Changeset 10419 for NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_C.tex
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NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_C.tex
r10368 r10419 1 \documentclass[../tex_main/NEMO_manual]{subfiles} 1 \documentclass[../main/NEMO_manual]{subfiles} 2 2 3 \begin{document} 3 4 % ================================================================ … … 6 7 \chapter{Discrete Invariants of the Equations} 7 8 \label{apdx:C} 9 8 10 \minitoc 9 11 … … 14 16 15 17 \newpage 16 $\ $\newline % force a new ligne17 18 18 19 % ================================================================ … … 25 26 26 27 fluxes at the faces of a $T$-box: 27 \ begin{equation*}28 U = e_{2u}\,e_{3u}\; u \qquad V = e_{1v}\,e_{3v}\; v \qquad W = e_{1w}\,e_{2w}\; \omega \\ 29 \ end{equation*}28 \[ 29 U = e_{2u}\,e_{3u}\; u \qquad V = e_{1v}\,e_{3v}\; v \qquad W = e_{1w}\,e_{2w}\; \omega 30 \] 30 31 31 32 volume of cells at $u$-, $v$-, and $T$-points: 32 \ begin{equation*}33 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} \\ 34 \ end{equation*}33 \[ 34 b_u = e_{1u}\,e_{2u}\,e_{3u} \qquad b_v = e_{1v}\,e_{2v}\,e_{3v} \qquad b_t = e_{1t}\,e_{2t}\,e_{3t} 35 \] 35 36 36 37 partial derivative notation: $\partial_\bullet = \frac{\partial}{\partial \bullet}$ … … 42 43 ($i.e.$ $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth). 43 44 \begin{flalign*} 44 z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s} e_3(\tilde{k}) \;d\tilde{k}45 = \eta - \int\limits_k^{k_s} e_3 \;d\tilde{k}45 z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s} e_3(\tilde{k}) \;d\tilde{k} 46 = \eta - \int\limits_k^{k_s} e_3 \;d\tilde{k} 46 47 \end{flalign*} 47 48 48 49 Continuity equation with the above notation: 49 \ begin{equation*}50 \frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t} \biggl\{ \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 051 \ end{equation*}50 \[ 51 \frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t} \biggl\{ \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0 52 \] 52 53 53 54 A quantity, $Q$ is conserved when its domain averaged time change is zero, that is when: 54 \ begin{equation*}55 \partial_t \left( \int_D{ Q\;dv } \right) =056 \ end{equation*}55 \[ 56 \partial_t \left( \int_D{ Q\;dv } \right) =0 57 \] 57 58 Noting that the coordinate system used .... blah blah 58 \ begin{equation*}59 \partial_t \left( \int_D {Q\;dv} \right) = \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk }60 61 \ end{equation*}59 \[ 60 \partial_t \left( \int_D {Q\;dv} \right) = \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } 61 = \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0 62 \] 62 63 equation of evolution of $Q$ written as 63 64 the time evolution of the vertical content of $Q$ like for tracers, or momentum in flux form, 64 65 the quadratic quantity $\frac{1}{2}Q^2$ is conserved when: 65 66 \begin{flalign*} 66 \partial_t \left( \int_D{ \frac{1}{2} \,Q^2\;dv } \right)67 =& \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 } \\68 =& \int_D { Q \;\partial_t\left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } 69 - \int_D { \frac{1}{2} Q^2 \,\partial_t (e_3) \;e_1e_2\;di\,dj\,dk } \\67 \partial_t \left( \int_D{ \frac{1}{2} \,Q^2\;dv } \right) 68 =& \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 } \\ 69 =& \int_D { Q \;\partial_t\left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } 70 - \int_D { \frac{1}{2} Q^2 \,\partial_t (e_3) \;e_1e_2\;di\,dj\,dk } \\ 70 71 \end{flalign*} 71 72 that is in a more compact form : 72 \begin{flalign} \label{eq:Q2_flux} 73 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 74 =& \int_D { \frac{Q}{e_3} \partial_t \left( e_3 \, Q \right) dv } 73 \begin{flalign} 74 \label{eq:Q2_flux} 75 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 76 =& \int_D { \frac{Q}{e_3} \partial_t \left( e_3 \, Q \right) dv } 75 77 - \frac{1}{2} \int_D { \frac{Q^2}{e_3} \partial_t (e_3) \;dv } 76 78 \end{flalign} … … 78 80 the quadratic quantity $\frac{1}{2}Q^2$ is conserved when: 79 81 \begin{flalign*} 80 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right)81 =& \int_D { \frac{1}{2} \partial_t \left( e_3 \, Q^2 \right) \;e_1e_2\;di\,dj\,dk } \\82 =& \int_D { Q \partial_t Q \;e_1e_2e_3\;di\,dj\,dk } 83 + \int_D { \frac{1}{2} Q^2 \, \partial_t e_3 \;e_1e_2\;di\,dj\,dk } \\82 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 83 =& \int_D { \frac{1}{2} \partial_t \left( e_3 \, Q^2 \right) \;e_1e_2\;di\,dj\,dk } \\ 84 =& \int_D { Q \partial_t Q \;e_1e_2e_3\;di\,dj\,dk } 85 + \int_D { \frac{1}{2} Q^2 \, \partial_t e_3 \;e_1e_2\;di\,dj\,dk } \\ 84 86 \end{flalign*} 85 87 that is in a more compact form: 86 \begin{flalign} \label{eq:Q2_vect} 87 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 88 =& \int_D { Q \,\partial_t Q \;dv } 89 + \frac{1}{2} \int_D { \frac{1}{e_3} Q^2 \partial_t e_3 \;dv } 88 \begin{flalign} 89 \label{eq:Q2_vect} 90 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 91 =& \int_D { Q \,\partial_t Q \;dv } 92 + \frac{1}{2} \int_D { \frac{1}{e_3} Q^2 \partial_t e_3 \;dv } 90 93 \end{flalign} 91 92 94 93 95 % ================================================================ … … 97 99 \label{sec:C.1} 98 100 99 100 101 The discretization of pimitive equation in $s$-coordinate ($i.e.$ time and space varying vertical coordinate) 101 102 must be chosen so that the discrete equation of the model satisfy integral constrains on energy and enstrophy. 102 103 103 104 104 Let us first establish those constraint in the continuous world. 105 105 The total energy ($i.e.$ kinetic plus potential energies) is conserved: 106 \begin{flalign} \label{eq:Tot_Energy} 106 \begin{flalign} 107 \label{eq:Tot_Energy} 107 108 \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 + \rho \, g \, z \right) \;dv \right) = & 0 108 109 \end{flalign} … … 118 119 \autoref{eq:Tot_Energy} for the latter form leads to: 119 120 120 \begin{subequations} \label{eq:E_tot} 121 121 % \label{eq:E_tot} 122 122 advection term (vector invariant form): 123 \begin{equation} \label{eq:E_tot_vect_vor_1} 124 \int\limits_D \zeta \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ 125 \end{equation} 123 \[ 124 % \label{eq:E_tot_vect_vor_1} 125 \int\limits_D \zeta \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ 126 \] 126 127 % 127 \ begin{equation} \label{eq:E_tot_vect_adv_1}128 \int\limits_D \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right) dv129 + \int\limits_D \textbf{U}_h \cdot \nabla_z \textbf{U}_h \;dv 130 - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 \\ 131 \end{equation} 132 128 \[ 129 % \label{eq:E_tot_vect_adv_1} 130 \int\limits_D \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right) dv 131 + \int\limits_D \textbf{U}_h \cdot \nabla_z \textbf{U}_h \;dv 132 - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 133 \] 133 134 advection term (flux form): 134 \begin{equation} \label{eq:E_tot_flux_metric} 135 \int\limits_D \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1 \right)\; 136 \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ 137 \end{equation} 138 139 \begin{equation} \label{eq:E_tot_flux_adv} 140 \int\limits_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} 141 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 142 \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ \end{array}} } \right) \;dv 143 + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 \\ 144 \end{equation} 145 135 \[ 136 % \label{eq:E_tot_flux_metric} 137 \int\limits_D \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1 \right)\; 138 \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 139 \] 140 \[ 141 % \label{eq:E_tot_flux_adv} 142 \int\limits_D \textbf{U}_h \cdot \left( {{ 143 \begin{array} {*{20}c} 144 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 145 \nabla \cdot \left( \textbf{U}\,v \right) \hfill 146 \end{array}} 147 } \right) \;dv 148 + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 149 \] 146 150 coriolis term 147 \ begin{equation} \label{eq:E_tot_cor}148 \int\limits_D f \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ 149 \end{equation} 150 151 \[ 152 % \label{eq:E_tot_cor} 153 \int\limits_D f \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 154 \] 151 155 pressure gradient: 152 \begin{equation} \label{eq:E_tot_pg_1} 153 - \int\limits_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 154 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 155 + \int\limits_D g\, \rho \; \partial_t z \;dv \\ 156 \end{equation} 156 \[ 157 % \label{eq:E_tot_pg_1} 158 - \int\limits_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 159 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 160 + \int\limits_D g\, \rho \; \partial_t z \;dv 161 \] 162 163 where $\nabla_h = \left. \nabla \right|_k$ is the gradient along the $s$-surfaces. 164 165 blah blah.... 166 167 The prognostic ocean dynamics equation can be summarized as follows: 168 \[ 169 \text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } 170 {\text{COR} + \text{ADV} } 171 + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} 172 \] 173 174 Vector invariant form: 175 % \label{eq:E_tot_vect} 176 \[ 177 % \label{eq:E_tot_vect_vor_2} 178 \int\limits_D \textbf{U}_h \cdot \text{VOR} \;dv = 0 179 \] 180 \[ 181 % \label{eq:E_tot_vect_adv_2} 182 \int\limits_D \textbf{U}_h \cdot \text{KEG} \;dv 183 + \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 184 - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 185 \] 186 \[ 187 % \label{eq:E_tot_pg_2} 188 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 189 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 190 + \int\limits_D g\, \rho \; \partial_t z \;dv 191 \] 192 193 Flux form: 194 \begin{subequations} 195 \label{eq:E_tot_flux} 196 \[ 197 % \label{eq:E_tot_flux_metric_2} 198 \int\limits_D \textbf{U}_h \cdot \text {COR} \; dv = 0 199 \] 200 \[ 201 % \label{eq:E_tot_flux_adv_2} 202 \int\limits_D \textbf{U}_h \cdot \text{ADV} \;dv 203 + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 204 \] 205 \begin{equation} 206 \label{eq:E_tot_pg_3} 207 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 208 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 209 + \int\limits_D g\, \rho \; \partial_t z \;dv 210 \end{equation} 157 211 \end{subequations} 158 159 where $\nabla_h = \left. \nabla \right|_k$ is the gradient along the $s$-surfaces.160 161 blah blah....162 $\ $\newline % force a new ligne163 The prognostic ocean dynamics equation can be summarized as follows:164 \begin{equation*}165 \text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} }166 {\text{COR} + \text{ADV} }167 + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}168 \end{equation*}169 $\ $\newline % force a new ligne170 171 Vector invariant form:172 \begin{subequations} \label{eq:E_tot_vect}173 \begin{equation} \label{eq:E_tot_vect_vor_2}174 \int\limits_D \textbf{U}_h \cdot \text{VOR} \;dv = 0 \\175 \end{equation}176 \begin{equation} \label{eq:E_tot_vect_adv_2}177 \int\limits_D \textbf{U}_h \cdot \text{KEG} \;dv178 + \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv179 - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 \\180 \end{equation}181 \begin{equation} \label{eq:E_tot_pg_2}182 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv183 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv184 + \int\limits_D g\, \rho \; \partial_t z \;dv \\185 \end{equation}186 \end{subequations}187 188 Flux form:189 \begin{subequations} \label{eq:E_tot_flux}190 \begin{equation} \label{eq:E_tot_flux_metric_2}191 \int\limits_D \textbf{U}_h \cdot \text {COR} \; dv = 0 \\192 \end{equation}193 \begin{equation} \label{eq:E_tot_flux_adv_2}194 \int\limits_D \textbf{U}_h \cdot \text{ADV} \;dv195 + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 \\196 \end{equation}197 \begin{equation} \label{eq:E_tot_pg_3}198 - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv199 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv200 + \int\limits_D g\, \rho \; \partial_t z \;dv \\201 \end{equation}202 \end{subequations}203 204 205 $\ $\newline % force a new ligne206 207 212 208 213 \autoref{eq:E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. 209 214 Indeed the left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 210 215 \begin{flalign*} 211 \partial_t \left( \int\limits_D { \rho \, g \, z \;dv} \right) 212 &= + \int\limits_D \frac{1}{e_3} \partial_t (e_3\,\rho) \;g\;z\;\;dv213 214 &= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv215 216 &= + \int\limits_D \rho \,g \left( \textbf {U}_h \cdot \nabla_h z + \omega \frac{1}{e_3} \partial_k z \right) \;dv217 218 &= + \int\limits_D \rho \,g \left( \omega + \partial_t z + \textbf {U}_h \cdot \nabla_h z \right) \;dv &&&\\219 &=+ \int\limits_D g\, \rho \; w \; dv &&&\\216 \partial_t \left( \int\limits_D { \rho \, g \, z \;dv} \right) 217 &= + \int\limits_D \frac{1}{e_3} \partial_t (e_3\,\rho) \;g\;z\;\;dv 218 + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\ 219 &= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 220 + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\ 221 &= + \int\limits_D \rho \,g \left( \textbf {U}_h \cdot \nabla_h z + \omega \frac{1}{e_3} \partial_k z \right) \;dv 222 + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\ 223 &= + \int\limits_D \rho \,g \left( \omega + \partial_t z + \textbf {U}_h \cdot \nabla_h z \right) \;dv &&&\\ 224 &=+ \int\limits_D g\, \rho \; w \; dv &&&\\ 220 225 \end{flalign*} 221 226 where the last equality is obtained by noting that the brackets is exactly the expression of $w$, … … 224 229 The left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 225 230 \begin{flalign*} 226 - \int\limits_D \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv 227 = - \int\limits_D \left( \nabla_h p + \rho \, g \nabla_h z \right) \cdot \textbf{U}_h \;dv &&&\\ 228 \allowdisplaybreaks 229 &= - \int\limits_D \nabla_h p \cdot \textbf{U}_h \;dv - \int\limits_D \rho \, g \, \nabla_h z \cdot \textbf{U}_h \;dv &&&\\ 230 \allowdisplaybreaks 231 &= +\int\limits_D p \,\nabla_h \cdot \textbf{U}_h \;dv + \int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ 232 \allowdisplaybreaks 233 &= -\int\limits_D p \left( \frac{1}{e_3} \partial_t e_3 + \frac{1}{e_3} \partial_k \omega \right) \;dv 234 +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ 235 \allowdisplaybreaks 236 &= -\int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 237 +\int\limits_D \frac{1}{e_3} \partial_k p\; \omega \;dv 238 +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ 239 &= -\int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 240 -\int\limits_D \rho \, g \, \omega \;dv 241 +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ 242 &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \; \;dv 243 - \int\limits_D \rho \, g \, w \;dv 244 + \int\limits_D \rho \, g \, \partial_t z \;dv &&&\\ 245 \allowdisplaybreaks 246 \intertext{introducing the hydrostatic balance $\partial_k p=-\rho \,g\,e_3$ in the last term, 247 it becomes:} 248 &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 249 - \int\limits_D \rho \, g \, w \;dv 250 - \int\limits_D \frac{1}{e_3} \partial_k p\, \partial_t z \;dv &&&\\ 251 &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 252 - \int\limits_D \rho \, g \, w \;dv 253 + \int\limits_D \,\frac{p}{e_3}\partial_t ( \partial_k z ) dv &&&\\ 254 % 255 &= - \int\limits_D \rho \, g \, w \;dv &&&\\ 256 \end{flalign*} 257 231 - \int\limits_D \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv 232 = - \int\limits_D \left( \nabla_h p + \rho \, g \nabla_h z \right) \cdot \textbf{U}_h \;dv &&&\\ 233 \allowdisplaybreaks 234 &= - \int\limits_D \nabla_h p \cdot \textbf{U}_h \;dv - \int\limits_D \rho \, g \, \nabla_h z \cdot \textbf{U}_h \;dv &&&\\ 235 \allowdisplaybreaks 236 &= +\int\limits_D p \,\nabla_h \cdot \textbf{U}_h \;dv + \int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ 237 \allowdisplaybreaks 238 &= -\int\limits_D p \left( \frac{1}{e_3} \partial_t e_3 + \frac{1}{e_3} \partial_k \omega \right) \;dv 239 +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ 240 \allowdisplaybreaks 241 &= -\int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 242 +\int\limits_D \frac{1}{e_3} \partial_k p\; \omega \;dv 243 +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ 244 &= -\int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 245 -\int\limits_D \rho \, g \, \omega \;dv 246 +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ 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 \rho \, g \, \partial_t z \;dv &&&\\ 250 \allowdisplaybreaks 251 \intertext{introducing the hydrostatic balance $\partial_k p=-\rho \,g\,e_3$ in the last term, 252 it becomes:} 253 &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 254 - \int\limits_D \rho \, g \, w \;dv 255 - \int\limits_D \frac{1}{e_3} \partial_k p\, \partial_t z \;dv &&&\\ 256 &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 257 - \int\limits_D \rho \, g \, w \;dv 258 + \int\limits_D \,\frac{p}{e_3}\partial_t ( \partial_k z ) dv &&&\\ 259 % 260 &= - \int\limits_D \rho \, g \, w \;dv &&&\\ 261 \end{flalign*} 258 262 259 263 %gm comment … … 262 266 The last equality comes from the following equation, 263 267 \begin{flalign*} 264 \int\limits_D p \frac{1}{e_3} \frac{\partial e_3}{\partial t}\; \;dv 265 = \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv \quad, \\268 \int\limits_D p \frac{1}{e_3} \frac{\partial e_3}{\partial t}\; \;dv 269 = \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv \quad, 266 270 \end{flalign*} 267 271 that can be demonstrated as follows: 268 272 269 273 \begin{flalign*} 270 \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv 271 &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv 274 \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv 275 &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv 272 276 - \int\limits_D \rho \, g \, \frac{\partial}{\partial t} \left( \int\limits_k^{k_s} e_3 \;d\tilde{k} \right) \;dv &&&\\ 273 &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv 277 &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv 274 278 - \int\limits_D \rho \, g \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \;dv &&&\\ 275 % 276 \allowdisplaybreaks 277 \intertext{The second term of the right hand side can be transformed by applying the integration by part rule: 278 $\left[ a\,b \right]_{k_b}^{k_s} = \int_{k_b}^{k_s} a\,\frac{\partial b}{\partial k} \;dk 279 + \int_{k_b}^{k_s} \frac{\partial a}{\partial k} \,b \;dk $ 280 to the following function: $a= \int_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k}$ 281 and $b= \int_k^{k_s} \rho \, e_3 \;d\tilde{k}$ 282 (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). 283 This leads to: } 284 \end{flalign*} 285 \begin{flalign*} 286 &\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} 287 =-\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 288 -\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 289 &&&\\ 290 \allowdisplaybreaks 291 \intertext{Noting that $\frac{\partial \eta}{\partial t} 292 = \frac{\partial}{\partial t} \left( \int_{k_b}^{k_s} e_3 \;d\tilde{k} \right) 293 = \int_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k}$ 294 and 295 $p(k) = \int_k^{k_s} \rho \,g \, e_3 \;d\tilde{k} $, 296 but also that $\frac{\partial \eta}{\partial t}$ does not depends on $k$, it comes: 297 } 298 & - \int\limits_{k_b}^{k_s} \rho \, \frac{\partial \eta}{\partial t} \, e_3 \;dk 299 = - \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 300 - \int\limits_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \frac{p}{g} \;dk &&&\\ 279 % 280 \allowdisplaybreaks 281 \intertext{The second term of the right hand side can be transformed by applying the integration by part rule: 282 $\left[ a\,b \right]_{k_b}^{k_s} = \int_{k_b}^{k_s} a\,\frac{\partial b}{\partial k} \;dk 283 + \int_{k_b}^{k_s} \frac{\partial a}{\partial k} \,b \;dk $ 284 to the following function: $a= \int_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k}$ 285 and $b= \int_k^{k_s} \rho \, e_3 \;d\tilde{k}$ 286 (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). 287 This leads to: } 288 \end{flalign*} 289 \begin{flalign*} 290 &\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} 291 =-\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 292 -\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 &&&\\ 293 \allowdisplaybreaks 294 \intertext{Noting that $\frac{\partial \eta}{\partial t} 295 = \frac{\partial}{\partial t} \left( \int_{k_b}^{k_s} e_3 \;d\tilde{k} \right) 296 = \int_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k}$ 297 and 298 $p(k) = \int_k^{k_s} \rho \,g \, e_3 \;d\tilde{k} $, 299 but also that $\frac{\partial \eta}{\partial t}$ does not depends on $k$, it comes: 300 } 301 & - \int\limits_{k_b}^{k_s} \rho \, \frac{\partial \eta}{\partial t} \, e_3 \;dk 302 = - \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 303 - \int\limits_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \frac{p}{g} \;dk &&&\\ 301 304 \end{flalign*} 302 305 Mutliplying by $g$ and integrating over the $(i,j)$ domain it becomes: 303 306 \begin{flalign*} 304 \int\limits_D \rho \, g \, \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \;dv305 = \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} dv307 \int\limits_D \rho \, g \, \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \;dv 308 = \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} dv 306 309 - \int\limits_D \frac{p}{e_3}\frac{\partial e_3}{\partial t} \;dv 307 310 \end{flalign*} 308 311 Using this property, we therefore have: 309 312 \begin{flalign*} 310 \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv 311 &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv 313 \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv 314 &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv 312 315 - \left( \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} dv 313 314 %315 &=\int\limits_D \frac{p}{e_3} \frac{\partial (e_3\,\rho)}{\partial t}\; \;dv 316 - \int\limits_D \frac{p}{e_3}\frac{\partial e_3}{\partial t} \;dv \right) &&&\\ 317 % 318 &=\int\limits_D \frac{p}{e_3} \frac{\partial (e_3\,\rho)}{\partial t}\; \;dv 316 319 \end{flalign*} 317 320 % end gm comment … … 319 322 % 320 323 321 322 324 % ================================================================ 323 325 % Discrete Total energy Conservation : vector invariant form … … 334 336 The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: 335 337 \begin{flalign*} 336 \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 \\ 338 \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) &=0 337 339 \end{flalign*} 338 340 which in vector invariant forms, it leads to: 339 \begin{equation} \label{eq:KE+PE_vect_discrete} \begin{split} 340 \sum\limits_{i,j,k} \biggl\{ u\, \partial_t u \;b_u 341 + v\, \partial_t v \;b_v \biggr\} 342 + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{e_{3u}}\partial_t e_{3u} \;b_u 343 + \frac{v^2}{e_{3v}}\partial_t e_{3v} \;b_v \biggr\} \\ 344 = - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}}\partial_t (e_{3t} \rho) \, g \, z_t \;b_t \biggr\} 345 - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} 346 \end{split} \end{equation} 341 \begin{equation} 342 \label{eq:KE+PE_vect_discrete} 343 \begin{split} 344 \sum\limits_{i,j,k} \biggl\{ u\, \partial_t u \;b_u 345 + v\, \partial_t v \;b_v \biggr\} 346 + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{e_{3u}}\partial_t e_{3u} \;b_u 347 + \frac{v^2}{e_{3v}}\partial_t e_{3v} \;b_v \biggr\} \\ 348 = - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}}\partial_t (e_{3t} \rho) \, g \, z_t \;b_t \biggr\} 349 - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} 350 \end{split} 351 \end{equation} 347 352 348 353 Substituting the discrete expression of the time derivative of the velocity either in vector invariant, … … 365 370 366 371 For the ENE scheme, the two components of the vorticity term are given by: 367 \begin{equation*} 368 - e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h \equiv 369 \left( {{ \begin{array} {*{20}c} 370 + \frac{1} {e_{1u}} \; 371 \overline {\, q \ \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} \hfill \\ 372 - \frac{1} {e_{2v}} \; 373 \overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} \hfill \\ 374 \end{array}} } \right) 375 \end{equation*} 372 \[ 373 - e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h \equiv 374 \left( {{ 375 \begin{array} {*{20}c} 376 + \frac{1} {e_{1u}} \; 377 \overline {\, q \ \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} \hfill \\ 378 - \frac{1} {e_{2v}} \; 379 \overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} \hfill 380 \end{array} 381 } } \right) 382 \] 376 383 377 384 This formulation does not conserve the enstrophy but it does conserve the total kinetic energy. … … 379 386 averaged over the ocean domain can be transformed as follows: 380 387 \begin{flalign*} 381 &\int\limits_D - \left( e_3 \, q \;\textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv && \\ 382 & \qquad \qquad {\begin{array}{*{20}l} 383 &\equiv \sum\limits_{i,j,k} \biggl\{ 384 \frac{1} {e_{1u}} \overline { \,q\ \overline{ V }^{\,i+1/2}} ^{\,j} \, u \; b_u 385 - \frac{1} {e_{2v}}\overline { \, q\ \overline{ U }^{\,j+1/2}} ^{\,i} \, v \; b_v \; \biggr\} \\ 386 &\equiv \sum\limits_{i,j,k} \biggl\{ 387 \overline { \,q\ \overline{ V }^{\,i+1/2}}^{\,j} \; U 388 - \overline { \,q\ \overline{ U }^{\,j+1/2}}^{\,i} \; V \; \biggr\} \\ 389 &\equiv \sum\limits_{i,j,k} q \ \biggl\{ \overline{ V }^{\,i+1/2}\; \overline{ U }^{\,j+1/2} 390 - \overline{ U }^{\,j+1/2}\; \overline{ V }^{\,i+1/2} \biggr\} \quad \equiv 0 391 \end{array} } 388 &\int\limits_D - \left( e_3 \, q \;\textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv && \\ 389 & \qquad \qquad 390 { 391 \begin{array}{*{20}l} 392 &\equiv \sum\limits_{i,j,k} \biggl\{ 393 \frac{1} {e_{1u}} \overline { \,q\ \overline{ V }^{\,i+1/2}} ^{\,j} \, u \; b_u 394 - \frac{1} {e_{2v}}\overline { \, q\ \overline{ U }^{\,j+1/2}} ^{\,i} \, v \; b_v \; \biggr\} \\ 395 &\equiv \sum\limits_{i,j,k} \biggl\{ 396 \overline { \,q\ \overline{ V }^{\,i+1/2}}^{\,j} \; U 397 - \overline { \,q\ \overline{ U }^{\,j+1/2}}^{\,i} \; V \; \biggr\} \\ 398 &\equiv \sum\limits_{i,j,k} q \ \biggl\{ \overline{ V }^{\,i+1/2}\; \overline{ U }^{\,j+1/2} 399 - \overline{ U }^{\,j+1/2}\; \overline{ V }^{\,i+1/2} \biggr\} \quad \equiv 0 400 \end{array} 401 } 392 402 \end{flalign*} 393 403 In other words, the domain averaged kinetic energy does not change due to the vorticity term. 394 395 404 396 405 % ------------------------------------------------------------------------------------------------------------- … … 401 410 402 411 With the EEN scheme, the vorticity terms are represented as: 403 \begin{equation} \tag{\ref{eq:dynvor_een}} 404 \left\{ { \begin{aligned} 405 +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} 406 {^{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} \\ 407 - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} 408 {^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} \\ 409 \end{aligned} } \right. 412 \begin{equation} 413 \tag{\ref{eq:dynvor_een}} 414 \left\{ { 415 \begin{aligned} 416 +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} 417 {^{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} \\ 418 - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} 419 {^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} 420 \end{aligned} 421 } \right. 410 422 \end{equation} 411 423 where the indices $i_p$ and $j_p$ take the following value: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 412 424 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 413 \begin{equation} \tag{\ref{eq:Q_triads}} 414 _i^j \mathbb{Q}^{i_p}_{j_p} 415 = \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) 425 \begin{equation} 426 \tag{\ref{eq:Q_triads}} 427 _i^j \mathbb{Q}^{i_p}_{j_p} 428 = \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) 416 429 \end{equation} 417 430 … … 419 432 Indeed, 420 433 \begin{flalign*} 421 &\int\limits_D - \textbf{U}_h \cdot \left( \zeta \;\textbf{k} \times \textbf{U}_h \right) \; dv && \\ 422 \equiv \sum\limits_{i,j,k} & \biggl\{ 423 \left[ \sum_{\substack{i_p,\,k_p}} 424 {^{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} % &&\\ 425 - \left[ \sum_{\substack{i_p,\,k_p}} 426 {^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\} && \\ 427 \\ 428 \equiv \sum\limits_{i,j,k} & \sum_{\substack{i_p,\,k_p}} \biggl\{ \ \ 429 {^{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} % &&\\ 430 - {^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\} && \\ 431 % 432 \allowdisplaybreaks 433 \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} 434 % 435 \equiv \sum\limits_{i,j,k} & \biggl\{ \ \ 436 {^{i+1}_j }\mathbb{Q}^{-1/2}_{+1/2} \;V^{i+1}_{j+1/2} \; U^{\,i+1/2}_{j} 437 - {^i_{j}\quad}\mathbb{Q}^{-1/2}_{+1/2} \; U^{i-1/2}_{j} \; V^{\,i}_{j+1/2} && \\ 438 & + {^{i+1}_j }\mathbb{Q}^{-1/2}_{-1/2} \; V^{i+1}_{j-1/2} \; U^{\,i+1/2}_{j} 439 - {^i_{j+1} }\mathbb{Q}^{-1/2}_{-1/2} \; U^{i-1/2}_{j+1} \; V^{\,i}_{j+1/2} \biggr. && \\ 440 & + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2} \; U^{\,i+1/2}_{j} 441 - {^i_{j}\quad}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j} \; V^{\,i}_{j+1/2} \biggr. && \\ 442 & + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{-1/2} \; V^{i}_{j-1/2} \; U^{\,i+1/2}_{j} 443 - {^i_{j+1} }\mathbb{Q}^{+1/2}_{-1/2} \; U^{i+1/2}_{j+1}\; V^{\,i}_{j+1/2} \ \; \biggr\} && \\ 444 % 445 \allowdisplaybreaks 446 \intertext{The summation is done over all $i$ and $j$ indices, it is therefore possible to introduce 447 a shift of $-1$ either in $i$ or $j$ direction in some of the term of the summation (first term of the 448 first and second lines, second term of the second and fourth lines). By doning so, we can regroup 449 all the terms of the summation by triad at a ($i$,$j$) point. In other words, we regroup all the terms 450 in the neighbourhood that contain a triad at the same ($i$,$j$) indices. It becomes: } 451 \allowdisplaybreaks 452 % 453 \equiv \sum\limits_{i,j,k} & \biggl\{ \ \ 454 {^{i}_j}\mathbb{Q}^{-1/2}_{+1/2} \left[ V^{i}_{j+1/2}\, U^{\,i-1/2}_{j} 455 - U^{i-1/2}_{j} \, V^{\,i}_{j+1/2} \right] && \\ 456 & + {^{i}_j}\mathbb{Q}^{-1/2}_{-1/2} \left[ V^{i}_{j-1/2} \, U^{\,i-1/2}_{j} 457 - U^{i-1/2}_{j} \, V^{\,i}_{j-1/2} \right] \biggr. && \\ 458 & + {^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} \left[ V^{i}_{j+1/2} \, U^{\,i+1/2}_{j} 459 - U^{i+1/2}_{j} \, V^{\,i}_{j+1/2} \right] \biggr. && \\ 460 & + {^{i}_j}\mathbb{Q}^{+1/2}_{-1/2} \left[ V^{i}_{j-1/2} \, U^{\,i+1/2}_{j} 461 - U^{i+1/2}_{j-1} \, V^{\,i}_{j-1/2} \right] \ \; \biggr\} \qquad 462 \equiv \ 0 && 463 \end{flalign*} 464 434 &\int\limits_D - \textbf{U}_h \cdot \left( \zeta \;\textbf{k} \times \textbf{U}_h \right) \; dv && \\ 435 \equiv \sum\limits_{i,j,k} & \biggl\{ 436 \left[ \sum_{\substack{i_p,\,k_p}} 437 {^{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} % &&\\ 438 - \left[ \sum_{\substack{i_p,\,k_p}} 439 {^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\} && \\ \\ 440 \equiv \sum\limits_{i,j,k} & \sum_{\substack{i_p,\,k_p}} \biggl\{ \ \ 441 {^{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} % &&\\ 442 - {^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\} && \\ 443 % 444 \allowdisplaybreaks 445 \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} 446 % 447 \equiv \sum\limits_{i,j,k} & \biggl\{ \ \ 448 {^{i+1}_j }\mathbb{Q}^{-1/2}_{+1/2} \;V^{i+1}_{j+1/2} \; U^{\,i+1/2}_{j} 449 - {^i_{j}\quad}\mathbb{Q}^{-1/2}_{+1/2} \; U^{i-1/2}_{j} \; V^{\,i}_{j+1/2} && \\ 450 & + {^{i+1}_j }\mathbb{Q}^{-1/2}_{-1/2} \; V^{i+1}_{j-1/2} \; U^{\,i+1/2}_{j} 451 - {^i_{j+1} }\mathbb{Q}^{-1/2}_{-1/2} \; U^{i-1/2}_{j+1} \; V^{\,i}_{j+1/2} \biggr. && \\ 452 & + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2} \; U^{\,i+1/2}_{j} 453 - {^i_{j}\quad}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j} \; V^{\,i}_{j+1/2} \biggr. && \\ 454 & + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{-1/2} \; V^{i}_{j-1/2} \; U^{\,i+1/2}_{j} 455 - {^i_{j+1} }\mathbb{Q}^{+1/2}_{-1/2} \; U^{i+1/2}_{j+1}\; V^{\,i}_{j+1/2} \ \; \biggr\} && \\ 456 % 457 \allowdisplaybreaks 458 \intertext{The summation is done over all $i$ and $j$ indices, it is therefore possible to introduce 459 a shift of $-1$ either in $i$ or $j$ direction in some of the term of the summation (first term of the 460 first and second lines, second term of the second and fourth lines). By doning so, we can regroup 461 all the terms of the summation by triad at a ($i$,$j$) point. In other words, we regroup all the terms 462 in the neighbourhood that contain a triad at the same ($i$,$j$) indices. It becomes: } 463 \allowdisplaybreaks 464 % 465 \equiv \sum\limits_{i,j,k} & \biggl\{ \ \ 466 {^{i}_j}\mathbb{Q}^{-1/2}_{+1/2} \left[ V^{i}_{j+1/2}\, U^{\,i-1/2}_{j} 467 - U^{i-1/2}_{j} \, V^{\,i}_{j+1/2} \right] && \\ 468 & + {^{i}_j}\mathbb{Q}^{-1/2}_{-1/2} \left[ V^{i}_{j-1/2} \, U^{\,i-1/2}_{j} 469 - U^{i-1/2}_{j} \, V^{\,i}_{j-1/2} \right] \biggr. && \\ 470 & + {^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} \left[ V^{i}_{j+1/2} \, U^{\,i+1/2}_{j} 471 - U^{i+1/2}_{j} \, V^{\,i}_{j+1/2} \right] \biggr. && \\ 472 & + {^{i}_j}\mathbb{Q}^{+1/2}_{-1/2} \left[ V^{i}_{j-1/2} \, U^{\,i+1/2}_{j} 473 - U^{i+1/2}_{j-1} \, V^{\,i}_{j-1/2} \right] \ \; \biggr\} \qquad 474 \equiv \ 0 && 475 \end{flalign*} 465 476 466 477 % ------------------------------------------------------------------------------------------------------------- … … 471 482 472 483 The change of Kinetic Energy (KE) due to the vertical advection is exactly balanced by the change of KE due to the horizontal gradient of KE~: 473 \ begin{equation*}474 475 = - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv476 + \frac{1}{2} \int_D { \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv } \\477 \ end{equation*}484 \[ 485 \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv 486 = - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv 487 + \frac{1}{2} \int_D { \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv } 488 \] 478 489 Indeed, using successively \autoref{eq:DOM_di_adj} ($i.e.$ the skew symmetry property of the $\delta$ operator) 479 490 and the continuity equation, then \autoref{eq:DOM_di_adj} again, … … 482 493 applied in the horizontal and vertical directions, it becomes: 483 494 \begin{flalign*} 484 & - \int_D \textbf{U}_h \cdot \text{KEG}\;dv 485 = - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv &&&\\486 %487 \equiv & - \sum\limits_{i,j,k} \frac{1}{2} \biggl\{ 488 \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] u \ b_u 489 + \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] v \ b_v \biggr\} &&& \\490 %491 \equiv & + \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right)\;492 \biggl\{ \delta_{i} \left[ U \right] + \delta_{j} \left[ V \right] \biggr\} &&& \\493 \allowdisplaybreaks494 %495 \equiv & - \sum\limits_{i,j,k} \frac{1}{2}496 \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) \; 497 498 \allowdisplaybreaks499 %500 \equiv & + \sum\limits_{i,j,k} \frac{1}{2} \delta_{k+1/2} \left[ \overline{ u^2}^{\,i} + \overline{ v^2}^{\,j} \right] \; W 501 502 \allowdisplaybreaks503 %504 \equiv & + \sum\limits_{i,j,k} \frac{1} {2} \left( \overline{\delta_{k+1/2} \left[ u^2 \right]}^{\,i} 505 + \overline{\delta_{k+1/2} \left[ v^2 \right]}^{\,j} \right) \; W 506 507 508 \allowdisplaybreaks509 \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 time510 derivative of these two equations is satisfied, it becomes:}511 %512 \equiv & \sum\limits_{i,j,k} \frac{1} {2}513 \biggl\{ \; \overline{W}^{\,i+1/2}\;\delta_{k+1/2} \left[ u^2 \right] 514 + \overline{W}^{\,j+1/2}\;\delta_{k+1/2} \left[ v^2 \right] \; \biggr\} 515 516 517 \allowdisplaybreaks518 % 519 \equiv & \sum\limits_{i,j,k} 520 \biggl\{ \; \overline{W}^{\,i+1/2}\; \overline {u}^{\,k+1/2}\; \delta_{k+1/2}[ u ] 521 + \overline{W}^{\,j+1/2}\; \overline {v}^{\,k+1/2}\; \delta_{k+1/2}[ v ] \; \biggr\} 522 523 524 %525 \allowdisplaybreaks526 \equiv & \sum\limits_{i,j,k} 527 \biggl\{ \; \frac{1} {b_u } \; \overline { \overline{W}^{\,i+1/2}\,\delta_{k+1/2} \left[ u \right] }^{\,k} \;u\;b_u 528 + \frac{1} {b_v } \; \overline { \overline{W}^{\,j+1/2} \delta_{k+1/2} \left[ v \right] }^{\,k} \;v\;b_v \; \biggr\}529 530 531 %532 \intertext{The first term provides the discrete expression for the vertical advection of momentum (ZAD), 533 while the second term corresponds exactly to \autoref{eq:KE+PE_vect_discrete}, therefore:}534 \equiv& \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 535 536 \equiv& \int\limits_D \textbf{U}_h \cdot w \partial_k \textbf{U}_h \;dv 537 495 & - \int_D \textbf{U}_h \cdot \text{KEG}\;dv 496 = - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv &&&\\ 497 % 498 \equiv & - \sum\limits_{i,j,k} \frac{1}{2} \biggl\{ 499 \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] u \ b_u 500 + \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] v \ b_v \biggr\} &&& \\ 501 % 502 \equiv & + \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right)\; 503 \biggl\{ \delta_{i} \left[ U \right] + \delta_{j} \left[ V \right] \biggr\} &&& \\ 504 \allowdisplaybreaks 505 % 506 \equiv & - \sum\limits_{i,j,k} \frac{1}{2} 507 \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) \; 508 \biggl\{ \frac{b_t}{e_{3t}} \partial_t (e_{3t}) + \delta_k \left[ W \right] \biggr\} &&&\\ 509 \allowdisplaybreaks 510 % 511 \equiv & + \sum\limits_{i,j,k} \frac{1}{2} \delta_{k+1/2} \left[ \overline{ u^2}^{\,i} + \overline{ v^2}^{\,j} \right] \; W 512 - \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) \;\partial_t b_t &&& \\ 513 \allowdisplaybreaks 514 % 515 \equiv & + \sum\limits_{i,j,k} \frac{1} {2} \left( \overline{\delta_{k+1/2} \left[ u^2 \right]}^{\,i} 516 + \overline{\delta_{k+1/2} \left[ v^2 \right]}^{\,j} \right) \; W 517 - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t \overline{b_t}^{\,{i+1/2}} 518 + \frac{v^2}{2}\,\partial_t \overline{b_t}^{\,{j+1/2}} \right) &&& \\ 519 \allowdisplaybreaks 520 \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 521 derivative of these two equations is satisfied, it becomes:} 522 % 523 \equiv & \sum\limits_{i,j,k} \frac{1} {2} 524 \biggl\{ \; \overline{W}^{\,i+1/2}\;\delta_{k+1/2} \left[ u^2 \right] 525 + \overline{W}^{\,j+1/2}\;\delta_{k+1/2} \left[ v^2 \right] \; \biggr\} 526 - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t b_u 527 + \frac{v^2}{2}\,\partial_t b_v \right) &&& \\ 528 \allowdisplaybreaks 529 % 530 \equiv & \sum\limits_{i,j,k} 531 \biggl\{ \; \overline{W}^{\,i+1/2}\; \overline {u}^{\,k+1/2}\; \delta_{k+1/2}[ u ] 532 + \overline{W}^{\,j+1/2}\; \overline {v}^{\,k+1/2}\; \delta_{k+1/2}[ v ] \; \biggr\} 533 - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t b_u 534 + \frac{v^2}{2}\,\partial_t b_v \right) &&& \\ 535 % 536 \allowdisplaybreaks 537 \equiv & \sum\limits_{i,j,k} 538 \biggl\{ \; \frac{1} {b_u } \; \overline { \overline{W}^{\,i+1/2}\,\delta_{k+1/2} \left[ u \right] }^{\,k} \;u\;b_u 539 + \frac{1} {b_v } \; \overline { \overline{W}^{\,j+1/2} \delta_{k+1/2} \left[ v \right] }^{\,k} \;v\;b_v \; \biggr\} 540 - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t b_u 541 + \frac{v^2}{2}\,\partial_t b_v \right) &&& \\ 542 % 543 \intertext{The first term provides the discrete expression for the vertical advection of momentum (ZAD), 544 while the second term corresponds exactly to \autoref{eq:KE+PE_vect_discrete}, therefore:} 545 \equiv& \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 546 + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t (e_3) \;dv } &&&\\ 547 \equiv& \int\limits_D \textbf{U}_h \cdot w \partial_k \textbf{U}_h \;dv 548 + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t (e_3) \;dv } &&&\\ 538 549 \end{flalign*} 539 550 … … 544 555 For example KE can also be discretized as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. 545 556 This leads to the following expression for the vertical advection: 546 \begin{equation*} 547 \frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h 548 \equiv \left( {{\begin{array} {*{20}c} 549 \frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \; \overline{\overline {e_{1t}\,e_{2t} \,\omega\;\delta_{k+1/2} 550 \left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k} \hfill \\ 551 \frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \; \overline{\overline {e_{1t}\,e_{2t} \,\omega \;\delta_{k+1/2} 552 \left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill \\ 553 \end{array}} } \right) 554 \end{equation*} 557 \[ 558 \frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h 559 \equiv \left( {{ 560 \begin{array} {*{20}c} 561 \frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \; \overline{\overline {e_{1t}\,e_{2t} \,\omega\;\delta_{k+1/2} 562 \left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k} \hfill \\ 563 \frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \; \overline{\overline {e_{1t}\,e_{2t} \,\omega \;\delta_{k+1/2} 564 \left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill 565 \end{array} 566 } } \right) 567 \] 555 568 a formulation that requires an additional horizontal mean in contrast with the one used in NEMO. 556 569 Nine velocity points have to be used instead of 3. … … 560 573 an extra constraint arises on the time derivative of the volume at $u$- and $v$-points: 561 574 \begin{flalign*} 562 e_{1u}\,e_{2u}\,\partial_t (e_{3u}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,i+1/2} \\563 e_{1v}\,e_{2v}\,\partial_t (e_{3v}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,j+1/2}575 e_{1u}\,e_{2u}\,\partial_t (e_{3u}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,i+1/2} \\ 576 e_{1v}\,e_{2v}\,\partial_t (e_{3v}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,j+1/2} 564 577 \end{flalign*} 565 578 which is (over-)satified by defining the vertical scale factor as follows: 566 \begin{flalign} \label{eq:e3u-e3v} 567 e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2} \\ 568 e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2} 569 \end{flalign} 579 \begin{flalign*} 580 % \label{eq:e3u-e3v} 581 e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2} \\ 582 e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2} 583 \end{flalign*} 570 584 571 585 Blah blah required on the the step representation of bottom topography..... … … 588 602 the change of KE due to the work of pressure forces is balanced by 589 603 the change of potential energy due to buoyancy forces: 590 \ begin{equation*}591 - \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 592 = - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right) \,g\,z \;dv604 \[ 605 - \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 606 = - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right) \,g\,z \;dv 593 607 + \int_D g\, \rho \; \partial_t (z) \;dv 594 \ end{equation*}608 \] 595 609 596 610 This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates. … … 599 613 the work of pressure forces can be written as: 600 614 \begin{flalign*} 601 &- \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 602 \equiv \sum\limits_{i,j,k} \biggl\{ \; - \frac{1} {e_{1u}} \Bigl( 603 \delta_{i+1/2} [p_t] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} [z_t] \Bigr) \; u\;b_u 604 && \\& \qquad \qquad \qquad \qquad \qquad \quad \ \,605 - \frac{1} {e_{2v}} \Bigl( 606 \delta_{j+1/2} [p_t] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} [z_t] \Bigr) \; v\;b_v \; \biggr\} && \\ 607 %608 \allowdisplaybreaks609 \intertext{Using successively \autoref{eq:DOM_di_adj}, $i.e.$ the skew symmetry property of 610 the $\delta$ operator, \autoref{eq:wzv}, the continuity equation, \autoref{eq:dynhpg_sco}, 611 the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $,612 which comes from the definition of $z_t$, it becomes: }613 \allowdisplaybreaks614 %615 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 616 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 617 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 618 619 %620 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 621 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 622 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 623 - \left( \frac{b_t}{e_{3t}} \partial_t (e_{3t}) + \delta_k \left[ W \right] \right) \frac{p_t}{g} \biggr\} &&&\\ 624 %625 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 626 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 627 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 628 + \frac{W}{g}\;\delta_{k+1/2} [p_t] 629 630 %631 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 632 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 633 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 634 - W\;e_{3w} \overline \rho^{\,k+1/2} 635 636 %637 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 638 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 639 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 640 + W\; \overline \rho^{\,k+1/2}\;\delta_{k+1/2} [z_t] 641 642 %643 \allowdisplaybreaks644 %645 \equiv& - \sum\limits_{i,j,k} g \; z_t \biggl\{ 646 647 648 + \delta_k \left[ W\; \overline \rho^{\,k+1/2} \right] \biggr\} 649 - \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\650 %651 \equiv& + \sum\limits_{i,j,k} g \; z_t \biggl\{ \partial_t ( e_{3t} \,\rho) \biggr\} \; b_t 652 - \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\653 %615 &- \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 616 \equiv \sum\limits_{i,j,k} \biggl\{ \; - \frac{1} {e_{1u}} \Bigl( 617 \delta_{i+1/2} [p_t] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} [z_t] \Bigr) \; u\;b_u && \\ 618 & \qquad \qquad \qquad \qquad \qquad \quad \ \, 619 - \frac{1} {e_{2v}} \Bigl( 620 \delta_{j+1/2} [p_t] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} [z_t] \Bigr) \; v\;b_v \; \biggr\} && \\ 621 % 622 \allowdisplaybreaks 623 \intertext{Using successively \autoref{eq:DOM_di_adj}, $i.e.$ the skew symmetry property of 624 the $\delta$ operator, \autoref{eq:wzv}, the continuity equation, \autoref{eq:dynhpg_sco}, 625 the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $, 626 which comes from the definition of $z_t$, it becomes: } 627 \allowdisplaybreaks 628 % 629 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 630 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 631 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 632 +\Bigl( \delta_i[U] + \delta_j [V] \Bigr)\;\frac{p_t}{g} \biggr\} &&\\ 633 % 634 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 635 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 636 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 637 - \left( \frac{b_t}{e_{3t}} \partial_t (e_{3t}) + \delta_k \left[ W \right] \right) \frac{p_t}{g} \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 + \frac{W}{g}\;\delta_{k+1/2} [p_t] 643 - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\ 644 % 645 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 646 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 647 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 648 - W\;e_{3w} \overline \rho^{\,k+1/2} 649 - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\ 650 % 651 \equiv& + \sum\limits_{i,j,k} g \biggl\{ 652 \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 653 + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 654 + W\; \overline \rho^{\,k+1/2}\;\delta_{k+1/2} [z_t] 655 - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\ 656 % 657 \allowdisplaybreaks 658 % 659 \equiv& - \sum\limits_{i,j,k} g \; z_t \biggl\{ 660 \delta_i \left[ U\; \overline \rho^{\,i+1/2} \right] 661 + \delta_j \left[ V\; \overline \rho^{\,j+1/2} \right] 662 + \delta_k \left[ W\; \overline \rho^{\,k+1/2} \right] \biggr\} 663 - \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ 664 % 665 \equiv& + \sum\limits_{i,j,k} g \; z_t \biggl\{ \partial_t ( e_{3t} \,\rho) \biggr\} \; b_t 666 - \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ 667 % 654 668 \end{flalign*} 655 669 The first term is exactly the first term of the right-hand-side of \autoref{eq:KE+PE_vect_discrete}. … … 659 673 In other words, the following property must be satisfied: 660 674 \begin{flalign*} 661 \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\}662 \equiv \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} 675 \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} 676 \equiv \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} 663 677 \end{flalign*} 664 678 … … 667 681 668 682 \begin{flalign*} 669 \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\}670 &\equiv - \sum\limits_{i,j,k} \biggl\{ \delta_k [p_w]\,\partial_t (z_t) \,e_{1t}\,e_{2t} \biggr\} &&&\\671 %672 &\equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \delta_{k+1/2} [\partial_t (z_t)] \,e_{1t}\,e_{2t} \biggr\} 683 \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} 684 &\equiv - \sum\limits_{i,j,k} \biggl\{ \delta_k [p_w]\,\partial_t (z_t) \,e_{1t}\,e_{2t} \biggr\} &&&\\ 685 % 686 &\equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \delta_{k+1/2} [\partial_t (z_t)] \,e_{1t}\,e_{2t} \biggr\} 673 687 \equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (e_{3w}) \,e_{1t}\,e_{2t} \biggr\} &&&\\ 674 &\equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (b_w) \biggr\} 675 %676 % & \equiv \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w \right) \biggr\} &&&\\677 % & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t \left( \delta_k [z_t] \right) e_{1w}\,e_{2w} \biggr\} &&&\\678 % & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t b_w \biggr\} 688 &\equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (b_w) \biggr\} 689 % 690 % & \equiv \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w \right) \biggr\} &&&\\ 691 % & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t \left( \delta_k [z_t] \right) e_{1w}\,e_{2w} \biggr\} &&&\\ 692 % & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t b_w \biggr\} 679 693 \end{flalign*} 680 694 therefore, the balance to be satisfied is: 681 695 \begin{flalign*} 682 \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\}696 \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\} 683 697 \end{flalign*} 684 698 which is a purely vertical balance: 685 699 \begin{flalign*} 686 \sum\limits_{k} \biggl\{ p_t\;\partial_t (e_{3t}) \biggr\} \equiv \sum\limits_{k} \biggl\{ p_w\, \partial_t (e_{3w}) \biggr\}700 \sum\limits_{k} \biggl\{ p_t\;\partial_t (e_{3t}) \biggr\} \equiv \sum\limits_{k} \biggl\{ p_w\, \partial_t (e_{3w}) \biggr\} 687 701 \end{flalign*} 688 702 Defining $p_w = \overline{p_t}^{\,k+1/2}$ … … 690 704 %gm comment 691 705 \gmcomment{ 692 \begin{flalign*} 693 \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ 694 % 695 & \equiv \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w \biggr\} &&&\\ 696 & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t \left( \delta_{k+1/2} [z_t] \right) e_{1w}\,e_{2w} \biggr\} &&&\\ 697 & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t b_w \biggr\} 698 \end{flalign*} 699 700 701 \begin{flalign*} 702 \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv 703 \equiv& \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} p \biggr\} \; b_t &&&\\ 704 \equiv& \sum\limits_{i,j,k} \biggl\{ \biggr\} \; b_t &&&\\ 705 \end{flalign*} 706 707 % 708 \begin{flalign*} 709 \equiv& - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 710 + \int\limits_D g\, \rho \; \frac{\partial z}{\partial t} \;dv &&& \\ 711 \end{flalign*} 712 % 706 \begin{flalign*} 707 \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ 708 % 709 & \equiv \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w \biggr\} &&&\\ 710 & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t \left( \delta_{k+1/2} [z_t] \right) e_{1w}\,e_{2w} \biggr\} &&&\\ 711 & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t b_w \biggr\} 712 \end{flalign*} 713 714 \begin{flalign*} 715 \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv 716 \equiv& \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} p \biggr\} \; b_t &&&\\ 717 \equiv& \sum\limits_{i,j,k} \biggl\{ \biggr\} \; b_t &&&\\ 718 \end{flalign*} 719 720 % 721 \begin{flalign*} 722 \equiv& - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 723 + \int\limits_D g\, \rho \; \frac{\partial z}{\partial t} \;dv &&& \\ 724 \end{flalign*} 725 % 713 726 } 714 727 %end gm comment 715 716 728 717 729 Note that this property strongly constrains the discrete expression of both the depth of $T-$points and … … 719 731 Nevertheless, it is almost never satisfied since a linear equation of state is rarely used. 720 732 721 722 723 724 725 726 727 733 % ================================================================ 728 734 % Discrete Total energy Conservation : flux form … … 739 745 The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: 740 746 \begin{flalign*} 741 \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 \\747 \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 \\ 742 748 \end{flalign*} 743 749 which in flux form, it leads to: 744 750 \begin{flalign*} 745 \sum\limits_{i,j,k} \biggl\{ \frac{u }{e_{3u}}\,\frac{\partial (e_{3u}u)}{\partial t} \,b_u746 747 & - \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{e_{3u}}\frac{\partial e_{3u} }{\partial t} \,b_u 748 + \frac{v^2}{e_{3v}}\frac{\partial e_{3v} }{\partial t} \,b_v \biggr\}\\749 &= - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_3t}\frac{\partial e_{3t} \rho}{\partial t} \, g \, z_t \,b_t \biggr\} 750 - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\frac{\partial z_t}{\partial t} \,b_t \biggr\}\\751 \sum\limits_{i,j,k} \biggl\{ \frac{u }{e_{3u}}\,\frac{\partial (e_{3u}u)}{\partial t} \,b_u 752 + \frac{v }{e_{3v}}\,\frac{\partial (e_{3v}v)}{\partial t} \,b_v \biggr\} 753 & - \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{e_{3u}}\frac{\partial e_{3u} }{\partial t} \,b_u 754 + \frac{v^2}{e_{3v}}\frac{\partial e_{3v} }{\partial t} \,b_v \biggr\} \\ 755 &= - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_3t}\frac{\partial e_{3t} \rho}{\partial t} \, g \, z_t \,b_t \biggr\} 756 - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\frac{\partial z_t}{\partial t} \,b_t \biggr\} \\ 751 757 \end{flalign*} 752 758 … … 771 777 This altered Coriolis parameter is discretised at an f-point. 772 778 It is given by: 773 \ begin{equation*}774 f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\;775 \equiv \;776 f+\frac{1} {e_{1f}\,e_{2f}} \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] 777 778 \ end{equation*}779 \[ 780 f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\; 781 \equiv \; 782 f+\frac{1} {e_{1f}\,e_{2f}} \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] 783 -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u} \right] \right) 784 \] 779 785 780 786 Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. … … 793 799 Because of the centered second order scheme, it conserves the horizontal kinetic energy, that is: 794 800 795 \begin{equation} \label{eq:C_ADV_KE_flux} 796 - \int_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} 797 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 798 \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ \end{array}} } \right) \;dv 799 - \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv } =\;0 801 \begin{equation} 802 \label{eq:C_ADV_KE_flux} 803 - \int_D \textbf{U}_h \cdot \left( {{ 804 \begin{array} {*{20}c} 805 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 806 \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ 807 \end{array} 808 } } \right) \;dv 809 - \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv } =\;0 800 810 \end{equation} 801 811 … … 803 813 ($i.e.$ just the the terms associated with the i-component of the advection): 804 814 \begin{flalign*} 805 & - \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv \\806 %807 \equiv& - \sum\limits_{i,j,k} \biggl\{ \frac{1} {b_u} \biggl( 808 \delta_{i+1/2} \left[ \overline {U}^{\,i} \;\overline u^{\,i} \right]809 + \delta_j \left[ \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2} \right]810 + \delta_k \left[ \overline {W}^{\,i+1/2}\;\overline u^{\,k+1/2} \right] \biggr) \; \biggr\} \, b_u \;u &&& \\811 %812 \equiv& - \sum\limits_{i,j,k} 813 \biggl\{ 814 815 816 817 818 %819 \equiv& + \sum\limits_{i,j,k}820 \biggl\{ 821 \overline {U}^{\,i}\; \overline u^{\,i} \delta_i \left[ u \right] 822 + \overline {V}^{\,i+1/2}\; \overline u^{\,j+1/2} \delta_{j+1/2} \left[ u \right]823 824 %825 \equiv& + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ 826 \overline{U}^{\,i} \delta_i \left[ u^2 \right] 827 828 829 %830 \equiv& - \sum\limits_{i,j,k} \frac{1}{2} \bigg\{ 831 832 833 834 %835 \equiv& - \sum\limits_{i,j,k} \frac{1}{2} \overline {u^2}^{\,i} \biggl\{ 836 837 838 839 %840 \equiv& + \sum\limits_{i,j,k} \frac{1}{2} \overline {u^2}^{\,i} 841 815 & - \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv \\ 816 % 817 \equiv& - \sum\limits_{i,j,k} \biggl\{ \frac{1} {b_u} \biggl( 818 \delta_{i+1/2} \left[ \overline {U}^{\,i} \;\overline u^{\,i} \right] 819 + \delta_j \left[ \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] 820 + \delta_k \left[ \overline {W}^{\,i+1/2}\;\overline u^{\,k+1/2} \right] \biggr) \; \biggr\} \, b_u \;u &&& \\ 821 % 822 \equiv& - \sum\limits_{i,j,k} 823 \biggl\{ 824 \delta_{i+1/2} \left[ \overline {U}^{\,i}\; \overline u^{\,i} \right] 825 + \delta_j \left[ \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] 826 + \delta_k \left[ \overline {W}^{\,i+12}\;\overline u^{\,k+1/2} \right] 827 \; \biggr\} \; u \\ 828 % 829 \equiv& + \sum\limits_{i,j,k} 830 \biggl\{ 831 \overline {U}^{\,i}\; \overline u^{\,i} \delta_i \left[ u \right] 832 + \overline {V}^{\,i+1/2}\; \overline u^{\,j+1/2} \delta_{j+1/2} \left[ u \right] 833 + \overline {W}^{\,i+1/2}\; \overline u^{\,k+1/2} \delta_{k+1/2} \left[ u \right] \biggr\} && \\ 834 % 835 \equiv& + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ 836 \overline{U}^{\,i} \delta_i \left[ u^2 \right] 837 + \overline{V}^{\,i+1/2} \delta_{j+/2} \left[ u^2 \right] 838 + \overline{W}^{\,i+1/2} \delta_{k+1/2} \left[ u^2 \right] \biggr\} && \\ 839 % 840 \equiv& - \sum\limits_{i,j,k} \frac{1}{2} \bigg\{ 841 U \; \delta_{i+1/2} \left[ \overline {u^2}^{\,i} \right] 842 + V \; \delta_{j+1/2} \left[ \overline {u^2}^{\,i} \right] 843 + W \; \delta_{k+1/2} \left[ \overline {u^2}^{\,i} \right] \biggr\} &&& \\ 844 % 845 \equiv& - \sum\limits_{i,j,k} \frac{1}{2} \overline {u^2}^{\,i} \biggl\{ 846 \delta_{i+1/2} \left[ U \right] 847 + \delta_{j+1/2} \left[ V \right] 848 + \delta_{k+1/2} \left[ W \right] \biggr\} &&& \\ 849 % 850 \equiv& + \sum\limits_{i,j,k} \frac{1}{2} \overline {u^2}^{\,i} 851 \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} &&& \\ 842 852 \end{flalign*} 843 853 Applying similar manipulation applied to the second term of the scalar product leads to: 844 \begin{equation*} 845 - \int_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} 846 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 847 \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ \end{array}} } \right) \;dv 848 \equiv + \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) 849 \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} 850 \end{equation*} 854 \[ 855 - \int_D \textbf{U}_h \cdot \left( {{ 856 \begin{array} {*{20}c} 857 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 858 \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ 859 \end{array} 860 } } \right) \;dv 861 \equiv + \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) 862 \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} 863 \] 851 864 which is the discrete form of $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv $. 852 865 \autoref{eq:C_ADV_KE_flux} is thus satisfied. 853 854 866 855 867 When the UBS scheme is used to evaluate the flux form momentum advection, … … 857 869 The horizontal kinetic energy is not conserved, but forced to decay ($i.e.$ the scheme is diffusive). 858 870 859 860 861 862 863 864 865 866 867 868 871 % ================================================================ 869 872 % Discrete Enstrophy Conservation … … 872 875 \label{sec:C.4} 873 876 874 875 877 % ------------------------------------------------------------------------------------------------------------- 876 878 % Vorticity Term with ENS scheme … … 880 882 881 883 In the ENS scheme, the vorticity term is descretized as follows: 882 \begin{equation} \tag{\ref{eq:dynvor_ens}} 883 \left\{ \begin{aligned} 884 +\frac{1}{e_{1u}} & \overline{q}^{\,i} & {\overline{ \overline{\left( e_{1v}\,e_{3v}\; v \right) } } }^{\,i, j+1/2} \\ 885 - \frac{1}{e_{2v}} & \overline{q}^{\,j} & {\overline{ \overline{\left( e_{2u}\,e_{3u}\; u \right) } } }^{\,i+1/2, j} 886 \end{aligned} \right. 884 \begin{equation} 885 \tag{\ref{eq:dynvor_ens}} 886 \left\{ 887 \begin{aligned} 888 +\frac{1}{e_{1u}} & \overline{q}^{\,i} & {\overline{ \overline{\left( e_{1v}\,e_{3v}\; v \right) } } }^{\,i, j+1/2} \\ 889 - \frac{1}{e_{2v}} & \overline{q}^{\,j} & {\overline{ \overline{\left( e_{2u}\,e_{3u}\; u \right) } } }^{\,i+1/2, j} 890 \end{aligned} 891 \right. 887 892 \end{equation} 888 893 … … 892 897 ( \autoref{eq:DOM_mi_adj} and \autoref{eq:DOM_di_adj}), 893 898 it can be shown that: 894 \begin{equation} \label{eq:C_1.1} 895 \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 0 899 \begin{equation} 900 \label{eq:C_1.1} 901 \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 0 896 902 \end{equation} 897 903 where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. 898 904 Indeed, using \autoref{eq:dynvor_ens}, 899 905 the discrete form of the right hand side of \autoref{eq:C_1.1} can be transformed as follow: 900 \begin{flalign*} 901 &\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 902 \left( e_3 \, q \; \textbf{k} \times \textbf{U}_h \right)\; dv \\ 903 % 904 & \qquad {\begin{array}{*{20}l} 905 &\equiv \sum\limits_{i,j,k} 906 q \ \left\{ \delta_{i+1/2} \left[ - \,\overline {q}^{\,i}\; \overline{\overline U }^{\,i,j+1/ 2} \right] 907 - \delta_{j+1/2} \left[ \overline {q}^{\,j}\; \overline{\overline V }^{\,i+1/2, j} \right] \right\} \\ 908 % 909 &\equiv \sum\limits_{i,j,k} 910 \left\{ \delta_i [q] \; \overline{q}^{\,i} \; \overline{ \overline U }^{\,i,j+1/2} 911 + \delta_j [q] \; \overline{q}^{\,j} \; \overline{\overline V }^{\,i+1/2,j} \right\} && \\ 912 % 913 &\equiv \,\frac{1} {2} \sum\limits_{i,j,k} 914 \left\{ \delta_i \left[ q^2 \right] \; \overline{\overline U }^{\,i,j+1/2} 915 + \delta_j \left[ q^2 \right] \; \overline{\overline V }^{\,i+1/2,j} \right\} && \\ 916 % 917 &\equiv - \frac{1} {2} \sum\limits_{i,j,k} q^2 \; 918 \left\{ \delta_{i+1/2} \left[ \overline{\overline{ U }}^{\,i,j+1/2} \right] 919 + \delta_{j+1/2} \left[ \overline{\overline{ V }}^{\,i+1/2,j} \right] \right\} && \\ 920 \end{array} } 921 % 922 \allowdisplaybreaks 923 \intertext{ Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2} 924 \left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$, 925 and introducing the horizontal divergence $\chi $, it becomes: } 926 \allowdisplaybreaks 927 % 928 & \qquad {\begin{array}{*{20}l} 929 &\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} 930 \quad \equiv 0 && 931 \end{array} } 906 \begin{flalign*} 907 &\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 908 \left( e_3 \, q \; \textbf{k} \times \textbf{U}_h \right)\; dv \\ 909 % 910 & \qquad 911 { 912 \begin{array}{*{20}l} 913 &\equiv \sum\limits_{i,j,k} 914 q \ \left\{ \delta_{i+1/2} \left[ - \,\overline {q}^{\,i}\; \overline{\overline U }^{\,i,j+1/ 2} \right] 915 - \delta_{j+1/2} \left[ \overline {q}^{\,j}\; \overline{\overline V }^{\,i+1/2, j} \right] \right\} \\ 916 % 917 &\equiv \sum\limits_{i,j,k} 918 \left\{ \delta_i [q] \; \overline{q}^{\,i} \; \overline{ \overline U }^{\,i,j+1/2} 919 + \delta_j [q] \; \overline{q}^{\,j} \; \overline{\overline V }^{\,i+1/2,j} \right\} && \\ 920 % 921 &\equiv \,\frac{1} {2} \sum\limits_{i,j,k} 922 \left\{ \delta_i \left[ q^2 \right] \; \overline{\overline U }^{\,i,j+1/2} 923 + \delta_j \left[ q^2 \right] \; \overline{\overline V }^{\,i+1/2,j} \right\} && \\ 924 % 925 &\equiv - \frac{1} {2} \sum\limits_{i,j,k} q^2 \; 926 \left\{ \delta_{i+1/2} \left[ \overline{\overline{ U }}^{\,i,j+1/2} \right] 927 + \delta_{j+1/2} \left[ \overline{\overline{ V }}^{\,i+1/2,j} \right] \right\} && \\ 928 \end{array} 929 } 930 % 931 \allowdisplaybreaks 932 \intertext{ Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2} 933 \left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$, 934 and introducing the horizontal divergence $\chi $, it becomes: } 935 \allowdisplaybreaks 936 % 937 & \qquad { 938 \begin{array}{*{20}l} 939 &\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} 940 \quad \equiv 0 && 941 \end{array} 942 } 932 943 \end{flalign*} 933 944 The later equality is obtain only when the flow is horizontally non-divergent, $i.e.$ $\chi$=$0$. 934 935 945 936 946 % ------------------------------------------------------------------------------------------------------------- … … 941 951 942 952 With the EEN scheme, the vorticity terms are represented as: 943 \begin{equation} \tag{\ref{eq:dynvor_een}} 944 \left\{ { \begin{aligned} 945 +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} 946 {^{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} \\ 947 - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} 948 {^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} \\ 949 \end{aligned} } \right. 953 \begin{equation} 954 \tag{\ref{eq:dynvor_een}} 955 \left\{ { 956 \begin{aligned} 957 +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} 958 {^{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} \\ 959 - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} 960 {^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} \\ 961 \end{aligned} 962 } \right. 950 963 \end{equation} 951 964 where the indices $i_p$ and $k_p$ take the following values: 952 965 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 953 966 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 954 \begin{equation} \tag{\ref{eq:Q_triads}} 955 _i^j \mathbb{Q}^{i_p}_{j_p} 956 = \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) 967 \begin{equation} 968 \tag{\ref{eq:Q_triads}} 969 _i^j \mathbb{Q}^{i_p}_{j_p} 970 = \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) 957 971 \end{equation} 958 959 972 960 973 This formulation does conserve the potential enstrophy for a horizontally non-divergent flow ($i.e.$ $\chi=0$). … … 965 978 this triad only can be transformed as follow: 966 979 967 \begin{flalign*} 968 &\int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \\ 969 % 970 \equiv& \sum\limits_{i,j,k} 971 {q} \ \biggl\{ \;\; 972 \delta_{i+1/2} \left[ -\, {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}} \right] 973 - \delta_{j+1/2} \left[ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \right] 974 \;\;\biggr\} && \\ 975 % 976 \equiv& \sum\limits_{i,j,k} 977 \biggl\{ \delta_i [q] \ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}} 978 + \delta_j [q] \ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \biggr\} 979 && \\ 980 % 981 ... & &&\\ 982 &Demonstation \ to \ be \ done... &&\\ 983 ... & &&\\ 984 % 985 \equiv& \frac{1} {2} \sum\limits_{i,j,k} 986 \biggl\{ \delta_i \Bigl[ \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \Bigr]\; 987 \overline{\overline {U}}^{\,i,j+1/2} 988 + \delta_j \Bigl[ \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \Bigr]\; 989 \overline{\overline {V}}^{\,i+1/2,j} 990 \biggr\} 991 && \\ 992 % 993 \equiv& - \frac{1} {2} \sum\limits_{i,j,k} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2\; 994 \biggl\{ \delta_{i+1/2} 995 \left[ \overline{\overline {U}}^{\,i,j+1/2} \right] 996 + \delta_{j+1/2} 997 \left[ \overline{\overline {V}}^{\,i+1/2,j} \right] 998 \biggr\} && \\ 999 % 1000 \equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 1001 \; \overline{\overline{ b_t^{}\, \chi}}^{\,i+1/2,\,j+1/2} &&\\ 1002 % 1003 \ \ \equiv& \ 0 &&\\ 1004 \end{flalign*} 1005 1006 1007 1008 980 \begin{flalign*} 981 &\int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \\ 982 % 983 \equiv& \sum\limits_{i,j,k} 984 {q} \ \biggl\{ \;\; 985 \delta_{i+1/2} \left[ -\, {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}} \right] 986 - \delta_{j+1/2} \left[ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \right] 987 \;\;\biggr\} && \\ 988 % 989 \equiv& \sum\limits_{i,j,k} 990 \biggl\{ \delta_i [q] \ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}} 991 + \delta_j [q] \ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \biggr\} 992 && \\ 993 % 994 ... & &&\\ 995 &Demonstation \ to \ be \ done... &&\\ 996 ... & &&\\ 997 % 998 \equiv& \frac{1} {2} \sum\limits_{i,j,k} 999 \biggl\{ \delta_i \Bigl[ \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \Bigr]\; 1000 \overline{\overline {U}}^{\,i,j+1/2} 1001 + \delta_j \Bigl[ \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \Bigr]\; 1002 \overline{\overline {V}}^{\,i+1/2,j} 1003 \biggr\} 1004 && \\ 1005 % 1006 \equiv& - \frac{1} {2} \sum\limits_{i,j,k} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2\; 1007 \biggl\{ \delta_{i+1/2} 1008 \left[ \overline{\overline {U}}^{\,i,j+1/2} \right] 1009 + \delta_{j+1/2} 1010 \left[ \overline{\overline {V}}^{\,i+1/2,j} \right] 1011 \biggr\} && \\ 1012 % 1013 \equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 1014 \; \overline{\overline{ b_t^{}\, \chi}}^{\,i+1/2,\,j+1/2} &&\\ 1015 % 1016 \ \ \equiv& \ 0 &&\\ 1017 \end{flalign*} 1009 1018 1010 1019 % ================================================================ … … 1013 1022 \section{Conservation properties on tracers} 1014 1023 \label{sec:C.5} 1015 1016 1024 1017 1025 All the numerical schemes used in NEMO are written such that the tracer content is conserved by … … 1032 1040 1033 1041 conservation of a tracer, $T$: 1034 \ begin{equation*}1035 \frac{\partial }{\partial t} \left( \int_D {T\;dv} \right) 1036 = \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=01037 \ end{equation*}1042 \[ 1043 \frac{\partial }{\partial t} \left( \int_D {T\;dv} \right) 1044 = \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=0 1045 \] 1038 1046 1039 1047 conservation of its variance: 1040 \begin{flalign*} 1041 \frac{\partial }{\partial t} \left( \int_D {\frac{1}{2} T^2\;dv} \right) 1042 =& \int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv } 1043 - \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv } 1044 \end{flalign*} 1045 1048 \begin{flalign*} 1049 \frac{\partial }{\partial t} \left( \int_D {\frac{1}{2} T^2\;dv} \right) 1050 =& \int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv } 1051 - \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv } 1052 \end{flalign*} 1046 1053 1047 1054 Whatever the advection scheme considered it conserves of the tracer content as … … 1051 1058 the conservation of the tracer content due to the advection tendency is obtained as follows: 1052 1059 \begin{flalign*} 1053 &\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 &&&\\1054 &\equiv - \sum\limits_{i,j,k} \biggl\{1055 1056 + \delta_j \left[ V \;\tau_v \right] \right) 1057 1058 %1059 &\equiv - \sum\limits_{i,j,k} \left\{1060 1061 1060 &\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 &&&\\ 1061 &\equiv - \sum\limits_{i,j,k} \biggl\{ 1062 \frac{1} {b_t} \left( \delta_i \left[ U \;\tau_u \right] 1063 + \delta_j \left[ V \;\tau_v \right] \right) 1064 + \frac{1} {e_{3t}} \delta_k \left[ w\;\tau_w \right] \biggl\} b_t &&&\\ 1065 % 1066 &\equiv - \sum\limits_{i,j,k} \left\{ 1067 \delta_i \left[ U \;\tau_u \right] 1068 + \delta_j \left[ V \;\tau_v \right] 1062 1069 + \delta_k \left[ W \;\tau_w \right] \right\} && \\ 1063 &\equiv 0 &&&1070 &\equiv 0 &&& 1064 1071 \end{flalign*} 1065 1072 … … 1068 1075 It can be demonstarted as follows: 1069 1076 \begin{flalign*} 1070 &\int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv }1071 = - \int\limits_D \tau\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\1072 \equiv& - \sum\limits_{i,j,k} T\;1073 1074 1075 1076 1077 \equiv& + \sum\limits_{i,j,k} 1078 \left\{ U \overline T^{\,i+1/2} \,\delta_{i+1/2} \left[ T \right] 1079 1080 1081 \equiv& + \frac{1} {2} \sum\limits_{i,j,k}1082 1083 1084 + W \;\delta_{k+1/2} \left[ T^2 \right] \Bigr\} && \\1085 \equiv& - \frac{1} {2} \sum\limits_{i,j,k} T^21086 1087 1088 1089 \equiv& + \frac{1} {2} \sum\limits_{i,j,k} T^21090 1077 &\int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv } 1078 = - \int\limits_D \tau\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\ 1079 \equiv& - \sum\limits_{i,j,k} T\; 1080 \left\{ 1081 \delta_i \left[ U \overline T^{\,i+1/2} \right] 1082 + \delta_j \left[ V \overline T^{\,j+1/2} \right] 1083 + \delta_k \left[ W \overline T^{\,k+1/2} \right] \right\} && \\ 1084 \equiv& + \sum\limits_{i,j,k} 1085 \left\{ U \overline T^{\,i+1/2} \,\delta_{i+1/2} \left[ T \right] 1086 + V \overline T^{\,j+1/2} \;\delta_{j+1/2} \left[ T \right] 1087 + W \overline T^{\,k+1/2}\;\delta_{k+1/2} \left[ T \right] \right\} &&\\ 1088 \equiv& + \frac{1} {2} \sum\limits_{i,j,k} 1089 \Bigl\{ U \;\delta_{i+1/2} \left[ T^2 \right] 1090 + V \;\delta_{j+1/2} \left[ T^2 \right] 1091 + W \;\delta_{k+1/2} \left[ T^2 \right] \Bigr\} && \\ 1092 \equiv& - \frac{1} {2} \sum\limits_{i,j,k} T^2 1093 \Bigl\{ \delta_i \left[ U \right] 1094 + \delta_j \left[ V \right] 1095 + \delta_k \left[ W \right] \Bigr\} &&& \\ 1096 \equiv& + \frac{1} {2} \sum\limits_{i,j,k} T^2 1097 \Bigl\{ \frac{1}{e_{3t}} \frac{\partial e_{3t}\,T }{\partial t} \Bigr\} &&& \\ 1091 1098 \end{flalign*} 1092 1099 which is the discrete form of $ \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv }$. … … 1097 1104 \section{Conservation properties on lateral momentum physics} 1098 1105 \label{sec:dynldf_properties} 1099 1100 1106 1101 1107 The discrete formulation of the horizontal diffusion of momentum ensures … … 1122 1128 The lateral momentum diffusion term conserves the potential vorticity: 1123 1129 \begin{flalign*} 1124 &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 1125 1126 - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv \\ 1127 %\end{flalign*}1128 %%%%%%%%%%recheck here.... (gm)1129 %\begin{flalign*}1130 =& \int \limits_D -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times 1131 \Bigl[ \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv \\ 1132 %\end{flalign*}1133 %\begin{flalign*}1134 \equiv& \sum\limits_{i,j}1135 1136 1137 1138 \right\} \\ 1139 %1140 \intertext{Using \autoref{eq:DOM_di_adj}, it follows:}1141 %1142 \equiv& \sum\limits_{i,j,k} 1143 1144 1130 &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 1131 \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right) 1132 - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv \\ 1133 % \end{flalign*} 1134 %%%%%%%%%% recheck here.... (gm) 1135 % \begin{flalign*} 1136 =& \int \limits_D -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times 1137 \Bigl[ \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv \\ 1138 % \end{flalign*} 1139 % \begin{flalign*} 1140 \equiv& \sum\limits_{i,j} 1141 \left\{ 1142 \delta_{i+1/2} \left[ \frac {e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] \right] 1143 + \delta_{j+1/2} \left[ \frac {e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] \right] 1144 \right\} \\ 1145 % 1146 \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} 1147 % 1148 \equiv& \sum\limits_{i,j,k} 1149 -\,\left\{ 1150 \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_i \left[ 1\right] 1145 1151 + \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_j \left[ 1\right] 1146 \right\} \quad \equiv 0 1147 \\ 1152 \right\} \quad \equiv 0 1153 \\ 1148 1154 \end{flalign*} 1149 1155 … … 1156 1162 The lateral momentum diffusion term dissipates the horizontal kinetic energy: 1157 1163 %\begin{flalign*} 1158 \ begin{equation*}1159 \begin{split}1160 \int_D \textbf{U}_h \cdot 1161 \left[ \nabla_h \right. & \left. \left( A^{\,lm}\;\chi \right) 1162 1163 \\ %%%1164 \equiv& \sum\limits_{i,j,k} 1165 1166 1167 1168 1169 &\;\; + \left\{1170 \frac{1} {e_{2u}} \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right] 1171 + \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] 1172 \right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v \qquad \\ 1173 \\ %%%1174 \equiv& \sum\limits_{i,j,k} 1175 1176 1177 1178 \Bigl\} 1179 \\ 1180 &\;\; + \Bigl\{1181 1182 1183 \Bigl\} \\ 1184 \\ %%%1185 \equiv& \sum\limits_{i,j,k} 1186 1187 1188 + \delta_j \left[ e_{1v}\,e_{3v} \;v \right] 1189 \Bigr) \; A_T^{\,lm} \chi \\ 1190 &\;\; - \Bigl(1191 1192 - \delta_{j+1/2} \left[ e_{1u} \;u \right] 1193 \Bigr)\; A_f^{\,lm} e_{3f} \zeta \\ 1194 \\ %%%1195 \equiv& \sum\limits_{i,j,k} 1196 1197 - A_f ^{\,lm} \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f} 1198 \quad \leq 0 \\1199 \end{split}1200 \ end{equation*}1164 \[ 1165 \begin{split} 1166 \int_D \textbf{U}_h \cdot 1167 \left[ \nabla_h \right. & \left. \left( A^{\,lm}\;\chi \right) 1168 - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right] \; dv \\ 1169 \\ %%% 1170 \equiv& \sum\limits_{i,j,k} 1171 \left\{ 1172 \frac{1} {e_{1u}} \delta_{i+1/2} \left[ A_T^{\,lm} \chi \right] 1173 - \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] 1174 \right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u \\ 1175 &\;\; + \left\{ 1176 \frac{1} {e_{2u}} \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right] 1177 + \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] 1178 \right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v \qquad \\ 1179 \\ %%% 1180 \equiv& \sum\limits_{i,j,k} 1181 \Bigl\{ 1182 e_{2u}\,e_{3u} \;u\; \delta_{i+1/2} \left[ A_T^{\,lm} \chi \right] 1183 - e_{1u} \;u\; \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] 1184 \Bigl\} 1185 \\ 1186 &\;\; + \Bigl\{ 1187 e_{1v}\,e_{3v} \;v\; \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right] 1188 + e_{2v} \;v\; \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] 1189 \Bigl\} \\ 1190 \\ %%% 1191 \equiv& \sum\limits_{i,j,k} 1192 - \Bigl( 1193 \delta_i \left[ e_{2u}\,e_{3u} \;u \right] 1194 + \delta_j \left[ e_{1v}\,e_{3v} \;v \right] 1195 \Bigr) \; A_T^{\,lm} \chi \\ 1196 &\;\; - \Bigl( 1197 \delta_{i+1/2} \left[ e_{2v} \;v \right] 1198 - \delta_{j+1/2} \left[ e_{1u} \;u \right] 1199 \Bigr)\; A_f^{\,lm} e_{3f} \zeta \\ 1200 \\ %%% 1201 \equiv& \sum\limits_{i,j,k} 1202 - A_T^{\,lm} \,\chi^2 \;e_{1t}\,e_{2t}\,e_{3t} 1203 - A_f ^{\,lm} \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f} 1204 \quad \leq 0 \\ 1205 \end{split} 1206 \] 1201 1207 1202 1208 % ------------------------------------------------------------------------------------------------------------- … … 1208 1214 The lateral momentum diffusion term dissipates the enstrophy when the eddy coefficients are horizontally uniform: 1209 1215 \begin{flalign*} 1210 &\int\limits_D \zeta \; \textbf{k} \cdot \nabla \times 1211 1212 1213 &\quad = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 1214 1215 &\quad \equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f} 1216 1217 + \delta_{j+1/2} \left[ \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right] \right\} &&&\\ 1218 %1219 \intertext{Using \autoref{eq:DOM_di_adj}, it follows:}1220 %1221 &\quad \equiv - A^{\,lm} \sum\limits_{i,j,k} 1222 1223 1216 &\int\limits_D \zeta \; \textbf{k} \cdot \nabla \times 1217 \left[ \nabla_h \left( A^{\,lm}\;\chi \right) 1218 - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \right]\;dv &&&\\ 1219 &\quad = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 1220 \left[ \nabla_h \times \left( \zeta \; \textbf{k} \right) \right]\;dv &&&\\ 1221 &\quad \equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f} 1222 \left\{ \delta_{i+1/2} \left[ \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right] 1223 + \delta_{j+1/2} \left[ \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right] \right\} &&&\\ 1224 % 1225 \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} 1226 % 1227 &\quad \equiv - A^{\,lm} \sum\limits_{i,j,k} 1228 \left\{ \left( \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right)^2 b_v 1229 + \left( \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right)^2 b_u \right\} \quad \leq \;0 &&&\\ 1224 1230 \end{flalign*} 1225 1231 … … 1234 1240 The resulting term conserves the $\chi$ and dissipates $\chi^2$ when the eddy coefficients are horizontally uniform. 1235 1241 \begin{flalign*} 1236 & \int\limits_D \nabla_h \cdot 1237 1238 1239 = \int\limits_D \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi \right) dv \\1240 %1241 &\equiv \sum\limits_{i,j,k} 1242 1243 + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] \right\} \\ 1244 %1245 \intertext{Using \autoref{eq:DOM_di_adj}, it follows:}1246 %1247 &\equiv \sum\limits_{i,j,k} 1248 - \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] 1249 + \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\} 1250 \quad \equiv 0 \\ 1242 & \int\limits_D \nabla_h \cdot 1243 \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right) 1244 - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \Bigr] dv 1245 = \int\limits_D \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi \right) dv \\ 1246 % 1247 &\equiv \sum\limits_{i,j,k} 1248 \left\{ \delta_i \left[ A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] 1249 + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] \right\} \\ 1250 % 1251 \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} 1252 % 1253 &\equiv \sum\limits_{i,j,k} 1254 - \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] 1255 + \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\} 1256 \quad \equiv 0 1251 1257 \end{flalign*} 1252 1258 … … 1258 1264 1259 1265 \begin{flalign*} 1260 &\int\limits_D \chi \;\nabla_h \cdot 1261 1262 1263 = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\; dv \\1264 %1265 &\equiv A^{\,lm} \sum\limits_{i,j,k} \frac{1} {e_{1t}\,e_{2t}\,e_{3t}} \chi 1266 1267 1268 1269 \right\} \; e_{1t}\,e_{2t}\,e_{3t} \\ 1270 %1271 \intertext{Using \autoref{eq:DOM_di_adj}, it turns out to be:}1272 %1273 &\equiv - A^{\,lm} \sum\limits_{i,j,k}1274 1275 + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 b_v \right\}1276 \quad \leq 0 \\ 1266 &\int\limits_D \chi \;\nabla_h \cdot 1267 \left[ \nabla_h \left( A^{\,lm}\;\chi \right) 1268 - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right]\; dv 1269 = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\; dv \\ 1270 % 1271 &\equiv A^{\,lm} \sum\limits_{i,j,k} \frac{1} {e_{1t}\,e_{2t}\,e_{3t}} \chi 1272 \left\{ 1273 \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] 1274 + \delta_j \left[ \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] 1275 \right\} \; e_{1t}\,e_{2t}\,e_{3t} \\ 1276 % 1277 \intertext{Using \autoref{eq:DOM_di_adj}, it turns out to be:} 1278 % 1279 &\equiv - A^{\,lm} \sum\limits_{i,j,k} 1280 \left\{ \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right)^2 b_u 1281 + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 b_v \right\} 1282 \quad \leq 0 1277 1283 \end{flalign*} 1278 1284 … … 1287 1293 The first two are associated with the conservation of momentum and the dissipation of horizontal kinetic energy: 1288 1294 \begin{align*} 1289 \int\limits_D \frac{1} {e_3 }\; \frac{\partial } {\partial k}1290 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv1291 \qquad \quad &= \vec{\textbf{0}} \\ 1292 %1293 \intertext{and}1294 %1295 \int\limits_D 1296 1297 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv \quad &\leq 0 \\ 1295 \int\limits_D \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1296 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv 1297 \qquad \quad &= \vec{\textbf{0}} 1298 % 1299 \intertext{and} 1300 % 1301 \int\limits_D 1302 \textbf{U}_h \cdot \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1303 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv \quad &\leq 0 1298 1304 \end{align*} 1299 1305 … … 1301 1307 The second results from: 1302 1308 \begin{flalign*} 1303 \int\limits_D 1304 1305 1306 \end{flalign*} 1307 \begin{flalign*} 1308 &\equiv \sum\limits_{i,j,k} 1309 \left( 1310 u\; \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ u \right] \right]\; e_{1u}\,e_{2u} 1311 + v\; \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right]\; e_{1v}\,e_{2v} \right) &&&\\ 1312 %1313 \intertext{since the horizontal scale factor does not depend on $k$, it follows:}1314 %1315 &\equiv - \sum\limits_{i,j,k} 1316 \left( \frac{A_u^{\,vm}} {e_{3uw}} \left( \delta_{k+1/2} \left[ u \right] \right)^2\; e_{1u}\,e_{2u} 1317 1318 \quad \leq 0 &&&\\ 1309 \int\limits_D 1310 \textbf{U}_h \cdot \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1311 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\;dv &&&\\ 1312 \end{flalign*} 1313 \begin{flalign*} 1314 &\equiv \sum\limits_{i,j,k} 1315 \left( 1316 u\; \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ u \right] \right]\; e_{1u}\,e_{2u} 1317 + v\; \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right]\; e_{1v}\,e_{2v} \right) &&& 1318 % 1319 \intertext{since the horizontal scale factor does not depend on $k$, it follows:} 1320 % 1321 &\equiv - \sum\limits_{i,j,k} 1322 \left( \frac{A_u^{\,vm}} {e_{3uw}} \left( \delta_{k+1/2} \left[ u \right] \right)^2\; e_{1u}\,e_{2u} 1323 + \frac{A_v^{\,vm}} {e_{3vw}} \left( \delta_{k+1/2} \left[ v \right] \right)^2\; e_{1v}\,e_{2v} \right) 1324 \quad \leq 0 &&& 1319 1325 \end{flalign*} 1320 1326 … … 1322 1328 Indeed: 1323 1329 \begin{flalign*} 1324 \int \limits_D 1325 \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 1326 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} \left( 1327 \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k} 1328 \right) \right)\; dv &&&\\ 1329 \end{flalign*} 1330 \begin{flalign*} 1331 \equiv \sum\limits_{i,j,k} \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}}1332 1333 \delta_{i+1/2} 1334 1335 \biggl. 1336 - \delta_{j+1/2} 1337 1338 1339 e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0 && \\ 1330 \int \limits_D 1331 \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 1332 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} \left( 1333 \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k} 1334 \right) \right)\; dv &&& 1335 \end{flalign*} 1336 \begin{flalign*} 1337 \equiv \sum\limits_{i,j,k} \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}} 1338 \bigg\{ \biggr. \quad 1339 \delta_{i+1/2} 1340 &\left( \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right] \right) &&\\ 1341 \biggl. 1342 - \delta_{j+1/2} 1343 &\left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{1} {e_{3uw}}\delta_{k+1/2} \left[ u \right] \right] \right) 1344 \biggr\} \; 1345 e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0 && 1340 1346 \end{flalign*} 1341 1347 1342 1348 If the vertical diffusion coefficient is uniform over the whole domain, the enstrophy is dissipated, $i.e.$ 1343 1349 \begin{flalign*} 1344 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 1345 1346 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\ 1350 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 1351 \left( \frac{1} {e_3}\; \frac{\partial } {\partial k} 1352 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&& 1347 1353 \end{flalign*} 1348 1354 1349 1355 This property is only satisfied in $z$-coordinates: 1350 1356 \begin{flalign*} 1351 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 1352 1353 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv &&& \\ 1354 \end{flalign*} 1355 \begin{flalign*} 1356 \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \;1357 1358 \delta_{i+1/2} 1359 1360 - \delta_{j+1/2} 1361 1362 \left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) \biggr\} &&\\ 1363 \end{flalign*} 1364 \begin{flalign*} 1365 \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} 1366 1367 \frac{1} {e_{3v}} \delta_k 1368 &\left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ \delta_{i+1/2} \left[ e_{2v}\,v \right] \right] \right] &&\\ 1369 \biggl.1370 - \frac{1} {e_{3u}} \delta_k 1371 &\left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ \delta_{j+1/2} \left[ e_{1u}\,u \right] \right] \right] \biggr\} &&\\ 1357 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 1358 \left( \frac{1} {e_3}\; \frac{\partial } {\partial k} 1359 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv &&& 1360 \end{flalign*} 1361 \begin{flalign*} 1362 \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \; 1363 \biggl\{ \biggr. \quad 1364 \delta_{i+1/2} 1365 &\left( \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}[v] \right] \right) &&\\ 1366 - \delta_{j+1/2} 1367 &\biggl. 1368 \left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) \biggr\} && 1369 \end{flalign*} 1370 \begin{flalign*} 1371 \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} 1372 \biggl\{ \biggr. \quad 1373 \frac{1} {e_{3v}} \delta_k 1374 &\left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ \delta_{i+1/2} \left[ e_{2v}\,v \right] \right] \right] &&\\ 1375 \biggl. 1376 - \frac{1} {e_{3u}} \delta_k 1377 &\left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ \delta_{j+1/2} \left[ e_{1u}\,u \right] \right] \right] \biggr\} && 1372 1378 \end{flalign*} 1373 1379 Using the fact that the vertical diffusion coefficients are uniform, … … 1375 1381 $e_{3f} =e_{3u} =e_{3v} =e_{3t} $ and $e_{3w} =e_{3uw} =e_{3vw} $, it follows: 1376 1382 \begin{flalign*} 1377 \equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k 1378 1379 - \delta_{j+1/ 2} \left[ e_{1u}\,u \right] \Bigr] \right] &&&\\ 1380 \end{flalign*} 1381 \begin{flalign*} 1382 \equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}}1383 \left( \delta_{k+1/2} \left[ \zeta \right] \right)^2 \; e_{1f}\,e_{2f} \; \leq 0 &&&\\ 1383 \equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k 1384 \left[ \frac{1} {e_{3w}} \delta_{k+1/2} \Bigl[ \delta_{i+1/2} \left[ e_{2v}\,v \right] 1385 - \delta_{j+1/ 2} \left[ e_{1u}\,u \right] \Bigr] \right] &&& 1386 \end{flalign*} 1387 \begin{flalign*} 1388 \equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}} 1389 \left( \delta_{k+1/2} \left[ \zeta \right] \right)^2 \; e_{1f}\,e_{2f} \; \leq 0 &&& 1384 1390 \end{flalign*} 1385 1391 Similarly, the horizontal divergence is obviously conserved: 1386 1392 1387 1393 \begin{flalign*} 1388 \int\limits_D \nabla \cdot 1389 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}1390 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\ 1394 \int\limits_D \nabla \cdot 1395 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1396 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&& 1391 1397 \end{flalign*} 1392 1398 and the square of the horizontal divergence decreases ($i.e.$ the horizontal divergence is dissipated) if … … 1394 1400 1395 1401 \begin{flalign*} 1396 \int\limits_D \chi \;\nabla \cdot 1397 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}1398 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\ 1402 \int\limits_D \chi \;\nabla \cdot 1403 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1404 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&& 1399 1405 \end{flalign*} 1400 1406 This property is only satisfied in the $z$-coordinate: 1401 1407 \begin{flalign*} 1402 \int\limits_D \chi \;\nabla \cdot 1403 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}1404 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv &&&\\ 1405 \end{flalign*} 1406 \begin{flalign*} 1407 \equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1t}\,e_{2t}}1408 1409 \delta_{i+1/2} 1410 &\left( \frac{e_{2u}} {e_{3u}} \delta_k 1411 \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) &&\\ 1412 \Biggl.1413 + \delta_{j+1/2} 1414 &\left( \frac{e_{1v}} {e_{3v}} \delta_k 1415 1416 \Biggr\} \; e_{1t}\,e_{2t}\,e_{3t} &&\\ 1417 \end{flalign*} 1418 1419 \begin{flalign*} 1420 \equiv A^{\,vm} \sum\limits_{i,j,k} \chi \,1421 1422 1423 &\left( 1424 \delta_k \left[ 1425 \frac{1} {e_{3uw}} \delta_{k+1/2} \left[ e_{2u}\,u \right] \right] \right) && \\ 1426 \biggl. 1427 + \delta_{j+1/2} 1428 &\left( \delta_k \left[ 1429 \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ e_{1v}\,v \right] \right] \right) \biggr\} && \\ 1430 \end{flalign*} 1431 1432 \begin{flalign*} 1433 \equiv -A^{\,vm} \sum\limits_{i,j,k} 1434 \frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\; \biggl\{ 1435 1436 1437 + \delta_{j+1/2} \left[ e_{1v}\,v \right] \Bigr] \biggr\} &&&\\ 1438 \end{flalign*} 1439 1440 \begin{flalign*} 1441 \equiv -A^{\,vm} \sum\limits_{i,j,k}1442 \frac{1} {e_{3w}} \delta_{k+1/2} \left[ \chi \right]\; \delta_{k+1/2} \left[ e_{1t}\,e_{2t} \;\chi \right] &&&\\1443 \end{flalign*} 1444 1445 \begin{flalign*} 1446 \equiv -A^{\,vm} \sum\limits_{i,j,k} 1447 \frac{e_{1t}\,e_{2t}} {e_{3w}}\; \left( \delta_{k+1/2} \left[ \chi \right] \right)^2 \quad \equiv 0 &&&\\ 1408 \int\limits_D \chi \;\nabla \cdot 1409 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} 1410 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv &&& 1411 \end{flalign*} 1412 \begin{flalign*} 1413 \equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1t}\,e_{2t}} 1414 \biggl\{ \Biggr. \quad 1415 \delta_{i+1/2} 1416 &\left( \frac{e_{2u}} {e_{3u}} \delta_k 1417 \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) &&\\ 1418 \Biggl. 1419 + \delta_{j+1/2} 1420 &\left( \frac{e_{1v}} {e_{3v}} \delta_k 1421 \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} [v] \right] \right) 1422 \Biggr\} \; e_{1t}\,e_{2t}\,e_{3t} && 1423 \end{flalign*} 1424 1425 \begin{flalign*} 1426 \equiv A^{\,vm} \sum\limits_{i,j,k} \chi \, 1427 \biggl\{ \biggr. \quad 1428 \delta_{i+1/2} 1429 &\left( 1430 \delta_k \left[ 1431 \frac{1} {e_{3uw}} \delta_{k+1/2} \left[ e_{2u}\,u \right] \right] \right) && \\ 1432 \biggl. 1433 + \delta_{j+1/2} 1434 &\left( \delta_k \left[ 1435 \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ e_{1v}\,v \right] \right] \right) \biggr\} && 1436 \end{flalign*} 1437 1438 \begin{flalign*} 1439 \equiv -A^{\,vm} \sum\limits_{i,j,k} 1440 \frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\; \biggl\{ 1441 \delta_{k+1/2} \Bigl[ 1442 \delta_{i+1/2} \left[ e_{2u}\,u \right] 1443 + \delta_{j+1/2} \left[ e_{1v}\,v \right] \Bigr] \biggr\} &&& 1444 \end{flalign*} 1445 1446 \begin{flalign*} 1447 \equiv -A^{\,vm} \sum\limits_{i,j,k} 1448 \frac{1} {e_{3w}} \delta_{k+1/2} \left[ \chi \right]\; \delta_{k+1/2} \left[ e_{1t}\,e_{2t} \;\chi \right] &&& 1449 \end{flalign*} 1450 1451 \begin{flalign*} 1452 \equiv -A^{\,vm} \sum\limits_{i,j,k} 1453 \frac{e_{1t}\,e_{2t}} {e_{3w}}\; \left( \delta_{k+1/2} \left[ \chi \right] \right)^2 \quad \equiv 0 &&& 1448 1454 \end{flalign*} 1449 1455 … … 1468 1474 constraint of conservation of tracers: 1469 1475 \begin{flalign*} 1470 &\int\limits_D \nabla \cdot \left( A\;\nabla T \right)\;dv &&&\\ 1471 \\ 1472 &\equiv \sum\limits_{i,j,k} 1473 \biggl\{ \biggr. 1474 \delta_i 1475 \left[ 1476 A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} 1477 \left[ T \right] 1478 \right] 1479 + \delta_j 1480 \left[ 1481 A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} 1482 \left[ T \right] 1483 \right] 1484 &&\\ & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\; 1485 + \delta_k 1486 \left[ 1487 A_w^{\,vT} \frac{e_{1t}\,e_{2t}} {e_{3t}} \delta_{k+1/2} 1488 \left[ T \right] 1489 \right] 1490 \biggr\} \quad \equiv 0 1491 &&\\ 1476 &\int\limits_D \nabla \cdot \left( A\;\nabla T \right)\;dv &&& \\ \\ 1477 &\equiv \sum\limits_{i,j,k} 1478 \biggl\{ \biggr. 1479 \delta_i 1480 \left[ 1481 A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} 1482 \left[ T \right] 1483 \right] 1484 + \delta_j 1485 \left[ 1486 A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} 1487 \left[ T \right] 1488 \right] && \\ 1489 & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\; 1490 + \delta_k 1491 \left[ 1492 A_w^{\,vT} \frac{e_{1t}\,e_{2t}} {e_{3t}} \delta_{k+1/2} 1493 \left[ T \right] 1494 \right] 1495 \biggr\} \quad \equiv 0 1496 && 1492 1497 \end{flalign*} 1493 1498 … … 1502 1507 constraint on the dissipation of tracer variance: 1503 1508 \begin{flalign*} 1504 \int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv &&&\\ 1505 &\equiv \sum\limits_{i,j,k} \; T 1506 \biggl\{ \biggr. 1507 \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right] 1508 & + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right] 1509 \quad&& \\ 1510 \biggl. 1511 &&+ \delta_k \left[A_w^{\,vT}\frac{e_{1t}\,e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right] 1512 \biggr\} && 1513 \end{flalign*} 1514 \begin{flalign*} 1515 \equiv - \sum\limits_{i,j,k} 1516 \biggl\{ \biggr. \quad 1517 & A_u^{\,lT} \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ T \right] \right)^2 e_{1u}\,e_{2u}\,e_{3u} && \\ 1518 & + A_v^{\,lT} \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ T \right] \right)^2 e_{1v}\,e_{2v}\,e_{3v} && \\ \biggl. 1519 & + A_w^{\,vT} \left( \frac{1} {e_{3w}} \delta_{k+1/2} \left[ T \right] \right)^2 e_{1w}\,e_{2w}\,e_{3w} \biggr\} 1520 \quad \leq 0 && \\ 1521 \end{flalign*} 1522 1509 \int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv &&&\\ 1510 &\equiv \sum\limits_{i,j,k} \; T 1511 \biggl\{ \biggr. 1512 \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right] 1513 & + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right] 1514 \quad&& \\ 1515 \biggl. 1516 &&+ \delta_k \left[A_w^{\,vT}\frac{e_{1t}\,e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right] 1517 \biggr\} && 1518 \end{flalign*} 1519 \begin{flalign*} 1520 \equiv - \sum\limits_{i,j,k} 1521 \biggl\{ \biggr. \quad 1522 & A_u^{\,lT} \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ T \right] \right)^2 e_{1u}\,e_{2u}\,e_{3u} && \\ 1523 & + A_v^{\,lT} \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ T \right] \right)^2 e_{1v}\,e_{2v}\,e_{3v} && \\ \biggl. 1524 & + A_w^{\,vT} \left( \frac{1} {e_{3w}} \delta_{k+1/2} \left[ T \right] \right)^2 e_{1w}\,e_{2w}\,e_{3w} \biggr\} 1525 \quad \leq 0 && 1526 \end{flalign*} 1523 1527 1524 1528 %%%% end of appendix in gm comment 1525 1529 %} 1530 \biblio 1531 1526 1532 \end{document}
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