Changeset 9414 for branches/2017/dev_merge_2017/DOC/tex_sub
- Timestamp:
- 2018-03-21T15:39:48+01:00 (6 years ago)
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- branches/2017/dev_merge_2017/DOC/tex_sub
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branches/2017/dev_merge_2017/DOC/tex_sub/annex_A.tex
r9408 r9414 16 16 % ================================================================ 17 17 \section{Chain rule for $s-$coordinates} 18 \label{sec:A_c ontinuity}18 \label{sec:A_chain} 19 19 20 20 In order to establish the set of Primitive Equation in curvilinear $s$-coordinates … … 382 382 Applying similar manipulation to the second component and replacing 383 383 $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes: 384 \begin{equation} \label{apdx:A_grad_p }384 \begin{equation} \label{apdx:A_grad_p_1} 385 385 \begin{split} 386 386 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z … … 394 394 \end{equation} 395 395 396 An additional term appears in (\autoref{apdx:A_grad_p }) which accounts for the396 An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for the 397 397 tilt of $s-$surfaces with respect to geopotential $z-$surfaces. 398 398 … … 416 416 \end{equation*} 417 417 418 Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p } and using the definition of418 Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and using the definition of 419 419 the density anomaly it comes the expression in two parts: 420 \begin{equation} \label{apdx:A_grad_p }420 \begin{equation} \label{apdx:A_grad_p_2} 421 421 \begin{split} 422 422 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z … … 430 430 \end{equation} 431 431 This formulation of the pressure gradient is characterised by the appearance of a term depending on the 432 the sea surface height only (last term on the right hand side of expression \autoref{apdx:A_grad_p }).432 the sea surface height only (last term on the right hand side of expression \autoref{apdx:A_grad_p_2}). 433 433 This term will be loosely termed \textit{surface pressure gradient} 434 434 whereas the first term will be termed the -
branches/2017/dev_merge_2017/DOC/tex_sub/annex_C.tex
r9408 r9414 122 122 123 123 advection term (vector invariant form): 124 \begin{equation} \label{eq:E_tot_vect_vor }124 \begin{equation} \label{eq: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 \end{equation} 127 127 % 128 \begin{equation} \label{eq:E_tot_vect_adv }128 \begin{equation} \label{eq:E_tot_vect_adv_1} 129 129 \int\limits_D \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right) dv 130 130 + \int\limits_D \textbf{U}_h \cdot \nabla_z \textbf{U}_h \;dv … … 151 151 152 152 pressure gradient: 153 \begin{equation} \label{eq:E_tot_pg }153 \begin{equation} \label{eq:E_tot_pg_1} 154 154 - \int\limits_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 155 155 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv … … 172 172 Vector invariant form: 173 173 \begin{subequations} \label{eq:E_tot_vect} 174 \begin{equation} \label{eq:E_tot_vect_vor }174 \begin{equation} \label{eq:E_tot_vect_vor_2} 175 175 \int\limits_D \textbf{U}_h \cdot \text{VOR} \;dv = 0 \\ 176 176 \end{equation} 177 \begin{equation} \label{eq:E_tot_vect_adv }177 \begin{equation} \label{eq:E_tot_vect_adv_2} 178 178 \int\limits_D \textbf{U}_h \cdot \text{KEG} \;dv 179 179 + \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 180 180 - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 \\ 181 181 \end{equation} 182 \begin{equation} \label{eq:E_tot_pg }182 \begin{equation} \label{eq:E_tot_pg_2} 183 183 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 184 184 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv … … 189 189 Flux form: 190 190 \begin{subequations} \label{eq:E_tot_flux} 191 \begin{equation} \label{eq:E_tot_flux_metric }191 \begin{equation} \label{eq:E_tot_flux_metric_2} 192 192 \int\limits_D \textbf{U}_h \cdot \text {COR} \; dv = 0 \\ 193 193 \end{equation} 194 \begin{equation} \label{eq:E_tot_flux_adv }194 \begin{equation} \label{eq:E_tot_flux_adv_2} 195 195 \int\limits_D \textbf{U}_h \cdot \text{ADV} \;dv 196 196 + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 \\ 197 197 \end{equation} 198 \begin{equation} \label{eq:E_tot_pg }198 \begin{equation} \label{eq:E_tot_pg_3} 199 199 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 200 200 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv … … 207 207 208 208 209 \autoref{eq:E_tot_pg } is the balance between the conversion KE to PE and PE to KE.210 Indeed the left hand side of \autoref{eq:E_tot_pg } can be transformed as follows:209 \autoref{eq:E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. 210 Indeed the left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 211 211 \begin{flalign*} 212 212 \partial_t \left( \int\limits_D { \rho \, g \, z \;dv} \right) … … 223 223 the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{apdx:A_w_s}). 224 224 225 The left hand side of \autoref{eq:E_tot_pg } can be transformed as follows:225 The left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 226 226 \begin{flalign*} 227 227 - \int\limits_D \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv … … 325 325 % ================================================================ 326 326 \section{Discrete total energy conservation: vector invariant form} 327 \label{sec:C. 1}327 \label{sec:C.2} 328 328 329 329 % ------------------------------------------------------------------------------------------------------------- … … 331 331 % ------------------------------------------------------------------------------------------------------------- 332 332 \subsection{Total energy conservation} 333 \label{subsec:C_KE+PE }333 \label{subsec:C_KE+PE_vect} 334 334 335 335 The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: … … 401 401 % ------------------------------------------------------------------------------------------------------------- 402 402 \subsubsection{Vorticity term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 403 \label{subsec:C_vorEEN }403 \label{subsec:C_vorEEN_vect} 404 404 405 405 With the EEN scheme, the vorticity terms are represented as: 406 \begin{equation} \ label{eq:dynvor_een}406 \begin{equation} \tag{\ref{eq:dynvor_een}} 407 407 \left\{ { \begin{aligned} 408 408 +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} … … 415 415 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 416 416 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 417 \begin{equation} \ label{eq:Q_triads}417 \begin{equation} \tag{\ref{eq:Q_triads}} 418 418 _i^j \mathbb{Q}^{i_p}_{j_p} 419 419 = \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right) … … 580 580 % ------------------------------------------------------------------------------------------------------------- 581 581 \subsection{Pressure gradient term} 582 \label{subsec:C. 1.4}582 \label{subsec:C.2.6} 583 583 584 584 \gmcomment{ … … 733 733 % ================================================================ 734 734 \section{Discrete total energy conservation: flux form} 735 \label{sec:C. 1}735 \label{sec:C.3} 736 736 737 737 % ------------------------------------------------------------------------------------------------------------- … … 739 739 % ------------------------------------------------------------------------------------------------------------- 740 740 \subsection{Total energy conservation} 741 \label{subsec:C_KE+PE }741 \label{subsec:C_KE+PE_flux} 742 742 743 743 The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: … … 763 763 % ------------------------------------------------------------------------------------------------------------- 764 764 \subsection{Coriolis and advection terms: flux form} 765 \label{subsec:C. 1.3}765 \label{subsec:C.3.2} 766 766 767 767 % ------------------------------------------------------------------------------------------------------------- … … 769 769 % ------------------------------------------------------------------------------------------------------------- 770 770 \subsubsection{Coriolis plus ``metric'' term} 771 \label{subsec:C. 1.3.1}771 \label{subsec:C.3.3} 772 772 773 773 In flux from the vorticity term reduces to a Coriolis term in which the Coriolis … … 789 789 % ------------------------------------------------------------------------------------------------------------- 790 790 \subsubsection{Flux form advection} 791 \label{subsec:C. 1.3.2}791 \label{subsec:C.3.4} 792 792 793 793 The flux form operator of the momentum advection is evaluated using a … … 877 877 % ================================================================ 878 878 \section{Discrete enstrophy conservation} 879 \label{sec:C. 1}879 \label{sec:C.4} 880 880 881 881 … … 887 887 888 888 In the ENS scheme, the vorticity term is descretized as follows: 889 \begin{equation} \ label{eq:dynvor_ens}889 \begin{equation} \tag{\ref{eq:dynvor_ens}} 890 890 \left\{ \begin{aligned} 891 891 +\frac{1}{e_{1u}} & \overline{q}^{\,i} & {\overline{ \overline{\left( e_{1v}\,e_{3v}\; v \right) } } }^{\,i, j+1/2} \\ … … 947 947 948 948 With the EEN scheme, the vorticity terms are represented as: 949 \begin{equation} \ label{eq:dynvor_een}949 \begin{equation} \tag{\ref{eq:dynvor_een}} 950 950 \left\{ { \begin{aligned} 951 951 +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} … … 958 958 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 959 959 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 960 \begin{equation} \ label{eq:Q_triads}960 \begin{equation} \tag{\ref{eq:Q_triads}} 961 961 _i^j \mathbb{Q}^{i_p}_{j_p} 962 962 = \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right) … … 1017 1017 % ================================================================ 1018 1018 \section{Conservation properties on tracers} 1019 \label{sec:C. 2}1019 \label{sec:C.5} 1020 1020 1021 1021 … … 1033 1033 % ------------------------------------------------------------------------------------------------------------- 1034 1034 \subsection{Advection term} 1035 \label{subsec:C. 2.1}1035 \label{subsec:C.5.1} 1036 1036 1037 1037 conservation of a tracer, $T$: … … 1123 1123 % ------------------------------------------------------------------------------------------------------------- 1124 1124 \subsection{Conservation of potential vorticity} 1125 \label{subsec:C. 3.1}1125 \label{subsec:C.6.1} 1126 1126 1127 1127 The lateral momentum diffusion term conserves the potential vorticity : … … 1157 1157 % ------------------------------------------------------------------------------------------------------------- 1158 1158 \subsection{Dissipation of horizontal kinetic energy} 1159 \label{subsec:C. 3.2}1159 \label{subsec:C.6.2} 1160 1160 1161 1161 The lateral momentum diffusion term dissipates the horizontal kinetic energy: … … 1209 1209 % ------------------------------------------------------------------------------------------------------------- 1210 1210 \subsection{Dissipation of enstrophy} 1211 \label{subsec:C. 3.3}1211 \label{subsec:C.6.3} 1212 1212 1213 1213 The lateral momentum diffusion term dissipates the enstrophy when the eddy … … 1234 1234 % ------------------------------------------------------------------------------------------------------------- 1235 1235 \subsection{Conservation of horizontal divergence} 1236 \label{subsec:C. 3.4}1236 \label{subsec:C.6.4} 1237 1237 1238 1238 When the horizontal divergence of the horizontal diffusion of momentum … … 1263 1263 % ------------------------------------------------------------------------------------------------------------- 1264 1264 \subsection{Dissipation of horizontal divergence variance} 1265 \label{subsec:C. 3.5}1265 \label{subsec:C.6.5} 1266 1266 1267 1267 \begin{flalign*} … … 1289 1289 % ================================================================ 1290 1290 \section{Conservation properties on vertical momentum physics} 1291 \label{sec:C _4}1291 \label{sec:C.7} 1292 1292 1293 1293 As for the lateral momentum physics, the continuous form of the vertical diffusion … … 1461 1461 % ================================================================ 1462 1462 \section{Conservation properties on tracer physics} 1463 \label{sec:C. 5}1463 \label{sec:C.8} 1464 1464 1465 1465 The numerical schemes used for tracer subgridscale physics are written such … … 1473 1473 % ------------------------------------------------------------------------------------------------------------- 1474 1474 \subsection{Conservation of tracers} 1475 \label{subsec:C. 5.1}1475 \label{subsec:C.8.1} 1476 1476 1477 1477 constraint of conservation of tracers: … … 1507 1507 % ------------------------------------------------------------------------------------------------------------- 1508 1508 \subsection{Dissipation of tracer variance} 1509 \label{subsec:C. 5.2}1509 \label{subsec:C.8.2} 1510 1510 1511 1511 constraint on the dissipation of tracer variance: -
branches/2017/dev_merge_2017/DOC/tex_sub/annex_iso.tex
r9407 r9414 61 61 The iso-neutral second order tracer diffusive operator for small 62 62 angles between iso-neutral surfaces and geopotentials is given by 63 \autoref{eq: PE_iso_tensor}:64 \begin{subequations} \label{eq: PE_iso_tensor}63 \autoref{eq:iso_tensor_1}: 64 \begin{subequations} \label{eq:iso_tensor_1} 65 65 \begin{equation} 66 66 D^{lT}=-\Div\vect{f}^{lT}\equiv … … 73 73 \end{equation} 74 74 \begin{equation} 75 \label{eq: PE_iso_tensor:c}75 \label{eq:iso_tensor_2} 76 76 \mbox{with}\quad \;\;\Re = 77 77 \begin{pmatrix} … … 118 118 119 119 The off-diagonal terms of the small angle diffusion tensor 120 \autoref{eq: PE_iso_tensor}, \autoref{eq:PE_iso_tensor:c} produce skew-fluxes along the120 \autoref{eq:iso_tensor_1}, \autoref{eq:iso_tensor_2} produce skew-fluxes along the 121 121 $i$- and $j$-directions resulting from the vertical tracer gradient: 122 122 \begin{align} -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_ASM.tex
r9407 r9414 90 90 % Divergence damping description %%% 91 91 \section{Divergence damping initialisation} 92 \label{sec:ASM_d etails}92 \label{sec:ASM_div_dmp} 93 93 94 94 The velocity increments may be initialized by the iterative application of -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_DIA.tex
r9413 r9414 1617 1617 A non-Boussinesq fluid conserves mass. It satisfies the following relations: 1618 1618 1619 \begin{equation} \label{eq:MV_nBq} 1620 \begin{split} 1619 \[ \begin{split} 1621 1620 \mathcal{M} &= \mathcal{V} \;\bar{\rho} \\ 1622 1621 \mathcal{V} &= \mathcal{A} \;\bar{\eta} 1623 \end{split} 1624 \ end{equation}1622 \end{split} \label{eq:MV_nBq} 1623 \] 1625 1624 1626 1625 Temporal changes in total mass is obtained from the density conservation equation : -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_DOM.tex
r9407 r9414 557 557 Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level functions for 558 558 (a) T-point depth and (b) the associated scale factor as computed 559 from \autoref{eq:DOM_zgr_ana } using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.}559 from \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.} 560 560 \end{center} \end{figure} 561 561 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 579 579 near the ocean surface. The following function is proposed as a standard for a 580 580 $z$-coordinate (with either full or partial steps): 581 \begin{equation} \label{eq:DOM_zgr_ana }581 \begin{equation} \label{eq:DOM_zgr_ana_1} 582 582 \begin{split} 583 583 z_0 (k) &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\ … … 592 592 If the ice shelf cavities are opened (\np{ln\_isfcav}\forcode{ = .true.}), the definition of $z_0$ is the same. 593 593 However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to: 594 \begin{equation} \label{eq:DOM_zgr_ana }594 \begin{equation} \label{eq:DOM_zgr_ana_2} 595 595 \begin{split} 596 596 e_3^T(k) &= z_W (k+1) - z_W (k) \\ … … 616 616 With the choice of the stretching $h_{cr} =3$ and the number of levels 617 617 \jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in 618 \autoref{eq:DOM_zgr_ana } have been determined such that \autoref{eq:DOM_zgr_coef} is618 \autoref{eq:DOM_zgr_ana_2} have been determined such that \autoref{eq:DOM_zgr_coef} is 619 619 satisfied, through an optimisation procedure using a bisection method. For the first 620 620 standard ORCA2 vertical grid this led to the following values: $h_{sur} =4762.96$, … … 677 677 \caption{ \protect\label{tab:orca_zgr} 678 678 Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed 679 from \autoref{eq:DOM_zgr_ana } using the coefficients given in \autoref{eq:DOM_zgr_coef}}679 from \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef}} 680 680 \end{table} 681 681 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 761 761 \begin{equation} 762 762 s = -\frac{k}{n-1} \quad \text{ and } \quad 0 \leq k \leq n-1 763 \label{eq: s}763 \label{eq:DOM_s} 764 764 \end{equation} 765 765 -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_DYN.tex
r9407 r9414 414 414 Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een}) 415 415 schemes can be used to compute the product of the Coriolis parameter and the 416 vorticity. However, the energy-conserving scheme 416 vorticity. However, the energy-conserving scheme (\autoref{eq:dynvor_een}) has 417 417 exclusively been used to date. This term is evaluated using a leapfrog scheme, 418 418 $i.e.$ the velocity is centred in time (\textit{now} velocity). -
branches/2017/dev_merge_2017/DOC/tex_sub/chap_model_basics.tex
r9407 r9414 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:PE_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. \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}.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:PE_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:PE_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. … … 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_vector} 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_vector} 727 727 \frac{\partial v }{\partial t}= 728 728 - \left( {\zeta +f} \right)\,u … … 735 735 736 736 \vspace{0.5cm} 737 $\bullet$ Vector invariantform of the momentum equation :738 \begin{multline} \label{eq:PE_sco_u }737 $\bullet$ Flux form of the momentum equation : 738 \begin{multline} \label{eq:PE_sco_u_flux} 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_flux} 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} … … 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 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
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