[9389] | 1 | \documentclass[../tex_main/NEMO_manual]{subfiles} |
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[6997] | 2 | \begin{document} |
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[707] | 3 | % ================================================================ |
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| 4 | % Chapter Ñ Appendix C : Discrete Invariants of the Equations |
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| 5 | % ================================================================ |
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[817] | 6 | \chapter{Discrete Invariants of the Equations} |
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[9407] | 7 | \label{apdx:C} |
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[707] | 8 | \minitoc |
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| 9 | |
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[817] | 10 | %%% Appendix put in gmcomment as it has not been updated for z* and s coordinate |
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[2282] | 11 | %I'm writting this appendix. It will be available in a forthcoming release of the documentation |
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[817] | 12 | |
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[994] | 13 | %\gmcomment{ |
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[817] | 14 | |
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[2282] | 15 | \newpage |
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| 16 | $\ $\newline % force a new ligne |
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| 17 | |
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[707] | 18 | % ================================================================ |
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[2282] | 19 | % Introduction / Notations |
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[707] | 20 | % ================================================================ |
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[2282] | 21 | \section{Introduction / Notations} |
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[9407] | 22 | \label{sec:C.0} |
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[707] | 23 | |
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[2282] | 24 | Notation used in this appendix in the demonstations : |
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[707] | 25 | |
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[2282] | 26 | fluxes at the faces of a $T$-box: |
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| 27 | \begin{equation*} |
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| 28 | U = e_{2u}\,e_{3u}\; u \qquad V = e_{1v}\,e_{3v}\; v \qquad W = e_{1w}\,e_{2w}\; \omega \\ |
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| 29 | \end{equation*} |
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[707] | 30 | |
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[2282] | 31 | volume of cells at $u$-, $v$-, and $T$-points: |
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| 32 | \begin{equation*} |
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| 33 | b_u = e_{1u}\,e_{2u}\,e_{3u} \qquad b_v = e_{1v}\,e_{2v}\,e_{3v} \qquad b_t = e_{1t}\,e_{2t}\,e_{3t} \\ |
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| 34 | \end{equation*} |
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[707] | 35 | |
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[2282] | 36 | partial derivative notation: $\partial_\bullet = \frac{\partial}{\partial \bullet}$ |
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[707] | 37 | |
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[2282] | 38 | $dv=e_1\,e_2\,e_3 \,di\,dj\,dk$ is the volume element, with only $e_3$ that depends on time. |
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| 39 | $D$ and $S$ are the ocean domain volume and surface, respectively. |
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| 40 | No wetting/drying is allow ($i.e.$ $\frac{\partial S}{\partial t} = 0$) |
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| 41 | Let $k_s$ and $k_b$ be the ocean surface and bottom, resp. |
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| 42 | ($i.e.$ $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth). |
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[707] | 43 | \begin{flalign*} |
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[2282] | 44 | z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s} e_3(\tilde{k}) \;d\tilde{k} |
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| 45 | = \eta - \int\limits_k^{k_s} e_3 \;d\tilde{k} |
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[707] | 46 | \end{flalign*} |
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[2282] | 47 | |
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| 48 | Continuity equation with the above notation: |
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| 49 | \begin{equation*} |
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| 50 | \frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t} \biggl\{ \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0 |
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| 51 | \end{equation*} |
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| 52 | |
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| 53 | A quantity, $Q$ is conserved when its domain averaged time change is zero, that is when: |
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| 54 | \begin{equation*} |
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| 55 | \partial_t \left( \int_D{ Q\;dv } \right) =0 |
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| 56 | \end{equation*} |
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| 57 | Noting that the coordinate system used .... blah blah |
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| 58 | \begin{equation*} |
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| 59 | \partial_t \left( \int_D {Q\;dv} \right) = \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } |
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| 60 | = \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0 |
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| 61 | \end{equation*} |
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| 62 | equation of evolution of $Q$ written as the time evolution of the vertical content of $Q$ |
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| 63 | like for tracers, or momentum in flux form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when : |
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[707] | 64 | \begin{flalign*} |
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[2282] | 65 | \partial_t \left( \int_D{ \frac{1}{2} \,Q^2\;dv } \right) |
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| 66 | =& \int_D{ \frac{1}{2} \partial_t \left( \frac{1}{e_3}\left( e_3 \, Q \right)^2 \right) e_1e_2\;di\,dj\,dk } \\ |
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| 67 | =& \int_D { Q \;\partial_t\left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } |
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| 68 | - \int_D { \frac{1}{2} Q^2 \,\partial_t (e_3) \;e_1e_2\;di\,dj\,dk } \\ |
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[707] | 69 | \end{flalign*} |
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[2282] | 70 | that is in a more compact form : |
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[9407] | 71 | \begin{flalign} \label{eq:Q2_flux} |
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[2282] | 72 | \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) |
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| 73 | =& \int_D { \frac{Q}{e_3} \partial_t \left( e_3 \, Q \right) dv } |
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| 74 | - \frac{1}{2} \int_D { \frac{Q^2}{e_3} \partial_t (e_3) \;dv } |
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| 75 | \end{flalign} |
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| 76 | equation of evolution of $Q$ written as the time evolution of $Q$ |
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| 77 | like for momentum in vector invariant form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when : |
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[707] | 78 | \begin{flalign*} |
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[2282] | 79 | \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) |
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| 80 | =& \int_D { \frac{1}{2} \partial_t \left( e_3 \, Q^2 \right) \;e_1e_2\;di\,dj\,dk } \\ |
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| 81 | =& \int_D { Q \partial_t Q \;e_1e_2e_3\;di\,dj\,dk } |
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| 82 | + \int_D { \frac{1}{2} Q^2 \, \partial_t e_3 \;e_1e_2\;di\,dj\,dk } \\ |
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[707] | 83 | \end{flalign*} |
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[2282] | 84 | that is in a more compact form : |
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[9407] | 85 | \begin{flalign} \label{eq:Q2_vect} |
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[2282] | 86 | \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) |
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| 87 | =& \int_D { Q \,\partial_t Q \;dv } |
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| 88 | + \frac{1}{2} \int_D { \frac{1}{e_3} Q^2 \partial_t e_3 \;dv } |
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| 89 | \end{flalign} |
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[707] | 90 | |
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| 91 | |
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[2282] | 92 | % ================================================================ |
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| 93 | % Continuous Total energy Conservation |
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| 94 | % ================================================================ |
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| 95 | \section{Continuous conservation} |
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[9407] | 96 | \label{sec:C.1} |
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[2282] | 97 | |
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| 98 | |
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| 99 | The discretization of pimitive equation in $s$-coordinate ($i.e.$ time and space varying |
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| 100 | vertical coordinate) must be chosen so that the discrete equation of the model satisfy |
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| 101 | integral constrains on energy and enstrophy. |
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| 102 | |
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| 103 | |
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| 104 | Let us first establish those constraint in the continuous world. |
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| 105 | The total energy ($i.e.$ kinetic plus potential energies) is conserved : |
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[9407] | 106 | \begin{flalign} \label{eq:Tot_Energy} |
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[2282] | 107 | \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 + \rho \, g \, z \right) \;dv \right) = & 0 |
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| 108 | \end{flalign} |
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| 109 | under the following assumptions: no dissipation, no forcing |
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| 110 | (wind, buoyancy flux, atmospheric pressure variations), mass |
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| 111 | conservation, and closed domain. |
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| 112 | |
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| 113 | This equation can be transformed to obtain several sub-equalities. |
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| 114 | The transformation for the advection term depends on whether |
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| 115 | the vector invariant form or the flux form is used for the momentum equation. |
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[9407] | 116 | Using \autoref{eq:Q2_vect} and introducing \autoref{apdx:A_dyn_vect} in \autoref{eq:Tot_Energy} |
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[2282] | 117 | for the former form and |
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[9407] | 118 | Using \autoref{eq:Q2_flux} and introducing \autoref{apdx:A_dyn_flux} in \autoref{eq:Tot_Energy} |
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[2282] | 119 | for the latter form leads to: |
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| 120 | |
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[9407] | 121 | \begin{subequations} \label{eq:E_tot} |
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[2282] | 122 | |
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| 123 | advection term (vector invariant form): |
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[9414] | 124 | \begin{equation} \label{eq:E_tot_vect_vor_1} |
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[2282] | 125 | \int\limits_D \zeta \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ |
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| 126 | \end{equation} |
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| 127 | % |
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[9414] | 128 | \begin{equation} \label{eq:E_tot_vect_adv_1} |
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[2282] | 129 | \int\limits_D \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right) dv |
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| 130 | + \int\limits_D \textbf{U}_h \cdot \nabla_z \textbf{U}_h \;dv |
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| 131 | - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 \\ |
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| 132 | \end{equation} |
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| 133 | |
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| 134 | advection term (flux form): |
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[9407] | 135 | \begin{equation} \label{eq:E_tot_flux_metric} |
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[2282] | 136 | \int\limits_D \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1 \right)\; |
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| 137 | \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ |
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| 138 | \end{equation} |
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| 139 | |
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[9407] | 140 | \begin{equation} \label{eq:E_tot_flux_adv} |
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[2282] | 141 | \int\limits_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} |
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| 142 | \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ |
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| 143 | \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ \end{array}} } \right) \;dv |
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| 144 | + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 \\ |
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| 145 | \end{equation} |
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| 146 | |
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| 147 | coriolis term |
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[9407] | 148 | \begin{equation} \label{eq:E_tot_cor} |
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[2282] | 149 | \int\limits_D f \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ |
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| 150 | \end{equation} |
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| 151 | |
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| 152 | pressure gradient: |
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[9414] | 153 | \begin{equation} \label{eq:E_tot_pg_1} |
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[2282] | 154 | - \int\limits_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv |
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| 155 | = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv |
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| 156 | + \int\limits_D g\, \rho \; \partial_t z \;dv \\ |
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| 157 | \end{equation} |
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| 158 | \end{subequations} |
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| 159 | |
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| 160 | where $\nabla_h = \left. \nabla \right|_k$ is the gradient along the $s$-surfaces. |
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| 161 | |
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| 162 | blah blah.... |
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| 163 | $\ $\newline % force a new ligne |
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| 164 | The prognostic ocean dynamics equation can be summarized as follows: |
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[707] | 165 | \begin{equation*} |
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[2282] | 166 | \text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } |
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| 167 | {\text{COR} + \text{ADV} } |
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| 168 | + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} |
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[707] | 169 | \end{equation*} |
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[2282] | 170 | $\ $\newline % force a new ligne |
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[707] | 171 | |
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[2282] | 172 | Vector invariant form: |
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[9407] | 173 | \begin{subequations} \label{eq:E_tot_vect} |
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[9414] | 174 | \begin{equation} \label{eq:E_tot_vect_vor_2} |
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[2282] | 175 | \int\limits_D \textbf{U}_h \cdot \text{VOR} \;dv = 0 \\ |
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| 176 | \end{equation} |
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[9414] | 177 | \begin{equation} \label{eq:E_tot_vect_adv_2} |
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[2282] | 178 | \int\limits_D \textbf{U}_h \cdot \text{KEG} \;dv |
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| 179 | + \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv |
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| 180 | - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 \\ |
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| 181 | \end{equation} |
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[9414] | 182 | \begin{equation} \label{eq:E_tot_pg_2} |
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[2282] | 183 | - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv |
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| 184 | = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv |
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| 185 | + \int\limits_D g\, \rho \; \partial_t z \;dv \\ |
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| 186 | \end{equation} |
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| 187 | \end{subequations} |
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[707] | 188 | |
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[2282] | 189 | Flux form: |
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[9407] | 190 | \begin{subequations} \label{eq:E_tot_flux} |
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[9414] | 191 | \begin{equation} \label{eq:E_tot_flux_metric_2} |
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[2282] | 192 | \int\limits_D \textbf{U}_h \cdot \text {COR} \; dv = 0 \\ |
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| 193 | \end{equation} |
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[9414] | 194 | \begin{equation} \label{eq:E_tot_flux_adv_2} |
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[2282] | 195 | \int\limits_D \textbf{U}_h \cdot \text{ADV} \;dv |
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| 196 | + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 \\ |
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| 197 | \end{equation} |
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[9414] | 198 | \begin{equation} \label{eq:E_tot_pg_3} |
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[2282] | 199 | - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv |
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| 200 | = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv |
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| 201 | + \int\limits_D g\, \rho \; \partial_t z \;dv \\ |
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| 202 | \end{equation} |
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| 203 | \end{subequations} |
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[707] | 204 | |
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[2282] | 205 | |
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| 206 | $\ $\newline % force a new ligne |
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| 207 | |
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| 208 | |
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[9414] | 209 | \autoref{eq:E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. |
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| 210 | Indeed the left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: |
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[2282] | 211 | \begin{flalign*} |
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| 212 | \partial_t \left( \int\limits_D { \rho \, g \, z \;dv} \right) |
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| 213 | &= + \int\limits_D \frac{1}{e_3} \partial_t (e_3\,\rho) \;g\;z\;\;dv |
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| 214 | + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\ |
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| 215 | &= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv |
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| 216 | + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\ |
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| 217 | &= + \int\limits_D \rho \,g \left( \textbf {U}_h \cdot \nabla_h z + \omega \frac{1}{e_3} \partial_k z \right) \;dv |
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| 218 | + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\ |
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| 219 | &= + \int\limits_D \rho \,g \left( \omega + \partial_t z + \textbf {U}_h \cdot \nabla_h z \right) \;dv &&&\\ |
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| 220 | &=+ \int\limits_D g\, \rho \; w \; dv &&&\\ |
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| 221 | \end{flalign*} |
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| 222 | where the last equality is obtained by noting that the brackets is exactly the expression of $w$, |
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[9407] | 223 | the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{apdx:A_w_s}). |
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[2282] | 224 | |
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[9414] | 225 | The left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: |
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[2282] | 226 | \begin{flalign*} |
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| 227 | - \int\limits_D \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv |
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| 228 | = - \int\limits_D \left( \nabla_h p + \rho \, g \nabla_h z \right) \cdot \textbf{U}_h \;dv &&&\\ |
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| 229 | \allowdisplaybreaks |
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| 230 | &= - \int\limits_D \nabla_h p \cdot \textbf{U}_h \;dv - \int\limits_D \rho \, g \, \nabla_h z \cdot \textbf{U}_h \;dv &&&\\ |
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| 231 | \allowdisplaybreaks |
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| 232 | &= +\int\limits_D p \,\nabla_h \cdot \textbf{U}_h \;dv + \int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ |
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| 233 | \allowdisplaybreaks |
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| 234 | &= -\int\limits_D p \left( \frac{1}{e_3} \partial_t e_3 + \frac{1}{e_3} \partial_k \omega \right) \;dv |
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| 235 | +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ |
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| 236 | \allowdisplaybreaks |
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| 237 | &= -\int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv |
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| 238 | +\int\limits_D \frac{1}{e_3} \partial_k p\; \omega \;dv |
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| 239 | +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ |
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| 240 | &= -\int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv |
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| 241 | -\int\limits_D \rho \, g \, \omega \;dv |
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| 242 | +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ |
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| 243 | &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \; \;dv |
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| 244 | - \int\limits_D \rho \, g \, w \;dv |
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| 245 | + \int\limits_D \rho \, g \, \partial_t z \;dv &&&\\ |
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| 246 | \allowdisplaybreaks |
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| 247 | \intertext{introducing the hydrostatic balance $\partial_k p=-\rho \,g\,e_3$ in the last term, |
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| 248 | it becomes:} |
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| 249 | &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv |
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| 250 | - \int\limits_D \rho \, g \, w \;dv |
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| 251 | - \int\limits_D \frac{1}{e_3} \partial_k p\, \partial_t z \;dv &&&\\ |
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| 252 | &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv |
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| 253 | - \int\limits_D \rho \, g \, w \;dv |
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| 254 | + \int\limits_D \,\frac{p}{e_3}\partial_t ( \partial_k z ) dv &&&\\ |
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| 255 | % |
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| 256 | &= - \int\limits_D \rho \, g \, w \;dv &&&\\ |
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| 257 | \end{flalign*} |
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| 258 | |
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| 259 | |
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| 260 | %gm comment |
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| 261 | \gmcomment{ |
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[817] | 262 | % |
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[2282] | 263 | The last equality comes from the following equation, |
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| 264 | \begin{flalign*} |
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| 265 | \int\limits_D p \frac{1}{e_3} \frac{\partial e_3}{\partial t}\; \;dv |
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| 266 | = \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv \quad, \\ |
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| 267 | \end{flalign*} |
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| 268 | that can be demonstrated as follows: |
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| 269 | |
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| 270 | \begin{flalign*} |
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| 271 | \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv |
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| 272 | &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv |
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| 273 | - \int\limits_D \rho \, g \, \frac{\partial}{\partial t} \left( \int\limits_k^{k_s} e_3 \;d\tilde{k} \right) \;dv &&&\\ |
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| 274 | &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv |
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| 275 | - \int\limits_D \rho \, g \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \;dv &&&\\ |
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[817] | 276 | % |
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[2282] | 277 | \allowdisplaybreaks |
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| 278 | \intertext{The second term of the right hand side can be transformed by applying the integration by part rule: |
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| 279 | $\left[ a\,b \right]_{k_b}^{k_s} = \int_{k_b}^{k_s} a\,\frac{\partial b}{\partial k} \;dk |
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| 280 | + \int_{k_b}^{k_s} \frac{\partial a}{\partial k} \,b \;dk $ |
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| 281 | to the following function: $a= \int_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k}$ |
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| 282 | and $b= \int_k^{k_s} \rho \, e_3 \;d\tilde{k}$ |
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| 283 | (note that $\frac{\partial}{\partial k} \left( \int_k^{k_s} a \;d\tilde{k} \right) = - a$ as $k$ is the lower bound of the integral). |
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| 284 | This leads to: } |
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| 285 | \end{flalign*} |
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| 286 | \begin{flalign*} |
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| 287 | &\left[ \int\limits_{k}^{k_s} \frac{\partial e_3}{\partial t} \,dk \cdot \int\limits_{k}^{k_s} \rho \, e_3 \,dk \right]_{k_b}^{k_s} |
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| 288 | =-\int\limits_{k_b}^{k_s} \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \rho \,e_3 \;dk |
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| 289 | -\int\limits_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \left( \int\limits_k^{k_s} \rho \, e_3 \;d\tilde{k} \right) dk |
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| 290 | &&&\\ |
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| 291 | \allowdisplaybreaks |
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| 292 | \intertext{Noting that $\frac{\partial \eta}{\partial t} |
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| 293 | = \frac{\partial}{\partial t} \left( \int_{k_b}^{k_s} e_3 \;d\tilde{k} \right) |
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| 294 | = \int_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k}$ |
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| 295 | and |
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| 296 | $p(k) = \int_k^{k_s} \rho \,g \, e_3 \;d\tilde{k} $, |
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| 297 | but also that $\frac{\partial \eta}{\partial t}$ does not depends on $k$, it comes: |
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| 298 | } |
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| 299 | & - \int\limits_{k_b}^{k_s} \rho \, \frac{\partial \eta}{\partial t} \, e_3 \;dk |
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| 300 | = - \int\limits_{k_b}^{k_s} \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \, \rho \, g e_3\;dk |
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| 301 | - \int\limits_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \frac{p}{g} \;dk &&&\\ |
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| 302 | \end{flalign*} |
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| 303 | Mutliplying by $g$ and integrating over the $(i,j)$ domain it becomes: |
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| 304 | \begin{flalign*} |
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| 305 | \int\limits_D \rho \, g \, \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \;dv |
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| 306 | = \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} dv |
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| 307 | - \int\limits_D \frac{p}{e_3}\frac{\partial e_3}{\partial t} \;dv |
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| 308 | \end{flalign*} |
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| 309 | Using this property, we therefore have: |
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| 310 | \begin{flalign*} |
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| 311 | \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv |
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| 312 | &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv |
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| 313 | - \left( \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} dv |
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| 314 | - \int\limits_D \frac{p}{e_3}\frac{\partial e_3}{\partial t} \;dv \right) &&&\\ |
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[817] | 315 | % |
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[2282] | 316 | &=\int\limits_D \frac{p}{e_3} \frac{\partial (e_3\,\rho)}{\partial t}\; \;dv |
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| 317 | \end{flalign*} |
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| 318 | % end gm comment |
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| 319 | } |
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[817] | 320 | % |
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[2282] | 321 | |
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| 322 | |
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| 323 | % ================================================================ |
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| 324 | % Discrete Total energy Conservation : vector invariant form |
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| 325 | % ================================================================ |
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[9393] | 326 | \section{Discrete total energy conservation: vector invariant form} |
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[9414] | 327 | \label{sec:C.2} |
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[2282] | 328 | |
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| 329 | % ------------------------------------------------------------------------------------------------------------- |
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| 330 | % Total energy conservation |
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| 331 | % ------------------------------------------------------------------------------------------------------------- |
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| 332 | \subsection{Total energy conservation} |
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[9414] | 333 | \label{subsec:C_KE+PE_vect} |
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[2282] | 334 | |
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[9407] | 335 | The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: |
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[2282] | 336 | \begin{flalign*} |
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| 337 | \partial_t \left( \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v + \rho \, g \, z_t \,b_t \biggr\} \right) &=0 \\ |
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[707] | 338 | \end{flalign*} |
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[2282] | 339 | which in vector invariant forms, it leads to: |
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[9407] | 340 | \begin{equation} \label{eq:KE+PE_vect_discrete} \begin{split} |
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[2282] | 341 | \sum\limits_{i,j,k} \biggl\{ u\, \partial_t u \;b_u |
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| 342 | + v\, \partial_t v \;b_v \biggr\} |
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| 343 | + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{e_{3u}}\partial_t e_{3u} \;b_u |
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| 344 | + \frac{v^2}{e_{3v}}\partial_t e_{3v} \;b_v \biggr\} \\ |
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| 345 | = - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}}\partial_t (e_{3t} \rho) \, g \, z_t \;b_t \biggr\} |
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| 346 | - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} |
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| 347 | \end{split} \end{equation} |
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[707] | 348 | |
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[2282] | 349 | Substituting the discrete expression of the time derivative of the velocity either in vector invariant, |
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[9407] | 350 | leads to the discrete equivalent of the four equations \autoref{eq:E_tot_flux}. |
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[2282] | 351 | |
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| 352 | % ------------------------------------------------------------------------------------------------------------- |
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| 353 | % Vorticity term (coriolis + vorticity part of the advection) |
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| 354 | % ------------------------------------------------------------------------------------------------------------- |
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| 355 | \subsection{Vorticity term (coriolis + vorticity part of the advection)} |
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[9407] | 356 | \label{subsec:C_vor} |
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[2282] | 357 | |
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| 358 | Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$), or |
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| 359 | the planetary ($q=f/e_{3f}$), or the total potential vorticity ($q=(\zeta +f) /e_{3f}$). |
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| 360 | Two discretisation of the vorticity term (ENE and EEN) allows the conservation of |
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| 361 | the kinetic energy. |
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| 362 | % ------------------------------------------------------------------------------------------------------------- |
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| 363 | % Vorticity Term with ENE scheme |
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| 364 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 365 | \subsubsection{Vorticity term with ENE scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})} |
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[9407] | 366 | \label{subsec:C_vorENE} |
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[2282] | 367 | |
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| 368 | For the ENE scheme, the two components of the vorticity term are given by : |
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[707] | 369 | \begin{equation*} |
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[2282] | 370 | - e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h \equiv |
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| 371 | \left( {{ \begin{array} {*{20}c} |
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[707] | 372 | + \frac{1} {e_{1u}} \; |
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[2282] | 373 | \overline {\, q \ \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} \hfill \\ |
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[707] | 374 | - \frac{1} {e_{2v}} \; |
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[2282] | 375 | \overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} \hfill \\ |
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| 376 | \end{array}} } \right) |
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[707] | 377 | \end{equation*} |
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| 378 | |
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[1223] | 379 | This formulation does not conserve the enstrophy but it does conserve the |
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[2282] | 380 | total kinetic energy. Indeed, the kinetic energy tendency associated to the |
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| 381 | vorticity term and averaged over the ocean domain can be transformed as |
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| 382 | follows: |
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[707] | 383 | \begin{flalign*} |
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[2282] | 384 | &\int\limits_D - \left( e_3 \, q \;\textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv && \\ |
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| 385 | & \qquad \qquad {\begin{array}{*{20}l} |
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| 386 | &\equiv \sum\limits_{i,j,k} \biggl\{ |
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| 387 | \frac{1} {e_{1u}} \overline { \,q\ \overline{ V }^{\,i+1/2}} ^{\,j} \, u \; b_u |
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| 388 | - \frac{1} {e_{2v}}\overline { \, q\ \overline{ U }^{\,j+1/2}} ^{\,i} \, v \; b_v \; \biggr\} \\ |
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| 389 | &\equiv \sum\limits_{i,j,k} \biggl\{ |
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| 390 | \overline { \,q\ \overline{ V }^{\,i+1/2}}^{\,j} \; U |
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| 391 | - \overline { \,q\ \overline{ U }^{\,j+1/2}}^{\,i} \; V \; \biggr\} \\ |
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| 392 | &\equiv \sum\limits_{i,j,k} q \ \biggl\{ \overline{ V }^{\,i+1/2}\; \overline{ U }^{\,j+1/2} |
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| 393 | - \overline{ U }^{\,j+1/2}\; \overline{ V }^{\,i+1/2} \biggr\} \quad \equiv 0 |
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| 394 | \end{array} } |
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| 395 | \end{flalign*} |
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| 396 | In other words, the domain averaged kinetic energy does not change due to the vorticity term. |
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| 397 | |
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| 398 | |
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| 399 | % ------------------------------------------------------------------------------------------------------------- |
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| 400 | % Vorticity Term with EEN scheme |
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| 401 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 402 | \subsubsection{Vorticity term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} |
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[9414] | 403 | \label{subsec:C_vorEEN_vect} |
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[2282] | 404 | |
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| 405 | With the EEN scheme, the vorticity terms are represented as: |
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[9414] | 406 | \begin{equation} \tag{\ref{eq:dynvor_een}} |
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[2282] | 407 | \left\{ { \begin{aligned} |
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| 408 | +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} |
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| 409 | {^{i+1/2-i_p}_j} \mathbb{Q}^{i_p}_{j_p} \left( e_{1v} e_{3v} \ v \right)^{i+i_p-1/2}_{j+j_p} \\ |
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| 410 | - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} |
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| 411 | {^i_{j+1/2-j_p}} \mathbb{Q}^{i_p}_{j_p} \left( e_{2u} e_{3u} \ u \right)^{i+i_p}_{j+j_p-1/2} \\ |
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| 412 | \end{aligned} } \right. |
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| 413 | \end{equation} |
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[6140] | 414 | where the indices $i_p$ and $j_p$ take the following value: |
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[2282] | 415 | $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, |
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| 416 | and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: |
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[9414] | 417 | \begin{equation} \tag{\ref{eq:Q_triads}} |
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[2282] | 418 | _i^j \mathbb{Q}^{i_p}_{j_p} |
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| 419 | = \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right) |
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| 420 | \end{equation} |
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| 421 | |
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| 422 | This formulation does conserve the total kinetic energy. Indeed, |
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| 423 | \begin{flalign*} |
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[994] | 424 | &\int\limits_D - \textbf{U}_h \cdot \left( \zeta \;\textbf{k} \times \textbf{U}_h \right) \; dv && \\ |
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[2282] | 425 | \equiv \sum\limits_{i,j,k} & \biggl\{ |
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| 426 | \left[ \sum_{\substack{i_p,\,k_p}} |
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| 427 | {^{i+1/2-i_p}_j}\mathbb{Q}^{i_p}_{j_p} \; V^{i+1/2-i_p}_{j+j_p} \right] U^{i+1/2}_{j} % &&\\ |
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| 428 | - \left[ \sum_{\substack{i_p,\,k_p}} |
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| 429 | {^i_{j+1/2-j_p}}\mathbb{Q}^{i_p}_{j_p} \; U^{i+i_p}_{j+1/2-j_p} \right] V^{i}_{j+1/2} \biggr\} && \\ |
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[817] | 430 | \\ |
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[2282] | 431 | \equiv \sum\limits_{i,j,k} & \sum_{\substack{i_p,\,k_p}} \biggl\{ \ \ |
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| 432 | {^{i+1/2-i_p}_j}\mathbb{Q}^{i_p}_{j_p} \; V^{i+1/2-i_p}_{j+j_p} \, U^{i+1/2}_{j} % &&\\ |
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| 433 | - {^i_{j+1/2-j_p}}\mathbb{Q}^{i_p}_{j_p} \; U^{i+i_p}_{j+1/2-j_p} \, V^{i}_{j+1/2} \ \; \biggr\} && \\ |
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| 434 | % |
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| 435 | \allowdisplaybreaks |
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| 436 | \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} |
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| 437 | % |
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| 438 | \equiv \sum\limits_{i,j,k} & \biggl\{ \ \ |
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| 439 | {^{i+1}_j }\mathbb{Q}^{-1/2}_{+1/2} \;V^{i+1}_{j+1/2} \; U^{\,i+1/2}_{j} |
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| 440 | - {^i_{j}\quad}\mathbb{Q}^{-1/2}_{+1/2} \; U^{i-1/2}_{j} \; V^{\,i}_{j+1/2} && \\ |
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| 441 | & + {^{i+1}_j }\mathbb{Q}^{-1/2}_{-1/2} \; V^{i+1}_{j-1/2} \; U^{\,i+1/2}_{j} |
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| 442 | - {^i_{j+1} }\mathbb{Q}^{-1/2}_{-1/2} \; U^{i-1/2}_{j+1} \; V^{\,i}_{j+1/2} \biggr. && \\ |
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| 443 | & + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2} \; U^{\,i+1/2}_{j} |
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| 444 | - {^i_{j}\quad}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j} \; V^{\,i}_{j+1/2} \biggr. && \\ |
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| 445 | & + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{-1/2} \; V^{i}_{j-1/2} \; U^{\,i+1/2}_{j} |
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| 446 | - {^i_{j+1} }\mathbb{Q}^{+1/2}_{-1/2} \; U^{i+1/2}_{j+1}\; V^{\,i}_{j+1/2} \ \; \biggr\} && \\ |
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| 447 | % |
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| 448 | \allowdisplaybreaks |
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| 449 | \intertext{The summation is done over all $i$ and $j$ indices, it is therefore possible to introduce |
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| 450 | a shift of $-1$ either in $i$ or $j$ direction in some of the term of the summation (first term of the |
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| 451 | first and second lines, second term of the second and fourth lines). By doning so, we can regroup |
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| 452 | all the terms of the summation by triad at a ($i$,$j$) point. In other words, we regroup all the terms |
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| 453 | in the neighbourhood that contain a triad at the same ($i$,$j$) indices. It becomes: } |
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| 454 | \allowdisplaybreaks |
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| 455 | % |
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| 456 | \equiv \sum\limits_{i,j,k} & \biggl\{ \ \ |
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| 457 | {^{i}_j}\mathbb{Q}^{-1/2}_{+1/2} \left[ V^{i}_{j+1/2}\, U^{\,i-1/2}_{j} |
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| 458 | - U^{i-1/2}_{j} \, V^{\,i}_{j+1/2} \right] && \\ |
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| 459 | & + {^{i}_j}\mathbb{Q}^{-1/2}_{-1/2} \left[ V^{i}_{j-1/2} \, U^{\,i-1/2}_{j} |
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| 460 | - U^{i-1/2}_{j} \, V^{\,i}_{j-1/2} \right] \biggr. && \\ |
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| 461 | & + {^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} \left[ V^{i}_{j+1/2} \, U^{\,i+1/2}_{j} |
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| 462 | - U^{i+1/2}_{j} \, V^{\,i}_{j+1/2} \right] \biggr. && \\ |
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| 463 | & + {^{i}_j}\mathbb{Q}^{+1/2}_{-1/2} \left[ V^{i}_{j-1/2} \, U^{\,i+1/2}_{j} |
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| 464 | - U^{i+1/2}_{j-1} \, V^{\,i}_{j-1/2} \right] \ \; \biggr\} \qquad |
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| 465 | \equiv \ 0 && |
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[707] | 466 | \end{flalign*} |
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| 467 | |
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| 468 | |
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| 469 | % ------------------------------------------------------------------------------------------------------------- |
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| 470 | % Gradient of Kinetic Energy / Vertical Advection |
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| 471 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 472 | \subsubsection{Gradient of kinetic energy / Vertical advection} |
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[9407] | 473 | \label{subsec:C_zad} |
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[707] | 474 | |
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| 475 | The change of Kinetic Energy (KE) due to the vertical advection is exactly |
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| 476 | balanced by the change of KE due to the horizontal gradient of KE~: |
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| 477 | \begin{equation*} |
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[2282] | 478 | \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv |
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| 479 | = - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv |
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| 480 | + \frac{1}{2} \int_D { \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv } \\ |
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[707] | 481 | \end{equation*} |
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[9407] | 482 | Indeed, using successively \autoref{eq:DOM_di_adj} ($i.e.$ the skew symmetry |
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[2282] | 483 | property of the $\delta$ operator) and the continuity equation, then |
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[9407] | 484 | \autoref{eq:DOM_di_adj} again, then the commutativity of operators |
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| 485 | $\overline {\,\cdot \,}$ and $\delta$, and finally \autoref{eq:DOM_mi_adj} |
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[1223] | 486 | ($i.e.$ the symmetry property of the $\overline {\,\cdot \,}$ operator) |
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| 487 | applied in the horizontal and vertical directions, it becomes: |
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[707] | 488 | \begin{flalign*} |
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[2282] | 489 | & - \int_D \textbf{U}_h \cdot \text{KEG}\;dv |
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| 490 | = - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv &&&\\ |
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[817] | 491 | % |
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[2282] | 492 | \equiv & - \sum\limits_{i,j,k} \frac{1}{2} \biggl\{ |
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| 493 | \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] u \ b_u |
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| 494 | + \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] v \ b_v \biggr\} &&& \\ |
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[817] | 495 | % |
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[2282] | 496 | \equiv & + \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right)\; |
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| 497 | \biggl\{ \delta_{i} \left[ U \right] + \delta_{j} \left[ V \right] \biggr\} &&& \\ |
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| 498 | \allowdisplaybreaks |
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[817] | 499 | % |
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[2282] | 500 | \equiv & - \sum\limits_{i,j,k} \frac{1}{2} |
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| 501 | \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) \; |
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| 502 | \biggl\{ \frac{b_t}{e_{3t}} \partial_t (e_{3t}) + \delta_k \left[ W \right] \biggr\} &&&\\ |
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| 503 | \allowdisplaybreaks |
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| 504 | % |
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| 505 | \equiv & + \sum\limits_{i,j,k} \frac{1}{2} \delta_{k+1/2} \left[ \overline{ u^2}^{\,i} + \overline{ v^2}^{\,j} \right] \; W |
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| 506 | - \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) \;\partial_t b_t &&& \\ |
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| 507 | \allowdisplaybreaks |
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| 508 | % |
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| 509 | \equiv & + \sum\limits_{i,j,k} \frac{1} {2} \left( \overline{\delta_{k+1/2} \left[ u^2 \right]}^{\,i} |
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| 510 | + \overline{\delta_{k+1/2} \left[ v^2 \right]}^{\,j} \right) \; W |
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| 511 | - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t \overline{b_t}^{\,{i+1/2}} |
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| 512 | + \frac{v^2}{2}\,\partial_t \overline{b_t}^{\,{j+1/2}} \right) &&& \\ |
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| 513 | \allowdisplaybreaks |
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| 514 | \intertext{Assuming that $b_u= \overline{b_t}^{\,i+1/2}$ and $b_v= \overline{b_t}^{\,j+1/2}$, or at least that the time |
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| 515 | derivative of these two equations is satisfied, it becomes:} |
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| 516 | % |
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| 517 | \equiv & \sum\limits_{i,j,k} \frac{1} {2} |
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| 518 | \biggl\{ \; \overline{W}^{\,i+1/2}\;\delta_{k+1/2} \left[ u^2 \right] |
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| 519 | + \overline{W}^{\,j+1/2}\;\delta_{k+1/2} \left[ v^2 \right] \; \biggr\} |
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| 520 | - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t b_u |
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| 521 | + \frac{v^2}{2}\,\partial_t b_v \right) &&& \\ |
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| 522 | \allowdisplaybreaks |
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[817] | 523 | % |
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[2282] | 524 | \equiv & \sum\limits_{i,j,k} |
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| 525 | \biggl\{ \; \overline{W}^{\,i+1/2}\; \overline {u}^{\,k+1/2}\; \delta_{k+1/2}[ u ] |
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| 526 | + \overline{W}^{\,j+1/2}\; \overline {v}^{\,k+1/2}\; \delta_{k+1/2}[ v ] \; \biggr\} |
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| 527 | - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t b_u |
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| 528 | + \frac{v^2}{2}\,\partial_t b_v \right) &&& \\ |
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[817] | 529 | % |
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[2282] | 530 | \allowdisplaybreaks |
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| 531 | \equiv & \sum\limits_{i,j,k} |
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| 532 | \biggl\{ \; \frac{1} {b_u } \; \overline { \overline{W}^{\,i+1/2}\,\delta_{k+1/2} \left[ u \right] }^{\,k} \;u\;b_u |
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| 533 | + \frac{1} {b_v } \; \overline { \overline{W}^{\,j+1/2} \delta_{k+1/2} \left[ v \right] }^{\,k} \;v\;b_v \; \biggr\} |
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| 534 | - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t b_u |
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| 535 | + \frac{v^2}{2}\,\partial_t b_v \right) &&& \\ |
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| 536 | % |
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| 537 | \intertext{The first term provides the discrete expression for the vertical advection of momentum (ZAD), |
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[9407] | 538 | while the second term corresponds exactly to \autoref{eq:KE+PE_vect_discrete}, therefore:} |
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[2282] | 539 | \equiv& \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv |
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| 540 | + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t (e_3) \;dv } &&&\\ |
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| 541 | \equiv& \int\limits_D \textbf{U}_h \cdot w \partial_k \textbf{U}_h \;dv |
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| 542 | + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t (e_3) \;dv } &&&\\ |
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[707] | 543 | \end{flalign*} |
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| 544 | |
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[2282] | 545 | There is two main points here. First, the satisfaction of this property links the choice of |
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[1223] | 546 | the discrete formulation of the vertical advection and of the horizontal gradient |
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| 547 | of KE. Choosing one imposes the other. For example KE can also be discretized |
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| 548 | as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. This leads to the following |
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| 549 | expression for the vertical advection: |
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[707] | 550 | \begin{equation*} |
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[2282] | 551 | \frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h |
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[707] | 552 | \equiv \left( {{\begin{array} {*{20}c} |
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[2282] | 553 | \frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \; \overline{\overline {e_{1t}\,e_{2t} \,\omega\;\delta_{k+1/2} |
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[707] | 554 | \left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k} \hfill \\ |
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[2282] | 555 | \frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \; \overline{\overline {e_{1t}\,e_{2t} \,\omega \;\delta_{k+1/2} |
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[707] | 556 | \left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill \\ |
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| 557 | \end{array}} } \right) |
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| 558 | \end{equation*} |
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[1223] | 559 | a formulation that requires an additional horizontal mean in contrast with |
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| 560 | the one used in NEMO. Nine velocity points have to be used instead of 3. |
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| 561 | This is the reason why it has not been chosen. |
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[707] | 562 | |
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[2282] | 563 | Second, as soon as the chosen $s$-coordinate depends on time, an extra constraint |
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| 564 | arises on the time derivative of the volume at $u$- and $v$-points: |
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| 565 | \begin{flalign*} |
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| 566 | e_{1u}\,e_{2u}\,\partial_t (e_{3u}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,i+1/2} \\ |
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| 567 | e_{1v}\,e_{2v}\,\partial_t (e_{3v}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,j+1/2} |
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| 568 | \end{flalign*} |
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| 569 | which is (over-)satified by defining the vertical scale factor as follows: |
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[9407] | 570 | \begin{flalign} \label{eq:e3u-e3v} |
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[2282] | 571 | e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2} \\ |
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| 572 | e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2} |
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| 573 | \end{flalign} |
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| 574 | |
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| 575 | Blah blah required on the the step representation of bottom topography..... |
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| 576 | |
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| 577 | |
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[707] | 578 | % ------------------------------------------------------------------------------------------------------------- |
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[2282] | 579 | % Pressure Gradient Term |
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| 580 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 581 | \subsection{Pressure gradient term} |
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[9414] | 582 | \label{subsec:C.2.6} |
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[2282] | 583 | |
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| 584 | \gmcomment{ |
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| 585 | A pressure gradient has no contribution to the evolution of the vorticity as the |
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| 586 | curl of a gradient is zero. In the $z$-coordinate, this property is satisfied locally |
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[9407] | 587 | on a C-grid with 2nd order finite differences (property \autoref{eq:DOM_curl_grad}). |
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[2282] | 588 | } |
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| 589 | |
---|
| 590 | When the equation of state is linear ($i.e.$ when an advection-diffusion equation |
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| 591 | for density can be derived from those of temperature and salinity) the change of |
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| 592 | KE due to the work of pressure forces is balanced by the change of potential |
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| 593 | energy due to buoyancy forces: |
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| 594 | \begin{equation*} |
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| 595 | - \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv |
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| 596 | = - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right) \,g\,z \;dv |
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| 597 | + \int_D g\, \rho \; \partial_t (z) \;dv |
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| 598 | \end{equation*} |
---|
| 599 | |
---|
| 600 | This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates. |
---|
| 601 | Indeed, defining the depth of a $T$-point, $z_t$, as the sum of the vertical scale |
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| 602 | factors at $w$-points starting from the surface, the work of pressure forces can be |
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| 603 | written as: |
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| 604 | \begin{flalign*} |
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| 605 | &- \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv |
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| 606 | \equiv \sum\limits_{i,j,k} \biggl\{ \; - \frac{1} {e_{1u}} \Bigl( |
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| 607 | \delta_{i+1/2} [p_t] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} [z_t] \Bigr) \; u\;b_u |
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| 608 | && \\ & \qquad \qquad \qquad \qquad \qquad \quad \ \, |
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| 609 | - \frac{1} {e_{2v}} \Bigl( |
---|
| 610 | \delta_{j+1/2} [p_t] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} [z_t] \Bigr) \; v\;b_v \; \biggr\} && \\ |
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| 611 | % |
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| 612 | \allowdisplaybreaks |
---|
[9407] | 613 | \intertext{Using successively \autoref{eq:DOM_di_adj}, $i.e.$ the skew symmetry property of |
---|
[9408] | 614 | the $\delta$ operator, \autoref{eq:wzv}, the continuity equation, \autoref{eq:dynhpg_sco}, |
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[2282] | 615 | the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $, |
---|
| 616 | which comes from the definition of $z_t$, it becomes: } |
---|
| 617 | \allowdisplaybreaks |
---|
| 618 | % |
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| 619 | \equiv& + \sum\limits_{i,j,k} g \biggl\{ |
---|
| 620 | \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] |
---|
| 621 | + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] |
---|
| 622 | +\Bigl( \delta_i[U] + \delta_j [V] \Bigr)\;\frac{p_t}{g} \biggr\} &&\\ |
---|
| 623 | % |
---|
| 624 | \equiv& + \sum\limits_{i,j,k} g \biggl\{ |
---|
| 625 | \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] |
---|
| 626 | + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] |
---|
| 627 | - \left( \frac{b_t}{e_{3t}} \partial_t (e_{3t}) + \delta_k \left[ W \right] \right) \frac{p_t}{g} \biggr\} &&&\\ |
---|
| 628 | % |
---|
| 629 | \equiv& + \sum\limits_{i,j,k} g \biggl\{ |
---|
| 630 | \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] |
---|
| 631 | + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] |
---|
| 632 | + \frac{W}{g}\;\delta_{k+1/2} [p_t] |
---|
| 633 | - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\ |
---|
| 634 | % |
---|
| 635 | \equiv& + \sum\limits_{i,j,k} g \biggl\{ |
---|
| 636 | \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] |
---|
| 637 | + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] |
---|
| 638 | - W\;e_{3w} \overline \rho^{\,k+1/2} |
---|
| 639 | - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\ |
---|
| 640 | % |
---|
| 641 | \equiv& + \sum\limits_{i,j,k} g \biggl\{ |
---|
| 642 | \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] |
---|
| 643 | + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] |
---|
| 644 | + W\; \overline \rho^{\,k+1/2}\;\delta_{k+1/2} [z_t] |
---|
| 645 | - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\ |
---|
| 646 | % |
---|
| 647 | \allowdisplaybreaks |
---|
| 648 | % |
---|
| 649 | \equiv& - \sum\limits_{i,j,k} g \; z_t \biggl\{ |
---|
| 650 | \delta_i \left[ U\; \overline \rho^{\,i+1/2} \right] |
---|
| 651 | + \delta_j \left[ V\; \overline \rho^{\,j+1/2} \right] |
---|
| 652 | + \delta_k \left[ W\; \overline \rho^{\,k+1/2} \right] \biggr\} |
---|
| 653 | - \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ |
---|
| 654 | % |
---|
| 655 | \equiv& + \sum\limits_{i,j,k} g \; z_t \biggl\{ \partial_t ( e_{3t} \,\rho) \biggr\} \; b_t |
---|
| 656 | - \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ |
---|
| 657 | % |
---|
| 658 | \end{flalign*} |
---|
[9407] | 659 | The first term is exactly the first term of the right-hand-side of \autoref{eq:KE+PE_vect_discrete}. |
---|
[2282] | 660 | It remains to demonstrate that the last term, which is obviously a discrete analogue of |
---|
[9407] | 661 | $\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to the last term of \autoref{eq:KE+PE_vect_discrete}. |
---|
[2282] | 662 | In other words, the following property must be satisfied: |
---|
| 663 | \begin{flalign*} |
---|
| 664 | \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} |
---|
| 665 | \equiv \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} |
---|
| 666 | \end{flalign*} |
---|
| 667 | |
---|
| 668 | Let introduce $p_w$ the pressure at $w$-point such that $\delta_k [p_w] = - \rho \,g\,e_{3t}$. |
---|
| 669 | The right-hand-side of the above equation can be transformed as follows: |
---|
| 670 | |
---|
| 671 | \begin{flalign*} |
---|
| 672 | \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} |
---|
| 673 | &\equiv - \sum\limits_{i,j,k} \biggl\{ \delta_k [p_w]\,\partial_t (z_t) \,e_{1t}\,e_{2t} \biggr\} &&&\\ |
---|
| 674 | % |
---|
| 675 | &\equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \delta_{k+1/2} [\partial_t (z_t)] \,e_{1t}\,e_{2t} \biggr\} |
---|
| 676 | \equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (e_{3w}) \,e_{1t}\,e_{2t} \biggr\} &&&\\ |
---|
| 677 | &\equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (b_w) \biggr\} |
---|
| 678 | % |
---|
| 679 | % & \equiv \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w \right) \biggr\} &&&\\ |
---|
| 680 | % & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t \left( \delta_k [z_t] \right) e_{1w}\,e_{2w} \biggr\} &&&\\ |
---|
| 681 | % & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t b_w \biggr\} |
---|
| 682 | \end{flalign*} |
---|
| 683 | therefore, the balance to be satisfied is: |
---|
| 684 | \begin{flalign*} |
---|
| 685 | \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t (b_t) \biggr\} \equiv \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (b_w) \biggr\} |
---|
| 686 | \end{flalign*} |
---|
| 687 | which is a purely vertical balance: |
---|
| 688 | \begin{flalign*} |
---|
| 689 | \sum\limits_{k} \biggl\{ p_t\;\partial_t (e_{3t}) \biggr\} \equiv \sum\limits_{k} \biggl\{ p_w\, \partial_t (e_{3w}) \biggr\} |
---|
| 690 | \end{flalign*} |
---|
| 691 | Defining $p_w = \overline{p_t}^{\,k+1/2}$ |
---|
| 692 | |
---|
| 693 | %gm comment |
---|
| 694 | \gmcomment{ |
---|
| 695 | \begin{flalign*} |
---|
| 696 | \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ |
---|
| 697 | % |
---|
| 698 | & \equiv \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w \biggr\} &&&\\ |
---|
| 699 | & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t \left( \delta_{k+1/2} [z_t] \right) e_{1w}\,e_{2w} \biggr\} &&&\\ |
---|
| 700 | & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t b_w \biggr\} |
---|
| 701 | \end{flalign*} |
---|
| 702 | |
---|
| 703 | |
---|
| 704 | \begin{flalign*} |
---|
| 705 | \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv |
---|
| 706 | \equiv& \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} p \biggr\} \; b_t &&&\\ |
---|
| 707 | \equiv& \sum\limits_{i,j,k} \biggl\{ \biggr\} \; b_t &&&\\ |
---|
| 708 | \end{flalign*} |
---|
| 709 | |
---|
| 710 | % |
---|
| 711 | \begin{flalign*} |
---|
| 712 | \equiv& - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv |
---|
| 713 | + \int\limits_D g\, \rho \; \frac{\partial z}{\partial t} \;dv &&& \\ |
---|
| 714 | \end{flalign*} |
---|
| 715 | % |
---|
| 716 | } |
---|
| 717 | %end gm comment |
---|
| 718 | |
---|
| 719 | |
---|
| 720 | Note that this property strongly constrains the discrete expression of both |
---|
| 721 | the depth of $T-$points and of the term added to the pressure gradient in the |
---|
| 722 | $s$-coordinate. Nevertheless, it is almost never satisfied since a linear equation |
---|
| 723 | of state is rarely used. |
---|
| 724 | |
---|
| 725 | |
---|
| 726 | |
---|
| 727 | |
---|
| 728 | |
---|
| 729 | |
---|
| 730 | |
---|
| 731 | % ================================================================ |
---|
| 732 | % Discrete Total energy Conservation : flux form |
---|
| 733 | % ================================================================ |
---|
[9393] | 734 | \section{Discrete total energy conservation: flux form} |
---|
[9414] | 735 | \label{sec:C.3} |
---|
[2282] | 736 | |
---|
| 737 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 738 | % Total energy conservation |
---|
| 739 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 740 | \subsection{Total energy conservation} |
---|
[9414] | 741 | \label{subsec:C_KE+PE_flux} |
---|
[2282] | 742 | |
---|
[9407] | 743 | The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: |
---|
[2282] | 744 | \begin{flalign*} |
---|
| 745 | \partial_t \left( \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v + \rho \, g \, z_t \,b_t \biggr\} \right) &=0 \\ |
---|
| 746 | \end{flalign*} |
---|
| 747 | which in flux form, it leads to: |
---|
| 748 | \begin{flalign*} |
---|
| 749 | \sum\limits_{i,j,k} \biggl\{ \frac{u }{e_{3u}}\,\frac{\partial (e_{3u}u)}{\partial t} \,b_u |
---|
| 750 | + \frac{v }{e_{3v}}\,\frac{\partial (e_{3v}v)}{\partial t} \,b_v \biggr\} |
---|
| 751 | & - \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{e_{3u}}\frac{\partial e_{3u} }{\partial t} \,b_u |
---|
| 752 | + \frac{v^2}{e_{3v}}\frac{\partial e_{3v} }{\partial t} \,b_v \biggr\} \\ |
---|
| 753 | &= - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_3t}\frac{\partial e_{3t} \rho}{\partial t} \, g \, z_t \,b_t \biggr\} |
---|
| 754 | - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\frac{\partial z_t}{\partial t} \,b_t \biggr\} \\ |
---|
| 755 | \end{flalign*} |
---|
| 756 | |
---|
| 757 | Substituting the discrete expression of the time derivative of the velocity either in vector invariant or in flux form, |
---|
| 758 | leads to the discrete equivalent of the |
---|
| 759 | |
---|
| 760 | |
---|
| 761 | % ------------------------------------------------------------------------------------------------------------- |
---|
[707] | 762 | % Coriolis and advection terms: flux form |
---|
| 763 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 764 | \subsection{Coriolis and advection terms: flux form} |
---|
[9414] | 765 | \label{subsec:C.3.2} |
---|
[707] | 766 | |
---|
| 767 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 768 | % Coriolis plus ``metric'' Term |
---|
| 769 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 770 | \subsubsection{Coriolis plus ``metric'' term} |
---|
[9414] | 771 | \label{subsec:C.3.3} |
---|
[707] | 772 | |
---|
[1223] | 773 | In flux from the vorticity term reduces to a Coriolis term in which the Coriolis |
---|
| 774 | parameter has been modified to account for the ``metric'' term. This altered |
---|
| 775 | Coriolis parameter is discretised at an f-point. It is given by: |
---|
[707] | 776 | \begin{equation*} |
---|
[2282] | 777 | f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\; |
---|
[707] | 778 | \equiv \; |
---|
[2282] | 779 | f+\frac{1} {e_{1f}\,e_{2f}} \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] |
---|
| 780 | -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u} \right] \right) |
---|
[707] | 781 | \end{equation*} |
---|
| 782 | |
---|
[2282] | 783 | Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. |
---|
[1223] | 784 | It therefore conserves the total KE. The derivation is the same as for the |
---|
[9407] | 785 | vorticity term in the vector invariant form (\autoref{subsec:C_vor}). |
---|
[707] | 786 | |
---|
| 787 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 788 | % Flux form advection |
---|
| 789 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 790 | \subsubsection{Flux form advection} |
---|
[9414] | 791 | \label{subsec:C.3.4} |
---|
[707] | 792 | |
---|
[1223] | 793 | The flux form operator of the momentum advection is evaluated using a |
---|
| 794 | centered second order finite difference scheme. Because of the flux form, |
---|
| 795 | the discrete operator does not contribute to the global budget of linear |
---|
| 796 | momentum. Because of the centered second order scheme, it conserves |
---|
| 797 | the horizontal kinetic energy, that is : |
---|
[707] | 798 | |
---|
[9407] | 799 | \begin{equation} \label{eq:C_ADV_KE_flux} |
---|
[2282] | 800 | - \int_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} |
---|
[707] | 801 | \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ |
---|
[2282] | 802 | \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ \end{array}} } \right) \;dv |
---|
| 803 | - \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv } =\;0 |
---|
[707] | 804 | \end{equation} |
---|
| 805 | |
---|
[2282] | 806 | Let us first consider the first term of the scalar product ($i.e.$ just the the terms |
---|
| 807 | associated with the i-component of the advection) : |
---|
[707] | 808 | \begin{flalign*} |
---|
[2282] | 809 | & - \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv \\ |
---|
[817] | 810 | % |
---|
[2282] | 811 | \equiv& - \sum\limits_{i,j,k} \biggl\{ \frac{1} {b_u} \biggl( |
---|
| 812 | \delta_{i+1/2} \left[ \overline {U}^{\,i} \;\overline u^{\,i} \right] |
---|
| 813 | + \delta_j \left[ \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] |
---|
| 814 | + \delta_k \left[ \overline {W}^{\,i+1/2}\;\overline u^{\,k+1/2} \right] \biggr) \; \biggr\} \, b_u \;u &&& \\ |
---|
[817] | 815 | % |
---|
[2282] | 816 | \equiv& - \sum\limits_{i,j,k} |
---|
[707] | 817 | \biggl\{ |
---|
[2282] | 818 | \delta_{i+1/2} \left[ \overline {U}^{\,i}\; \overline u^{\,i} \right] |
---|
| 819 | + \delta_j \left[ \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] |
---|
| 820 | + \delta_k \left[ \overline {W}^{\,i+12}\;\overline u^{\,k+1/2} \right] |
---|
| 821 | \; \biggr\} \; u \\ |
---|
[817] | 822 | % |
---|
[2282] | 823 | \equiv& + \sum\limits_{i,j,k} |
---|
[707] | 824 | \biggl\{ |
---|
[2282] | 825 | \overline {U}^{\,i}\; \overline u^{\,i} \delta_i \left[ u \right] |
---|
| 826 | + \overline {V}^{\,i+1/2}\; \overline u^{\,j+1/2} \delta_{j+1/2} \left[ u \right] |
---|
| 827 | + \overline {W}^{\,i+1/2}\; \overline u^{\,k+1/2} \delta_{k+1/2} \left[ u \right] \biggr\} && \\ |
---|
[817] | 828 | % |
---|
[2282] | 829 | \equiv& + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ |
---|
| 830 | \overline{U}^{\,i} \delta_i \left[ u^2 \right] |
---|
| 831 | + \overline{V}^{\,i+1/2} \delta_{j+/2} \left[ u^2 \right] |
---|
| 832 | + \overline{W}^{\,i+1/2} \delta_{k+1/2} \left[ u^2 \right] \biggr\} && \\ |
---|
[817] | 833 | % |
---|
[2282] | 834 | \equiv& - \sum\limits_{i,j,k} \frac{1}{2} \bigg\{ |
---|
| 835 | U \; \delta_{i+1/2} \left[ \overline {u^2}^{\,i} \right] |
---|
| 836 | + V \; \delta_{j+1/2} \left[ \overline {u^2}^{\,i} \right] |
---|
| 837 | + W \; \delta_{k+1/2} \left[ \overline {u^2}^{\,i} \right] \biggr\} &&& \\ |
---|
[817] | 838 | % |
---|
[2282] | 839 | \equiv& - \sum\limits_{i,j,k} \frac{1}{2} \overline {u^2}^{\,i} \biggl\{ |
---|
| 840 | \delta_{i+1/2} \left[ U \right] |
---|
| 841 | + \delta_{j+1/2} \left[ V \right] |
---|
| 842 | + \delta_{k+1/2} \left[ W \right] \biggr\} &&& \\ |
---|
| 843 | % |
---|
| 844 | \equiv& + \sum\limits_{i,j,k} \frac{1}{2} \overline {u^2}^{\,i} |
---|
| 845 | \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} &&& \\ |
---|
[707] | 846 | \end{flalign*} |
---|
[2282] | 847 | Applying similar manipulation applied to the second term of the scalar product |
---|
| 848 | leads to : |
---|
| 849 | \begin{equation*} |
---|
| 850 | - \int_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} |
---|
| 851 | \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ |
---|
| 852 | \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ \end{array}} } \right) \;dv |
---|
| 853 | \equiv + \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) |
---|
| 854 | \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} |
---|
| 855 | \end{equation*} |
---|
| 856 | which is the discrete form of |
---|
| 857 | $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv $. |
---|
[9407] | 858 | \autoref{eq:C_ADV_KE_flux} is thus satisfied. |
---|
[707] | 859 | |
---|
[2282] | 860 | |
---|
[1223] | 861 | When the UBS scheme is used to evaluate the flux form momentum advection, |
---|
| 862 | the discrete operator does not contribute to the global budget of linear momentum |
---|
| 863 | (flux form). The horizontal kinetic energy is not conserved, but forced to decay |
---|
| 864 | ($i.e.$ the scheme is diffusive). |
---|
[707] | 865 | |
---|
[2282] | 866 | |
---|
| 867 | |
---|
| 868 | |
---|
| 869 | |
---|
| 870 | |
---|
| 871 | |
---|
| 872 | |
---|
| 873 | |
---|
| 874 | |
---|
| 875 | % ================================================================ |
---|
| 876 | % Discrete Enstrophy Conservation |
---|
| 877 | % ================================================================ |
---|
| 878 | \section{Discrete enstrophy conservation} |
---|
[9414] | 879 | \label{sec:C.4} |
---|
[2282] | 880 | |
---|
| 881 | |
---|
[707] | 882 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 883 | % Vorticity Term with ENS scheme |
---|
[707] | 884 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 885 | \subsubsection{Vorticity term with ENS scheme (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})} |
---|
[9407] | 886 | \label{subsec:C_vorENS} |
---|
[707] | 887 | |
---|
[2282] | 888 | In the ENS scheme, the vorticity term is descretized as follows: |
---|
[9414] | 889 | \begin{equation} \tag{\ref{eq:dynvor_ens}} |
---|
[2282] | 890 | \left\{ \begin{aligned} |
---|
| 891 | +\frac{1}{e_{1u}} & \overline{q}^{\,i} & {\overline{ \overline{\left( e_{1v}\,e_{3v}\; v \right) } } }^{\,i, j+1/2} \\ |
---|
| 892 | - \frac{1}{e_{2v}} & \overline{q}^{\,j} & {\overline{ \overline{\left( e_{2u}\,e_{3u}\; u \right) } } }^{\,i+1/2, j} |
---|
| 893 | \end{aligned} \right. |
---|
| 894 | \end{equation} |
---|
[707] | 895 | |
---|
[2282] | 896 | The scheme does not allow but the conservation of the total kinetic energy but the conservation |
---|
| 897 | of $q^2$, the potential enstrophy for a horizontally non-divergent flow ($i.e.$ when $\chi$=$0$). |
---|
[9407] | 898 | Indeed, using the symmetry or skew symmetry properties of the operators ( \autoref{eq:DOM_mi_adj} |
---|
| 899 | and \autoref{eq:DOM_di_adj}), it can be shown that: |
---|
| 900 | \begin{equation} \label{eq:C_1.1} |
---|
[2282] | 901 | \int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0 |
---|
| 902 | \end{equation} |
---|
| 903 | where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. Indeed, using |
---|
[9407] | 904 | \autoref{eq:dynvor_ens}, the discrete form of the right hand side of \autoref{eq:C_1.1} |
---|
[2282] | 905 | can be transformed as follow: |
---|
[817] | 906 | \begin{flalign*} |
---|
[2282] | 907 | &\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times |
---|
| 908 | \left( e_3 \, q \; \textbf{k} \times \textbf{U}_h \right)\; dv \\ |
---|
[817] | 909 | % |
---|
[2282] | 910 | & \qquad {\begin{array}{*{20}l} |
---|
| 911 | &\equiv \sum\limits_{i,j,k} |
---|
| 912 | q \ \left\{ \delta_{i+1/2} \left[ - \,\overline {q}^{\,i}\; \overline{\overline U }^{\,i,j+1/ 2} \right] |
---|
| 913 | - \delta_{j+1/2} \left[ \overline {q}^{\,j}\; \overline{\overline V }^{\,i+1/2, j} \right] \right\} \\ |
---|
[817] | 914 | % |
---|
[2282] | 915 | &\equiv \sum\limits_{i,j,k} |
---|
| 916 | \left\{ \delta_i [q] \; \overline{q}^{\,i} \; \overline{ \overline U }^{\,i,j+1/2} |
---|
| 917 | + \delta_j [q] \; \overline{q}^{\,j} \; \overline{\overline V }^{\,i+1/2,j} \right\} && \\ |
---|
[817] | 918 | % |
---|
[2282] | 919 | &\equiv \,\frac{1} {2} \sum\limits_{i,j,k} |
---|
| 920 | \left\{ \delta_i \left[ q^2 \right] \; \overline{\overline U }^{\,i,j+1/2} |
---|
| 921 | + \delta_j \left[ q^2 \right] \; \overline{\overline V }^{\,i+1/2,j} \right\} && \\ |
---|
[817] | 922 | % |
---|
[2282] | 923 | &\equiv - \frac{1} {2} \sum\limits_{i,j,k} q^2 \; |
---|
| 924 | \left\{ \delta_{i+1/2} \left[ \overline{\overline{ U }}^{\,i,j+1/2} \right] |
---|
| 925 | + \delta_{j+1/2} \left[ \overline{\overline{ V }}^{\,i+1/2,j} \right] \right\} && \\ |
---|
| 926 | \end{array} } |
---|
| 927 | % |
---|
| 928 | \allowdisplaybreaks |
---|
| 929 | \intertext{ Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2} |
---|
| 930 | \left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$, |
---|
| 931 | and introducing the horizontal divergence $\chi $, it becomes: } |
---|
| 932 | \allowdisplaybreaks |
---|
| 933 | % |
---|
| 934 | & \qquad {\begin{array}{*{20}l} |
---|
| 935 | &\equiv \sum\limits_{i,j,k} - \frac{1} {2} q^2 \; \overline{\overline{ e_{1t}\,e_{2t}\,e_{3t}^{}\, \chi}}^{\,i+1/2,j+1/2} |
---|
| 936 | \quad \equiv 0 && |
---|
| 937 | \end{array} } |
---|
[707] | 938 | \end{flalign*} |
---|
[2282] | 939 | The later equality is obtain only when the flow is horizontally non-divergent, $i.e.$ $\chi$=$0$. |
---|
[707] | 940 | |
---|
| 941 | |
---|
| 942 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 943 | % Vorticity Term with EEN scheme |
---|
[707] | 944 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 945 | \subsubsection{Vorticity Term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} |
---|
[9407] | 946 | \label{subsec:C_vorEEN} |
---|
[707] | 947 | |
---|
[2282] | 948 | With the EEN scheme, the vorticity terms are represented as: |
---|
[9414] | 949 | \begin{equation} \tag{\ref{eq:dynvor_een}} |
---|
[2282] | 950 | \left\{ { \begin{aligned} |
---|
| 951 | +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} |
---|
| 952 | {^{i+1/2-i_p}_j} \mathbb{Q}^{i_p}_{j_p} \left( e_{1v} e_{3v} \ v \right)^{i+i_p-1/2}_{j+j_p} \\ |
---|
| 953 | - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} |
---|
| 954 | {^i_{j+1/2-j_p}} \mathbb{Q}^{i_p}_{j_p} \left( e_{2u} e_{3u} \ u \right)^{i+i_p}_{j+j_p-1/2} \\ |
---|
| 955 | \end{aligned} } \right. |
---|
| 956 | \end{equation} |
---|
| 957 | where the indices $i_p$ and $k_p$ take the following value: |
---|
| 958 | $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, |
---|
| 959 | and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: |
---|
[9414] | 960 | \begin{equation} \tag{\ref{eq:Q_triads}} |
---|
[2282] | 961 | _i^j \mathbb{Q}^{i_p}_{j_p} |
---|
| 962 | = \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right) |
---|
| 963 | \end{equation} |
---|
[707] | 964 | |
---|
| 965 | |
---|
[2282] | 966 | This formulation does conserve the potential enstrophy for a horizontally non-divergent flow ($i.e.$ $\chi=0$). |
---|
| 967 | |
---|
| 968 | Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $, |
---|
| 969 | similar manipulation can be done for the 3 others. The discrete form of the right hand |
---|
[9407] | 970 | side of \autoref{eq:C_1.1} applied to this triad only can be transformed as follow: |
---|
[2282] | 971 | |
---|
| 972 | \begin{flalign*} |
---|
| 973 | &\int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \\ |
---|
[817] | 974 | % |
---|
[2282] | 975 | \equiv& \sum\limits_{i,j,k} |
---|
| 976 | {q} \ \biggl\{ \;\; |
---|
| 977 | \delta_{i+1/2} \left[ -\, {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}} \right] |
---|
| 978 | - \delta_{j+1/2} \left[ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \right] |
---|
| 979 | \;\;\biggr\} && \\ |
---|
[817] | 980 | % |
---|
[2282] | 981 | \equiv& \sum\limits_{i,j,k} |
---|
| 982 | \biggl\{ \delta_i [q] \ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}} |
---|
| 983 | + \delta_j [q] \ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \biggr\} |
---|
| 984 | && \\ |
---|
[817] | 985 | % |
---|
[2282] | 986 | ... & &&\\ |
---|
| 987 | &Demonstation \ to \ be \ done... &&\\ |
---|
| 988 | ... & &&\\ |
---|
[817] | 989 | % |
---|
[2282] | 990 | \equiv& \frac{1} {2} \sum\limits_{i,j,k} |
---|
| 991 | \biggl\{ \delta_i \Bigl[ \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \Bigr]\; |
---|
| 992 | \overline{\overline {U}}^{\,i,j+1/2} |
---|
| 993 | + \delta_j \Bigl[ \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \Bigr]\; |
---|
| 994 | \overline{\overline {V}}^{\,i+1/2,j} |
---|
[707] | 995 | \biggr\} |
---|
[2282] | 996 | && \\ |
---|
[817] | 997 | % |
---|
[2282] | 998 | \equiv& - \frac{1} {2} \sum\limits_{i,j,k} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2\; |
---|
| 999 | \biggl\{ \delta_{i+1/2} |
---|
| 1000 | \left[ \overline{\overline {U}}^{\,i,j+1/2} \right] |
---|
| 1001 | + \delta_{j+1/2} |
---|
| 1002 | \left[ \overline{\overline {V}}^{\,i+1/2,j} \right] |
---|
| 1003 | \biggr\} && \\ |
---|
[817] | 1004 | % |
---|
[2282] | 1005 | \equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 |
---|
| 1006 | \; \overline{\overline{ b_t^{}\, \chi}}^{\,i+1/2,\,j+1/2} &&\\ |
---|
| 1007 | % |
---|
| 1008 | \ \ \equiv& \ 0 &&\\ |
---|
[707] | 1009 | \end{flalign*} |
---|
| 1010 | |
---|
| 1011 | |
---|
[2282] | 1012 | |
---|
| 1013 | |
---|
| 1014 | |
---|
[707] | 1015 | % ================================================================ |
---|
| 1016 | % Conservation Properties on Tracers |
---|
| 1017 | % ================================================================ |
---|
[9393] | 1018 | \section{Conservation properties on tracers} |
---|
[9414] | 1019 | \label{sec:C.5} |
---|
[707] | 1020 | |
---|
| 1021 | |
---|
[1223] | 1022 | All the numerical schemes used in NEMO are written such that the tracer content |
---|
| 1023 | is conserved by the internal dynamics and physics (equations in flux form). |
---|
| 1024 | For advection, only the CEN2 scheme ($i.e.$ $2^{nd}$ order finite different scheme) |
---|
| 1025 | conserves the global variance of tracer. Nevertheless the other schemes ensure |
---|
| 1026 | that the global variance decreases ($i.e.$ they are at least slightly diffusive). |
---|
| 1027 | For diffusion, all the schemes ensure the decrease of the total tracer variance, |
---|
| 1028 | except the iso-neutral operator. There is generally no strict conservation of mass, |
---|
| 1029 | as the equation of state is non linear with respect to $T$ and $S$. In practice, |
---|
| 1030 | the mass is conserved to a very high accuracy. |
---|
[707] | 1031 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1032 | % Advection Term |
---|
| 1033 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 1034 | \subsection{Advection term} |
---|
[9414] | 1035 | \label{subsec:C.5.1} |
---|
[707] | 1036 | |
---|
[2282] | 1037 | conservation of a tracer, $T$: |
---|
| 1038 | \begin{equation*} |
---|
| 1039 | \frac{\partial }{\partial t} \left( \int_D {T\;dv} \right) |
---|
| 1040 | = \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=0 |
---|
| 1041 | \end{equation*} |
---|
| 1042 | |
---|
| 1043 | conservation of its variance: |
---|
| 1044 | \begin{flalign*} |
---|
| 1045 | \frac{\partial }{\partial t} \left( \int_D {\frac{1}{2} T^2\;dv} \right) |
---|
| 1046 | =& \int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv } |
---|
| 1047 | - \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv } |
---|
| 1048 | \end{flalign*} |
---|
| 1049 | |
---|
| 1050 | |
---|
[1223] | 1051 | Whatever the advection scheme considered it conserves of the tracer content as all |
---|
[2282] | 1052 | the scheme are written in flux form. Indeed, let $T$ be the tracer and $\tau_u$, $\tau_v$, |
---|
| 1053 | and $\tau_w$ its interpolated values at velocity point (whatever the interpolation is), |
---|
| 1054 | the conservation of the tracer content due to the advection tendency is obtained as follows: |
---|
[707] | 1055 | \begin{flalign*} |
---|
[2282] | 1056 | &\int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv } = - \int_D \nabla \cdot \left( T \textbf{U} \right)\;dv &&&\\ |
---|
| 1057 | &\equiv - \sum\limits_{i,j,k} \biggl\{ |
---|
| 1058 | \frac{1} {b_t} \left( \delta_i \left[ U \;\tau_u \right] |
---|
| 1059 | + \delta_j \left[ V \;\tau_v \right] \right) |
---|
| 1060 | + \frac{1} {e_{3t}} \delta_k \left[ w\;\tau_w \right] \biggl\} b_t &&&\\ |
---|
[817] | 1061 | % |
---|
[2282] | 1062 | &\equiv - \sum\limits_{i,j,k} \left\{ |
---|
| 1063 | \delta_i \left[ U \;\tau_u \right] |
---|
| 1064 | + \delta_j \left[ V \;\tau_v \right] |
---|
| 1065 | + \delta_k \left[ W \;\tau_w \right] \right\} && \\ |
---|
[817] | 1066 | &\equiv 0 &&& |
---|
[707] | 1067 | \end{flalign*} |
---|
| 1068 | |
---|
[2282] | 1069 | The conservation of the variance of tracer due to the advection tendency |
---|
| 1070 | can be achieved only with the CEN2 scheme, $i.e.$ when |
---|
| 1071 | $\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$. |
---|
[1223] | 1072 | It can be demonstarted as follows: |
---|
[707] | 1073 | \begin{flalign*} |
---|
[2282] | 1074 | &\int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv } |
---|
| 1075 | = - \int\limits_D \tau\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\ |
---|
| 1076 | \equiv& - \sum\limits_{i,j,k} T\; |
---|
[707] | 1077 | \left\{ |
---|
[2282] | 1078 | \delta_i \left[ U \overline T^{\,i+1/2} \right] |
---|
| 1079 | + \delta_j \left[ V \overline T^{\,j+1/2} \right] |
---|
| 1080 | + \delta_k \left[ W \overline T^{\,k+1/2} \right] \right\} && \\ |
---|
| 1081 | \equiv& + \sum\limits_{i,j,k} |
---|
| 1082 | \left\{ U \overline T^{\,i+1/2} \,\delta_{i+1/2} \left[ T \right] |
---|
| 1083 | + V \overline T^{\,j+1/2} \;\delta_{j+1/2} \left[ T \right] |
---|
| 1084 | + W \overline T^{\,k+1/2}\;\delta_{k+1/2} \left[ T \right] \right\} &&\\ |
---|
| 1085 | \equiv& + \frac{1} {2} \sum\limits_{i,j,k} |
---|
| 1086 | \Bigl\{ U \;\delta_{i+1/2} \left[ T^2 \right] |
---|
| 1087 | + V \;\delta_{j+1/2} \left[ T^2 \right] |
---|
| 1088 | + W \;\delta_{k+1/2} \left[ T^2 \right] \Bigr\} && \\ |
---|
| 1089 | \equiv& - \frac{1} {2} \sum\limits_{i,j,k} T^2 |
---|
| 1090 | \Bigl\{ \delta_i \left[ U \right] |
---|
| 1091 | + \delta_j \left[ V \right] |
---|
| 1092 | + \delta_k \left[ W \right] \Bigr\} &&& \\ |
---|
| 1093 | \equiv& + \frac{1} {2} \sum\limits_{i,j,k} T^2 |
---|
| 1094 | \Bigl\{ \frac{1}{e_{3t}} \frac{\partial e_{3t}\,T }{\partial t} \Bigr\} &&& \\ |
---|
[707] | 1095 | \end{flalign*} |
---|
[2282] | 1096 | which is the discrete form of $ \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv }$. |
---|
[707] | 1097 | |
---|
| 1098 | % ================================================================ |
---|
| 1099 | % Conservation Properties on Lateral Momentum Physics |
---|
| 1100 | % ================================================================ |
---|
[9393] | 1101 | \section{Conservation properties on lateral momentum physics} |
---|
[9407] | 1102 | \label{sec:dynldf_properties} |
---|
[707] | 1103 | |
---|
| 1104 | |
---|
| 1105 | The discrete formulation of the horizontal diffusion of momentum ensures the |
---|
[1223] | 1106 | conservation of potential vorticity and the horizontal divergence, and the |
---|
[6289] | 1107 | dissipation of the square of these quantities ($i.e.$ enstrophy and the |
---|
[707] | 1108 | variance of the horizontal divergence) as well as the dissipation of the |
---|
| 1109 | horizontal kinetic energy. In particular, when the eddy coefficients are |
---|
| 1110 | horizontally uniform, it ensures a complete separation of vorticity and |
---|
| 1111 | horizontal divergence fields, so that diffusion (dissipation) of vorticity |
---|
| 1112 | (enstrophy) does not generate horizontal divergence (variance of the |
---|
| 1113 | horizontal divergence) and \textit{vice versa}. |
---|
| 1114 | |
---|
[1223] | 1115 | These properties of the horizontal diffusion operator are a direct consequence |
---|
[9407] | 1116 | of properties \autoref{eq:DOM_curl_grad} and \autoref{eq:DOM_div_curl}. |
---|
[1223] | 1117 | When the vertical curl of the horizontal diffusion of momentum (discrete sense) |
---|
| 1118 | is taken, the term associated with the horizontal gradient of the divergence is |
---|
| 1119 | locally zero. |
---|
[707] | 1120 | |
---|
| 1121 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1122 | % Conservation of Potential Vorticity |
---|
| 1123 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 1124 | \subsection{Conservation of potential vorticity} |
---|
[9414] | 1125 | \label{subsec:C.6.1} |
---|
[707] | 1126 | |
---|
| 1127 | The lateral momentum diffusion term conserves the potential vorticity : |
---|
| 1128 | \begin{flalign*} |
---|
[817] | 1129 | &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times |
---|
[2282] | 1130 | \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right) |
---|
[6289] | 1131 | - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv \\ |
---|
| 1132 | %\end{flalign*} |
---|
[817] | 1133 | %%%%%%%%%% recheck here.... (gm) |
---|
[6289] | 1134 | %\begin{flalign*} |
---|
| 1135 | =& \int \limits_D -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times |
---|
| 1136 | \Bigl[ \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv \\ |
---|
| 1137 | %\end{flalign*} |
---|
| 1138 | %\begin{flalign*} |
---|
[707] | 1139 | \equiv& \sum\limits_{i,j} |
---|
| 1140 | \left\{ |
---|
[6289] | 1141 | \delta_{i+1/2} \left[ \frac {e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] \right] |
---|
| 1142 | + \delta_{j+1/2} \left[ \frac {e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] \right] |
---|
| 1143 | \right\} \\ |
---|
[817] | 1144 | % |
---|
[9407] | 1145 | \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} |
---|
[817] | 1146 | % |
---|
[707] | 1147 | \equiv& \sum\limits_{i,j,k} |
---|
| 1148 | -\,\left\{ |
---|
[6289] | 1149 | \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_i \left[ 1\right] |
---|
| 1150 | + \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_j \left[ 1\right] |
---|
[707] | 1151 | \right\} \quad \equiv 0 |
---|
[6289] | 1152 | \\ |
---|
[707] | 1153 | \end{flalign*} |
---|
| 1154 | |
---|
| 1155 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1156 | % Dissipation of Horizontal Kinetic Energy |
---|
| 1157 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 1158 | \subsection{Dissipation of horizontal kinetic energy} |
---|
[9414] | 1159 | \label{subsec:C.6.2} |
---|
[707] | 1160 | |
---|
[817] | 1161 | The lateral momentum diffusion term dissipates the horizontal kinetic energy: |
---|
| 1162 | %\begin{flalign*} |
---|
| 1163 | \begin{equation*} |
---|
| 1164 | \begin{split} |
---|
[707] | 1165 | \int_D \textbf{U}_h \cdot |
---|
[817] | 1166 | \left[ \nabla_h \right. & \left. \left( A^{\,lm}\;\chi \right) |
---|
| 1167 | - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right] \; dv \\ |
---|
| 1168 | \\ %%% |
---|
| 1169 | \equiv& \sum\limits_{i,j,k} |
---|
| 1170 | \left\{ |
---|
| 1171 | \frac{1} {e_{1u}} \delta_{i+1/2} \left[ A_T^{\,lm} \chi \right] |
---|
| 1172 | - \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 1173 | \right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u \\ |
---|
| 1174 | &\;\; + \left\{ |
---|
| 1175 | \frac{1} {e_{2u}} \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right] |
---|
| 1176 | + \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 1177 | \right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v \qquad \\ |
---|
| 1178 | \\ %%% |
---|
| 1179 | \equiv& \sum\limits_{i,j,k} |
---|
| 1180 | \Bigl\{ |
---|
| 1181 | e_{2u}\,e_{3u} \;u\; \delta_{i+1/2} \left[ A_T^{\,lm} \chi \right] |
---|
| 1182 | - e_{1u} \;u\; \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 1183 | \Bigl\} |
---|
| 1184 | \\ |
---|
| 1185 | &\;\; + \Bigl\{ |
---|
| 1186 | e_{1v}\,e_{3v} \;v\; \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right] |
---|
| 1187 | + e_{2v} \;v\; \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 1188 | \Bigl\} \\ |
---|
| 1189 | \\ %%% |
---|
| 1190 | \equiv& \sum\limits_{i,j,k} |
---|
| 1191 | - \Bigl( |
---|
| 1192 | \delta_i \left[ e_{2u}\,e_{3u} \;u \right] |
---|
| 1193 | + \delta_j \left[ e_{1v}\,e_{3v} \;v \right] |
---|
| 1194 | \Bigr) \; A_T^{\,lm} \chi \\ |
---|
| 1195 | &\;\; - \Bigl( |
---|
| 1196 | \delta_{i+1/2} \left[ e_{2v} \;v \right] |
---|
| 1197 | - \delta_{j+1/2} \left[ e_{1u} \;u \right] |
---|
| 1198 | \Bigr)\; A_f^{\,lm} e_{3f} \zeta \\ |
---|
| 1199 | \\ %%% |
---|
| 1200 | \equiv& \sum\limits_{i,j,k} |
---|
[2282] | 1201 | - A_T^{\,lm} \,\chi^2 \;e_{1t}\,e_{2t}\,e_{3t} |
---|
[817] | 1202 | - A_f ^{\,lm} \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f} |
---|
| 1203 | \quad \leq 0 \\ |
---|
| 1204 | \end{split} |
---|
| 1205 | \end{equation*} |
---|
[707] | 1206 | |
---|
| 1207 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1208 | % Dissipation of Enstrophy |
---|
| 1209 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 1210 | \subsection{Dissipation of enstrophy} |
---|
[9414] | 1211 | \label{subsec:C.6.3} |
---|
[707] | 1212 | |
---|
| 1213 | The lateral momentum diffusion term dissipates the enstrophy when the eddy |
---|
| 1214 | coefficients are horizontally uniform: |
---|
| 1215 | \begin{flalign*} |
---|
[817] | 1216 | &\int\limits_D \zeta \; \textbf{k} \cdot \nabla \times |
---|
[2282] | 1217 | \left[ \nabla_h \left( A^{\,lm}\;\chi \right) |
---|
| 1218 | - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \right]\;dv &&&\\ |
---|
[6289] | 1219 | &\quad = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times |
---|
[2282] | 1220 | \left[ \nabla_h \times \left( \zeta \; \textbf{k} \right) \right]\;dv &&&\\ |
---|
[6289] | 1221 | &\quad \equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f} |
---|
[2282] | 1222 | \left\{ \delta_{i+1/2} \left[ \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right] |
---|
| 1223 | + \delta_{j+1/2} \left[ \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right] \right\} &&&\\ |
---|
[817] | 1224 | % |
---|
[9407] | 1225 | \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} |
---|
[817] | 1226 | % |
---|
[6289] | 1227 | &\quad \equiv - A^{\,lm} \sum\limits_{i,j,k} |
---|
[2282] | 1228 | \left\{ \left( \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right)^2 b_v |
---|
[6289] | 1229 | + \left( \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right)^2 b_u \right\} \quad \leq \;0 &&&\\ |
---|
[707] | 1230 | \end{flalign*} |
---|
| 1231 | |
---|
| 1232 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1233 | % Conservation of Horizontal Divergence |
---|
| 1234 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 1235 | \subsection{Conservation of horizontal divergence} |
---|
[9414] | 1236 | \label{subsec:C.6.4} |
---|
[707] | 1237 | |
---|
| 1238 | When the horizontal divergence of the horizontal diffusion of momentum |
---|
[1223] | 1239 | (discrete sense) is taken, the term associated with the vertical curl of the |
---|
[9407] | 1240 | vorticity is zero locally, due to \autoref{eq:DOM_div_curl}. |
---|
[6289] | 1241 | The resulting term conserves the $\chi$ and dissipates $\chi^2$ |
---|
| 1242 | when the eddy coefficients are horizontally uniform. |
---|
[707] | 1243 | \begin{flalign*} |
---|
[817] | 1244 | & \int\limits_D \nabla_h \cdot |
---|
[2282] | 1245 | \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right) |
---|
| 1246 | - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \Bigr] dv |
---|
[6289] | 1247 | = \int\limits_D \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi \right) dv \\ |
---|
[2282] | 1248 | % |
---|
[817] | 1249 | &\equiv \sum\limits_{i,j,k} |
---|
[2282] | 1250 | \left\{ \delta_i \left[ A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] |
---|
[6289] | 1251 | + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] \right\} \\ |
---|
[817] | 1252 | % |
---|
[9407] | 1253 | \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} |
---|
[817] | 1254 | % |
---|
| 1255 | &\equiv \sum\limits_{i,j,k} |
---|
[2282] | 1256 | - \left\{ \frac{e_{2u}\,e_{3u}} {e_{1u}} A_u^{\,lm} \delta_{i+1/2} \left[ \chi \right] \delta_{i+1/2} \left[ 1 \right] |
---|
[6289] | 1257 | + \frac{e_{1v}\,e_{3v}} {e_{2v}} A_v^{\,lm} \delta_{j+1/2} \left[ \chi \right] \delta_{j+1/2} \left[ 1 \right] \right\} |
---|
| 1258 | \quad \equiv 0 \\ |
---|
[707] | 1259 | \end{flalign*} |
---|
| 1260 | |
---|
| 1261 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1262 | % Dissipation of Horizontal Divergence Variance |
---|
| 1263 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 1264 | \subsection{Dissipation of horizontal divergence variance} |
---|
[9414] | 1265 | \label{subsec:C.6.5} |
---|
[707] | 1266 | |
---|
| 1267 | \begin{flalign*} |
---|
[817] | 1268 | &\int\limits_D \chi \;\nabla_h \cdot |
---|
| 1269 | \left[ \nabla_h \left( A^{\,lm}\;\chi \right) |
---|
| 1270 | - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right]\; dv |
---|
[6289] | 1271 | = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\; dv \\ |
---|
[817] | 1272 | % |
---|
[2282] | 1273 | &\equiv A^{\,lm} \sum\limits_{i,j,k} \frac{1} {e_{1t}\,e_{2t}\,e_{3t}} \chi |
---|
[707] | 1274 | \left\{ |
---|
[817] | 1275 | \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] |
---|
| 1276 | + \delta_j \left[ \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] |
---|
[6289] | 1277 | \right\} \; e_{1t}\,e_{2t}\,e_{3t} \\ |
---|
[817] | 1278 | % |
---|
[9407] | 1279 | \intertext{Using \autoref{eq:DOM_di_adj}, it turns out to be:} |
---|
[817] | 1280 | % |
---|
| 1281 | &\equiv - A^{\,lm} \sum\limits_{i,j,k} |
---|
[2282] | 1282 | \left\{ \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right)^2 b_u |
---|
[6289] | 1283 | + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 b_v \right\} |
---|
| 1284 | \quad \leq 0 \\ |
---|
[707] | 1285 | \end{flalign*} |
---|
| 1286 | |
---|
| 1287 | % ================================================================ |
---|
| 1288 | % Conservation Properties on Vertical Momentum Physics |
---|
| 1289 | % ================================================================ |
---|
[9393] | 1290 | \section{Conservation properties on vertical momentum physics} |
---|
[9414] | 1291 | \label{sec:C.7} |
---|
[707] | 1292 | |
---|
[1223] | 1293 | As for the lateral momentum physics, the continuous form of the vertical diffusion |
---|
| 1294 | of momentum satisfies several integral constraints. The first two are associated |
---|
| 1295 | with the conservation of momentum and the dissipation of horizontal kinetic energy: |
---|
[817] | 1296 | \begin{align*} |
---|
[2282] | 1297 | \int\limits_D \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1298 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv |
---|
| 1299 | \qquad \quad &= \vec{\textbf{0}} \\ |
---|
[817] | 1300 | % |
---|
| 1301 | \intertext{and} |
---|
| 1302 | % |
---|
[707] | 1303 | \int\limits_D |
---|
[2282] | 1304 | \textbf{U}_h \cdot \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1305 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv \quad &\leq 0 \\ |
---|
[817] | 1306 | \end{align*} |
---|
[6289] | 1307 | |
---|
[707] | 1308 | The first property is obvious. The second results from: |
---|
| 1309 | \begin{flalign*} |
---|
| 1310 | \int\limits_D |
---|
[2282] | 1311 | \textbf{U}_h \cdot \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1312 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\;dv &&&\\ |
---|
[707] | 1313 | \end{flalign*} |
---|
| 1314 | \begin{flalign*} |
---|
[817] | 1315 | &\equiv \sum\limits_{i,j,k} |
---|
[707] | 1316 | \left( |
---|
[2282] | 1317 | u\; \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ u \right] \right]\; e_{1u}\,e_{2u} |
---|
| 1318 | + v\; \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right]\; e_{1v}\,e_{2v} \right) &&&\\ |
---|
[817] | 1319 | % |
---|
[1223] | 1320 | \intertext{since the horizontal scale factor does not depend on $k$, it follows:} |
---|
[817] | 1321 | % |
---|
| 1322 | &\equiv - \sum\limits_{i,j,k} |
---|
[2282] | 1323 | \left( \frac{A_u^{\,vm}} {e_{3uw}} \left( \delta_{k+1/2} \left[ u \right] \right)^2\; e_{1u}\,e_{2u} |
---|
| 1324 | + \frac{A_v^{\,vm}} {e_{3vw}} \left( \delta_{k+1/2} \left[ v \right] \right)^2\; e_{1v}\,e_{2v} \right) |
---|
| 1325 | \quad \leq 0 &&&\\ |
---|
[707] | 1326 | \end{flalign*} |
---|
[817] | 1327 | |
---|
[707] | 1328 | The vorticity is also conserved. Indeed: |
---|
| 1329 | \begin{flalign*} |
---|
| 1330 | \int \limits_D |
---|
| 1331 | \frac{1} {e_3 } \textbf{k} \cdot \nabla \times |
---|
[2282] | 1332 | \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} \left( |
---|
| 1333 | \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k} |
---|
| 1334 | \right) \right)\; dv &&&\\ |
---|
[707] | 1335 | \end{flalign*} |
---|
| 1336 | \begin{flalign*} |
---|
| 1337 | \equiv \sum\limits_{i,j,k} \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}} |
---|
| 1338 | \bigg\{ \biggr. \quad |
---|
| 1339 | \delta_{i+1/2} |
---|
[2282] | 1340 | &\left( \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right] \right) &&\\ |
---|
[707] | 1341 | \biggl. |
---|
| 1342 | - \delta_{j+1/2} |
---|
[2282] | 1343 | &\left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{1} {e_{3uw}}\delta_{k+1/2} \left[ u \right] \right] \right) |
---|
[707] | 1344 | \biggr\} \; |
---|
[2282] | 1345 | e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0 && \\ |
---|
[707] | 1346 | \end{flalign*} |
---|
[6289] | 1347 | |
---|
[707] | 1348 | If the vertical diffusion coefficient is uniform over the whole domain, the |
---|
[1223] | 1349 | enstrophy is dissipated, $i.e.$ |
---|
[707] | 1350 | \begin{flalign*} |
---|
| 1351 | \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times |
---|
[2282] | 1352 | \left( \frac{1} {e_3}\; \frac{\partial } {\partial k} |
---|
| 1353 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\ |
---|
[707] | 1354 | \end{flalign*} |
---|
[6289] | 1355 | |
---|
[707] | 1356 | This property is only satisfied in $z$-coordinates: |
---|
| 1357 | \begin{flalign*} |
---|
| 1358 | \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times |
---|
[2282] | 1359 | \left( \frac{1} {e_3}\; \frac{\partial } {\partial k} |
---|
| 1360 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv &&& \\ |
---|
[707] | 1361 | \end{flalign*} |
---|
| 1362 | \begin{flalign*} |
---|
| 1363 | \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \; |
---|
| 1364 | \biggl\{ \biggr. \quad |
---|
| 1365 | \delta_{i+1/2} |
---|
[2282] | 1366 | &\left( \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}[v] \right] \right) &&\\ |
---|
[707] | 1367 | - \delta_{j+1/2} |
---|
| 1368 | &\biggl. |
---|
[2282] | 1369 | \left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) \biggr\} &&\\ |
---|
[707] | 1370 | \end{flalign*} |
---|
| 1371 | \begin{flalign*} |
---|
| 1372 | \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} |
---|
| 1373 | \biggl\{ \biggr. \quad |
---|
| 1374 | \frac{1} {e_{3v}} \delta_k |
---|
[2282] | 1375 | &\left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ \delta_{i+1/2} \left[ e_{2v}\,v \right] \right] \right] &&\\ |
---|
[707] | 1376 | \biggl. |
---|
| 1377 | - \frac{1} {e_{3u}} \delta_k |
---|
[2282] | 1378 | &\left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ \delta_{j+1/2} \left[ e_{1u}\,u \right] \right] \right] \biggr\} &&\\ |
---|
[707] | 1379 | \end{flalign*} |
---|
[1223] | 1380 | Using the fact that the vertical diffusion coefficients are uniform, and that in |
---|
| 1381 | $z$-coordinate, the vertical scale factors do not depend on $i$ and $j$ so |
---|
[2282] | 1382 | that: $e_{3f} =e_{3u} =e_{3v} =e_{3t} $ and $e_{3w} =e_{3uw} =e_{3vw} $, |
---|
[1223] | 1383 | it follows: |
---|
[707] | 1384 | \begin{flalign*} |
---|
| 1385 | \equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k |
---|
[2282] | 1386 | \left[ \frac{1} {e_{3w}} \delta_{k+1/2} \Bigl[ \delta_{i+1/2} \left[ e_{2v}\,v \right] |
---|
| 1387 | - \delta_{j+1/ 2} \left[ e_{1u}\,u \right] \Bigr] \right] &&&\\ |
---|
[707] | 1388 | \end{flalign*} |
---|
| 1389 | \begin{flalign*} |
---|
| 1390 | \equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}} |
---|
[2282] | 1391 | \left( \delta_{k+1/2} \left[ \zeta \right] \right)^2 \; e_{1f}\,e_{2f} \; \leq 0 &&&\\ |
---|
[707] | 1392 | \end{flalign*} |
---|
| 1393 | Similarly, the horizontal divergence is obviously conserved: |
---|
| 1394 | |
---|
| 1395 | \begin{flalign*} |
---|
| 1396 | \int\limits_D \nabla \cdot |
---|
[2282] | 1397 | \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1398 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\ |
---|
[707] | 1399 | \end{flalign*} |
---|
[1223] | 1400 | and the square of the horizontal divergence decreases ($i.e.$ the horizontal |
---|
| 1401 | divergence is dissipated) if the vertical diffusion coefficient is uniform over the |
---|
| 1402 | whole domain: |
---|
[707] | 1403 | |
---|
| 1404 | \begin{flalign*} |
---|
| 1405 | \int\limits_D \chi \;\nabla \cdot |
---|
[2282] | 1406 | \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1407 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\ |
---|
[707] | 1408 | \end{flalign*} |
---|
[1223] | 1409 | This property is only satisfied in the $z$-coordinate: |
---|
[707] | 1410 | \begin{flalign*} |
---|
| 1411 | \int\limits_D \chi \;\nabla \cdot |
---|
[2282] | 1412 | \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1413 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv &&&\\ |
---|
[707] | 1414 | \end{flalign*} |
---|
| 1415 | \begin{flalign*} |
---|
[2282] | 1416 | \equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1t}\,e_{2t}} |
---|
[707] | 1417 | \biggl\{ \Biggr. \quad |
---|
| 1418 | \delta_{i+1/2} |
---|
[2282] | 1419 | &\left( \frac{e_{2u}} {e_{3u}} \delta_k |
---|
| 1420 | \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) &&\\ |
---|
[707] | 1421 | \Biggl. |
---|
| 1422 | + \delta_{j+1/2} |
---|
[2282] | 1423 | &\left( \frac{e_{1v}} {e_{3v}} \delta_k |
---|
| 1424 | \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} [v] \right] \right) |
---|
| 1425 | \Biggr\} \; e_{1t}\,e_{2t}\,e_{3t} &&\\ |
---|
[707] | 1426 | \end{flalign*} |
---|
| 1427 | |
---|
| 1428 | \begin{flalign*} |
---|
| 1429 | \equiv A^{\,vm} \sum\limits_{i,j,k} \chi \, |
---|
| 1430 | \biggl\{ \biggr. \quad |
---|
| 1431 | \delta_{i+1/2} |
---|
| 1432 | &\left( |
---|
[2282] | 1433 | \delta_k \left[ |
---|
| 1434 | \frac{1} {e_{3uw}} \delta_{k+1/2} \left[ e_{2u}\,u \right] \right] \right) && \\ |
---|
[707] | 1435 | \biggl. |
---|
| 1436 | + \delta_{j+1/2} |
---|
[2282] | 1437 | &\left( \delta_k \left[ |
---|
| 1438 | \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ e_{1v}\,v \right] \right] \right) \biggr\} && \\ |
---|
[707] | 1439 | \end{flalign*} |
---|
| 1440 | |
---|
| 1441 | \begin{flalign*} |
---|
| 1442 | \equiv -A^{\,vm} \sum\limits_{i,j,k} |
---|
[2282] | 1443 | \frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\; \biggl\{ |
---|
| 1444 | \delta_{k+1/2} \Bigl[ |
---|
| 1445 | \delta_{i+1/2} \left[ e_{2u}\,u \right] |
---|
| 1446 | + \delta_{j+1/2} \left[ e_{1v}\,v \right] \Bigr] \biggr\} &&&\\ |
---|
[707] | 1447 | \end{flalign*} |
---|
| 1448 | |
---|
| 1449 | \begin{flalign*} |
---|
| 1450 | \equiv -A^{\,vm} \sum\limits_{i,j,k} |
---|
[2282] | 1451 | \frac{1} {e_{3w}} \delta_{k+1/2} \left[ \chi \right]\; \delta_{k+1/2} \left[ e_{1t}\,e_{2t} \;\chi \right] &&&\\ |
---|
[707] | 1452 | \end{flalign*} |
---|
| 1453 | |
---|
| 1454 | \begin{flalign*} |
---|
| 1455 | \equiv -A^{\,vm} \sum\limits_{i,j,k} |
---|
[2282] | 1456 | \frac{e_{1t}\,e_{2t}} {e_{3w}}\; \left( \delta_{k+1/2} \left[ \chi \right] \right)^2 \quad \equiv 0 &&&\\ |
---|
[707] | 1457 | \end{flalign*} |
---|
| 1458 | |
---|
| 1459 | % ================================================================ |
---|
| 1460 | % Conservation Properties on Tracer Physics |
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| 1461 | % ================================================================ |
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[9393] | 1462 | \section{Conservation properties on tracer physics} |
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[9414] | 1463 | \label{sec:C.8} |
---|
[707] | 1464 | |
---|
[1223] | 1465 | The numerical schemes used for tracer subgridscale physics are written such |
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[6289] | 1466 | that the heat and salt contents are conserved (equations in flux form). |
---|
| 1467 | Since a flux form is used to compute the temperature and salinity, |
---|
| 1468 | the quadratic form of these quantities ($i.e.$ their variance) globally tends to diminish. |
---|
| 1469 | As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear. |
---|
[707] | 1470 | |
---|
| 1471 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1472 | % Conservation of Tracers |
---|
| 1473 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 1474 | \subsection{Conservation of tracers} |
---|
[9414] | 1475 | \label{subsec:C.8.1} |
---|
[707] | 1476 | |
---|
| 1477 | constraint of conservation of tracers: |
---|
| 1478 | \begin{flalign*} |
---|
[994] | 1479 | &\int\limits_D \nabla \cdot \left( A\;\nabla T \right)\;dv &&&\\ |
---|
[817] | 1480 | \\ |
---|
| 1481 | &\equiv \sum\limits_{i,j,k} |
---|
[707] | 1482 | \biggl\{ \biggr. |
---|
| 1483 | \delta_i |
---|
| 1484 | \left[ |
---|
| 1485 | A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} |
---|
| 1486 | \left[ T \right] |
---|
| 1487 | \right] |
---|
| 1488 | + \delta_j |
---|
[817] | 1489 | \left[ |
---|
[707] | 1490 | A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} |
---|
| 1491 | \left[ T \right] |
---|
| 1492 | \right] |
---|
[817] | 1493 | &&\\ & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\; |
---|
[707] | 1494 | + \delta_k |
---|
[817] | 1495 | \left[ |
---|
[2282] | 1496 | A_w^{\,vT} \frac{e_{1t}\,e_{2t}} {e_{3t}} \delta_{k+1/2} |
---|
[707] | 1497 | \left[ T \right] |
---|
| 1498 | \right] |
---|
[817] | 1499 | \biggr\} \quad \equiv 0 |
---|
[707] | 1500 | &&\\ |
---|
| 1501 | \end{flalign*} |
---|
| 1502 | |
---|
[1223] | 1503 | In fact, this property simply results from the flux form of the operator. |
---|
[707] | 1504 | |
---|
| 1505 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1506 | % Dissipation of Tracer Variance |
---|
| 1507 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 1508 | \subsection{Dissipation of tracer variance} |
---|
[9414] | 1509 | \label{subsec:C.8.2} |
---|
[707] | 1510 | |
---|
[1223] | 1511 | constraint on the dissipation of tracer variance: |
---|
[707] | 1512 | \begin{flalign*} |
---|
[994] | 1513 | \int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv &&&\\ |
---|
| 1514 | &\equiv \sum\limits_{i,j,k} \; T |
---|
| 1515 | \biggl\{ \biggr. |
---|
| 1516 | \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right] |
---|
| 1517 | & + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right] |
---|
| 1518 | \quad&& \\ |
---|
| 1519 | \biggl. |
---|
[2282] | 1520 | &&+ \delta_k \left[A_w^{\,vT}\frac{e_{1t}\,e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right] |
---|
[994] | 1521 | \biggr\} && |
---|
[707] | 1522 | \end{flalign*} |
---|
| 1523 | \begin{flalign*} |
---|
| 1524 | \equiv - \sum\limits_{i,j,k} |
---|
| 1525 | \biggl\{ \biggr. \quad |
---|
[2282] | 1526 | & A_u^{\,lT} \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ T \right] \right)^2 e_{1u}\,e_{2u}\,e_{3u} && \\ |
---|
| 1527 | & + A_v^{\,lT} \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ T \right] \right)^2 e_{1v}\,e_{2v}\,e_{3v} && \\ \biggl. |
---|
| 1528 | & + A_w^{\,vT} \left( \frac{1} {e_{3w}} \delta_{k+1/2} \left[ T \right] \right)^2 e_{1w}\,e_{2w}\,e_{3w} \biggr\} |
---|
| 1529 | \quad \leq 0 && \\ |
---|
[707] | 1530 | \end{flalign*} |
---|
| 1531 | |
---|
[817] | 1532 | |
---|
| 1533 | %%%% end of appendix in gm comment |
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[994] | 1534 | %} |
---|
[6997] | 1535 | \end{document} |
---|