_3$ is now a function of $(i,j,k,t)$ ), 664 664 and the slopes in the $(i,j)$ directions between $s$- and $z$-surfaces by: 665 665 \begin{equation} 666 \label{eq: PE_sco_slope}666 \label{eq:MB_sco_slope} 667 667 \sigma_1 = \frac{1}{e_1} \; \lt. \pd[z]{i} \rt|_s \quad \text{and} \quad 668 668 \sigma_2 = \frac{1}{e_2} \; \lt. \pd[z]{j} \rt|_s … … 671 671 relative to the moving $s$-surfaces and normal to them: 672 672 \[ 673 % \label{eq: PE_sco_w}673 % \label{eq:MB_sco_w} 674 674 \omega = w - \, \lt. \pd[z]{t} \rt|_s - \sigma_1 \, u - \sigma_2 \, v 675 675 \] 676 676 677 The equations solved by the ocean model \autoref{eq: PE} in $s$-coordinate can be written as follows678 (see \autoref{sec: A_momentum}):677 The equations solved by the ocean model \autoref{eq:MB_PE} in $s$-coordinate can be written as follows 678 (see \autoref{sec:SCOORD_momentum}): 679 679 680 680 \begin{itemize} 681 681 \item \textbf{Vector invariant form of the momentum equation}: 682 682 \begin{multline*} 683 % \label{eq: PE_sco_u_vector}683 % \label{eq:MB_sco_u_vector} 684 684 \pd[u]{t} = + (\zeta + f) \, v - \frac{1}{2 \, e_1} \pd[]{i} (u^2 + v^2) - \frac{1}{e_3} \omega \pd[u]{k} \\ 685 685 - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_1 … … 687 687 \end{multline*} 688 688 \begin{multline*} 689 % \label{eq: PE_sco_v_vector}689 % \label{eq:MB_sco_v_vector} 690 690 \pd[v]{t} = - (\zeta + f) \, u - \frac{1}{2 \, e_2} \pd[]{j}(u^2 + v^2) - \frac{1}{e_3} \omega \pd[v]{k} \\ 691 691 - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_2 … … 694 694 \item \textbf{Flux form of the momentum equation}: 695 695 \begin{multline*} 696 % \label{eq: PE_sco_u_flux}696 % \label{eq:MB_sco_u_flux} 697 697 \frac{1}{e_3} \pd[(e_3 \, u)]{t} = + \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v \\ 698 698 - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[(e_2 \, e_3 \, u \, u)]{i} + \pd[(e_1 \, e_3 \, v \, u)]{j} \rt) \\ … … 702 702 \end{multline*} 703 703 \begin{multline*} 704 % \label{eq: PE_sco_v_flux}704 % \label{eq:MB_sco_v_flux} 705 705 \frac{1}{e_3} \pd[(e_3 \, v)]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u \\ 706 706 - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[( e_2 \; e_3 \, u \, v)]{i} + \pd[(e_1 \; e_3 \, v \, v)]{j} \rt) \\ … … 712 712 and the hydrostatic pressure have the same expressions as in $z$-coordinates although 713 713 they do not represent exactly the same quantities. 714 $\omega$ is provided by the continuity equation (see \autoref{apdx: A}):714 $\omega$ is provided by the continuity equation (see \autoref{apdx:SCOORD}): 715 715 \[ 716 % \label{eq: PE_sco_continuity}716 % \label{eq:MB_sco_continuity} 717 717 \pd[e_3]{t} + e_3 \; \chi + \pd[\omega]{s} = 0 \quad \text{with} \quad 718 718 \chi = \frac{1}{e_1 e_2 e_3} \lt( \pd[(e_2 e_3 \, u)]{i} + \pd[(e_1 e_3 \, v)]{j} \rt) … … 720 720 \item \textit{tracer equations}: 721 721 \begin{multline*} 722 % \label{eq: PE_sco_t}722 % \label{eq:MB_sco_t} 723 723 \frac{1}{e_3} \pd[(e_3 \, T)]{t} = - \frac{1}{e_1 e_2 e_3} \lt( \pd[(e_2 e_3 \, u \, T)]{i} 724 724 + \pd[(e_1 e_3 \, v \, T)]{j} \rt) \\ … … 726 726 \end{multline*} 727 727 \begin{multline} 728 % \label{eq: PE_sco_s}728 % \label{eq:MB_sco_s} 729 729 \frac{1}{e_3} \pd[(e_3 \, S)]{t} = - \frac{1}{e_1 e_2 e_3} \lt( \pd[(e_2 e_3 \, u \, S)]{i} 730 730 + \pd[(e_1 e_3 \, v \, S)]{j} \rt) \\ … … 745 745 % ------------------------------------------------------------------------------------------------------------- 746 746 \subsection{Curvilinear \zstar-coordinate system} 747 \label{subsec: PE_zco_star}747 \label{subsec:MB_zco_star} 748 748 749 749 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 752 752 \includegraphics[width=\textwidth]{Fig_z_zstar} 753 753 \caption{ 754 \protect\label{fig: z_zstar}754 \protect\label{fig:MB_z_zstar} 755 755 (a) $z$-coordinate in linear free-surface case ; 756 756 (b) $z$-coordinate in non-linear free surface case ; … … 771 771 as in the $z$-coordinate formulation, but is equally distributed over the full water column. 772 772 Thus vertical levels naturally follow sea-surface variations, with a linear attenuation with depth, 773 as illustrated by \autoref{fig: z_zstar}.774 Note that with a flat bottom, such as in \autoref{fig: z_zstar}, the bottom-following $z$ coordinate and \zstar are equivalent.773 as illustrated by \autoref{fig:MB_z_zstar}. 774 Note that with a flat bottom, such as in \autoref{fig:MB_z_zstar}, the bottom-following $z$ coordinate and \zstar are equivalent. 775 775 The definition and modified oceanic equations for the rescaled vertical coordinate \zstar, 776 776 including the treatment of fresh-water flux at the surface, are detailed in Adcroft and Campin (2004). … … 778 778 The position (\zstar) and vertical discretization (\zstar) are expressed as: 779 779 \[ 780 % \label{eq: PE_z-star}780 % \label{eq:MB_z-star} 781 781 H + \zstar = (H + z) / r \quad \text{and} \quad \delta \zstar 782 782 = \delta z / r \quad \text{with} \quad r … … 785 785 Simple re-organisation of the above expressions gives 786 786 \[ 787 % \label{eq: PE_zstar_2}787 % \label{eq:MB_zstar_2} 788 788 \zstar = H \lt( \frac{z - \eta}{H + \eta} \rt) . 789 789 \] … … 806 806 it is clear that surfaces constant \zstar are very similar to the depth surfaces. 807 807 These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to 808 terrain following sigma models discussed in \autoref{subsec: PE_sco}.808 terrain following sigma models discussed in \autoref{subsec:MB_sco}. 809 809 Additionally, since $\zstar = z$ when $\eta = 0$, 810 810 no flow is spontaneously generated in an unforced ocean starting from rest, regardless the bottom topography. … … 839 839 % ------------------------------------------------------------------------------------------------------------- 840 840 \subsection{Curvilinear terrain-following \textit{s}--coordinate} 841 \label{subsec: PE_sco}841 \label{subsec:MB_sco} 842 842 843 843 % ------------------------------------------------------------------------------------------------------------- … … 851 851 For example, the topographic $\beta$-effect is usually larger than the planetary one along continental slopes. 852 852 Topographic Rossby waves can be excited and can interact with the mean current. 853 In the $z$-coordinate system presented in the previous section (\autoref{sec: PE_zco}),853 In the $z$-coordinate system presented in the previous section (\autoref{sec:MB_zco}), 854 854 $z$-surfaces are geopotential surfaces. 855 855 The bottom topography is discretised by steps. … … 875 875 The main two problems come from the truncation error in the horizontal pressure gradient and 876 876 a possibly increased diapycnal diffusion. 877 The horizontal pressure force in $s$-coordinate consists of two terms (see \autoref{apdx: A}),877 The horizontal pressure force in $s$-coordinate consists of two terms (see \autoref{apdx:SCOORD}), 878 878 879 879 \begin{equation} 880 \label{eq: PE_p_sco}880 \label{eq:MB_p_sco} 881 881 \nabla p |_z = \nabla p |_s - \frac{1}{e_3} \pd[p]{s} \nabla z |_s 882 882 \end{equation} 883 883 884 The second term in \autoref{eq: PE_p_sco} depends on the tilt of the coordinate surface and884 The second term in \autoref{eq:MB_p_sco} depends on the tilt of the coordinate surface and 885 885 leads to a truncation error that is not present in a $z$-model. 886 886 In the special case of a $\sigma$-coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), … … 898 898 an envelope topography is defined in $s$-coordinate on which a full or 899 899 partial step bottom topography is then applied in order to adjust the model depth to the observed one 900 (see \autoref{s ec:DOM_zgr}.900 (see \autoref{subsec:DOM_zgr}. 901 901 902 902 For numerical reasons a minimum of diffusion is required along the coordinate surfaces of … … 915 915 the strongly stratified portion of the water column (\ie\ the main thermocline) \citep{madec.delecluse.ea_JPO96}. 916 916 An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces 917 (see \autoref{subsec: PE_ldf}).917 (see \autoref{subsec:MB_ldf}). 918 918 Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large, 919 919 strongly exceeding the stability limit of such a operator when it is discretized (see \autoref{chap:LDF}). … … 928 928 % ------------------------------------------------------------------------------------------------------------- 929 929 \subsection{\texorpdfstring{Curvilinear \ztilde-coordinate}{}} 930 \label{subsec: PE_zco_tilde}930 \label{subsec:MB_zco_tilde} 931 931 932 932 The \ztilde -coordinate has been developed by \citet{leclair.madec_OM11}. … … 941 941 % ================================================================ 942 942 \section{Subgrid scale physics} 943 \label{sec: PE_zdf_ldf}943 \label{sec:MB_zdf_ldf} 944 944 945 945 The hydrostatic primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than … … 958 958 The control exerted by gravity on the flow induces a strong anisotropy between the lateral and vertical motions. 959 959 Therefore subgrid-scale physics \textbf{D}$^{\vect U}$, $D^{S}$ and $D^{T}$ in 960 \autoref{eq: PE_dyn}, \autoref{eq:PE_tra_T} and \autoref{eq:PE_tra_S} are divided into960 \autoref{eq:MB_PE_dyn}, \autoref{eq:MB_PE_tra_T} and \autoref{eq:MB_PE_tra_S} are divided into 961 961 a lateral part \textbf{D}$^{l \vect U}$, $D^{l S}$ and $D^{l T}$ and 962 962 a vertical part \textbf{D}$^{v \vect U}$, $D^{v S}$ and $D^{v T}$. … … 967 967 % ------------------------------------------------------------------------------------------------------------- 968 968 \subsection{Vertical subgrid scale physics} 969 \label{subsec: PE_zdf}969 \label{subsec:MB_zdf} 970 970 971 971 The model resolution is always larger than the scale at which the major sources of vertical turbulence occur … … 981 981 The resulting vertical momentum and tracer diffusive operators are of second order: 982 982 \begin{equation} 983 \label{eq: PE_zdf}983 \label{eq:MB_zdf} 984 984 \begin{gathered} 985 985 \vect D^{v \vect U} = \pd[]{z} \lt( A^{vm} \pd[\vect U_h]{z} \rt) \ , \\ … … 1001 1001 % ------------------------------------------------------------------------------------------------------------- 1002 1002 \subsection{Formulation of the lateral diffusive and viscous operators} 1003 \label{subsec: PE_ldf}1003 \label{subsec:MB_ldf} 1004 1004 1005 1005 Lateral turbulence can be roughly divided into a mesoscale turbulence associated with eddies … … 1055 1055 \subsubsection{Lateral laplacian tracer diffusive operator} 1056 1056 1057 The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx: B}):1057 The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx:DIFFOPERS}): 1058 1058 \begin{equation} 1059 \label{eq: PE_iso_tensor}1059 \label{eq:MB_iso_tensor} 1060 1060 D^{lT} = \nabla \vect . \lt( A^{lT} \; \Re \; \nabla T \rt) \quad \text{with} \quad 1061 1061 \Re = … … 1068 1068 where $r_1$ and $r_2$ are the slopes between the surface along which the diffusive operator acts and 1069 1069 the model level (\eg\ $z$- or $s$-surfaces). 1070 Note that the formulation \autoref{eq: PE_iso_tensor} is exact for1070 Note that the formulation \autoref{eq:MB_iso_tensor} is exact for 1071 1071 the rotation between geopotential and $s$-surfaces, 1072 1072 while it is only an approximation for the rotation between isoneutral and $z$- or $s$-surfaces. 1073 Indeed, in the latter case, two assumptions are made to simplify \autoref{eq: PE_iso_tensor} \citep{cox_OM87}.1073 Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:MB_iso_tensor} \citep{cox_OM87}. 1074 1074 First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and 1075 1075 dia-neutral diffusive coefficients is known to be several orders of magnitude smaller than unity. 1076 1076 Second, the two isoneutral directions of diffusion are assumed to be independent since 1077 the slopes are generally less than $10^{-2}$ in the ocean (see \autoref{apdx: B}).1077 the slopes are generally less than $10^{-2}$ in the ocean (see \autoref{apdx:DIFFOPERS}). 1078 1078 1079 1079 For \textit{iso-level} diffusion, $r_1$ and $r_2 $ are zero. … … 1082 1082 For \textit{geopotential} diffusion, 1083 1083 $r_1$ and $r_2 $ are the slopes between the geopotential and computational surfaces: 1084 they are equal to $\sigma_1$ and $\sigma_2$, respectively (see \autoref{eq: PE_sco_slope}).1084 they are equal to $\sigma_1$ and $\sigma_2$, respectively (see \autoref{eq:MB_sco_slope}). 1085 1085 1086 1086 For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces. … … 1088 1088 In $z$-coordinates: 1089 1089 \begin{equation} 1090 \label{eq: PE_iso_slopes}1090 \label{eq:MB_iso_slopes} 1091 1091 r_1 = \frac{e_3}{e_1} \lt( \pd[\rho]{i} \rt) \lt( \pd[\rho]{k} \rt)^{-1} \quad 1092 1092 r_2 = \frac{e_3}{e_2} \lt( \pd[\rho]{j} \rt) \lt( \pd[\rho]{k} \rt)^{-1} … … 1099 1099 an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: 1100 1100 \[ 1101 % \label{eq: PE_iso+eiv}1101 % \label{eq:MB_iso+eiv} 1102 1102 D^{lT} = \nabla \cdot \lt( A^{lT} \; \Re \; \nabla T \rt) + \nabla \cdot \lt( \vect U^\ast \, T \rt) 1103 1103 \] … … 1105 1105 eddy-induced transport velocity. This velocity field is defined by: 1106 1106 \begin{gather} 1107 % \label{eq: PE_eiv}1107 % \label{eq:MB_eiv} 1108 1108 u^\ast = \frac{1}{e_3} \pd[]{k} \lt( A^{eiv} \; \tilde{r}_1 \rt) \\ 1109 1109 v^\ast = \frac{1}{e_3} \pd[]{k} \lt( A^{eiv} \; \tilde{r}_2 \rt) \\ … … 1116 1116 Their values are thus independent of the vertical coordinate, but their expression depends on the coordinate: 1117 1117 \begin{align} 1118 \label{eq: PE_slopes_eiv}1118 \label{eq:MB_slopes_eiv} 1119 1119 \tilde{r}_n = 1120 1120 \begin{cases} … … 1134 1134 The lateral bilaplacian tracer diffusive operator is defined by: 1135 1135 \[ 1136 % \label{eq: PE_bilapT}1136 % \label{eq:MB_bilapT} 1137 1137 D^{lT}= - \Delta \; (\Delta T) \quad \text{where} \quad 1138 1138 \Delta \bullet = \nabla \lt( \sqrt{B^{lT}} \; \Re \; \nabla \bullet \rt) 1139 1139 \] 1140 It is the Laplacian operator given by \autoref{eq: PE_iso_tensor} applied twice with1140 It is the Laplacian operator given by \autoref{eq:MB_iso_tensor} applied twice with 1141 1141 the harmonic eddy diffusion coefficient set to the square root of the biharmonic one. 1142 1142 … … 1144 1144 1145 1145 The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by 1146 applying \autoref{eq: PE_lap_vector} to the horizontal velocity vector (see \autoref{apdx:B}):1146 applying \autoref{eq:MB_lap_vector} to the horizontal velocity vector (see \autoref{apdx:DIFFOPERS}): 1147 1147 \begin{align*} 1148 % \label{eq: PE_lapU}1148 % \label{eq:MB_lapU} 1149 1149 \vect D^{l \vect U} &= \nabla_h \big( A^{lm} \chi \big) 1150 1150 - \nabla_h \times \big( A^{lm} \, \zeta \; \vect k \big) \\ … … 1156 1156 1157 1157 Such a formulation ensures a complete separation between the vorticity and horizontal divergence fields 1158 (see \autoref{apdx: C}).1158 (see \autoref{apdx:INVARIANTS}). 1159 1159 Unfortunately, it is only available in \textit{iso-level} direction. 1160 1160 When a rotation is required … … 1162 1162 the $u$ and $v$-fields are considered as independent scalar fields, so that the diffusive operator is given by: 1163 1163 \begin{gather*} 1164 % \label{eq: PE_lapU_iso}1164 % \label{eq:MB_lapU_iso} 1165 1165 D_u^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla u \rt) \\ 1166 1166 D_v^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla v \rt) 1167 1167 \end{gather*} 1168 where $\Re$ is given by \autoref{eq: PE_iso_tensor}.1168 where $\Re$ is given by \autoref{eq:MB_iso_tensor}. 1169 1169 It is the same expression as those used for diffusive operator on tracers. 1170 1170 It must be emphasised that such a formulation is only exact in a Cartesian coordinate system, -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex
r11537 r11543 40 40 the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. 41 41 These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to 42 terrain following sigma models discussed in \autoref{subsec: PE_sco}.42 terrain following sigma models discussed in \autoref{subsec:MB_sco}. 43 43 Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in 44 44 an unforced ocean starting from rest, regardless the bottom topography. … … 81 81 %------------------------------------------------------------------------------------------------------------ 82 82 Options are defined through the \nam{\_dynspg} namelist variables. 83 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec: PE_hor_pg}).83 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}). 84 84 The main distinction is between the fixed volume case (linear free surface or rigid lid) and 85 85 the variable volume case (nonlinear free surface, \key{vvl} is active). 86 In the linear free surface case (\autoref{subsec: PE_free_surface}) and rigid lid (\autoref{PE_rigid_lid}),86 In the linear free surface case (\autoref{subsec:MB_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), 87 87 the vertical scale factors $e_{3}$ are fixed in time, 88 while in the nonlinear case (\autoref{subsec: PE_free_surface}) they are time-dependent.88 while in the nonlinear case (\autoref{subsec:MB_free_surface}) they are time-dependent. 89 89 With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 90 90 which imposes a very small time step when an explicit time stepping is used. 91 91 Two methods are proposed to allow a longer time step for the three-dimensional equations: 92 the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq: PE_flt?}),92 the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:MB_flt?}), 93 93 and the split-explicit free surface described below. 94 94 The extra term introduced in the filtered method is calculated implicitly, … … 116 116 and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. 117 117 The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, 118 (\ie\ the velocity appearing in (\autoref{eq: dynspg_ssh}) is centred in time (\textit{now} velocity).118 (\ie\ the velocity appearing in (\autoref{eq:DYN_spg_ssh}) is centred in time (\textit{now} velocity). 119 119 120 120 The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by: … … 130 130 131 131 Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in 132 (\autoref{eq: dynspg_exp}).132 (\autoref{eq:DYN_spg_exp}). 133 133 134 134 %------------------------------------------------------------- … … 316 316 This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. 317 317 The three time-stepping methods (explicit, split-explicit and filtered) are the same as in 318 \autoref{ DYN_spg_linear} except that the ocean depth is now time-dependent.318 \autoref{?:DYN_spg_linear?} except that the ocean depth is now time-dependent. 319 319 In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step. 320 320 -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_time_domain.tex
r11537 r11543 6 6 % Chapter 2 ——— Time Domain (step.F90) 7 7 % ================================================================ 8 \chapter{Time Domain (STP)}9 \label{chap: STP}8 \chapter{Time Domain} 9 \label{chap:TD} 10 10 \chaptertoc 11 11 … … 19 19 \newpage 20 20 21 Having defined the continuous equations in \autoref{chap: PE}, we need now to choose a time discretization,21 Having defined the continuous equations in \autoref{chap:MB}, we need now to choose a time discretization, 22 22 a key feature of an ocean model as it exerts a strong influence on the structure of the computer code 23 23 (\ie\ on its flowchart). … … 29 29 % ================================================================ 30 30 \section{Time stepping environment} 31 \label{sec: STP_environment}31 \label{sec:TD_environment} 32 32 33 33 The time stepping used in \NEMO\ is a three level scheme that can be represented as follows: 34 34 \begin{equation} 35 \label{eq: STP}35 \label{eq:TD} 36 36 x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t - \rdt, \, t, \, t + \rdt} 37 37 \end{equation} … … 52 52 The third array, although referred to as $x_a$ (after) in the code, 53 53 is usually not the variable at the after time step; 54 but rather it is used to store the time derivative (RHS in \autoref{eq: STP}) prior to time-stepping the equation.54 but rather it is used to store the time derivative (RHS in \autoref{eq:TD}) prior to time-stepping the equation. 55 55 The time stepping itself is performed once at each time step where implicit vertical diffusion is computed, \ie\ in the \mdl{trazdf} and \mdl{dynzdf} modules. 56 56 … … 59 59 % ------------------------------------------------------------------------------------------------------------- 60 60 \section{Non-diffusive part --- Leapfrog scheme} 61 \label{sec: STP_leap_frog}61 \label{sec:TD_leap_frog} 62 62 63 63 The time stepping used for processes other than diffusion is the well-known leapfrog scheme 64 64 \citep{mesinger.arakawa_bk76}. 65 65 This scheme is widely used for advection processes in low-viscosity fluids. 66 It is a time centred scheme, \ie\ the RHS in \autoref{eq: STP} is evaluated at time step $t$, the now time step.66 It is a time centred scheme, \ie\ the RHS in \autoref{eq:TD} is evaluated at time step $t$, the now time step. 67 67 It may be used for momentum and tracer advection, pressure gradient, and Coriolis terms, 68 68 but not for diffusion terms. … … 81 81 is a kind of laplacian diffusion in time that mixes odd and even time steps: 82 82 \begin{equation} 83 \label{eq: STP_asselin}83 \label{eq:TD_asselin} 84 84 x_F^t = x^t + \gamma \, \lt[ x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt] 85 85 \end{equation} 86 86 where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin coefficient. 87 87 $\gamma$ is initialized as \np{rn\_atfp} (namelist parameter). 88 Its default value is \np{rn\_atfp}\forcode{ =10.e-3} (see \autoref{sec:STP_mLF}),88 Its default value is \np{rn\_atfp}\forcode{ = 10.e-3} (see \autoref{sec:TD_mLF}), 89 89 causing only a weak dissipation of high frequency motions (\citep{farge-coulombier_phd87}). 90 90 The addition of a time filter degrades the accuracy of the calculation from second to first order. … … 102 102 % ------------------------------------------------------------------------------------------------------------- 103 103 \section{Diffusive part --- Forward or backward scheme} 104 \label{sec: STP_forward_imp}104 \label{sec:TD_forward_imp} 105 105 106 106 The leapfrog differencing scheme is unsuitable for the representation of diffusion and damping processes. … … 108 108 (when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme is used : 109 109 \[ 110 %\label{eq: STP_euler}110 %\label{eq:TD_euler} 111 111 x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ D_x^{t - \rdt} 112 112 \] … … 115 115 The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{griffies_bk04}: 116 116 \begin{equation} 117 \label{eq: STP_euler_stability}117 \label{eq:TD_euler_stability} 118 118 A^h < 119 119 \begin{cases} … … 123 123 \end{equation} 124 124 where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient. 125 The linear constraint \autoref{eq: STP_euler_stability} is a necessary condition, but not sufficient.125 The linear constraint \autoref{eq:TD_euler_stability} is a necessary condition, but not sufficient. 126 126 If it is not satisfied, even mildly, then the model soon becomes wildly unstable. 127 127 The instability can be removed by either reducing the length of the time steps or reducing the mixing coefficient. … … 131 131 backward (or implicit) time differencing scheme is used. This scheme is unconditionally stable but diffusive and can be written as follows: 132 132 \begin{equation} 133 \label{eq: STP_imp}133 \label{eq:TD_imp} 134 134 x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t + \rdt} 135 135 \end{equation} … … 141 141 This scheme is rather time consuming since it requires a matrix inversion. For example, the finite difference approximation of the temperature equation is: 142 142 \[ 143 % \label{eq: STP_imp_zdf}143 % \label{eq:TD_imp_zdf} 144 144 \frac{T(k)^{t + 1} - T(k)^{t - 1}}{2 \; \rdt} 145 145 \equiv … … 147 147 \] 148 148 where RHS is the right hand side of the equation except for the vertical diffusion term. 149 We rewrite \autoref{eq: STP_imp} as:150 \begin{equation} 151 \label{eq: STP_imp_mat}149 We rewrite \autoref{eq:TD_imp} as: 150 \begin{equation} 151 \label{eq:TD_imp_mat} 152 152 -c(k + 1) \; T^{t + 1}(k + 1) + d(k) \; T^{t + 1}(k) - \; c(k) \; T^{t + 1}(k - 1) \equiv b(k) 153 153 \end{equation} … … 159 159 \end{align*} 160 160 161 \autoref{eq: STP_imp_mat} is a linear system of equations with an associated matrix which is tridiagonal.161 \autoref{eq:TD_imp_mat} is a linear system of equations with an associated matrix which is tridiagonal. 162 162 Moreover, 163 163 $c(k)$ and $d(k)$ are positive and the diagonal term is greater than the sum of the two extra-diagonal terms, … … 169 169 % ------------------------------------------------------------------------------------------------------------- 170 170 \section{Surface pressure gradient} 171 \label{sec: STP_spg_ts}171 \label{sec:TD_spg_ts} 172 172 173 173 The leapfrog environment supports a centred in time computation of the surface pressure, \ie\ evaluated … … 177 177 (\np{ln\_dynspg\_ts}\forcode{=.true.}) in which barotropic and baroclinic dynamical equations are solved separately with ad-hoc 178 178 time steps. The use of the time-splitting (in combination with non-linear free surface) imposes some constraints on the design of 179 the overall flowchart, in particular to ensure exact tracer conservation (see \autoref{fig:T imeStep_flowchart}).179 the overall flowchart, in particular to ensure exact tracer conservation (see \autoref{fig:TD_TimeStep_flowchart}). 180 180 181 181 Compared to the former use of the filtered free surface in \NEMO\ v3.6 (\citet{roullet.madec_JGR00}), the use of a split-explicit free surface is advantageous … … 189 189 \includegraphics[width=\textwidth]{Fig_TimeStepping_flowchart_v4} 190 190 \caption{ 191 \protect\label{fig:T imeStep_flowchart}191 \protect\label{fig:TD_TimeStep_flowchart} 192 192 Sketch of the leapfrog time stepping sequence in \NEMO\ with split-explicit free surface. The latter combined 193 193 with non-linear free surface requires the dynamical tendency being updated prior tracers tendency to ensure … … 205 205 % ------------------------------------------------------------------------------------------------------------- 206 206 \section{Modified Leapfrog -- Asselin filter scheme} 207 \label{sec: STP_mLF}207 \label{sec:TD_mLF} 208 208 209 209 Significant changes have been introduced by \cite{leclair.madec_OM09} in the LF-RA scheme in order to … … 214 214 \ie\ it is time-stepped over a $2 \rdt$ period: 215 215 $x^t = x^t + 2 \rdt Q^t$ where $Q$ is the forcing applied to $x$, 216 and the time filter is given by \autoref{eq: STP_asselin} so that $Q$ is redistributed over several time step.216 and the time filter is given by \autoref{eq:TD_asselin} so that $Q$ is redistributed over several time step. 217 217 In the modified LF-RA environment, these two formulations have been replaced by: 218 218 \begin{gather} 219 \label{eq: STP_forcing}219 \label{eq:TD_forcing} 220 220 x^{t + \rdt} = x^{t - \rdt} + \rdt \lt( Q^{t - \rdt / 2} + Q^{t + \rdt / 2} \rt) \\ 221 \label{eq: STP_RA}221 \label{eq:TD_RA} 222 222 x_F^t = x^t + \gamma \, \lt( x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt) 223 223 - \gamma \, \rdt \, \lt( Q^{t + \rdt / 2} - Q^{t - \rdt / 2} \rt) 224 224 \end{gather} 225 The change in the forcing formulation given by \autoref{eq: STP_forcing} (see \autoref{fig:MLF_forcing})225 The change in the forcing formulation given by \autoref{eq:TD_forcing} (see \autoref{fig:TD_MLF_forcing}) 226 226 has a significant effect: 227 227 the forcing term no longer excites the divergence of odd and even time steps \citep{leclair.madec_OM09}. … … 231 231 Indeed, time filtering is no longer required on the forcing part. 232 232 The influence of the Asselin filter on the forcing is explicitly removed by adding a new term in the filter 233 (last term in \autoref{eq: STP_RA} compared to \autoref{eq:STP_asselin}).233 (last term in \autoref{eq:TD_RA} compared to \autoref{eq:TD_asselin}). 234 234 Since the filtering of the forcing was the source of non-conservation in the classical LF-RA scheme, 235 235 the modified formulation becomes conservative \citep{leclair.madec_OM09}. 236 236 Second, the LF-RA becomes a truly quasi -second order scheme. 237 Indeed, \autoref{eq: STP_forcing} used in combination with a careful treatment of static instability237 Indeed, \autoref{eq:TD_forcing} used in combination with a careful treatment of static instability 238 238 (\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}) 239 239 (the two other main sources of time step divergence), … … 242 242 Note that the forcing is now provided at the middle of a time step: 243 243 $Q^{t + \rdt / 2}$ is the forcing applied over the $[t,t + \rdt]$ time interval. 244 This and the change in the time filter, \autoref{eq: STP_RA},244 This and the change in the time filter, \autoref{eq:TD_RA}, 245 245 allows for an exact evaluation of the contribution due to the forcing term between any two time steps, 246 246 even if separated by only $\rdt$ since the time filter is no longer applied to the forcing term. … … 251 251 \includegraphics[width=\textwidth]{Fig_MLF_forcing} 252 252 \caption{ 253 \protect\label{fig: MLF_forcing}253 \protect\label{fig:TD_MLF_forcing} 254 254 Illustration of forcing integration methods. 255 255 (top) ''Traditional'' formulation: … … 268 268 % ------------------------------------------------------------------------------------------------------------- 269 269 \section{Start/Restart strategy} 270 \label{sec: STP_rst}270 \label{sec:TD_rst} 271 271 272 272 %--------------------------------------------namrun------------------------------------------- … … 277 277 (Euler time integration): 278 278 \[ 279 % \label{eq: DOM_euler}279 % \label{eq:TD_DOM_euler} 280 280 x^1 = x^0 + \rdt \ \text{RHS}^0 281 281 \] 282 This is done simply by keeping the leapfrog environment (\ie\ the \autoref{eq: STP} three level time stepping) but282 This is done simply by keeping the leapfrog environment (\ie\ the \autoref{eq:TD} three level time stepping) but 283 283 setting all $x^0$ (\textit{before}) and $x^1$ (\textit{now}) fields equal at the first time step and 284 284 using half the value of a leapfrog time step ($2 \rdt$). … … 314 314 % ------------------------------------------------------------------------------------------------------------- 315 315 \subsection{Time domain} 316 \label{subsec: STP_time}316 \label{subsec:TD_time} 317 317 %--------------------------------------------namrun------------------------------------------- 318 318 -
NEMO/trunk/doc/latex/global/coding_rules.tex
r11515 r11543 1 1 2 2 \chapter{Coding Rules} 3 \label{apdx: coding}3 \label{apdx:CODING} 4 4 5 5 \chaptertoc -
NEMO/trunk/doc/latex/global/document.tex
r11524 r11543 11 11 12 12 %% Document layout 13 \documentclass[ draft,fontsize = 10pt,13 \documentclass[fontsize = 10pt, 14 14 twoside = semi, abstract = on, 15 15 open = right]{scrreprt} … … 46 46 \input{../../global/info_page} 47 47 48 \listoffigures \listoftables %\listoflistings %% \listoflistings not working 49 48 50 \clearpage 49 51 50 52 \pagenumbering{roman} 53 \ofoot[]{\engine\ Reference Manual} \ifoot[]{\pagemark} 54 51 55 \input{introduction} 52 56 53 57 %% Table of Contents 54 58 \tableofcontents 55 \listoffigures \listoftables \listoflistings56 59 57 60 \clearpage … … 73 76 74 77 \appendix %% Chapter numbering with letters by now 78 \lohead{Apdx\ \thechapter\ \leftmark} 75 79 \include{appendices} 76 80 … … 78 82 \input{../../global/coding_rules} 79 83 84 \clearpage 80 85 81 86 %% Backmatter
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