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branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_C.tex
r1223 r2282 7 7 8 8 %%% Appendix put in gmcomment as it has not been updated for z* and s coordinate 9 I'm writting this appendix. It will be available in a forthcoming release of the documentation9 %I'm writting this appendix. It will be available in a forthcoming release of the documentation 10 10 11 11 %\gmcomment{ 12 12 13 % ================================================================ 14 % Conservation Properties on Ocean Dynamics 15 % ================================================================ 16 \section{Conservation Properties on Ocean Dynamics} 13 \newpage 14 $\ $\newline % force a new ligne 15 16 % ================================================================ 17 % Introduction / Notations 18 % ================================================================ 19 \section{Introduction / Notations} 20 \label{Apdx_C.0} 21 22 Notation used in this appendix in the demonstations : 23 24 fluxes at the faces of a $T$-box: 25 \begin{equation*} 26 U = e_{2u}\,e_{3u}\; u \qquad V = e_{1v}\,e_{3v}\; v \qquad W = e_{1w}\,e_{2w}\; \omega \\ 27 \end{equation*} 28 29 volume of cells at $u$-, $v$-, and $T$-points: 30 \begin{equation*} 31 b_u = e_{1u}\,e_{2u}\,e_{3u} \qquad b_v = e_{1v}\,e_{2v}\,e_{3v} \qquad b_t = e_{1t}\,e_{2t}\,e_{3t} \\ 32 \end{equation*} 33 34 partial derivative notation: $\partial_\bullet = \frac{\partial}{\partial \bullet}$ 35 36 $dv=e_1\,e_2\,e_3 \,di\,dj\,dk$ is the volume element, with only $e_3$ that depends on time. 37 $D$ and $S$ are the ocean domain volume and surface, respectively. 38 No wetting/drying is allow ($i.e.$ $\frac{\partial S}{\partial t} = 0$) 39 Let $k_s$ and $k_b$ be the ocean surface and bottom, resp. 40 ($i.e.$ $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth). 41 \begin{flalign*} 42 z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s} e_3(\tilde{k}) \;d\tilde{k} 43 = \eta - \int\limits_k^{k_s} e_3 \;d\tilde{k} 44 \end{flalign*} 45 46 Continuity equation with the above notation: 47 \begin{equation*} 48 \frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t} \biggl\{ \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0 49 \end{equation*} 50 51 A quantity, $Q$ is conserved when its domain averaged time change is zero, that is when: 52 \begin{equation*} 53 \partial_t \left( \int_D{ Q\;dv } \right) =0 54 \end{equation*} 55 Noting that the coordinate system used .... blah blah 56 \begin{equation*} 57 \partial_t \left( \int_D {Q\;dv} \right) = \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } 58 = \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0 59 \end{equation*} 60 equation of evolution of $Q$ written as the time evolution of the vertical content of $Q$ 61 like for tracers, or momentum in flux form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when : 62 \begin{flalign*} 63 \partial_t \left( \int_D{ \frac{1}{2} \,Q^2\;dv } \right) 64 =& \int_D{ \frac{1}{2} \partial_t \left( \frac{1}{e_3}\left( e_3 \, Q \right)^2 \right) e_1e_2\;di\,dj\,dk } \\ 65 =& \int_D { Q \;\partial_t\left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } 66 - \int_D { \frac{1}{2} Q^2 \,\partial_t (e_3) \;e_1e_2\;di\,dj\,dk } \\ 67 \end{flalign*} 68 that is in a more compact form : 69 \begin{flalign} \label{Eq_Q2_flux} 70 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 71 =& \int_D { \frac{Q}{e_3} \partial_t \left( e_3 \, Q \right) dv } 72 - \frac{1}{2} \int_D { \frac{Q^2}{e_3} \partial_t (e_3) \;dv } 73 \end{flalign} 74 equation of evolution of $Q$ written as the time evolution of $Q$ 75 like for momentum in vector invariant form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when : 76 \begin{flalign*} 77 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 78 =& \int_D { \frac{1}{2} \partial_t \left( e_3 \, Q^2 \right) \;e_1e_2\;di\,dj\,dk } \\ 79 =& \int_D { Q \partial_t Q \;e_1e_2e_3\;di\,dj\,dk } 80 + \int_D { \frac{1}{2} Q^2 \, \partial_t e_3 \;e_1e_2\;di\,dj\,dk } \\ 81 \end{flalign*} 82 that is in a more compact form : 83 \begin{flalign} \label{Eq_Q2_vect} 84 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 85 =& \int_D { Q \,\partial_t Q \;dv } 86 + \frac{1}{2} \int_D { \frac{1}{e_3} Q^2 \partial_t e_3 \;dv } 87 \end{flalign} 88 89 90 % ================================================================ 91 % Continuous Total energy Conservation 92 % ================================================================ 93 \section{Continuous conservation} 17 94 \label{Apdx_C.1} 18 95 19 First, the boundary condition on the vertical velocity (no flux through the surface 20 and the bottom) is established for the discrete set of momentum equations. 21 Then, it is shown that the non-linear terms of the momentum equation are written 22 such that the potential enstrophy of a horizontally non-divergent flow is preserved 23 while all the other non-diffusive terms preserve the kinetic energy; in practice the 24 energy is also preserved. In addition, an option is also offered for the vorticity term 25 discretization which provides a total kinetic energy conserving discretization for 26 that term. 27 28 Nota Bene: these properties are established here in the rigid-lid case and for the 29 2nd order centered scheme. A forthcoming update will be their generalisation to 30 the free surface case and higher order scheme. 31 32 % ------------------------------------------------------------------------------------------------------------- 33 % Bottom Boundary Condition on Vertical Velocity Field 34 % ------------------------------------------------------------------------------------------------------------- 35 \subsection{Bottom Boundary Condition on Vertical Velocity Field} 36 \label{Apdx_C.1.1} 37 38 39 The discrete set of momentum equations used in the rigid-lid approximation 40 automatically satisfies the surface and bottom boundary conditions 41 (no flux through the surface and the bottom: $w_{surface} =w_{bottom} =~0$). 42 Indeed, taking the discrete horizontal divergence of the vertical sum of the 43 horizontal momentum equations (!!!Eqs. (II.2.1) and (II.2.2)!!!) weighted by the 44 vertical scale factors, it becomes: 45 \begin{flalign*} 46 \frac{\partial } {\partial t} \left( \sum\limits_k \chi \right) 47 \equiv 48 \frac{\partial } {\partial t} \left( w_{surface} -w_{bottom} \right)&&&\\ 49 \end{flalign*} 50 \begin{flalign*} 51 \equiv \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} 52 \biggl\{ \quad 53 \delta_i 54 &\left[ 55 e_{2u}\,H_u 56 \left( 57 M_u - M_u - \frac{1} {H_u\,e_{2u}} \delta_j 58 \left[ \partial_t\, \psi \right] 59 \right) 60 \right] && 61 \biggr. \\ 62 \biggl. 63 + \delta_j 64 &\left[ 65 e_{1v}\,H_v 66 \left( M_v - M_v - \frac{1} {H_v\,e_{1v}} \delta_i 67 \left[ \partial_i\, \psi \right] 68 \right) 69 \right] 70 \biggr\}&& \\ 71 \end{flalign*} 72 \begin{flalign*} 73 \equiv \frac{1} {e_{1T} \,e_{2T} \,e_{3T}} \; 74 \biggl\{ 75 - \delta_i 76 \Bigl[ 77 \delta_j 78 \left[ \partial_t \psi \right] 79 \Bigr] 80 + \delta_j 81 \Bigl[ 82 \delta_i 83 \left[ \partial_t \psi \right] 84 \Bigr]\; 85 \biggr\}\; 86 \equiv 0 87 &&&\\ 88 \end{flalign*} 89 90 91 The surface boundary condition associated with the rigid lid approximation 92 ($w_{surface} =0)$ is imposed in the computation of the vertical velocity (!!! II.2.5!!!!). 93 Therefore, it turns out to be: 96 97 The discretization of pimitive equation in $s$-coordinate ($i.e.$ time and space varying 98 vertical coordinate) must be chosen so that the discrete equation of the model satisfy 99 integral constrains on energy and enstrophy. 100 101 102 Let us first establish those constraint in the continuous world. 103 The total energy ($i.e.$ kinetic plus potential energies) is conserved : 104 \begin{flalign} \label{Eq_Tot_Energy} 105 \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 + \rho \, g \, z \right) \;dv \right) = & 0 106 \end{flalign} 107 under the following assumptions: no dissipation, no forcing 108 (wind, buoyancy flux, atmospheric pressure variations), mass 109 conservation, and closed domain. 110 111 This equation can be transformed to obtain several sub-equalities. 112 The transformation for the advection term depends on whether 113 the vector invariant form or the flux form is used for the momentum equation. 114 Using \eqref{Eq_Q2_vect} and introducing \eqref{Apdx_A_dyn_vect} in \eqref{Eq_Tot_Energy} 115 for the former form and 116 Using \eqref{Eq_Q2_flux} and introducing \eqref{Apdx_A_dyn_flux} in \eqref{Eq_Tot_Energy} 117 for the latter form leads to: 118 119 \begin{subequations} \label{E_tot} 120 121 advection term (vector invariant form): 122 \begin{equation} \label{E_tot_vect_vor} 123 \int\limits_D \zeta \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ 124 \end{equation} 125 % 126 \begin{equation} \label{E_tot_vect_adv} 127 \int\limits_D \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right) dv 128 + \int\limits_D \textbf{U}_h \cdot \nabla_z \textbf{U}_h \;dv 129 - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 \\ 130 \end{equation} 131 132 advection term (flux form): 133 \begin{equation} \label{E_tot_flux_metric} 134 \int\limits_D \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1 \right)\; 135 \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ 136 \end{equation} 137 138 \begin{equation} \label{E_tot_flux_adv} 139 \int\limits_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} 140 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 141 \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ \end{array}} } \right) \;dv 142 + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 \\ 143 \end{equation} 144 145 coriolis term 146 \begin{equation} \label{E_tot_cor} 147 \int\limits_D f \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ 148 \end{equation} 149 150 pressure gradient: 151 \begin{equation} \label{E_tot_pg} 152 - \int\limits_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 153 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 154 + \int\limits_D g\, \rho \; \partial_t z \;dv \\ 155 \end{equation} 156 \end{subequations} 157 158 where $\nabla_h = \left. \nabla \right|_k$ is the gradient along the $s$-surfaces. 159 160 blah blah.... 161 $\ $\newline % force a new ligne 162 The prognostic ocean dynamics equation can be summarized as follows: 94 163 \begin{equation*} 95 \frac{\partial } {\partial t}w_{bottom} \equiv 0 164 \text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } 165 {\text{COR} + \text{ADV} } 166 + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} 96 167 \end{equation*} 97 As the bottom velocity is initially set to zero, it remains zero all the time. 98 Symmetrically, if $w_{bottom} =0$ is used in the computation of the vertical 99 velocity (upward integral of the horizontal divergence), the same computation 100 leads to $w_{surface} =0$ as soon as the surface vertical velocity is initially 101 set to zero. 102 103 % ------------------------------------------------------------------------------------------------------------- 104 % Coriolis and advection terms: vector invariant form 105 % ------------------------------------------------------------------------------------------------------------- 106 \subsection{Coriolis and advection terms: vector invariant form} 107 \label{Apdx_C_vor_zad} 108 109 % ------------------------------------------------------------------------------------------------------------- 110 % Vorticity Term 111 % ------------------------------------------------------------------------------------------------------------- 112 \subsubsection{Vorticity Term} 113 \label{Apdx_C_vor} 114 115 Potential vorticity is located at $f$-points and defined as: $\zeta / e_{3f}$. 116 The standard discrete formulation of the relative vorticity term obviously 117 conserves potential vorticity (ENS scheme). It also conserves the potential 118 enstrophy for a horizontally non-divergent flow (i.e. $\chi $=0) but not the 119 total kinetic energy. Indeed, using the symmetry or skew symmetry properties 120 of the operators (Eqs \eqref{DOM_mi_adj} and \eqref{DOM_di_adj}), it can 121 be shown that: 122 \begin{equation} \label{Apdx_C_1.1} 123 \int_D {\zeta / e_3\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {\zeta \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0 168 $\ $\newline % force a new ligne 169 170 Vector invariant form: 171 \begin{subequations} \label{E_tot_vect} 172 \begin{equation} \label{E_tot_vect_vor} 173 \int\limits_D \textbf{U}_h \cdot \text{VOR} \;dv = 0 \\ 124 174 \end{equation} 125 where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. Indeed, using 126 \eqref{Eq_dynvor_ens}, the discrete form of the right hand side of \eqref{Apdx_C_1.1} 127 can be transformed as follow: 128 \begin{flalign*} 129 &\int_D \zeta / e_3\,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 130 \left( 131 \zeta \; \textbf{k} \times \textbf{U}_h 132 \right)\; 133 dv 134 &&& \displaybreak[0] \\ 135 % 136 \equiv& \sum\limits_{i,j,k} 137 \frac{\zeta / e_{3f}} {e_{1f}\,e_{2f}\,e_{3f}} 138 \biggl\{ \quad 139 \delta_{i+1/2} 140 \left[ 141 - \overline {\left( {\zeta / e_{3f}} \right)}^{\,i}\; 142 \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/ 2} 143 \right] 144 && \\ & \qquad \qquad \qquad \;\; 145 - \delta_{j+1/2} 146 \left[ \;\;\; 147 \overline {\left( \zeta / e_{3f} \right)}^{\,j}\; 148 \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 149 \right] 150 \;\;\biggr\} \; e_{1f}\,e_{2f}\,e_{3f} && \displaybreak[0] \\ 151 % 152 \equiv& \sum\limits_{i,j,k} 153 \biggl\{ \delta_i \left[ \frac{\zeta} {e_{3f}} \right] \; 154 \overline{ \left( \frac{\zeta} {e_{3f}} \right) }^{\,i}\; 155 \overline{ \overline{ \left( e_{1u}\,e_{3u}\,u \right) } }^{\,i,j+1/2} 156 + \delta_j \left[ \frac{\zeta} {e_{3f}} \right] \; 157 \overline{ \left( \frac{\zeta} {e_{3f}} \right) }^{\,j} \; 158 \overline{\overline {\left( e_{2v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} \biggr\} 159 &&&& \displaybreak[0] \\ 160 % 161 \equiv& \frac{1} {2} \sum\limits_{i,j,k} 162 \biggl\{ \delta_i \Bigl[ \left( \frac{\zeta} {e_{3f}} \right)^2 \Bigr]\; 163 \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 164 + \delta_j \Bigl[ \left( \zeta / e_{3f} \right)^2 \Bigr]\; 165 \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 166 \biggr\} 167 && \displaybreak[0] \\ 168 % 169 \equiv& - \frac{1} {2} \sum\limits_{i,j,k} \left( \frac{\zeta} {e_{3f}} \right)^2\; 170 \biggl\{ \delta_{i+1/2} 171 \left[ \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} \right] 172 + \delta_{j+1/2} 173 \left[ \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} \right] 174 \biggr\} && \\ 175 \end{flalign*} 176 Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2} 177 \left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$, 178 and introducing the horizontal divergence $\chi $, it becomes: 179 \begin{align*} 180 \equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left( \frac{\zeta} {e_{3f}} \right)^2 \; \overline{\overline{ e_{1T}\,e_{2T}\,e_{3T}\, \chi}}^{\,i+1/2,j+1/2} \;\;\equiv 0 181 \qquad \qquad \qquad \qquad \qquad \qquad \qquad &&&&\\ 182 \end{align*} 183 184 Note that the derivation is demonstrated here for the relative potential 185 vorticity but it applies also to the planetary ($f/e_3$) and the total 186 potential vorticity $((\zeta +f) /e_3 )$. Another formulation of the two 187 components of the vorticity term is optionally offered (ENE scheme) : 175 \begin{equation} \label{E_tot_vect_adv} 176 \int\limits_D \textbf{U}_h \cdot \text{KEG} \;dv 177 + \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 178 - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 \\ 179 \end{equation} 180 \begin{equation} \label{E_tot_pg} 181 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 182 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 183 + \int\limits_D g\, \rho \; \partial_t z \;dv \\ 184 \end{equation} 185 \end{subequations} 186 187 Flux form: 188 \begin{subequations} \label{E_tot_flux} 189 \begin{equation} \label{E_tot_flux_metric} 190 \int\limits_D \textbf{U}_h \cdot \text {COR} \; dv = 0 \\ 191 \end{equation} 192 \begin{equation} \label{E_tot_flux_adv} 193 \int\limits_D \textbf{U}_h \cdot \text{ADV} \;dv 194 + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 \\ 195 \end{equation} 196 \begin{equation} \label{E_tot_pg} 197 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 198 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 199 + \int\limits_D g\, \rho \; \partial_t z \;dv \\ 200 \end{equation} 201 \end{subequations} 202 203 204 $\ $\newline % force a new ligne 205 206 207 \eqref{E_tot_pg} is the balance between the conversion KE to PE and PE to KE. 208 Indeed the left hand side of \eqref{E_tot_pg} can be transformed as follows: 209 \begin{flalign*} 210 \partial_t \left( \int\limits_D { \rho \, g \, z \;dv} \right) 211 &= + \int\limits_D \frac{1}{e_3} \partial_t (e_3\,\rho) \;g\;z\;\;dv 212 + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\ 213 &= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 214 + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\ 215 &= + \int\limits_D \rho \,g \left( \textbf {U}_h \cdot \nabla_h z + \omega \frac{1}{e_3} \partial_k z \right) \;dv 216 + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\ 217 &= + \int\limits_D \rho \,g \left( \omega + \partial_t z + \textbf {U}_h \cdot \nabla_h z \right) \;dv &&&\\ 218 &=+ \int\limits_D g\, \rho \; w \; dv &&&\\ 219 \end{flalign*} 220 where the last equality is obtained by noting that the brackets is exactly the expression of $w$, 221 the vertical velocity referenced to the fixe $z$-coordinate system (see \eqref{Apdx_A_w_s}). 222 223 The left hand side of \eqref{E_tot_pg} can be transformed as follows: 224 \begin{flalign*} 225 - \int\limits_D \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv 226 = - \int\limits_D \left( \nabla_h p + \rho \, g \nabla_h z \right) \cdot \textbf{U}_h \;dv &&&\\ 227 \allowdisplaybreaks 228 &= - \int\limits_D \nabla_h p \cdot \textbf{U}_h \;dv - \int\limits_D \rho \, g \, \nabla_h z \cdot \textbf{U}_h \;dv &&&\\ 229 \allowdisplaybreaks 230 &= +\int\limits_D p \,\nabla_h \cdot \textbf{U}_h \;dv + \int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ 231 \allowdisplaybreaks 232 &= -\int\limits_D p \left( \frac{1}{e_3} \partial_t e_3 + \frac{1}{e_3} \partial_k \omega \right) \;dv 233 +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ 234 \allowdisplaybreaks 235 &= -\int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 236 +\int\limits_D \frac{1}{e_3} \partial_k p\; \omega \;dv 237 +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ 238 &= -\int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 239 -\int\limits_D \rho \, g \, \omega \;dv 240 +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ 241 &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \; \;dv 242 - \int\limits_D \rho \, g \, w \;dv 243 + \int\limits_D \rho \, g \, \partial_t z \;dv &&&\\ 244 \allowdisplaybreaks 245 \intertext{introducing the hydrostatic balance $\partial_k p=-\rho \,g\,e_3$ in the last term, 246 it becomes:} 247 &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 248 - \int\limits_D \rho \, g \, w \;dv 249 - \int\limits_D \frac{1}{e_3} \partial_k p\, \partial_t z \;dv &&&\\ 250 &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 251 - \int\limits_D \rho \, g \, w \;dv 252 + \int\limits_D \,\frac{p}{e_3}\partial_t ( \partial_k z ) dv &&&\\ 253 % 254 &= - \int\limits_D \rho \, g \, w \;dv &&&\\ 255 \end{flalign*} 256 257 258 %gm comment 259 \gmcomment{ 260 % 261 The last equality comes from the following equation, 262 \begin{flalign*} 263 \int\limits_D p \frac{1}{e_3} \frac{\partial e_3}{\partial t}\; \;dv 264 = \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv \quad, \\ 265 \end{flalign*} 266 that can be demonstrated as follows: 267 268 \begin{flalign*} 269 \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv 270 &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv 271 - \int\limits_D \rho \, g \, \frac{\partial}{\partial t} \left( \int\limits_k^{k_s} e_3 \;d\tilde{k} \right) \;dv &&&\\ 272 &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv 273 - \int\limits_D \rho \, g \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \;dv &&&\\ 274 % 275 \allowdisplaybreaks 276 \intertext{The second term of the right hand side can be transformed by applying the integration by part rule: 277 $\left[ a\,b \right]_{k_b}^{k_s} = \int_{k_b}^{k_s} a\,\frac{\partial b}{\partial k} \;dk 278 + \int_{k_b}^{k_s} \frac{\partial a}{\partial k} \,b \;dk $ 279 to the following function: $a= \int_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k}$ 280 and $b= \int_k^{k_s} \rho \, e_3 \;d\tilde{k}$ 281 (note that $\frac{\partial}{\partial k} \left( \int_k^{k_s} a \;d\tilde{k} \right) = - a$ as $k$ is the lower bound of the integral). 282 This leads to: } 283 \end{flalign*} 284 \begin{flalign*} 285 &\left[ \int\limits_{k}^{k_s} \frac{\partial e_3}{\partial t} \,dk \cdot \int\limits_{k}^{k_s} \rho \, e_3 \,dk \right]_{k_b}^{k_s} 286 =-\int\limits_{k_b}^{k_s} \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \rho \,e_3 \;dk 287 -\int\limits_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \left( \int\limits_k^{k_s} \rho \, e_3 \;d\tilde{k} \right) dk 288 &&&\\ 289 \allowdisplaybreaks 290 \intertext{Noting that $\frac{\partial \eta}{\partial t} 291 = \frac{\partial}{\partial t} \left( \int_{k_b}^{k_s} e_3 \;d\tilde{k} \right) 292 = \int_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k}$ 293 and 294 $p(k) = \int_k^{k_s} \rho \,g \, e_3 \;d\tilde{k} $, 295 but also that $\frac{\partial \eta}{\partial t}$ does not depends on $k$, it comes: 296 } 297 & - \int\limits_{k_b}^{k_s} \rho \, \frac{\partial \eta}{\partial t} \, e_3 \;dk 298 = - \int\limits_{k_b}^{k_s} \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \, \rho \, g e_3\;dk 299 - \int\limits_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \frac{p}{g} \;dk &&&\\ 300 \end{flalign*} 301 Mutliplying by $g$ and integrating over the $(i,j)$ domain it becomes: 302 \begin{flalign*} 303 \int\limits_D \rho \, g \, \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \;dv 304 = \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} dv 305 - \int\limits_D \frac{p}{e_3}\frac{\partial e_3}{\partial t} \;dv 306 \end{flalign*} 307 Using this property, we therefore have: 308 \begin{flalign*} 309 \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv 310 &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv 311 - \left( \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} dv 312 - \int\limits_D \frac{p}{e_3}\frac{\partial e_3}{\partial t} \;dv \right) &&&\\ 313 % 314 &=\int\limits_D \frac{p}{e_3} \frac{\partial (e_3\,\rho)}{\partial t}\; \;dv 315 \end{flalign*} 316 % end gm comment 317 } 318 % 319 320 321 % ================================================================ 322 % Discrete Total energy Conservation : vector invariant form 323 % ================================================================ 324 \section{Discrete total energy conservation : vector invariant form} 325 \label{Apdx_C.1} 326 327 % ------------------------------------------------------------------------------------------------------------- 328 % Total energy conservation 329 % ------------------------------------------------------------------------------------------------------------- 330 \subsection{Total energy conservation} 331 \label{Apdx_C_KE+PE} 332 333 The discrete form of the total energy conservation, \eqref{Eq_Tot_Energy}, is given by: 334 \begin{flalign*} 335 \partial_t \left( \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v + \rho \, g \, z_t \,b_t \biggr\} \right) &=0 \\ 336 \end{flalign*} 337 which in vector invariant forms, it leads to: 338 \begin{equation} \label{KE+PE_vect_discrete} \begin{split} 339 \sum\limits_{i,j,k} \biggl\{ u\, \partial_t u \;b_u 340 + v\, \partial_t v \;b_v \biggr\} 341 + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{e_{3u}}\partial_t e_{3u} \;b_u 342 + \frac{v^2}{e_{3v}}\partial_t e_{3v} \;b_v \biggr\} \\ 343 = - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}}\partial_t (e_{3t} \rho) \, g \, z_t \;b_t \biggr\} 344 - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} 345 \end{split} \end{equation} 346 347 Substituting the discrete expression of the time derivative of the velocity either in vector invariant, 348 leads to the discrete equivalent of the four equations \eqref{E_tot_flux}. 349 350 % ------------------------------------------------------------------------------------------------------------- 351 % Vorticity term (coriolis + vorticity part of the advection) 352 % ------------------------------------------------------------------------------------------------------------- 353 \subsection{Vorticity term (coriolis + vorticity part of the advection)} 354 \label{Apdx_C_vor} 355 356 Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$), or 357 the planetary ($q=f/e_{3f}$), or the total potential vorticity ($q=(\zeta +f) /e_{3f}$). 358 Two discretisation of the vorticity term (ENE and EEN) allows the conservation of 359 the kinetic energy. 360 % ------------------------------------------------------------------------------------------------------------- 361 % Vorticity Term with ENE scheme 362 % ------------------------------------------------------------------------------------------------------------- 363 \subsubsection{Vorticity Term with ENE scheme (\np{ln\_dynvor\_ene}=.true.)} 364 \label{Apdx_C_vorENE} 365 366 For the ENE scheme, the two components of the vorticity term are given by : 188 367 \begin{equation*} 189 - \zeta \;{\textbf{k}}\times {\textbf {U}}_h 190 \equiv 191 \left( {{ 192 \begin{array} {*{20}c} 368 - e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h \equiv 369 \left( {{ \begin{array} {*{20}c} 193 370 + \frac{1} {e_{1u}} \; 194 \overline {\left( \zeta / e_{3f} \right) 195 \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} 196 \hfill \\ 371 \overline {\, q \ \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} \hfill \\ 197 372 - \frac{1} {e_{2v}} \; 198 \overline {\left( \zeta / e_{3f} \right) 199 \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} 200 \hfill \\ 201 \end{array}} } 202 \right) 373 \overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} \hfill \\ 374 \end{array}} } \right) 203 375 \end{equation*} 204 376 205 377 This formulation does not conserve the enstrophy but it does conserve the 206 total kinetic energy. It is also possible to mix the two formulations in order 207 to conserve enstrophy on the relative vorticity term and energy on the 208 Coriolis term. 378 total kinetic energy. Indeed, the kinetic energy tendency associated to the 379 vorticity term and averaged over the ocean domain can be transformed as 380 follows: 381 \begin{flalign*} 382 &\int\limits_D - \left( e_3 \, q \;\textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv && \\ 383 & \qquad \qquad {\begin{array}{*{20}l} 384 &\equiv \sum\limits_{i,j,k} \biggl\{ 385 \frac{1} {e_{1u}} \overline { \,q\ \overline{ V }^{\,i+1/2}} ^{\,j} \, u \; b_u 386 - \frac{1} {e_{2v}}\overline { \, q\ \overline{ U }^{\,j+1/2}} ^{\,i} \, v \; b_v \; \biggr\} \\ 387 &\equiv \sum\limits_{i,j,k} \biggl\{ 388 \overline { \,q\ \overline{ V }^{\,i+1/2}}^{\,j} \; U 389 - \overline { \,q\ \overline{ U }^{\,j+1/2}}^{\,i} \; V \; \biggr\} \\ 390 &\equiv \sum\limits_{i,j,k} q \ \biggl\{ \overline{ V }^{\,i+1/2}\; \overline{ U }^{\,j+1/2} 391 - \overline{ U }^{\,j+1/2}\; \overline{ V }^{\,i+1/2} \biggr\} \quad \equiv 0 392 \end{array} } 393 \end{flalign*} 394 In other words, the domain averaged kinetic energy does not change due to the vorticity term. 395 396 397 % ------------------------------------------------------------------------------------------------------------- 398 % Vorticity Term with EEN scheme 399 % ------------------------------------------------------------------------------------------------------------- 400 \subsubsection{Vorticity Term with EEN scheme (\np{ln\_dynvor\_een}=.true.)} 401 \label{Apdx_C_vorEEN} 402 403 With the EEN scheme, the vorticity terms are represented as: 404 \begin{equation} \label{Eq_dynvor_een} 405 \left\{ { \begin{aligned} 406 +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} 407 {^{i+1/2-i_p}_j} \mathbb{Q}^{i_p}_{j_p} \left( e_{1v} e_{3v} \ v \right)^{i+i_p-1/2}_{j+j_p} \\ 408 - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} 409 {^i_{j+1/2-j_p}} \mathbb{Q}^{i_p}_{j_p} \left( e_{2u} e_{3u} \ u \right)^{i+i_p}_{j+j_p-1/2} \\ 410 \end{aligned} } \right. 411 \end{equation} 412 where the indices $i_p$ and $k_p$ take the following value: 413 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 414 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 415 \begin{equation} \label{Q_triads} 416 _i^j \mathbb{Q}^{i_p}_{j_p} 417 = \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) 418 \end{equation} 419 420 This formulation does conserve the total kinetic energy. Indeed, 209 421 \begin{flalign*} 210 422 &\int\limits_D - \textbf{U}_h \cdot \left( \zeta \;\textbf{k} \times \textbf{U}_h \right) \; dv && \\ 211 \equiv& \sum\limits_{i,j,k} \biggl\{ 212 \overline {\left( \frac{\zeta} {e_{3f}} \right) 213 \overline {\left( e_{1v}e_{3v}v \right)}^{\,i+1/2}} ^{\,j} \, e_{2u}e_{3u}u 214 - \overline {\left( \frac{\zeta} {e_{3f}} \right) 215 \overline {\left( e_{2u}e_{3u}u \right)}^{\,j+1/2}} ^{\,i} \, e_{1v}e_{3v}v \; 216 \biggr\} 423 \equiv \sum\limits_{i,j,k} & \biggl\{ 424 \left[ \sum_{\substack{i_p,\,k_p}} 425 {^{i+1/2-i_p}_j}\mathbb{Q}^{i_p}_{j_p} \; V^{i+1/2-i_p}_{j+j_p} \right] U^{i+1/2}_{j} % &&\\ 426 - \left[ \sum_{\substack{i_p,\,k_p}} 427 {^i_{j+1/2-j_p}}\mathbb{Q}^{i_p}_{j_p} \; U^{i+i_p}_{j+1/2-j_p} \right] V^{i}_{j+1/2} \biggr\} && \\ 217 428 \\ 218 \equiv& \sum\limits_{i,j,k} \frac{\zeta} {e_{3f}} 219 \biggl\{ \overline {\left( e_{1v}e_{3v} v \right)}^{\,i+1/2}\; 220 \overline {\left( e_{2u}e_{3u} u \right)}^{\,j+1/2} 221 - \overline {\left( e_{2u}e_{3u} u \right)}^{\,j+1/2}\; 222 \overline {\left( e_{1v}e_{3v} v \right)}^{\,i+1/2} 223 \biggr\} 224 \equiv 0 429 \equiv \sum\limits_{i,j,k} & \sum_{\substack{i_p,\,k_p}} \biggl\{ \ \ 430 {^{i+1/2-i_p}_j}\mathbb{Q}^{i_p}_{j_p} \; V^{i+1/2-i_p}_{j+j_p} \, U^{i+1/2}_{j} % &&\\ 431 - {^i_{j+1/2-j_p}}\mathbb{Q}^{i_p}_{j_p} \; U^{i+i_p}_{j+1/2-j_p} \, V^{i}_{j+1/2} \ \; \biggr\} && \\ 432 % 433 \allowdisplaybreaks 434 \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} 435 % 436 \equiv \sum\limits_{i,j,k} & \biggl\{ \ \ 437 {^{i+1}_j }\mathbb{Q}^{-1/2}_{+1/2} \;V^{i+1}_{j+1/2} \; U^{\,i+1/2}_{j} 438 - {^i_{j}\quad}\mathbb{Q}^{-1/2}_{+1/2} \; U^{i-1/2}_{j} \; V^{\,i}_{j+1/2} && \\ 439 & + {^{i+1}_j }\mathbb{Q}^{-1/2}_{-1/2} \; V^{i+1}_{j-1/2} \; U^{\,i+1/2}_{j} 440 - {^i_{j+1} }\mathbb{Q}^{-1/2}_{-1/2} \; U^{i-1/2}_{j+1} \; V^{\,i}_{j+1/2} \biggr. && \\ 441 & + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2} \; U^{\,i+1/2}_{j} 442 - {^i_{j}\quad}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j} \; V^{\,i}_{j+1/2} \biggr. && \\ 443 & + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{-1/2} \; V^{i}_{j-1/2} \; U^{\,i+1/2}_{j} 444 - {^i_{j+1} }\mathbb{Q}^{+1/2}_{-1/2} \; U^{i+1/2}_{j+1}\; V^{\,i}_{j+1/2} \ \; \biggr\} && \\ 445 % 446 \allowdisplaybreaks 447 \intertext{The summation is done over all $i$ and $j$ indices, it is therefore possible to introduce 448 a shift of $-1$ either in $i$ or $j$ direction in some of the term of the summation (first term of the 449 first and second lines, second term of the second and fourth lines). By doning so, we can regroup 450 all the terms of the summation by triad at a ($i$,$j$) point. In other words, we regroup all the terms 451 in the neighbourhood that contain a triad at the same ($i$,$j$) indices. It becomes: } 452 \allowdisplaybreaks 453 % 454 \equiv \sum\limits_{i,j,k} & \biggl\{ \ \ 455 {^{i}_j}\mathbb{Q}^{-1/2}_{+1/2} \left[ V^{i}_{j+1/2}\, U^{\,i-1/2}_{j} 456 - U^{i-1/2}_{j} \, V^{\,i}_{j+1/2} \right] && \\ 457 & + {^{i}_j}\mathbb{Q}^{-1/2}_{-1/2} \left[ V^{i}_{j-1/2} \, U^{\,i-1/2}_{j} 458 - U^{i-1/2}_{j} \, V^{\,i}_{j-1/2} \right] \biggr. && \\ 459 & + {^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} \left[ V^{i}_{j+1/2} \, U^{\,i+1/2}_{j} 460 - U^{i+1/2}_{j} \, V^{\,i}_{j+1/2} \right] \biggr. && \\ 461 & + {^{i}_j}\mathbb{Q}^{+1/2}_{-1/2} \left[ V^{i}_{j-1/2} \, U^{\,i+1/2}_{j} 462 - U^{i+1/2}_{j-1} \, V^{\,i}_{j-1/2} \right] \ \; \biggr\} \qquad 463 \equiv \ 0 && 225 464 \end{flalign*} 226 465 … … 235 474 balanced by the change of KE due to the horizontal gradient of KE~: 236 475 \begin{equation*} 237 \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv 238 = - \int_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv 476 \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv 477 = - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv 478 + \frac{1}{2} \int_D { \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv } \\ 239 479 \end{equation*} 240 480 Indeed, using successively \eqref{DOM_di_adj} ($i.e.$ the skew symmetry 241 property of the $\delta$ operator) and the incompressibility, then481 property of the $\delta$ operator) and the continuity equation, then 242 482 \eqref{DOM_di_adj} again, then the commutativity of operators 243 483 $\overline {\,\cdot \,}$ and $\delta$, and finally \eqref{DOM_mi_adj} … … 245 485 applied in the horizontal and vertical directions, it becomes: 246 486 \begin{flalign*} 247 &\int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv &&&\\ 248 \equiv& \frac{1}{2} \sum\limits_{i,j,k} 249 \biggl\{ 250 \frac{1} {e_{1u}} \delta_{i+1/2} 251 \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] u\,e_{1u}e_{2u}e_{3u} 252 + \frac{1} {e_{2v}} \delta_{j+1/2} 253 \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] v\,e_{1v}e_{2v}e_{3v} 254 \biggr\} 255 &&& \displaybreak[0] \\ 256 % 257 \equiv& \frac{1}{2} \sum\limits_{i,j,k} 258 \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right)\; 259 \delta_k \left[ e_{1T}\,e_{2T} \,w \right] 260 % 261 \;\; \equiv -\frac{1}{2} \sum\limits_{i,j,k} \delta_{k+1/2} 262 \left[ 263 \overline{ u^2}^{\,i} 264 + \overline{ v^2}^{\,j} 265 \right] \; 266 e_{1v}\,e_{2v}\,w 267 &&& \displaybreak[0]\\ 268 % 269 \equiv &\frac{1} {2} \sum\limits_{i,j,k} 270 \left( \overline {\delta_{k+1/2} \left[ u^2 \right]}^{\,i} 271 + \overline {\delta_{k+1/2} \left[ v^2 \right]}^{\,j} \right) \; e_{1T}\,e_{2T} \,w 272 && \displaybreak[0] \\ 487 & - \int_D \textbf{U}_h \cdot \text{KEG}\;dv 488 = - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv &&&\\ 489 % 490 \equiv & - \sum\limits_{i,j,k} \frac{1}{2} \biggl\{ 491 \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] u \ b_u 492 + \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] v \ b_v \biggr\} &&& \\ 493 % 494 \equiv & + \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right)\; 495 \biggl\{ \delta_{i} \left[ U \right] + \delta_{j} \left[ V \right] \biggr\} &&& \\ 496 \allowdisplaybreaks 497 % 498 \equiv & - \sum\limits_{i,j,k} \frac{1}{2} 499 \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) \; 500 \biggl\{ \frac{b_t}{e_{3t}} \partial_t (e_{3t}) + \delta_k \left[ W \right] \biggr\} &&&\\ 501 \allowdisplaybreaks 502 % 503 \equiv & + \sum\limits_{i,j,k} \frac{1}{2} \delta_{k+1/2} \left[ \overline{ u^2}^{\,i} + \overline{ v^2}^{\,j} \right] \; W 504 - \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) \;\partial_t b_t &&& \\ 505 \allowdisplaybreaks 506 % 507 \equiv & + \sum\limits_{i,j,k} \frac{1} {2} \left( \overline{\delta_{k+1/2} \left[ u^2 \right]}^{\,i} 508 + \overline{\delta_{k+1/2} \left[ v^2 \right]}^{\,j} \right) \; W 509 - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t \overline{b_t}^{\,{i+1/2}} 510 + \frac{v^2}{2}\,\partial_t \overline{b_t}^{\,{j+1/2}} \right) &&& \\ 511 \allowdisplaybreaks 512 \intertext{Assuming that $b_u= \overline{b_t}^{\,i+1/2}$ and $b_v= \overline{b_t}^{\,j+1/2}$, or at least that the time 513 derivative of these two equations is satisfied, it becomes:} 514 % 515 \equiv & \sum\limits_{i,j,k} \frac{1} {2} 516 \biggl\{ \; \overline{W}^{\,i+1/2}\;\delta_{k+1/2} \left[ u^2 \right] 517 + \overline{W}^{\,j+1/2}\;\delta_{k+1/2} \left[ v^2 \right] \; \biggr\} 518 - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t b_u 519 + \frac{v^2}{2}\,\partial_t b_v \right) &&& \\ 520 \allowdisplaybreaks 273 521 % 274 \equiv &\frac{1} {2} \sum\limits_{i,j,k} 275 \biggl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\;2 276 \overline {u}^{\,k+1/2}\; \delta_{k+1/2} \left[ u \right] %&&& \\ 277 + \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2}\;2 \overline {v}^{\,k+1/2}\; \delta_{k+1/2} \left[ v \right] \; 278 \biggr\} 279 &&\displaybreak[0] \\ 280 % 281 \equiv& -\sum\limits_{i,j,k} 282 \biggl\{ 283 \quad \frac{1} {b_u } \; 284 \overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\,\delta_{k+1/2} 285 \left[ u \right] 286 \Bigr\} }^{\,k} \;u\;e_{1u}\,e_{2u}\,e_{3u} 287 && \\ 288 &\qquad \quad\; + \frac{1} {b_v } \; 289 \overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2} \delta_{k+1/2} 290 \left[ v \right] 291 \Bigr\} }^{\,k} \;v\;e_{1v}\,e_{2v}\,e_{3v} \; 292 \biggr\} 293 && \\ 294 \equiv& -\int\limits_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv &&&\\ 295 \end{flalign*} 296 297 The main point here is that the satisfaction of this property links the choice of 522 \equiv & \sum\limits_{i,j,k} 523 \biggl\{ \; \overline{W}^{\,i+1/2}\; \overline {u}^{\,k+1/2}\; \delta_{k+1/2}[ u ] 524 + \overline{W}^{\,j+1/2}\; \overline {v}^{\,k+1/2}\; \delta_{k+1/2}[ v ] \; \biggr\} 525 - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t b_u 526 + \frac{v^2}{2}\,\partial_t b_v \right) &&& \\ 527 % 528 \allowdisplaybreaks 529 \equiv & \sum\limits_{i,j,k} 530 \biggl\{ \; \frac{1} {b_u } \; \overline { \overline{W}^{\,i+1/2}\,\delta_{k+1/2} \left[ u \right] }^{\,k} \;u\;b_u 531 + \frac{1} {b_v } \; \overline { \overline{W}^{\,j+1/2} \delta_{k+1/2} \left[ v \right] }^{\,k} \;v\;b_v \; \biggr\} 532 - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t b_u 533 + \frac{v^2}{2}\,\partial_t b_v \right) &&& \\ 534 % 535 \intertext{The first term provides the discrete expression for the vertical advection of momentum (ZAD), 536 while the second term corresponds exactly to \eqref{KE+PE_vect_discrete}, therefore:} 537 \equiv& \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 538 + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t (e_3) \;dv } &&&\\ 539 \equiv& \int\limits_D \textbf{U}_h \cdot w \partial_k \textbf{U}_h \;dv 540 + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t (e_3) \;dv } &&&\\ 541 \end{flalign*} 542 543 There is two main points here. First, the satisfaction of this property links the choice of 298 544 the discrete formulation of the vertical advection and of the horizontal gradient 299 545 of KE. Choosing one imposes the other. For example KE can also be discretized … … 301 547 expression for the vertical advection: 302 548 \begin{equation*} 303 \frac{1} {e_3 }\; w\; \frac{\partial \textbf{U}_h } {\partial k}549 \frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h 304 550 \equiv \left( {{\begin{array} {*{20}c} 305 \frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \; 306 \overline{\overline {e_{1T}\,e_{2T} \,w\;\delta_{k+1/2} 551 \frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \; \overline{\overline {e_{1t}\,e_{2t} \,\omega\;\delta_{k+1/2} 307 552 \left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k} \hfill \\ 308 \frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \; 309 \overline{\overline {e_{1T}\,e_{2T} \,w\;\delta_{k+1/2} 553 \frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \; \overline{\overline {e_{1t}\,e_{2t} \,\omega \;\delta_{k+1/2} 310 554 \left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill \\ 311 555 \end{array}} } \right) … … 315 559 This is the reason why it has not been chosen. 316 560 561 Second, as soon as the chosen $s$-coordinate depends on time, an extra constraint 562 arises on the time derivative of the volume at $u$- and $v$-points: 563 \begin{flalign*} 564 e_{1u}\,e_{2u}\,\partial_t (e_{3u}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,i+1/2} \\ 565 e_{1v}\,e_{2v}\,\partial_t (e_{3v}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,j+1/2} 566 \end{flalign*} 567 which is (over-)satified by defining the vertical scale factor as follows: 568 \begin{flalign} \label{e3u-e3v} 569 e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2} \\ 570 e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2} 571 \end{flalign} 572 573 Blah blah required on the the step representation of bottom topography..... 574 575 576 % ------------------------------------------------------------------------------------------------------------- 577 % Pressure Gradient Term 578 % ------------------------------------------------------------------------------------------------------------- 579 \subsection{Pressure Gradient Term} 580 \label{Apdx_C.1.4} 581 582 \gmcomment{ 583 A pressure gradient has no contribution to the evolution of the vorticity as the 584 curl of a gradient is zero. In the $z$-coordinate, this property is satisfied locally 585 on a C-grid with 2nd order finite differences (property \eqref{Eq_DOM_curl_grad}). 586 } 587 588 When the equation of state is linear ($i.e.$ when an advection-diffusion equation 589 for density can be derived from those of temperature and salinity) the change of 590 KE due to the work of pressure forces is balanced by the change of potential 591 energy due to buoyancy forces: 592 \begin{equation*} 593 - \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 594 = - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right) \,g\,z \;dv 595 + \int_D g\, \rho \; \partial_t (z) \;dv 596 \end{equation*} 597 598 This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates. 599 Indeed, defining the depth of a $T$-point, $z_t$, as the sum of the vertical scale 600 factors at $w$-points starting from the surface, the work of pressure forces can be 601 written as: 602 \begin{flalign*} 603 &- \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 604 \equiv \sum\limits_{i,j,k} \biggl\{ \; - \frac{1} {e_{1u}} \Bigl( 605 \delta_{i+1/2} [p_t] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} [z_t] \Bigr) \; u\;b_u 606 && \\ & \qquad \qquad \qquad \qquad \qquad \quad \ \, 607 - \frac{1} {e_{2v}} \Bigl( 608 \delta_{j+1/2} [p_t] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} [z_t] \Bigr) \; v\;b_v \; \biggr\} && \\ 609 % 610 \allowdisplaybreaks 611 \intertext{Using successively \eqref{DOM_di_adj}, $i.e.$ the skew symmetry property of 612 the $\delta$ operator, \eqref{Eq_wzv}, the continuity equation, \eqref{Eq_dynhpg_sco}, 613 the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $, 614 which comes from the definition of $z_t$, it becomes: } 615 \allowdisplaybreaks 616 % 617 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 618 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 619 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 620 +\Bigl( \delta_i[U] + \delta_j [V] \Bigr)\;\frac{p_t}{g} \biggr\} &&\\ 621 % 622 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 623 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 624 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 625 - \left( \frac{b_t}{e_{3t}} \partial_t (e_{3t}) + \delta_k \left[ W \right] \right) \frac{p_t}{g} \biggr\} &&&\\ 626 % 627 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 628 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 629 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 630 + \frac{W}{g}\;\delta_{k+1/2} [p_t] 631 - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\ 632 % 633 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 634 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 635 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 636 - W\;e_{3w} \overline \rho^{\,k+1/2} 637 - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\ 638 % 639 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 640 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 641 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 642 + W\; \overline \rho^{\,k+1/2}\;\delta_{k+1/2} [z_t] 643 - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\ 644 % 645 \allowdisplaybreaks 646 % 647 \equiv& - \sum\limits_{i,j,k} g \; z_t \biggl\{ 648 \delta_i \left[ U\; \overline \rho^{\,i+1/2} \right] 649 + \delta_j \left[ V\; \overline \rho^{\,j+1/2} \right] 650 + \delta_k \left[ W\; \overline \rho^{\,k+1/2} \right] \biggr\} 651 - \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ 652 % 653 \equiv& + \sum\limits_{i,j,k} g \; z_t \biggl\{ \partial_t ( e_{3t} \,\rho) \biggr\} \; b_t 654 - \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ 655 % 656 \end{flalign*} 657 The first term is exactly the first term of the right-hand-side of \eqref{KE+PE_vect_discrete}. 658 It remains to demonstrate that the last term, which is obviously a discrete analogue of 659 $\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to the last term of \eqref{KE+PE_vect_discrete}. 660 In other words, the following property must be satisfied: 661 \begin{flalign*} 662 \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} 663 \equiv \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} 664 \end{flalign*} 665 666 Let introduce $p_w$ the pressure at $w$-point such that $\delta_k [p_w] = - \rho \,g\,e_{3t}$. 667 The right-hand-side of the above equation can be transformed as follows: 668 669 \begin{flalign*} 670 \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} 671 &\equiv - \sum\limits_{i,j,k} \biggl\{ \delta_k [p_w]\,\partial_t (z_t) \,e_{1t}\,e_{2t} \biggr\} &&&\\ 672 % 673 &\equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \delta_{k+1/2} [\partial_t (z_t)] \,e_{1t}\,e_{2t} \biggr\} 674 \equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (e_{3w}) \,e_{1t}\,e_{2t} \biggr\} &&&\\ 675 &\equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (b_w) \biggr\} 676 % 677 % & \equiv \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w \right) \biggr\} &&&\\ 678 % & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t \left( \delta_k [z_t] \right) e_{1w}\,e_{2w} \biggr\} &&&\\ 679 % & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t b_w \biggr\} 680 \end{flalign*} 681 therefore, the balance to be satisfied is: 682 \begin{flalign*} 683 \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t (b_t) \biggr\} \equiv \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (b_w) \biggr\} 684 \end{flalign*} 685 which is a purely vertical balance: 686 \begin{flalign*} 687 \sum\limits_{k} \biggl\{ p_t\;\partial_t (e_{3t}) \biggr\} \equiv \sum\limits_{k} \biggl\{ p_w\, \partial_t (e_{3w}) \biggr\} 688 \end{flalign*} 689 Defining $p_w = \overline{p_t}^{\,k+1/2}$ 690 691 %gm comment 692 \gmcomment{ 693 \begin{flalign*} 694 \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ 695 % 696 & \equiv \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w \biggr\} &&&\\ 697 & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t \left( \delta_{k+1/2} [z_t] \right) e_{1w}\,e_{2w} \biggr\} &&&\\ 698 & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t b_w \biggr\} 699 \end{flalign*} 700 701 702 \begin{flalign*} 703 \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv 704 \equiv& \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} p \biggr\} \; b_t &&&\\ 705 \equiv& \sum\limits_{i,j,k} \biggl\{ \biggr\} \; b_t &&&\\ 706 \end{flalign*} 707 708 % 709 \begin{flalign*} 710 \equiv& - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 711 + \int\limits_D g\, \rho \; \frac{\partial z}{\partial t} \;dv &&& \\ 712 \end{flalign*} 713 % 714 } 715 %end gm comment 716 717 718 Note that this property strongly constrains the discrete expression of both 719 the depth of $T-$points and of the term added to the pressure gradient in the 720 $s$-coordinate. Nevertheless, it is almost never satisfied since a linear equation 721 of state is rarely used. 722 723 724 725 726 727 728 729 % ================================================================ 730 % Discrete Total energy Conservation : flux form 731 % ================================================================ 732 \section{Discrete total energy conservation : flux form} 733 \label{Apdx_C.1} 734 735 % ------------------------------------------------------------------------------------------------------------- 736 % Total energy conservation 737 % ------------------------------------------------------------------------------------------------------------- 738 \subsection{Total energy conservation} 739 \label{Apdx_C_KE+PE} 740 741 The discrete form of the total energy conservation, \eqref{Eq_Tot_Energy}, is given by: 742 \begin{flalign*} 743 \partial_t \left( \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v + \rho \, g \, z_t \,b_t \biggr\} \right) &=0 \\ 744 \end{flalign*} 745 which in flux form, it leads to: 746 \begin{flalign*} 747 \sum\limits_{i,j,k} \biggl\{ \frac{u }{e_{3u}}\,\frac{\partial (e_{3u}u)}{\partial t} \,b_u 748 + \frac{v }{e_{3v}}\,\frac{\partial (e_{3v}v)}{\partial t} \,b_v \biggr\} 749 & - \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{e_{3u}}\frac{\partial e_{3u} }{\partial t} \,b_u 750 + \frac{v^2}{e_{3v}}\frac{\partial e_{3v} }{\partial t} \,b_v \biggr\} \\ 751 &= - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_3t}\frac{\partial e_{3t} \rho}{\partial t} \, g \, z_t \,b_t \biggr\} 752 - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\frac{\partial z_t}{\partial t} \,b_t \biggr\} \\ 753 \end{flalign*} 754 755 Substituting the discrete expression of the time derivative of the velocity either in vector invariant or in flux form, 756 leads to the discrete equivalent of the 757 758 317 759 % ------------------------------------------------------------------------------------------------------------- 318 760 % Coriolis and advection terms: flux form … … 331 773 Coriolis parameter is discretised at an f-point. It is given by: 332 774 \begin{equation*} 333 f+\frac{1} {e_1 e_2 } 334 \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\; 775 f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\; 335 776 \equiv \; 336 f+\frac{1} {e_{1f}\,e_{2f}} 337 \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] 338 -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u} \right] \right) 777 f+\frac{1} {e_{1f}\,e_{2f}} \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] 778 -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u} \right] \right) 339 779 \end{equation*} 340 780 341 The ENEscheme is then applied to obtain the vorticity term in flux form.781 Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. 342 782 It therefore conserves the total KE. The derivation is the same as for the 343 783 vorticity term in the vector invariant form (\S\ref{Apdx_C_vor}). … … 355 795 the horizontal kinetic energy, that is : 356 796 357 \begin{equation} \label{Apdx_C_I.3.10} 358 \int_D \textbf{U}_h \cdot 359 \left( {{\begin{array} {*{20}c} 797 \begin{equation} \label{Apdx_C_ADV_KE_flux} 798 - \int_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} 360 799 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 361 \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ 362 \end{array}} } \right)\;dv=\;0800 \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ \end{array}} } \right) \;dv 801 - \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv } =\;0 363 802 \end{equation} 364 803 365 Let us demonstrate this property for the first term of the scalar product 366 ($i.e.$ considering just the the terms associated with the i-component of 367 the advection): 368 \begin{flalign*} 369 &\int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv &&&\\ 370 % 371 \equiv& \sum\limits_{i,j,k} 372 \biggl\{ \frac{1} {e_{1u}\, e_{2u}\,e_{3u}} \biggl( 373 \delta_{i+1/2} \left[ \overline {e_{2u}\,e_{3u}\,u}^{\,i} \;\overline u^{\,i} \right] 374 + \delta_j \left[ \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] 375 &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad 376 + \delta_k \left[ \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\;\overline u^{\,k+1/2} \right] 377 \biggr) \; \biggr\} \, e_{1u}\,e_{2u}\,e_{3u} \;u 378 &&& \displaybreak[0] \\ 379 % 380 \equiv& \sum\limits_{i,j,k} 804 Let us first consider the first term of the scalar product ($i.e.$ just the the terms 805 associated with the i-component of the advection) : 806 \begin{flalign*} 807 & - \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv \\ 808 % 809 \equiv& - \sum\limits_{i,j,k} \biggl\{ \frac{1} {b_u} \biggl( 810 \delta_{i+1/2} \left[ \overline {U}^{\,i} \;\overline u^{\,i} \right] 811 + \delta_j \left[ \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] 812 + \delta_k \left[ \overline {W}^{\,i+1/2}\;\overline u^{\,k+1/2} \right] \biggr) \; \biggr\} \, b_u \;u &&& \\ 813 % 814 \equiv& - \sum\limits_{i,j,k} 381 815 \biggl\{ 382 \delta_{i+1/2} \left[ \overline {e_{2u}\,e_{3u}\,u}^{\,i}\; \overline u^{\,i} \right] 383 + \delta_j \left[ \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] 384 &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 385 + \delta_k \left[ \overline {e_{1w}\,e_{2w}\,w}^{\,i+12}\;\overline u^{\,k+1/2} \right] 386 \; \biggr\} \; u &&& \displaybreak[0] \\ 387 % 388 \equiv& - \sum\limits_{i,j,k} 816 \delta_{i+1/2} \left[ \overline {U}^{\,i}\; \overline u^{\,i} \right] 817 + \delta_j \left[ \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] 818 + \delta_k \left[ \overline {W}^{\,i+12}\;\overline u^{\,k+1/2} \right] 819 \; \biggr\} \; u \\ 820 % 821 \equiv& + \sum\limits_{i,j,k} 389 822 \biggl\{ 390 \overline {e_{2u}\,e_{3u}\,u}^{\,i}\; \overline u^{\,i} \delta_i 391 \left[ u \right] 392 + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\; \overline u^{\,j+1/2} \delta_{j+1/2} 393 \left[ u \right] 394 &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 395 + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\; \overline u^{\,k+1/2} \delta_{k+1/2} \left[ u \right] \biggr\} && \displaybreak[0] \\ 396 % 397 \equiv& - \sum\limits_{i,j,k} 398 \biggl\{ 399 \overline {e_{2u}\,e_{3u}\,u}^{\,i} \delta_i \left[ u^2 \right] 400 + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2} \delta_{j+/2} \left[ u^2 \right] 401 + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2} \delta_{k+1/2} \left[ u^2 \right] 402 \biggr\} 403 && \displaybreak[0] \\ 404 % 405 \equiv& \sum\limits_{i,j,k} 406 \bigg\{ 407 e_{2u}\,e_{3u}\,u\; \delta_{i+1/2} \left[ \overline {u^2}^{\,i} \right] 408 + e_{1u}\,e_{3u}\,v\; \delta_{j+1/2} \; \left[ \overline {u^2}^{\,i} \right] 409 + e_{1w}\,e_{2w}\,w\; \delta_{k+1/2} \left[ \overline {u^2}^{\,i} \right] 410 \biggr\} 411 && \displaybreak[0] \\ 412 % 413 \equiv& \sum\limits_{i,j,k} 414 \overline {u^2}^{\,i} 415 \biggl\{ 416 \delta_{i+1/2} \left[ e_{2u}\,e_{3u}\,u \right] 417 + \delta_{j+1/2} \left[ e_{1u}\,e_{3u}\,v \right] 418 + \delta_{k+1/2} \left[ e_{1w}\,e_{2w}\,w \right] 419 \biggr\} \;\; \equiv 0 420 &&& \\ 421 \end{flalign*} 823 \overline {U}^{\,i}\; \overline u^{\,i} \delta_i \left[ u \right] 824 + \overline {V}^{\,i+1/2}\; \overline u^{\,j+1/2} \delta_{j+1/2} \left[ u \right] 825 + \overline {W}^{\,i+1/2}\; \overline u^{\,k+1/2} \delta_{k+1/2} \left[ u \right] \biggr\} && \\ 826 % 827 \equiv& + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ 828 \overline{U}^{\,i} \delta_i \left[ u^2 \right] 829 + \overline{V}^{\,i+1/2} \delta_{j+/2} \left[ u^2 \right] 830 + \overline{W}^{\,i+1/2} \delta_{k+1/2} \left[ u^2 \right] \biggr\} && \\ 831 % 832 \equiv& - \sum\limits_{i,j,k} \frac{1}{2} \bigg\{ 833 U \; \delta_{i+1/2} \left[ \overline {u^2}^{\,i} \right] 834 + V \; \delta_{j+1/2} \left[ \overline {u^2}^{\,i} \right] 835 + W \; \delta_{k+1/2} \left[ \overline {u^2}^{\,i} \right] \biggr\} &&& \\ 836 % 837 \equiv& - \sum\limits_{i,j,k} \frac{1}{2} \overline {u^2}^{\,i} \biggl\{ 838 \delta_{i+1/2} \left[ U \right] 839 + \delta_{j+1/2} \left[ V \right] 840 + \delta_{k+1/2} \left[ W \right] \biggr\} &&& \\ 841 % 842 \equiv& + \sum\limits_{i,j,k} \frac{1}{2} \overline {u^2}^{\,i} 843 \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} &&& \\ 844 \end{flalign*} 845 Applying similar manipulation applied to the second term of the scalar product 846 leads to : 847 \begin{equation*} 848 - \int_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} 849 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 850 \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ \end{array}} } \right) \;dv 851 \equiv + \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) 852 \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} 853 \end{equation*} 854 which is the discrete form of 855 $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv $. 856 \eqref{Apdx_C_ADV_KE_flux} is thus satisfied. 857 422 858 423 859 When the UBS scheme is used to evaluate the flux form momentum advection, … … 426 862 ($i.e.$ the scheme is diffusive). 427 863 428 % ------------------------------------------------------------------------------------------------------------- 429 % Hydrostatic Pressure Gradient Term 430 % ------------------------------------------------------------------------------------------------------------- 431 \subsection{Hydrostatic Pressure Gradient Term} 432 \label{Apdx_C.1.4} 433 434 435 A pressure gradient has no contribution to the evolution of the vorticity as the 436 curl of a gradient is zero. In the $z$-coordinate, this property is satisfied locally 437 on a C-grid with 2nd order finite differences (property \eqref{Eq_DOM_curl_grad}). 438 When the equation of state is linear ($i.e.$ when an advection-diffusion equation 439 for density can be derived from those of temperature and salinity) the change of 440 KE due to the work of pressure forces is balanced by the change of potential 441 energy due to buoyancy forces: 442 \begin{equation*} 443 \int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv 444 = \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 445 \end{equation*} 446 447 This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates. 448 Indeed, defining the depth of a $T$-point, $z_T$, as the sum of the vertical scale 449 factors at $w$-points starting from the surface, the work of pressure forces can be 450 written as: 451 \begin{flalign*} 452 &\int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv &&& \\ 453 \equiv& \sum\limits_{i,j,k} \biggl\{ \; - \frac{1} {\rho_o e_{1u}} \Bigl( 454 \delta_{i+1/2} \left[ p^h \right] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] 455 \Bigr) \; u\;e_{1u}\,e_{2u}\,e_{3u} 456 && \\ & \qquad \qquad 457 - \frac{1} {\rho_o e_{2v}} \Bigl( 458 \delta_{j+1/2} \left[ p^h \right] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} \left[ z_T \right] 459 \Bigr) \; v\;e_{1v}\,e_{2v}\,e_{3v} \; 460 \biggr\} && \\ 461 \end{flalign*} 462 463 Using \eqref{DOM_di_adj}, $i.e.$ the skew symmetry property of the $\delta$ 464 operator, \eqref{Eq_wzv}, the continuity equation), and \eqref{Eq_dynhpg_sco}, 465 the hydrostatic equation in the $s$-coordinate, it becomes: 864 865 866 867 868 869 870 871 872 873 % ================================================================ 874 % Discrete Enstrophy Conservation 875 % ================================================================ 876 \section{Discrete enstrophy conservation} 877 \label{Apdx_C.1} 878 879 880 % ------------------------------------------------------------------------------------------------------------- 881 % Vorticity Term with ENS scheme 882 % ------------------------------------------------------------------------------------------------------------- 883 \subsubsection{Vorticity Term with ENS scheme (\np{ln\_dynvor\_ens}=.true.)} 884 \label{Apdx_C_vorENS} 885 886 In the ENS scheme, the vorticity term is descretized as follows: 887 \begin{equation} \label{Eq_dynvor_ens} 888 \left\{ \begin{aligned} 889 +\frac{1}{e_{1u}} & \overline{q}^{\,i} & {\overline{ \overline{\left( e_{1v}\,e_{3v}\; v \right) } } }^{\,i, j+1/2} \\ 890 - \frac{1}{e_{2v}} & \overline{q}^{\,j} & {\overline{ \overline{\left( e_{2u}\,e_{3u}\; u \right) } } }^{\,i+1/2, j} 891 \end{aligned} \right. 892 \end{equation} 893 894 The scheme does not allow but the conservation of the total kinetic energy but the conservation 895 of $q^2$, the potential enstrophy for a horizontally non-divergent flow ($i.e.$ when $\chi$=$0$). 896 Indeed, using the symmetry or skew symmetry properties of the operators (Eqs \eqref{DOM_mi_adj} 897 and \eqref{DOM_di_adj}), it can be shown that: 898 \begin{equation} \label{Apdx_C_1.1} 899 \int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0 900 \end{equation} 901 where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. Indeed, using 902 \eqref{Eq_dynvor_ens}, the discrete form of the right hand side of \eqref{Apdx_C_1.1} 903 can be transformed as follow: 466 904 \begin{flalign*} 467 \equiv& \frac{1} {\rho_o} \sum\limits_{i,j,k} \biggl\{ 468 e_{2u}\,e_{3u} \;u\;g\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2}[ z_T] 469 + e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2}\;\delta_{j+1/2}[ z_T] 470 && \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, 471 +\Bigl( \delta_i[ e_{2u}\,e_{3u}\,u] + \delta_j [ e_{1v}\,e_{3v}\,v] \Bigr)\;p^h \biggr\} &&\\ 472 % 473 \equiv& \frac{1} {\rho_o } \sum\limits_{i,j,k} 474 \biggl\{ 475 e_{2u}\,e_{3u} \;u\;g\; \overline \rho^{\,i+1/2} \delta_{i+1/2} \left[ z_T \right] 476 + e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2} \delta_{j+1/2} \left[ z_T \right] 477 &&&\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, 478 - \delta_k \left[ e_{1w} e_{2w}\,w \right]\;p^h \biggr\} &&&\\ 479 % 480 \equiv& \frac{1} {\rho_o } \sum\limits_{i,j,k} 481 \biggl\{ 482 e_{2u}\,e_{3u} \;u\;g\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] 483 + e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2} \;\delta_{j+1/2} \left[ z_T \right] 484 &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, 485 + e_{1w} e_{2w} \;w\;\delta_{k+1/2} \left[ p_h \right] 486 \biggr\} &&&\\ 487 % 488 \equiv& \frac{g} {\rho_o} \sum\limits_{i,j,k} 489 \biggl\{ 490 e_{2u}\,e_{3u} \;u\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] 491 + e_{1v}\,e_{3v} \;v\; \overline \rho^{\,j+1/2}\;\delta_{j+1/2} \left[ z_T \right] 492 &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, 493 - e_{1w} e_{2w} \;w\;e_{3w} \overline \rho^{\,k+1/2} 494 \biggr\} &&&\\ 495 \end{flalign*} 496 noting that by definition of $z_T$, $\delta_{k+1/2} \left[ z_T \right] \equiv - e_{3w} $, thus: 497 \begin{multline*} 498 \equiv \frac{g} {\rho_o} \sum\limits_{i,j,k} 499 \biggl\{ 500 e_{2u}\,e_{3u} \;u\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] 501 + e_{1v}\,e_{3v} \;v\; \overline \rho^{\,j+1/2} \delta_{j+1/2} \left[ z_T \right] 502 \biggr. \\ 503 \shoveright{ 504 \biggl. 505 + e_{1w} e_{2w} \;w\; \overline \rho^{\,k+1/2}\;\delta_{k+1/2} \left[ z_T \right] 506 \biggr\} } \\ 507 \end{multline*} 508 Using \eqref{DOM_di_adj}, it becomes: 509 \begin{flalign*} 510 \equiv& - \frac{g} {\rho_o} \sum\limits_{i,j,k} z_T 511 \biggl\{ 512 \delta_i \left[ e_{2u}\,e_{3u}\,u\; \overline \rho^{\,i+1/2} \right] 513 + \delta_j \left[ e_{1v}\,e_{3v}\,v\; \overline \rho^{\,j+1/2} \right] 514 + \delta_k \left[ e_{1w} e_{2w}\,w\; \overline \rho^{\,k+1/2} \right] 905 &\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 906 \left( e_3 \, q \; \textbf{k} \times \textbf{U}_h \right)\; dv \\ 907 % 908 & \qquad {\begin{array}{*{20}l} 909 &\equiv \sum\limits_{i,j,k} 910 q \ \left\{ \delta_{i+1/2} \left[ - \,\overline {q}^{\,i}\; \overline{\overline U }^{\,i,j+1/ 2} \right] 911 - \delta_{j+1/2} \left[ \overline {q}^{\,j}\; \overline{\overline V }^{\,i+1/2, j} \right] \right\} \\ 912 % 913 &\equiv \sum\limits_{i,j,k} 914 \left\{ \delta_i [q] \; \overline{q}^{\,i} \; \overline{ \overline U }^{\,i,j+1/2} 915 + \delta_j [q] \; \overline{q}^{\,j} \; \overline{\overline V }^{\,i+1/2,j} \right\} && \\ 916 % 917 &\equiv \,\frac{1} {2} \sum\limits_{i,j,k} 918 \left\{ \delta_i \left[ q^2 \right] \; \overline{\overline U }^{\,i,j+1/2} 919 + \delta_j \left[ q^2 \right] \; \overline{\overline V }^{\,i+1/2,j} \right\} && \\ 920 % 921 &\equiv - \frac{1} {2} \sum\limits_{i,j,k} q^2 \; 922 \left\{ \delta_{i+1/2} \left[ \overline{\overline{ U }}^{\,i,j+1/2} \right] 923 + \delta_{j+1/2} \left[ \overline{\overline{ V }}^{\,i+1/2,j} \right] \right\} && \\ 924 \end{array} } 925 % 926 \allowdisplaybreaks 927 \intertext{ Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2} 928 \left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$, 929 and introducing the horizontal divergence $\chi $, it becomes: } 930 \allowdisplaybreaks 931 % 932 & \qquad {\begin{array}{*{20}l} 933 &\equiv \sum\limits_{i,j,k} - \frac{1} {2} q^2 \; \overline{\overline{ e_{1t}\,e_{2t}\,e_{3t}^{}\, \chi}}^{\,i+1/2,j+1/2} 934 \quad \equiv 0 && 935 \end{array} } 936 \end{flalign*} 937 The later equality is obtain only when the flow is horizontally non-divergent, $i.e.$ $\chi$=$0$. 938 939 940 % ------------------------------------------------------------------------------------------------------------- 941 % Vorticity Term with EEN scheme 942 % ------------------------------------------------------------------------------------------------------------- 943 \subsubsection{Vorticity Term with EEN scheme (\np{ln\_dynvor\_een}=.true.)} 944 \label{Apdx_C_vorEEN} 945 946 With the EEN scheme, the vorticity terms are represented as: 947 \begin{equation} \label{Eq_dynvor_een} 948 \left\{ { \begin{aligned} 949 +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} 950 {^{i+1/2-i_p}_j} \mathbb{Q}^{i_p}_{j_p} \left( e_{1v} e_{3v} \ v \right)^{i+i_p-1/2}_{j+j_p} \\ 951 - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} 952 {^i_{j+1/2-j_p}} \mathbb{Q}^{i_p}_{j_p} \left( e_{2u} e_{3u} \ u \right)^{i+i_p}_{j+j_p-1/2} \\ 953 \end{aligned} } \right. 954 \end{equation} 955 where the indices $i_p$ and $k_p$ take the following value: 956 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 957 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 958 \begin{equation} \label{Q_triads} 959 _i^j \mathbb{Q}^{i_p}_{j_p} 960 = \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) 961 \end{equation} 962 963 964 This formulation does conserve the potential enstrophy for a horizontally non-divergent flow ($i.e.$ $\chi=0$). 965 966 Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $, 967 similar manipulation can be done for the 3 others. The discrete form of the right hand 968 side of \eqref{Apdx_C_1.1} applied to this triad only can be transformed as follow: 969 970 \begin{flalign*} 971 &\int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \\ 972 % 973 \equiv& \sum\limits_{i,j,k} 974 {q} \ \biggl\{ \;\; 975 \delta_{i+1/2} \left[ -\, {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}} \right] 976 - \delta_{j+1/2} \left[ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \right] 977 \;\;\biggr\} && \\ 978 % 979 \equiv& \sum\limits_{i,j,k} 980 \biggl\{ \delta_i [q] \ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}} 981 + \delta_j [q] \ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \biggr\} 982 && \\ 983 % 984 ... & &&\\ 985 &Demonstation \ to \ be \ done... &&\\ 986 ... & &&\\ 987 % 988 \equiv& \frac{1} {2} \sum\limits_{i,j,k} 989 \biggl\{ \delta_i \Bigl[ \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \Bigr]\; 990 \overline{\overline {U}}^{\,i,j+1/2} 991 + \delta_j \Bigl[ \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \Bigr]\; 992 \overline{\overline {V}}^{\,i+1/2,j} 515 993 \biggr\} 516 &&& \\ 517 % 518 \equiv& -\int_D \nabla \cdot \left( \rho \, \textbf{U} \right)\;g\;z\;\;dv &&& \\ 519 \end{flalign*} 520 521 Note that this property strongly constrains the discrete expression of both 522 the depth of $T-$points and of the term added to the pressure gradient in the 523 $s$-coordinate. Nevertheless, it is almost never satisfied since a linear equation 524 of state is rarely used. 525 526 % ------------------------------------------------------------------------------------------------------------- 527 % Surface Pressure Gradient Term 528 % ------------------------------------------------------------------------------------------------------------- 529 \subsection{Surface Pressure Gradient Term} 530 \label{Apdx_C.1.5} 531 532 533 The surface pressure gradient has no contribution to the evolution of the vorticity. 534 This property is trivially satisfied locally since the equation verified by $\psi$ has 535 been derived from the discrete formulation of the momentum equation and of the curl. 536 But it has to be noted that since the elliptic equation satisfied by $\psi$ is solved 537 numerically by an iterative solver (preconditioned conjugate gradient or successive 538 over relaxation), the property is only satisfied at the precision requested for the 539 solver used. 540 541 With the rigid-lid approximation, the change of KE due to the work of surface 542 pressure forces is exactly zero. This is satisfied in discrete form, at the precision 543 requested for the elliptic solver used to solve this equation. This can be 544 demonstrated as follows: 545 \begin{flalign*} 546 \int\limits_D - \frac{1} {\rho_o} \nabla_h \left( p_s \right) \cdot \textbf{U}_h \;dv% &&& \\ 547 % 548 &\equiv \sum\limits_{i,j,k} \biggl\{ \; 549 \left( - M_u - \frac{1} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] \right)\; 550 u\;e_{1u}\,e_{2u}\,e_{3u} 551 &&&\\& \qquad \;\;\, 552 + \left( - M_v + \frac{1} {H_v \,e_{1v}} \delta_i \left[ \partial_t \psi \right] \right)\; 553 v\;e_{1v}\,e_{2v}\,e_{3v} \; \biggr\} 554 &&&\\ 555 \\ 556 % 557 &\equiv \sum\limits_{i,j} \Biggl\{ \; 558 \biggl( - M_u - \frac{1} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] \biggr) 559 \biggl( \sum\limits_k u\;e_{3u} \biggr)\; e_{1u}\,e_{2u} 560 &&&\\& \qquad \;\;\, 561 + \biggl( - M_v + \frac{1} {H_v \,e_{1v}} \delta_i \left[ \partial_t \psi \right] \biggr) 562 \biggl( \sum\limits_k v\;e_{3v} \biggr)\; e_{1v}\,e_{2v} \; \Biggr\} 563 && \\ 564 % 565 \intertext{using the relation between \textit{$\psi $} and the vertical sum of the velocity, it becomes:} 566 % 567 &\equiv \sum\limits_{i,j} 568 \biggl\{ \; 569 \left( \;\;\, 570 M_u + \frac{1} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] 571 \right)\; 572 e_{1u} \,\delta_j 573 \left[ \partial_t \psi \right] 574 && \\ & \qquad \;\;\, 575 + \left( 576 - M_v + \frac{1} {H_v \,e_{1v}} \delta_i \left[ \partial_t \psi \right] 577 \right)\; 578 e_{2v} \,\delta_i \left[ \partial_t \psi \right] \; 579 \biggr\} 580 && \\ 581 % 582 \intertext{applying the adjoint of the $\delta$ operator, it is now:} 583 % 584 &\equiv \sum\limits_{i,j} - \partial_t \psi \; 585 \biggl\{ \; 586 \delta_{j+1/2} \left[ e_{1u} M_u \right] 587 - \delta_{i+1/2} \left[ e_{1v} M_v \right] 588 && \\ & \qquad \;\;\, 589 + \delta_{i+1/2} 590 \left[ \frac{e_{2v}} {H_v \,e_{2v}} \delta_i \left[ \partial_t \psi \right] 591 \right] 592 + \delta_{j+1/2} 593 \left[ \frac{e_{1u}} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] 594 \right] 595 \biggr\} &&&\\ 596 &\equiv 0 && \\ 597 \end{flalign*} 598 599 The last equality is obtained using \eqref{Eq_dynspg_rl}, the discrete barotropic 600 streamfunction time evolution equation. By the way, this shows that 601 \eqref{Eq_dynspg_rl} is the only way to compute the streamfunction, 602 otherwise the surface pressure forces will do work. Nevertheless, since 603 the elliptic equation satisfied by $\psi $ is solved numerically by an iterative 604 solver, the property is only satisfied at the precision requested for the solver. 994 && \\ 995 % 996 \equiv& - \frac{1} {2} \sum\limits_{i,j,k} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2\; 997 \biggl\{ \delta_{i+1/2} 998 \left[ \overline{\overline {U}}^{\,i,j+1/2} \right] 999 + \delta_{j+1/2} 1000 \left[ \overline{\overline {V}}^{\,i+1/2,j} \right] 1001 \biggr\} && \\ 1002 % 1003 \equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 1004 \; \overline{\overline{ b_t^{}\, \chi}}^{\,i+1/2,\,j+1/2} &&\\ 1005 % 1006 \ \ \equiv& \ 0 &&\\ 1007 \end{flalign*} 1008 1009 1010 1011 605 1012 606 1013 % ================================================================ … … 626 1033 \label{Apdx_C.2.1} 627 1034 1035 conservation of a tracer, $T$: 1036 \begin{equation*} 1037 \frac{\partial }{\partial t} \left( \int_D {T\;dv} \right) 1038 = \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=0 1039 \end{equation*} 1040 1041 conservation of its variance: 1042 \begin{flalign*} 1043 \frac{\partial }{\partial t} \left( \int_D {\frac{1}{2} T^2\;dv} \right) 1044 =& \int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv } 1045 - \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv } 1046 \end{flalign*} 1047 1048 628 1049 Whatever the advection scheme considered it conserves of the tracer content as all 629 the scheme are written in flux form. Let $\tau$ be the tracer interpolated at velocity point 630 (whatever the interpolation is). The conservation of the tracer content is obtained as follows: 631 \begin{flalign*} 632 &\int_D \nabla \cdot \left( T \textbf{U} \right)\;dv &&&\\ 633 &\equiv \sum\limits_{i,j,k} \biggl\{ 634 \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} 635 \left( \delta_i \left[ e_{2u}\,e_{3u}\; u \;\tau_u \right] 636 + \delta_j \left[ e_{1v}\,e_{3v}\; v \;\tau_v \right] \right) 637 &&&\\& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 638 + \frac{1} {e_{3T}} \delta_k \left[ w\;\tau \right] \biggl\} e_{1T}\,e_{2T}\,e_{3T} &&&\\ 639 % 640 &\equiv \sum\limits_{i,j,k} \left\{ 641 \delta_i \left[ e_{2u}\,e_{3u} \,\overline T^{\,i+1/2}\,u \right] 642 + \delta_j \left[ e_{1v}\,e_{3v} \,\overline T^{\,j+1/2}\,v \right] 643 + \delta_k \left[ e_{1T}\,e_{2T} \,\overline T^{\,k+1/2}\,w \right] \right\} 644 && \\ 1050 the scheme are written in flux form. Indeed, let $T$ be the tracer and $\tau_u$, $\tau_v$, 1051 and $\tau_w$ its interpolated values at velocity point (whatever the interpolation is), 1052 the conservation of the tracer content due to the advection tendency is obtained as follows: 1053 \begin{flalign*} 1054 &\int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv } = - \int_D \nabla \cdot \left( T \textbf{U} \right)\;dv &&&\\ 1055 &\equiv - \sum\limits_{i,j,k} \biggl\{ 1056 \frac{1} {b_t} \left( \delta_i \left[ U \;\tau_u \right] 1057 + \delta_j \left[ V \;\tau_v \right] \right) 1058 + \frac{1} {e_{3t}} \delta_k \left[ w\;\tau_w \right] \biggl\} b_t &&&\\ 1059 % 1060 &\equiv - \sum\limits_{i,j,k} \left\{ 1061 \delta_i \left[ U \;\tau_u \right] 1062 + \delta_j \left[ V \;\tau_v \right] 1063 + \delta_k \left[ W \;\tau_w \right] \right\} && \\ 645 1064 &\equiv 0 &&& 646 1065 \end{flalign*} 647 1066 648 The conservation of the variance of tracer can be achieved only with the CEN2 scheme. 1067 The conservation of the variance of tracer due to the advection tendency 1068 can be achieved only with the CEN2 scheme, $i.e.$ when 1069 $\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$. 649 1070 It can be demonstarted as follows: 650 1071 \begin{flalign*} 651 &\int\limits_D T\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\ 652 \equiv& \sum\limits_{i,j,k} T\; 1072 &\int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv } 1073 = - \int\limits_D \tau\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\ 1074 \equiv& - \sum\limits_{i,j,k} T\; 653 1075 \left\{ 654 \delta_i \left[ e_{2u}\,e_{3u} \overline T^{\,i+1/2}\,u \right] 655 + \delta_j \left[ e_{1v}\,e_{3v} \overline T^{\,j+1/2}\,v \right] 656 + \delta_k \left[ e_{1T}\,e_{2T} \overline T^{\,k+1/2}w \right] 657 \right\} 658 && \\ 659 \equiv& \sum\limits_{i,j,k} 660 \left\{ 661 - e_{2u}\,e_{3u} \overline T^{\,i+1/2}\,u\,\delta_{i+1/2} \left[ T \right] \right. 662 - e_{1v}\,e_{3v} \overline T^{\,j+1/2}\,v\;\delta_{j+1/2} \left[ T \right] 663 &&&\\& \qquad \qquad \qquad \qquad \qquad \qquad \quad \; 664 - \left. e_{1T}\,e_{2T} \overline T^{\,k+1/2}w\;\delta_{k+1/2} \left[ T \right] 665 \right\} 666 &&\\ 667 \equiv& -\frac{1} {2} \sum\limits_{i,j,k} 668 \Bigl\{ 669 e_{2u}\,e_{3u} \; u\;\delta_{i+1/2} \left[ T^2 \right] 670 + e_{1v}\,e_{3v} \; v\;\delta_{j+1/2} \left[ T^2 \right] 671 + e_{1T}\,e_{2T} \;w\;\delta_{k+1/2} \left[ T^2 \right] 672 \Bigr\} 673 && \\ 674 \equiv& \frac{1} {2} \sum\limits_{i,j,k} T^2 675 \Bigl\{ 676 \delta_i \left[ e_{2u}\,e_{3u}\,u \right] 677 + \delta_j \left[ e_{1v}\,e_{3v}\,v \right] 678 + \delta_k \left[ e_{1T}\,e_{2T}\,w \right] 679 \Bigr\} 680 \quad \equiv 0 &&& 681 \end{flalign*} 682 1076 \delta_i \left[ U \overline T^{\,i+1/2} \right] 1077 + \delta_j \left[ V \overline T^{\,j+1/2} \right] 1078 + \delta_k \left[ W \overline T^{\,k+1/2} \right] \right\} && \\ 1079 \equiv& + \sum\limits_{i,j,k} 1080 \left\{ U \overline T^{\,i+1/2} \,\delta_{i+1/2} \left[ T \right] 1081 + V \overline T^{\,j+1/2} \;\delta_{j+1/2} \left[ T \right] 1082 + W \overline T^{\,k+1/2}\;\delta_{k+1/2} \left[ T \right] \right\} &&\\ 1083 \equiv& + \frac{1} {2} \sum\limits_{i,j,k} 1084 \Bigl\{ U \;\delta_{i+1/2} \left[ T^2 \right] 1085 + V \;\delta_{j+1/2} \left[ T^2 \right] 1086 + W \;\delta_{k+1/2} \left[ T^2 \right] \Bigr\} && \\ 1087 \equiv& - \frac{1} {2} \sum\limits_{i,j,k} T^2 1088 \Bigl\{ \delta_i \left[ U \right] 1089 + \delta_j \left[ V \right] 1090 + \delta_k \left[ W \right] \Bigr\} &&& \\ 1091 \equiv& + \frac{1} {2} \sum\limits_{i,j,k} T^2 1092 \Bigl\{ \frac{1}{e_{3t}} \frac{\partial e_{3t}\,T }{\partial t} \Bigr\} &&& \\ 1093 \end{flalign*} 1094 which is the discrete form of $ \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv }$. 683 1095 684 1096 % ================================================================ … … 714 1126 \begin{flalign*} 715 1127 &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 716 \Bigl[ \nabla_h 717 \left( A^{\,lm}\;\chi \right) 718 - \nabla_h \times 719 \left( A^{\,lm}\;\zeta \; \textbf{k} \right) 720 \Bigr]\;dv = 0 1128 \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right) 1129 - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv = 0 721 1130 \end{flalign*} 722 1131 %%%%%%%%%% recheck here.... (gm) 723 1132 \begin{flalign*} 724 1133 = \int \limits_D -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times 725 \Bigl[ \nabla_h \times 726 \left( A^{\,lm}\;\zeta \; \textbf{k} \right) 727 \Bigr]\;dv &&& \\ 1134 \Bigl[ \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv &&& \\ 728 1135 \end{flalign*} 729 1136 \begin{flalign*} … … 802 1209 \\ %%% 803 1210 \equiv& \sum\limits_{i,j,k} 804 - A_T^{\,lm} \,\chi^2 \;e_{1 T}\,e_{2T}\,e_{3T}1211 - A_T^{\,lm} \,\chi^2 \;e_{1t}\,e_{2t}\,e_{3t} 805 1212 - A_f ^{\,lm} \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f} 806 1213 \quad \leq 0 \\ … … 819 1226 \begin{flalign*} 820 1227 &\int\limits_D \zeta \; \textbf{k} \cdot \nabla \times 821 \left[ 822 \nabla_h 823 \left( A^{\,lm}\;\chi \right) 824 -\nabla_h \times 825 \left( A^{\,lm}\;\zeta \; \textbf{k} \right) 826 \right]\;dv &&&\\ 1228 \left[ \nabla_h \left( A^{\,lm}\;\chi \right) 1229 - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \right]\;dv &&&\\ 827 1230 &= A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 828 \left[ 829 \nabla_h \times 830 \left( \zeta \; \textbf{k} \right) 831 \right]\;dv &&&\displaybreak[0]\\ 1231 \left[ \nabla_h \times \left( \zeta \; \textbf{k} \right) \right]\;dv &&&\\ 832 1232 &\equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f} 833 \left\{ 834 \delta_{i+1/2} 835 \left[ 836 \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i 837 \left[ e_{3f} \zeta \right] 838 \right] 839 + \delta_{j+1/2} 840 \left[ 841 \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j 842 \left[ e_{3f} \zeta \right] 843 \right] 844 \right\} 845 &&&\\ 1233 \left\{ \delta_{i+1/2} \left[ \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right] 1234 + \delta_{j+1/2} \left[ \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right] \right\} &&&\\ 846 1235 % 847 1236 \intertext{Using \eqref{DOM_di_adj}, it follows:} 848 1237 % 849 1238 &\equiv - A^{\,lm} \sum\limits_{i,j,k} 850 \left\{ 851 \left( 852 \frac{1} {e_{1v}\,e_{3v}} \delta_i 853 \left[ e_{3f} \zeta \right] 854 \right)^2 e_{1v}\,e_{2v}\,e_{3v} 855 + \left( 856 \frac{1} {e_{2u}\,e_{3u}} \delta_j 857 \left[ e_{3f} \zeta \right] 858 \right)^2 e_{1u}\,e_{2u}\,e_{3u} 859 \right\} &&&\\ 1239 \left\{ \left( \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right)^2 b_v 1240 + \left( \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right)^2 b_u \right\} &&&\\ 860 1241 & \leq \;0 &&&\\ 861 1242 \end{flalign*} … … 874 1255 \begin{flalign*} 875 1256 & \int\limits_D \nabla_h \cdot 876 \Bigl[ 877 \nabla_h 878 \left( A^{\,lm}\;\chi \right) 879 - \nabla_h \times 880 \left( A^{\,lm}\;\zeta \;\textbf{k} \right) 881 \Bigr] 882 dv 883 = \int\limits_D \nabla_h \cdot \nabla_h 884 \left( A^{\,lm}\;\chi \right) 885 dv 886 &&&\\ 1257 \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right) 1258 - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \Bigr] dv 1259 = \int\limits_D \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi \right) dv &&&\\ 1260 % 887 1261 &\equiv \sum\limits_{i,j,k} 888 \left\{ 889 \delta_i 890 \left[ 891 A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} 892 \left[ \chi \right] 893 \right] 894 + \delta_j 895 \left[ 896 A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} 897 \left[ \chi \right] 898 \right] 899 \right\} 900 &&&\\ 1262 \left\{ \delta_i \left[ A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] 1263 + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] \right\} &&&\\ 901 1264 % 902 1265 \intertext{Using \eqref{DOM_di_adj}, it follows:} 903 1266 % 904 1267 &\equiv \sum\limits_{i,j,k} 905 - \left\{ 906 \frac{e_{2u}\,e_{3u}} {e_{1u}} A_u^{\,lm} \delta_{i+1/2} 907 \left[ \chi \right] 908 \delta_{i+1/2} 909 \left[ 1 \right] 910 + \frac{e_{1v}\,e_{3v}} {e_{2v}} A_v^{\,lm} \delta_{j+1/2} 911 \left[ \chi \right] 912 \delta_{j+1/2} 913 \left[ 1 \right] 914 \right\} \; 915 \equiv 0 916 &&& \\ 1268 - \left\{ \frac{e_{2u}\,e_{3u}} {e_{1u}} A_u^{\,lm} \delta_{i+1/2} \left[ \chi \right] \delta_{i+1/2} \left[ 1 \right] 1269 + \frac{e_{1v}\,e_{3v}} {e_{2v}} A_v^{\,lm} \delta_{j+1/2} \left[ \chi \right] \delta_{j+1/2} \left[ 1 \right] \right\} 1270 \qquad \equiv 0 &&& \\ 917 1271 \end{flalign*} 918 1272 … … 929 1283 = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\; dv &&&\\ 930 1284 % 931 &\equiv A^{\,lm} \sum\limits_{i,j,k} \frac{1} {e_{1 T}\,e_{2T}\,e_{3T}} \chi1285 &\equiv A^{\,lm} \sum\limits_{i,j,k} \frac{1} {e_{1t}\,e_{2t}\,e_{3t}} \chi 932 1286 \left\{ 933 1287 \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] 934 1288 + \delta_j \left[ \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] 935 \right\} \; e_{1 T}\,e_{2T}\,e_{3T} &&&\\1289 \right\} \; e_{1t}\,e_{2t}\,e_{3t} &&&\\ 936 1290 % 937 1291 \intertext{Using \eqref{DOM_di_adj}, it turns out to be:} 938 1292 % 939 1293 &\equiv - A^{\,lm} \sum\limits_{i,j,k} 940 \left\{ 941 \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right)^2 e_{1u}\,e_{2u}\,e_{3u} 942 + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 e_{1v}\,e_{2v}\,e_{3v} 943 \right\} \; &&&\\ 1294 \left\{ \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right)^2 b_u 1295 + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 b_v \right\} \; &&&\\ 1296 % 944 1297 &\leq 0 &&&\\ 945 1298 \end{flalign*} … … 956 1309 with the conservation of momentum and the dissipation of horizontal kinetic energy: 957 1310 \begin{align*} 958 \int\limits_D 959 \frac{1} {e_3 }\; \frac{\partial } {\partial k} 960 \left( 961 \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 962 \right)\; 963 dv \qquad \quad &= \vec{\textbf{0}} 964 \\ 1311 \int\limits_D \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1312 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv 1313 \qquad \quad &= \vec{\textbf{0}} \\ 965 1314 % 966 1315 \intertext{and} 967 1316 % 968 1317 \int\limits_D 969 \textbf{U}_h \cdot 970 \frac{1} {e_3 }\; \frac{\partial } {\partial k} 971 \left( 972 \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 973 \right)\; 974 dv \quad &\leq 0 975 \\ 1318 \textbf{U}_h \cdot \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1319 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv \quad &\leq 0 \\ 976 1320 \end{align*} 977 1321 The first property is obvious. The second results from: … … 979 1323 \begin{flalign*} 980 1324 \int\limits_D 981 \textbf{U}_h \cdot 982 \frac{1} {e_3 }\; \frac{\partial } {\partial k} 983 \left( 984 \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 985 \right)\;dv 986 &&&\\ 1325 \textbf{U}_h \cdot \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1326 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\;dv &&&\\ 987 1327 \end{flalign*} 988 1328 \begin{flalign*} 989 1329 &\equiv \sum\limits_{i,j,k} 990 1330 \left( 991 u\; \delta_k 992 \left[ 993 \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 994 \left[ u \right] 995 \right]\; 996 e_{1u}\,e_{2u} 997 + v\;\delta_k 998 \left[ 999 \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 1000 \left[ v \right] 1001 \right]\; 1002 e_{1v}\,e_{2v} 1003 \right) 1004 &&&\\ 1331 u\; \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ u \right] \right]\; e_{1u}\,e_{2u} 1332 + v\; \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right]\; e_{1v}\,e_{2v} \right) &&&\\ 1005 1333 % 1006 1334 \intertext{since the horizontal scale factor does not depend on $k$, it follows:} 1007 1335 % 1008 1336 &\equiv - \sum\limits_{i,j,k} 1009 \left( 1010 \frac{A_u^{\,vm}} {e_{3uw}} 1011 \left( 1012 \delta_{k+1/2} 1013 \left[ u \right] 1014 \right)^2\; 1015 e_{1u}\,e_{2u} 1016 + \frac{A_v^{\,vm}} {e_{3vw}} 1017 \left( \delta_{k+1/2} 1018 \left[ v \right] 1019 \right)^2\; 1020 e_{1v}\,e_{2v} 1021 \right) 1022 \quad \leq 0 1023 &&&\\ 1337 \left( \frac{A_u^{\,vm}} {e_{3uw}} \left( \delta_{k+1/2} \left[ u \right] \right)^2\; e_{1u}\,e_{2u} 1338 + \frac{A_v^{\,vm}} {e_{3vw}} \left( \delta_{k+1/2} \left[ v \right] \right)^2\; e_{1v}\,e_{2v} \right) 1339 \quad \leq 0 &&&\\ 1024 1340 \end{flalign*} 1025 1341 … … 1028 1344 \int \limits_D 1029 1345 \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 1030 \left( 1031 \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1032 \left( 1033 \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k} 1034 \right) 1035 \right)\; 1036 dv 1037 &&&\\ 1346 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} \left( 1347 \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k} 1348 \right) \right)\; dv &&&\\ 1038 1349 \end{flalign*} 1039 1350 \begin{flalign*} … … 1041 1352 \bigg\{ \biggr. \quad 1042 1353 \delta_{i+1/2} 1043 &\left( 1044 \frac{e_{2v}} {e_{3v}} \delta_k 1045 \left[ 1046 \frac{1} {e_{3vw}} \delta_{k+1/2} 1047 \left[ v \right] 1048 \right] 1049 \right) 1050 &&\\ 1354 &\left( \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right] \right) &&\\ 1051 1355 \biggl. 1052 1356 - \delta_{j+1/2} 1053 &\left( 1054 \frac{e_{1u}} {e_{3u}} \delta_k 1055 \left[ 1056 \frac{1} {e_{3uw}}\delta_{k+1/2} 1057 \left[ u \right] 1058 \right] 1059 \right) 1357 &\left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{1} {e_{3uw}}\delta_{k+1/2} \left[ u \right] \right] \right) 1060 1358 \biggr\} \; 1061 e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0 1062 && \\ 1359 e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0 && \\ 1063 1360 \end{flalign*} 1064 1361 If the vertical diffusion coefficient is uniform over the whole domain, the … … 1066 1363 \begin{flalign*} 1067 1364 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 1068 \left( 1069 \frac{1} {e_3}\; \frac{\partial } {\partial k} 1070 \left( 1071 \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 1072 \right) 1073 \right)\; 1074 dv = 0 1075 &&&\\ 1365 \left( \frac{1} {e_3}\; \frac{\partial } {\partial k} 1366 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\ 1076 1367 \end{flalign*} 1077 1368 This property is only satisfied in $z$-coordinates: … … 1079 1370 \begin{flalign*} 1080 1371 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 1081 \left( 1082 \frac{1} {e_3}\; \frac{\partial } {\partial k} 1083 \left( 1084 \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 1085 \right) 1086 \right)\; 1087 dv 1088 &&& \\ 1372 \left( \frac{1} {e_3}\; \frac{\partial } {\partial k} 1373 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv &&& \\ 1089 1374 \end{flalign*} 1090 1375 \begin{flalign*} … … 1092 1377 \biggl\{ \biggr. \quad 1093 1378 \delta_{i+1/2} 1094 &\left( 1095 \frac{e_{2v}} {e_{3v}} \delta_k 1096 \left[ 1097 \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 1098 \left[ v \right] 1099 \right] 1100 \right) 1101 &&\\ 1379 &\left( \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}[v] \right] \right) &&\\ 1102 1380 - \delta_{j+1/2} 1103 1381 &\biggl. 1104 \left( 1105 \frac{e_{1u}} {e_{3u}} \delta_k 1106 \left[ 1107 \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 1108 \left[ u \right] 1109 \right] 1110 \right) 1111 \biggr\} 1112 &&\\ 1382 \left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) \biggr\} &&\\ 1113 1383 \end{flalign*} 1114 1384 \begin{flalign*} … … 1116 1386 \biggl\{ \biggr. \quad 1117 1387 \frac{1} {e_{3v}} \delta_k 1118 &\left[ 1119 \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 1120 \left[ \delta_{i+1/2} 1121 \left[ e_{2v}\,v \right] 1122 \right] 1123 \right] 1124 &&\\ 1388 &\left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ \delta_{i+1/2} \left[ e_{2v}\,v \right] \right] \right] &&\\ 1125 1389 \biggl. 1126 1390 - \frac{1} {e_{3u}} \delta_k 1127 &\left[ 1128 \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 1129 \left[ \delta_{j+1/2} 1130 \left[ e_{1u}\,u \right] 1131 \right] 1132 \right] 1133 \biggr\} 1134 &&\\ 1391 &\left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ \delta_{j+1/2} \left[ e_{1u}\,u \right] \right] \right] \biggr\} &&\\ 1135 1392 \end{flalign*} 1136 1393 Using the fact that the vertical diffusion coefficients are uniform, and that in 1137 1394 $z$-coordinate, the vertical scale factors do not depend on $i$ and $j$ so 1138 that: $e_{3f} =e_{3u} =e_{3v} =e_{3 T} $ and $e_{3w} =e_{3uw} =e_{3vw} $,1395 that: $e_{3f} =e_{3u} =e_{3v} =e_{3t} $ and $e_{3w} =e_{3uw} =e_{3vw} $, 1139 1396 it follows: 1140 1397 \begin{flalign*} 1141 1398 \equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k 1142 \left[ 1143 \frac{1} {e_{3w}} \delta_{k+1/2} 1144 \Bigl[ 1145 \delta_{i+1/2} 1146 \left[ e_{2v}\,v \right] 1147 - \delta_{j+1/ 2} 1148 \left[ e_{1u}\,u \right] 1149 \Bigr] 1150 \right] 1151 &&&\\ 1399 \left[ \frac{1} {e_{3w}} \delta_{k+1/2} \Bigl[ \delta_{i+1/2} \left[ e_{2v}\,v \right] 1400 - \delta_{j+1/ 2} \left[ e_{1u}\,u \right] \Bigr] \right] &&&\\ 1152 1401 \end{flalign*} 1153 1402 \begin{flalign*} 1154 1403 \equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}} 1155 \left( 1156 \delta_{k+1/2} 1157 \left[ \zeta \right] 1158 \right)^2 \; 1159 e_{1f}\,e_{2f} 1160 \; \leq 0 1161 &&&\\ 1404 \left( \delta_{k+1/2} \left[ \zeta \right] \right)^2 \; e_{1f}\,e_{2f} \; \leq 0 &&&\\ 1162 1405 \end{flalign*} 1163 1406 Similarly, the horizontal divergence is obviously conserved: … … 1165 1408 \begin{flalign*} 1166 1409 \int\limits_D \nabla \cdot 1167 \left( 1168 \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1169 \left( 1170 \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 1171 \right) 1172 \right)\; 1173 dv = 0 1174 &&&\\ 1410 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1411 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\ 1175 1412 \end{flalign*} 1176 1413 and the square of the horizontal divergence decreases ($i.e.$ the horizontal … … 1180 1417 \begin{flalign*} 1181 1418 \int\limits_D \chi \;\nabla \cdot 1182 \left( 1183 \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1184 \left( 1185 \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 1186 \right) 1187 \right)\; 1188 dv = 0 1189 &&&\\ 1419 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1420 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\ 1190 1421 \end{flalign*} 1191 1422 This property is only satisfied in the $z$-coordinate: 1192 1423 \begin{flalign*} 1193 1424 \int\limits_D \chi \;\nabla \cdot 1194 \left( 1195 \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1196 \left( 1197 \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 1198 \right) 1199 \right)\; 1200 dv 1201 &&&\\ 1202 \end{flalign*} 1203 \begin{flalign*} 1204 \equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1T}\,e_{2T}} 1425 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1426 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv &&&\\ 1427 \end{flalign*} 1428 \begin{flalign*} 1429 \equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1t}\,e_{2t}} 1205 1430 \biggl\{ \Biggr. \quad 1206 1431 \delta_{i+1/2} 1207 &\left( 1208 \frac{e_{2u}} {e_{3u}} \delta_k 1209 \left[ 1210 \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 1211 \left[ u \right] 1212 \right] 1213 \right) 1214 &&\\ 1432 &\left( \frac{e_{2u}} {e_{3u}} \delta_k 1433 \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) &&\\ 1215 1434 \Biggl. 1216 1435 + \delta_{j+1/2} 1217 &\left( 1218 \frac{e_{1v}} {e_{3v}} \delta_k 1219 \left[ 1220 \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 1221 \left[ v \right] 1222 \right] 1223 \right) 1224 \Biggr\} \; 1225 e_{1T}\,e_{2T}\,e_{3T} 1226 &&\\ 1436 &\left( \frac{e_{1v}} {e_{3v}} \delta_k 1437 \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} [v] \right] \right) 1438 \Biggr\} \; e_{1t}\,e_{2t}\,e_{3t} &&\\ 1227 1439 \end{flalign*} 1228 1440 … … 1232 1444 \delta_{i+1/2} 1233 1445 &\left( 1234 \delta_k 1235 \left[ 1236 \frac{1} {e_{3uw}} \delta_{k+1/2} 1237 \left[ e_{2u}\,u \right] 1238 \right] 1239 \right) 1240 && \\ 1446 \delta_k \left[ 1447 \frac{1} {e_{3uw}} \delta_{k+1/2} \left[ e_{2u}\,u \right] \right] \right) && \\ 1241 1448 \biggl. 1242 1449 + \delta_{j+1/2} 1243 &\left( 1244 \delta_k 1245 \left[ 1246 \frac{1} {e_{3vw}} \delta_{k+1/2} 1247 \left[ e_{1v}\,v \right] 1248 \right] 1249 \right) 1250 \biggr\} 1251 && \\ 1450 &\left( \delta_k \left[ 1451 \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ e_{1v}\,v \right] \right] \right) \biggr\} && \\ 1252 1452 \end{flalign*} 1253 1453 1254 1454 \begin{flalign*} 1255 1455 \equiv -A^{\,vm} \sum\limits_{i,j,k} 1256 \frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\; 1257 \biggl\{ 1258 \delta_{k+1/2} 1259 \Bigl[ 1260 \delta_{i+1/2} 1261 \left[ e_{2u}\,u \right] 1262 + \delta_{j+1/2} 1263 \left[ e_{1v}\,v \right] 1264 \Bigr] 1265 \biggr\} 1266 &&&\\ 1456 \frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\; \biggl\{ 1457 \delta_{k+1/2} \Bigl[ 1458 \delta_{i+1/2} \left[ e_{2u}\,u \right] 1459 + \delta_{j+1/2} \left[ e_{1v}\,v \right] \Bigr] \biggr\} &&&\\ 1267 1460 \end{flalign*} 1268 1461 1269 1462 \begin{flalign*} 1270 1463 \equiv -A^{\,vm} \sum\limits_{i,j,k} 1271 \frac{1} {e_{3w}} 1272 \delta_{k+1/2} 1273 \left[ \chi \right]\; 1274 \delta_{k+1/2} 1275 \left[ e_{1T}\,e_{2T} \;\chi \right] 1276 &&&\\ 1464 \frac{1} {e_{3w}} \delta_{k+1/2} \left[ \chi \right]\; \delta_{k+1/2} \left[ e_{1t}\,e_{2t} \;\chi \right] &&&\\ 1277 1465 \end{flalign*} 1278 1466 1279 1467 \begin{flalign*} 1280 1468 \equiv -A^{\,vm} \sum\limits_{i,j,k} 1281 \frac{e_{1T}\,e_{2T}} {e_{3w}}\; 1282 \left( 1283 \delta_{k+1/2} 1284 \left[ \chi \right] 1285 \right)^2 1286 \quad \equiv 0 1287 &&&\\ 1469 \frac{e_{1t}\,e_{2t}} {e_{3w}}\; \left( \delta_{k+1/2} \left[ \chi \right] \right)^2 \quad \equiv 0 &&&\\ 1288 1470 \end{flalign*} 1289 1471 … … 1327 1509 + \delta_k 1328 1510 \left[ 1329 A_w^{\,vT} \frac{e_{1 T}\,e_{2T}} {e_{3T}} \delta_{k+1/2}1511 A_w^{\,vT} \frac{e_{1t}\,e_{2t}} {e_{3t}} \delta_{k+1/2} 1330 1512 \left[ T \right] 1331 1513 \right] … … 1351 1533 \quad&& \\ 1352 1534 \biggl. 1353 &&+ \delta_k \left[A_w^{\,vT}\frac{e_{1 T}\,e_{2T}} {e_{3T}}\delta_{k+1/2}\left[T\right]\right]1535 &&+ \delta_k \left[A_w^{\,vT}\frac{e_{1t}\,e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right] 1354 1536 \biggr\} && 1355 1537 \end{flalign*} … … 1357 1539 \equiv - \sum\limits_{i,j,k} 1358 1540 \biggl\{ \biggr. \quad 1359 & A_u^{\,lT} 1360 \left( 1361 \frac{1} {e_{1u}} \delta_{i+1/2} 1362 \left[ T \right] 1363 \right)^2 1364 e_{1u}\,e_{2u}\,e_{3u} 1365 && \\ 1366 & + A_v^{\,lT} 1367 \left( 1368 \frac{1} {e_{2v}} \delta_{j+1/2} 1369 \left[ T \right] 1370 \right)^2 1371 e_{1v}\,e_{2v}\,e_{3v} 1372 && \\ 1373 \biggl. 1374 & + A_w^{\,vT} 1375 \left( 1376 \frac{1} {e_{3w}} \delta_{k+1/2} 1377 \left[ T \right] 1378 \right)^2 1379 e_{1w}\,e_{2w}\,e_{3w} 1380 \biggr\} 1381 \quad \leq 0 1382 && \\ 1541 & A_u^{\,lT} \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ T \right] \right)^2 e_{1u}\,e_{2u}\,e_{3u} && \\ 1542 & + A_v^{\,lT} \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ T \right] \right)^2 e_{1v}\,e_{2v}\,e_{3v} && \\ \biggl. 1543 & + A_w^{\,vT} \left( \frac{1} {e_{3w}} \delta_{k+1/2} \left[ T \right] \right)^2 e_{1w}\,e_{2w}\,e_{3w} \biggr\} 1544 \quad \leq 0 && \\ 1383 1545 \end{flalign*} 1384 1546
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