[707] | 1 | % ================================================================ |
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| 2 | % Chapter Ñ Appendix C : Discrete Invariants of the Equations |
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| 3 | % ================================================================ |
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[817] | 4 | \chapter{Discrete Invariants of the Equations} |
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[707] | 5 | \label{Apdx_C} |
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| 6 | \minitoc |
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| 7 | |
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[817] | 8 | %%% Appendix put in gmcomment as it has not been updated for z* and s coordinate |
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| 9 | I'm writting this appendix. It will be available in a forthcoming release of the documentation |
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| 10 | |
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[994] | 11 | %\gmcomment{ |
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[817] | 12 | |
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[707] | 13 | % ================================================================ |
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| 14 | % Conservation Properties on Ocean Dynamics |
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| 15 | % ================================================================ |
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| 16 | \section{Conservation Properties on Ocean Dynamics} |
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| 17 | \label{Apdx_C.1} |
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| 18 | |
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[1223] | 19 | First, the boundary condition on the vertical velocity (no flux through the surface |
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| 20 | and the bottom) is established for the discrete set of momentum equations. |
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| 21 | Then, it is shown that the non-linear terms of the momentum equation are written |
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| 22 | such that the potential enstrophy of a horizontally non-divergent flow is preserved |
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| 23 | while all the other non-diffusive terms preserve the kinetic energy; in practice the |
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| 24 | energy is also preserved. In addition, an option is also offered for the vorticity term |
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| 25 | discretization which provides a total kinetic energy conserving discretization for |
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| 26 | that term. |
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[707] | 27 | |
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[1223] | 28 | Nota Bene: these properties are established here in the rigid-lid case and for the |
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| 29 | 2nd order centered scheme. A forthcoming update will be their generalisation to |
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| 30 | the free surface case and higher order scheme. |
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[707] | 31 | |
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| 32 | % ------------------------------------------------------------------------------------------------------------- |
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| 33 | % Bottom Boundary Condition on Vertical Velocity Field |
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| 34 | % ------------------------------------------------------------------------------------------------------------- |
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| 35 | \subsection{Bottom Boundary Condition on Vertical Velocity Field} |
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| 36 | \label{Apdx_C.1.1} |
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| 37 | |
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| 38 | |
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[1223] | 39 | The discrete set of momentum equations used in the rigid-lid approximation |
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[707] | 40 | automatically satisfies the surface and bottom boundary conditions |
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[817] | 41 | (no flux through the surface and the bottom: $w_{surface} =w_{bottom} =~0$). |
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[707] | 42 | Indeed, taking the discrete horizontal divergence of the vertical sum of the |
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[1223] | 43 | horizontal momentum equations (!!!Eqs. (II.2.1) and (II.2.2)!!!) weighted by the |
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[707] | 44 | vertical scale factors, it becomes: |
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| 45 | \begin{flalign*} |
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[817] | 46 | \frac{\partial } {\partial t} \left( \sum\limits_k \chi \right) |
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| 47 | \equiv |
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| 48 | \frac{\partial } {\partial t} \left( w_{surface} -w_{bottom} \right)&&&\\ |
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[707] | 49 | \end{flalign*} |
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| 50 | \begin{flalign*} |
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| 51 | \equiv \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} |
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| 52 | \biggl\{ \quad |
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| 53 | \delta_i |
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| 54 | &\left[ |
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| 55 | e_{2u}\,H_u |
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| 56 | \left( |
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| 57 | M_u - M_u - \frac{1} {H_u\,e_{2u}} \delta_j |
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| 58 | \left[ \partial_t\, \psi \right] |
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| 59 | \right) |
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| 60 | \right] && |
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| 61 | \biggr. \\ |
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| 62 | \biggl. |
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| 63 | + \delta_j |
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| 64 | &\left[ |
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| 65 | e_{1v}\,H_v |
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| 66 | \left( M_v - M_v - \frac{1} {H_v\,e_{1v}} \delta_i |
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| 67 | \left[ \partial_i\, \psi \right] |
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| 68 | \right) |
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| 69 | \right] |
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| 70 | \biggr\}&& \\ |
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| 71 | \end{flalign*} |
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| 72 | \begin{flalign*} |
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| 73 | \equiv \frac{1} {e_{1T} \,e_{2T} \,e_{3T}} \; |
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| 74 | \biggl\{ |
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| 75 | - \delta_i |
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| 76 | \Bigl[ |
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| 77 | \delta_j |
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| 78 | \left[ \partial_t \psi \right] |
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| 79 | \Bigr] |
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| 80 | + \delta_j |
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| 81 | \Bigl[ |
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| 82 | \delta_i |
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| 83 | \left[ \partial_t \psi \right] |
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| 84 | \Bigr]\; |
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| 85 | \biggr\}\; |
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| 86 | \equiv 0 |
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| 87 | &&&\\ |
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| 88 | \end{flalign*} |
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| 89 | |
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| 90 | |
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[1223] | 91 | The surface boundary condition associated with the rigid lid approximation |
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| 92 | ($w_{surface} =0)$ is imposed in the computation of the vertical velocity (!!! II.2.5!!!!). |
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| 93 | Therefore, it turns out to be: |
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[707] | 94 | \begin{equation*} |
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| 95 | \frac{\partial } {\partial t}w_{bottom} \equiv 0 |
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| 96 | \end{equation*} |
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[1223] | 97 | As the bottom velocity is initially set to zero, it remains zero all the time. |
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| 98 | Symmetrically, if $w_{bottom} =0$ is used in the computation of the vertical |
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| 99 | velocity (upward integral of the horizontal divergence), the same computation |
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| 100 | leads to $w_{surface} =0$ as soon as the surface vertical velocity is initially |
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| 101 | set to zero. |
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[707] | 102 | |
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| 103 | % ------------------------------------------------------------------------------------------------------------- |
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| 104 | % Coriolis and advection terms: vector invariant form |
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| 105 | % ------------------------------------------------------------------------------------------------------------- |
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| 106 | \subsection{Coriolis and advection terms: vector invariant form} |
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[817] | 107 | \label{Apdx_C_vor_zad} |
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[707] | 108 | |
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| 109 | % ------------------------------------------------------------------------------------------------------------- |
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| 110 | % Vorticity Term |
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| 111 | % ------------------------------------------------------------------------------------------------------------- |
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| 112 | \subsubsection{Vorticity Term} |
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[817] | 113 | \label{Apdx_C_vor} |
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[707] | 114 | |
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[1223] | 115 | Potential vorticity is located at $f$-points and defined as: $\zeta / e_{3f}$. |
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| 116 | The standard discrete formulation of the relative vorticity term obviously |
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| 117 | conserves potential vorticity (ENS scheme). It also conserves the potential |
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| 118 | enstrophy for a horizontally non-divergent flow (i.e. $\chi $=0) but not the |
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| 119 | total kinetic energy. Indeed, using the symmetry or skew symmetry properties |
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| 120 | of the operators (Eqs \eqref{DOM_mi_adj} and \eqref{DOM_di_adj}), it can |
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| 121 | be shown that: |
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[707] | 122 | \begin{equation} \label{Apdx_C_1.1} |
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[817] | 123 | \int_D {\zeta / e_3\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {\zeta \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0 |
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[707] | 124 | \end{equation} |
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[817] | 125 | where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. Indeed, using |
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| 126 | \eqref{Eq_dynvor_ens}, the discrete form of the right hand side of \eqref{Apdx_C_1.1} |
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| 127 | can be transformed as follow: |
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[707] | 128 | \begin{flalign*} |
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[817] | 129 | &\int_D \zeta / e_3\,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times |
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[707] | 130 | \left( |
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| 131 | \zeta \; \textbf{k} \times \textbf{U}_h |
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| 132 | \right)\; |
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| 133 | dv |
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[817] | 134 | &&& \displaybreak[0] \\ |
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| 135 | % |
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| 136 | \equiv& \sum\limits_{i,j,k} |
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[707] | 137 | \frac{\zeta / e_{3f}} {e_{1f}\,e_{2f}\,e_{3f}} |
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| 138 | \biggl\{ \quad |
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| 139 | \delta_{i+1/2} |
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[817] | 140 | \left[ |
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[707] | 141 | - \overline {\left( {\zeta / e_{3f}} \right)}^{\,i}\; |
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| 142 | \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/ 2} |
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| 143 | \right] |
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[817] | 144 | && \\ & \qquad \qquad \qquad \;\; |
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[707] | 145 | - \delta_{j+1/2} |
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[817] | 146 | \left[ \;\;\; |
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[707] | 147 | \overline {\left( \zeta / e_{3f} \right)}^{\,j}\; |
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| 148 | \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} |
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| 149 | \right] |
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[817] | 150 | \;\;\biggr\} \; e_{1f}\,e_{2f}\,e_{3f} && \displaybreak[0] \\ |
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| 151 | % |
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| 152 | \equiv& \sum\limits_{i,j,k} |
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| 153 | \biggl\{ \delta_i \left[ \frac{\zeta} {e_{3f}} \right] \; |
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| 154 | \overline{ \left( \frac{\zeta} {e_{3f}} \right) }^{\,i}\; |
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| 155 | \overline{ \overline{ \left( e_{1u}\,e_{3u}\,u \right) } }^{\,i,j+1/2} |
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| 156 | + \delta_j \left[ \frac{\zeta} {e_{3f}} \right] \; |
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| 157 | \overline{ \left( \frac{\zeta} {e_{3f}} \right) }^{\,j} \; |
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| 158 | \overline{\overline {\left( e_{2v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} \biggr\} |
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| 159 | &&&& \displaybreak[0] \\ |
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| 160 | % |
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| 161 | \equiv& \frac{1} {2} \sum\limits_{i,j,k} |
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| 162 | \biggl\{ \delta_i \Bigl[ \left( \frac{\zeta} {e_{3f}} \right)^2 \Bigr]\; |
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| 163 | \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} |
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| 164 | + \delta_j \Bigl[ \left( \zeta / e_{3f} \right)^2 \Bigr]\; |
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| 165 | \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} |
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[707] | 166 | \biggr\} |
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[817] | 167 | && \displaybreak[0] \\ |
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| 168 | % |
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| 169 | \equiv& - \frac{1} {2} \sum\limits_{i,j,k} \left( \frac{\zeta} {e_{3f}} \right)^2\; |
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| 170 | \biggl\{ \delta_{i+1/2} |
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| 171 | \left[ \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} \right] |
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| 172 | + \delta_{j+1/2} |
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| 173 | \left[ \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} \right] |
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| 174 | \biggr\} && \\ |
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[707] | 175 | \end{flalign*} |
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| 176 | Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2} |
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| 177 | \left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$, |
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| 178 | and introducing the horizontal divergence $\chi $, it becomes: |
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[817] | 179 | \begin{align*} |
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[994] | 180 | \equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left( \frac{\zeta} {e_{3f}} \right)^2 \; \overline{\overline{ e_{1T}\,e_{2T}\,e_{3T}\, \chi}}^{\,i+1/2,j+1/2} \;\;\equiv 0 |
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[817] | 181 | \qquad \qquad \qquad \qquad \qquad \qquad \qquad &&&&\\ |
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| 182 | \end{align*} |
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[707] | 183 | |
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[1223] | 184 | Note that the derivation is demonstrated here for the relative potential |
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| 185 | vorticity but it applies also to the planetary ($f/e_3$) and the total |
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| 186 | potential vorticity $((\zeta +f) /e_3 )$. Another formulation of the two |
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| 187 | components of the vorticity term is optionally offered (ENE scheme) : |
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[707] | 188 | \begin{equation*} |
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[994] | 189 | - \zeta \;{\textbf{k}}\times {\textbf {U}}_h |
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[707] | 190 | \equiv |
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| 191 | \left( {{ |
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| 192 | \begin{array} {*{20}c} |
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| 193 | + \frac{1} {e_{1u}} \; |
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| 194 | \overline {\left( \zeta / e_{3f} \right) |
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| 195 | \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} |
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| 196 | \hfill \\ |
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| 197 | - \frac{1} {e_{2v}} \; |
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| 198 | \overline {\left( \zeta / e_{3f} \right) |
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| 199 | \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} |
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| 200 | \hfill \\ |
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| 201 | \end{array}} } |
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| 202 | \right) |
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| 203 | \end{equation*} |
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| 204 | |
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[1223] | 205 | This formulation does not conserve the enstrophy but it does conserve the |
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| 206 | total kinetic energy. It is also possible to mix the two formulations in order |
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| 207 | to conserve enstrophy on the relative vorticity term and energy on the |
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| 208 | Coriolis term. |
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[707] | 209 | \begin{flalign*} |
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[994] | 210 | &\int\limits_D - \textbf{U}_h \cdot \left( \zeta \;\textbf{k} \times \textbf{U}_h \right) \; dv && \\ |
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[817] | 211 | \equiv& \sum\limits_{i,j,k} \biggl\{ |
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| 212 | \overline {\left( \frac{\zeta} {e_{3f}} \right) |
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| 213 | \overline {\left( e_{1v}e_{3v}v \right)}^{\,i+1/2}} ^{\,j} \, e_{2u}e_{3u}u |
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| 214 | - \overline {\left( \frac{\zeta} {e_{3f}} \right) |
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| 215 | \overline {\left( e_{2u}e_{3u}u \right)}^{\,j+1/2}} ^{\,i} \, e_{1v}e_{3v}v \; |
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| 216 | \biggr\} |
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| 217 | \\ |
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| 218 | \equiv& \sum\limits_{i,j,k} \frac{\zeta} {e_{3f}} |
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| 219 | \biggl\{ \overline {\left( e_{1v}e_{3v} v \right)}^{\,i+1/2}\; |
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| 220 | \overline {\left( e_{2u}e_{3u} u \right)}^{\,j+1/2} |
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| 221 | - \overline {\left( e_{2u}e_{3u} u \right)}^{\,j+1/2}\; |
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| 222 | \overline {\left( e_{1v}e_{3v} v \right)}^{\,i+1/2} |
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| 223 | \biggr\} |
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| 224 | \equiv 0 |
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[707] | 225 | \end{flalign*} |
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| 226 | |
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| 227 | |
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| 228 | % ------------------------------------------------------------------------------------------------------------- |
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| 229 | % Gradient of Kinetic Energy / Vertical Advection |
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| 230 | % ------------------------------------------------------------------------------------------------------------- |
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| 231 | \subsubsection{Gradient of Kinetic Energy / Vertical Advection} |
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[817] | 232 | \label{Apdx_C_zad} |
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[707] | 233 | |
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| 234 | The change of Kinetic Energy (KE) due to the vertical advection is exactly |
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| 235 | balanced by the change of KE due to the horizontal gradient of KE~: |
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| 236 | \begin{equation*} |
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[817] | 237 | \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv |
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[707] | 238 | = - \int_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv |
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| 239 | \end{equation*} |
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[1223] | 240 | Indeed, using successively \eqref{DOM_di_adj} ($i.e.$ the skew symmetry |
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| 241 | property of the $\delta$ operator) and the incompressibility, then |
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| 242 | \eqref{DOM_di_adj} again, then the commutativity of operators |
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| 243 | $\overline {\,\cdot \,}$ and $\delta$, and finally \eqref{DOM_mi_adj} |
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| 244 | ($i.e.$ the symmetry property of the $\overline {\,\cdot \,}$ operator) |
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| 245 | applied in the horizontal and vertical directions, it becomes: |
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[707] | 246 | \begin{flalign*} |
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[817] | 247 | &\int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv &&&\\ |
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| 248 | \equiv& \frac{1}{2} \sum\limits_{i,j,k} |
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| 249 | \biggl\{ |
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| 250 | \frac{1} {e_{1u}} \delta_{i+1/2} |
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| 251 | \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] u\,e_{1u}e_{2u}e_{3u} |
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| 252 | + \frac{1} {e_{2v}} \delta_{j+1/2} |
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| 253 | \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] v\,e_{1v}e_{2v}e_{3v} |
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[707] | 254 | \biggr\} |
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[817] | 255 | &&& \displaybreak[0] \\ |
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| 256 | % |
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| 257 | \equiv& \frac{1}{2} \sum\limits_{i,j,k} |
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| 258 | \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right)\; |
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| 259 | \delta_k \left[ e_{1T}\,e_{2T} \,w \right] |
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| 260 | % |
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| 261 | \;\; \equiv -\frac{1}{2} \sum\limits_{i,j,k} \delta_{k+1/2} |
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[707] | 262 | \left[ |
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| 263 | \overline{ u^2}^{\,i} |
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| 264 | + \overline{ v^2}^{\,j} |
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| 265 | \right] \; |
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| 266 | e_{1v}\,e_{2v}\,w |
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[817] | 267 | &&& \displaybreak[0]\\ |
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| 268 | % |
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| 269 | \equiv &\frac{1} {2} \sum\limits_{i,j,k} |
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| 270 | \left( \overline {\delta_{k+1/2} \left[ u^2 \right]}^{\,i} |
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| 271 | + \overline {\delta_{k+1/2} \left[ v^2 \right]}^{\,j} \right) \; e_{1T}\,e_{2T} \,w |
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| 272 | && \displaybreak[0] \\ |
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| 273 | % |
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| 274 | \equiv &\frac{1} {2} \sum\limits_{i,j,k} |
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| 275 | \biggl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\;2 |
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| 276 | \overline {u}^{\,k+1/2}\; \delta_{k+1/2} \left[ u \right] %&&& \\ |
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[1223] | 277 | + \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2}\;2 \overline {v}^{\,k+1/2}\; \delta_{k+1/2} \left[ v \right] \; |
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[707] | 278 | \biggr\} |
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[817] | 279 | &&\displaybreak[0] \\ |
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| 280 | % |
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| 281 | \equiv& -\sum\limits_{i,j,k} |
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| 282 | \biggl\{ |
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| 283 | \quad \frac{1} {b_u } \; |
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[707] | 284 | \overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\,\delta_{k+1/2} |
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| 285 | \left[ u \right] |
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| 286 | \Bigr\} }^{\,k} \;u\;e_{1u}\,e_{2u}\,e_{3u} |
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[817] | 287 | && \\ |
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| 288 | &\qquad \quad\; + \frac{1} {b_v } \; |
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[707] | 289 | \overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2} \delta_{k+1/2} |
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| 290 | \left[ v \right] |
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| 291 | \Bigr\} }^{\,k} \;v\;e_{1v}\,e_{2v}\,e_{3v} \; |
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| 292 | \biggr\} |
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| 293 | && \\ |
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[817] | 294 | \equiv& -\int\limits_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv &&&\\ |
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[707] | 295 | \end{flalign*} |
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| 296 | |
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[1223] | 297 | The main point here is that the satisfaction of this property links the choice of |
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| 298 | the discrete formulation of the vertical advection and of the horizontal gradient |
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| 299 | of KE. Choosing one imposes the other. For example KE can also be discretized |
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| 300 | as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. This leads to the following |
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| 301 | expression for the vertical advection: |
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[707] | 302 | \begin{equation*} |
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| 303 | \frac{1} {e_3 }\; w\; \frac{\partial \textbf{U}_h } {\partial k} |
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| 304 | \equiv \left( {{\begin{array} {*{20}c} |
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| 305 | \frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \; |
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| 306 | \overline{\overline {e_{1T}\,e_{2T} \,w\;\delta_{k+1/2} |
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| 307 | \left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k} \hfill \\ |
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| 308 | \frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \; |
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| 309 | \overline{\overline {e_{1T}\,e_{2T} \,w\;\delta_{k+1/2} |
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| 310 | \left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill \\ |
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| 311 | \end{array}} } \right) |
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| 312 | \end{equation*} |
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[1223] | 313 | a formulation that requires an additional horizontal mean in contrast with |
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| 314 | the one used in NEMO. Nine velocity points have to be used instead of 3. |
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| 315 | This is the reason why it has not been chosen. |
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[707] | 316 | |
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| 317 | % ------------------------------------------------------------------------------------------------------------- |
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| 318 | % Coriolis and advection terms: flux form |
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| 319 | % ------------------------------------------------------------------------------------------------------------- |
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| 320 | \subsection{Coriolis and advection terms: flux form} |
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| 321 | \label{Apdx_C.1.3} |
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| 322 | |
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| 323 | % ------------------------------------------------------------------------------------------------------------- |
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| 324 | % Coriolis plus ``metric'' Term |
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| 325 | % ------------------------------------------------------------------------------------------------------------- |
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| 326 | \subsubsection{Coriolis plus ``metric'' Term} |
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| 327 | \label{Apdx_C.1.3.1} |
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| 328 | |
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[1223] | 329 | In flux from the vorticity term reduces to a Coriolis term in which the Coriolis |
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| 330 | parameter has been modified to account for the ``metric'' term. This altered |
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| 331 | Coriolis parameter is discretised at an f-point. It is given by: |
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[707] | 332 | \begin{equation*} |
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| 333 | f+\frac{1} {e_1 e_2 } |
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| 334 | \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\; |
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| 335 | \equiv \; |
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| 336 | f+\frac{1} {e_{1f}\,e_{2f}} |
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| 337 | \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] |
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| 338 | -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u} \right] \right) |
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| 339 | \end{equation*} |
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| 340 | |
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[1223] | 341 | The ENE scheme is then applied to obtain the vorticity term in flux form. |
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| 342 | It therefore conserves the total KE. The derivation is the same as for the |
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| 343 | vorticity term in the vector invariant form (\S\ref{Apdx_C_vor}). |
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[707] | 344 | |
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| 345 | % ------------------------------------------------------------------------------------------------------------- |
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| 346 | % Flux form advection |
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| 347 | % ------------------------------------------------------------------------------------------------------------- |
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| 348 | \subsubsection{Flux form advection} |
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| 349 | \label{Apdx_C.1.3.2} |
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| 350 | |
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[1223] | 351 | The flux form operator of the momentum advection is evaluated using a |
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| 352 | centered second order finite difference scheme. Because of the flux form, |
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| 353 | the discrete operator does not contribute to the global budget of linear |
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| 354 | momentum. Because of the centered second order scheme, it conserves |
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| 355 | the horizontal kinetic energy, that is : |
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[707] | 356 | |
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| 357 | \begin{equation} \label{Apdx_C_I.3.10} |
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| 358 | \int_D \textbf{U}_h \cdot |
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| 359 | \left( {{\begin{array} {*{20}c} |
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| 360 | \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ |
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| 361 | \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ |
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| 362 | \end{array}} } \right)\;dv =\;0 |
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| 363 | \end{equation} |
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| 364 | |
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[1223] | 365 | Let us demonstrate this property for the first term of the scalar product |
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| 366 | ($i.e.$ considering just the the terms associated with the i-component of |
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| 367 | the advection): |
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[707] | 368 | \begin{flalign*} |
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[817] | 369 | &\int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv &&&\\ |
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| 370 | % |
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| 371 | \equiv& \sum\limits_{i,j,k} |
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| 372 | \biggl\{ \frac{1} {e_{1u}\, e_{2u}\,e_{3u}} \biggl( |
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| 373 | \delta_{i+1/2} \left[ \overline {e_{2u}\,e_{3u}\,u}^{\,i} \;\overline u^{\,i} \right] |
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| 374 | + \delta_j \left[ \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] |
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| 375 | &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad |
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| 376 | + \delta_k \left[ \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\;\overline u^{\,k+1/2} \right] |
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| 377 | \biggr) \; \biggr\} \, e_{1u}\,e_{2u}\,e_{3u} \;u |
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| 378 | &&& \displaybreak[0] \\ |
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| 379 | % |
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| 380 | \equiv& \sum\limits_{i,j,k} |
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[707] | 381 | \biggl\{ |
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[817] | 382 | \delta_{i+1/2} \left[ \overline {e_{2u}\,e_{3u}\,u}^{\,i}\; \overline u^{\,i} \right] |
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| 383 | + \delta_j \left[ \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] |
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| 384 | &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad |
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| 385 | + \delta_k \left[ \overline {e_{1w}\,e_{2w}\,w}^{\,i+12}\;\overline u^{\,k+1/2} \right] |
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[994] | 386 | \; \biggr\} \; u &&& \displaybreak[0] \\ |
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[817] | 387 | % |
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[994] | 388 | \equiv& - \sum\limits_{i,j,k} |
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[707] | 389 | \biggl\{ |
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| 390 | \overline {e_{2u}\,e_{3u}\,u}^{\,i}\; \overline u^{\,i} \delta_i |
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| 391 | \left[ u \right] |
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[817] | 392 | + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\; \overline u^{\,j+1/2} \delta_{j+1/2} |
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[707] | 393 | \left[ u \right] |
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[817] | 394 | &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad |
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| 395 | + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\; \overline u^{\,k+1/2} \delta_{k+1/2} \left[ u \right] \biggr\} && \displaybreak[0] \\ |
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| 396 | % |
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[994] | 397 | \equiv& - \sum\limits_{i,j,k} |
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[707] | 398 | \biggl\{ |
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| 399 | \overline {e_{2u}\,e_{3u}\,u}^{\,i} \delta_i \left[ u^2 \right] |
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[817] | 400 | + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2} \delta_{j+/2} \left[ u^2 \right] |
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| 401 | + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2} \delta_{k+1/2} \left[ u^2 \right] |
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[707] | 402 | \biggr\} |
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[817] | 403 | && \displaybreak[0] \\ |
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| 404 | % |
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| 405 | \equiv& \sum\limits_{i,j,k} |
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[707] | 406 | \bigg\{ |
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| 407 | e_{2u}\,e_{3u}\,u\; \delta_{i+1/2} \left[ \overline {u^2}^{\,i} \right] |
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[817] | 408 | + e_{1u}\,e_{3u}\,v\; \delta_{j+1/2} \; \left[ \overline {u^2}^{\,i} \right] |
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| 409 | + e_{1w}\,e_{2w}\,w\; \delta_{k+1/2} \left[ \overline {u^2}^{\,i} \right] |
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[707] | 410 | \biggr\} |
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[817] | 411 | && \displaybreak[0] \\ |
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| 412 | % |
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| 413 | \equiv& \sum\limits_{i,j,k} |
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[707] | 414 | \overline {u^2}^{\,i} |
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| 415 | \biggl\{ |
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| 416 | \delta_{i+1/2} \left[ e_{2u}\,e_{3u}\,u \right] |
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| 417 | + \delta_{j+1/2} \left[ e_{1u}\,e_{3u}\,v \right] |
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| 418 | + \delta_{k+1/2} \left[ e_{1w}\,e_{2w}\,w \right] |
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[817] | 419 | \biggr\} \;\; \equiv 0 |
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[707] | 420 | &&& \\ |
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| 421 | \end{flalign*} |
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| 422 | |
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[1223] | 423 | When the UBS scheme is used to evaluate the flux form momentum advection, |
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| 424 | the discrete operator does not contribute to the global budget of linear momentum |
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| 425 | (flux form). The horizontal kinetic energy is not conserved, but forced to decay |
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| 426 | ($i.e.$ the scheme is diffusive). |
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[707] | 427 | |
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| 428 | % ------------------------------------------------------------------------------------------------------------- |
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| 429 | % Hydrostatic Pressure Gradient Term |
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| 430 | % ------------------------------------------------------------------------------------------------------------- |
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| 431 | \subsection{Hydrostatic Pressure Gradient Term} |
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| 432 | \label{Apdx_C.1.4} |
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| 433 | |
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| 434 | |
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[1223] | 435 | A pressure gradient has no contribution to the evolution of the vorticity as the |
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| 436 | curl of a gradient is zero. In the $z$-coordinate, this property is satisfied locally |
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| 437 | on a C-grid with 2nd order finite differences (property \eqref{Eq_DOM_curl_grad}). |
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| 438 | When the equation of state is linear ($i.e.$ when an advection-diffusion equation |
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| 439 | for density can be derived from those of temperature and salinity) the change of |
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| 440 | KE due to the work of pressure forces is balanced by the change of potential |
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| 441 | energy due to buoyancy forces: |
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[707] | 442 | \begin{equation*} |
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| 443 | \int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv |
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| 444 | = \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv |
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| 445 | \end{equation*} |
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| 446 | |
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[1223] | 447 | This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates. |
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| 448 | Indeed, defining the depth of a $T$-point, $z_T$, as the sum of the vertical scale |
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| 449 | factors at $w$-points starting from the surface, the work of pressure forces can be |
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| 450 | written as: |
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[707] | 451 | \begin{flalign*} |
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[817] | 452 | &\int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv &&& \\ |
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| 453 | \equiv& \sum\limits_{i,j,k} \biggl\{ \; - \frac{1} {\rho_o e_{1u}} \Bigl( |
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| 454 | \delta_{i+1/2} \left[ p^h \right] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] |
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| 455 | \Bigr) \; u\;e_{1u}\,e_{2u}\,e_{3u} |
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| 456 | && \\ & \qquad \qquad |
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| 457 | - \frac{1} {\rho_o e_{2v}} \Bigl( |
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| 458 | \delta_{j+1/2} \left[ p^h \right] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} \left[ z_T \right] |
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| 459 | \Bigr) \; v\;e_{1v}\,e_{2v}\,e_{3v} \; |
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| 460 | \biggr\} && \\ |
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[707] | 461 | \end{flalign*} |
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| 462 | |
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[1223] | 463 | Using \eqref{DOM_di_adj}, $i.e.$ the skew symmetry property of the $\delta$ |
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| 464 | operator, \eqref{Eq_wzv}, the continuity equation), and \eqref{Eq_dynhpg_sco}, |
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| 465 | the hydrostatic equation in the $s$-coordinate, it becomes: |
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[817] | 466 | \begin{flalign*} |
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| 467 | \equiv& \frac{1} {\rho_o} \sum\limits_{i,j,k} \biggl\{ |
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[707] | 468 | e_{2u}\,e_{3u} \;u\;g\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2}[ z_T] |
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[817] | 469 | + e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2}\;\delta_{j+1/2}[ z_T] |
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| 470 | && \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, |
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| 471 | +\Bigl( \delta_i[ e_{2u}\,e_{3u}\,u] + \delta_j [ e_{1v}\,e_{3v}\,v] \Bigr)\;p^h \biggr\} &&\\ |
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| 472 | % |
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| 473 | \equiv& \frac{1} {\rho_o } \sum\limits_{i,j,k} |
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[707] | 474 | \biggl\{ |
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| 475 | e_{2u}\,e_{3u} \;u\;g\; \overline \rho^{\,i+1/2} \delta_{i+1/2} \left[ z_T \right] |
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| 476 | + e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2} \delta_{j+1/2} \left[ z_T \right] |
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[817] | 477 | &&&\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, |
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| 478 | - \delta_k \left[ e_{1w} e_{2w}\,w \right]\;p^h \biggr\} &&&\\ |
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| 479 | % |
---|
| 480 | \equiv& \frac{1} {\rho_o } \sum\limits_{i,j,k} |
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[707] | 481 | \biggl\{ |
---|
| 482 | e_{2u}\,e_{3u} \;u\;g\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] |
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[817] | 483 | + e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2} \;\delta_{j+1/2} \left[ z_T \right] |
---|
| 484 | &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, |
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[707] | 485 | + e_{1w} e_{2w} \;w\;\delta_{k+1/2} \left[ p_h \right] |
---|
[817] | 486 | \biggr\} &&&\\ |
---|
| 487 | % |
---|
| 488 | \equiv& \frac{g} {\rho_o} \sum\limits_{i,j,k} |
---|
[707] | 489 | \biggl\{ |
---|
| 490 | e_{2u}\,e_{3u} \;u\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] |
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[817] | 491 | + e_{1v}\,e_{3v} \;v\; \overline \rho^{\,j+1/2}\;\delta_{j+1/2} \left[ z_T \right] |
---|
| 492 | &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, |
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[707] | 493 | - e_{1w} e_{2w} \;w\;e_{3w} \overline \rho^{\,k+1/2} |
---|
[817] | 494 | \biggr\} &&&\\ |
---|
| 495 | \end{flalign*} |
---|
[707] | 496 | noting that by definition of $z_T$, $\delta_{k+1/2} \left[ z_T \right] \equiv - e_{3w} $, thus: |
---|
| 497 | \begin{multline*} |
---|
| 498 | \equiv \frac{g} {\rho_o} \sum\limits_{i,j,k} |
---|
| 499 | \biggl\{ |
---|
| 500 | e_{2u}\,e_{3u} \;u\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] |
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| 501 | + e_{1v}\,e_{3v} \;v\; \overline \rho^{\,j+1/2} \delta_{j+1/2} \left[ z_T \right] |
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| 502 | \biggr. \\ |
---|
| 503 | \shoveright{ |
---|
| 504 | \biggl. |
---|
| 505 | + e_{1w} e_{2w} \;w\; \overline \rho^{\,k+1/2}\;\delta_{k+1/2} \left[ z_T \right] |
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| 506 | \biggr\} } \\ |
---|
| 507 | \end{multline*} |
---|
[817] | 508 | Using \eqref{DOM_di_adj}, it becomes: |
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[707] | 509 | \begin{flalign*} |
---|
[817] | 510 | \equiv& - \frac{g} {\rho_o} \sum\limits_{i,j,k} z_T |
---|
[707] | 511 | \biggl\{ |
---|
| 512 | \delta_i \left[ e_{2u}\,e_{3u}\,u\; \overline \rho^{\,i+1/2} \right] |
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| 513 | + \delta_j \left[ e_{1v}\,e_{3v}\,v\; \overline \rho^{\,j+1/2} \right] |
---|
| 514 | + \delta_k \left[ e_{1w} e_{2w}\,w\; \overline \rho^{\,k+1/2} \right] |
---|
| 515 | \biggr\} |
---|
| 516 | &&& \\ |
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[817] | 517 | % |
---|
| 518 | \equiv& -\int_D \nabla \cdot \left( \rho \, \textbf{U} \right)\;g\;z\;\;dv &&& \\ |
---|
[707] | 519 | \end{flalign*} |
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| 520 | |
---|
[1223] | 521 | Note that this property strongly constrains the discrete expression of both |
---|
| 522 | the depth of $T-$points and of the term added to the pressure gradient in the |
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| 523 | $s$-coordinate. Nevertheless, it is almost never satisfied since a linear equation |
---|
| 524 | of state is rarely used. |
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[707] | 525 | |
---|
| 526 | % ------------------------------------------------------------------------------------------------------------- |
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| 527 | % Surface Pressure Gradient Term |
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| 528 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 529 | \subsection{Surface Pressure Gradient Term} |
---|
| 530 | \label{Apdx_C.1.5} |
---|
| 531 | |
---|
| 532 | |
---|
[1223] | 533 | The surface pressure gradient has no contribution to the evolution of the vorticity. |
---|
| 534 | This property is trivially satisfied locally since the equation verified by $\psi$ has |
---|
| 535 | been derived from the discrete formulation of the momentum equation and of the curl. |
---|
| 536 | But it has to be noted that since the elliptic equation satisfied by $\psi$ is solved |
---|
| 537 | numerically by an iterative solver (preconditioned conjugate gradient or successive |
---|
| 538 | over relaxation), the property is only satisfied at the precision requested for the |
---|
| 539 | solver used. |
---|
[707] | 540 | |
---|
[1223] | 541 | With the rigid-lid approximation, the change of KE due to the work of surface |
---|
| 542 | pressure forces is exactly zero. This is satisfied in discrete form, at the precision |
---|
| 543 | requested for the elliptic solver used to solve this equation. This can be |
---|
| 544 | demonstrated as follows: |
---|
[707] | 545 | \begin{flalign*} |
---|
[817] | 546 | \int\limits_D - \frac{1} {\rho_o} \nabla_h \left( p_s \right) \cdot \textbf{U}_h \;dv% &&& \\ |
---|
| 547 | % |
---|
| 548 | &\equiv \sum\limits_{i,j,k} \biggl\{ \; |
---|
| 549 | \left( - M_u - \frac{1} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] \right)\; |
---|
| 550 | u\;e_{1u}\,e_{2u}\,e_{3u} |
---|
| 551 | &&&\\& \qquad \;\;\, |
---|
| 552 | + \left( - M_v + \frac{1} {H_v \,e_{1v}} \delta_i \left[ \partial_t \psi \right] \right)\; |
---|
| 553 | v\;e_{1v}\,e_{2v}\,e_{3v} \; \biggr\} |
---|
| 554 | &&&\\ |
---|
| 555 | \\ |
---|
| 556 | % |
---|
| 557 | &\equiv \sum\limits_{i,j} \Biggl\{ \; |
---|
| 558 | \biggl( - M_u - \frac{1} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] \biggr) |
---|
| 559 | \biggl( \sum\limits_k u\;e_{3u} \biggr)\; e_{1u}\,e_{2u} |
---|
| 560 | &&&\\& \qquad \;\;\, |
---|
| 561 | + \biggl( - M_v + \frac{1} {H_v \,e_{1v}} \delta_i \left[ \partial_t \psi \right] \biggr) |
---|
| 562 | \biggl( \sum\limits_k v\;e_{3v} \biggr)\; e_{1v}\,e_{2v} \; \Biggr\} |
---|
[707] | 563 | && \\ |
---|
[817] | 564 | % |
---|
[1223] | 565 | \intertext{using the relation between \textit{$\psi $} and the vertical sum of the velocity, it becomes:} |
---|
[817] | 566 | % |
---|
| 567 | &\equiv \sum\limits_{i,j} |
---|
| 568 | \biggl\{ \; |
---|
| 569 | \left( \;\;\, |
---|
[707] | 570 | M_u + \frac{1} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] |
---|
| 571 | \right)\; |
---|
| 572 | e_{1u} \,\delta_j |
---|
| 573 | \left[ \partial_t \psi \right] |
---|
[817] | 574 | && \\ & \qquad \;\;\, |
---|
| 575 | + \left( |
---|
[707] | 576 | - M_v + \frac{1} {H_v \,e_{1v}} \delta_i \left[ \partial_t \psi \right] |
---|
| 577 | \right)\; |
---|
| 578 | e_{2v} \,\delta_i \left[ \partial_t \psi \right] \; |
---|
| 579 | \biggr\} |
---|
| 580 | && \\ |
---|
[817] | 581 | % |
---|
| 582 | \intertext{applying the adjoint of the $\delta$ operator, it is now:} |
---|
| 583 | % |
---|
| 584 | &\equiv \sum\limits_{i,j} - \partial_t \psi \; |
---|
| 585 | \biggl\{ \; |
---|
| 586 | \delta_{j+1/2} \left[ e_{1u} M_u \right] |
---|
[707] | 587 | - \delta_{i+1/2} \left[ e_{1v} M_v \right] |
---|
[817] | 588 | && \\ & \qquad \;\;\, |
---|
| 589 | + \delta_{i+1/2} |
---|
[707] | 590 | \left[ \frac{e_{2v}} {H_v \,e_{2v}} \delta_i \left[ \partial_t \psi \right] |
---|
| 591 | \right] |
---|
| 592 | + \delta_{j+1/2} |
---|
| 593 | \left[ \frac{e_{1u}} {H_u \,e_{2u}} \delta_j \left[ \partial_t \psi \right] |
---|
| 594 | \right] |
---|
[817] | 595 | \biggr\} &&&\\ |
---|
| 596 | &\equiv 0 && \\ |
---|
[707] | 597 | \end{flalign*} |
---|
| 598 | |
---|
[1223] | 599 | The last equality is obtained using \eqref{Eq_dynspg_rl}, the discrete barotropic |
---|
| 600 | streamfunction time evolution equation. By the way, this shows that |
---|
| 601 | \eqref{Eq_dynspg_rl} is the only way to compute the streamfunction, |
---|
| 602 | otherwise the surface pressure forces will do work. Nevertheless, since |
---|
| 603 | the elliptic equation satisfied by $\psi $ is solved numerically by an iterative |
---|
| 604 | solver, the property is only satisfied at the precision requested for the solver. |
---|
[707] | 605 | |
---|
| 606 | % ================================================================ |
---|
| 607 | % Conservation Properties on Tracers |
---|
| 608 | % ================================================================ |
---|
| 609 | \section{Conservation Properties on Tracers} |
---|
| 610 | \label{Apdx_C.2} |
---|
| 611 | |
---|
| 612 | |
---|
[1223] | 613 | All the numerical schemes used in NEMO are written such that the tracer content |
---|
| 614 | is conserved by the internal dynamics and physics (equations in flux form). |
---|
| 615 | For advection, only the CEN2 scheme ($i.e.$ $2^{nd}$ order finite different scheme) |
---|
| 616 | conserves the global variance of tracer. Nevertheless the other schemes ensure |
---|
| 617 | that the global variance decreases ($i.e.$ they are at least slightly diffusive). |
---|
| 618 | For diffusion, all the schemes ensure the decrease of the total tracer variance, |
---|
| 619 | except the iso-neutral operator. There is generally no strict conservation of mass, |
---|
| 620 | as the equation of state is non linear with respect to $T$ and $S$. In practice, |
---|
| 621 | the mass is conserved to a very high accuracy. |
---|
[707] | 622 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 623 | % Advection Term |
---|
| 624 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 625 | \subsection{Advection Term} |
---|
| 626 | \label{Apdx_C.2.1} |
---|
| 627 | |
---|
[1223] | 628 | Whatever the advection scheme considered it conserves of the tracer content as all |
---|
| 629 | the scheme are written in flux form. Let $\tau$ be the tracer interpolated at velocity point |
---|
| 630 | (whatever the interpolation is). The conservation of the tracer content is obtained as follows: |
---|
[707] | 631 | \begin{flalign*} |
---|
[817] | 632 | &\int_D \nabla \cdot \left( T \textbf{U} \right)\;dv &&&\\ |
---|
| 633 | &\equiv \sum\limits_{i,j,k} \biggl\{ |
---|
[707] | 634 | \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} |
---|
[817] | 635 | \left( \delta_i \left[ e_{2u}\,e_{3u}\; u \;\tau_u \right] |
---|
| 636 | + \delta_j \left[ e_{1v}\,e_{3v}\; v \;\tau_v \right] \right) |
---|
| 637 | &&&\\& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad |
---|
| 638 | + \frac{1} {e_{3T}} \delta_k \left[ w\;\tau \right] \biggl\} e_{1T}\,e_{2T}\,e_{3T} &&&\\ |
---|
| 639 | % |
---|
| 640 | &\equiv \sum\limits_{i,j,k} \left\{ |
---|
[707] | 641 | \delta_i \left[ e_{2u}\,e_{3u} \,\overline T^{\,i+1/2}\,u \right] |
---|
| 642 | + \delta_j \left[ e_{1v}\,e_{3v} \,\overline T^{\,j+1/2}\,v \right] |
---|
| 643 | + \delta_k \left[ e_{1T}\,e_{2T} \,\overline T^{\,k+1/2}\,w \right] \right\} |
---|
[817] | 644 | && \\ |
---|
| 645 | &\equiv 0 &&& |
---|
[707] | 646 | \end{flalign*} |
---|
| 647 | |
---|
[1223] | 648 | The conservation of the variance of tracer can be achieved only with the CEN2 scheme. |
---|
| 649 | It can be demonstarted as follows: |
---|
[707] | 650 | \begin{flalign*} |
---|
[817] | 651 | &\int\limits_D T\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\ |
---|
[707] | 652 | \equiv& \sum\limits_{i,j,k} T\; |
---|
| 653 | \left\{ |
---|
| 654 | \delta_i \left[ e_{2u}\,e_{3u} \overline T^{\,i+1/2}\,u \right] |
---|
| 655 | + \delta_j \left[ e_{1v}\,e_{3v} \overline T^{\,j+1/2}\,v \right] |
---|
| 656 | + \delta_k \left[ e_{1T}\,e_{2T} \overline T^{\,k+1/2}w \right] |
---|
| 657 | \right\} |
---|
| 658 | && \\ |
---|
[817] | 659 | \equiv& \sum\limits_{i,j,k} |
---|
[707] | 660 | \left\{ |
---|
| 661 | - e_{2u}\,e_{3u} \overline T^{\,i+1/2}\,u\,\delta_{i+1/2} \left[ T \right] \right. |
---|
[817] | 662 | - e_{1v}\,e_{3v} \overline T^{\,j+1/2}\,v\;\delta_{j+1/2} \left[ T \right] |
---|
| 663 | &&&\\& \qquad \qquad \qquad \qquad \qquad \qquad \quad \; |
---|
| 664 | - \left. e_{1T}\,e_{2T} \overline T^{\,k+1/2}w\;\delta_{k+1/2} \left[ T \right] |
---|
[707] | 665 | \right\} |
---|
| 666 | &&\\ |
---|
| 667 | \equiv& -\frac{1} {2} \sum\limits_{i,j,k} |
---|
| 668 | \Bigl\{ |
---|
| 669 | e_{2u}\,e_{3u} \; u\;\delta_{i+1/2} \left[ T^2 \right] |
---|
| 670 | + e_{1v}\,e_{3v} \; v\;\delta_{j+1/2} \left[ T^2 \right] |
---|
| 671 | + e_{1T}\,e_{2T} \;w\;\delta_{k+1/2} \left[ T^2 \right] |
---|
| 672 | \Bigr\} |
---|
| 673 | && \\ |
---|
| 674 | \equiv& \frac{1} {2} \sum\limits_{i,j,k} T^2 |
---|
| 675 | \Bigl\{ |
---|
| 676 | \delta_i \left[ e_{2u}\,e_{3u}\,u \right] |
---|
| 677 | + \delta_j \left[ e_{1v}\,e_{3v}\,v \right] |
---|
| 678 | + \delta_k \left[ e_{1T}\,e_{2T}\,w \right] |
---|
| 679 | \Bigr\} |
---|
[817] | 680 | \quad \equiv 0 &&& |
---|
[707] | 681 | \end{flalign*} |
---|
| 682 | |
---|
| 683 | |
---|
| 684 | % ================================================================ |
---|
| 685 | % Conservation Properties on Lateral Momentum Physics |
---|
| 686 | % ================================================================ |
---|
| 687 | \section{Conservation Properties on Lateral Momentum Physics} |
---|
[999] | 688 | \label{Apdx_dynldf_properties} |
---|
[707] | 689 | |
---|
| 690 | |
---|
| 691 | The discrete formulation of the horizontal diffusion of momentum ensures the |
---|
[1223] | 692 | conservation of potential vorticity and the horizontal divergence, and the |
---|
[707] | 693 | dissipation of the square of these quantities (i.e. enstrophy and the |
---|
| 694 | variance of the horizontal divergence) as well as the dissipation of the |
---|
| 695 | horizontal kinetic energy. In particular, when the eddy coefficients are |
---|
| 696 | horizontally uniform, it ensures a complete separation of vorticity and |
---|
| 697 | horizontal divergence fields, so that diffusion (dissipation) of vorticity |
---|
| 698 | (enstrophy) does not generate horizontal divergence (variance of the |
---|
| 699 | horizontal divergence) and \textit{vice versa}. |
---|
| 700 | |
---|
[1223] | 701 | These properties of the horizontal diffusion operator are a direct consequence |
---|
| 702 | of properties \eqref{Eq_DOM_curl_grad} and \eqref{Eq_DOM_div_curl}. |
---|
| 703 | When the vertical curl of the horizontal diffusion of momentum (discrete sense) |
---|
| 704 | is taken, the term associated with the horizontal gradient of the divergence is |
---|
| 705 | locally zero. |
---|
[707] | 706 | |
---|
| 707 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 708 | % Conservation of Potential Vorticity |
---|
| 709 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 710 | \subsection{Conservation of Potential Vorticity} |
---|
| 711 | \label{Apdx_C.3.1} |
---|
| 712 | |
---|
| 713 | The lateral momentum diffusion term conserves the potential vorticity : |
---|
| 714 | \begin{flalign*} |
---|
[817] | 715 | &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times |
---|
[707] | 716 | \Bigl[ \nabla_h |
---|
| 717 | \left( A^{\,lm}\;\chi \right) |
---|
| 718 | - \nabla_h \times |
---|
| 719 | \left( A^{\,lm}\;\zeta \; \textbf{k} \right) |
---|
[817] | 720 | \Bigr]\;dv = 0 |
---|
[707] | 721 | \end{flalign*} |
---|
[817] | 722 | %%%%%%%%%% recheck here.... (gm) |
---|
[707] | 723 | \begin{flalign*} |
---|
| 724 | = \int \limits_D -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times |
---|
| 725 | \Bigl[ \nabla_h \times |
---|
| 726 | \left( A^{\,lm}\;\zeta \; \textbf{k} \right) |
---|
| 727 | \Bigr]\;dv &&& \\ |
---|
| 728 | \end{flalign*} |
---|
| 729 | \begin{flalign*} |
---|
| 730 | \equiv& \sum\limits_{i,j} |
---|
| 731 | \left\{ |
---|
| 732 | \delta_{i+1/2} |
---|
| 733 | \left[ |
---|
| 734 | \frac {e_{2v}} {e_{1v}\,e_{3v}} \delta_i |
---|
| 735 | \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 736 | \right] |
---|
| 737 | + \delta_{j+1/2} |
---|
| 738 | \left[ |
---|
| 739 | \frac {e_{1u}} {e_{2u}\,e_{3u}} \delta_j |
---|
| 740 | \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 741 | \right] |
---|
| 742 | \right\} |
---|
| 743 | && \\ |
---|
[817] | 744 | % |
---|
| 745 | \intertext{Using \eqref{DOM_di_adj}, it follows:} |
---|
| 746 | % |
---|
[707] | 747 | \equiv& \sum\limits_{i,j,k} |
---|
| 748 | -\,\left\{ |
---|
| 749 | \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i |
---|
| 750 | \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_i \left[ 1\right] |
---|
| 751 | + \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j |
---|
| 752 | \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_j \left[ 1\right] |
---|
| 753 | \right\} \quad \equiv 0 |
---|
| 754 | && \\ |
---|
| 755 | \end{flalign*} |
---|
| 756 | |
---|
| 757 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 758 | % Dissipation of Horizontal Kinetic Energy |
---|
| 759 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 760 | \subsection{Dissipation of Horizontal Kinetic Energy} |
---|
| 761 | \label{Apdx_C.3.2} |
---|
| 762 | |
---|
| 763 | |
---|
[817] | 764 | The lateral momentum diffusion term dissipates the horizontal kinetic energy: |
---|
| 765 | %\begin{flalign*} |
---|
| 766 | \begin{equation*} |
---|
| 767 | \begin{split} |
---|
[707] | 768 | \int_D \textbf{U}_h \cdot |
---|
[817] | 769 | \left[ \nabla_h \right. & \left. \left( A^{\,lm}\;\chi \right) |
---|
| 770 | - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right] \; dv \\ |
---|
| 771 | \\ %%% |
---|
| 772 | \equiv& \sum\limits_{i,j,k} |
---|
| 773 | \left\{ |
---|
| 774 | \frac{1} {e_{1u}} \delta_{i+1/2} \left[ A_T^{\,lm} \chi \right] |
---|
| 775 | - \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 776 | \right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u \\ |
---|
| 777 | &\;\; + \left\{ |
---|
| 778 | \frac{1} {e_{2u}} \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right] |
---|
| 779 | + \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 780 | \right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v \qquad \\ |
---|
| 781 | \\ %%% |
---|
| 782 | \equiv& \sum\limits_{i,j,k} |
---|
| 783 | \Bigl\{ |
---|
| 784 | e_{2u}\,e_{3u} \;u\; \delta_{i+1/2} \left[ A_T^{\,lm} \chi \right] |
---|
| 785 | - e_{1u} \;u\; \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 786 | \Bigl\} |
---|
| 787 | \\ |
---|
| 788 | &\;\; + \Bigl\{ |
---|
| 789 | e_{1v}\,e_{3v} \;v\; \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right] |
---|
| 790 | + e_{2v} \;v\; \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 791 | \Bigl\} \\ |
---|
| 792 | \\ %%% |
---|
| 793 | \equiv& \sum\limits_{i,j,k} |
---|
| 794 | - \Bigl( |
---|
| 795 | \delta_i \left[ e_{2u}\,e_{3u} \;u \right] |
---|
| 796 | + \delta_j \left[ e_{1v}\,e_{3v} \;v \right] |
---|
| 797 | \Bigr) \; A_T^{\,lm} \chi \\ |
---|
| 798 | &\;\; - \Bigl( |
---|
| 799 | \delta_{i+1/2} \left[ e_{2v} \;v \right] |
---|
| 800 | - \delta_{j+1/2} \left[ e_{1u} \;u \right] |
---|
| 801 | \Bigr)\; A_f^{\,lm} e_{3f} \zeta \\ |
---|
| 802 | \\ %%% |
---|
| 803 | \equiv& \sum\limits_{i,j,k} |
---|
| 804 | - A_T^{\,lm} \,\chi^2 \;e_{1T}\,e_{2T}\,e_{3T} |
---|
| 805 | - A_f ^{\,lm} \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f} |
---|
| 806 | \quad \leq 0 \\ |
---|
| 807 | \end{split} |
---|
| 808 | \end{equation*} |
---|
[707] | 809 | |
---|
| 810 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 811 | % Dissipation of Enstrophy |
---|
| 812 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 813 | \subsection{Dissipation of Enstrophy} |
---|
| 814 | \label{Apdx_C.3.3} |
---|
| 815 | |
---|
| 816 | |
---|
| 817 | The lateral momentum diffusion term dissipates the enstrophy when the eddy |
---|
| 818 | coefficients are horizontally uniform: |
---|
| 819 | \begin{flalign*} |
---|
[817] | 820 | &\int\limits_D \zeta \; \textbf{k} \cdot \nabla \times |
---|
[707] | 821 | \left[ |
---|
| 822 | \nabla_h |
---|
| 823 | \left( A^{\,lm}\;\chi \right) |
---|
| 824 | -\nabla_h \times |
---|
| 825 | \left( A^{\,lm}\;\zeta \; \textbf{k} \right) |
---|
| 826 | \right]\;dv &&&\\ |
---|
[817] | 827 | &= A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times |
---|
[707] | 828 | \left[ |
---|
| 829 | \nabla_h \times |
---|
| 830 | \left( \zeta \; \textbf{k} \right) |
---|
[817] | 831 | \right]\;dv &&&\displaybreak[0]\\ |
---|
| 832 | &\equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f} |
---|
[707] | 833 | \left\{ |
---|
| 834 | \delta_{i+1/2} |
---|
| 835 | \left[ |
---|
| 836 | \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i |
---|
| 837 | \left[ e_{3f} \zeta \right] |
---|
| 838 | \right] |
---|
| 839 | + \delta_{j+1/2} |
---|
| 840 | \left[ |
---|
| 841 | \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j |
---|
| 842 | \left[ e_{3f} \zeta \right] |
---|
| 843 | \right] |
---|
| 844 | \right\} |
---|
| 845 | &&&\\ |
---|
[817] | 846 | % |
---|
| 847 | \intertext{Using \eqref{DOM_di_adj}, it follows:} |
---|
| 848 | % |
---|
| 849 | &\equiv - A^{\,lm} \sum\limits_{i,j,k} |
---|
[707] | 850 | \left\{ |
---|
| 851 | \left( |
---|
| 852 | \frac{1} {e_{1v}\,e_{3v}} \delta_i |
---|
| 853 | \left[ e_{3f} \zeta \right] |
---|
| 854 | \right)^2 e_{1v}\,e_{2v}\,e_{3v} |
---|
| 855 | + \left( |
---|
| 856 | \frac{1} {e_{2u}\,e_{3u}} \delta_j |
---|
| 857 | \left[ e_{3f} \zeta \right] |
---|
| 858 | \right)^2 e_{1u}\,e_{2u}\,e_{3u} |
---|
[817] | 859 | \right\} &&&\\ |
---|
| 860 | & \leq \;0 &&&\\ |
---|
[707] | 861 | \end{flalign*} |
---|
| 862 | |
---|
| 863 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 864 | % Conservation of Horizontal Divergence |
---|
| 865 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 866 | \subsection{Conservation of Horizontal Divergence} |
---|
| 867 | \label{Apdx_C.3.4} |
---|
| 868 | |
---|
| 869 | When the horizontal divergence of the horizontal diffusion of momentum |
---|
[1223] | 870 | (discrete sense) is taken, the term associated with the vertical curl of the |
---|
| 871 | vorticity is zero locally, due to (!!! II.1.8 !!!!!). The resulting term conserves the |
---|
[707] | 872 | $\chi$ and dissipates $\chi^2$ when the eddy coefficients are |
---|
| 873 | horizontally uniform. |
---|
| 874 | \begin{flalign*} |
---|
[817] | 875 | & \int\limits_D \nabla_h \cdot |
---|
[707] | 876 | \Bigl[ |
---|
| 877 | \nabla_h |
---|
| 878 | \left( A^{\,lm}\;\chi \right) |
---|
| 879 | - \nabla_h \times |
---|
| 880 | \left( A^{\,lm}\;\zeta \;\textbf{k} \right) |
---|
| 881 | \Bigr] |
---|
| 882 | dv |
---|
| 883 | = \int\limits_D \nabla_h \cdot \nabla_h |
---|
| 884 | \left( A^{\,lm}\;\chi \right) |
---|
| 885 | dv |
---|
| 886 | &&&\\ |
---|
[817] | 887 | &\equiv \sum\limits_{i,j,k} |
---|
[707] | 888 | \left\{ |
---|
| 889 | \delta_i |
---|
| 890 | \left[ |
---|
| 891 | A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} |
---|
| 892 | \left[ \chi \right] |
---|
| 893 | \right] |
---|
| 894 | + \delta_j |
---|
| 895 | \left[ |
---|
| 896 | A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} |
---|
| 897 | \left[ \chi \right] |
---|
| 898 | \right] |
---|
| 899 | \right\} |
---|
| 900 | &&&\\ |
---|
[817] | 901 | % |
---|
| 902 | \intertext{Using \eqref{DOM_di_adj}, it follows:} |
---|
| 903 | % |
---|
| 904 | &\equiv \sum\limits_{i,j,k} |
---|
[707] | 905 | - \left\{ |
---|
| 906 | \frac{e_{2u}\,e_{3u}} {e_{1u}} A_u^{\,lm} \delta_{i+1/2} |
---|
| 907 | \left[ \chi \right] |
---|
| 908 | \delta_{i+1/2} |
---|
| 909 | \left[ 1 \right] |
---|
| 910 | + \frac{e_{1v}\,e_{3v}} {e_{2v}} A_v^{\,lm} \delta_{j+1/2} |
---|
| 911 | \left[ \chi \right] |
---|
| 912 | \delta_{j+1/2} |
---|
| 913 | \left[ 1 \right] |
---|
| 914 | \right\} \; |
---|
| 915 | \equiv 0 |
---|
| 916 | &&& \\ |
---|
| 917 | \end{flalign*} |
---|
| 918 | |
---|
| 919 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 920 | % Dissipation of Horizontal Divergence Variance |
---|
| 921 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 922 | \subsection{Dissipation of Horizontal Divergence Variance} |
---|
| 923 | \label{Apdx_C.3.5} |
---|
| 924 | |
---|
| 925 | \begin{flalign*} |
---|
[817] | 926 | &\int\limits_D \chi \;\nabla_h \cdot |
---|
| 927 | \left[ \nabla_h \left( A^{\,lm}\;\chi \right) |
---|
| 928 | - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right]\; dv |
---|
| 929 | = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\; dv &&&\\ |
---|
| 930 | % |
---|
| 931 | &\equiv A^{\,lm} \sum\limits_{i,j,k} \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} \chi |
---|
[707] | 932 | \left\{ |
---|
[817] | 933 | \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] |
---|
| 934 | + \delta_j \left[ \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] |
---|
| 935 | \right\} \; e_{1T}\,e_{2T}\,e_{3T} &&&\\ |
---|
| 936 | % |
---|
| 937 | \intertext{Using \eqref{DOM_di_adj}, it turns out to be:} |
---|
| 938 | % |
---|
| 939 | &\equiv - A^{\,lm} \sum\limits_{i,j,k} |
---|
[707] | 940 | \left\{ |
---|
[817] | 941 | \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right)^2 e_{1u}\,e_{2u}\,e_{3u} |
---|
| 942 | + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 e_{1v}\,e_{2v}\,e_{3v} |
---|
| 943 | \right\} \; &&&\\ |
---|
| 944 | &\leq 0 &&&\\ |
---|
[707] | 945 | \end{flalign*} |
---|
| 946 | |
---|
| 947 | % ================================================================ |
---|
| 948 | % Conservation Properties on Vertical Momentum Physics |
---|
| 949 | % ================================================================ |
---|
| 950 | \section{Conservation Properties on Vertical Momentum Physics} |
---|
| 951 | \label{Apdx_C_4} |
---|
| 952 | |
---|
| 953 | |
---|
[1223] | 954 | As for the lateral momentum physics, the continuous form of the vertical diffusion |
---|
| 955 | of momentum satisfies several integral constraints. The first two are associated |
---|
| 956 | with the conservation of momentum and the dissipation of horizontal kinetic energy: |
---|
[817] | 957 | \begin{align*} |
---|
[707] | 958 | \int\limits_D |
---|
| 959 | \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 960 | \left( |
---|
| 961 | \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} |
---|
| 962 | \right)\; |
---|
[817] | 963 | dv \qquad \quad &= \vec{\textbf{0}} |
---|
| 964 | \\ |
---|
| 965 | % |
---|
| 966 | \intertext{and} |
---|
| 967 | % |
---|
[707] | 968 | \int\limits_D |
---|
| 969 | \textbf{U}_h \cdot |
---|
| 970 | \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 971 | \left( |
---|
| 972 | \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} |
---|
| 973 | \right)\; |
---|
[817] | 974 | dv \quad &\leq 0 |
---|
| 975 | \\ |
---|
| 976 | \end{align*} |
---|
[707] | 977 | The first property is obvious. The second results from: |
---|
| 978 | |
---|
| 979 | \begin{flalign*} |
---|
| 980 | \int\limits_D |
---|
| 981 | \textbf{U}_h \cdot |
---|
| 982 | \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 983 | \left( |
---|
| 984 | \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} |
---|
| 985 | \right)\;dv |
---|
| 986 | &&&\\ |
---|
| 987 | \end{flalign*} |
---|
| 988 | \begin{flalign*} |
---|
[817] | 989 | &\equiv \sum\limits_{i,j,k} |
---|
[707] | 990 | \left( |
---|
| 991 | u\; \delta_k |
---|
| 992 | \left[ |
---|
| 993 | \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} |
---|
| 994 | \left[ u \right] |
---|
| 995 | \right]\; |
---|
| 996 | e_{1u}\,e_{2u} |
---|
| 997 | + v\;\delta_k |
---|
| 998 | \left[ |
---|
| 999 | \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} |
---|
| 1000 | \left[ v \right] |
---|
| 1001 | \right]\; |
---|
| 1002 | e_{1v}\,e_{2v} |
---|
| 1003 | \right) |
---|
| 1004 | &&&\\ |
---|
[817] | 1005 | % |
---|
[1223] | 1006 | \intertext{since the horizontal scale factor does not depend on $k$, it follows:} |
---|
[817] | 1007 | % |
---|
| 1008 | &\equiv - \sum\limits_{i,j,k} |
---|
[707] | 1009 | \left( |
---|
| 1010 | \frac{A_u^{\,vm}} {e_{3uw}} |
---|
| 1011 | \left( |
---|
| 1012 | \delta_{k+1/2} |
---|
| 1013 | \left[ u \right] |
---|
| 1014 | \right)^2\; |
---|
| 1015 | e_{1u}\,e_{2u} |
---|
| 1016 | + \frac{A_v^{\,vm}} {e_{3vw}} |
---|
| 1017 | \left( \delta_{k+1/2} |
---|
| 1018 | \left[ v \right] |
---|
| 1019 | \right)^2\; |
---|
| 1020 | e_{1v}\,e_{2v} |
---|
| 1021 | \right) |
---|
| 1022 | \quad \leq 0 |
---|
| 1023 | &&&\\ |
---|
| 1024 | \end{flalign*} |
---|
[817] | 1025 | |
---|
[707] | 1026 | The vorticity is also conserved. Indeed: |
---|
| 1027 | \begin{flalign*} |
---|
| 1028 | \int \limits_D |
---|
| 1029 | \frac{1} {e_3 } \textbf{k} \cdot \nabla \times |
---|
| 1030 | \left( |
---|
| 1031 | \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1032 | \left( |
---|
| 1033 | \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k} |
---|
| 1034 | \right) |
---|
| 1035 | \right)\; |
---|
| 1036 | dv |
---|
| 1037 | &&&\\ |
---|
| 1038 | \end{flalign*} |
---|
| 1039 | \begin{flalign*} |
---|
| 1040 | \equiv \sum\limits_{i,j,k} \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}} |
---|
| 1041 | \bigg\{ \biggr. \quad |
---|
| 1042 | \delta_{i+1/2} |
---|
| 1043 | &\left( |
---|
| 1044 | \frac{e_{2v}} {e_{3v}} \delta_k |
---|
| 1045 | \left[ |
---|
| 1046 | \frac{1} {e_{3vw}} \delta_{k+1/2} |
---|
| 1047 | \left[ v \right] |
---|
| 1048 | \right] |
---|
| 1049 | \right) |
---|
| 1050 | &&\\ |
---|
| 1051 | \biggl. |
---|
| 1052 | - \delta_{j+1/2} |
---|
| 1053 | &\left( |
---|
| 1054 | \frac{e_{1u}} {e_{3u}} \delta_k |
---|
| 1055 | \left[ |
---|
| 1056 | \frac{1} {e_{3uw}}\delta_{k+1/2} |
---|
| 1057 | \left[ u \right] |
---|
| 1058 | \right] |
---|
| 1059 | \right) |
---|
| 1060 | \biggr\} \; |
---|
| 1061 | e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0 |
---|
| 1062 | && \\ |
---|
| 1063 | \end{flalign*} |
---|
| 1064 | If the vertical diffusion coefficient is uniform over the whole domain, the |
---|
[1223] | 1065 | enstrophy is dissipated, $i.e.$ |
---|
[707] | 1066 | \begin{flalign*} |
---|
| 1067 | \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times |
---|
| 1068 | \left( |
---|
| 1069 | \frac{1} {e_3}\; \frac{\partial } {\partial k} |
---|
| 1070 | \left( |
---|
| 1071 | \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} |
---|
| 1072 | \right) |
---|
| 1073 | \right)\; |
---|
| 1074 | dv = 0 |
---|
| 1075 | &&&\\ |
---|
| 1076 | \end{flalign*} |
---|
| 1077 | This property is only satisfied in $z$-coordinates: |
---|
| 1078 | |
---|
| 1079 | \begin{flalign*} |
---|
| 1080 | \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times |
---|
| 1081 | \left( |
---|
| 1082 | \frac{1} {e_3}\; \frac{\partial } {\partial k} |
---|
| 1083 | \left( |
---|
| 1084 | \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} |
---|
| 1085 | \right) |
---|
| 1086 | \right)\; |
---|
| 1087 | dv |
---|
| 1088 | &&& \\ |
---|
| 1089 | \end{flalign*} |
---|
| 1090 | \begin{flalign*} |
---|
| 1091 | \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \; |
---|
| 1092 | \biggl\{ \biggr. \quad |
---|
| 1093 | \delta_{i+1/2} |
---|
| 1094 | &\left( |
---|
| 1095 | \frac{e_{2v}} {e_{3v}} \delta_k |
---|
| 1096 | \left[ |
---|
| 1097 | \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} |
---|
| 1098 | \left[ v \right] |
---|
| 1099 | \right] |
---|
| 1100 | \right) |
---|
| 1101 | &&\\ |
---|
| 1102 | - \delta_{j+1/2} |
---|
| 1103 | &\biggl. |
---|
| 1104 | \left( |
---|
| 1105 | \frac{e_{1u}} {e_{3u}} \delta_k |
---|
| 1106 | \left[ |
---|
| 1107 | \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} |
---|
| 1108 | \left[ u \right] |
---|
| 1109 | \right] |
---|
| 1110 | \right) |
---|
| 1111 | \biggr\} |
---|
| 1112 | &&\\ |
---|
| 1113 | \end{flalign*} |
---|
| 1114 | \begin{flalign*} |
---|
| 1115 | \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} |
---|
| 1116 | \biggl\{ \biggr. \quad |
---|
| 1117 | \frac{1} {e_{3v}} \delta_k |
---|
| 1118 | &\left[ |
---|
| 1119 | \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} |
---|
| 1120 | \left[ \delta_{i+1/2} |
---|
| 1121 | \left[ e_{2v}\,v \right] |
---|
| 1122 | \right] |
---|
| 1123 | \right] |
---|
| 1124 | &&\\ |
---|
| 1125 | \biggl. |
---|
| 1126 | - \frac{1} {e_{3u}} \delta_k |
---|
| 1127 | &\left[ |
---|
| 1128 | \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} |
---|
| 1129 | \left[ \delta_{j+1/2} |
---|
| 1130 | \left[ e_{1u}\,u \right] |
---|
| 1131 | \right] |
---|
| 1132 | \right] |
---|
| 1133 | \biggr\} |
---|
| 1134 | &&\\ |
---|
| 1135 | \end{flalign*} |
---|
[1223] | 1136 | Using the fact that the vertical diffusion coefficients are uniform, and that in |
---|
| 1137 | $z$-coordinate, the vertical scale factors do not depend on $i$ and $j$ so |
---|
| 1138 | that: $e_{3f} =e_{3u} =e_{3v} =e_{3T} $ and $e_{3w} =e_{3uw} =e_{3vw} $, |
---|
| 1139 | it follows: |
---|
[707] | 1140 | \begin{flalign*} |
---|
| 1141 | \equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k |
---|
| 1142 | \left[ |
---|
| 1143 | \frac{1} {e_{3w}} \delta_{k+1/2} |
---|
| 1144 | \Bigl[ |
---|
| 1145 | \delta_{i+1/2} |
---|
| 1146 | \left[ e_{2v}\,v \right] |
---|
| 1147 | - \delta_{j+1/ 2} |
---|
| 1148 | \left[ e_{1u}\,u \right] |
---|
| 1149 | \Bigr] |
---|
| 1150 | \right] |
---|
| 1151 | &&&\\ |
---|
| 1152 | \end{flalign*} |
---|
| 1153 | \begin{flalign*} |
---|
| 1154 | \equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}} |
---|
| 1155 | \left( |
---|
| 1156 | \delta_{k+1/2} |
---|
| 1157 | \left[ \zeta \right] |
---|
| 1158 | \right)^2 \; |
---|
| 1159 | e_{1f}\,e_{2f} |
---|
| 1160 | \; \leq 0 |
---|
| 1161 | &&&\\ |
---|
| 1162 | \end{flalign*} |
---|
| 1163 | Similarly, the horizontal divergence is obviously conserved: |
---|
| 1164 | |
---|
| 1165 | \begin{flalign*} |
---|
| 1166 | \int\limits_D \nabla \cdot |
---|
| 1167 | \left( |
---|
| 1168 | \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1169 | \left( |
---|
| 1170 | \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} |
---|
| 1171 | \right) |
---|
| 1172 | \right)\; |
---|
| 1173 | dv = 0 |
---|
| 1174 | &&&\\ |
---|
| 1175 | \end{flalign*} |
---|
[1223] | 1176 | and the square of the horizontal divergence decreases ($i.e.$ the horizontal |
---|
| 1177 | divergence is dissipated) if the vertical diffusion coefficient is uniform over the |
---|
| 1178 | whole domain: |
---|
[707] | 1179 | |
---|
| 1180 | \begin{flalign*} |
---|
| 1181 | \int\limits_D \chi \;\nabla \cdot |
---|
| 1182 | \left( |
---|
| 1183 | \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1184 | \left( |
---|
| 1185 | \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} |
---|
| 1186 | \right) |
---|
| 1187 | \right)\; |
---|
| 1188 | dv = 0 |
---|
| 1189 | &&&\\ |
---|
| 1190 | \end{flalign*} |
---|
[1223] | 1191 | This property is only satisfied in the $z$-coordinate: |
---|
[707] | 1192 | \begin{flalign*} |
---|
| 1193 | \int\limits_D \chi \;\nabla \cdot |
---|
| 1194 | \left( |
---|
| 1195 | \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1196 | \left( |
---|
| 1197 | \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} |
---|
| 1198 | \right) |
---|
| 1199 | \right)\; |
---|
| 1200 | dv |
---|
| 1201 | &&&\\ |
---|
| 1202 | \end{flalign*} |
---|
| 1203 | \begin{flalign*} |
---|
| 1204 | \equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1T}\,e_{2T}} |
---|
| 1205 | \biggl\{ \Biggr. \quad |
---|
| 1206 | \delta_{i+1/2} |
---|
| 1207 | &\left( |
---|
| 1208 | \frac{e_{2u}} {e_{3u}} \delta_k |
---|
| 1209 | \left[ |
---|
| 1210 | \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} |
---|
| 1211 | \left[ u \right] |
---|
| 1212 | \right] |
---|
| 1213 | \right) |
---|
| 1214 | &&\\ |
---|
| 1215 | \Biggl. |
---|
| 1216 | + \delta_{j+1/2} |
---|
| 1217 | &\left( |
---|
| 1218 | \frac{e_{1v}} {e_{3v}} \delta_k |
---|
| 1219 | \left[ |
---|
| 1220 | \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} |
---|
| 1221 | \left[ v \right] |
---|
| 1222 | \right] |
---|
| 1223 | \right) |
---|
| 1224 | \Biggr\} \; |
---|
| 1225 | e_{1T}\,e_{2T}\,e_{3T} |
---|
| 1226 | &&\\ |
---|
| 1227 | \end{flalign*} |
---|
| 1228 | |
---|
| 1229 | \begin{flalign*} |
---|
| 1230 | \equiv A^{\,vm} \sum\limits_{i,j,k} \chi \, |
---|
| 1231 | \biggl\{ \biggr. \quad |
---|
| 1232 | \delta_{i+1/2} |
---|
| 1233 | &\left( |
---|
| 1234 | \delta_k |
---|
| 1235 | \left[ |
---|
| 1236 | \frac{1} {e_{3uw}} \delta_{k+1/2} |
---|
| 1237 | \left[ e_{2u}\,u \right] |
---|
| 1238 | \right] |
---|
| 1239 | \right) |
---|
| 1240 | && \\ |
---|
| 1241 | \biggl. |
---|
| 1242 | + \delta_{j+1/2} |
---|
| 1243 | &\left( |
---|
| 1244 | \delta_k |
---|
| 1245 | \left[ |
---|
| 1246 | \frac{1} {e_{3vw}} \delta_{k+1/2} |
---|
| 1247 | \left[ e_{1v}\,v \right] |
---|
| 1248 | \right] |
---|
| 1249 | \right) |
---|
| 1250 | \biggr\} |
---|
| 1251 | && \\ |
---|
| 1252 | \end{flalign*} |
---|
| 1253 | |
---|
| 1254 | \begin{flalign*} |
---|
| 1255 | \equiv -A^{\,vm} \sum\limits_{i,j,k} |
---|
| 1256 | \frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\; |
---|
| 1257 | \biggl\{ |
---|
| 1258 | \delta_{k+1/2} |
---|
| 1259 | \Bigl[ |
---|
| 1260 | \delta_{i+1/2} |
---|
| 1261 | \left[ e_{2u}\,u \right] |
---|
| 1262 | + \delta_{j+1/2} |
---|
| 1263 | \left[ e_{1v}\,v \right] |
---|
| 1264 | \Bigr] |
---|
| 1265 | \biggr\} |
---|
| 1266 | &&&\\ |
---|
| 1267 | \end{flalign*} |
---|
| 1268 | |
---|
| 1269 | \begin{flalign*} |
---|
| 1270 | \equiv -A^{\,vm} \sum\limits_{i,j,k} |
---|
| 1271 | \frac{1} {e_{3w}} |
---|
| 1272 | \delta_{k+1/2} |
---|
| 1273 | \left[ \chi \right]\; |
---|
| 1274 | \delta_{k+1/2} |
---|
| 1275 | \left[ e_{1T}\,e_{2T} \;\chi \right] |
---|
| 1276 | &&&\\ |
---|
| 1277 | \end{flalign*} |
---|
| 1278 | |
---|
| 1279 | \begin{flalign*} |
---|
| 1280 | \equiv -A^{\,vm} \sum\limits_{i,j,k} |
---|
| 1281 | \frac{e_{1T}\,e_{2T}} {e_{3w}}\; |
---|
| 1282 | \left( |
---|
| 1283 | \delta_{k+1/2} |
---|
| 1284 | \left[ \chi \right] |
---|
| 1285 | \right)^2 |
---|
| 1286 | \quad \equiv 0 |
---|
| 1287 | &&&\\ |
---|
| 1288 | \end{flalign*} |
---|
| 1289 | |
---|
| 1290 | % ================================================================ |
---|
| 1291 | % Conservation Properties on Tracer Physics |
---|
| 1292 | % ================================================================ |
---|
| 1293 | \section{Conservation Properties on Tracer Physics} |
---|
| 1294 | \label{Apdx_C.5} |
---|
| 1295 | |
---|
[1223] | 1296 | The numerical schemes used for tracer subgridscale physics are written such |
---|
| 1297 | that the heat and salt contents are conserved (equations in flux form, second |
---|
| 1298 | order centered finite differences). Since a flux form is used to compute the |
---|
| 1299 | temperature and salinity, the quadratic form of these quantities (i.e. their variance) |
---|
| 1300 | globally tends to diminish. As for the advection term, there is generally no strict |
---|
| 1301 | conservation of mass, even if in practice the mass is conserved to a very high |
---|
| 1302 | accuracy. |
---|
[707] | 1303 | |
---|
| 1304 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1305 | % Conservation of Tracers |
---|
| 1306 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1307 | \subsection{Conservation of Tracers} |
---|
| 1308 | \label{Apdx_C.5.1} |
---|
| 1309 | |
---|
| 1310 | constraint of conservation of tracers: |
---|
| 1311 | \begin{flalign*} |
---|
[994] | 1312 | &\int\limits_D \nabla \cdot \left( A\;\nabla T \right)\;dv &&&\\ |
---|
[817] | 1313 | \\ |
---|
| 1314 | &\equiv \sum\limits_{i,j,k} |
---|
[707] | 1315 | \biggl\{ \biggr. |
---|
| 1316 | \delta_i |
---|
| 1317 | \left[ |
---|
| 1318 | A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} |
---|
| 1319 | \left[ T \right] |
---|
| 1320 | \right] |
---|
| 1321 | + \delta_j |
---|
[817] | 1322 | \left[ |
---|
[707] | 1323 | A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} |
---|
| 1324 | \left[ T \right] |
---|
| 1325 | \right] |
---|
[817] | 1326 | &&\\ & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\; |
---|
[707] | 1327 | + \delta_k |
---|
[817] | 1328 | \left[ |
---|
[707] | 1329 | A_w^{\,vT} \frac{e_{1T}\,e_{2T}} {e_{3T}} \delta_{k+1/2} |
---|
| 1330 | \left[ T \right] |
---|
| 1331 | \right] |
---|
[817] | 1332 | \biggr\} \quad \equiv 0 |
---|
[707] | 1333 | &&\\ |
---|
| 1334 | \end{flalign*} |
---|
| 1335 | |
---|
[1223] | 1336 | In fact, this property simply results from the flux form of the operator. |
---|
[707] | 1337 | |
---|
| 1338 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1339 | % Dissipation of Tracer Variance |
---|
| 1340 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1341 | \subsection{Dissipation of Tracer Variance} |
---|
| 1342 | \label{Apdx_C.5.2} |
---|
| 1343 | |
---|
[1223] | 1344 | constraint on the dissipation of tracer variance: |
---|
[707] | 1345 | \begin{flalign*} |
---|
[994] | 1346 | \int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv &&&\\ |
---|
| 1347 | &\equiv \sum\limits_{i,j,k} \; T |
---|
| 1348 | \biggl\{ \biggr. |
---|
| 1349 | \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right] |
---|
| 1350 | & + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right] |
---|
| 1351 | \quad&& \\ |
---|
| 1352 | \biggl. |
---|
| 1353 | &&+ \delta_k \left[A_w^{\,vT}\frac{e_{1T}\,e_{2T}} {e_{3T}}\delta_{k+1/2}\left[T\right]\right] |
---|
| 1354 | \biggr\} && |
---|
[707] | 1355 | \end{flalign*} |
---|
| 1356 | \begin{flalign*} |
---|
| 1357 | \equiv - \sum\limits_{i,j,k} |
---|
| 1358 | \biggl\{ \biggr. \quad |
---|
| 1359 | & A_u^{\,lT} |
---|
| 1360 | \left( |
---|
| 1361 | \frac{1} {e_{1u}} \delta_{i+1/2} |
---|
| 1362 | \left[ T \right] |
---|
| 1363 | \right)^2 |
---|
| 1364 | e_{1u}\,e_{2u}\,e_{3u} |
---|
| 1365 | && \\ |
---|
| 1366 | & + A_v^{\,lT} |
---|
| 1367 | \left( |
---|
| 1368 | \frac{1} {e_{2v}} \delta_{j+1/2} |
---|
| 1369 | \left[ T \right] |
---|
| 1370 | \right)^2 |
---|
| 1371 | e_{1v}\,e_{2v}\,e_{3v} |
---|
| 1372 | && \\ |
---|
| 1373 | \biggl. |
---|
| 1374 | & + A_w^{\,vT} |
---|
| 1375 | \left( |
---|
| 1376 | \frac{1} {e_{3w}} \delta_{k+1/2} |
---|
| 1377 | \left[ T \right] |
---|
| 1378 | \right)^2 |
---|
| 1379 | e_{1w}\,e_{2w}\,e_{3w} |
---|
| 1380 | \biggr\} |
---|
| 1381 | \quad \leq 0 |
---|
| 1382 | && \\ |
---|
| 1383 | \end{flalign*} |
---|
| 1384 | |
---|
[817] | 1385 | |
---|
| 1386 | %%%% end of appendix in gm comment |
---|
[994] | 1387 | %} |
---|