[707] | 1 | % ================================================================ |
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[2282] | 2 | % Appendix E : Note on some algorithms |
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[707] | 3 | % ================================================================ |
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[2282] | 4 | \chapter{Note on some algorithms} |
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| 5 | \label{Apdx_E} |
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[707] | 6 | \minitoc |
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| 7 | |
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[2282] | 8 | \newpage |
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| 9 | $\ $\newline % force a new ligne |
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| 10 | |
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| 11 | This appendix some on going consideration on algorithms used or planned to be used |
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| 12 | in \NEMO. |
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[707] | 13 | |
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[2282] | 14 | $\ $\newline % force a new ligne |
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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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| 19 | \section{Upstream Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=T)} |
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| 20 | \label{TRA_adv_ubs} |
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| 21 | |
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| 22 | The UBS advection scheme is an upstream biased third order scheme based on |
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[996] | 23 | an upstream-biased parabolic interpolation. It is also known as Cell Averaged |
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| 24 | QUICK scheme (Quadratic Upstream Interpolation for Convective |
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[707] | 25 | Kinematics). For example, in the $i$-direction : |
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| 26 | \begin{equation} \label{Eq_tra_adv_ubs2} |
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| 27 | \tau _u^{ubs} = \left\{ \begin{aligned} |
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| 28 | & \tau _u^{cen4} + \frac{1}{12} \,\tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ |
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| 29 | & \tau _u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 |
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| 30 | \end{aligned} \right. |
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| 31 | \end{equation} |
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| 32 | or equivalently, the advective flux is |
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| 33 | \begin{equation} \label{Eq_tra_adv_ubs2} |
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| 34 | U_{i+1/2} \ \tau _u^{ubs} |
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| 35 | =U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2} |
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| 36 | - \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
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| 37 | \end{equation} |
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[1223] | 38 | where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and |
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| 39 | $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$. |
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| 40 | By choosing this expression for $\tau "$ we consider a fourth order approximation |
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| 41 | of $\partial_i^2$ with a constant i-grid spacing ($\Delta i=1$). |
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[707] | 42 | |
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[1223] | 43 | Alternative choice: introduce the scale factors: |
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| 44 | $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta _{i+1/2}[\tau] \right]$. |
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[707] | 45 | |
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| 46 | |
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| 47 | This results in a dissipatively dominant (i.e. hyper-diffusive) truncation |
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[2282] | 48 | error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of the |
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[707] | 49 | advection scheme is similar to that reported in \cite{Farrow1995}. |
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| 50 | It is a relatively good compromise between accuracy and smoothness. It is |
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| 51 | not a \emph{positive} scheme meaning false extrema are permitted but the |
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| 52 | amplitude of such are significantly reduced over the centred second order |
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| 53 | method. Nevertheless it is not recommended to apply it to a passive tracer |
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| 54 | that requires positivity. |
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| 55 | |
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| 56 | The intrinsic diffusion of UBS makes its use risky in the vertical direction |
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| 57 | where the control of artificial diapycnal fluxes is of paramount importance. |
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| 58 | It has therefore been preferred to evaluate the vertical flux using the TVD |
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| 59 | scheme when \np{ln\_traadv\_ubs}=T. |
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| 60 | |
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[1223] | 61 | For stability reasons, in \eqref{Eq_tra_adv_ubs}, the first term which corresponds |
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| 62 | to a second order centred scheme is evaluated using the \textit{now} velocity |
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| 63 | (centred in time) while the second term which is the diffusive part of the scheme, |
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| 64 | is evaluated using the \textit{before} velocity (forward in time. This is discussed |
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[2282] | 65 | by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme. UBS and QUICK |
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[707] | 66 | schemes only differ by one coefficient. Substituting 1/6 with 1/8 in |
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[2282] | 67 | (\ref{Eq_tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. |
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[1223] | 68 | This option is not available through a namelist parameter, since the 1/6 |
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[707] | 69 | coefficient is hard coded. Nevertheless it is quite easy to make the |
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| 70 | substitution in \mdl{traadv\_ubs} module and obtain a QUICK scheme |
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| 71 | |
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| 72 | NB 1: When a high vertical resolution $O(1m)$ is used, the model stability can |
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| 73 | be controlled by vertical advection (not vertical diffusion which is usually |
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| 74 | solved using an implicit scheme). Computer time can be saved by using a |
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| 75 | time-splitting technique on vertical advection. This possibility have been |
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| 76 | implemented and validated in ORCA05-L301. It is not currently offered in the |
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| 77 | current reference version. |
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| 78 | |
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| 79 | NB 2 : In a forthcoming release four options will be proposed for the |
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| 80 | vertical component used in the UBS scheme. $\tau _w^{ubs}$ will be |
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[1223] | 81 | evaluated using either \textit{(a)} a centred $2^{nd}$ order scheme , |
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| 82 | or \textit{(b)} a TVD scheme, or \textit{(c)} an interpolation based on conservative |
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[2282] | 83 | parabolic splines following \citet{Shchepetkin_McWilliams_OM05} implementation of UBS in ROMS, |
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[707] | 84 | or \textit{(d)} an UBS. The $3^{rd}$ case has dispersion properties similar to an |
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| 85 | eight-order accurate conventional scheme. |
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| 86 | |
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| 87 | NB 3 : It is straight forward to rewrite \eqref{Eq_tra_adv_ubs} as follows: |
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| 88 | \begin{equation} \label{Eq_tra_adv_ubs2} |
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| 89 | \tau _u^{ubs} = \left\{ \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} \right. |
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| 93 | \end{equation} |
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| 94 | or equivalently |
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| 95 | \begin{equation} \label{Eq_tra_adv_ubs2} |
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| 96 | \begin{split} |
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| 97 | e_{2u} e_{3u}\,u_{i+1/2} \ \tau _u^{ubs} |
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| 98 | &= e_{2u} e_{3u}\,u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\tau"_i }^{\,i+1/2} \\ |
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| 99 | & - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
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| 100 | \end{split} |
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| 101 | \end{equation} |
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[1223] | 102 | \eqref{Eq_tra_adv_ubs2} has several advantages. First it clearly evidence that |
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| 103 | the UBS scheme is based on the fourth order scheme to which is added an |
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| 104 | upstream biased diffusive term. Second, this emphasises that the $4^{th}$ order |
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| 105 | part have to be evaluated at \emph{now} time step, not only the $2^{th}$ order |
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| 106 | part as stated above using \eqref{Eq_tra_adv_ubs}. Third, the diffusive term is |
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| 107 | in fact a biharmonic operator with a eddy coefficient with is simply proportional |
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| 108 | to the velocity. |
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[707] | 109 | |
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| 110 | laplacian diffusion: |
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| 111 | \begin{equation} \label{Eq_tra_ldf_lap} |
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| 112 | \begin{split} |
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| 113 | D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\; e_{3T} } &\left[ {\quad \delta _i |
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| 114 | \left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2} |
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| 115 | \left[ T \right]} \right]} \right. |
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| 116 | \\ |
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| 117 | &\ \left. {+\; \delta _j \left[ |
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| 118 | {A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta _{j+1/2} \left[ T |
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| 119 | \right]} \right)} \right]\quad } \right] |
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| 120 | \end{split} |
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| 121 | \end{equation} |
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| 122 | |
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| 123 | bilaplacian: |
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| 124 | \begin{equation} \label{Eq_tra_ldf_lap} |
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| 125 | \begin{split} |
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| 126 | D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ |
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| 127 | & \delta _i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta _{i+1/2} |
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| 128 | \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} |
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| 129 | \delta _i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta _{i+1/2} |
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| 130 | [T] \right] \right] \right] |
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| 131 | \end{split} |
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| 132 | \end{equation} |
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| 133 | with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$, |
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| 134 | $i.e.$ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$ |
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| 135 | it comes : |
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| 136 | \begin{equation} \label{Eq_tra_ldf_lap} |
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| 137 | \begin{split} |
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| 138 | D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ |
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| 139 | & \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta _{i+1/2} |
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| 140 | \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} |
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| 141 | \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta _{i+1/2} |
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| 142 | [T] \right] \right] \right] |
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| 143 | \end{split} |
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| 144 | \end{equation} |
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| 145 | if the velocity is uniform ($i.e.$ $|u|=cst$) then the diffusive flux is |
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| 146 | \begin{equation} \label{Eq_tra_ldf_lap} |
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| 147 | \begin{split} |
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| 148 | F_u^{lT} = - \frac{1}{12} |
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| 149 | e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta _{i+1/2} |
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| 150 | \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} |
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| 151 | \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta _{i+1/2} |
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| 152 | [T] \right] \right] |
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| 153 | \end{split} |
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| 154 | \end{equation} |
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| 155 | beurk.... reverte the logic: starting from the diffusive part of the advective flux it comes: |
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| 156 | |
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| 157 | \begin{equation} \label{Eq_tra_adv_ubs2} |
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| 158 | \begin{split} |
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| 159 | F_u^{lT} |
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| 160 | &= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
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| 161 | \end{split} |
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| 162 | \end{equation} |
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| 163 | if the velocity is uniform ($i.e.$ $|u|=cst$) and 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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| 164 | |
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| 165 | sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$): |
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| 166 | \begin{equation} \label{Eq_tra_adv_ubs2} |
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| 167 | \begin{split} |
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| 168 | F_u^{lT} |
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| 169 | &= - \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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| 170 | \end{split} |
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| 171 | \end{equation} |
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| 172 | which leads to ${A_T^{lT}}^2 = \frac{1}{12} {e_{1T}}^3\ \overline{|u|}^{\,i+1/2}$ |
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| 173 | |
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| 174 | sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$ |
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| 175 | \begin{equation} \label{Eq_tra_adv_ubs2} |
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| 176 | \begin{split} |
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| 177 | F_u^{lT} |
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| 178 | &= - \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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| 179 | &= - \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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| 180 | \end{split} |
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| 181 | \end{equation} |
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| 182 | which leads to ${A_u^{lT}} = \frac{1}{12} {e_{1u}}^3\ |u|$ |
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| 183 | |
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| 184 | |
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| 185 | % ------------------------------------------------------------------------------------------------------------- |
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| 186 | % Leap-Frog energetic |
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| 187 | % ------------------------------------------------------------------------------------------------------------- |
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| 188 | \section{Leap-Frog energetic } |
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| 189 | \label{LF} |
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| 190 | |
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| 191 | We adopt the following semi-discrete notation for time derivative. Given the values of a variable $q$ at successive time step, the time derivation and averaging operators at the mid time step are: |
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| 192 | \begin{subequations} \label{dt_mt} |
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| 193 | \begin{align} |
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[2282] | 194 | \delta _{t+\rdt/2} [q] &= \ \ \, q^{t+\rdt} - q^{t} \\ |
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| 195 | \overline q^{\,t+\rdt/2} &= \left\{ q^{t+\rdt} + q^{t} \right\} \; / \; 2 |
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[707] | 196 | \end{align} |
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| 197 | \end{subequations} |
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[1223] | 198 | As for space operator, the adjoint of the derivation and averaging time operators are |
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[2282] | 199 | $\delta_t^*=\delta_{t+\rdt/2}$ and $\overline{\cdot}^{\,t\,*}= \overline{\cdot}^{\,t+\Delta/2}$ |
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[1223] | 200 | , respectively. |
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[707] | 201 | |
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| 202 | The Leap-frog time stepping given by \eqref{Eq_DOM_nxt} can be defined as: |
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| 203 | \begin{equation} \label{LF} |
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| 204 | \frac{\partial q}{\partial t} |
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[2282] | 205 | \equiv \frac{1}{\rdt} \overline{ \delta _{t+\rdt/2}[q]}^{\,t} |
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| 206 | = \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} |
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[707] | 207 | \end{equation} |
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[2282] | 208 | Note that \eqref{LF} shows that the leapfrog time step is $\rdt$, not $2\rdt$ |
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[1223] | 209 | as it can be found sometime in literature. |
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| 210 | The leap-Frog time stepping is a second order centered scheme. As such it respects |
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| 211 | the quadratic invariant in integral forms, $i.e.$ the following continuous property, |
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[707] | 212 | \begin{equation} \label{Energy} |
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| 213 | \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} |
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| 214 | =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} |
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| 215 | = \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) , |
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| 216 | \end{equation} |
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| 217 | is satisfied in discrete form. Indeed, |
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| 218 | \begin{equation} \begin{split} |
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| 219 | \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} |
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| 220 | &\equiv \sum\limits_{0}^{N} |
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[2282] | 221 | {\frac{1}{\rdt} q^t \ \overline{ \delta _{t+\rdt/2}[q]}^{\,t} \ \rdt} |
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| 222 | \equiv \sum\limits_{0}^{N} { q^t \ \overline{ \delta _{t+\rdt/2}[q]}^{\,t} } \\ |
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| 223 | &\equiv \sum\limits_{0}^{N} { \overline{q}^{\,t+\Delta/2}{ \delta _{t+\rdt/2}[q]}} |
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| 224 | \equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta _{t+\rdt/2}[q^2] }\\ |
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| 225 | &\equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta _{t+\rdt/2}[q^2] } |
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[707] | 226 | \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) |
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| 227 | \end{split} \end{equation} |
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[1223] | 228 | NB here pb of boundary condition when applying the adjoin! In space, setting to 0 |
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| 229 | the quantity in land area is sufficient to get rid of the boundary condition |
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| 230 | (equivalently of the boundary value of the integration by part). In time this boundary |
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| 231 | condition is not physical and \textbf{add something here!!!} |
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[707] | 232 | |
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| 233 | |
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| 234 | |
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[2282] | 235 | |
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| 236 | |
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| 237 | |
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| 238 | % ================================================================ |
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| 239 | % Iso-neutral diffusion : |
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| 240 | % ================================================================ |
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| 241 | |
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| 242 | \section{Lateral diffusion operator} |
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| 243 | |
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| 244 | % ================================================================ |
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| 245 | % Griffies' iso-neutral diffusion operator : |
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| 246 | % ================================================================ |
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| 247 | \subsection{Griffies' iso-neutral diffusion operator} |
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| 248 | |
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| 249 | Let try to define a scheme that get its inspiration from the \citet{Griffies_al_JPO98} |
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| 250 | scheme, but is formulated within the \NEMO framework ($i.e.$ using scale |
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| 251 | factors rather than grid-size and having a position of $T$-points that is not |
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| 252 | necessary in the middle of vertical velocity points, see Fig.~\ref{Fig_zgr_e3}). |
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| 253 | |
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| 254 | In the formulation \eqref{Eq_tra_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO, |
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| 255 | the off-diagonal terms of the small angle diffusion tensor contain several double |
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| 256 | spatial averages of a gradient, for example $\overline{\overline{\delta_k \cdot}}^{\,i,k}$. |
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| 257 | It is apparent that the combination of a $k$ average and a $k$ derivative of the tracer |
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| 258 | allows for the presence of grid point oscillation structures that will be invisible |
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| 259 | to the operator. These structures are \textit{computational modes}. They |
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| 260 | will not be damped by the iso-neutral operator, and even possibly amplified by it. |
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| 261 | In other word, the operator applied to a tracer does not warranties the decrease of |
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| 262 | its global average variance. To circumvent this, we have introduced a smoothing of |
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| 263 | the slopes of the iso-neutral surfaces (see \S\ref{LDF}). Nevertheless, this technique |
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| 264 | works fine for $T$ and $S$ as they are active tracers ($i.e.$ they enter the computation |
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| 265 | of density), but it does not work for a passive tracer. \citep{Griffies_al_JPO98} introduce |
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| 266 | a different way to discretise the off-diagonal terms that nicely solve the problem. |
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| 267 | The idea is to get rid of combinations of an averaged in one direction combined |
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| 268 | with a derivative in the same direction by considering triads. For example in the |
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| 269 | (\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows: |
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| 270 | \begin{equation} \label{Gf_triads} |
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| 271 | _i^k \mathbb{T}_{i_p}^{k_p} (T) |
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| 272 | = \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \left( |
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| 273 | \frac{ \delta_{i + i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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| 274 | -\ {_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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| 275 | \right) |
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| 276 | \end{equation} |
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| 277 | where the indices $i_p$ and $k_p$ define the four triads and take the following value: |
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| 278 | $i_p = -1/2$ or $1/2$ and $k_p = -1/2$ or $1/2$, |
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| 279 | $b_u= e_{1u}\,e_{2u}\,e_{3u}$ is the volume of $u$-cells, |
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| 280 | $A_i^k$ is the lateral eddy diffusivity coefficient defined at $T$-point, |
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| 281 | and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad : |
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| 282 | \begin{equation} \label{Gf_slopes} |
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| 283 | _i^k \mathbb{R}_{i_p}^{k_p} |
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| 284 | =\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac |
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| 285 | {\left(\alpha / \beta \right)_i^k \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] } |
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| 286 | {\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] } |
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| 287 | \end{equation} |
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| 288 | Note that in \eqref{Gf_slopes} we use the ratio $\alpha / \beta$ instead of |
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| 289 | multiplying the temperature derivative by $\alpha$ and the salinity derivative |
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| 290 | by $\beta$. This is more efficient as the ratio $\alpha / \beta$ can to be |
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| 291 | evaluated directly. |
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| 292 | |
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| 293 | Note that in \eqref{Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of |
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| 294 | ${b_{uw}}_{\,i+i_p}^{\,k+k_p}$. This choice has been motivated by the decrease |
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| 295 | of tracer variance and the presence of partial cell at the ocean bottom |
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| 296 | (see Appendix~\ref{Apdx_Gf_operator}). |
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| 297 | |
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| 298 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 299 | \begin{figure}[!ht] \label{Fig_ISO_triad} |
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| 300 | \begin{center} |
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| 301 | \includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_ISO_triad.pdf} |
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[2376] | 302 | \caption{ \label{Fig_ISO_triad} |
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| 303 | Triads used in the Griffies's like iso-neutral diffision scheme for |
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[2282] | 304 | $u$-component (upper panel) and $w$-component (lower panel).} |
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| 305 | \end{center} |
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| 306 | \end{figure} |
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| 307 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 308 | |
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| 309 | The four iso-neutral fluxes associated with the triads are defined at $T$-point. |
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| 310 | They take the following expression : |
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| 311 | \begin{flalign} \label{Gf_fluxes} |
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| 312 | \begin{split} |
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| 313 | {_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) |
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| 314 | &= \ \; \qquad \quad { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+i_p}^{\,k}} \\ |
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| 315 | {_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
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| 316 | &= -\; { _i^k \mathbb{R}_{i_p}^{k_p} } |
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| 317 | \ \; { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+k_p}} |
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| 318 | \end{split} |
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| 319 | \end{flalign} |
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| 320 | |
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| 321 | The resulting iso-neutral fluxes at $u$- and $w$-points are then given by the |
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| 322 | sum of the fluxes that cross the $u$- and $w$-face (Fig.~\ref{Fig_ISO_triad}): |
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| 323 | \begin{flalign} \label{Eq_iso_flux} |
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| 324 | \textbf{F}_{iso}(T) |
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| 325 | &\equiv \sum_{\substack{i_p,\,k_p}} |
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| 326 | \begin{pmatrix} |
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| 327 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ |
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| 328 | \\ |
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| 329 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) \\ |
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| 330 | \end{pmatrix} \notag \\ |
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| 331 | & \notag \\ |
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| 332 | &\equiv \sum_{\substack{i_p,\,k_p}} |
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| 333 | \begin{pmatrix} |
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| 334 | && { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+1/2}^{\,k} } \\ |
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| 335 | \\ |
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| 336 | & -\; { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } |
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| 337 | & {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+1/2} } \\ |
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| 338 | \end{pmatrix} % \\ |
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| 339 | % &\\ |
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| 340 | % &\equiv \sum_{\substack{i_p,\,k_p}} |
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| 341 | % \begin{pmatrix} |
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| 342 | % \qquad \qquad \qquad |
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| 343 | % \frac{1}{ {e_{1u}}_{\,i+1/2}^{\,k} } \ \; |
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| 344 | % { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T)\\ |
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| 345 | % \\ |
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| 346 | % -\frac{1}{ {e_{3w}}_{\,i}^{\,k+1/2} } \ \; |
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| 347 | % { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; |
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| 348 | % {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T)\\ |
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| 349 | % \end{pmatrix} |
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| 350 | \end{flalign} |
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| 351 | resulting in a iso-neutral diffusion tendency on temperature given by the divergence |
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| 352 | of the sum of all the four triad fluxes : |
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| 353 | \begin{equation} \label{Gf_operator} |
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| 354 | D_l^T = \frac{1}{b_T} \sum_{\substack{i_p,\,k_p}} \left\{ |
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| 355 | \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
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| 356 | + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} |
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| 357 | \end{equation} |
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| 358 | where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. |
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| 359 | |
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| 360 | This expression of the iso-neutral diffusion has been chosen in order to satisfy |
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| 361 | the following six properties: |
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| 362 | \begin{description} |
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| 363 | \item[$\bullet$ horizontal diffusion] The discretization of the diffusion operator |
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| 364 | recovers the traditional five-point Laplacian in the limit of flat iso-neutral direction : |
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| 365 | \begin{equation} \label{Gf_property1a} |
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| 366 | D_l^T = \frac{1}{b_T} \ \delta_{i} |
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| 367 | \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] |
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| 368 | \qquad \text{when} \quad |
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| 369 | { _i^k \mathbb{R}_{i_p}^{k_p} }=0 |
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| 370 | \end{equation} |
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| 371 | |
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| 372 | \item[$\bullet$ implicit treatment in the vertical] In the diagonal term associated |
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| 373 | with the vertical divergence of the iso-neutral fluxes (i.e. the term associated |
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| 374 | with a second order vertical derivative) appears only tracer values associated |
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| 375 | with a single water column. This is of paramount importance since it means |
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| 376 | that the implicit in time algorithm for solving the vertical diffusion equation can |
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| 377 | be used to evaluate this term. It is a necessity since the vertical eddy diffusivity |
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| 378 | associated with this term, |
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| 379 | \begin{equation} |
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| 380 | \sum_{\substack{i_p, \,k_p}} \left\{ |
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| 381 | A_i^k \; \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
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| 382 | \right\} |
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| 383 | \end{equation} |
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| 384 | can be quite large. |
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| 385 | |
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| 386 | \item[$\bullet$ pure iso-neutral operator] The iso-neutral flux of locally referenced |
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| 387 | potential density is zero, $i.e.$ |
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| 388 | \begin{align} \label{Gf_property2} |
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| 389 | \begin{matrix} |
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| 390 | &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)} |
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| 391 | &= &\alpha_i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) |
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| 392 | &- \ \; \beta _i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (S) & = \ 0 \\ |
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| 393 | &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)} |
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| 394 | &= &\alpha_i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
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| 395 | &- \ \; \beta _i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (S) &= \ 0 |
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| 396 | \end{matrix} |
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| 397 | \end{align} |
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| 398 | This result is trivially obtained using the \eqref{Gf_triads} applied to $T$ and $S$ |
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| 399 | and the definition of the triads' slopes \eqref{Gf_slopes}. |
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| 400 | |
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| 401 | \item[$\bullet$ conservation of tracer] The iso-neutral diffusion term conserve the |
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| 402 | total tracer content, $i.e.$ |
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| 403 | \begin{equation} \label{Gf_property1} |
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| 404 | \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 |
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| 405 | \end{equation} |
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| 406 | This property is trivially satisfied since the iso-neutral diffusive operator |
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| 407 | is written in flux form. |
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| 408 | |
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| 409 | \item[$\bullet$ decrease of tracer variance] The iso-neutral diffusion term does |
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| 410 | not increase the total tracer variance, $i.e.$ |
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| 411 | \begin{equation} \label{Gf_property1} |
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| 412 | \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 |
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| 413 | \end{equation} |
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| 414 | The property is demonstrated in the Appendix~\ref{Apdx_Gf_operator}. It is a |
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| 415 | key property for a diffusion term. It means that the operator is also a dissipation |
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| 416 | term, $i.e.$ it is a sink term for the square of the quantity on which it is applied. |
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| 417 | It therfore ensure that, when the diffusivity coefficient is large enough, the field |
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| 418 | on which it is applied become free of grid-point noise. |
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| 419 | |
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| 420 | \item[$\bullet$ self-adjoint operator] The iso-neutral diffusion operator is self-adjoint, |
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| 421 | $i.e.$ |
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| 422 | \begin{equation} \label{Gf_property1} |
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| 423 | \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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| 424 | \end{equation} |
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| 425 | In other word, there is no needs to develop a specific routine from the adjoint of this |
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| 426 | operator. We just have to apply the same routine. This properties can be demonstrated |
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| 427 | quite easily in a similar way the "non increase of tracer variance" property has been |
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| 428 | proved (see Appendix~\ref{Apdx_Gf_operator}). |
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| 429 | \end{description} |
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| 430 | |
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| 431 | |
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| 432 | $\ $\newline %force an empty line |
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| 433 | % ================================================================ |
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| 434 | % Skew flux formulation for Eddy Induced Velocity : |
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| 435 | % ================================================================ |
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| 436 | \subsection{ Eddy induced velocity and Skew flux formulation} |
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| 437 | |
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| 438 | When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), |
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| 439 | an additional advection term is added. The associated velocity is the so called |
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| 440 | eddy induced velocity, the formulation of which depends on the slopes of iso- |
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| 441 | neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used |
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| 442 | here are referenced to the geopotential surfaces, $i.e.$ \eqref{Eq_ldfslp_geo} |
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| 443 | is used in $z$-coordinate, and the sum \eqref{Eq_ldfslp_geo} |
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| 444 | + \eqref{Eq_ldfslp_iso} in $z^*$ or $s$-coordinates. |
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| 445 | |
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| 446 | The eddy induced velocity is given by: |
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| 447 | \begin{equation} \label{Eq_eiv_v} |
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| 448 | \begin{split} |
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| 449 | u^* & = - \frac{1}{e_2\,e_{3}} \;\partial_k \left( e_2 \, A_e \; r_i \right) |
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| 450 | = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_i \right) \\ |
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| 451 | v^* & = - \frac{1}{e_1\,e_3}\; \partial_k \left( e_1 \, A_e \; r_j \right) |
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| 452 | = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_j \right) \\ |
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| 453 | w^* & = \frac{1}{e_1\,e_2}\; \left\{ \partial_i \left( e_2 \, A_e \; r_i \right) |
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| 454 | + \partial_j \left( e_1 \, A_e \;r_j \right) \right\} \\ |
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| 455 | \end{split} |
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| 456 | \end{equation} |
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| 457 | where $A_{e}$ is the eddy induced velocity coefficient, and $r_i$ and $r_j$ the |
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| 458 | slopes between the iso-neutral and the geopotential surfaces. |
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| 459 | %%gm wrong: to be modified with 2 2D streamfunctions |
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| 460 | In other words, |
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| 461 | the eddy induced velocity can be derived from a vector streamfuntion, $\phi$, which |
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| 462 | is given by $\phi = A_e\,\textbf{r}$ as $\textbf{U}^* = \textbf{k} \times \nabla \phi$ |
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| 463 | %%end gm |
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| 464 | |
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| 465 | A traditional way to implement this additional advection is to add it to the eulerian |
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| 466 | velocity prior to compute the tracer advection. This allows us to take advantage of |
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| 467 | all the advection schemes offered for the tracers (see \S\ref{TRA_adv}) and not just |
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| 468 | a $2^{nd}$ order advection scheme. This is particularly useful for passive tracers |
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| 469 | where \emph{positivity} of the advection scheme is of paramount importance. |
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| 470 | % give here the expression using the triads. It is different from the one given in \eqref{Eq_ldfeiv} |
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| 471 | % see just below a copy of this equation: |
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| 472 | %\begin{equation} \label{Eq_ldfeiv} |
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| 473 | %\begin{split} |
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| 474 | % 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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| 475 | % 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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| 476 | %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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| 477 | %\end{split} |
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| 478 | %\end{equation} |
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| 479 | \begin{equation} \label{Eq_eiv_vd} |
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| 480 | \textbf{F}_{eiv}^T \equiv \left( \begin{aligned} |
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| 481 | \sum_{\substack{i_p,\,k_p}} & |
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| 482 | +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k} |
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| 483 | \ \ \ { _{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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| 484 | \\ |
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| 485 | \sum_{\substack{i_p,\,k_p}} & |
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| 486 | - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p} |
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| 487 | \ \ { _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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| 488 | \end{aligned} \right) |
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| 489 | \end{equation} |
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| 490 | |
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| 491 | \ref{Griffies_JPO98} introduces another way to implement the eddy induced advection, |
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| 492 | the so-called skew form. It is based on a transformation of the advective fluxes |
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| 493 | using the non-divergent nature of the eddy induced velocity. |
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| 494 | For example in the (\textbf{i},\textbf{k}) plane, the tracer advective fluxes can be |
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| 495 | transformed as follows: |
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| 496 | \begin{flalign*} |
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| 497 | \begin{split} |
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| 498 | \textbf{F}_{eiv}^T = |
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| 499 | \begin{pmatrix} |
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| 500 | {e_{2}\,e_{3}\; u^*} \\ |
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| 501 | {e_{1}\,e_{2}\; w^*} \\ |
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| 502 | \end{pmatrix} \; T |
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| 503 | &= |
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| 504 | \begin{pmatrix} |
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| 505 | { - \partial_k \left( e_{2} \, A_{e} \; r_i \right) \; T \;} \\ |
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| 506 | {+ \partial_i \left( e_{2} \, A_{e} \; r_i \right) \; T \;} \\ |
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| 507 | \end{pmatrix} \\ |
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| 508 | &= |
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| 509 | \begin{pmatrix} |
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| 510 | { - \partial_k \left( e_{2} \, A_{e} \; r_i \; T \right) \;} \\ |
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| 511 | {+ \partial_i \left( e_{2} \, A_{e} \; r_i \; T \right) \;} \\ |
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| 512 | \end{pmatrix} |
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| 513 | + |
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| 514 | \begin{pmatrix} |
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| 515 | {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ |
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| 516 | { - e_{2} \, A_{e} \; r_i \; \partial_i T} \\ |
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| 517 | \end{pmatrix} |
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| 518 | \end{split} |
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| 519 | \end{flalign*} |
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| 520 | and since the eddy induces velocity field is no-divergent, we end up with the skew |
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| 521 | form of the eddy induced advective fluxes: |
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| 522 | \begin{equation} \label{Eq_eiv_skew_continuous} |
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| 523 | \textbf{F}_{eiv}^T = \begin{pmatrix} |
---|
| 524 | {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ |
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| 525 | { - e_{2} \, A_{e} \; r_i \; \partial_i T} \\ |
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| 526 | \end{pmatrix} |
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| 527 | \end{equation} |
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| 528 | The tendency associated with eddy induced velocity is then simply the divergence |
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| 529 | of the \eqref{Eq_eiv_skew_continuous} fluxes. It naturally conserves the tracer |
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| 530 | content, as it is expressed in flux form and, as the advective form, it preserve the |
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| 531 | tracer variance. Another interesting property of \eqref{Eq_eiv_skew_continuous} |
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| 532 | form is that when $A=A_e$, a simplification occurs in the sum of the iso-neutral |
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| 533 | diffusion and eddy induced velocity terms: |
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| 534 | \begin{flalign} \label{Eq_eiv_skew+eiv_continuous} |
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| 535 | \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &= |
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| 536 | \begin{pmatrix} |
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| 537 | + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T - e_2 \, A \; r_i \;\partial_k T \\ |
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| 538 | - e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T \\ |
---|
| 539 | \end{pmatrix} |
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| 540 | + |
---|
| 541 | \begin{pmatrix} |
---|
| 542 | {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ |
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| 543 | { - e_{2} \, A_{e} \; r_i \; \partial_i T} \\ |
---|
| 544 | \end{pmatrix} \\ |
---|
| 545 | &= \begin{pmatrix} |
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| 546 | + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T \\ |
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| 547 | - 2\; e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T \\ |
---|
| 548 | \end{pmatrix} |
---|
| 549 | \end{flalign} |
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| 550 | The horizontal component reduces to the one use for an horizontal laplacian |
---|
| 551 | operator and the vertical one keep the same complexity, but not more. This property |
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| 552 | has been used to reduce the computational time \citep{Griffies_JPO98}, but it is |
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| 553 | not of practical use as usually $A \neq A_e$. Nevertheless this property can be used to |
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| 554 | choose a discret form of \eqref{Eq_eiv_skew_continuous} which is consistent with the |
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| 555 | iso-neutral operator \eqref{Gf_operator}. Using the slopes \eqref{Gf_slopes} |
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| 556 | and defining $A_e$ at $T$-point($i.e.$ as $A$, the eddy diffusivity coefficient), |
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| 557 | the resulting discret form is given by: |
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| 558 | \begin{equation} \label{Eq_eiv_skew} |
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| 559 | \textbf{F}_{eiv}^T \equiv \frac{1}{4} \left( \begin{aligned} |
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| 560 | \sum_{\substack{i_p,\,k_p}} & |
---|
| 561 | +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k} |
---|
| 562 | \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ |
---|
| 563 | \\ |
---|
| 564 | \sum_{\substack{i_p,\,k_p}} & |
---|
| 565 | - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p} |
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| 566 | \ \ { _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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| 567 | \end{aligned} \right) |
---|
| 568 | \end{equation} |
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| 569 | Note that \eqref{Eq_eiv_skew} is valid in $z$-coordinate with or without partial cells. |
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| 570 | In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces |
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| 571 | must be added to $\mathbb{R}$ for the discret form to be exact. |
---|
| 572 | |
---|
| 573 | Such a choice of discretisation is consistent with the iso-neutral operator as it uses the |
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| 574 | same definition for the slopes. It also ensures the conservation of the tracer variance |
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| 575 | (see Appendix \ref{Apdx_eiv_skew}), $i.e.$ it does not include a diffusive component |
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| 576 | but is a "pure" advection term. |
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| 577 | |
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| 578 | |
---|
| 579 | |
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| 580 | |
---|
| 581 | $\ $\newpage %force an empty line |
---|
| 582 | % ================================================================ |
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| 583 | % Discrete Invariants of the iso-neutral diffrusion |
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| 584 | % ================================================================ |
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| 585 | \subsection{Discrete Invariants of the iso-neutral diffrusion} |
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| 586 | \label{Apdx_Gf_operator} |
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| 587 | |
---|
| 588 | Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane. |
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| 589 | |
---|
| 590 | This part will be moved in an Appendix. |
---|
| 591 | |
---|
| 592 | The continuous property to be demonstrated is : |
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| 593 | \begin{align*} |
---|
| 594 | \int_D D_l^T \; T \;dv \leq 0 |
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| 595 | \end{align*} |
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| 596 | The discrete form of its left hand side is obtained using \eqref{Eq_iso_flux} |
---|
| 597 | |
---|
| 598 | \begin{align*} |
---|
| 599 | &\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\ |
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| 600 | &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
| 601 | \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
---|
| 602 | + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\ |
---|
| 603 | &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
| 604 | {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] |
---|
| 605 | + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\ |
---|
| 606 | &\equiv -\sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
| 607 | \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] |
---|
| 608 | - { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; |
---|
| 609 | \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] |
---|
| 610 | \right\} \\ |
---|
| 611 | % |
---|
| 612 | \allowdisplaybreaks |
---|
| 613 | \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} |
---|
| 614 | % |
---|
| 615 | &\equiv -\sum_{i,k} |
---|
| 616 | \begin{Bmatrix} |
---|
| 617 | &\ \ \Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{-1/2} (T) } |
---|
| 618 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 619 | & -\ \ {_{i}^{k+1} \mathbb{R}_{-1/2}^{-1/2}} |
---|
| 620 | & {_{i}^{k+1} \mathbb{T}_{-1/2}^{-1/2} (T) } |
---|
| 621 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
---|
| 622 | & \\ |
---|
| 623 | &+\Bigl( \ \;\; { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } |
---|
| 624 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 625 | & -\ \ {_i^{k+1} \mathbb{R}_{+1/2}^{-1/2}} |
---|
| 626 | & { _i^{k+1} \mathbb{T}_{+1/2}^{-1/2} (T) } |
---|
| 627 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
---|
| 628 | & \\ |
---|
| 629 | &+\Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } |
---|
| 630 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 631 | & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{-1/2}^{+1/2}} |
---|
| 632 | & \ \;\;{_{i}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } |
---|
| 633 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
---|
| 634 | & \\ |
---|
| 635 | &+\Bigl( \ \;\; { _{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } |
---|
| 636 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 637 | & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{+1/2}^{+1/2}} |
---|
| 638 | & \ \;\;{_{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } |
---|
| 639 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) \\ |
---|
| 640 | \end{Bmatrix} |
---|
| 641 | % |
---|
| 642 | \allowdisplaybreaks |
---|
| 643 | \intertext{The summation is done over all $i$ and $k$ indices, it is therefore possible to introduce a shift of $-1$ either in $i$ or $k$ direction in order to regroup all the terms of the summation by triad at a ($i$,$k$) point. In other words, we regroup all the terms in the neighbourhood that contain a triad at the same ($i$,$k$) indices. It becomes: } |
---|
| 644 | % |
---|
| 645 | &\equiv -\sum_{i,k} |
---|
| 646 | \begin{Bmatrix} |
---|
| 647 | &\ \ \Bigl( {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } |
---|
| 648 | &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} |
---|
| 649 | & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} |
---|
| 650 | & {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } |
---|
| 651 | &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr) |
---|
| 652 | & \\ |
---|
| 653 | &+\Bigl( { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } |
---|
| 654 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 655 | & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} |
---|
| 656 | & { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } |
---|
| 657 | &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr) |
---|
| 658 | & \\ |
---|
| 659 | &+\Bigl( {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } |
---|
| 660 | &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} |
---|
| 661 | & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} |
---|
| 662 | & {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } |
---|
| 663 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
---|
| 664 | & \\ |
---|
| 665 | &+\Bigl( { _i^k \mathbb{T}_{+1/2}^{+1/2} (T) } |
---|
| 666 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 667 | & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} |
---|
| 668 | & {_i^k \mathbb{T}_{+1/2}^{+1/2} (T) } |
---|
| 669 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) \\ |
---|
| 670 | \end{Bmatrix} \\ |
---|
| 671 | % |
---|
| 672 | \allowdisplaybreaks |
---|
| 673 | \intertext{Then outing in factor the triad in each of the four terms of the summation and substituting the triads by their expression given in \eqref{Gf_triads}. It becomes: } |
---|
| 674 | % |
---|
| 675 | &\equiv -\sum_{i,k} |
---|
| 676 | \begin{Bmatrix} |
---|
| 677 | &\ \ \Bigl( \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} |
---|
| 678 | & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} |
---|
| 679 | &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)^2 |
---|
| 680 | & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k} \ A_i^k |
---|
| 681 | & \\ |
---|
| 682 | &+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 683 | & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} |
---|
| 684 | &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)^2 |
---|
| 685 | & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k} \ A_i^k |
---|
| 686 | & \\ |
---|
| 687 | &+\Bigl( \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} |
---|
| 688 | & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} |
---|
| 689 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)^2 |
---|
| 690 | & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k} \ A_i^k |
---|
| 691 | & \\ |
---|
| 692 | &+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
| 693 | & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} |
---|
| 694 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)^2 |
---|
| 695 | & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k} \ A_i^k \\ |
---|
| 696 | \end{Bmatrix} \\ |
---|
| 697 | & \\ |
---|
| 698 | % |
---|
| 699 | &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
| 700 | \begin{matrix} |
---|
| 701 | &\Bigl( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} |
---|
| 702 | & -\ \ {_i^k \mathbb{R}_{i_p}^{k_p}} |
---|
| 703 | &\frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}} \Bigr)^2 |
---|
| 704 | & \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \ \ |
---|
| 705 | \end{matrix} |
---|
| 706 | \right\} |
---|
| 707 | \quad \leq 0 |
---|
| 708 | \end{align*} |
---|
| 709 | The last inequality is obviously obtained as we succeed in obtaining a negative summation of square quantities. |
---|
| 710 | |
---|
| 711 | Note that, if instead of multiplying $D_l^T$ by $T$, we were using another tracer field, let say $S$, then the previous demonstration would have let to: |
---|
| 712 | \begin{align*} |
---|
| 713 | \int_D S \; D_l^T \;dv &\equiv \sum_{i,k} \left\{ S \ D_l^T \ b_T \right\} \\ |
---|
| 714 | &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
| 715 | \left( \frac{ \delta_{i +i_p} [S] }{{e_{1u} }_{\,i+i_p}^{\,k}} |
---|
| 716 | - {_i^k \mathbb{R}_{i_p}^{k_p}} |
---|
| 717 | \frac{ \delta_{k+k_p} [S] }{{e_{3w}}_{\,i}^{\,k+k_p}} \right) \right. |
---|
| 718 | \\ & \qquad \qquad \qquad \ \left. |
---|
| 719 | \left( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} |
---|
| 720 | - {_i^k \mathbb{R}_{i_p}^{k_p}} |
---|
| 721 | \frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}} \right) |
---|
| 722 | \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \ |
---|
| 723 | \right\} |
---|
| 724 | % |
---|
| 725 | \allowdisplaybreaks |
---|
| 726 | \intertext{which, by applying the same operation as before but in reverse order, leads to: } |
---|
| 727 | % |
---|
| 728 | &\equiv \sum_{i,k} \left\{ D_l^S \ T \ b_T \right\} |
---|
| 729 | \end{align*} |
---|
| 730 | This means that the iso-neutral operator is self-adjoint. There is no need to develop a specific to obtain it. |
---|
| 731 | |
---|
| 732 | |
---|
| 733 | |
---|
| 734 | $\ $\newpage %force an empty line |
---|
| 735 | % ================================================================ |
---|
| 736 | % Discrete Invariants of the skew flux formulation |
---|
| 737 | % ================================================================ |
---|
| 738 | \subsection{Discrete Invariants of the skew flux formulation} |
---|
| 739 | \label{Apdx_eiv_skew} |
---|
| 740 | |
---|
| 741 | |
---|
| 742 | Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. |
---|
| 743 | |
---|
| 744 | This have to be moved in an Appendix. |
---|
| 745 | |
---|
| 746 | The continuous property to be demonstrated is : |
---|
| 747 | \begin{align*} |
---|
| 748 | \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv \equiv 0 |
---|
| 749 | \end{align*} |
---|
| 750 | The discrete form of its left hand side is obtained using \eqref{Eq_eiv_skew} |
---|
| 751 | \begin{align*} |
---|
| 752 | \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; |
---|
| 753 | \delta_i &\left[ |
---|
| 754 | {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} |
---|
| 755 | \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] |
---|
| 756 | \right] \; T_i^k \\ |
---|
| 757 | - \delta_k &\left[ |
---|
| 758 | {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} |
---|
| 759 | \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] |
---|
| 760 | \right] \; T_i^k \ \Biggr\} |
---|
| 761 | \end{align*} |
---|
| 762 | apply the adjoint of delta operator, it becomes |
---|
| 763 | \begin{align*} |
---|
| 764 | \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; |
---|
| 765 | &\left( |
---|
| 766 | {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} |
---|
| 767 | \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] |
---|
| 768 | \right) \; \delta_{i+1/2}[T^{k}] \\ |
---|
| 769 | - &\left( |
---|
| 770 | {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} |
---|
| 771 | \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] |
---|
| 772 | \right) \; \delta_{k+1/2}[T_{i}] \ \Biggr\} |
---|
| 773 | \end{align*} |
---|
| 774 | Expending the summation on $i_p$ and $k_p$, it becomes: |
---|
| 775 | \begin{align*} |
---|
| 776 | \begin{matrix} |
---|
| 777 | &\sum\limits_{i,k} \Bigl\{ |
---|
| 778 | &+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} |
---|
| 779 | &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{-1/2}} &\delta_{k-1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
| 780 | &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} |
---|
| 781 | &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{-1/2}} &\delta_{k-1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
| 782 | &&+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} |
---|
| 783 | &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{+1/2}} &\delta_{k+1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
| 784 | &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} |
---|
| 785 | &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{k+1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
| 786 | % |
---|
| 787 | &&-{e_{2u}}_i^{k+1} &{A_{e}}_i^{k+1} |
---|
| 788 | &{_i^{k+1} \mathbb{R}_{-1/2}^{- 1/2}} &\delta_{i-1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\ |
---|
| 789 | &&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} |
---|
| 790 | &{\ \ \;_i^k \mathbb{R}_{-1/2}^{+1/2}} &\delta_{i-1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] &\\ |
---|
| 791 | &&-{e_{2u}}_i^{k+1 } &{A_{e}}_i^{k+1} |
---|
| 792 | &{_i^{k+1} \mathbb{R}_{+1/2}^{- 1/2}} &\delta_{i+1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\ |
---|
| 793 | &&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} |
---|
| 794 | &{\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{i+1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] |
---|
| 795 | &\Bigr\} \\ |
---|
| 796 | \end{matrix} |
---|
| 797 | \end{align*} |
---|
| 798 | The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the |
---|
| 799 | same but of opposite signs, they cancel out. |
---|
| 800 | Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{-1/2}}$. |
---|
| 801 | The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{-1/2}}$ are the |
---|
| 802 | same but both of opposite signs and shifted by 1 in $k$ direction. When summing over $k$ |
---|
| 803 | they cancel out with the neighbouring grid points. |
---|
| 804 | Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{+1/2}}$ in the |
---|
| 805 | $i$ direction. Therefore the sum over the domain is zero, $i.e.$ the variance of the |
---|
| 806 | tracer is preserved by the discretisation of the skew fluxes. |
---|
| 807 | |
---|