[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
---|
| 2 | |
---|
[6997] | 3 | \begin{document} |
---|
[707] | 4 | % ================================================================ |
---|
[2282] | 5 | % Appendix E : Note on some algorithms |
---|
[707] | 6 | % ================================================================ |
---|
[2282] | 7 | \chapter{Note on some algorithms} |
---|
[11543] | 8 | \label{apdx:ALGOS} |
---|
[10414] | 9 | |
---|
[11435] | 10 | \chaptertoc |
---|
[707] | 11 | |
---|
[2282] | 12 | \newpage |
---|
[707] | 13 | |
---|
[11543] | 14 | This appendix some on going consideration on algorithms used or planned to be used in \NEMO. |
---|
[10354] | 15 | |
---|
[707] | 16 | % ------------------------------------------------------------------------------------------------------------- |
---|
[11543] | 17 | % UBS scheme |
---|
[707] | 18 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 19 | \section{Upstream Biased Scheme (UBS) (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})} |
---|
[11544] | 20 | \label{sec:ALGOS_tra_adv_ubs} |
---|
[707] | 21 | |
---|
[10354] | 22 | The UBS advection scheme is an upstream biased third order scheme based on |
---|
| 23 | an upstream-biased parabolic interpolation. |
---|
| 24 | It is also known as Cell Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective Kinematics). |
---|
| 25 | For example, in the $i$-direction: |
---|
[10414] | 26 | \begin{equation} |
---|
[11544] | 27 | \label{eq:ALGOS_tra_adv_ubs2} |
---|
[10414] | 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. |
---|
[707] | 34 | \end{equation} |
---|
| 35 | or equivalently, the advective flux is |
---|
[10414] | 36 | \begin{equation} |
---|
[11544] | 37 | \label{eq:ALGOS_tra_adv_ubs2} |
---|
[10414] | 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] |
---|
[707] | 41 | \end{equation} |
---|
[10354] | 42 | where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and |
---|
[10406] | 43 | $\tau "_i =\delta_i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$. |
---|
[10354] | 44 | By choosing this expression for $\tau "$ we consider a fourth order approximation of $\partial_i^2$ with |
---|
| 45 | a constant i-grid spacing ($\Delta i=1$). |
---|
[707] | 46 | |
---|
[11543] | 47 | Alternative choice: introduce the scale factors: |
---|
[10406] | 48 | $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta_{i+1/2}[\tau] \right]$. |
---|
[707] | 49 | |
---|
[11435] | 50 | This results in a dissipatively dominant (\ie\ hyper-diffusive) truncation error |
---|
[11123] | 51 | \citep{shchepetkin.mcwilliams_OM05}. |
---|
| 52 | The overall performance of the advection scheme is similar to that reported in \cite{farrow.stevens_JPO95}. |
---|
[10354] | 53 | It is a relatively good compromise between accuracy and smoothness. |
---|
| 54 | It is not a \emph{positive} scheme meaning false extrema are permitted but |
---|
| 55 | the amplitude of such are significantly reduced over the centred second order method. |
---|
[11543] | 56 | Nevertheless it is not recommended to apply it to a passive tracer that requires positivity. |
---|
[707] | 57 | |
---|
[10354] | 58 | The intrinsic diffusion of UBS makes its use risky in the vertical direction where |
---|
| 59 | the control of artificial diapycnal fluxes is of paramount importance. |
---|
| 60 | It has therefore been preferred to evaluate the vertical flux using the TVD scheme when |
---|
| 61 | \np{ln\_traadv\_ubs}\forcode{ = .true.}. |
---|
[707] | 62 | |
---|
[11543] | 63 | For stability reasons, in \autoref{eq:TRA_adv_ubs}, the first term which corresponds to |
---|
[10354] | 64 | a second order centred scheme is evaluated using the \textit{now} velocity (centred in time) while |
---|
| 65 | the second term which is the diffusive part of the scheme, is evaluated using the \textit{before} velocity |
---|
| 66 | (forward in time). |
---|
[11123] | 67 | This is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the Quick advection scheme. |
---|
[10354] | 68 | UBS and QUICK schemes only differ by one coefficient. |
---|
[11543] | 69 | Substituting 1/6 with 1/8 in (\autoref{eq:TRA_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. |
---|
[10354] | 70 | This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded. |
---|
| 71 | Nevertheless it is quite easy to make the substitution in \mdl{traadv\_ubs} module and obtain a QUICK scheme. |
---|
[707] | 72 | |
---|
[10354] | 73 | NB 1: When a high vertical resolution $O(1m)$ is used, the model stability can be controlled by vertical advection |
---|
| 74 | (not vertical diffusion which is usually solved using an implicit scheme). |
---|
| 75 | Computer time can be saved by using a time-splitting technique on vertical advection. |
---|
| 76 | This possibility have been implemented and validated in ORCA05-L301. |
---|
[11543] | 77 | It is not currently offered in the current reference version. |
---|
[707] | 78 | |
---|
[10354] | 79 | NB 2: In a forthcoming release four options will be proposed for the vertical component used in the UBS scheme. |
---|
[10406] | 80 | $\tau_w^{ubs}$ will be evaluated using either \textit{(a)} a centered $2^{nd}$ order scheme, |
---|
[10354] | 81 | or \textit{(b)} a TVD scheme, or \textit{(c)} an interpolation based on conservative parabolic splines following |
---|
[11123] | 82 | \citet{shchepetkin.mcwilliams_OM05} implementation of UBS in ROMS, or \textit{(d)} an UBS. |
---|
[10354] | 83 | The $3^{rd}$ case has dispersion properties similar to an eight-order accurate conventional scheme. |
---|
[707] | 84 | |
---|
[11543] | 85 | NB 3: It is straight forward to rewrite \autoref{eq:TRA_adv_ubs} as follows: |
---|
[10414] | 86 | \begin{equation} |
---|
[11544] | 87 | \label{eq:ALGOS_tra_adv_ubs2} |
---|
[10414] | 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. |
---|
[707] | 94 | \end{equation} |
---|
[11543] | 95 | or equivalently |
---|
[10414] | 96 | \begin{equation} |
---|
[11544] | 97 | \label{eq:ALGOS_tra_adv_ubs2} |
---|
[10414] | 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} |
---|
[707] | 103 | \end{equation} |
---|
[11543] | 104 | \autoref{eq:TRA_adv_ubs2} has several advantages. |
---|
[10354] | 105 | First it clearly evidences that the UBS scheme is based on the fourth order scheme to which |
---|
| 106 | is added an upstream biased diffusive term. |
---|
| 107 | Second, this emphasises that the $4^{th}$ order part have to be evaluated at \emph{now} time step, |
---|
[11543] | 108 | not only the $2^{th}$ order part as stated above using \autoref{eq:TRA_adv_ubs}. |
---|
[10354] | 109 | Third, the diffusive term is in fact a biharmonic operator with a eddy coefficient which |
---|
| 110 | is simply proportional to the velocity. |
---|
[707] | 111 | |
---|
| 112 | laplacian diffusion: |
---|
[10414] | 113 | \begin{equation} |
---|
[11544] | 114 | \label{eq:ALGOS_tra_ldf_lap} |
---|
[10414] | 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} |
---|
[707] | 123 | \end{equation} |
---|
| 124 | |
---|
| 125 | bilaplacian: |
---|
[10414] | 126 | \begin{equation} |
---|
[11544] | 127 | \label{eq:ALGOS_tra_ldf_lap} |
---|
[10414] | 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} |
---|
[707] | 135 | \end{equation} |
---|
[11543] | 136 | with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$, |
---|
[11435] | 137 | \ie\ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$ |
---|
[10354] | 138 | it comes: |
---|
[10414] | 139 | \begin{equation} |
---|
[11544] | 140 | \label{eq:ALGOS_tra_ldf_lap} |
---|
[10414] | 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} |
---|
[707] | 148 | \end{equation} |
---|
[11435] | 149 | if the velocity is uniform (\ie\ $|u|=cst$) then the diffusive flux is |
---|
[10414] | 150 | \begin{equation} |
---|
[11544] | 151 | \label{eq:ALGOS_tra_ldf_lap} |
---|
[10414] | 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} |
---|
[707] | 159 | \end{equation} |
---|
| 160 | beurk.... reverte the logic: starting from the diffusive part of the advective flux it comes: |
---|
| 161 | |
---|
[10414] | 162 | \begin{equation} |
---|
[11544] | 163 | \label{eq:ALGOS_tra_adv_ubs2} |
---|
[10414] | 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} |
---|
[707] | 167 | \end{equation} |
---|
[11435] | 168 | if the velocity is uniform (\ie\ $|u|=cst$) and |
---|
[10406] | 169 | choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right]$ |
---|
[707] | 170 | |
---|
| 171 | sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$): |
---|
[10414] | 172 | \begin{equation} |
---|
[11544] | 173 | \label{eq:ALGOS_tra_adv_ubs2} |
---|
[10414] | 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} |
---|
[707] | 177 | \end{equation} |
---|
| 178 | which leads to ${A_T^{lT}}^2 = \frac{1}{12} {e_{1T}}^3\ \overline{|u|}^{\,i+1/2}$ |
---|
| 179 | |
---|
| 180 | sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$ |
---|
[10414] | 181 | \begin{equation} |
---|
[11544] | 182 | \label{eq:ALGOS_tra_adv_ubs2} |
---|
[10414] | 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} |
---|
[707] | 187 | \end{equation} |
---|
| 188 | which leads to ${A_u^{lT}} = \frac{1}{12} {e_{1u}}^3\ |u|$ |
---|
| 189 | |
---|
| 190 | % ------------------------------------------------------------------------------------------------------------- |
---|
[11543] | 191 | % Leap-Frog energetic |
---|
[707] | 192 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 193 | \section{Leapfrog energetic} |
---|
[11544] | 194 | \label{sec:ALGOS_LF} |
---|
[707] | 195 | |
---|
[10354] | 196 | We adopt the following semi-discrete notation for time derivative. |
---|
| 197 | Given the values of a variable $q$ at successive time step, |
---|
| 198 | the time derivation and averaging operators at the mid time step are: |
---|
[10414] | 199 | \[ |
---|
[11544] | 200 | % \label{eq:ALGOS_dt_mt} |
---|
[10414] | 201 | \begin{split} |
---|
| 202 | \delta_{t+\rdt/2} [q] &= \ \ \, q^{t+\rdt} - q^{t} \\ |
---|
| 203 | \overline q^{\,t+\rdt/2} &= \left\{ q^{t+\rdt} + q^{t} \right\} \; / \; 2 |
---|
| 204 | \end{split} |
---|
| 205 | \] |
---|
[10354] | 206 | As for space operator, |
---|
| 207 | the adjoint of the derivation and averaging time operators are $\delta_t^*=\delta_{t+\rdt/2}$ and |
---|
| 208 | $\overline{\cdot}^{\,t\,*}= \overline{\cdot}^{\,t+\Delta/2}$, respectively. |
---|
[707] | 209 | |
---|
[9407] | 210 | The Leap-frog time stepping given by \autoref{eq:DOM_nxt} can be defined as: |
---|
[10414] | 211 | \[ |
---|
[11544] | 212 | % \label{eq:ALGOS_LF} |
---|
[10414] | 213 | \frac{\partial q}{\partial t} |
---|
| 214 | \equiv \frac{1}{\rdt} \overline{ \delta_{t+\rdt/2}[q]}^{\,t} |
---|
| 215 | = \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} |
---|
[11543] | 216 | \] |
---|
[10354] | 217 | Note that \autoref{chap:LF} shows that the leapfrog time step is $\rdt$, |
---|
| 218 | not $2\rdt$ as it can be found sometimes in literature. |
---|
| 219 | The leap-Frog time stepping is a second order centered scheme. |
---|
[11435] | 220 | As such it respects the quadratic invariant in integral forms, \ie\ the following continuous property, |
---|
[10414] | 221 | \[ |
---|
[11544] | 222 | % \label{eq:ALGOS_Energy} |
---|
[10414] | 223 | \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} |
---|
| 224 | =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} |
---|
| 225 | = \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) , |
---|
| 226 | \] |
---|
[10354] | 227 | is satisfied in discrete form. |
---|
[11543] | 228 | Indeed, |
---|
[10414] | 229 | \[ |
---|
| 230 | \begin{split} |
---|
| 231 | \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} |
---|
| 232 | &\equiv \sum\limits_{0}^{N} |
---|
| 233 | {\frac{1}{\rdt} q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} \ \rdt} |
---|
| 234 | \equiv \sum\limits_{0}^{N} { q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} } \\ |
---|
| 235 | &\equiv \sum\limits_{0}^{N} { \overline{q}^{\,t+\Delta/2}{ \delta_{t+\rdt/2}[q]}} |
---|
| 236 | \equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta_{t+\rdt/2}[q^2] }\\ |
---|
| 237 | &\equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta_{t+\rdt/2}[q^2] } |
---|
| 238 | \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) |
---|
| 239 | \end{split} |
---|
| 240 | \] |
---|
[10354] | 241 | NB here pb of boundary condition when applying the adjoint! |
---|
[11543] | 242 | In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition |
---|
[10354] | 243 | (equivalently of the boundary value of the integration by part). |
---|
| 244 | In time this boundary condition is not physical and \textbf{add something here!!!} |
---|
[707] | 245 | |
---|
[2282] | 246 | % ================================================================ |
---|
[11543] | 247 | % Iso-neutral diffusion : |
---|
[2282] | 248 | % ================================================================ |
---|
| 249 | |
---|
| 250 | \section{Lateral diffusion operator} |
---|
| 251 | |
---|
| 252 | % ================================================================ |
---|
[11543] | 253 | % Griffies' iso-neutral diffusion operator : |
---|
[2282] | 254 | % ================================================================ |
---|
[9393] | 255 | \subsection{Griffies iso-neutral diffusion operator} |
---|
[2282] | 256 | |
---|
[11123] | 257 | Let try to define a scheme that get its inspiration from the \citet{griffies.gnanadesikan.ea_JPO98} scheme, |
---|
[11435] | 258 | but is formulated within the \NEMO\ framework |
---|
| 259 | (\ie\ using scale factors rather than grid-size and having a position of $T$-points that |
---|
[11543] | 260 | is not necessary in the middle of vertical velocity points, see \autoref{fig:DOM_zgr_e3}). |
---|
[2282] | 261 | |
---|
[11543] | 262 | In the formulation \autoref{eq:TRA_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO, |
---|
[10354] | 263 | the off-diagonal terms of the small angle diffusion tensor contain several double spatial averages of a gradient, |
---|
| 264 | for example $\overline{\overline{\delta_k \cdot}}^{\,i,k}$. |
---|
| 265 | It is apparent that the combination of a $k$ average and a $k$ derivative of the tracer allows for |
---|
| 266 | the presence of grid point oscillation structures that will be invisible to the operator. |
---|
| 267 | These structures are \textit{computational modes}. |
---|
| 268 | They will not be damped by the iso-neutral operator, and even possibly amplified by it. |
---|
| 269 | In other word, the operator applied to a tracer does not warranties the decrease of its global average variance. |
---|
| 270 | To circumvent this, we have introduced a smoothing of the slopes of the iso-neutral surfaces |
---|
| 271 | (see \autoref{chap:LDF}). |
---|
| 272 | Nevertheless, this technique works fine for $T$ and $S$ as they are active tracers |
---|
[11435] | 273 | (\ie\ they enter the computation of density), but it does not work for a passive tracer. |
---|
[11123] | 274 | \citep{griffies.gnanadesikan.ea_JPO98} introduce a different way to discretise the off-diagonal terms that |
---|
[10354] | 275 | nicely solve the problem. |
---|
| 276 | The idea is to get rid of combinations of an averaged in one direction combined with |
---|
| 277 | a derivative in the same direction by considering triads. |
---|
| 278 | For example in the (\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows: |
---|
[10414] | 279 | \begin{equation} |
---|
[11544] | 280 | \label{eq:ALGOS_Gf_triads} |
---|
[10414] | 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) |
---|
[2282] | 286 | \end{equation} |
---|
[10354] | 287 | where the indices $i_p$ and $k_p$ define the four triads and take the following value: |
---|
| 288 | $i_p = -1/2$ or $1/2$ and $k_p = -1/2$ or $1/2$, |
---|
| 289 | $b_u= e_{1u}\,e_{2u}\,e_{3u}$ is the volume of $u$-cells, |
---|
[2282] | 290 | $A_i^k$ is the lateral eddy diffusivity coefficient defined at $T$-point, |
---|
[10354] | 291 | and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad: |
---|
[10414] | 292 | \begin{equation} |
---|
[11544] | 293 | \label{eq:ALGOS_Gf_slopes} |
---|
[10414] | 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 ] } |
---|
[2282] | 298 | \end{equation} |
---|
[11544] | 299 | Note that in \autoref{eq:ALGOS_Gf_slopes} we use the ratio $\alpha / \beta$ instead of |
---|
[10354] | 300 | multiplying the temperature derivative by $\alpha$ and the salinity derivative by $\beta$. |
---|
| 301 | This is more efficient as the ratio $\alpha / \beta$ can to be evaluated directly. |
---|
[2282] | 302 | |
---|
[11544] | 303 | Note that in \autoref{eq:ALGOS_Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of ${b_{uw}}_{\,i+i_p}^{\,k+k_p}$. |
---|
[10354] | 304 | This choice has been motivated by the decrease of tracer variance and |
---|
[11544] | 305 | the presence of partial cell at the ocean bottom (see \autoref{subsec:ALGOS_Gf_operator}). |
---|
[2282] | 306 | |
---|
| 307 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[10414] | 308 | \begin{figure}[!ht] |
---|
[11558] | 309 | \centering |
---|
[11561] | 310 | \includegraphics[width=0.66\textwidth]{Fig_ISO_triad} |
---|
[11558] | 311 | \caption[Triads used in the Griffies's like iso-neutral diffision scheme for |
---|
| 312 | $u$- and $w$-components)]{ |
---|
| 313 | Triads used in the Griffies's like iso-neutral diffision scheme for |
---|
| 314 | $u$-component (upper panel) and $w$-component (lower panel).} |
---|
| 315 | \label{fig:ALGOS_ISO_triad} |
---|
[2282] | 316 | \end{figure} |
---|
| 317 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 318 | |
---|
[11543] | 319 | The four iso-neutral fluxes associated with the triads are defined at $T$-point. |
---|
[10354] | 320 | They take the following expression: |
---|
[10414] | 321 | \begin{flalign*} |
---|
[11544] | 322 | % \label{eq:ALGOS_Gf_fluxes} |
---|
[10414] | 323 | \begin{split} |
---|
| 324 | {_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) |
---|
| 325 | &= \ \; \qquad \quad { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+i_p}^{\,k}} \\ |
---|
| 326 | {_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
---|
| 327 | &= -\; { _i^k \mathbb{R}_{i_p}^{k_p} } |
---|
| 328 | \ \; { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+k_p}} |
---|
| 329 | \end{split} |
---|
| 330 | \end{flalign*} |
---|
[2282] | 331 | |
---|
[10354] | 332 | The resulting iso-neutral fluxes at $u$- and $w$-points are then given by |
---|
[11543] | 333 | the sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:TRIADS_ISO_triad}): |
---|
[10414] | 334 | \begin{flalign} |
---|
[11544] | 335 | \label{eq:ALGOS_iso_flux} |
---|
[10414] | 336 | \textbf{F}_{iso}(T) |
---|
| 337 | &\equiv \sum_{\substack{i_p,\,k_p}} |
---|
| 338 | \begin{pmatrix} |
---|
| 339 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\ |
---|
| 340 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
---|
| 341 | \end{pmatrix} |
---|
| 342 | \notag \\ |
---|
| 343 | & \notag \\ |
---|
| 344 | &\equiv \sum_{\substack{i_p,\,k_p}} |
---|
| 345 | \begin{pmatrix} |
---|
| 346 | && { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+1/2}^{\,k} } \\ \\ |
---|
| 347 | & -\; { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } |
---|
| 348 | & {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+1/2} } |
---|
| 349 | \end{pmatrix} % \\ |
---|
| 350 | % &\\ |
---|
| 351 | % &\equiv \sum_{\substack{i_p,\,k_p}} |
---|
| 352 | % \begin{pmatrix} |
---|
| 353 | % \qquad \qquad \qquad |
---|
| 354 | % \frac{1}{ {e_{1u}}_{\,i+1/2}^{\,k} } \ \; |
---|
| 355 | % { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T)\\ |
---|
| 356 | % \\ |
---|
| 357 | % -\frac{1}{ {e_{3w}}_{\,i}^{\,k+1/2} } \ \; |
---|
| 358 | % { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; |
---|
| 359 | % {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T)\\ |
---|
| 360 | % \end{pmatrix} |
---|
[2282] | 361 | \end{flalign} |
---|
[10354] | 362 | resulting in a iso-neutral diffusion tendency on temperature given by |
---|
| 363 | the divergence of the sum of all the four triad fluxes: |
---|
[10414] | 364 | \begin{equation} |
---|
[11544] | 365 | \label{eq:ALGOS_Gf_operator} |
---|
[10414] | 366 | D_l^T = \frac{1}{b_T} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
| 367 | \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
---|
| 368 | + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} |
---|
[2282] | 369 | \end{equation} |
---|
[11543] | 370 | where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. |
---|
[2282] | 371 | |
---|
[10354] | 372 | This expression of the iso-neutral diffusion has been chosen in order to satisfy the following six properties: |
---|
[2282] | 373 | \begin{description} |
---|
[10354] | 374 | \item[$\bullet$ horizontal diffusion] |
---|
| 375 | The discretization of the diffusion operator recovers the traditional five-point Laplacian in |
---|
| 376 | the limit of flat iso-neutral direction: |
---|
[10414] | 377 | \[ |
---|
[11544] | 378 | % \label{eq:ALGOS_Gf_property1a} |
---|
[10414] | 379 | D_l^T = \frac{1}{b_T} \ \delta_{i} |
---|
| 380 | \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] |
---|
| 381 | \qquad \text{when} \quad |
---|
| 382 | { _i^k \mathbb{R}_{i_p}^{k_p} }=0 |
---|
| 383 | \] |
---|
[2282] | 384 | |
---|
[10354] | 385 | \item[$\bullet$ implicit treatment in the vertical] |
---|
| 386 | In the diagonal term associated with the vertical divergence of the iso-neutral fluxes |
---|
[11435] | 387 | \ie\ the term associated with a second order vertical derivative) |
---|
[10354] | 388 | appears only tracer values associated with a single water column. |
---|
| 389 | This is of paramount importance since it means that |
---|
| 390 | the implicit in time algorithm for solving the vertical diffusion equation can be used to evaluate this term. |
---|
[10414] | 391 | It is a necessity since the vertical eddy diffusivity associated with this term, |
---|
| 392 | \[ |
---|
| 393 | \sum_{\substack{i_p, \,k_p}} \left\{ |
---|
[2282] | 394 | A_i^k \; \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
---|
[10414] | 395 | \right\} |
---|
| 396 | \] |
---|
| 397 | can be quite large. |
---|
[2282] | 398 | |
---|
[10354] | 399 | \item[$\bullet$ pure iso-neutral operator] |
---|
[10442] | 400 | The iso-neutral flux of locally referenced potential density is zero, \ie |
---|
[10414] | 401 | \begin{align*} |
---|
[11544] | 402 | % \label{eq:ALGOS_Gf_property2} |
---|
[10414] | 403 | \begin{matrix} |
---|
| 404 | &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)} |
---|
| 405 | &= &\alpha_i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) |
---|
| 406 | &- \ \; \beta _i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (S) & = \ 0 \\ |
---|
| 407 | &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)} |
---|
| 408 | &= &\alpha_i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
---|
| 409 | &- \ \; \beta _i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (S) &= \ 0 |
---|
| 410 | \end{matrix} |
---|
| 411 | \end{align*} |
---|
[11544] | 412 | This result is trivially obtained using the \autoref{eq:ALGOS_Gf_triads} applied to $T$ and $S$ and |
---|
| 413 | the definition of the triads' slopes \autoref{eq:ALGOS_Gf_slopes}. |
---|
[2282] | 414 | |
---|
[10354] | 415 | \item[$\bullet$ conservation of tracer] |
---|
[10442] | 416 | The iso-neutral diffusion term conserve the total tracer content, \ie |
---|
[10414] | 417 | \[ |
---|
[11544] | 418 | % \label{eq:ALGOS_Gf_property1} |
---|
[10414] | 419 | \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 |
---|
| 420 | \] |
---|
[10354] | 421 | This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form. |
---|
[2282] | 422 | |
---|
[10354] | 423 | \item[$\bullet$ decrease of tracer variance] |
---|
[10442] | 424 | The iso-neutral diffusion term does not increase the total tracer variance, \ie |
---|
[10414] | 425 | \[ |
---|
[11544] | 426 | % \label{eq:ALGOS_Gf_property1} |
---|
[10414] | 427 | \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 |
---|
| 428 | \] |
---|
[11544] | 429 | The property is demonstrated in the \autoref{subsec:ALGOS_Gf_operator}. |
---|
[10354] | 430 | It is a key property for a diffusion term. |
---|
| 431 | It means that the operator is also a dissipation term, |
---|
[11435] | 432 | \ie\ it is a sink term for the square of the quantity on which it is applied. |
---|
[10354] | 433 | It therfore ensures that, when the diffusivity coefficient is large enough, |
---|
| 434 | the field on which it is applied become free of grid-point noise. |
---|
[2282] | 435 | |
---|
[10354] | 436 | \item[$\bullet$ self-adjoint operator] |
---|
[10442] | 437 | The iso-neutral diffusion operator is self-adjoint, \ie |
---|
[10414] | 438 | \[ |
---|
[11544] | 439 | % \label{eq:ALGOS_Gf_property1} |
---|
[10414] | 440 | \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} |
---|
| 441 | \] |
---|
[10354] | 442 | In other word, there is no needs to develop a specific routine from the adjoint of this operator. |
---|
| 443 | We just have to apply the same routine. |
---|
| 444 | This properties can be demonstrated quite easily in a similar way the "non increase of tracer variance" property |
---|
[11544] | 445 | has been proved (see \autoref{apdx:ALGOS_Gf_operator}). |
---|
[2282] | 446 | \end{description} |
---|
| 447 | |
---|
| 448 | % ================================================================ |
---|
[11543] | 449 | % Skew flux formulation for Eddy Induced Velocity : |
---|
[2282] | 450 | % ================================================================ |
---|
[9393] | 451 | \subsection{Eddy induced velocity and skew flux formulation} |
---|
[2282] | 452 | |
---|
[10354] | 453 | When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), |
---|
| 454 | an additional advection term is added. |
---|
| 455 | The associated velocity is the so called eddy induced velocity, |
---|
| 456 | the formulation of which depends on the slopes of iso-neutral surfaces. |
---|
| 457 | Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, |
---|
[11543] | 458 | \ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinate, |
---|
| 459 | and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $z^*$ or $s$-coordinates. |
---|
[2282] | 460 | |
---|
[11543] | 461 | The eddy induced velocity is given by: |
---|
[10414] | 462 | \begin{equation} |
---|
[11544] | 463 | \label{eq:ALGOS_eiv_v} |
---|
[10414] | 464 | \begin{split} |
---|
| 465 | u^* & = - \frac{1}{e_2\,e_{3}} \;\partial_k \left( e_2 \, A_e \; r_i \right) |
---|
| 466 | = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_i \right) \\ |
---|
| 467 | v^* & = - \frac{1}{e_1\,e_3}\; \partial_k \left( e_1 \, A_e \; r_j \right) |
---|
| 468 | = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_j \right) \\ |
---|
| 469 | w^* & = \frac{1}{e_1\,e_2}\; \left\{ \partial_i \left( e_2 \, A_e \; r_i \right) |
---|
| 470 | + \partial_j \left( e_1 \, A_e \;r_j \right) \right\} |
---|
| 471 | \end{split} |
---|
[2282] | 472 | \end{equation} |
---|
[10354] | 473 | where $A_{e}$ is the eddy induced velocity coefficient, |
---|
| 474 | and $r_i$ and $r_j$ the slopes between the iso-neutral and the geopotential surfaces. |
---|
[2282] | 475 | %%gm wrong: to be modified with 2 2D streamfunctions |
---|
[10354] | 476 | In other words, the eddy induced velocity can be derived from a vector streamfuntion, $\phi$, |
---|
| 477 | which is given by $\phi = A_e\,\textbf{r}$ as $\textbf{U}^* = \textbf{k} \times \nabla \phi$. |
---|
[2282] | 478 | %%end gm |
---|
| 479 | |
---|
[10354] | 480 | A traditional way to implement this additional advection is to add it to the eulerian velocity prior to |
---|
| 481 | compute the tracer advection. |
---|
| 482 | This allows us to take advantage of all the advection schemes offered for the tracers |
---|
| 483 | (see \autoref{sec:TRA_adv}) and not just a $2^{nd}$ order advection scheme. |
---|
| 484 | This is particularly useful for passive tracers where |
---|
[11543] | 485 | \emph{positivity} of the advection scheme is of paramount importance. |
---|
| 486 | % give here the expression using the triads. It is different from the one given in \autoref{eq:LDF_eiv} |
---|
[2282] | 487 | % see just below a copy of this equation: |
---|
[11544] | 488 | %\begin{equation} \label{eq:ALGOS_ldfeiv} |
---|
[2282] | 489 | %\begin{split} |
---|
| 490 | % u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ |
---|
| 491 | % v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\ |
---|
| 492 | %w^* & = \frac{1}{e_{1w}e_{2w}}\; \left\{ \delta_i \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right] + %\delta_j \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right] \right\} \\ |
---|
| 493 | %\end{split} |
---|
| 494 | %\end{equation} |
---|
[10414] | 495 | \[ |
---|
[11544] | 496 | % \label{eq:ALGOS_eiv_vd} |
---|
[10414] | 497 | \textbf{F}_{eiv}^T \equiv \left( |
---|
| 498 | \begin{aligned} |
---|
| 499 | \sum_{\substack{i_p,\,k_p}} & |
---|
| 500 | +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k} |
---|
| 501 | \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ \\ |
---|
| 502 | \sum_{\substack{i_p,\,k_p}} & |
---|
| 503 | - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p} |
---|
| 504 | \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \delta_{i+i_p}[T^{k+1/2-k_p}] |
---|
| 505 | \end{aligned} |
---|
| 506 | \right) |
---|
| 507 | \] |
---|
[2282] | 508 | |
---|
[11123] | 509 | \citep{griffies_JPO98} introduces another way to implement the eddy induced advection, the so-called skew form. |
---|
[10354] | 510 | It is based on a transformation of the advective fluxes using the non-divergent nature of the eddy induced velocity. |
---|
| 511 | For example in the (\textbf{i},\textbf{k}) plane, the tracer advective fluxes can be transformed as follows: |
---|
[2282] | 512 | \begin{flalign*} |
---|
[10414] | 513 | \begin{split} |
---|
| 514 | \textbf{F}_{eiv}^T = |
---|
| 515 | \begin{pmatrix} |
---|
| 516 | {e_{2}\,e_{3}\; u^*} \\ |
---|
| 517 | {e_{1}\,e_{2}\; w^*} |
---|
| 518 | \end{pmatrix} |
---|
| 519 | \; T |
---|
| 520 | &= |
---|
| 521 | \begin{pmatrix} |
---|
| 522 | { - \partial_k \left( e_{2} \, A_{e} \; r_i \right) \; T \;} \\ |
---|
| 523 | {+ \partial_i \left( e_{2} \, A_{e} \; r_i \right) \; T \;} |
---|
| 524 | \end{pmatrix} |
---|
| 525 | \\ |
---|
| 526 | &= |
---|
| 527 | \begin{pmatrix} |
---|
| 528 | { - \partial_k \left( e_{2} \, A_{e} \; r_i \; T \right) \;} \\ |
---|
| 529 | {+ \partial_i \left( e_{2} \, A_{e} \; r_i \; T \right) \;} |
---|
| 530 | \end{pmatrix} |
---|
| 531 | + |
---|
| 532 | \begin{pmatrix} |
---|
| 533 | {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ |
---|
| 534 | { - e_{2} \, A_{e} \; r_i \; \partial_i T} |
---|
| 535 | \end{pmatrix} |
---|
| 536 | \end{split} |
---|
[2282] | 537 | \end{flalign*} |
---|
[10354] | 538 | and since the eddy induces velocity field is no-divergent, |
---|
| 539 | we end up with the skew form of the eddy induced advective fluxes: |
---|
[10414] | 540 | \begin{equation} |
---|
[11544] | 541 | \label{eq:ALGOS_eiv_skew_continuous} |
---|
[10414] | 542 | \textbf{F}_{eiv}^T = |
---|
| 543 | \begin{pmatrix} |
---|
| 544 | {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ |
---|
| 545 | { - e_{2} \, A_{e} \; r_i \; \partial_i T} |
---|
| 546 | \end{pmatrix} |
---|
[2282] | 547 | \end{equation} |
---|
[10354] | 548 | The tendency associated with eddy induced velocity is then simply the divergence of |
---|
[11544] | 549 | the \autoref{eq:ALGOS_eiv_skew_continuous} fluxes. |
---|
[10354] | 550 | It naturally conserves the tracer content, as it is expressed in flux form and, |
---|
| 551 | as the advective form, it preserves the tracer variance. |
---|
[11544] | 552 | Another interesting property of \autoref{eq:ALGOS_eiv_skew_continuous} form is that when $A=A_e$, |
---|
[10354] | 553 | a simplification occurs in the sum of the iso-neutral diffusion and eddy induced velocity terms: |
---|
[10414] | 554 | \begin{flalign*} |
---|
[11544] | 555 | % \label{eq:ALGOS_eiv_skew+eiv_continuous} |
---|
[10414] | 556 | \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &= |
---|
| 557 | \begin{pmatrix} |
---|
| 558 | + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T - e_2 \, A \; r_i \;\partial_k T \\ |
---|
| 559 | - e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T |
---|
| 560 | \end{pmatrix} |
---|
| 561 | + |
---|
| 562 | \begin{pmatrix} |
---|
| 563 | {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ |
---|
| 564 | { - e_{2} \, A_{e} \; r_i \; \partial_i T} |
---|
| 565 | \end{pmatrix} |
---|
| 566 | \\ |
---|
| 567 | &= |
---|
| 568 | \begin{pmatrix} |
---|
| 569 | + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T \\ |
---|
| 570 | - 2\; e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T |
---|
| 571 | \end{pmatrix} |
---|
| 572 | \end{flalign*} |
---|
[10354] | 573 | The horizontal component reduces to the one use for an horizontal laplacian operator and |
---|
| 574 | the vertical one keeps the same complexity, but not more. |
---|
[11123] | 575 | This property has been used to reduce the computational time \citep{griffies_JPO98}, |
---|
[10354] | 576 | but it is not of practical use as usually $A \neq A_e$. |
---|
[11544] | 577 | Nevertheless this property can be used to choose a discret form of \autoref{eq:ALGOS_eiv_skew_continuous} which |
---|
| 578 | is consistent with the iso-neutral operator \autoref{eq:ALGOS_Gf_operator}. |
---|
| 579 | Using the slopes \autoref{eq:ALGOS_Gf_slopes} and defining $A_e$ at $T$-point(\ie\ as $A$, |
---|
[10354] | 580 | the eddy diffusivity coefficient), the resulting discret form is given by: |
---|
[10414] | 581 | \begin{equation} |
---|
[11544] | 582 | \label{eq:ALGOS_eiv_skew} |
---|
[10414] | 583 | \textbf{F}_{eiv}^T \equiv \frac{1}{4} \left( |
---|
| 584 | \begin{aligned} |
---|
| 585 | \sum_{\substack{i_p,\,k_p}} & |
---|
| 586 | +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k} |
---|
| 587 | \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ \\ |
---|
| 588 | \sum_{\substack{i_p,\,k_p}} & |
---|
| 589 | - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p} |
---|
| 590 | \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \delta_{i+i_p}[T^{k+1/2-k_p}] |
---|
| 591 | \end{aligned} |
---|
| 592 | \right) |
---|
[2282] | 593 | \end{equation} |
---|
[11544] | 594 | Note that \autoref{eq:ALGOS_eiv_skew} is valid in $z$-coordinate with or without partial cells. |
---|
[10354] | 595 | In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces must be added to |
---|
[11543] | 596 | $\mathbb{R}$ for the discret form to be exact. |
---|
[2282] | 597 | |
---|
[10354] | 598 | Such a choice of discretisation is consistent with the iso-neutral operator as |
---|
| 599 | it uses the same definition for the slopes. |
---|
[11544] | 600 | It also ensures the conservation of the tracer variance (see \autoref{subsec:ALGOS_eiv_skew}), |
---|
[11435] | 601 | \ie\ it does not include a diffusive component but is a "pure" advection term. |
---|
[2282] | 602 | |
---|
| 603 | $\ $\newpage %force an empty line |
---|
| 604 | % ================================================================ |
---|
[11543] | 605 | % Discrete Invariants of the iso-neutral diffrusion |
---|
[2282] | 606 | % ================================================================ |
---|
[9393] | 607 | \subsection{Discrete invariants of the iso-neutral diffrusion} |
---|
[11544] | 608 | \label{subsec:ALGOS_Gf_operator} |
---|
[2282] | 609 | |
---|
[11543] | 610 | Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane. |
---|
[2282] | 611 | |
---|
| 612 | This part will be moved in an Appendix. |
---|
| 613 | |
---|
[10354] | 614 | The continuous property to be demonstrated is: |
---|
[10414] | 615 | \[ |
---|
| 616 | \int_D D_l^T \; T \;dv \leq 0 |
---|
| 617 | \] |
---|
[11543] | 618 | The discrete form of its left hand side is obtained using \autoref{eq:TRIADS_iso_flux} |
---|
[2282] | 619 | |
---|
| 620 | \begin{align*} |
---|
[10414] | 621 | &\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\ |
---|
| 622 | &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
| 623 | \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
---|
| 624 | + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\ |
---|
| 625 | &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
| 626 | {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] |
---|
| 627 | + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\ |
---|
| 628 | &\equiv -\sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
| 629 | \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] |
---|
| 630 | - { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; |
---|
| 631 | \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] |
---|
| 632 | \right\} \\ |
---|
| 633 | % |
---|
| 634 | \allowdisplaybreaks |
---|
| 635 | \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} |
---|
| 636 | % |
---|
| 637 | &\equiv -\sum_{i,k} |
---|
| 638 | \begin{Bmatrix} |
---|
| 639 | &\ \ \Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{-1/2} (T) } |
---|
| 640 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 641 | & -\ \ {_{i}^{k+1} \mathbb{R}_{-1/2}^{-1/2}} |
---|
| 642 | & {_{i}^{k+1} \mathbb{T}_{-1/2}^{-1/2} (T) } |
---|
| 643 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
---|
| 644 | & \\ |
---|
| 645 | &+\Bigl( \ \;\; { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } |
---|
| 646 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 647 | & -\ \ {_i^{k+1} \mathbb{R}_{+1/2}^{-1/2}} |
---|
| 648 | & { _i^{k+1} \mathbb{T}_{+1/2}^{-1/2} (T) } |
---|
| 649 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
---|
| 650 | & \\ |
---|
| 651 | &+\Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } |
---|
| 652 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 653 | & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{-1/2}^{+1/2}} |
---|
| 654 | & \ \;\;{_{i}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } |
---|
| 655 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
---|
| 656 | & \\ |
---|
| 657 | &+\Bigl( \ \;\; { _{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } |
---|
| 658 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 659 | & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{+1/2}^{+1/2}} |
---|
| 660 | & \ \;\;{_{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } |
---|
| 661 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) \\ |
---|
| 662 | \end{Bmatrix} |
---|
| 663 | % |
---|
| 664 | \allowdisplaybreaks |
---|
| 665 | \intertext{ |
---|
| 666 | The summation is done over all $i$ and $k$ indices, |
---|
[10354] | 667 | it is therefore possible to introduce a shift of $-1$ either in $i$ or $k$ direction in order to |
---|
| 668 | regroup all the terms of the summation by triad at a ($i$,$k$) point. |
---|
| 669 | In other words, we regroup all the terms in the neighbourhood that contain a triad at the same ($i$,$k$) indices. |
---|
[10414] | 670 | It becomes: |
---|
| 671 | } |
---|
| 672 | % |
---|
| 673 | &\equiv -\sum_{i,k} |
---|
| 674 | \begin{Bmatrix} |
---|
| 675 | &\ \ \Bigl( {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } |
---|
| 676 | &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} |
---|
| 677 | & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} |
---|
| 678 | & {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } |
---|
| 679 | &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr) |
---|
| 680 | & \\ |
---|
| 681 | &+\Bigl( { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } |
---|
| 682 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 683 | & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} |
---|
| 684 | & { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } |
---|
| 685 | &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr) |
---|
| 686 | & \\ |
---|
| 687 | &+\Bigl( {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } |
---|
| 688 | &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} |
---|
| 689 | & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} |
---|
| 690 | & {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } |
---|
| 691 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
---|
| 692 | & \\ |
---|
| 693 | &+\Bigl( { _i^k \mathbb{T}_{+1/2}^{+1/2} (T) } |
---|
| 694 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 695 | & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} |
---|
| 696 | & {_i^k \mathbb{T}_{+1/2}^{+1/2} (T) } |
---|
| 697 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) \\ |
---|
| 698 | \end{Bmatrix} \\ |
---|
| 699 | % |
---|
| 700 | \allowdisplaybreaks |
---|
| 701 | \intertext{ |
---|
| 702 | Then outing in factor the triad in each of the four terms of the summation and |
---|
[11544] | 703 | substituting the triads by their expression given in \autoref{eq:ALGOS_Gf_triads}. |
---|
[10414] | 704 | It becomes: |
---|
| 705 | } |
---|
| 706 | % |
---|
| 707 | &\equiv -\sum_{i,k} |
---|
| 708 | \begin{Bmatrix} |
---|
| 709 | &\ \ \Bigl( \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} |
---|
| 710 | & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} |
---|
| 711 | &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)^2 |
---|
| 712 | & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k} \ A_i^k |
---|
| 713 | & \\ |
---|
| 714 | &+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 715 | & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} |
---|
| 716 | &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)^2 |
---|
| 717 | & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k} \ A_i^k |
---|
| 718 | & \\ |
---|
| 719 | &+\Bigl( \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} |
---|
| 720 | & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} |
---|
| 721 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)^2 |
---|
| 722 | & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k} \ A_i^k |
---|
| 723 | & \\ |
---|
| 724 | &+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 725 | & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} |
---|
| 726 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)^2 |
---|
| 727 | & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k} \ A_i^k \\ |
---|
| 728 | \end{Bmatrix} |
---|
| 729 | \\ |
---|
| 730 | & \\ |
---|
| 731 | % |
---|
| 732 | &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
| 733 | \begin{matrix} |
---|
| 734 | &\Bigl( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} |
---|
| 735 | & -\ \ {_i^k \mathbb{R}_{i_p}^{k_p}} |
---|
| 736 | &\frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}} \Bigr)^2 |
---|
| 737 | & \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \ \ |
---|
| 738 | \end{matrix} |
---|
| 739 | \right\} |
---|
| 740 | \quad \leq 0 |
---|
[11543] | 741 | \end{align*} |
---|
[2282] | 742 | The last inequality is obviously obtained as we succeed in obtaining a negative summation of square quantities. |
---|
| 743 | |
---|
[10354] | 744 | Note that, if instead of multiplying $D_l^T$ by $T$, we were using another tracer field, let say $S$, |
---|
| 745 | then the previous demonstration would have let to: |
---|
[2282] | 746 | \begin{align*} |
---|
[10414] | 747 | \int_D S \; D_l^T \;dv &\equiv \sum_{i,k} \left\{ S \ D_l^T \ b_T \right\} \\ |
---|
| 748 | &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
| 749 | \left( \frac{ \delta_{i +i_p} [S] }{{e_{1u} }_{\,i+i_p}^{\,k}} |
---|
| 750 | - {_i^k \mathbb{R}_{i_p}^{k_p}} |
---|
| 751 | \frac{ \delta_{k+k_p} [S] }{{e_{3w}}_{\,i}^{\,k+k_p}} \right) \right. \\ |
---|
| 752 | & \qquad \qquad \qquad \ \left. |
---|
| 753 | \left( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} |
---|
| 754 | - {_i^k \mathbb{R}_{i_p}^{k_p}} |
---|
| 755 | \frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}} \right) |
---|
| 756 | \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \ |
---|
| 757 | \right\} |
---|
| 758 | % |
---|
| 759 | \allowdisplaybreaks |
---|
| 760 | \intertext{ |
---|
| 761 | which, by applying the same operation as before but in reverse order, leads to: |
---|
| 762 | } |
---|
| 763 | % |
---|
| 764 | &\equiv \sum_{i,k} \left\{ D_l^S \ T \ b_T \right\} |
---|
[11543] | 765 | \end{align*} |
---|
[10354] | 766 | This means that the iso-neutral operator is self-adjoint. |
---|
| 767 | There is no need to develop a specific to obtain it. |
---|
[2282] | 768 | |
---|
[10414] | 769 | \newpage |
---|
[2282] | 770 | |
---|
| 771 | % ================================================================ |
---|
| 772 | % Discrete Invariants of the skew flux formulation |
---|
| 773 | % ================================================================ |
---|
[9393] | 774 | \subsection{Discrete invariants of the skew flux formulation} |
---|
[11544] | 775 | \label{subsec:ALGOS_eiv_skew} |
---|
[2282] | 776 | |
---|
[11543] | 777 | Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. |
---|
[2282] | 778 | |
---|
| 779 | This have to be moved in an Appendix. |
---|
| 780 | |
---|
[10354] | 781 | The continuous property to be demonstrated is: |
---|
[2282] | 782 | \begin{align*} |
---|
[10414] | 783 | \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv \equiv 0 |
---|
[2282] | 784 | \end{align*} |
---|
[11544] | 785 | The discrete form of its left hand side is obtained using \autoref{eq:ALGOS_eiv_skew} |
---|
[2282] | 786 | \begin{align*} |
---|
[10414] | 787 | \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; |
---|
| 788 | \delta_i &\left[ |
---|
| 789 | {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} |
---|
| 790 | \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] |
---|
| 791 | \right] \; T_i^k \\ |
---|
| 792 | - \delta_k &\left[ |
---|
| 793 | {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} |
---|
| 794 | \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] |
---|
| 795 | \right] \; T_i^k \ \Biggr\} |
---|
[2282] | 796 | \end{align*} |
---|
| 797 | apply the adjoint of delta operator, it becomes |
---|
| 798 | \begin{align*} |
---|
[10414] | 799 | \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; |
---|
| 800 | &\left( |
---|
| 801 | {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} |
---|
| 802 | \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] |
---|
| 803 | \right) \; \delta_{i+1/2}[T^{k}] \\ |
---|
| 804 | - &\left( |
---|
| 805 | {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} |
---|
| 806 | \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] |
---|
| 807 | \right) \; \delta_{k+1/2}[T_{i}] \ \Biggr\} |
---|
[2282] | 808 | \end{align*} |
---|
| 809 | Expending the summation on $i_p$ and $k_p$, it becomes: |
---|
| 810 | \begin{align*} |
---|
[10414] | 811 | \begin{matrix} |
---|
| 812 | &\sum\limits_{i,k} \Bigl\{ |
---|
| 813 | &+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} |
---|
| 814 | &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{-1/2}} &\delta_{k-1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
| 815 | &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} |
---|
| 816 | &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{-1/2}} &\delta_{k-1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
| 817 | &&+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} |
---|
| 818 | &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{+1/2}} &\delta_{k+1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
| 819 | &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} |
---|
[2282] | 820 | &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{k+1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
[10414] | 821 | % |
---|
| 822 | &&-{e_{2u}}_i^{k+1} &{A_{e}}_i^{k+1} |
---|
| 823 | &{_i^{k+1} \mathbb{R}_{-1/2}^{- 1/2}} &\delta_{i-1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\ |
---|
| 824 | &&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} |
---|
| 825 | &{\ \ \;_i^k \mathbb{R}_{-1/2}^{+1/2}} &\delta_{i-1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] &\\ |
---|
| 826 | &&-{e_{2u}}_i^{k+1 } &{A_{e}}_i^{k+1} |
---|
| 827 | &{_i^{k+1} \mathbb{R}_{+1/2}^{- 1/2}} &\delta_{i+1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\ |
---|
| 828 | &&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} |
---|
| 829 | &{\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{i+1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] |
---|
| 830 | &\Bigr\} \\ |
---|
[11543] | 831 | \end{matrix} |
---|
[2282] | 832 | \end{align*} |
---|
[10354] | 833 | The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the same but of opposite signs, |
---|
[11543] | 834 | they cancel out. |
---|
[10354] | 835 | Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{-1/2}}$. |
---|
| 836 | The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{-1/2}}$ are the same but both of opposite signs and |
---|
| 837 | shifted by 1 in $k$ direction. |
---|
| 838 | When summing over $k$ they cancel out with the neighbouring grid points. |
---|
| 839 | Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{+1/2}}$ in the $i$ direction. |
---|
| 840 | Therefore the sum over the domain is zero, |
---|
[11435] | 841 | \ie\ the variance of the tracer is preserved by the discretisation of the skew fluxes. |
---|
[2282] | 842 | |
---|
[10414] | 843 | \biblio |
---|
| 844 | |
---|
[10442] | 845 | \pindex |
---|
| 846 | |
---|
[6997] | 847 | \end{document} |
---|