Changeset 2285
- Timestamp:
- 2010-10-17T17:08:15+02:00 (14 years ago)
- Location:
- branches/nemo_v3_3_beta/DOC/TexFiles
- Files:
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- 1 added
- 4 edited
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Chapters/Annex_ISO.tex (modified) (1 diff)
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Chapters/Chap_DOM.tex (modified) (2 diffs)
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Chapters/Chap_DYN.tex (modified) (11 diffs)
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Chapters/Chap_TRA.tex (modified) (4 diffs)
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math_abbrev.sty (added)
Legend:
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branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_ISO.tex
r2282 r2285 2 2 % Iso-neutral diffusion : 3 3 % ================================================================ 4 \chapter{Griffies's iso-neutral diffusion and \\ 5 eddy-induced advection} 4 \chapter{Griffies's iso-neutral diffusion} 6 5 \label{Apdx_C} 7 6 \minitoc 8 7 9 \section{Griffies's formulation of iso neutral diffusion}8 \section{Griffies's formulation of iso-neutral diffusion} 10 9 11 10 \subsection{Introduction} -
branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_DOM.tex
r2282 r2285 522 522 %%% 523 523 524 Generally, the arrays describing the grid point depths and vertical scale factors 525 are three dimensional arrays $(i,j,k)$. In the special case of $z$-coordinates with 526 full step bottom topography, it is possible to define those arrays as one-dimensional, 527 in order to save memory. This is performed by defining the \key{zco} 528 C-Pre-Processor (CPP) key. To improve the code readability while providing this 529 flexibility, the vertical coordinate and scale factors are defined as functions of 530 $(i,j,k)$ with "fs" as prefix (examples: \textit{fsdept, fse3t,} etc) that can be either 531 three-dimensional arrays, or a one dimensional array when \key{zco} is defined. 524 The arrays describing the grid point depths and vertical scale factors 525 are three dimensional arrays $(i,j,k)$ even in the case of $z$-coordinate with 526 full step bottom topography. In non-linear free surface (\key{vvl}), their knowledge 527 is required at \textit{before}, \textit{now} and \textit{after} time step, while they 528 do not vary in time in linear free surface case. 529 To improve the code readability while providing this flexibility, the vertical coordinate 530 and scale factors are defined as functions of 531 $(i,j,k)$ with "fs" as prefix (examples: \textit{fse3t\_b, fse3t\_n, fse3t\_a,} 532 for the \textit{before}, \textit{now} and \textit{after} scale factors at $t$-point) 533 that can be either three different arrays when \key{vvl} is defined, or a single fixed arrays. 532 534 These functions are defined in the file \hf{domzgr\_substitute} of the DOM directory. 533 535 They are used throughout the code, and replaced by the corresponding arrays at … … 568 570 % z-coordinate and reference coordinate transformation 569 571 % ------------------------------------------------------------------------------------------------------------- 570 \subsection[$z$-coordinate (\np{ln\_zco} or \key{zco})]571 {$z$-coordinate (\np{ln\_zco}= .true. or \key{zco}) and reference coordinate}572 \subsection[$z$-coordinate (\np{ln\_zco}] 573 {$z$-coordinate (\np{ln\_zco}=true) and reference coordinate} 572 574 \label{DOM_zco} 573 575 -
branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_DYN.tex
r2282 r2285 25 25 \end{equation*} 26 26 NXT stands for next, referring to the time-stepping. The first group of terms on 27 the rhs of th e this equation corresponds to the Coriolis and advection28 terms that are decomposed into a vorticity part (VOR), a kinetic energy part (KEG)29 and , eithera vertical advection part (ZAD) in the vector invariant formulation, or a Coriolis27 the rhs of this equation corresponds to the Coriolis and advection 28 terms that are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG) 29 and a vertical advection part (ZAD) in the vector invariant formulation, or a Coriolis 30 30 and advection part (COR+ADV) in the flux formulation. The terms following these 31 31 are the pressure gradient contributions (HPG, Hydrostatic Pressure Gradient, … … 86 86 \end{equation} 87 87 88 Note that in the $z$-coordinate with full step (when \key{zco} is defined), 89 $e_{3u}$=$e_{3v}$=$e_{3f}$ so that these metric terms cancel in \eqref{Eq_divcur_div}. 90 91 Note also that although the vorticity has the same discrete expression in $z$- 88 Note that although the vorticity has the same discrete expression in $z$- 92 89 and $s$-coordinates, its physical meaning is not identical. $\zeta$ is a pseudo 93 90 vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along … … 113 110 \begin{aligned} 114 111 \frac{\partial \eta }{\partial t} 115 &\equiv \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left (\delta _i \left[ {e_{2u}\,e_{3u}\;u} \right]116 +\delta _j \left[ {e_{1v}\,e_{3v}\;v} \right] \right )}112 &\equiv \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{ \delta _i \left[ {e_{2u}\,e_{3u}\;u} \right] 113 +\delta _j \left[ {e_{1v}\,e_{3v}\;v} \right] \right\} } 117 114 - \frac{\textit{emp}}{\rho _w } \\ 118 115 &\equiv \sum\limits_k {\chi \ e_{3t}} - \frac{\textit{emp}}{\rho _w } … … 120 117 \end{equation} 121 118 where \textit{emp} is the surface freshwater budget (evaporation minus precipitation), 122 expressed in Kg/m$^2$/s (which is equal to mm/s), and $\rho _w$=1,000~Kg/m$^3$ 123 is the density of pure water. If river runoff is expressed as a surface freshwater 124 flux (see \S\ref{SBC}) then \textit{emp} can be written as the evaporation minus 125 precipitation, minus the river runoff. The sea-surface height is evaluated 126 using exactly the same time stepping scheme as the tracer equation \eqref{Eq_tra_nxt}: 119 expressed in Kg/m$^2$/s (which is equal to mm/s), and $\rho _w$=1,035~Kg/m$^3$ 120 is the reference density of sea water (Boussinesq approximation). If river runoff is 121 expressed as a surface freshwater flux (see \S\ref{SBC}) then \textit{emp} can be 122 written as the evaporation minus precipitation, minus the river runoff. 123 The sea-surface height is evaluated using exactly the same time stepping scheme 124 as the tracer equation \eqref{Eq_tra_nxt}: 127 125 a leapfrog scheme in combination with an Asselin time filter, $i.e.$ the velocity appearing 128 126 in \eqref{Eq_dynspg_ssh} is centred in time (\textit{now} velocity). … … 133 131 The vertical velocity is computed by an upward integration of the horizontal 134 132 divergence starting at the bottom, taking into account the change of the thickness of the levels : 135 136 133 \begin{equation} \label{Eq_wzv} 137 134 \left\{ \begin{aligned} 138 &\left. w \right|_{ 3/2} \quad= 0\\139 &\left. w \right|_{k+1/2} = \left. w \right|_{k-1/2} + e_{3t}\; \left. \chi \right|_k140 - \frac{ e_{3t}^{t+1} - e_{3t}^{t-1} } {2 \rdt}135 &\left. w \right|_{k_b-1/2} \quad= 0 \qquad \text{where } k_b \text{ is the level just above the sea floor } \\ 136 &\left. w \right|_{k+1/2} = \left. w \right|_{k-1/2} + \left. e_{3t} \right|_{k}\; \left. \chi \right|_k 137 - \frac{1} {2 \rdt} \left( \left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right) 141 138 \end{aligned} \right. 142 139 \end{equation} 143 \sgacomment{should e3t involve k in this equation?}144 140 145 141 In the case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$, 146 142 as changes in the divergence of the barotropic transport are absorbed into the change 147 143 of the level thicknesses, re-orientated downward. 144 \gmcomment{not sure of this... to be modified with the change in emp setting} 148 145 In the case of a linear free surface, the time derivative in \eqref{Eq_wzv} disappears. 149 146 The upper boundary condition applies at a fixed level $z=0$. The top vertical velocity … … 193 190 term (MIX scheme) ; or conserving both the potential enstrophy of horizontally non-divergent 194 191 flow and horizontal kinetic energy (ENE scheme) (see Appendix~\ref{Apdx_C_vor_zad}). 195 The vorticity terms are given below for the general case, but note that in the full step 196 $z$-coordinate (\key{zco} is defined), $e_{3u}$=$e_{3v}$=$e_{3f}$ so that the vertical scale 197 factors disappear. The vorticity terms are all computed in dedicated routines that can be found in 192 The vorticity terms are all computed in dedicated routines that can be found in 198 193 the \mdl{dynvor} module. 199 194 … … 270 265 that will be at least partly damped by the momentum diffusion operator ($i.e.$ the 271 266 subgrid-scale advection), but not by the resolved advection term. The ENS and ENE schemes 272 therefore do not contribute to any grid point noise in the horizontal velocity field. 273 Such noise would result in more noise in the vertical velocity field, an undesirable feature. This is a well-known 274 characteristic of $C$-grid discretization where $u$ and $v$ are located at different grid points, 275 a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid. 267 therefore do not contribute to dump any grid point noise in the horizontal velocity field. 268 Such noise would result in more noise in the vertical velocity field, an undesirable feature. 269 This is a well-known characteristic of $C$-grid discretization where $u$ and $v$ are located 270 at different grid points, a price worth paying to avoid a double averaging in the pressure 271 gradient term as in the $B$-grid. 276 272 \gmcomment{ To circumvent this, Adcroft (ADD REF HERE) 277 278 273 Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....} 279 274 280 A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}. The idea is281 to get rid of the double averaging by considering triad combinations of vorticity.275 A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}. 276 The idea is to get rid of the double averaging by considering triad combinations of vorticity. 282 277 It is noteworthy that this solution is conceptually quite similar to the one proposed by 283 \citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator .278 \citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see App.\ref{Apdx_C}). 284 279 285 280 The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified … … 311 306 extends by continuity the value of $e_{3f}$ into the land areas. This feature is essential for 312 307 the $z$-coordinate with partial steps. 313 314 308 315 309 Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as … … 337 331 It conserves both total energy and potential enstrophy in the limit of horizontally 338 332 nondivergent flow ($i.e.$ $\chi$=$0$) (see Appendix~\ref{Apdx_C_vor_zad}). 339 Applied to a realistic ocean configuration, it has been shown that it 340 leads to a significantreduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}.333 Applied to a realistic ocean configuration, it has been shown that it leads to a significant 334 reduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}. 341 335 Furthermore, used in combination with a partial steps representation of bottom topography, 342 336 it improves the interaction between current and topography, leading to a larger … … 959 953 and curl of the vorticity) preserves symmetry and ensures a complete 960 954 separation between the vorticity and divergence parts of the momentum diffusion. 961 Note that in the full step $z$-coordinate (\key{zco} is defined), $e_{3u} =e_{3v} =e_{3f}$962 so that they cancel in the rotational part of \eqref{Eq_dynldf_lap}.963 955 964 956 %-------------------------------------------------------------------------------------------------------------- … … 1114 1106 Both of which will be introduced into the reference version soon. 1115 1107 1108 \gmcomment{atmospheric pressure is there!!!! include its description } 1109 1116 1110 % ================================================================ 1117 1111 % Time evolution term -
branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_TRA.tex
r2282 r2285 48 48 In the present chapter we also describe the diagnostic equations used to compute 49 49 the sea-water properties (density, Brunt-Vais\"{a}l\"{a} frequency, specific heat and 50 freezing point with associated modules \mdl{eosbn2} , \mdl{ocfzpt}and \mdl{phycst}).50 freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}). 51 51 52 52 The different options available to the user are managed by namelist logicals or … … 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 In pure $z$-coordinate (\key{zco} is defined), it reduces to : 82 \begin{equation} \label{Eq_tra_adv_zco} 83 ADV_\tau = - \frac{1}{e_{1t}\,e_{2t}} \left( \; \delta_i \left[ e_{2u} \;u \;\tau_u \right] 84 + \delta_j \left[ e_{1v} \;v \;\tau_v \right] \; \right) 85 - \frac{1}{e_{3t}} \delta_k \left[ w \;\tau_w \right] 86 \end{equation} 87 since the vertical scale factors are functions of $k$ only, and thus 88 $e_{3u} =e_{3v} =e_{3t} $. The flux form in \eqref{Eq_tra_adv} 81 The flux form in \eqref{Eq_tra_adv} 89 82 implicitly requires the use of the continuity equation. Indeed, it is obtained 90 83 by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ 91 84 which results from the use of the continuity equation, $\nabla \cdot \vect{U}=0$ or 92 $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ in constant volume (default option) 93 or variable volume (\key{vvl} defined) case, respectively. 85 $ \partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ in constant or variable volume case, respectively. 94 86 Therefore it is of paramount importance to design the discrete analogue of the 95 87 advection tendency so that it is consistent with the continuity equation in order to … … 481 473 It is therefore not recommended. 482 474 483 Note that 484 (a) In the pure $z$-coordinate (\key{zco} is defined), $e_{3u}$=$e_{3v}$=$e_{3t}$, 485 so that the vertical scale factors disappear from (\ref{Eq_tra_ldf_lap}) ; 486 (b) In the partial step $z$-coordinate (\np{ln\_zps}=true), tracers in horizontally 475 Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), tracers in horizontally 487 476 adjacent cells are located at different depths in the vicinity of the bottom. 488 477 In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level … … 1096 1085 structure in equilibrium with its physics. 1097 1086 The choice of the shape of the Newtonian damping is controlled by two 1098 namelist parameters \np{ ??} and \np{nn\_zdmp}. The former allows us to specify: the1087 namelist parameters \np{nn\_hdmp} and \np{nn\_zdmp}. The former allows us to specify: the 1099 1088 width of the equatorial band in which no damping is applied; a decrease 1100 1089 in the vicinity of the coast; and a damping everywhere in the Red and Med Seas.
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