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branches/2017/dev_merge_2017/DOC/tex_sub/chap_model_basics.tex
r9393 r9407 6 6 7 7 \chapter{Model Basics} 8 \label{ PE}8 \label{chap:PE} 9 9 \minitoc 10 10 … … 16 16 % ================================================================ 17 17 \section{Primitive equations} 18 \label{ PE_PE}18 \label{sec:PE_PE} 19 19 20 20 % ------------------------------------------------------------------------------------------------------------- … … 23 23 24 24 \subsection{Vector invariant formulation} 25 \label{ PE_Vector}25 \label{subsec:PE_Vector} 26 26 27 27 … … 61 61 hydrostatic equilibrium, the incompressibility equation, the heat and salt conservation 62 62 equations and an equation of state): 63 \begin{subequations} \label{ Eq_PE}64 \begin{equation} \label{ Eq_PE_dyn}63 \begin{subequations} \label{eq:PE} 64 \begin{equation} \label{eq:PE_dyn} 65 65 \frac{\partial {\rm {\bf U}}_h }{\partial t}= 66 66 -\left[ {\left( {\nabla \times {\rm {\bf U}}} \right)\times {\rm {\bf U}} … … 69 69 -\frac{1}{\rho _o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}} 70 70 \end{equation} 71 \begin{equation} \label{ Eq_PE_hydrostatic}71 \begin{equation} \label{eq:PE_hydrostatic} 72 72 \frac{\partial p }{\partial z} = - \rho \ g 73 73 \end{equation} 74 \begin{equation} \label{ Eq_PE_continuity}74 \begin{equation} \label{eq:PE_continuity} 75 75 \nabla \cdot {\bf U}= 0 76 76 \end{equation} 77 \begin{equation} \label{ Eq_PE_tra_T}77 \begin{equation} \label{eq:PE_tra_T} 78 78 \frac{\partial T}{\partial t} = - \nabla \cdot \left( T \ \rm{\bf U} \right) + D^T + F^T 79 79 \end{equation} 80 \begin{equation} \label{ Eq_PE_tra_S}80 \begin{equation} \label{eq:PE_tra_S} 81 81 \frac{\partial S}{\partial t} = - \nabla \cdot \left( S \ \rm{\bf U} \right) + D^S + F^S 82 82 \end{equation} 83 \begin{equation} \label{ Eq_PE_eos}83 \begin{equation} \label{eq:PE_eos} 84 84 \rho = \rho \left( T,S,p \right) 85 85 \end{equation} … … 87 87 where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions, 88 88 $t$ is the time, $z$ is the vertical coordinate, $\rho $ is the \textit{in situ} density given by 89 the equation of state (\ ref{Eq_PE_eos}), $\rho_o$ is a reference density, $p$ the pressure,89 the equation of state (\autoref{eq:PE_eos}), $\rho_o$ is a reference density, $p$ the pressure, 90 90 $f=2 \bf \Omega \cdot \bf k$ is the Coriolis acceleration (where $\bf \Omega$ is the Earth's 91 91 angular velocity vector), and $g$ is the gravitational acceleration. … … 93 93 physics for momentum, temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ 94 94 and $F^S$ surface forcing terms. Their nature and formulation are discussed in 95 \ S\ref{PE_zdf_ldf} and page \S\ref{PE_boundary_condition}.95 \autoref{sec:PE_zdf_ldf} and \autoref{subsec:PE_boundary_condition}. 96 96 97 97 . … … 101 101 % ------------------------------------------------------------------------------------------------------------- 102 102 \subsection{Boundary conditions} 103 \label{ PE_boundary_condition}103 \label{subsec:PE_boundary_condition} 104 104 105 105 An ocean is bounded by complex coastlines, bottom topography at its base and an air-sea … … 107 107 and $z=\eta(i,j,k,t)$, where $H$ is the depth of the ocean bottom and $\eta$ is the height 108 108 of the sea surface. Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$, 109 chosen as a mean sea surface ( Fig.~\ref{Fig_ocean_bc}). Through these two boundaries,109 chosen as a mean sea surface (\autoref{fig:ocean_bc}). Through these two boundaries, 110 110 the ocean can exchange fluxes of heat, fresh water, salt, and momentum with the solid earth, 111 111 the continental margins, the sea ice and the atmosphere. However, some of these fluxes are … … 117 117 \begin{figure}[!ht] \begin{center} 118 118 \includegraphics[width=0.90\textwidth]{Fig_I_ocean_bc} 119 \caption{ \protect\label{ Fig_ocean_bc}119 \caption{ \protect\label{fig:ocean_bc} 120 120 The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,t)$, where $H$ 121 121 is the depth of the sea floor and $\eta$ the height of the sea surface. … … 137 137 \footnote{In fact, it has been shown that the heat flux associated with the solid Earth cooling 138 138 ($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world 139 ocean (see \ ref{TRA_bbc}).}.139 ocean (see \autoref{subsec:TRA_bbc}).}. 140 140 The boundary condition is thus set to no flux of heat and salt across solid boundaries. 141 141 For momentum, the situation is different. There is no flow across solid boundaries, … … 143 143 the bottom velocity is parallel to solid boundaries). This kinematic boundary condition 144 144 can be expressed as: 145 \begin{equation} \label{ Eq_PE_w_bbc}145 \begin{equation} \label{eq:PE_w_bbc} 146 146 w = -{\rm {\bf U}}_h \cdot \nabla _h \left( H \right) 147 147 \end{equation} … … 150 150 in terms of turbulent fluxes using bottom and/or lateral boundary conditions. Its specification 151 151 depends on the nature of the physical parameterisation used for ${\rm {\bf D}}^{\rm {\bf U}}$ 152 in \ eqref{Eq_PE_dyn}. It is discussed in \S\ref{PE_zdf}, page~\pageref{PE_zdf}.% and Chap. III.6 to 9.152 in \autoref{eq:PE_dyn}. It is discussed in \autoref{eq:PE_zdf}.% and Chap. III.6 to 9. 153 153 \item[Atmosphere - ocean interface:] the kinematic surface condition plus the mass flux 154 154 of fresh water PE (the precipitation minus evaporation budget) leads to: 155 \begin{equation} \label{ Eq_PE_w_sbc}155 \begin{equation} \label{eq:PE_w_sbc} 156 156 w = \frac{\partial \eta }{\partial t} 157 157 + \left. {{\rm {\bf U}}_h } \right|_{z=\eta } \cdot \nabla _h \left( \eta \right) … … 176 176 % ================================================================ 177 177 \section{Horizontal pressure gradient } 178 \label{ PE_hor_pg}178 \label{sec:PE_hor_pg} 179 179 180 180 % ------------------------------------------------------------------------------------------------------------- … … 182 182 % ------------------------------------------------------------------------------------------------------------- 183 183 \subsection{Pressure formulation} 184 \label{ PE_p_formulation}184 \label{subsec:PE_p_formulation} 185 185 186 186 The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at a 187 187 reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that: 188 $p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\ ref{Eq_PE_hydrostatic}),189 assuming that pressure in decibars can be approximated by depth in meters in (\ ref{Eq_PE_eos}).188 $p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\autoref{eq:PE_hydrostatic}), 189 assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:PE_eos}). 190 190 The hydrostatic pressure is then given by: 191 \begin{equation} \label{ Eq_PE_pressure}191 \begin{equation} \label{eq:PE_pressure} 192 192 p_h \left( {i,j,z,t} \right) 193 193 = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma } … … 213 213 % ------------------------------------------------------------------------------------------------------------- 214 214 \subsection{Free surface formulation} 215 \label{ PE_free_surface}215 \label{subsec:PE_free_surface} 216 216 217 217 In the free surface formulation, a variable $\eta$, the sea-surface height, is introduced 218 218 which describes the shape of the air-sea interface. This variable is solution of a 219 219 prognostic equation which is established by forming the vertical average of the kinematic 220 surface condition (\ ref{Eq_PE_w_bbc}):221 \begin{equation} \label{ Eq_PE_ssh}220 surface condition (\autoref{eq:PE_w_bbc}): 221 \begin{equation} \label{eq:PE_ssh} 222 222 \frac{\partial \eta }{\partial t}=-D+P-E 223 223 \quad \text{where} \ 224 224 D=\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right] 225 225 \end{equation} 226 and using (\ ref{Eq_PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$.226 and using (\autoref{eq:PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$. 227 227 228 228 Allowing the air-sea interface to move introduces the external gravity waves (EGWs) … … 237 237 with the baroclinic structure of the ocean (internal waves) possibly in shallow seas, 238 238 then a non linear free surface is the most appropriate. This means that no 239 approximation is made in (\ ref{Eq_PE_ssh}) and that the variation of the ocean239 approximation is made in (\autoref{eq:PE_ssh}) and that the variation of the ocean 240 240 volume is fully taken into account. Note that in order to study the fast time scales 241 241 associated with EGWs it is necessary to minimize time filtering effects (use an 242 242 explicit time scheme with very small time step, or a split-explicit scheme with 243 reasonably small time step, see \ S\ref{DYN_spg_exp} or \S\ref{DYN_spg_ts}.243 reasonably small time step, see \autoref{subsec:DYN_spg_exp} or \autoref{subsec:DYN_spg_ts}. 244 244 245 245 $\bullet$ If one is not interested in EGW but rather sees them as high frequency … … 247 247 not altering the slow barotropic Rossby waves. If further, an approximative conservation 248 248 of heat and salt contents is sufficient for the problem solved, then it is 249 sufficient to solve a linearized version of (\ ref{Eq_PE_ssh}), which still allows249 sufficient to solve a linearized version of (\autoref{eq:PE_ssh}), which still allows 250 250 to take into account freshwater fluxes applied at the ocean surface \citep{Roullet_Madec_JGR00}. 251 251 Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost. … … 255 255 or the implicit scheme \citep{Dukowicz1994} or the addition of a filtering force in the momentum equation 256 256 \citep{Roullet_Madec_JGR00}. With the present release, \NEMO offers the choice between 257 an explicit free surface (see \ S\ref{DYN_spg_exp}) or a split-explicit scheme strongly258 inspired the one proposed by \citet{Shchepetkin_McWilliams_OM05} (see \ S\ref{DYN_spg_ts}).257 an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or a split-explicit scheme strongly 258 inspired the one proposed by \citet{Shchepetkin_McWilliams_OM05} (see \autoref{subsec:DYN_spg_ts}). 259 259 260 260 %\newpage … … 265 265 % ================================================================ 266 266 \section{Curvilinear \textit{z-}coordinate system} 267 \label{ PE_zco}267 \label{sec:PE_zco} 268 268 269 269 … … 272 272 % ------------------------------------------------------------------------------------------------------------- 273 273 \subsection{Tensorial formalism} 274 \label{ PE_tensorial}274 \label{subsec:PE_tensorial} 275 275 276 276 In many ocean circulation problems, the flow field has regions of enhanced dynamics … … 294 294 associated with the positively oriented orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) 295 295 linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are 296 two vectors orthogonal to \textbf{k}, $i.e.$ along geopotential surfaces ( Fig.\ref{Fig_referential}).296 two vectors orthogonal to \textbf{k}, $i.e.$ along geopotential surfaces (\autoref{fig:referential}). 297 297 Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined 298 298 by the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and the distance from the centre of 299 299 the earth $a+z(k)$ where $a$ is the earth's radius and $z$ the altitude above a reference sea 300 level ( Fig.\ref{Fig_referential}). The local deformation of the curvilinear coordinate system is300 level (\autoref{fig:referential}). The local deformation of the curvilinear coordinate system is 301 301 given by $e_1$, $e_2$ and $e_3$, the three scale factors: 302 \begin{equation} \label{ Eq_scale_factors}302 \begin{equation} \label{eq:scale_factors} 303 303 \begin{aligned} 304 304 e_1 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda … … 315 315 \begin{figure}[!tb] \begin{center} 316 316 \includegraphics[width=0.60\textwidth]{Fig_I_earth_referential} 317 \caption{ \protect\label{ Fig_referential}317 \caption{ \protect\label{fig:referential} 318 318 the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear 319 319 coordinate system (\textbf{i},\textbf{j},\textbf{k}). } … … 322 322 323 323 Since the ocean depth is far smaller than the earth's radius, $a+z$, can be replaced by 324 $a$ in (\ ref{Eq_scale_factors}) (thin-shell approximation). The resulting horizontal scale324 $a$ in (\autoref{eq:scale_factors}) (thin-shell approximation). The resulting horizontal scale 325 325 factors $e_1$, $e_2$ are independent of $k$ while the vertical scale factor is a single 326 326 function of $k$ as \textbf{k} is parallel to \textbf{z}. The scalar and vector operators that 327 appear in the primitive equations ( Eqs. \eqref{Eq_PE_dyn} to \eqref{Eq_PE_eos}) can327 appear in the primitive equations (\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can 328 328 be written in the tensorial form, invariant in any orthogonal horizontal curvilinear coordinate 329 329 system transformation: 330 \begin{subequations} \label{ Eq_PE_discrete_operators}331 \begin{equation} \label{ Eq_PE_grad}330 \begin{subequations} \label{eq:PE_discrete_operators} 331 \begin{equation} \label{eq:PE_grad} 332 332 \nabla q=\frac{1}{e_1 }\frac{\partial q}{\partial i}\;{\rm {\bf 333 333 i}}+\frac{1}{e_2 }\frac{\partial q}{\partial j}\;{\rm {\bf j}}+\frac{1}{e_3 334 334 }\frac{\partial q}{\partial k}\;{\rm {\bf k}} \\ 335 335 \end{equation} 336 \begin{equation} \label{ Eq_PE_div}336 \begin{equation} \label{eq:PE_div} 337 337 \nabla \cdot {\rm {\bf A}} 338 338 = \frac{1}{e_1 \; e_2} \left[ … … 341 341 + \frac{1}{e_3} \left[ \frac{\partial a_3}{\partial k } \right] 342 342 \end{equation} 343 \begin{equation} \label{ Eq_PE_curl}343 \begin{equation} \label{eq:PE_curl} 344 344 \begin{split} 345 345 \nabla \times \vect{A} = … … 352 352 \end{split} 353 353 \end{equation} 354 \begin{equation} \label{ Eq_PE_lap}354 \begin{equation} \label{eq:PE_lap} 355 355 \Delta q = \nabla \cdot \left( \nabla q \right) 356 356 \end{equation} 357 \begin{equation} \label{ Eq_PE_lap_vector}357 \begin{equation} \label{eq:PE_lap_vector} 358 358 \Delta {\rm {\bf A}} = 359 359 \nabla \left( \nabla \cdot {\rm {\bf A}} \right) … … 367 367 % ------------------------------------------------------------------------------------------------------------- 368 368 \subsection{Continuous model equations} 369 \label{ PE_zco_Eq}369 \label{subsec:PE_zco_Eq} 370 370 371 371 In order to express the Primitive Equations in tensorial formalism, it is necessary to compute 372 372 the horizontal component of the non-linear and viscous terms of the equation using 373 \ eqref{Eq_PE_grad}) to \eqref{Eq_PE_lap_vector}.373 \autoref{eq:PE_grad}) to \autoref{eq:PE_lap_vector}. 374 374 Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate 375 375 system and define the relative vorticity $\zeta$ and the divergence of the horizontal velocity 376 376 field $\chi$, by: 377 \begin{equation} \label{ Eq_PE_curl_Uh}377 \begin{equation} \label{eq:PE_curl_Uh} 378 378 \zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v} 379 379 \right)}{\partial i}-\frac{\partial \left( {e_1 \,u} \right)}{\partial j}} 380 380 \right] 381 381 \end{equation} 382 \begin{equation} \label{ Eq_PE_div_Uh}382 \begin{equation} \label{eq:PE_div_Uh} 383 383 \chi =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,u} 384 384 \right)}{\partial i}+\frac{\partial \left( {e_1 \,v} \right)}{\partial j}} … … 388 388 Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ 389 389 and that $e_3$ is a function of the single variable $k$, the nonlinear term of 390 \ eqref{Eq_PE_dyn} can be transformed as follows:390 \autoref{eq:PE_dyn} can be transformed as follows: 391 391 \begin{flalign*} 392 392 &\left[ {\left( { \nabla \times {\rm {\bf U}} } \right) \times {\rm {\bf U}} … … 427 427 428 428 The last term of the right hand side is obviously zero, and thus the nonlinear term of 429 \ eqref{Eq_PE_dyn} is written in the $(i,j,k)$ coordinate system:430 \begin{equation} \label{ Eq_PE_vector_form}429 \autoref{eq:PE_dyn} is written in the $(i,j,k)$ coordinate system: 430 \begin{equation} \label{eq:PE_vector_form} 431 431 \left[ {\left( { \nabla \times {\rm {\bf U}} } \right) \times {\rm {\bf U}} 432 432 +\frac{1}{2} \nabla \left( {{\rm {\bf U}}^2} \right)} \right]_h … … 440 440 For some purposes, it can be advantageous to write this term in the so-called flux form, 441 441 $i.e.$ to write it as the divergence of fluxes. For example, the first component of 442 \ eqref{Eq_PE_vector_form} (the $i$-component) is transformed as follows:442 \autoref{eq:PE_vector_form} (the $i$-component) is transformed as follows: 443 443 \begin{flalign*} 444 444 &{ \begin{array}{*{20}l} … … 509 509 510 510 The flux form of the momentum advection term is therefore given by: 511 \begin{multline} \label{ Eq_PE_flux_form}511 \begin{multline} \label{eq:PE_flux_form} 512 512 \left[ 513 513 \left( {\nabla \times {\rm {\bf U}}} \right) \times {\rm {\bf U}} … … 529 529 the curvilinear nature of the coordinate system used. The latter is called the \emph{metric} 530 530 term and can be viewed as a modification of the Coriolis parameter: 531 \begin{equation} \label{ Eq_PE_cor+metric}531 \begin{equation} \label{eq:PE_cor+metric} 532 532 f \to f + \frac{1}{e_1\;e_2} \left( v \frac{\partial e_2}{\partial i} 533 533 -u \frac{\partial e_1}{\partial j} \right) … … 547 547 $\bullet$ \textbf{Vector invariant form of the momentum equations} : 548 548 549 \begin{subequations} \label{ Eq_PE_dyn_vect}550 \begin{equation} \label{ Eq_PE_dyn_vect_u} \begin{split}549 \begin{subequations} \label{eq:PE_dyn_vect} 550 \begin{equation} \label{eq:PE_dyn_vect_u} \begin{split} 551 551 \frac{\partial u}{\partial t} 552 552 = + \left( {\zeta +f} \right)\,v … … 568 568 \vspace{+10pt} 569 569 $\bullet$ \textbf{flux form of the momentum equations} : 570 \begin{subequations} \label{ Eq_PE_dyn_flux}571 \begin{multline} \label{ Eq_PE_dyn_flux_u}570 \begin{subequations} \label{eq:PE_dyn_flux} 571 \begin{multline} \label{eq:PE_dyn_flux_u} 572 572 \frac{\partial u}{\partial t}= 573 573 + \left( { f + \frac{1}{e_1 \; e_2} … … 581 581 + D_u^{\vect{U}} + F_u^{\vect{U}} 582 582 \end{multline} 583 \begin{multline} \label{ Eq_PE_dyn_flux_v}583 \begin{multline} \label{eq:PE_dyn_flux_v} 584 584 \frac{\partial v}{\partial t}= 585 585 - \left( { f + \frac{1}{e_1 \; e_2} … … 594 594 \end{multline} 595 595 \end{subequations} 596 where $\zeta$, the relative vorticity, is given by \ eqref{Eq_PE_curl_Uh} and $p_s $,596 where $\zeta$, the relative vorticity, is given by \autoref{eq:PE_curl_Uh} and $p_s $, 597 597 the surface pressure, is given by: 598 \begin{equation} \label{ Eq_PE_spg}598 \begin{equation} \label{eq:PE_spg} 599 599 p_s = \rho \,g \,\eta 600 600 \end{equation} 601 with $\eta$ is solution of \ eqref{Eq_PE_ssh}601 with $\eta$ is solution of \autoref{eq:PE_ssh} 602 602 603 603 The vertical velocity and the hydrostatic pressure are diagnosed from the following equations: 604 \begin{equation} \label{ Eq_w_diag}604 \begin{equation} \label{eq:w_diag} 605 605 \frac{\partial w}{\partial k}=-\chi \;e_3 606 606 \end{equation} 607 \begin{equation} \label{ Eq_hp_diag}607 \begin{equation} \label{eq:hp_diag} 608 608 \frac{\partial p_h }{\partial k}=-\rho \;g\;e_3 609 609 \end{equation} 610 where the divergence of the horizontal velocity, $\chi$ is given by \ eqref{Eq_PE_div_Uh}.610 where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:PE_div_Uh}. 611 611 612 612 \vspace{+10pt} 613 613 $\bullet$ \textit{tracer equations} : 614 \begin{equation} \label{ Eq_S}614 \begin{equation} \label{eq:S} 615 615 \frac{\partial T}{\partial t} = 616 616 -\frac{1}{e_1 e_2 }\left[ { \frac{\partial \left( {e_2 T\,u} \right)}{\partial i} … … 618 618 -\frac{1}{e_3 }\frac{\partial \left( {T\,w} \right)}{\partial k} + D^T + F^T 619 619 \end{equation} 620 \begin{equation} \label{ Eq_T}620 \begin{equation} \label{eq:T} 621 621 \frac{\partial S}{\partial t} = 622 622 -\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 S\,u} \right)}{\partial i} … … 624 624 -\frac{1}{e_3 }\frac{\partial \left( {S\,w} \right)}{\partial k} + D^S + F^S 625 625 \end{equation} 626 \begin{equation} \label{ Eq_rho}626 \begin{equation} \label{eq:rho} 627 627 \rho =\rho \left( {T,S,z(k)} \right) 628 628 \end{equation} 629 629 630 630 The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale 631 parameterisation used. It will be defined in \ S\ref{PE_zdf}. The nature and formulation of631 parameterisation used. It will be defined in \autoref{eq:PE_zdf}. The nature and formulation of 632 632 ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms, are discussed 633 in Chapter~\ref{SBC}.633 in \autoref{chap:SBC}. 634 634 635 635 … … 640 640 % ================================================================ 641 641 \section{Curvilinear generalised vertical coordinate system} 642 \label{ PE_gco}642 \label{sec:PE_gco} 643 643 644 644 The ocean domain presents a huge diversity of situation in the vertical. First the ocean surface is a time dependent surface (moving surface). Second the ocean floor depends on the geographical position, varying from more than 6,000 meters in abyssal trenches to zero at the coast. Last but not least, the ocean stratification exerts a strong barrier to vertical motions and mixing. … … 648 648 649 649 In fact one is totally free to choose any space and time vertical coordinate by introducing an arbitrary vertical coordinate : 650 \begin{equation} \label{ Eq_s}650 \begin{equation} \label{eq:s} 651 651 s=s(i,j,k,t) 652 652 \end{equation} 653 with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \ eqref{Eq_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \eqref{Eq_s}.653 with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \autoref{eq:s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \autoref{eq:s}. 654 654 This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact an Arbitrary Lagrangian--Eulerian (ALE) coordinate. Indeed, choosing an expression for $s$ is an arbitrary choice that determines which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and which part will be used to move them (Lagrangian part). 655 655 The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09}, since the coordinate system is adapted in the course of the simulation. Its most often used implementation is via an ALE algorithm, in which a pure lagrangian step is followed by regridding and remapping steps, the later step implicitly embedding the vertical advection \citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}. Here we follow the \citep{Kasahara_MWR74} strategy : a regridding step (an update of the vertical coordinate) followed by an eulerian step with an explicit computation of vertical advection relative to the moving s-surfaces. … … 693 693 \subsection{\textit{S-}coordinate formulation} 694 694 695 Starting from the set of equations established in \ S\ref{PE_zco} for the special case $k=z$695 Starting from the set of equations established in \autoref{sec:PE_zco} for the special case $k=z$ 696 696 and thus $e_3=1$, we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$, which includes 697 697 $z$-, \textit{z*}- and $\sigma-$coordinates as special cases ($s=z$, $s=\textit{z*}$, and 698 698 $s=\sigma=z/H$ or $=z/\left(H+\eta \right)$, resp.). A formal derivation of the transformed 699 equations is given in Appendix~\ref{Apdx_A}. Let us define the vertical scale factor by699 equations is given in \autoref{apdx:A}. Let us define the vertical scale factor by 700 700 $e_3=\partial_s z$ ($e_3$ is now a function of $(i,j,k,t)$ ), and the slopes in the 701 701 (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by : 702 \begin{equation} \label{ Eq_PE_sco_slope}702 \begin{equation} \label{eq:PE_sco_slope} 703 703 \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s 704 704 \quad \text{, and } \quad … … 707 707 We also introduce $\omega $, a dia-surface velocity component, defined as the velocity 708 708 relative to the moving $s$-surfaces and normal to them: 709 \begin{equation} \label{ Eq_PE_sco_w}709 \begin{equation} \label{eq:PE_sco_w} 710 710 \omega = w - e_3 \, \frac{\partial s}{\partial t} - \sigma _1 \,u - \sigma _2 \,v \\ 711 711 \end{equation} 712 712 713 The equations solved by the ocean model \ eqref{Eq_PE} in $s-$coordinate can be written as follows (see Appendix~\ref{Apdx_A_momentum}):713 The equations solved by the ocean model \autoref{eq:PE} in $s-$coordinate can be written as follows (see \autoref{sec:A_momentum}): 714 714 715 715 \vspace{0.5cm} 716 716 $\bullet$ Vector invariant form of the momentum equation : 717 \begin{multline} \label{ Eq_PE_sco_u}717 \begin{multline} \label{eq:PE_sco_u} 718 718 \frac{\partial u }{\partial t}= 719 719 + \left( {\zeta +f} \right)\,v … … 724 724 + D_u^{\vect{U}} + F_u^{\vect{U}} \quad 725 725 \end{multline} 726 \begin{multline} \label{ Eq_PE_sco_v}726 \begin{multline} \label{eq:PE_sco_v} 727 727 \frac{\partial v }{\partial t}= 728 728 - \left( {\zeta +f} \right)\,u … … 736 736 \vspace{0.5cm} 737 737 $\bullet$ Vector invariant form of the momentum equation : 738 \begin{multline} \label{ Eq_PE_sco_u}738 \begin{multline} \label{eq:PE_sco_u} 739 739 \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t}= 740 740 + \left( { f + \frac{1}{e_1 \; e_2 } … … 749 749 + D_u^{\vect{U}} + F_u^{\vect{U}} \quad 750 750 \end{multline} 751 \begin{multline} \label{ Eq_PE_sco_v}751 \begin{multline} \label{eq:PE_sco_v} 752 752 \frac{1}{e_3} \frac{\partial \left( e_3\,v \right) }{\partial t}= 753 753 - \left( { f + \frac{1}{e_1 \; e_2} … … 766 766 pressure have the same expressions as in $z$-coordinates although they do not represent 767 767 exactly the same quantities. $\omega$ is provided by the continuity equation 768 (see Appendix~\ref{Apdx_A}):769 \begin{equation} \label{ Eq_PE_sco_continuity}768 (see \autoref{apdx:A}): 769 \begin{equation} \label{eq:PE_sco_continuity} 770 770 \frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0 771 771 \qquad \text{with }\;\; … … 777 777 \vspace{0.5cm} 778 778 $\bullet$ tracer equations: 779 \begin{multline} \label{ Eq_PE_sco_t}779 \begin{multline} \label{eq:PE_sco_t} 780 780 \frac{1}{e_3} \frac{\partial \left( e_3\,T \right) }{\partial t}= 781 781 -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,T} \right)}{\partial i} … … 784 784 \end{multline} 785 785 786 \begin{multline} \label{ Eq_PE_sco_s}786 \begin{multline} \label{eq:PE_sco_s} 787 787 \frac{1}{e_3} \frac{\partial \left( e_3\,S \right) }{\partial t}= 788 788 -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,S} \right)}{\partial i} … … 805 805 % ------------------------------------------------------------------------------------------------------------- 806 806 \subsection{Curvilinear \textit{z*}--coordinate system} 807 \label{ PE_zco_star}807 \label{subsec:PE_zco_star} 808 808 809 809 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 810 810 \begin{figure}[!b] \begin{center} 811 811 \includegraphics[width=1.0\textwidth]{Fig_z_zstar} 812 \caption{ \protect\label{ Fig_z_zstar}812 \caption{ \protect\label{fig:z_zstar} 813 813 (a) $z$-coordinate in linear free-surface case ; 814 814 (b) $z-$coordinate in non-linear free surface case ; … … 837 837 detailed in Adcroft and Campin (2004). The major points are summarized 838 838 here. The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as: 839 \begin{equation} \label{ Eq_z-star}839 \begin{equation} \label{eq:z-star} 840 840 H + \textit{z*} = (H + z) / r \quad \text{and} \ \delta \textit{z*} = \delta z / r \quad \text{with} \ r = \frac{H+\eta} {H} 841 841 \end{equation} … … 855 855 To overcome problems with vanishing surface and/or bottom cells, we consider the 856 856 zstar coordinate 857 \begin{equation} \label{ PE_}857 \begin{equation} \label{eq:PE_} 858 858 z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 859 859 \end{equation} … … 867 867 The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between 868 868 undulations of the bottom topography versus undulations in the surface height, it 869 is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \ S\ref{PE_sco}.869 is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \autoref{subsec:PE_sco}. 870 870 Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an 871 871 unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in … … 873 873 gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using 874 874 the same techniques as in $z$-models (see Chapters 13-16 of \cite{Griffies_Bk04}) for a 875 discussion of neutral physics in $z$-models, as well as Section \S\ref{LDF_slp}875 discussion of neutral physics in $z$-models, as well as \autoref{sec:LDF_slp} 876 876 in this document for treatment in \NEMO). 877 877 … … 902 902 % ------------------------------------------------------------------------------------------------------------- 903 903 \subsection{Curvilinear terrain-following \textit{s}--coordinate} 904 \label{ PE_sco}904 \label{subsec:PE_sco} 905 905 906 906 % ------------------------------------------------------------------------------------------------------------- … … 915 915 one along continental slopes. Topographic Rossby waves can be excited and can interact 916 916 with the mean current. In the $z-$coordinate system presented in the previous section 917 (\ S\ref{PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is917 (\autoref{sec:PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is 918 918 discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom 919 919 and to large localized depth gradients associated with large localized vertical velocities. … … 937 937 The main two problems come from the truncation error in the horizontal pressure 938 938 gradient and a possibly increased diapycnal diffusion. The horizontal pressure force 939 in $s$-coordinate consists of two terms (see Appendix~\ref{Apdx_A}),940 941 \begin{equation} \label{ Eq_PE_p_sco}939 in $s$-coordinate consists of two terms (see \autoref{apdx:A}), 940 941 \begin{equation} \label{eq:PE_p_sco} 942 942 \left. {\nabla p} \right|_z =\left. {\nabla p} \right|_s -\frac{\partial 943 943 p}{\partial s}\left. {\nabla z} \right|_s 944 944 \end{equation} 945 945 946 The second term in \ eqref{Eq_PE_p_sco} depends on the tilt of the coordinate surface946 The second term in \autoref{eq:PE_p_sco} depends on the tilt of the coordinate surface 947 947 and introduces a truncation error that is not present in a $z$-model. In the special case 948 948 of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), … … 958 958 topography: a envelope topography is defined in $s$-coordinate on which a full or 959 959 partial step bottom topography is then applied in order to adjust the model depth to 960 the observed one (see \ S\ref{DOM_zgr}.960 the observed one (see \autoref{sec:DOM_zgr}. 961 961 962 962 For numerical reasons a minimum of diffusion is required along the coordinate surfaces … … 973 973 the ocean will stay at rest in a $z$-model. As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below the strongly stratified portion of the water column 974 974 ($i.e.$ the main thermocline) \citep{Madec_al_JPO96}. An alternate solution consists of rotating 975 the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \ S\ref{PE_ldf}.975 the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \autoref{subsec:PE_ldf}. 976 976 Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large, 977 strongly exceeding the stability limit of such a operator when it is discretized (see Chapter~\ref{LDF}).977 strongly exceeding the stability limit of such a operator when it is discretized (see \autoref{chap:LDF}). 978 978 979 979 The $s-$coordinates introduced here \citep{Lott_al_OM90,Madec_al_JPO96} differ mainly in two … … 988 988 % ------------------------------------------------------------------------------------------------------------- 989 989 \subsection{\texorpdfstring{Curvilinear $\tilde{z}$--coordinate}{}} 990 \label{ PE_zco_tilde}990 \label{subsec:PE_zco_tilde} 991 991 992 992 The $\tilde{z}$-coordinate has been developed by \citet{Leclair_Madec_OM11}. … … 1000 1000 % ================================================================ 1001 1001 \section{Subgrid scale physics} 1002 \label{ PE_zdf_ldf}1002 \label{sec:PE_zdf_ldf} 1003 1003 1004 1004 The primitive equations describe the behaviour of a geophysical fluid at … … 1019 1019 The control exerted by gravity on the flow induces a strong anisotropy 1020 1020 between the lateral and vertical motions. Therefore subgrid-scale physics 1021 \textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$ in \ eqref{Eq_PE_dyn},1022 \ eqref{Eq_PE_tra_T} and \eqref{Eq_PE_tra_S} are divided into a lateral part1021 \textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$ in \autoref{eq:PE_dyn}, 1022 \autoref{eq:PE_tra_T} and \autoref{eq:PE_tra_S} are divided into a lateral part 1023 1023 \textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and a vertical part 1024 1024 \textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$. The formulation of these terms … … 1029 1029 % ------------------------------------------------------------------------------------------------------------- 1030 1030 \subsection{Vertical subgrid scale physics} 1031 \label{ PE_zdf}1031 \label{subsec:PE_zdf} 1032 1032 1033 1033 The model resolution is always larger than the scale at which the major … … 1044 1044 turbulent motions is simply impractical. The resulting vertical momentum and 1045 1045 tracer diffusive operators are of second order: 1046 \begin{equation} \label{ Eq_PE_zdf}1046 \begin{equation} \label{eq:PE_zdf} 1047 1047 \begin{split} 1048 1048 {\vect{D}}^{v \vect{U}} &=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\ … … 1054 1054 where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, 1055 1055 respectively. At the sea surface and at the bottom, turbulent fluxes of momentum, heat 1056 and salt must be specified (see Chap.~\ref{SBC} and \ref{ZDF} and \S\ref{TRA_bbl}).1056 and salt must be specified (see \autoref{chap:SBC} and \autoref{chap:ZDF} and \autoref{sec:TRA_bbl}). 1057 1057 All the vertical physics is embedded in the specification of the eddy coefficients. 1058 1058 They can be assumed to be either constant, or function of the local fluid properties 1059 1059 ($e.g.$ Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), or computed from a 1060 turbulent closure model. The choices available in \NEMO are discussed in \ S\ref{ZDF}).1060 turbulent closure model. The choices available in \NEMO are discussed in \autoref{chap:ZDF}). 1061 1061 1062 1062 % ------------------------------------------------------------------------------------------------------------- … … 1064 1064 % ------------------------------------------------------------------------------------------------------------- 1065 1065 \subsection{Formulation of the lateral diffusive and viscous operators} 1066 \label{ PE_ldf}1066 \label{subsec:PE_ldf} 1067 1067 1068 1068 Lateral turbulence can be roughly divided into a mesoscale turbulence … … 1124 1124 \subsubsection{Lateral laplacian tracer diffusive operator} 1125 1125 1126 The lateral Laplacian tracer diffusive operator is defined by (see Appendix~\ref{Apdx_B}):1127 \begin{equation} \label{ Eq_PE_iso_tensor}1126 The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx:B}): 1127 \begin{equation} \label{eq:PE_iso_tensor} 1128 1128 D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 1129 1129 \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} … … 1135 1135 where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along 1136 1136 which the diffusive operator acts and the model level ($e. g.$ $z$- or 1137 $s$-surfaces). Note that the formulation \ eqref{Eq_PE_iso_tensor} is exact for the1137 $s$-surfaces). Note that the formulation \autoref{eq:PE_iso_tensor} is exact for the 1138 1138 rotation between geopotential and $s$-surfaces, while it is only an approximation 1139 1139 for the rotation between isoneutral and $z$- or $s$-surfaces. Indeed, in the latter 1140 case, two assumptions are made to simplify \ eqref{Eq_PE_iso_tensor} \citep{Cox1987}.1140 case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{Cox1987}. 1141 1141 First, the horizontal contribution of the dianeutral mixing is neglected since the ratio 1142 1142 between iso and dia-neutral diffusive coefficients is known to be several orders of 1143 1143 magnitude smaller than unity. Second, the two isoneutral directions of diffusion are 1144 1144 assumed to be independent since the slopes are generally less than $10^{-2}$ in the 1145 ocean (see Appendix~\ref{Apdx_B}).1145 ocean (see \autoref{apdx:B}). 1146 1146 1147 1147 For \textit{iso-level} diffusion, $r_1$ and $r_2 $ are zero. $\Re $ reduces to the identity … … 1150 1150 For \textit{geopotential} diffusion, $r_1$ and $r_2 $ are the slopes between the 1151 1151 geopotential and computational surfaces: they are equal to $\sigma _1$ and $\sigma _2$, 1152 respectively (see \ eqref{Eq_PE_sco_slope}).1152 respectively (see \autoref{eq:PE_sco_slope}). 1153 1153 1154 1154 For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral 1155 1155 and computational surfaces. Therefore, they are different quantities, 1156 1156 but have similar expressions in $z$- and $s$-coordinates. In $z$-coordinates: 1157 \begin{equation} \label{ Eq_PE_iso_slopes}1157 \begin{equation} \label{eq:PE_iso_slopes} 1158 1158 r_1 =\frac{e_3 }{e_1 } \left( \pd[\rho]{i} \right) \left( \pd[\rho]{k} \right)^{-1} \, \quad 1159 1159 r_2 =\frac{e_3 }{e_2 } \left( \pd[\rho]{j} \right) \left( \pd[\rho]{k} \right)^{-1} \, … … 1164 1164 When the \textit{eddy induced velocity} parametrisation (eiv) \citep{Gent1990} is used, 1165 1165 an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: 1166 \begin{equation} \label{ Eq_PE_iso+eiv}1166 \begin{equation} \label{eq:PE_iso+eiv} 1167 1167 D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right) 1168 1168 +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right) … … 1170 1170 where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent, 1171 1171 eddy-induced transport velocity. This velocity field is defined by: 1172 \begin{equation} \label{ Eq_PE_eiv}1172 \begin{equation} \label{eq:PE_eiv} 1173 1173 \begin{split} 1174 1174 u^\ast &= +\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_1 } \right] \\ … … 1183 1183 between isoneutral and \emph{geopotential} surfaces. Their values are 1184 1184 thus independent of the vertical coordinate, but their expression depends on the coordinate: 1185 \begin{align} \label{ Eq_PE_slopes_eiv}1185 \begin{align} \label{eq:PE_slopes_eiv} 1186 1186 \tilde{r}_n = \begin{cases} 1187 1187 r_n & \text{in $z$-coordinate} \\ … … 1193 1193 The normal component of the eddy induced velocity is zero at all the boundaries. 1194 1194 This can be achieved in a model by tapering either the eddy coefficient or the slopes 1195 to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. Chap.~\ref{LDF}).1195 to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. \autoref{chap:LDF}). 1196 1196 1197 1197 \subsubsection{Lateral bilaplacian tracer diffusive operator} 1198 1198 1199 1199 The lateral bilaplacian tracer diffusive operator is defined by: 1200 \begin{equation} \label{ Eq_PE_bilapT}1200 \begin{equation} \label{eq:PE_bilapT} 1201 1201 D^{lT}= - \Delta \left( \;\Delta T \right) 1202 1202 \qquad \text{where} \;\; \Delta \bullet = \nabla \left( {\sqrt{B^{lT}\,}\;\Re \;\nabla \bullet} \right) 1203 1203 \end{equation} 1204 It is the Laplacian operator given by \ eqref{Eq_PE_iso_tensor} applied twice with1204 It is the Laplacian operator given by \autoref{eq:PE_iso_tensor} applied twice with 1205 1205 the harmonic eddy diffusion coefficient set to the square root of the biharmonic one. 1206 1206 … … 1209 1209 1210 1210 The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by 1211 applying \ eqref{Eq_PE_lap_vector} to the horizontal velocity vector (see Appendix~\ref{Apdx_B}):1212 \begin{equation} \label{ Eq_PE_lapU}1211 applying \autoref{eq:PE_lap_vector} to the horizontal velocity vector (see \autoref{apdx:B}): 1212 \begin{equation} \label{eq:PE_lapU} 1213 1213 \begin{split} 1214 1214 {\rm {\bf D}}^{l{\rm {\bf U}}} … … 1225 1225 1226 1226 Such a formulation ensures a complete separation between the vorticity and 1227 horizontal divergence fields (see Appendix~\ref{Apdx_C}).1227 horizontal divergence fields (see \autoref{apdx:C}). 1228 1228 Unfortunately, it is only available in \textit{iso-level} direction. 1229 1229 When a rotation is required ($i.e.$ geopotential diffusion in $s-$coordinates 1230 1230 or isoneutral diffusion in both $z$- and $s$-coordinates), the $u$ and $v-$fields 1231 1231 are considered as independent scalar fields, so that the diffusive operator is given by: 1232 \begin{equation} \label{ Eq_PE_lapU_iso}1232 \begin{equation} \label{eq:PE_lapU_iso} 1233 1233 \begin{split} 1234 1234 D_u^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla u} \right) \\ … … 1236 1236 \end{split} 1237 1237 \end{equation} 1238 where $\Re$ is given by \eqref{Eq_PE_iso_tensor}. It is the same expression as1238 where $\Re$ is given by \autoref{eq:PE_iso_tensor}. It is the same expression as 1239 1239 those used for diffusive operator on tracers. It must be emphasised that such a 1240 1240 formulation is only exact in a Cartesian coordinate system, $i.e.$ on a $f-$ or
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