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r3218 r3267 2 2 % Isoneutral diffusion : 3 3 % ================================================================ 4 \chapter{Griffies's isoneutral diffusion} 4 \chapter[Isoneutral diffusion and eddy advection using 5 triads]{Isoneutral diffusion and eddy advection using triads} 5 6 \label{sec:triad} 6 7 \minitoc 7 8 \section{Griffies's formulation of isoneutral diffusion} 8 \pagebreak 9 \section{Choice of namelist parameters} 10 %nam_traldf 11 \namdisplay{namtra_ldf} 12 % 13 If the namelist variable \np{ln\_traldf\_grif} is set true (and 14 \key{ldfslp} is set), \NEMO updates both active and passive tracers 15 using the Griffies triad representation of isoneutral diffusion and 16 the eddyinduced advective skew (GM) fluxes. Otherwise (by default) the 17 filtered version of Cox's original scheme is employed 18 (\S\ref{LDF_slp}). In the present implementation of the Griffies 19 scheme, the advective skew fluxes are implemented even if 20 \key{traldf\_eiv} is not set. 21 22 Values of isoneutral diffusivity and GM coefficient are set as 23 described in \S\ref{LDF_coef}. If none of the keys \key{traldf\_cNd}, 24 N=1,2,3 is set (the default), spatially constant isoneutral $A_l$ and 25 GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and 26 \np{rn\_aeiv\_0}. If 2Dvarying coefficients are set with 27 \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal 28 scale factor according to \eqref{Eq_title} \footnote{Except in global 29 $0.5^{\circ}$ runs (\key{orca\_r05}) with \key{traldf\_eiv}, where 30 $A_l$ is set like $A_e$ but with a minimum vale of 31 $100\;\mathrm{m}^2\;\mathrm{s}^{1}$}. In idealised setups with 32 \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} 33 is set in the global configurations \key{orca\_r2}, \key{orca\_r1} or 34 \key{orca\_r05} with \key{traldf\_c2d}, a horizontally varying $A_e$ is 35 instead set from the HeldLarichev parameterisation\footnote{In this 36 case, $A_e$ at low latitudes $\theta<20^{\circ}$ is further 37 reduced by a factor $f/f_{20}$, where $f_{20}$ is the value of $f$ 38 at $20^{\circ}$~N} (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored 39 unless it is zero. 40 41 The options specific to the Griffies scheme include: 42 \begin{description}[font=\normalfont] 43 \item[\np{ln\_traldf\_gdia}] Default value is false. See \S\ref{sec:triad:sfdiag}. If this is set true, timemean 44 eddyadvective (GM) velocities are output for diagnostic purposes, even 45 though the eddy advection is accomplished by means of the skew 46 fluxes. 47 \item[\np{ln\_traldf\_iso}] See \S\ref{sec:triad:taper}. If this is set false (the default), then 48 `isoneutral' mixing is accomplished within the surface mixedlayer 49 along slopes linearly decreasing with depth from the value immediately below 50 the mixedlayer to zero (flat) at the surface (\S\ref{sec:triad:lintaper}). This is the same 51 treatment as used in the default implementation 52 \S\ref{LDF_slp_iso}; Fig.~\ref{Fig_eiv_slp}. Where 53 \np{ln\_traldf\_iso} is set true, the vertical skew flux is further 54 reduced to ensure no vertical buoyancy flux, giving an almost pure 55 horizontal diffusive tracer flux within the mixed layer. This is similar to 56 the tapering suggested by \citet{Gerdes1991}. See \S\ref{sec:triad:Gerdestaper} 57 \item[\np{ln\_traldf\_botmix}] See \S\ref{sec:triad:iso_bdry}. If this 58 is set false (the default) then the lateral diffusive fluxes 59 associated with triads partly masked by topography are neglected. If 60 it is set true, however, then these lateral diffusive fluxes are 61 applied, giving smoother bottom tracer fields at the cost of 62 introducing diapycnal mixing. 63 \end{description} 64 \section{Triad formulation of isoneutral diffusion} 9 65 \label{sec:triad:iso} 10 11 We define a scheme inspired by \citet{Griffies_al_JPO98}, but formulated within the \NEMO 66 We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98}, but formulated within the \NEMO 12 67 framework, using scale factors rather than gridsizes. 13 68 … … 613 668 or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$point is 614 669 masked. The associated lateral fluxes (greyblack dashed line) are 615 masked if \n lv{ln\_botmix\_grif=.false.}, but left unmasked,616 giving bottom mixing, if \n lv{ln\_botmix\_grif=.true.}.617 618 The default option \n lv{ln\_botmix\_grif=.false.}is suitable when the619 bbl mixing option is enabled (\key{trabbl}, with \n lv{nn\_bbl\_ldf=1}),670 masked if \np{ln\_botmix\_grif}=false, but left unmasked, 671 giving bottom mixing, if \np{ln\_botmix\_grif}=true. 672 673 The default option \np{ln\_botmix\_grif}=false is suitable when the 674 bbl mixing option is enabled (\key{trabbl}, with \np{nn\_bbl\_ldf}=1), 620 675 or for simple idealized problems. For setups with topography without 621 bbl mixing, \n lv{ln\_botmix\_grif=.true.}may be necessary.676 bbl mixing, \np{ln\_botmix\_grif}=true may be necessary. 622 677 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 623 678 \begin{figure}[h] \begin{center} … … 636 691 or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$point 637 692 is masked. The associated lateral fluxes (greyblack dashed 638 line) are masked if \ smnlv{ln\_botmix\_grif=.false.}, but left639 unmasked, giving bottom mixing, if \ smnlv{ln\_botmix\_grif=.true.}}693 line) are masked if \np{botmix\_grif}=.false., but left 694 unmasked, giving bottom mixing, if \np{botmix\_grif}=.true.} 640 695 \end{center} \end{figure} 641 696 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 665 720 isoneutral density flux that drives dianeutral mixing. In particular this isoneutral density flux 666 721 is always downwards, and so acts to reduce gravitational potential energy. 667 \subsection{Tapering within the surface mixed layer} 722 \subsection{Tapering within the surface mixed layer}\label{sec:triad:taper} 723 668 724 Additional tapering of the isoneutral fluxes is necessary within the 669 725 surface mixed layer. When the Griffies triads are used, we offer two … … 671 727 \subsubsection{Linear slope tapering within the surface mixed layer}\label{sec:triad:lintaper} 672 728 This is the option activated by the default choice 673 \n lv{ln\_triad\_iso=.false.}. Slopes $\tilde{r}_i$ relative to729 \np{ln\_triad\_iso}=false. Slopes $\tilde{r}_i$ relative to 674 730 geopotentials are tapered linearly from their value immediately below the mixed layer to zero at the 675 731 surface, as described in option (c) of Fig.~\ref{Fig_eiv_slp}, to values … … 798 854 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 799 855 800 \subsubsection{Additional truncation of skew isoneutral flux components} 801 The alternative option is activated by setting \nlv{ln\_triad\_iso = 802 .true.}. This retains the same tapered slope $\rML$ described above for the 856 \subsubsection{Additional truncation of skew isoneutral flux 857 components} 858 \label{sec:triad:Gerdestaper} 859 The alternative option is activated by setting \np{ln\_triad\_iso} = 860 true. This retains the same tapered slope $\rML$ described above for the 803 861 calculation of the $_{33}$ term of the isoneutral diffusion tensor (the 804 862 vertical tracer flux driven by vertical tracer gradients), but … … 839 897 % Skew flux formulation for Eddy Induced Velocity : 840 898 % ================================================================ 841 \section{Eddy induced advection and its formulation as a skewflux}899 \section{Eddy induced advection formulated as a skew flux}\label{sec:triad:skewflux} 842 900 843 901 \subsection{The continuous skew flux formulation}\label{sec:triad:continuousskewflux} 844 902 845 When Gent and McWilliams's [1990] diffusion is used (\key{traldf\_eiv} defined),903 When Gent and McWilliams's [1990] diffusion is used, 846 904 an additional advection term is added. The associated velocity is the so called 847 905 eddy induced velocity, the formulation of which depends on the slopes of iso … … 852 910 853 911 The eddy induced velocity is given by: 854 \begin{equation} \label{eq:triad:eiv_v} 912 \begin{subequations} \label{eq:triad:eiv} 913 \begin{equation}\label{eq:triad:eiv_v} 855 914 \begin{split} 856 u^* & =  \frac{1}{e_{3}}\; \partial_ k \left( A_{e} \; \tilde{r}_1 \right)\\857 v^* & =  \frac{1}{e_{3}}\; \partial_ k \left( A_{e} \; \tilde{r}_2 \right)\\858 w^* & = \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i \left( e_{2} \, A_{e} \; \tilde{r}_1\right)859 + \partial_j \left( e_{1} \, A_{e} \;\tilde{r}_2 \right) \right\}915 u^* & =  \frac{1}{e_{3}}\; \partial_i\psi_1, \\ 916 v^* & =  \frac{1}{e_{3}}\; \partial_j\psi_2, \\ 917 w^* & = \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i \left( e_{2} \, \psi_1\right) 918 + \partial_j \left( e_{1} \, \psi_2\right) \right\}, 860 919 \end{split} 861 920 \end{equation} 862 where $A_{e}$ is the eddy induced velocity coefficient, and $\tilde{r}_1$ and $\tilde{r}_2$ the slopes between the isoneutral and the geopotential surfaces. 863 864 The traditional way to implement this additional advection is to add it to the Eulerian 865 velocity prior to computing the tracer advection. This allows us to take advantage of 866 all the advection schemes offered for the tracers (see \S\ref{TRA_adv}) and not just 867 a $2^{nd}$ order advection scheme. This is particularly useful for passive tracers 868 where \emph{positivity} of the advection scheme is of paramount importance. 869 870 \citet{Griffies_JPO98} introduces another way to implement the eddy induced advection, 871 the socalled skew form. It is based on a transformation of the advective fluxes 921 where the streamfunctions $\psi_i$ are given by 922 \begin{equation} \label{eq:triad:eiv_psi} 923 \begin{split} 924 \psi_1 & = A_{e} \; \tilde{r}_1, \\ 925 \psi_2 & = A_{e} \; \tilde{r}_2, 926 \end{split} 927 \end{equation} 928 \end{subequations} 929 with $A_{e}$ the eddy induced velocity coefficient, and $\tilde{r}_1$ and $\tilde{r}_2$ the slopes between the isoneutral and the geopotential surfaces. 930 931 The traditional way to implement this additional advection is to add 932 it to the Eulerian velocity prior to computing the tracer 933 advection. This is implemented if \key{traldf\_eiv} is set in the 934 default implementation, where \np{ln\_traldf\_grif} is set 935 false. This allows us to take advantage of all the advection schemes 936 offered for the tracers (see \S\ref{TRA_adv}) and not just a $2^{nd}$ 937 order advection scheme. This is particularly useful for passive 938 tracers where \emph{positivity} of the advection scheme is of 939 paramount importance. 940 941 However, when \np{ln\_traldf\_grif} is set true, \NEMO instead 942 implements eddy induced advection according to the socalled skew form 943 \citep{Griffies_JPO98}. It is based on a transformation of the advective fluxes 872 944 using the nondivergent nature of the eddy induced velocity. 873 945 For example in the (\textbf{i},\textbf{k}) plane, the tracer advective … … 883 955 &= 884 956 \begin{pmatrix} 885 {  \partial_k \left( e_{2} \, A_{e} \; \tilde{r}_1 \right) \; T \;} \\886 {+ \partial_i \left( e_{2} \, A_{e} \; \tilde{r}_1 \right) \; T \;} \\957 {  \partial_k \left( e_{2} \,\psi_1 \right) \; T \;} \\ 958 {+ \partial_i \left( e_{2} \, \psi_1 \right) \; T \;} \\ 887 959 \end{pmatrix} \\ 888 960 &= 889 961 \begin{pmatrix} 890 {  \partial_k \left( e_{2} \, A_{e} \; \tilde{r}_1 \; T \right) \;} \\891 {+ \partial_i \left( e_{2} \, A_{e} \; \tilde{r}_1\; T \right) \;} \\962 {  \partial_k \left( e_{2} \, \psi_1 \; T \right) \;} \\ 963 {+ \partial_i \left( e_{2} \,\psi_1 \; T \right) \;} \\ 892 964 \end{pmatrix} 893 965 + 894 966 \begin{pmatrix} 895 {+ e_{2} \, A_{e} \; \tilde{r}_1 \; \partial_k T} \\896 {  e_{2} \, A_{e} \; \tilde{r}_1 \; \partial_i T} \\967 {+ e_{2} \, \psi_1 \; \partial_k T} \\ 968 {  e_{2} \, \psi_1 \; \partial_i T} \\ 897 969 \end{pmatrix} 898 970 \end{split} … … 902 974 \begin{equation} \label{eq:triad:eiv_skew_ijk} 903 975 \textbf{F}_\mathrm{eiv}^T = \begin{pmatrix} 904 {+ e_{2} \, A_{e} \; \tilde{r}_1 \; \partial_k T} \\905 {  e_{2} \, A_{e} \; \tilde{r}_1 \; \partial_i T} \\976 {+ e_{2} \, \psi_1 \; \partial_k T} \\ 977 {  e_{2} \, \psi_1 \; \partial_i T} \\ 906 978 \end{pmatrix} 907 979 \end{equation} … … 909 981 \begin{equation}\label{eq:triad:eiv_skew_physical} 910 982 \begin{split} 911 f^*_1 & = \frac{1}{e_{3}}\; A_{e} \; \tilde{r}_1 \partial_k T \\912 f^*_2 & = \frac{1}{e_{3}}\; A_{e} \; \tilde{r}_2 \partial_k T \\913 f^*_3 & = \frac{1}{e_{1}e_{2}}\; A_{e} \left\{ e_{2} \tilde{r}_1 \partial_i T914 + e_{1} \ tilde{r}_2 \partial_j T \right\}. \\983 f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T \\ 984 f^*_2 & = \frac{1}{e_{3}}\; \psi_2 \partial_k T \\ 985 f^*_3 & = \frac{1}{e_{1}e_{2}}\; \left\{ e_{2} \psi_1 \partial_i T 986 + e_{1} \psi_2 \partial_j T \right\}. \\ 915 987 \end{split} 916 988 \end{equation} 917 989 Note that Eq.~ \eqref{eq:triad:eiv_skew_physical} takes the same form whatever the 918 990 vertical coordinate, though of course the slopes 919 $\tilde{r}_i$ are relative to geopotentials.991 $\tilde{r}_i$ which define the $\psi_i$ in \eqref{eq:triad:eiv_psi} are relative to geopotentials. 920 992 The tendency associated with eddy induced velocity is then simply the convergence 921 993 of the fluxes (\ref{eq:triad:eiv_skew_ijk}, \ref{eq:triad:eiv_skew_physical}), so 922 994 \begin{equation} \label{eq:triad:skew_eiv_conv} 923 995 \frac{\partial T}{\partial t}= \frac{1}{e_1 \, e_2 \, e_3 } \left[ 924 \frac{\partial}{\partial i} \left( e_2 A_{e} \; \tilde{r}_1 \partial_k T\right)925 + \frac{\partial}{\partial j} \left( e_1 A_{e}\;926 \ tilde{r}_2 \partial_k T\right)927  \frac{\partial}{\partial k} A_{e} \left( e_{2} \tilde{r}_1 \partial_i T928 + e_{1} \ tilde{r}_2 \partial_j T \right) \right]996 \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right) 997 + \frac{\partial}{\partial j} \left( e_1 \; 998 \psi_2 \partial_k T\right) 999  \frac{\partial}{\partial k} \left( e_{2} \psi_1 \partial_i T 1000 + e_{1} \psi_2 \partial_j T \right) \right] 929 1001 \end{equation} 930 1002 It naturally conserves the tracer content, as it is expressed in flux … … 976 1048 The discretization conserves tracer variance, $i.e.$ it does not 977 1049 include a diffusive component but is a `pure' advection term. This can 978 be seen either from Appendix \ref{Apdx_eiv_skew} or by considering the 1050 be seen 1051 %either from Appendix \ref{Apdx_eiv_skew} or 1052 by considering the 979 1053 fluxes associated with a given triad slope 980 1054 $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. For, following … … 1064 1138 and $\triadt{i+1}{k}{R}{1/2}{1/2}$ are masked when either of the 1065 1139 $i,k+1$ or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ 1066 $u$point is masked. The namelist parameter \n lv{ln\_botmix\_grif} has1140 $u$point is masked. The namelist parameter \np{ln\_botmix\_grif} has 1067 1141 no effect on the eddyinduced skewfluxes. 1068 1142 … … 1079 1153 option (c) of Fig.~\ref{Fig_eiv_slp}. This linear tapering for the 1080 1154 slopes used to calculate the eddyinduced fluxes is 1081 unaffected by the value of \n lv{ln\_triad\_iso}.1155 unaffected by the value of \np{ln\_triad\_iso}. 1082 1156 1083 1157 The justification for this linear slope tapering is that, for $A_e$ … … 1094 1168 1095 1169 \subsection{Streamfunction diagnostics}\label{sec:triad:sfdiag} 1096 Where the namelist parameter \n lv{ln\_botmix\_grif=.true.}, diagnosed1170 Where the namelist parameter \np{ln\_botmix\_grif}=true, diagnosed 1097 1171 mean eddyinduced velocities are output. Each time step, 1098 1172 streamfunctions are calculated in the $i$$k$ and $j$$k$ planes at … … 1104 1178 \begin{equation} 1105 1179 \label{eq:triad:sfdiagi} 1106 {\psi_{[i]}}_{i+1/2}^{k+1/2}={\quarter}\sum_{\substack{i_p,\,k_p}} 1107 {A_e}_{i+1/2i_p}^{k+1/2k_p}\:\triadd{i+1/2i_p}{k+1/2k_p}{R}{i_p}{k_p} 1108 \end{equation} 1109 1110 \newpage %force an empty line 1111 % ================================================================ 1112 % Discrete Invariants of the skew flux formulation 1113 % ================================================================ 1114 \subsection{Discrete Invariants of the skew flux formulation} 1115 \label{Apdx_eiv_skew} 1116 1117 1118 Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. 1119 1120 This have to be moved in an Appendix. 1121 1122 The continuous property to be demonstrated is : 1123 \begin{align*} 1124 \int_D \nabla \cdot \textbf{F}_\mathrm{eiv}(T) \; T \;dv \equiv 0 1125 \end{align*} 1126 The discrete form of its left hand side is obtained using \eqref{eq:triad:allskewflux} 1127 \begin{align*} 1128 \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; 1129 \delta_i &\left[ 1130 {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} 1131 \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] 1132 \right] \; T_i^k \\ 1133  \delta_k &\left[ 1134 {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} 1135 \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] 1136 \right] \; T_i^k \ \Biggr\} 1137 \end{align*} 1138 apply the adjoint of delta operator, it becomes 1139 \begin{align*} 1140 \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; 1141 &\left( 1142 {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} 1143 \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] 1144 \right) \; \delta_{i+1/2}[T^{k}] \\ 1145  &\left( 1146 {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} 1147 \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] 1148 \right) \; \delta_{k+1/2}[T_{i}] \ \Biggr\} 1149 \end{align*} 1150 Expending the summation on $i_p$ and $k_p$, it becomes: 1151 \begin{align*} 1152 \begin{matrix} 1153 &\sum\limits_{i,k} \Bigl\{ 1154 &+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} 1155 &\ {_{i+1}^k \mathbb{R}_{ 1/2}^{1/2}} &\delta_{k1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\ 1156 &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} 1157 &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{1/2}} &\delta_{k1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ 1158 &&+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} 1159 &\ {_{i+1}^k \mathbb{R}_{ 1/2}^{+1/2}} &\delta_{k+1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\ 1160 &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} 1161 &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{k+1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ 1162 % 1163 &&{e_{2u}}_i^{k+1} &{A_{e}}_i^{k+1} 1164 &{_i^{k+1} \mathbb{R}_{1/2}^{ 1/2}} &\delta_{i1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\ 1165 &&{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} 1166 &{\ \ \;_i^k \mathbb{R}_{1/2}^{+1/2}} &\delta_{i1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] &\\ 1167 &&{e_{2u}}_i^{k+1 } &{A_{e}}_i^{k+1} 1168 &{_i^{k+1} \mathbb{R}_{+1/2}^{ 1/2}} &\delta_{i+1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\ 1169 &&{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} 1170 &{\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{i+1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] 1171 &\Bigr\} \\ 1172 \end{matrix} 1173 \end{align*} 1174 The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the 1175 same but of opposite signs, they cancel out. 1176 Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{1/2}^{1/2}}$. 1177 The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{1/2}}$ are the 1178 same but both of opposite signs and shifted by 1 in $k$ direction. When summing over $k$ 1179 they cancel out with the neighbouring grid points. 1180 Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{1/2}^{+1/2}}$ in the 1181 $i$ direction. Therefore the sum over the domain is zero, $i.e.$ the variance of the 1182 tracer is preserved by the discretisation of the skew fluxes. 1183 1184 %%% Local Variables: 1185 %%% TeXmaster: "../../NEMO_bookluatex.tex" 1186 %%% End: 1180 {\psi_1}_{i+1/2}^{k+1/2}={\quarter}\sum_{\substack{i_p,\,k_p}} 1181 {A_e}_{i+1/2i_p}^{k+1/2k_p}\:\triadd{i+1/2i_p}{k+1/2k_p}{R}{i_p}{k_p}. 1182 \end{equation} 1183 The streamfunction $\psi_1$ is calculated similarly at $vw$ points. 1184 The eddyinduced velocities are then calculated from the 1185 straightforward discretisation of \eqref{eq:triad:eiv_v}: 1186 \begin{equation}\label{eq:triad:eiv_v} 1187 \begin{split} 1188 {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), \\ 1189 {v^*}_{j+1/2}^{k} & =  \frac{1}{{e_{3v}}_{j}^{k}}\left({\psi_2}_{j+1/2}^{k+1/2}{\psi_2}_{j+1/2}^{k+1/2}\right), \\ 1190 {w^*}_{i,j}^{k+1/2} & = \frac{1}{e_{1t}e_{2t}}\; \left\{ 1191 {e_{2u}}_{i+1/2}^{k+1/2} \,{\psi_1}_{i+1/2}^{k+1/2}  1192 {e_{2u}}_{i1/2}^{k+1/2} \,{\psi_1}_{i1/2}^{k+1/2} \right. + \\ 1193 \phantom{=} & \qquad\qquad\left. {e_{2v}}_{j+1/2}^{k+1/2} \,{\psi_2}_{j+1/2}^{k+1/2}  {e_{2v}}_{j1/2}^{k+1/2} \,{\psi_2}_{j1/2}^{k+1/2} \right\}, 1194 \end{split} 1195 \end{equation}
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