[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
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| 2 | |
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[6997] | 3 | \begin{document} |
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[707] | 4 | % ================================================================ |
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[2282] | 5 | % Appendix E : Note on some algorithms |
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[707] | 6 | % ================================================================ |
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[2282] | 7 | \chapter{Note on some algorithms} |
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[9407] | 8 | \label{apdx:E} |
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[10414] | 9 | |
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[707] | 10 | \minitoc |
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| 11 | |
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[2282] | 12 | \newpage |
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[707] | 13 | |
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[10354] | 14 | This appendix some on going consideration on algorithms used or planned to be used in \NEMO. |
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| 15 | |
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[707] | 16 | % ------------------------------------------------------------------------------------------------------------- |
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| 17 | % UBS scheme |
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| 18 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 19 | \section{Upstream Biased Scheme (UBS) (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})} |
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[9407] | 20 | \label{sec:TRA_adv_ubs} |
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[707] | 21 | |
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[10354] | 22 | The UBS advection scheme is an upstream biased third order scheme based on |
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| 23 | an upstream-biased parabolic interpolation. |
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| 24 | It is also known as Cell Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective Kinematics). |
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| 25 | For example, in the $i$-direction: |
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[10414] | 26 | \begin{equation} |
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| 27 | \label{eq:tra_adv_ubs2} |
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| 28 | \tau_u^{ubs} = \left\{ |
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| 29 | \begin{aligned} |
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| 30 | & \tau_u^{cen4} + \frac{1}{12} \,\tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ |
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| 31 | & \tau_u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 |
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| 32 | \end{aligned} |
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| 33 | \right. |
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[707] | 34 | \end{equation} |
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| 35 | or equivalently, the advective flux is |
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[10414] | 36 | \begin{equation} |
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| 37 | \label{eq:tra_adv_ubs2} |
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| 38 | U_{i+1/2} \ \tau_u^{ubs} |
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| 39 | =U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2} |
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| 40 | - \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
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[707] | 41 | \end{equation} |
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[10354] | 42 | where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and |
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[10406] | 43 | $\tau "_i =\delta_i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$. |
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[10354] | 44 | By choosing this expression for $\tau "$ we consider a fourth order approximation of $\partial_i^2$ with |
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| 45 | a constant i-grid spacing ($\Delta i=1$). |
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[707] | 46 | |
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[1223] | 47 | Alternative choice: introduce the scale factors: |
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[10406] | 48 | $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta_{i+1/2}[\tau] \right]$. |
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[707] | 49 | |
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[10354] | 50 | This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error |
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| 51 | \citep{Shchepetkin_McWilliams_OM05}. |
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| 52 | The overall performance of the advection scheme is similar to that reported in \cite{Farrow1995}. |
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| 53 | It is a relatively good compromise between accuracy and smoothness. |
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| 54 | It is not a \emph{positive} scheme meaning false extrema are permitted but |
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| 55 | the amplitude of such are significantly reduced over the centred second order method. |
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| 56 | Nevertheless it is not recommended to apply it to a passive tracer that requires positivity. |
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[707] | 57 | |
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[10354] | 58 | The intrinsic diffusion of UBS makes its use risky in the vertical direction where |
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| 59 | the control of artificial diapycnal fluxes is of paramount importance. |
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| 60 | It has therefore been preferred to evaluate the vertical flux using the TVD scheme when |
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| 61 | \np{ln\_traadv\_ubs}\forcode{ = .true.}. |
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[707] | 62 | |
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[10354] | 63 | For stability reasons, in \autoref{eq:tra_adv_ubs}, the first term which corresponds to |
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| 64 | a second order centred scheme is evaluated using the \textit{now} velocity (centred in time) while |
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| 65 | the second term which is the diffusive part of the scheme, is evaluated using the \textit{before} velocity |
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| 66 | (forward in time). |
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| 67 | This is discussed by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme. |
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| 68 | UBS and QUICK schemes only differ by one coefficient. |
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| 69 | Substituting 1/6 with 1/8 in (\autoref{eq:tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. |
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| 70 | This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded. |
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| 71 | Nevertheless it is quite easy to make the substitution in \mdl{traadv\_ubs} module and obtain a QUICK scheme. |
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[707] | 72 | |
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[10354] | 73 | NB 1: When a high vertical resolution $O(1m)$ is used, the model stability can be controlled by vertical advection |
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| 74 | (not vertical diffusion which is usually solved using an implicit scheme). |
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| 75 | Computer time can be saved by using a time-splitting technique on vertical advection. |
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| 76 | This possibility have been implemented and validated in ORCA05-L301. |
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| 77 | It is not currently offered in the current reference version. |
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[707] | 78 | |
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[10354] | 79 | NB 2: In a forthcoming release four options will be proposed for the vertical component used in the UBS scheme. |
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[10406] | 80 | $\tau_w^{ubs}$ will be evaluated using either \textit{(a)} a centered $2^{nd}$ order scheme, |
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[10354] | 81 | or \textit{(b)} a TVD scheme, or \textit{(c)} an interpolation based on conservative parabolic splines following |
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| 82 | \citet{Shchepetkin_McWilliams_OM05} implementation of UBS in ROMS, or \textit{(d)} an UBS. |
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| 83 | The $3^{rd}$ case has dispersion properties similar to an eight-order accurate conventional scheme. |
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[707] | 84 | |
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[10354] | 85 | NB 3: It is straight forward to rewrite \autoref{eq:tra_adv_ubs} as follows: |
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[10414] | 86 | \begin{equation} |
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| 87 | \label{eq:tra_adv_ubs2} |
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| 88 | \tau_u^{ubs} = \left\{ |
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| 89 | \begin{aligned} |
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| 90 | & \tau_u^{cen4} + \frac{1}{12} \tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ |
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| 91 | & \tau_u^{cen4} - \frac{1}{12} \tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 |
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| 92 | \end{aligned} |
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| 93 | \right. |
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[707] | 94 | \end{equation} |
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| 95 | or equivalently |
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[10414] | 96 | \begin{equation} |
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| 97 | \label{eq:tra_adv_ubs2} |
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| 98 | \begin{split} |
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| 99 | e_{2u} e_{3u}\,u_{i+1/2} \ \tau_u^{ubs} |
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| 100 | &= e_{2u} e_{3u}\,u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\tau"_i }^{\,i+1/2} \\ |
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| 101 | & - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
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| 102 | \end{split} |
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[707] | 103 | \end{equation} |
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[10354] | 104 | \autoref{eq:tra_adv_ubs2} has several advantages. |
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| 105 | First it clearly evidences that the UBS scheme is based on the fourth order scheme to which |
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| 106 | is added an upstream biased diffusive term. |
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| 107 | Second, this emphasises that the $4^{th}$ order part have to be evaluated at \emph{now} time step, |
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| 108 | not only the $2^{th}$ order part as stated above using \autoref{eq:tra_adv_ubs}. |
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| 109 | Third, the diffusive term is in fact a biharmonic operator with a eddy coefficient which |
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| 110 | is simply proportional to the velocity. |
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[707] | 111 | |
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| 112 | laplacian diffusion: |
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[10414] | 113 | \begin{equation} |
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| 114 | \label{eq:tra_ldf_lap} |
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| 115 | \begin{split} |
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| 116 | D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\; e_{3T} } &\left[ {\quad \delta_i |
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| 117 | \left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta_{i+1/2} |
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| 118 | \left[ T \right]} \right]} \right. \\ |
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| 119 | &\ \left. {+\; \delta_j \left[ |
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| 120 | {A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta_{j+1/2} \left[ T |
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| 121 | \right]} \right)} \right]\quad } \right] |
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| 122 | \end{split} |
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[707] | 123 | \end{equation} |
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| 124 | |
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| 125 | bilaplacian: |
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[10414] | 126 | \begin{equation} |
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| 127 | \label{eq:tra_ldf_lap} |
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| 128 | \begin{split} |
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| 129 | D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ |
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| 130 | & \delta_i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2} |
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| 131 | \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} |
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| 132 | \delta_i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2} |
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| 133 | [T] \right] \right] \right] |
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| 134 | \end{split} |
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[707] | 135 | \end{equation} |
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| 136 | with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$, |
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| 137 | $i.e.$ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$ |
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[10354] | 138 | it comes: |
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[10414] | 139 | \begin{equation} |
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| 140 | \label{eq:tra_ldf_lap} |
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| 141 | \begin{split} |
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| 142 | D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ |
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| 143 | & \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2} |
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| 144 | \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} |
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| 145 | \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2} |
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| 146 | [T] \right] \right] \right] |
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| 147 | \end{split} |
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[707] | 148 | \end{equation} |
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| 149 | if the velocity is uniform ($i.e.$ $|u|=cst$) then the diffusive flux is |
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[10414] | 150 | \begin{equation} |
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| 151 | \label{eq:tra_ldf_lap} |
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| 152 | \begin{split} |
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| 153 | F_u^{lT} = - \frac{1}{12} |
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| 154 | e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta_{i+1/2} |
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| 155 | \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} |
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| 156 | \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta_{i+1/2} |
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| 157 | [T] \right] \right] |
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| 158 | \end{split} |
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[707] | 159 | \end{equation} |
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| 160 | beurk.... reverte the logic: starting from the diffusive part of the advective flux it comes: |
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| 161 | |
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[10414] | 162 | \begin{equation} |
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| 163 | \label{eq:tra_adv_ubs2} |
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| 164 | \begin{split} |
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| 165 | F_u^{lT} &= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
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| 166 | \end{split} |
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[707] | 167 | \end{equation} |
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[10354] | 168 | if the velocity is uniform ($i.e.$ $|u|=cst$) and |
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[10406] | 169 | choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right]$ |
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[707] | 170 | |
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| 171 | sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$): |
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[10414] | 172 | \begin{equation} |
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| 173 | \label{eq:tra_adv_ubs2} |
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| 174 | \begin{split} |
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| 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] |
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| 176 | \end{split} |
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[707] | 177 | \end{equation} |
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| 178 | which leads to ${A_T^{lT}}^2 = \frac{1}{12} {e_{1T}}^3\ \overline{|u|}^{\,i+1/2}$ |
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| 179 | |
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| 180 | sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$ |
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[10414] | 181 | \begin{equation} |
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| 182 | \label{eq:tra_adv_ubs2} |
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| 183 | \begin{split} |
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| 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] \\ |
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| 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] |
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| 186 | \end{split} |
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[707] | 187 | \end{equation} |
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| 188 | which leads to ${A_u^{lT}} = \frac{1}{12} {e_{1u}}^3\ |u|$ |
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| 189 | |
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| 190 | % ------------------------------------------------------------------------------------------------------------- |
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| 191 | % Leap-Frog energetic |
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| 192 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 193 | \section{Leapfrog energetic} |
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[9407] | 194 | \label{sec:LF} |
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[707] | 195 | |
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[10354] | 196 | We adopt the following semi-discrete notation for time derivative. |
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| 197 | Given the values of a variable $q$ at successive time step, |
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| 198 | the time derivation and averaging operators at the mid time step are: |
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[10414] | 199 | \[ |
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| 200 | % \label{eq:dt_mt} |
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| 201 | \begin{split} |
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| 202 | \delta_{t+\rdt/2} [q] &= \ \ \, q^{t+\rdt} - q^{t} \\ |
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| 203 | \overline q^{\,t+\rdt/2} &= \left\{ q^{t+\rdt} + q^{t} \right\} \; / \; 2 |
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| 204 | \end{split} |
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| 205 | \] |
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[10354] | 206 | As for space operator, |
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| 207 | the adjoint of the derivation and averaging time operators are $\delta_t^*=\delta_{t+\rdt/2}$ and |
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| 208 | $\overline{\cdot}^{\,t\,*}= \overline{\cdot}^{\,t+\Delta/2}$, respectively. |
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[707] | 209 | |
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[9407] | 210 | The Leap-frog time stepping given by \autoref{eq:DOM_nxt} can be defined as: |
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[10414] | 211 | \[ |
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| 212 | % \label{eq:LF} |
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| 213 | \frac{\partial q}{\partial t} |
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| 214 | \equiv \frac{1}{\rdt} \overline{ \delta_{t+\rdt/2}[q]}^{\,t} |
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| 215 | = \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} |
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| 216 | \] |
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[10354] | 217 | Note that \autoref{chap:LF} shows that the leapfrog time step is $\rdt$, |
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| 218 | not $2\rdt$ as it can be found sometimes in literature. |
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| 219 | The leap-Frog time stepping is a second order centered scheme. |
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| 220 | As such it respects the quadratic invariant in integral forms, $i.e.$ the following continuous property, |
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[10414] | 221 | \[ |
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| 222 | % \label{eq:Energy} |
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| 223 | \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} |
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| 224 | =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} |
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| 225 | = \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) , |
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| 226 | \] |
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[10354] | 227 | is satisfied in discrete form. |
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| 228 | Indeed, |
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[10414] | 229 | \[ |
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| 230 | \begin{split} |
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| 231 | \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} |
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| 232 | &\equiv \sum\limits_{0}^{N} |
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| 233 | {\frac{1}{\rdt} q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} \ \rdt} |
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| 234 | \equiv \sum\limits_{0}^{N} { q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} } \\ |
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| 235 | &\equiv \sum\limits_{0}^{N} { \overline{q}^{\,t+\Delta/2}{ \delta_{t+\rdt/2}[q]}} |
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| 236 | \equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta_{t+\rdt/2}[q^2] }\\ |
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| 237 | &\equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta_{t+\rdt/2}[q^2] } |
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| 238 | \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) |
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| 239 | \end{split} |
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| 240 | \] |
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[10354] | 241 | NB here pb of boundary condition when applying the adjoint! |
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| 242 | In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition |
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| 243 | (equivalently of the boundary value of the integration by part). |
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| 244 | In time this boundary condition is not physical and \textbf{add something here!!!} |
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[707] | 245 | |
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[2282] | 246 | % ================================================================ |
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| 247 | % Iso-neutral diffusion : |
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| 248 | % ================================================================ |
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| 249 | |
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| 250 | \section{Lateral diffusion operator} |
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| 251 | |
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| 252 | % ================================================================ |
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| 253 | % Griffies' iso-neutral diffusion operator : |
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| 254 | % ================================================================ |
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[9393] | 255 | \subsection{Griffies iso-neutral diffusion operator} |
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[2282] | 256 | |
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[10354] | 257 | Let try to define a scheme that get its inspiration from the \citet{Griffies_al_JPO98} scheme, |
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| 258 | but is formulated within the \NEMO framework |
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| 259 | ($i.e.$ using scale factors rather than grid-size and having a position of $T$-points that |
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| 260 | is not necessary in the middle of vertical velocity points, see \autoref{fig:zgr_e3}). |
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[2282] | 261 | |
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[10354] | 262 | In the formulation \autoref{eq:tra_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO, |
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| 263 | the off-diagonal terms of the small angle diffusion tensor contain several double spatial averages of a gradient, |
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| 264 | for example $\overline{\overline{\delta_k \cdot}}^{\,i,k}$. |
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| 265 | It is apparent that the combination of a $k$ average and a $k$ derivative of the tracer allows for |
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| 266 | the presence of grid point oscillation structures that will be invisible to the operator. |
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| 267 | These structures are \textit{computational modes}. |
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| 268 | They will not be damped by the iso-neutral operator, and even possibly amplified by it. |
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| 269 | In other word, the operator applied to a tracer does not warranties the decrease of its global average variance. |
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| 270 | To circumvent this, we have introduced a smoothing of the slopes of the iso-neutral surfaces |
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| 271 | (see \autoref{chap:LDF}). |
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| 272 | Nevertheless, this technique works fine for $T$ and $S$ as they are active tracers |
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| 273 | ($i.e.$ they enter the computation of density), but it does not work for a passive tracer. |
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| 274 | \citep{Griffies_al_JPO98} introduce a different way to discretise the off-diagonal terms that |
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| 275 | nicely solve the problem. |
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| 276 | The idea is to get rid of combinations of an averaged in one direction combined with |
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| 277 | a derivative in the same direction by considering triads. |
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| 278 | For example in the (\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows: |
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[10414] | 279 | \begin{equation} |
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| 280 | \label{eq:Gf_triads} |
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| 281 | _i^k \mathbb{T}_{i_p}^{k_p} (T) |
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| 282 | = \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \left( |
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| 283 | \frac{ \delta_{i + i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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| 284 | -\ {_i^k \mathbb{R}_{i_p}^{k_p}} \ \frac{ \delta_{k+k_p} [T^i] }{ {e_{3w}}_{\,i}^{\,k+k_p} } |
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| 285 | \right) |
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[2282] | 286 | \end{equation} |
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[10354] | 287 | where the indices $i_p$ and $k_p$ define the four triads and take the following value: |
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| 288 | $i_p = -1/2$ or $1/2$ and $k_p = -1/2$ or $1/2$, |
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| 289 | $b_u= e_{1u}\,e_{2u}\,e_{3u}$ is the volume of $u$-cells, |
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[2282] | 290 | $A_i^k$ is the lateral eddy diffusivity coefficient defined at $T$-point, |
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[10354] | 291 | and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad: |
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[10414] | 292 | \begin{equation} |
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| 293 | \label{eq:Gf_slopes} |
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| 294 | _i^k \mathbb{R}_{i_p}^{k_p} |
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| 295 | =\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac |
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| 296 | {\left(\alpha / \beta \right)_i^k \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] } |
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| 297 | {\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] } |
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[2282] | 298 | \end{equation} |
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[10354] | 299 | Note that in \autoref{eq:Gf_slopes} we use the ratio $\alpha / \beta$ instead of |
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| 300 | multiplying the temperature derivative by $\alpha$ and the salinity derivative by $\beta$. |
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| 301 | This is more efficient as the ratio $\alpha / \beta$ can to be evaluated directly. |
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[2282] | 302 | |
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[10354] | 303 | Note that in \autoref{eq:Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of ${b_{uw}}_{\,i+i_p}^{\,k+k_p}$. |
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| 304 | This choice has been motivated by the decrease of tracer variance and |
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| 305 | the presence of partial cell at the ocean bottom (see \autoref{apdx:Gf_operator}). |
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[2282] | 306 | |
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| 307 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[10414] | 308 | \begin{figure}[!ht] |
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| 309 | \begin{center} |
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| 310 | \includegraphics[width=0.70\textwidth]{Fig_ISO_triad} |
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| 311 | \caption{ |
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| 312 | \protect\label{fig:ISO_triad} |
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| 313 | Triads used in the Griffies's like iso-neutral diffision scheme for |
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| 314 | $u$-component (upper panel) and $w$-component (lower panel). |
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| 315 | } |
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| 316 | \end{center} |
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[2282] | 317 | \end{figure} |
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| 318 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 319 | |
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| 320 | The four iso-neutral fluxes associated with the triads are defined at $T$-point. |
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[10354] | 321 | They take the following expression: |
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[10414] | 322 | \begin{flalign*} |
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| 323 | % \label{eq:Gf_fluxes} |
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| 324 | \begin{split} |
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| 325 | {_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) |
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| 326 | &= \ \; \qquad \quad { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+i_p}^{\,k}} \\ |
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| 327 | {_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
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| 328 | &= -\; { _i^k \mathbb{R}_{i_p}^{k_p} } |
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| 329 | \ \; { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+k_p}} |
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| 330 | \end{split} |
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| 331 | \end{flalign*} |
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[2282] | 332 | |
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[10354] | 333 | The resulting iso-neutral fluxes at $u$- and $w$-points are then given by |
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| 334 | the sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:ISO_triad}): |
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[10414] | 335 | \begin{flalign} |
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| 336 | \label{eq:iso_flux} |
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| 337 | \textbf{F}_{iso}(T) |
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| 338 | &\equiv \sum_{\substack{i_p,\,k_p}} |
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| 339 | \begin{pmatrix} |
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| 340 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\ |
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| 341 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
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| 342 | \end{pmatrix} |
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| 343 | \notag \\ |
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| 344 | & \notag \\ |
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| 345 | &\equiv \sum_{\substack{i_p,\,k_p}} |
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| 346 | \begin{pmatrix} |
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| 347 | && { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+1/2}^{\,k} } \\ \\ |
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| 348 | & -\; { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } |
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| 349 | & {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+1/2} } |
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| 350 | \end{pmatrix} % \\ |
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| 351 | % &\\ |
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| 352 | % &\equiv \sum_{\substack{i_p,\,k_p}} |
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| 353 | % \begin{pmatrix} |
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| 354 | % \qquad \qquad \qquad |
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| 355 | % \frac{1}{ {e_{1u}}_{\,i+1/2}^{\,k} } \ \; |
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| 356 | % { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T)\\ |
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| 357 | % \\ |
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| 358 | % -\frac{1}{ {e_{3w}}_{\,i}^{\,k+1/2} } \ \; |
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| 359 | % { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; |
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| 360 | % {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T)\\ |
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| 361 | % \end{pmatrix} |
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[2282] | 362 | \end{flalign} |
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[10354] | 363 | resulting in a iso-neutral diffusion tendency on temperature given by |
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| 364 | the divergence of the sum of all the four triad fluxes: |
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[10414] | 365 | \begin{equation} |
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| 366 | \label{eq:Gf_operator} |
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| 367 | D_l^T = \frac{1}{b_T} \sum_{\substack{i_p,\,k_p}} \left\{ |
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| 368 | \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
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| 369 | + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} |
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[2282] | 370 | \end{equation} |
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| 371 | where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. |
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| 372 | |
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[10354] | 373 | This expression of the iso-neutral diffusion has been chosen in order to satisfy the following six properties: |
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[2282] | 374 | \begin{description} |
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[10354] | 375 | \item[$\bullet$ horizontal diffusion] |
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| 376 | The discretization of the diffusion operator recovers the traditional five-point Laplacian in |
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| 377 | the limit of flat iso-neutral direction: |
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[10414] | 378 | \[ |
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| 379 | % \label{eq:Gf_property1a} |
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| 380 | D_l^T = \frac{1}{b_T} \ \delta_{i} |
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| 381 | \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] |
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| 382 | \qquad \text{when} \quad |
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| 383 | { _i^k \mathbb{R}_{i_p}^{k_p} }=0 |
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| 384 | \] |
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[2282] | 385 | |
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[10354] | 386 | \item[$\bullet$ implicit treatment in the vertical] |
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| 387 | In the diagonal term associated with the vertical divergence of the iso-neutral fluxes |
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| 388 | (i.e. the term associated with a second order vertical derivative) |
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| 389 | appears only tracer values associated with a single water column. |
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| 390 | This is of paramount importance since it means that |
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| 391 | the implicit in time algorithm for solving the vertical diffusion equation can be used to evaluate this term. |
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[10414] | 392 | It is a necessity since the vertical eddy diffusivity associated with this term, |
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| 393 | \[ |
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| 394 | \sum_{\substack{i_p, \,k_p}} \left\{ |
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[2282] | 395 | A_i^k \; \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
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[10414] | 396 | \right\} |
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| 397 | \] |
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| 398 | can be quite large. |
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[2282] | 399 | |
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[10354] | 400 | \item[$\bullet$ pure iso-neutral operator] |
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| 401 | The iso-neutral flux of locally referenced potential density is zero, $i.e.$ |
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[10414] | 402 | \begin{align*} |
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| 403 | % \label{eq:Gf_property2} |
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| 404 | \begin{matrix} |
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| 405 | &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)} |
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| 406 | &= &\alpha_i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) |
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| 407 | &- \ \; \beta _i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (S) & = \ 0 \\ |
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| 408 | &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)} |
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| 409 | &= &\alpha_i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
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| 410 | &- \ \; \beta _i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (S) &= \ 0 |
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| 411 | \end{matrix} |
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| 412 | \end{align*} |
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| 413 | This result is trivially obtained using the \autoref{eq:Gf_triads} applied to $T$ and $S$ and |
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| 414 | the definition of the triads' slopes \autoref{eq:Gf_slopes}. |
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[2282] | 415 | |
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[10354] | 416 | \item[$\bullet$ conservation of tracer] |
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| 417 | The iso-neutral diffusion term conserve the total tracer content, $i.e.$ |
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[10414] | 418 | \[ |
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| 419 | % \label{eq:Gf_property1} |
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| 420 | \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 |
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| 421 | \] |
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[10354] | 422 | This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form. |
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[2282] | 423 | |
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[10354] | 424 | \item[$\bullet$ decrease of tracer variance] |
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| 425 | The iso-neutral diffusion term does not increase the total tracer variance, $i.e.$ |
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[10414] | 426 | \[ |
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| 427 | % \label{eq:Gf_property1} |
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| 428 | \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 |
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| 429 | \] |
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[10354] | 430 | The property is demonstrated in the \autoref{apdx:Gf_operator}. |
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| 431 | It is a key property for a diffusion term. |
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| 432 | It means that the operator is also a dissipation term, |
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| 433 | $i.e.$ it is a sink term for the square of the quantity on which it is applied. |
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| 434 | It therfore ensures that, when the diffusivity coefficient is large enough, |
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| 435 | the field on which it is applied become free of grid-point noise. |
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[2282] | 436 | |
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[10354] | 437 | \item[$\bullet$ self-adjoint operator] |
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| 438 | The iso-neutral diffusion operator is self-adjoint, $i.e.$ |
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[10414] | 439 | \[ |
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| 440 | % \label{eq:Gf_property1} |
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| 441 | \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} |
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| 442 | \] |
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[10354] | 443 | In other word, there is no needs to develop a specific routine from the adjoint of this operator. |
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| 444 | We just have to apply the same routine. |
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| 445 | This properties can be demonstrated quite easily in a similar way the "non increase of tracer variance" property |
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| 446 | has been proved (see \autoref{apdx:Gf_operator}). |
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[2282] | 447 | \end{description} |
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| 448 | |
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| 449 | % ================================================================ |
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| 450 | % Skew flux formulation for Eddy Induced Velocity : |
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| 451 | % ================================================================ |
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[9393] | 452 | \subsection{Eddy induced velocity and skew flux formulation} |
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[2282] | 453 | |
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[10354] | 454 | When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), |
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| 455 | an additional advection term is added. |
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| 456 | The associated velocity is the so called eddy induced velocity, |
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| 457 | the formulation of which depends on the slopes of iso-neutral surfaces. |
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| 458 | Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, |
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| 459 | $i.e.$ \autoref{eq:ldfslp_geo} is used in $z$-coordinate, |
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| 460 | and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates. |
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[2282] | 461 | |
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| 462 | The eddy induced velocity is given by: |
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[10414] | 463 | \begin{equation} |
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| 464 | \label{eq:eiv_v} |
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| 465 | \begin{split} |
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| 466 | u^* & = - \frac{1}{e_2\,e_{3}} \;\partial_k \left( e_2 \, A_e \; r_i \right) |
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| 467 | = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_i \right) \\ |
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| 468 | v^* & = - \frac{1}{e_1\,e_3}\; \partial_k \left( e_1 \, A_e \; r_j \right) |
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| 469 | = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_j \right) \\ |
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| 470 | w^* & = \frac{1}{e_1\,e_2}\; \left\{ \partial_i \left( e_2 \, A_e \; r_i \right) |
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| 471 | + \partial_j \left( e_1 \, A_e \;r_j \right) \right\} |
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| 472 | \end{split} |
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[2282] | 473 | \end{equation} |
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[10354] | 474 | where $A_{e}$ is the eddy induced velocity coefficient, |
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| 475 | and $r_i$ and $r_j$ the slopes between the iso-neutral and the geopotential surfaces. |
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[2282] | 476 | %%gm wrong: to be modified with 2 2D streamfunctions |
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[10354] | 477 | In other words, the eddy induced velocity can be derived from a vector streamfuntion, $\phi$, |
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| 478 | which is given by $\phi = A_e\,\textbf{r}$ as $\textbf{U}^* = \textbf{k} \times \nabla \phi$. |
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[2282] | 479 | %%end gm |
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| 480 | |
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[10354] | 481 | A traditional way to implement this additional advection is to add it to the eulerian velocity prior to |
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| 482 | compute the tracer advection. |
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| 483 | This allows us to take advantage of all the advection schemes offered for the tracers |
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| 484 | (see \autoref{sec:TRA_adv}) and not just a $2^{nd}$ order advection scheme. |
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| 485 | This is particularly useful for passive tracers where |
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| 486 | \emph{positivity} of the advection scheme is of paramount importance. |
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[9407] | 487 | % give here the expression using the triads. It is different from the one given in \autoref{eq:ldfeiv} |
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[2282] | 488 | % see just below a copy of this equation: |
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[9407] | 489 | %\begin{equation} \label{eq:ldfeiv} |
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[2282] | 490 | %\begin{split} |
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| 491 | % u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ |
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| 492 | % v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\ |
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| 493 | %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\} \\ |
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| 494 | %\end{split} |
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| 495 | %\end{equation} |
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[10414] | 496 | \[ |
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| 497 | % \label{eq:eiv_vd} |
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| 498 | \textbf{F}_{eiv}^T \equiv \left( |
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| 499 | \begin{aligned} |
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| 500 | \sum_{\substack{i_p,\,k_p}} & |
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| 501 | +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k} |
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| 502 | \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ \\ |
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| 503 | \sum_{\substack{i_p,\,k_p}} & |
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| 504 | - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p} |
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| 505 | \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \delta_{i+i_p}[T^{k+1/2-k_p}] |
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| 506 | \end{aligned} |
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| 507 | \right) |
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| 508 | \] |
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[2282] | 509 | |
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[10354] | 510 | \citep{Griffies_JPO98} introduces another way to implement the eddy induced advection, the so-called skew form. |
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| 511 | It is based on a transformation of the advective fluxes using the non-divergent nature of the eddy induced velocity. |
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| 512 | For example in the (\textbf{i},\textbf{k}) plane, the tracer advective fluxes can be transformed as follows: |
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[2282] | 513 | \begin{flalign*} |
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[10414] | 514 | \begin{split} |
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| 515 | \textbf{F}_{eiv}^T = |
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| 516 | \begin{pmatrix} |
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| 517 | {e_{2}\,e_{3}\; u^*} \\ |
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| 518 | {e_{1}\,e_{2}\; w^*} |
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| 519 | \end{pmatrix} |
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| 520 | \; T |
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| 521 | &= |
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| 522 | \begin{pmatrix} |
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| 523 | { - \partial_k \left( e_{2} \, A_{e} \; r_i \right) \; T \;} \\ |
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| 524 | {+ \partial_i \left( e_{2} \, A_{e} \; r_i \right) \; T \;} |
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| 525 | \end{pmatrix} |
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| 526 | \\ |
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| 527 | &= |
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| 528 | \begin{pmatrix} |
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| 529 | { - \partial_k \left( e_{2} \, A_{e} \; r_i \; T \right) \;} \\ |
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| 530 | {+ \partial_i \left( e_{2} \, A_{e} \; r_i \; T \right) \;} |
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| 531 | \end{pmatrix} |
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| 532 | + |
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| 533 | \begin{pmatrix} |
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| 534 | {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ |
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| 535 | { - e_{2} \, A_{e} \; r_i \; \partial_i T} |
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| 536 | \end{pmatrix} |
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| 537 | \end{split} |
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[2282] | 538 | \end{flalign*} |
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[10354] | 539 | and since the eddy induces velocity field is no-divergent, |
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| 540 | we end up with the skew form of the eddy induced advective fluxes: |
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[10414] | 541 | \begin{equation} |
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| 542 | \label{eq:eiv_skew_continuous} |
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| 543 | \textbf{F}_{eiv}^T = |
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| 544 | \begin{pmatrix} |
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| 545 | {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ |
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| 546 | { - e_{2} \, A_{e} \; r_i \; \partial_i T} |
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| 547 | \end{pmatrix} |
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[2282] | 548 | \end{equation} |
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[10354] | 549 | The tendency associated with eddy induced velocity is then simply the divergence of |
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| 550 | the \autoref{eq:eiv_skew_continuous} fluxes. |
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| 551 | It naturally conserves the tracer content, as it is expressed in flux form and, |
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| 552 | as the advective form, it preserves the tracer variance. |
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| 553 | Another interesting property of \autoref{eq:eiv_skew_continuous} form is that when $A=A_e$, |
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| 554 | a simplification occurs in the sum of the iso-neutral diffusion and eddy induced velocity terms: |
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[10414] | 555 | \begin{flalign*} |
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| 556 | % \label{eq:eiv_skew+eiv_continuous} |
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| 557 | \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &= |
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| 558 | \begin{pmatrix} |
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| 559 | + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T - e_2 \, A \; r_i \;\partial_k T \\ |
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| 560 | - e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T |
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| 561 | \end{pmatrix} |
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| 562 | + |
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| 563 | \begin{pmatrix} |
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| 564 | {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ |
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| 565 | { - e_{2} \, A_{e} \; r_i \; \partial_i T} |
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| 566 | \end{pmatrix} |
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| 567 | \\ |
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| 568 | &= |
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| 569 | \begin{pmatrix} |
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| 570 | + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T \\ |
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| 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 |
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| 572 | \end{pmatrix} |
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| 573 | \end{flalign*} |
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[10354] | 574 | The horizontal component reduces to the one use for an horizontal laplacian operator and |
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| 575 | the vertical one keeps the same complexity, but not more. |
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| 576 | This property has been used to reduce the computational time \citep{Griffies_JPO98}, |
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| 577 | but it is not of practical use as usually $A \neq A_e$. |
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| 578 | Nevertheless this property can be used to choose a discret form of \autoref{eq:eiv_skew_continuous} which |
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| 579 | is consistent with the iso-neutral operator \autoref{eq:Gf_operator}. |
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| 580 | Using the slopes \autoref{eq:Gf_slopes} and defining $A_e$ at $T$-point($i.e.$ as $A$, |
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| 581 | the eddy diffusivity coefficient), the resulting discret form is given by: |
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[10414] | 582 | \begin{equation} |
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| 583 | \label{eq:eiv_skew} |
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| 584 | \textbf{F}_{eiv}^T \equiv \frac{1}{4} \left( |
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| 585 | \begin{aligned} |
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| 586 | \sum_{\substack{i_p,\,k_p}} & |
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| 587 | +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k} |
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| 588 | \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ \\ |
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| 589 | \sum_{\substack{i_p,\,k_p}} & |
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| 590 | - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p} |
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| 591 | \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \delta_{i+i_p}[T^{k+1/2-k_p}] |
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| 592 | \end{aligned} |
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| 593 | \right) |
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[2282] | 594 | \end{equation} |
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[9407] | 595 | Note that \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells. |
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[10354] | 596 | In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces must be added to |
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| 597 | $\mathbb{R}$ for the discret form to be exact. |
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[2282] | 598 | |
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[10354] | 599 | Such a choice of discretisation is consistent with the iso-neutral operator as |
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| 600 | it uses the same definition for the slopes. |
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| 601 | It also ensures the conservation of the tracer variance (see Appendix \autoref{apdx:eiv_skew}), |
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| 602 | $i.e.$ it does not include a diffusive component but is a "pure" advection term. |
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[2282] | 603 | |
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| 604 | $\ $\newpage %force an empty line |
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| 605 | % ================================================================ |
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| 606 | % Discrete Invariants of the iso-neutral diffrusion |
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| 607 | % ================================================================ |
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[9393] | 608 | \subsection{Discrete invariants of the iso-neutral diffrusion} |
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[9407] | 609 | \label{subsec:Gf_operator} |
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[2282] | 610 | |
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| 611 | Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane. |
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| 612 | |
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| 613 | This part will be moved in an Appendix. |
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| 614 | |
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[10354] | 615 | The continuous property to be demonstrated is: |
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[10414] | 616 | \[ |
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| 617 | \int_D D_l^T \; T \;dv \leq 0 |
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| 618 | \] |
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[9407] | 619 | The discrete form of its left hand side is obtained using \autoref{eq:iso_flux} |
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[2282] | 620 | |
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| 621 | \begin{align*} |
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[10414] | 622 | &\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\ |
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| 623 | &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
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| 624 | \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
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| 625 | + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\ |
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| 626 | &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
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| 627 | {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] |
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| 628 | + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\ |
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| 629 | &\equiv -\sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
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| 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] |
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| 631 | - { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; |
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| 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] |
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| 633 | \right\} \\ |
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| 634 | % |
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| 635 | \allowdisplaybreaks |
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| 636 | \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} |
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| 637 | % |
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| 638 | &\equiv -\sum_{i,k} |
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| 639 | \begin{Bmatrix} |
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| 640 | &\ \ \Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{-1/2} (T) } |
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| 641 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
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| 642 | & -\ \ {_{i}^{k+1} \mathbb{R}_{-1/2}^{-1/2}} |
---|
| 643 | & {_{i}^{k+1} \mathbb{T}_{-1/2}^{-1/2} (T) } |
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| 644 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
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| 645 | & \\ |
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| 646 | &+\Bigl( \ \;\; { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } |
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| 647 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
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| 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, |
---|
[10354] | 668 | it is therefore possible to introduce a shift of $-1$ either in $i$ or $k$ direction in order to |
---|
| 669 | regroup all the terms of the summation by triad at a ($i$,$k$) point. |
---|
| 670 | In other words, we regroup all the terms in the neighbourhood that contain a triad at the same ($i$,$k$) indices. |
---|
[10414] | 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 |
---|
[10354] | 704 | substituting the triads by their expression given in \autoref{eq:Gf_triads}. |
---|
[10414] | 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 |
---|
[2282] | 742 | \end{align*} |
---|
| 743 | The last inequality is obviously obtained as we succeed in obtaining a negative summation of square quantities. |
---|
| 744 | |
---|
[10354] | 745 | Note that, if instead of multiplying $D_l^T$ by $T$, we were using another tracer field, let say $S$, |
---|
| 746 | then the previous demonstration would have let to: |
---|
[2282] | 747 | \begin{align*} |
---|
[10414] | 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\} |
---|
[2282] | 766 | \end{align*} |
---|
[10354] | 767 | This means that the iso-neutral operator is self-adjoint. |
---|
| 768 | There is no need to develop a specific to obtain it. |
---|
[2282] | 769 | |
---|
[10414] | 770 | \newpage |
---|
[2282] | 771 | |
---|
| 772 | % ================================================================ |
---|
| 773 | % Discrete Invariants of the skew flux formulation |
---|
| 774 | % ================================================================ |
---|
[9393] | 775 | \subsection{Discrete invariants of the skew flux formulation} |
---|
[9407] | 776 | \label{subsec:eiv_skew} |
---|
[2282] | 777 | |
---|
| 778 | Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. |
---|
| 779 | |
---|
| 780 | This have to be moved in an Appendix. |
---|
| 781 | |
---|
[10354] | 782 | The continuous property to be demonstrated is: |
---|
[2282] | 783 | \begin{align*} |
---|
[10414] | 784 | \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv \equiv 0 |
---|
[2282] | 785 | \end{align*} |
---|
[9407] | 786 | The discrete form of its left hand side is obtained using \autoref{eq:eiv_skew} |
---|
[2282] | 787 | \begin{align*} |
---|
[10414] | 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\} |
---|
[2282] | 797 | \end{align*} |
---|
| 798 | apply the adjoint of delta operator, it becomes |
---|
| 799 | \begin{align*} |
---|
[10414] | 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\} |
---|
[2282] | 809 | \end{align*} |
---|
| 810 | Expending the summation on $i_p$ and $k_p$, it becomes: |
---|
| 811 | \begin{align*} |
---|
[10414] | 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\ \ \ \:} |
---|
[2282] | 821 | &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{k+1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
[10414] | 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} |
---|
[2282] | 833 | \end{align*} |
---|
[10354] | 834 | The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the same but of opposite signs, |
---|
| 835 | they cancel out. |
---|
| 836 | Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{-1/2}}$. |
---|
| 837 | The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{-1/2}}$ are the same but both of opposite signs and |
---|
| 838 | shifted by 1 in $k$ direction. |
---|
| 839 | When summing over $k$ they cancel out with the neighbouring grid points. |
---|
| 840 | Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{+1/2}}$ in the $i$ direction. |
---|
| 841 | Therefore the sum over the domain is zero, |
---|
| 842 | $i.e.$ the variance of the tracer is preserved by the discretisation of the skew fluxes. |
---|
[2282] | 843 | |
---|
[10414] | 844 | \biblio |
---|
| 845 | |
---|
[6997] | 846 | \end{document} |
---|