Changeset 10419 for NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_E.tex
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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{Note on some algorithms} 7 8 \label{apdx:E} 9 8 10 \minitoc 9 11 10 12 \newpage 11 $\ $\newline % force a new ligne12 13 13 14 This appendix some on going consideration on algorithms used or planned to be used in \NEMO. 14 15 $\ $\newline % force a new ligne16 15 17 16 % ------------------------------------------------------------------------------------------------------------- … … 25 24 It is also known as Cell Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective Kinematics). 26 25 For example, in the $i$-direction: 27 \begin{equation} \label{eq:tra_adv_ubs2} 28 \tau _u^{ubs} = \left\{ \begin{aligned} 29 & \tau _u^{cen4} + \frac{1}{12} \,\tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 30 & \tau _u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 31 \end{aligned} \right. 26 \begin{equation} 27 \label{eq:tra_adv_ubs2} 28 \tau_u^{ubs} = \left\{ 29 \begin{aligned} 30 & \tau_u^{cen4} + \frac{1}{12} \,\tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 31 & \tau_u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 32 \end{aligned} 33 \right. 32 34 \end{equation} 33 35 or equivalently, the advective flux is 34 \begin{equation} \label{eq:tra_adv_ubs2} 35 U_{i+1/2} \ \tau _u^{ubs} 36 =U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2} 37 - \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 36 \begin{equation} 37 \label{eq:tra_adv_ubs2} 38 U_{i+1/2} \ \tau_u^{ubs} 39 =U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2} 40 - \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 38 41 \end{equation} 39 42 where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and 40 $\tau "_i =\delta _i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$.43 $\tau "_i =\delta_i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$. 41 44 By choosing this expression for $\tau "$ we consider a fourth order approximation of $\partial_i^2$ with 42 45 a constant i-grid spacing ($\Delta i=1$). 43 46 44 47 Alternative choice: introduce the scale factors: 45 $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta _{i+1/2}[\tau] \right]$. 46 48 $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta_{i+1/2}[\tau] \right]$. 47 49 48 50 This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error … … 76 78 77 79 NB 2: In a forthcoming release four options will be proposed for the vertical component used in the UBS scheme. 78 $\tau 80 $\tau_w^{ubs}$ will be evaluated using either \textit{(a)} a centered $2^{nd}$ order scheme, 79 81 or \textit{(b)} a TVD scheme, or \textit{(c)} an interpolation based on conservative parabolic splines following 80 82 \citet{Shchepetkin_McWilliams_OM05} implementation of UBS in ROMS, or \textit{(d)} an UBS. … … 82 84 83 85 NB 3: It is straight forward to rewrite \autoref{eq:tra_adv_ubs} as follows: 84 \begin{equation} \label{eq:tra_adv_ubs2} 85 \tau _u^{ubs} = \left\{ \begin{aligned} 86 & \tau _u^{cen4} + \frac{1}{12} \tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 87 & \tau _u^{cen4} - \frac{1}{12} \tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 88 \end{aligned} \right. 86 \begin{equation} 87 \label{eq:tra_adv_ubs2} 88 \tau_u^{ubs} = \left\{ 89 \begin{aligned} 90 & \tau_u^{cen4} + \frac{1}{12} \tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 91 & \tau_u^{cen4} - \frac{1}{12} \tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 92 \end{aligned} 93 \right. 89 94 \end{equation} 90 95 or equivalently 91 \begin{equation} \label{eq:tra_adv_ubs2} 92 \begin{split} 93 e_{2u} e_{3u}\,u_{i+1/2} \ \tau _u^{ubs} 94 &= e_{2u} e_{3u}\,u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\tau"_i }^{\,i+1/2} \\ 95 & - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 96 \end{split} 96 \begin{equation} 97 \label{eq:tra_adv_ubs2} 98 \begin{split} 99 e_{2u} e_{3u}\,u_{i+1/2} \ \tau_u^{ubs} 100 &= e_{2u} e_{3u}\,u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\tau"_i }^{\,i+1/2} \\ 101 & - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 102 \end{split} 97 103 \end{equation} 98 104 \autoref{eq:tra_adv_ubs2} has several advantages. … … 105 111 106 112 laplacian diffusion: 107 \begin{equation} \label{eq:tra_ldf_lap}108 \begin{split}109 D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\; e_{3T} } &\left[ {\quad \delta _i 110 \left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2} 111 \left[ T \right]} \right]} \right. 112 \\113 &\ \left. {+\; \delta _j \left[ 114 {A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta _{j+1/2} \left[ T 115 \right]} \right)} \right]\quad } \right]116 \end{split}113 \begin{equation} 114 \label{eq:tra_ldf_lap} 115 \begin{split} 116 D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\; e_{3T} } &\left[ {\quad \delta_i 117 \left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta_{i+1/2} 118 \left[ T \right]} \right]} \right. \\ 119 &\ \left. {+\; \delta_j \left[ 120 {A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta_{j+1/2} \left[ T 121 \right]} \right)} \right]\quad } \right] 122 \end{split} 117 123 \end{equation} 118 124 119 125 bilaplacian: 120 \begin{equation} \label{eq:tra_ldf_lap} 121 \begin{split} 122 D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ 123 & \delta _i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta _{i+1/2} 124 \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 125 \delta _i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta _{i+1/2} 126 [T] \right] \right] \right] 127 \end{split} 126 \begin{equation} 127 \label{eq:tra_ldf_lap} 128 \begin{split} 129 D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ 130 & \delta_i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2} 131 \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 132 \delta_i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2} 133 [T] \right] \right] \right] 134 \end{split} 128 135 \end{equation} 129 136 with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$, 130 137 $i.e.$ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$ 131 138 it comes: 132 \begin{equation} \label{eq:tra_ldf_lap} 133 \begin{split} 134 D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ 135 & \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta _{i+1/2} 136 \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 137 \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta _{i+1/2} 138 [T] \right] \right] \right] 139 \end{split} 139 \begin{equation} 140 \label{eq:tra_ldf_lap} 141 \begin{split} 142 D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ 143 & \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2} 144 \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 145 \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2} 146 [T] \right] \right] \right] 147 \end{split} 140 148 \end{equation} 141 149 if the velocity is uniform ($i.e.$ $|u|=cst$) then the diffusive flux is 142 \begin{equation} \label{eq:tra_ldf_lap} 143 \begin{split} 144 F_u^{lT} = - \frac{1}{12} 145 e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta _{i+1/2} 146 \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 147 \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta _{i+1/2} 148 [T] \right] \right] 149 \end{split} 150 \begin{equation} 151 \label{eq:tra_ldf_lap} 152 \begin{split} 153 F_u^{lT} = - \frac{1}{12} 154 e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta_{i+1/2} 155 \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 156 \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta_{i+1/2} 157 [T] \right] \right] 158 \end{split} 150 159 \end{equation} 151 160 beurk.... reverte the logic: starting from the diffusive part of the advective flux it comes: 152 161 153 \begin{equation} \label{eq:tra_adv_ubs2}154 \begin{split}155 F_u^{lT}156 &= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]157 \end{split}162 \begin{equation} 163 \label{eq:tra_adv_ubs2} 164 \begin{split} 165 F_u^{lT} &= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 166 \end{split} 158 167 \end{equation} 159 168 if the velocity is uniform ($i.e.$ $|u|=cst$) and 160 choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right]$169 choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right]$ 161 170 162 171 sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$): 163 \begin{equation} \label{eq:tra_adv_ubs2}164 \begin{split}165 F_u^{lT}166 &= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right]167 \end{split}172 \begin{equation} 173 \label{eq:tra_adv_ubs2} 174 \begin{split} 175 F_u^{lT} &= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] 176 \end{split} 168 177 \end{equation} 169 178 which leads to ${A_T^{lT}}^2 = \frac{1}{12} {e_{1T}}^3\ \overline{|u|}^{\,i+1/2}$ 170 179 171 180 sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$ 172 \begin{equation} \label{eq:tra_adv_ubs2}173 \begin{split}174 F_u^{lT}175 &= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta _i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] \\176 &= - \frac{1}{12} e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{1T}\,e_{2T}\,e_{3T}}\,\delta _i \left[ e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u} }{e_{1u}} \delta_{i+1/2}[\tau] \right] \right]177 \end{split}181 \begin{equation} 182 \label{eq:tra_adv_ubs2} 183 \begin{split} 184 F_u^{lT} &= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta_i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] \\ 185 &= - \frac{1}{12} e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{1T}\,e_{2T}\,e_{3T}}\,\delta_i \left[ e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u} }{e_{1u}} \delta_{i+1/2}[\tau] \right] \right] 186 \end{split} 178 187 \end{equation} 179 188 which leads to ${A_u^{lT}} = \frac{1}{12} {e_{1u}}^3\ |u|$ 180 181 189 182 190 % ------------------------------------------------------------------------------------------------------------- … … 189 197 Given the values of a variable $q$ at successive time step, 190 198 the time derivation and averaging operators at the mid time step are: 191 \begin{subequations} \label{eq:dt_mt} 192 \begin{align} 193 \delta _{t+\rdt/2} [q] &= \ \ \, q^{t+\rdt} - q^{t} \\ 194 \overline q^{\,t+\rdt/2} &= \left\{ q^{t+\rdt} + q^{t} \right\} \; / \; 2 195 \end{align} 196 \end{subequations} 199 \[ 200 % \label{eq:dt_mt} 201 \begin{split} 202 \delta_{t+\rdt/2} [q] &= \ \ \, q^{t+\rdt} - q^{t} \\ 203 \overline q^{\,t+\rdt/2} &= \left\{ q^{t+\rdt} + q^{t} \right\} \; / \; 2 204 \end{split} 205 \] 197 206 As for space operator, 198 207 the adjoint of the derivation and averaging time operators are $\delta_t^*=\delta_{t+\rdt/2}$ and … … 200 209 201 210 The Leap-frog time stepping given by \autoref{eq:DOM_nxt} can be defined as: 202 \begin{equation} \label{eq:LF} 203 \frac{\partial q}{\partial t} 204 \equiv \frac{1}{\rdt} \overline{ \delta _{t+\rdt/2}[q]}^{\,t} 205 = \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} 206 \end{equation} 211 \[ 212 % \label{eq:LF} 213 \frac{\partial q}{\partial t} 214 \equiv \frac{1}{\rdt} \overline{ \delta_{t+\rdt/2}[q]}^{\,t} 215 = \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} 216 \] 207 217 Note that \autoref{chap:LF} shows that the leapfrog time step is $\rdt$, 208 218 not $2\rdt$ as it can be found sometimes in literature. 209 219 The leap-Frog time stepping is a second order centered scheme. 210 220 As such it respects the quadratic invariant in integral forms, $i.e.$ the following continuous property, 211 \begin{equation} \label{eq:Energy} 212 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 213 =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} 214 = \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) , 215 \end{equation} 221 \[ 222 % \label{eq:Energy} 223 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 224 =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} 225 = \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) , 226 \] 216 227 is satisfied in discrete form. 217 228 Indeed, 218 \begin{equation} \begin{split} 219 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 220 &\equiv \sum\limits_{0}^{N} 221 {\frac{1}{\rdt} q^t \ \overline{ \delta _{t+\rdt/2}[q]}^{\,t} \ \rdt} 222 \equiv \sum\limits_{0}^{N} { q^t \ \overline{ \delta _{t+\rdt/2}[q]}^{\,t} } \\ 223 &\equiv \sum\limits_{0}^{N} { \overline{q}^{\,t+\Delta/2}{ \delta _{t+\rdt/2}[q]}} 224 \equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta _{t+\rdt/2}[q^2] }\\ 225 &\equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta _{t+\rdt/2}[q^2] } 226 \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) 227 \end{split} \end{equation} 229 \[ 230 \begin{split} 231 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 232 &\equiv \sum\limits_{0}^{N} 233 {\frac{1}{\rdt} q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} \ \rdt} 234 \equiv \sum\limits_{0}^{N} { q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} } \\ 235 &\equiv \sum\limits_{0}^{N} { \overline{q}^{\,t+\Delta/2}{ \delta_{t+\rdt/2}[q]}} 236 \equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta_{t+\rdt/2}[q^2] }\\ 237 &\equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta_{t+\rdt/2}[q^2] } 238 \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) 239 \end{split} 240 \] 228 241 NB here pb of boundary condition when applying the adjoint! 229 242 In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition 230 243 (equivalently of the boundary value of the integration by part). 231 244 In time this boundary condition is not physical and \textbf{add something here!!!} 232 233 234 235 236 237 245 238 246 % ================================================================ … … 269 277 a derivative in the same direction by considering triads. 270 278 For example in the (\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows: 271 \begin{equation} \label{eq:Gf_triads} 272 _i^k \mathbb{T}_{i_p}^{k_p} (T) 273 = \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \left( 274 \frac{ \delta_{i + i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 275 -\ {_i^k \mathbb{R}_{i_p}^{k_p}} \ \frac{ \delta_{k+k_p} [T^i] }{ {e_{3w}}_{\,i}^{\,k+k_p} } 276 \right) 279 \begin{equation} 280 \label{eq:Gf_triads} 281 _i^k \mathbb{T}_{i_p}^{k_p} (T) 282 = \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \left( 283 \frac{ \delta_{i + i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 284 -\ {_i^k \mathbb{R}_{i_p}^{k_p}} \ \frac{ \delta_{k+k_p} [T^i] }{ {e_{3w}}_{\,i}^{\,k+k_p} } 285 \right) 277 286 \end{equation} 278 287 where the indices $i_p$ and $k_p$ define the four triads and take the following value: … … 281 290 $A_i^k$ is the lateral eddy diffusivity coefficient defined at $T$-point, 282 291 and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad: 283 \begin{equation} \label{eq:Gf_slopes} 284 _i^k \mathbb{R}_{i_p}^{k_p} 285 =\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac 286 {\left(\alpha / \beta \right)_i^k \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] } 287 {\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] } 292 \begin{equation} 293 \label{eq:Gf_slopes} 294 _i^k \mathbb{R}_{i_p}^{k_p} 295 =\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac 296 {\left(\alpha / \beta \right)_i^k \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] } 297 {\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] } 288 298 \end{equation} 289 299 Note that in \autoref{eq:Gf_slopes} we use the ratio $\alpha / \beta$ instead of … … 296 306 297 307 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 298 \begin{figure}[!ht] \begin{center} 299 \includegraphics[width=0.70\textwidth]{Fig_ISO_triad} 300 \caption{ \protect\label{fig:ISO_triad} 301 Triads used in the Griffies's like iso-neutral diffision scheme for 302 $u$-component (upper panel) and $w$-component (lower panel).} 303 \end{center} 308 \begin{figure}[!ht] 309 \begin{center} 310 \includegraphics[width=0.70\textwidth]{Fig_ISO_triad} 311 \caption{ 312 \protect\label{fig:ISO_triad} 313 Triads used in the Griffies's like iso-neutral diffision scheme for 314 $u$-component (upper panel) and $w$-component (lower panel). 315 } 316 \end{center} 304 317 \end{figure} 305 318 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 307 320 The four iso-neutral fluxes associated with the triads are defined at $T$-point. 308 321 They take the following expression: 309 \begin{flalign} \label{eq:Gf_fluxes} 310 \begin{split} 311 {_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) 312 &= \ \; \qquad \quad { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+i_p}^{\,k}} \\ 313 {_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) 314 &= -\; { _i^k \mathbb{R}_{i_p}^{k_p} } 315 \ \; { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+k_p}} 316 \end{split} 317 \end{flalign} 322 \begin{flalign*} 323 % \label{eq:Gf_fluxes} 324 \begin{split} 325 {_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) 326 &= \ \; \qquad \quad { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+i_p}^{\,k}} \\ 327 {_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) 328 &= -\; { _i^k \mathbb{R}_{i_p}^{k_p} } 329 \ \; { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+k_p}} 330 \end{split} 331 \end{flalign*} 318 332 319 333 The resulting iso-neutral fluxes at $u$- and $w$-points are then given by 320 334 the sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:ISO_triad}): 321 \begin{flalign} \label{eq:iso_flux}322 \textbf{F}_{iso}(T) 323 &\equiv \sum_{\substack{i_p,\,k_p}} 324 \begin{pmatrix}325 {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\326 327 {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) \\328 \end{pmatrix} \notag \\329 &\notag \\330 &\equiv \sum_{\substack{i_p,\,k_p}} 331 \begin{pmatrix}332 && { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+1/2}^{\,k} } \\333 334 335 & {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+1/2} } \\336 337 % &\\338 % &\equiv \sum_{\substack{i_p,\,k_p}} 339 % \begin{pmatrix} 340 % \qquad \qquad \qquad 341 % \frac{1}{ {e_{1u}}_{\,i+1/2}^{\,k} } \ \; 342 %{ _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T)\\343 %\\344 % -\frac{1}{ {e_{3w}}_{\,i}^{\,k+1/2} } \ \; 345 % { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; 346 % {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T)\\ 347 % \end{pmatrix} 335 \begin{flalign} 336 \label{eq:iso_flux} 337 \textbf{F}_{iso}(T) 338 &\equiv \sum_{\substack{i_p,\,k_p}} 339 \begin{pmatrix} 340 {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\ 341 {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) 342 \end{pmatrix} 343 \notag \\ 344 & \notag \\ 345 &\equiv \sum_{\substack{i_p,\,k_p}} 346 \begin{pmatrix} 347 && { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+1/2}^{\,k} } \\ \\ 348 & -\; { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } 349 & {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+1/2} } 350 \end{pmatrix} % \\ 351 % &\\ 352 % &\equiv \sum_{\substack{i_p,\,k_p}} 353 % \begin{pmatrix} 354 % \qquad \qquad \qquad 355 % \frac{1}{ {e_{1u}}_{\,i+1/2}^{\,k} } \ \; 356 % { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T)\\ 357 % \\ 358 % -\frac{1}{ {e_{3w}}_{\,i}^{\,k+1/2} } \ \; 359 % { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; 360 % {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T)\\ 361 % \end{pmatrix} 348 362 \end{flalign} 349 363 resulting in a iso-neutral diffusion tendency on temperature given by 350 364 the divergence of the sum of all the four triad fluxes: 351 \begin{equation} \label{eq:Gf_operator} 352 D_l^T = \frac{1}{b_T} \sum_{\substack{i_p,\,k_p}} \left\{ 353 \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] 354 + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} 365 \begin{equation} 366 \label{eq:Gf_operator} 367 D_l^T = \frac{1}{b_T} \sum_{\substack{i_p,\,k_p}} \left\{ 368 \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] 369 + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} 355 370 \end{equation} 356 371 where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. … … 361 376 The discretization of the diffusion operator recovers the traditional five-point Laplacian in 362 377 the limit of flat iso-neutral direction: 363 \begin{equation} \label{eq:Gf_property1a} 364 D_l^T = \frac{1}{b_T} \ \delta_{i} 365 \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] 366 \qquad \text{when} \quad 367 { _i^k \mathbb{R}_{i_p}^{k_p} }=0 368 \end{equation} 378 \[ 379 % \label{eq:Gf_property1a} 380 D_l^T = \frac{1}{b_T} \ \delta_{i} 381 \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] 382 \qquad \text{when} \quad 383 { _i^k \mathbb{R}_{i_p}^{k_p} }=0 384 \] 369 385 370 386 \item[$\bullet$ implicit treatment in the vertical] … … 374 390 This is of paramount importance since it means that 375 391 the implicit in time algorithm for solving the vertical diffusion equation can be used to evaluate this term. 376 It is a necessity since the vertical eddy diffusivity associated with this term, 377 \begin{equation} 378 \sum_{\substack{i_p, \,k_p}} \left\{ 392 It is a necessity since the vertical eddy diffusivity associated with this term, 393 \[ 394 \sum_{\substack{i_p, \,k_p}} \left\{ 379 395 A_i^k \; \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 380 \right\} 381 \end{equation} 382 can be quite large.396 \right\} 397 \] 398 can be quite large. 383 399 384 400 \item[$\bullet$ pure iso-neutral operator] 385 401 The iso-neutral flux of locally referenced potential density is zero, $i.e.$ 386 \begin{align} \label{eq:Gf_property2} 387 \begin{matrix} 388 &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)} 389 &= &\alpha_i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) 390 &- \ \; \beta _i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (S) & = \ 0 \\ 391 &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)} 392 &= &\alpha_i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) 393 &- \ \; \beta _i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (S) &= \ 0 394 \end{matrix} 395 \end{align} 396 This result is trivially obtained using the \autoref{eq:Gf_triads} applied to $T$ and $S$ and 397 the definition of the triads' slopes \autoref{eq:Gf_slopes}. 402 \begin{align*} 403 % \label{eq:Gf_property2} 404 \begin{matrix} 405 &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)} 406 &= &\alpha_i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) 407 &- \ \; \beta _i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (S) & = \ 0 \\ 408 &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)} 409 &= &\alpha_i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) 410 &- \ \; \beta _i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (S) &= \ 0 411 \end{matrix} 412 \end{align*} 413 This result is trivially obtained using the \autoref{eq:Gf_triads} applied to $T$ and $S$ and 414 the definition of the triads' slopes \autoref{eq:Gf_slopes}. 398 415 399 416 \item[$\bullet$ conservation of tracer] 400 417 The iso-neutral diffusion term conserve the total tracer content, $i.e.$ 401 \begin{equation} \label{eq:Gf_property1} 402 \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 403 \end{equation} 418 \[ 419 % \label{eq:Gf_property1} 420 \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 421 \] 404 422 This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form. 405 423 406 424 \item[$\bullet$ decrease of tracer variance] 407 425 The iso-neutral diffusion term does not increase the total tracer variance, $i.e.$ 408 \begin{equation} \label{eq:Gf_property1} 409 \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 410 \end{equation} 426 \[ 427 % \label{eq:Gf_property1} 428 \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 429 \] 411 430 The property is demonstrated in the \autoref{apdx:Gf_operator}. 412 431 It is a key property for a diffusion term. … … 418 437 \item[$\bullet$ self-adjoint operator] 419 438 The iso-neutral diffusion operator is self-adjoint, $i.e.$ 420 \begin{equation} \label{eq:Gf_property1} 421 \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 422 \end{equation} 439 \[ 440 % \label{eq:Gf_property1} 441 \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 442 \] 423 443 In other word, there is no needs to develop a specific routine from the adjoint of this operator. 424 444 We just have to apply the same routine. … … 427 447 \end{description} 428 448 429 430 $\ $\newline %force an empty line431 449 % ================================================================ 432 450 % Skew flux formulation for Eddy Induced Velocity : … … 443 461 444 462 The eddy induced velocity is given by: 445 \begin{equation} \label{eq:eiv_v} 446 \begin{split} 447 u^* & = - \frac{1}{e_2\,e_{3}} \;\partial_k \left( e_2 \, A_e \; r_i \right) 448 = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_i \right) \\ 449 v^* & = - \frac{1}{e_1\,e_3}\; \partial_k \left( e_1 \, A_e \; r_j \right) 450 = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_j \right) \\ 451 w^* & = \frac{1}{e_1\,e_2}\; \left\{ \partial_i \left( e_2 \, A_e \; r_i \right) 452 + \partial_j \left( e_1 \, A_e \;r_j \right) \right\} \\ 453 \end{split} 463 \begin{equation} 464 \label{eq:eiv_v} 465 \begin{split} 466 u^* & = - \frac{1}{e_2\,e_{3}} \;\partial_k \left( e_2 \, A_e \; r_i \right) 467 = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_i \right) \\ 468 v^* & = - \frac{1}{e_1\,e_3}\; \partial_k \left( e_1 \, A_e \; r_j \right) 469 = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_j \right) \\ 470 w^* & = \frac{1}{e_1\,e_2}\; \left\{ \partial_i \left( e_2 \, A_e \; r_i \right) 471 + \partial_j \left( e_1 \, A_e \;r_j \right) \right\} 472 \end{split} 454 473 \end{equation} 455 474 where $A_{e}$ is the eddy induced velocity coefficient, … … 475 494 %\end{split} 476 495 %\end{equation} 477 \begin{equation} \label{eq:eiv_vd} 478 \textbf{F}_{eiv}^T \equiv \left( \begin{aligned} 479 \sum_{\substack{i_p,\,k_p}} & 480 +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k} 481 \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ 482 \\ 483 \sum_{\substack{i_p,\,k_p}} & 484 - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p} 485 \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \delta_{i+i_p}[T^{k+1/2-k_p}] \\ 486 \end{aligned} \right) 487 \end{equation} 496 \[ 497 % \label{eq:eiv_vd} 498 \textbf{F}_{eiv}^T \equiv \left( 499 \begin{aligned} 500 \sum_{\substack{i_p,\,k_p}} & 501 +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k} 502 \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ \\ 503 \sum_{\substack{i_p,\,k_p}} & 504 - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p} 505 \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \delta_{i+i_p}[T^{k+1/2-k_p}] 506 \end{aligned} 507 \right) 508 \] 488 509 489 510 \citep{Griffies_JPO98} introduces another way to implement the eddy induced advection, the so-called skew form. … … 491 512 For example in the (\textbf{i},\textbf{k}) plane, the tracer advective fluxes can be transformed as follows: 492 513 \begin{flalign*} 493 \begin{split} 494 \textbf{F}_{eiv}^T = 495 \begin{pmatrix} 496 {e_{2}\,e_{3}\; u^*} \\ 497 {e_{1}\,e_{2}\; w^*} \\ 498 \end{pmatrix} \; T 499 &= 500 \begin{pmatrix} 501 { - \partial_k \left( e_{2} \, A_{e} \; r_i \right) \; T \;} \\ 502 {+ \partial_i \left( e_{2} \, A_{e} \; r_i \right) \; T \;} \\ 503 \end{pmatrix} \\ 504 &= 505 \begin{pmatrix} 506 { - \partial_k \left( e_{2} \, A_{e} \; r_i \; T \right) \;} \\ 507 {+ \partial_i \left( e_{2} \, A_{e} \; r_i \; T \right) \;} \\ 508 \end{pmatrix} 509 + 510 \begin{pmatrix} 511 {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ 512 { - e_{2} \, A_{e} \; r_i \; \partial_i T} \\ 513 \end{pmatrix} 514 \end{split} 514 \begin{split} 515 \textbf{F}_{eiv}^T = 516 \begin{pmatrix} 517 {e_{2}\,e_{3}\; u^*} \\ 518 {e_{1}\,e_{2}\; w^*} 519 \end{pmatrix} 520 \; T 521 &= 522 \begin{pmatrix} 523 { - \partial_k \left( e_{2} \, A_{e} \; r_i \right) \; T \;} \\ 524 {+ \partial_i \left( e_{2} \, A_{e} \; r_i \right) \; T \;} 525 \end{pmatrix} 526 \\ 527 &= 528 \begin{pmatrix} 529 { - \partial_k \left( e_{2} \, A_{e} \; r_i \; T \right) \;} \\ 530 {+ \partial_i \left( e_{2} \, A_{e} \; r_i \; T \right) \;} 531 \end{pmatrix} 532 + 533 \begin{pmatrix} 534 {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ 535 { - e_{2} \, A_{e} \; r_i \; \partial_i T} 536 \end{pmatrix} 537 \end{split} 515 538 \end{flalign*} 516 539 and since the eddy induces velocity field is no-divergent, 517 540 we end up with the skew form of the eddy induced advective fluxes: 518 \begin{equation} \label{eq:eiv_skew_continuous} 519 \textbf{F}_{eiv}^T = \begin{pmatrix} 520 {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ 521 { - e_{2} \, A_{e} \; r_i \; \partial_i T} \\ 522 \end{pmatrix} 541 \begin{equation} 542 \label{eq:eiv_skew_continuous} 543 \textbf{F}_{eiv}^T = 544 \begin{pmatrix} 545 {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ 546 { - e_{2} \, A_{e} \; r_i \; \partial_i T} 547 \end{pmatrix} 523 548 \end{equation} 524 549 The tendency associated with eddy induced velocity is then simply the divergence of … … 528 553 Another interesting property of \autoref{eq:eiv_skew_continuous} form is that when $A=A_e$, 529 554 a simplification occurs in the sum of the iso-neutral diffusion and eddy induced velocity terms: 530 \begin{flalign} \label{eq:eiv_skew+eiv_continuous} 531 \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &= 532 \begin{pmatrix} 533 + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T - e_2 \, A \; r_i \;\partial_k T \\ 534 - e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T \\ 535 \end{pmatrix} 536 + 537 \begin{pmatrix} 538 {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ 539 { - e_{2} \, A_{e} \; r_i \; \partial_i T} \\ 540 \end{pmatrix} \\ 541 &= \begin{pmatrix} 542 + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T \\ 543 - 2\; e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T \\ 544 \end{pmatrix} 545 \end{flalign} 555 \begin{flalign*} 556 % \label{eq:eiv_skew+eiv_continuous} 557 \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &= 558 \begin{pmatrix} 559 + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T - e_2 \, A \; r_i \;\partial_k T \\ 560 - e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T 561 \end{pmatrix} 562 + 563 \begin{pmatrix} 564 {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ 565 { - e_{2} \, A_{e} \; r_i \; \partial_i T} 566 \end{pmatrix} 567 \\ 568 &= 569 \begin{pmatrix} 570 + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T \\ 571 - 2\; e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T 572 \end{pmatrix} 573 \end{flalign*} 546 574 The horizontal component reduces to the one use for an horizontal laplacian operator and 547 575 the vertical one keeps the same complexity, but not more. … … 552 580 Using the slopes \autoref{eq:Gf_slopes} and defining $A_e$ at $T$-point($i.e.$ as $A$, 553 581 the eddy diffusivity coefficient), the resulting discret form is given by: 554 \begin{equation} \label{eq:eiv_skew} 555 \textbf{F}_{eiv}^T \equiv \frac{1}{4} \left( \begin{aligned} 556 \sum_{\substack{i_p,\,k_p}} & 557 +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k} 558 \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ 559 \\ 560 \sum_{\substack{i_p,\,k_p}} & 561 - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p} 562 \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \delta_{i+i_p}[T^{k+1/2-k_p}] \\ 563 \end{aligned} \right) 582 \begin{equation} 583 \label{eq:eiv_skew} 584 \textbf{F}_{eiv}^T \equiv \frac{1}{4} \left( 585 \begin{aligned} 586 \sum_{\substack{i_p,\,k_p}} & 587 +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k} 588 \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ \\ 589 \sum_{\substack{i_p,\,k_p}} & 590 - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p} 591 \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \delta_{i+i_p}[T^{k+1/2-k_p}] 592 \end{aligned} 593 \right) 564 594 \end{equation} 565 595 Note that \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells. … … 572 602 $i.e.$ it does not include a diffusive component but is a "pure" advection term. 573 603 574 575 576 577 604 $\ $\newpage %force an empty line 578 605 % ================================================================ … … 587 614 588 615 The continuous property to be demonstrated is: 616 \[ 617 \int_D D_l^T \; T \;dv \leq 0 618 \] 619 The discrete form of its left hand side is obtained using \autoref{eq:iso_flux} 620 589 621 \begin{align*} 590 \int_D D_l^T \; T \;dv \leq 0 591 \end{align*} 592 The discrete form of its left hand side is obtained using \autoref{eq:iso_flux} 593 594 \begin{align*} 595 &\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\ 596 &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 597 \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] 598 + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\ 599 &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 600 {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] 601 + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\ 602 &\equiv -\sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 603 \frac{ _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{1u}}_{\,i+1/2}^{\,k} } \ \delta_{i+1/2} [T] 604 - { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; 605 \frac{ _i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{3w}}_{\,i}^{\,k+1/2} } \ \delta_{k+1/2} [T] 606 \right\} \\ 607 % 608 \allowdisplaybreaks 609 \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} 610 % 611 &\equiv -\sum_{i,k} 612 \begin{Bmatrix} 613 &\ \ \Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{-1/2} (T) } 614 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 615 & -\ \ {_{i}^{k+1} \mathbb{R}_{-1/2}^{-1/2}} 616 & {_{i}^{k+1} \mathbb{T}_{-1/2}^{-1/2} (T) } 617 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) 618 & \\ 619 &+\Bigl( \ \;\; { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } 620 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 621 & -\ \ {_i^{k+1} \mathbb{R}_{+1/2}^{-1/2}} 622 & { _i^{k+1} \mathbb{T}_{+1/2}^{-1/2} (T) } 623 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) 624 & \\ 625 &+\Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } 626 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 627 & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{-1/2}^{+1/2}} 628 & \ \;\;{_{i}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } 629 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) 630 & \\ 631 &+\Bigl( \ \;\; { _{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } 632 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 633 & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{+1/2}^{+1/2}} 634 & \ \;\;{_{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } 635 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) \\ 636 \end{Bmatrix} 637 % 638 \allowdisplaybreaks 639 \intertext{The summation is done over all $i$ and $k$ indices, 622 &\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\ 623 &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 624 \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] 625 + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\ 626 &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 627 {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] 628 + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\ 629 &\equiv -\sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 630 \frac{ _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{1u}}_{\,i+1/2}^{\,k} } \ \delta_{i+1/2} [T] 631 - { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; 632 \frac{ _i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{3w}}_{\,i}^{\,k+1/2} } \ \delta_{k+1/2} [T] 633 \right\} \\ 634 % 635 \allowdisplaybreaks 636 \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} 637 % 638 &\equiv -\sum_{i,k} 639 \begin{Bmatrix} 640 &\ \ \Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{-1/2} (T) } 641 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 642 & -\ \ {_{i}^{k+1} \mathbb{R}_{-1/2}^{-1/2}} 643 & {_{i}^{k+1} \mathbb{T}_{-1/2}^{-1/2} (T) } 644 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) 645 & \\ 646 &+\Bigl( \ \;\; { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } 647 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 648 & -\ \ {_i^{k+1} \mathbb{R}_{+1/2}^{-1/2}} 649 & { _i^{k+1} \mathbb{T}_{+1/2}^{-1/2} (T) } 650 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) 651 & \\ 652 &+\Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } 653 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 654 & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{-1/2}^{+1/2}} 655 & \ \;\;{_{i}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } 656 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) 657 & \\ 658 &+\Bigl( \ \;\; { _{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } 659 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 660 & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{+1/2}^{+1/2}} 661 & \ \;\;{_{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } 662 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) \\ 663 \end{Bmatrix} 664 % 665 \allowdisplaybreaks 666 \intertext{ 667 The summation is done over all $i$ and $k$ indices, 640 668 it is therefore possible to introduce a shift of $-1$ either in $i$ or $k$ direction in order to 641 669 regroup all the terms of the summation by triad at a ($i$,$k$) point. 642 670 In other words, we regroup all the terms in the neighbourhood that contain a triad at the same ($i$,$k$) indices. 643 It becomes: } 644 % 645 &\equiv -\sum_{i,k} 646 \begin{Bmatrix} 647 &\ \ \Bigl( {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } 648 &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 649 & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} 650 & {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } 651 &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr) 652 & \\ 653 &+\Bigl( { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } 654 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 655 & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} 656 & { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } 657 &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr) 658 & \\ 659 &+\Bigl( {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } 660 &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 661 & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} 662 & {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } 663 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) 664 & \\ 665 &+\Bigl( { _i^k \mathbb{T}_{+1/2}^{+1/2} (T) } 666 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 667 & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} 668 & {_i^k \mathbb{T}_{+1/2}^{+1/2} (T) } 669 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) \\ 670 \end{Bmatrix} \\ 671 % 672 \allowdisplaybreaks 673 \intertext{Then outing in factor the triad in each of the four terms of the summation and 671 It becomes: 672 } 673 % 674 &\equiv -\sum_{i,k} 675 \begin{Bmatrix} 676 &\ \ \Bigl( {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } 677 &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 678 & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} 679 & {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } 680 &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr) 681 & \\ 682 &+\Bigl( { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } 683 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 684 & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} 685 & { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } 686 &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr) 687 & \\ 688 &+\Bigl( {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } 689 &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 690 & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} 691 & {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } 692 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) 693 & \\ 694 &+\Bigl( { _i^k \mathbb{T}_{+1/2}^{+1/2} (T) } 695 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 696 & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} 697 & {_i^k \mathbb{T}_{+1/2}^{+1/2} (T) } 698 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) \\ 699 \end{Bmatrix} \\ 700 % 701 \allowdisplaybreaks 702 \intertext{ 703 Then outing in factor the triad in each of the four terms of the summation and 674 704 substituting the triads by their expression given in \autoref{eq:Gf_triads}. 675 It becomes: } 676 % 677 &\equiv -\sum_{i,k} 678 \begin{Bmatrix} 679 &\ \ \Bigl( \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 680 & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} 681 &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)^2 682 & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k} \ A_i^k 683 & \\ 684 &+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 685 & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} 686 &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)^2 687 & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k} \ A_i^k 688 & \\ 689 &+\Bigl( \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 690 & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} 691 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)^2 692 & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k} \ A_i^k 693 & \\ 694 &+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 695 & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} 696 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)^2 697 & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k} \ A_i^k \\ 698 \end{Bmatrix} \\ 699 & \\ 700 % 701 &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 702 \begin{matrix} 703 &\Bigl( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} 704 & -\ \ {_i^k \mathbb{R}_{i_p}^{k_p}} 705 &\frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}} \Bigr)^2 706 & \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \ \ 707 \end{matrix} 708 \right\} 709 \quad \leq 0 705 It becomes: 706 } 707 % 708 &\equiv -\sum_{i,k} 709 \begin{Bmatrix} 710 &\ \ \Bigl( \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 711 & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} 712 &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)^2 713 & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k} \ A_i^k 714 & \\ 715 &+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 716 & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} 717 &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)^2 718 & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k} \ A_i^k 719 & \\ 720 &+\Bigl( \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 721 & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} 722 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)^2 723 & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k} \ A_i^k 724 & \\ 725 &+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 726 & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} 727 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)^2 728 & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k} \ A_i^k \\ 729 \end{Bmatrix} 730 \\ 731 & \\ 732 % 733 &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 734 \begin{matrix} 735 &\Bigl( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} 736 & -\ \ {_i^k \mathbb{R}_{i_p}^{k_p}} 737 &\frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}} \Bigr)^2 738 & \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \ \ 739 \end{matrix} 740 \right\} 741 \quad \leq 0 710 742 \end{align*} 711 743 The last inequality is obviously obtained as we succeed in obtaining a negative summation of square quantities. … … 714 746 then the previous demonstration would have let to: 715 747 \begin{align*} 716 \int_D S \; D_l^T \;dv &\equiv \sum_{i,k} \left\{ S \ D_l^T \ b_T \right\} \\ 717 &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 718 \left( \frac{ \delta_{i +i_p} [S] }{{e_{1u} }_{\,i+i_p}^{\,k}} 719 - {_i^k \mathbb{R}_{i_p}^{k_p}} 720 \frac{ \delta_{k+k_p} [S] }{{e_{3w}}_{\,i}^{\,k+k_p}} \right) \right. 721 \\ & \qquad \qquad \qquad \ \left. 722 \left( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} 723 - {_i^k \mathbb{R}_{i_p}^{k_p}} 724 \frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}} \right) 725 \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \ 726 \right\} 727 % 728 \allowdisplaybreaks 729 \intertext{which, by applying the same operation as before but in reverse order, leads to: } 730 % 731 &\equiv \sum_{i,k} \left\{ D_l^S \ T \ b_T \right\} 748 \int_D S \; D_l^T \;dv &\equiv \sum_{i,k} \left\{ S \ D_l^T \ b_T \right\} \\ 749 &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 750 \left( \frac{ \delta_{i +i_p} [S] }{{e_{1u} }_{\,i+i_p}^{\,k}} 751 - {_i^k \mathbb{R}_{i_p}^{k_p}} 752 \frac{ \delta_{k+k_p} [S] }{{e_{3w}}_{\,i}^{\,k+k_p}} \right) \right. \\ 753 & \qquad \qquad \qquad \ \left. 754 \left( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} 755 - {_i^k \mathbb{R}_{i_p}^{k_p}} 756 \frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}} \right) 757 \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \ 758 \right\} 759 % 760 \allowdisplaybreaks 761 \intertext{ 762 which, by applying the same operation as before but in reverse order, leads to: 763 } 764 % 765 &\equiv \sum_{i,k} \left\{ D_l^S \ T \ b_T \right\} 732 766 \end{align*} 733 767 This means that the iso-neutral operator is self-adjoint. 734 768 There is no need to develop a specific to obtain it. 735 769 736 737 738 $\ $\newpage %force an empty line 770 \newpage 771 739 772 % ================================================================ 740 773 % Discrete Invariants of the skew flux formulation … … 743 776 \label{subsec:eiv_skew} 744 777 745 746 778 Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. 747 779 … … 750 782 The continuous property to be demonstrated is: 751 783 \begin{align*} 752 \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv \equiv 0784 \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv \equiv 0 753 785 \end{align*} 754 786 The discrete form of its left hand side is obtained using \autoref{eq:eiv_skew} 755 787 \begin{align*} 756 \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\;757 \delta_i &\left[758 {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} 759 \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] 760 \right] \; T_i^k \\761 - \delta_k &\left[ 762 {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} 763 \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] 764 \right] \; T_i^k \ \Biggr\}788 \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; 789 \delta_i &\left[ 790 {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} 791 \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] 792 \right] \; T_i^k \\ 793 - \delta_k &\left[ 794 {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} 795 \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] 796 \right] \; T_i^k \ \Biggr\} 765 797 \end{align*} 766 798 apply the adjoint of delta operator, it becomes 767 799 \begin{align*} 768 \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\;769 &\left( 770 {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} 771 \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] 772 \right) \; \delta_{i+1/2}[T^{k}] \\773 - &\left( 774 {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} 775 \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] 776 \right) \; \delta_{k+1/2}[T_{i}] \ \Biggr\}800 \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; 801 &\left( 802 {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} 803 \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] 804 \right) \; \delta_{i+1/2}[T^{k}] \\ 805 - &\left( 806 {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} 807 \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] 808 \right) \; \delta_{k+1/2}[T_{i}] \ \Biggr\} 777 809 \end{align*} 778 810 Expending the summation on $i_p$ and $k_p$, it becomes: 779 811 \begin{align*} 780 \begin{matrix}781 &\sum\limits_{i,k} \Bigl\{ 782 &+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k}783 &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{-1/2}} &\delta_{k-1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\784 &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} 785 &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{-1/2}} &\delta_{k-1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\786 &&+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} 787 &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{+1/2}} &\delta_{k+1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\788 &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} 812 \begin{matrix} 813 &\sum\limits_{i,k} \Bigl\{ 814 &+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} 815 &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{-1/2}} &\delta_{k-1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\ 816 &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} 817 &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{-1/2}} &\delta_{k-1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ 818 &&+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} 819 &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{+1/2}} &\delta_{k+1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\ 820 &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} 789 821 &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{k+1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ 790 %791 &&-{e_{2u}}_i^{k+1} &{A_{e}}_i^{k+1} 792 &{_i^{k+1} \mathbb{R}_{-1/2}^{- 1/2}} &\delta_{i-1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\793 &&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} 794 &{\ \ \;_i^k \mathbb{R}_{-1/2}^{+1/2}} &\delta_{i-1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] &\\795 &&-{e_{2u}}_i^{k+1 } &{A_{e}}_i^{k+1} 796 &{_i^{k+1} \mathbb{R}_{+1/2}^{- 1/2}} &\delta_{i+1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\797 &&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} 798 &{\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{i+1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}]799 &\Bigr\} \\800 \end{matrix}822 % 823 &&-{e_{2u}}_i^{k+1} &{A_{e}}_i^{k+1} 824 &{_i^{k+1} \mathbb{R}_{-1/2}^{- 1/2}} &\delta_{i-1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\ 825 &&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} 826 &{\ \ \;_i^k \mathbb{R}_{-1/2}^{+1/2}} &\delta_{i-1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] &\\ 827 &&-{e_{2u}}_i^{k+1 } &{A_{e}}_i^{k+1} 828 &{_i^{k+1} \mathbb{R}_{+1/2}^{- 1/2}} &\delta_{i+1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\ 829 &&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} 830 &{\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{i+1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] 831 &\Bigr\} \\ 832 \end{matrix} 801 833 \end{align*} 802 834 The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the same but of opposite signs, … … 810 842 $i.e.$ the variance of the tracer is preserved by the discretisation of the skew fluxes. 811 843 844 \biblio 845 812 846 \end{document}
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