[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
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| 2 | |
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[6997] | 3 | \begin{document} |
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[11598] | 4 | |
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[817] | 5 | \chapter{Discrete Invariants of the Equations} |
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[11543] | 6 | \label{apdx:INVARIANTS} |
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[10414] | 7 | |
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[11598] | 8 | \thispagestyle{plain} |
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| 9 | |
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[11435] | 10 | \chaptertoc |
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[707] | 11 | |
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[11598] | 12 | \paragraph{Changes record} ~\\ |
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| 13 | |
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| 14 | {\footnotesize |
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| 15 | \begin{tabularx}{\textwidth}{l||X|X} |
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| 16 | Release & Author(s) & Modifications \\ |
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| 17 | \hline |
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| 18 | {\em 4.0} & {\em ...} & {\em ...} \\ |
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| 19 | {\em 3.6} & {\em ...} & {\em ...} \\ |
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| 20 | {\em 3.4} & {\em ...} & {\em ...} \\ |
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| 21 | {\em <=3.4} & {\em ...} & {\em ...} |
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| 22 | \end{tabularx} |
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| 23 | } |
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| 24 | |
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| 25 | \clearpage |
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| 26 | |
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[11693] | 27 | %%% Appendix put in cmtgm as it has not been updated for \zstar and s coordinate |
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[2282] | 28 | %I'm writting this appendix. It will be available in a forthcoming release of the documentation |
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[817] | 29 | |
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[11693] | 30 | %\cmtgm{ |
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[817] | 31 | |
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[11597] | 32 | %% ================================================================================================= |
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[2282] | 33 | \section{Introduction / Notations} |
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[11543] | 34 | \label{sec:INVARIANTS_0} |
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[707] | 35 | |
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[10354] | 36 | Notation used in this appendix in the demonstations: |
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[707] | 37 | |
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[2282] | 38 | fluxes at the faces of a $T$-box: |
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[10406] | 39 | \[ |
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[10414] | 40 | U = e_{2u}\,e_{3u}\; u \qquad V = e_{1v}\,e_{3v}\; v \qquad W = e_{1w}\,e_{2w}\; \omega |
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[10406] | 41 | \] |
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[707] | 42 | |
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[2282] | 43 | volume of cells at $u$-, $v$-, and $T$-points: |
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[10406] | 44 | \[ |
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[10414] | 45 | b_u = e_{1u}\,e_{2u}\,e_{3u} \qquad b_v = e_{1v}\,e_{2v}\,e_{3v} \qquad b_t = e_{1t}\,e_{2t}\,e_{3t} |
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[10406] | 46 | \] |
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[707] | 47 | |
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[2282] | 48 | partial derivative notation: $\partial_\bullet = \frac{\partial}{\partial \bullet}$ |
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[707] | 49 | |
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[2282] | 50 | $dv=e_1\,e_2\,e_3 \,di\,dj\,dk$ is the volume element, with only $e_3$ that depends on time. |
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[10354] | 51 | $D$ and $S$ are the ocean domain volume and surface, respectively. |
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[11435] | 52 | No wetting/drying is allow (\ie\ $\frac{\partial S}{\partial t} = 0$). |
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[10354] | 53 | Let $k_s$ and $k_b$ be the ocean surface and bottom, resp. |
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[11435] | 54 | (\ie\ $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth). |
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[707] | 55 | \begin{flalign*} |
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[10414] | 56 | z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s} e_3(\tilde{k}) \;d\tilde{k} |
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| 57 | = \eta - \int\limits_k^{k_s} e_3 \;d\tilde{k} |
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[707] | 58 | \end{flalign*} |
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[2282] | 59 | |
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| 60 | Continuity equation with the above notation: |
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[10406] | 61 | \[ |
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[10414] | 62 | \frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t} \biggl\{ \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0 |
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[10406] | 63 | \] |
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[2282] | 64 | |
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| 65 | A quantity, $Q$ is conserved when its domain averaged time change is zero, that is when: |
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[10406] | 66 | \[ |
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[10414] | 67 | \partial_t \left( \int_D{ Q\;dv } \right) =0 |
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[10406] | 68 | \] |
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[2282] | 69 | Noting that the coordinate system used .... blah blah |
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[10406] | 70 | \[ |
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[10414] | 71 | \partial_t \left( \int_D {Q\;dv} \right) = \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } |
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| 72 | = \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0 |
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[10406] | 73 | \] |
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[10354] | 74 | equation of evolution of $Q$ written as |
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| 75 | the time evolution of the vertical content of $Q$ like for tracers, or momentum in flux form, |
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| 76 | the quadratic quantity $\frac{1}{2}Q^2$ is conserved when: |
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[707] | 77 | \begin{flalign*} |
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[10414] | 78 | \partial_t \left( \int_D{ \frac{1}{2} \,Q^2\;dv } \right) |
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| 79 | =& \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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| 80 | =& \int_D { Q \;\partial_t\left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } |
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| 81 | - \int_D { \frac{1}{2} Q^2 \,\partial_t (e_3) \;e_1e_2\;di\,dj\,dk } \\ |
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[707] | 82 | \end{flalign*} |
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[2282] | 83 | that is in a more compact form : |
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[10414] | 84 | \begin{flalign} |
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[11543] | 85 | \label{eq:INVARIANTS_Q2_flux} |
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[10414] | 86 | \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) |
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| 87 | =& \int_D { \frac{Q}{e_3} \partial_t \left( e_3 \, Q \right) dv } |
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[2282] | 88 | - \frac{1}{2} \int_D { \frac{Q^2}{e_3} \partial_t (e_3) \;dv } |
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| 89 | \end{flalign} |
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[10354] | 90 | equation of evolution of $Q$ written as the time evolution of $Q$ like for momentum in vector invariant form, |
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| 91 | the quadratic quantity $\frac{1}{2}Q^2$ is conserved when: |
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[707] | 92 | \begin{flalign*} |
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[10414] | 93 | \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) |
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| 94 | =& \int_D { \frac{1}{2} \partial_t \left( e_3 \, Q^2 \right) \;e_1e_2\;di\,dj\,dk } \\ |
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| 95 | =& \int_D { Q \partial_t Q \;e_1e_2e_3\;di\,dj\,dk } |
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| 96 | + \int_D { \frac{1}{2} Q^2 \, \partial_t e_3 \;e_1e_2\;di\,dj\,dk } \\ |
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[707] | 97 | \end{flalign*} |
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[10354] | 98 | that is in a more compact form: |
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[10414] | 99 | \begin{flalign} |
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[11543] | 100 | \label{eq:INVARIANTS_Q2_vect} |
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[10414] | 101 | \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) |
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| 102 | =& \int_D { Q \,\partial_t Q \;dv } |
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| 103 | + \frac{1}{2} \int_D { \frac{1}{e_3} Q^2 \partial_t e_3 \;dv } |
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[2282] | 104 | \end{flalign} |
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[707] | 105 | |
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[11597] | 106 | %% ================================================================================================= |
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[2282] | 107 | \section{Continuous conservation} |
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[11543] | 108 | \label{sec:INVARIANTS_1} |
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[2282] | 109 | |
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[11435] | 110 | The discretization of pimitive equation in $s$-coordinate (\ie\ time and space varying vertical coordinate) |
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| 111 | must be chosen so that the discrete equation of the model satisfy integral constrains on energy and enstrophy. |
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[2282] | 112 | |
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| 113 | Let us first establish those constraint in the continuous world. |
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[11435] | 114 | The total energy (\ie\ kinetic plus potential energies) is conserved: |
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[10414] | 115 | \begin{flalign} |
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[11543] | 116 | \label{eq:INVARIANTS_Tot_Energy} |
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[2282] | 117 | \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 + \rho \, g \, z \right) \;dv \right) = & 0 |
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| 118 | \end{flalign} |
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[10354] | 119 | under the following assumptions: no dissipation, no forcing (wind, buoyancy flux, atmospheric pressure variations), |
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[11435] | 120 | mass conservation, and closed domain. |
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[2282] | 121 | |
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[10354] | 122 | This equation can be transformed to obtain several sub-equalities. |
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| 123 | The transformation for the advection term depends on whether the vector invariant form or |
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| 124 | the flux form is used for the momentum equation. |
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[11543] | 125 | Using \autoref{eq:INVARIANTS_Q2_vect} and introducing \autoref{eq:SCOORD_dyn_vect} in |
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| 126 | \autoref{eq:INVARIANTS_Tot_Energy} for the former form and |
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| 127 | using \autoref{eq:INVARIANTS_Q2_flux} and introducing \autoref{eq:SCOORD_dyn_flux} in |
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| 128 | \autoref{eq:INVARIANTS_Tot_Energy} for the latter form leads to: |
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[2282] | 129 | |
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[11543] | 130 | % \label{eq:INVARIANTS_E_tot} |
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[2282] | 131 | advection term (vector invariant form): |
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[10414] | 132 | \[ |
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[11543] | 133 | % \label{eq:INVARIANTS_E_tot_vect_vor_1} |
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[10414] | 134 | \int\limits_D \zeta \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\ |
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| 135 | \] |
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| 136 | \[ |
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[11543] | 137 | % \label{eq:INVARIANTS_E_tot_vect_adv_1} |
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[10414] | 138 | \int\limits_D \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right) dv |
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| 139 | + \int\limits_D \textbf{U}_h \cdot \nabla_z \textbf{U}_h \;dv |
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| 140 | - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 |
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| 141 | \] |
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[2282] | 142 | advection term (flux form): |
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[10414] | 143 | \[ |
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[11543] | 144 | % \label{eq:INVARIANTS_E_tot_flux_metric} |
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[10414] | 145 | \int\limits_D \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1 \right)\; |
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| 146 | \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 |
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| 147 | \] |
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| 148 | \[ |
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[11543] | 149 | % \label{eq:INVARIANTS_E_tot_flux_adv} |
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[10414] | 150 | \int\limits_D \textbf{U}_h \cdot \left( {{ |
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| 151 | \begin{array} {*{20}c} |
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| 152 | \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ |
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| 153 | \nabla \cdot \left( \textbf{U}\,v \right) \hfill |
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| 154 | \end{array}} |
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| 155 | } \right) \;dv |
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| 156 | + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 |
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| 157 | \] |
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[2282] | 158 | coriolis term |
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[10414] | 159 | \[ |
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[11543] | 160 | % \label{eq:INVARIANTS_E_tot_cor} |
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[10414] | 161 | \int\limits_D f \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 |
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| 162 | \] |
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[2282] | 163 | pressure gradient: |
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[10414] | 164 | \[ |
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[11543] | 165 | % \label{eq:INVARIANTS_E_tot_pg_1} |
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[10414] | 166 | - \int\limits_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv |
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| 167 | = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv |
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| 168 | + \int\limits_D g\, \rho \; \partial_t z \;dv |
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| 169 | \] |
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[2282] | 170 | |
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| 171 | where $\nabla_h = \left. \nabla \right|_k$ is the gradient along the $s$-surfaces. |
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| 172 | |
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| 173 | blah blah.... |
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[10414] | 174 | |
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[2282] | 175 | The prognostic ocean dynamics equation can be summarized as follows: |
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[10406] | 176 | \[ |
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[10414] | 177 | \text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } |
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| 178 | {\text{COR} + \text{ADV} } |
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| 179 | + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} |
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[10406] | 180 | \] |
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[707] | 181 | |
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[2282] | 182 | Vector invariant form: |
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[11543] | 183 | % \label{eq:INVARIANTS_E_tot_vect} |
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[10414] | 184 | \[ |
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[11543] | 185 | % \label{eq:INVARIANTS_E_tot_vect_vor_2} |
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[10414] | 186 | \int\limits_D \textbf{U}_h \cdot \text{VOR} \;dv = 0 |
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| 187 | \] |
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| 188 | \[ |
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[11543] | 189 | % \label{eq:INVARIANTS_E_tot_vect_adv_2} |
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[10414] | 190 | \int\limits_D \textbf{U}_h \cdot \text{KEG} \;dv |
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| 191 | + \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv |
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| 192 | - \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 |
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| 193 | \] |
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| 194 | \[ |
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[11543] | 195 | % \label{eq:INVARIANTS_E_tot_pg_2} |
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[10414] | 196 | - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv |
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| 197 | = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv |
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| 198 | + \int\limits_D g\, \rho \; \partial_t z \;dv |
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| 199 | \] |
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[707] | 200 | |
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[2282] | 201 | Flux form: |
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[10414] | 202 | \begin{subequations} |
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[11543] | 203 | \label{eq:INVARIANTS_E_tot_flux} |
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[10414] | 204 | \[ |
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[11543] | 205 | % \label{eq:INVARIANTS_E_tot_flux_metric_2} |
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[10414] | 206 | \int\limits_D \textbf{U}_h \cdot \text {COR} \; dv = 0 |
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| 207 | \] |
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| 208 | \[ |
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[11543] | 209 | % \label{eq:INVARIANTS_E_tot_flux_adv_2} |
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[10414] | 210 | \int\limits_D \textbf{U}_h \cdot \text{ADV} \;dv |
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| 211 | + \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 |
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| 212 | \] |
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| 213 | \begin{equation} |
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[11543] | 214 | \label{eq:INVARIANTS_E_tot_pg_3} |
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[10414] | 215 | - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv |
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| 216 | = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv |
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| 217 | + \int\limits_D g\, \rho \; \partial_t z \;dv |
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| 218 | \end{equation} |
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[2282] | 219 | \end{subequations} |
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[707] | 220 | |
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[11543] | 221 | \autoref{eq:INVARIANTS_E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. |
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| 222 | Indeed the left hand side of \autoref{eq:INVARIANTS_E_tot_pg_3} can be transformed as follows: |
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[2282] | 223 | \begin{flalign*} |
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[10414] | 224 | \partial_t \left( \int\limits_D { \rho \, g \, z \;dv} \right) |
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| 225 | &= + \int\limits_D \frac{1}{e_3} \partial_t (e_3\,\rho) \;g\;z\;\;dv |
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| 226 | + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\ |
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| 227 | &= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv |
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| 228 | + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\ |
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| 229 | &= + \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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| 230 | + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\ |
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| 231 | &= + \int\limits_D \rho \,g \left( \omega + \partial_t z + \textbf {U}_h \cdot \nabla_h z \right) \;dv &&&\\ |
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| 232 | &=+ \int\limits_D g\, \rho \; w \; dv &&&\\ |
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[2282] | 233 | \end{flalign*} |
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[11435] | 234 | where the last equality is obtained by noting that the brackets is exactly the expression of $w$, |
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[11543] | 235 | the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{eq:SCOORD_w_s}). |
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[11435] | 236 | |
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[11543] | 237 | The left hand side of \autoref{eq:INVARIANTS_E_tot_pg_3} can be transformed as follows: |
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[2282] | 238 | \begin{flalign*} |
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[10414] | 239 | - \int\limits_D \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv |
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| 240 | = - \int\limits_D \left( \nabla_h p + \rho \, g \nabla_h z \right) \cdot \textbf{U}_h \;dv &&&\\ |
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| 241 | \allowdisplaybreaks |
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| 242 | &= - \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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| 243 | \allowdisplaybreaks |
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| 244 | &= +\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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| 245 | \allowdisplaybreaks |
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| 246 | &= -\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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| 247 | +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ |
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| 248 | \allowdisplaybreaks |
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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 \frac{1}{e_3} \partial_k p\; \omega \;dv |
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| 251 | +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;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 \, \omega \;dv |
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| 254 | +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\ |
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| 255 | &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \; \;dv |
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| 256 | - \int\limits_D \rho \, g \, w \;dv |
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| 257 | + \int\limits_D \rho \, g \, \partial_t z \;dv &&&\\ |
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| 258 | \allowdisplaybreaks |
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| 259 | \intertext{introducing the hydrostatic balance $\partial_k p=-\rho \,g\,e_3$ in the last term, |
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| 260 | it becomes:} |
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| 261 | &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv |
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| 262 | - \int\limits_D \rho \, g \, w \;dv |
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| 263 | - \int\limits_D \frac{1}{e_3} \partial_k p\, \partial_t z \;dv &&&\\ |
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| 264 | &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv |
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| 265 | - \int\limits_D \rho \, g \, w \;dv |
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| 266 | + \int\limits_D \,\frac{p}{e_3}\partial_t ( \partial_k z ) dv &&&\\ |
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| 267 | % |
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| 268 | &= - \int\limits_D \rho \, g \, w \;dv &&&\\ |
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[2282] | 269 | \end{flalign*} |
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| 270 | |
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| 271 | %gm comment |
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[11693] | 272 | \cmtgm{ |
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[2282] | 273 | The last equality comes from the following equation, |
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| 274 | \begin{flalign*} |
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[10414] | 275 | \int\limits_D p \frac{1}{e_3} \frac{\partial e_3}{\partial t}\; \;dv |
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| 276 | = \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv \quad, |
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[2282] | 277 | \end{flalign*} |
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| 278 | that can be demonstrated as follows: |
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| 279 | |
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| 280 | \begin{flalign*} |
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[10414] | 281 | \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv |
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| 282 | &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv |
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[2282] | 283 | - \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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[10414] | 284 | &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv |
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[2282] | 285 | - \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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[10414] | 286 | % |
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| 287 | \allowdisplaybreaks |
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| 288 | \intertext{The second term of the right hand side can be transformed by applying the integration by part rule: |
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| 289 | $\left[ a\,b \right]_{k_b}^{k_s} = \int_{k_b}^{k_s} a\,\frac{\partial b}{\partial k} \;dk |
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| 290 | + \int_{k_b}^{k_s} \frac{\partial a}{\partial k} \,b \;dk $ |
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| 291 | to the following function: $a= \int_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k}$ |
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| 292 | and $b= \int_k^{k_s} \rho \, e_3 \;d\tilde{k}$ |
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| 293 | (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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| 294 | This leads to: } |
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[2282] | 295 | \end{flalign*} |
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| 296 | \begin{flalign*} |
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[10414] | 297 | &\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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| 298 | =-\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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| 299 | -\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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| 300 | \allowdisplaybreaks |
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| 301 | \intertext{Noting that $\frac{\partial \eta}{\partial t} |
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| 302 | = \frac{\partial}{\partial t} \left( \int_{k_b}^{k_s} e_3 \;d\tilde{k} \right) |
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| 303 | = \int_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k}$ |
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| 304 | and |
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| 305 | $p(k) = \int_k^{k_s} \rho \,g \, e_3 \;d\tilde{k} $, |
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| 306 | but also that $\frac{\partial \eta}{\partial t}$ does not depends on $k$, it comes: |
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| 307 | } |
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| 308 | & - \int\limits_{k_b}^{k_s} \rho \, \frac{\partial \eta}{\partial t} \, e_3 \;dk |
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| 309 | = - \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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| 310 | - \int\limits_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \frac{p}{g} \;dk &&&\\ |
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[2282] | 311 | \end{flalign*} |
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| 312 | Mutliplying by $g$ and integrating over the $(i,j)$ domain it becomes: |
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| 313 | \begin{flalign*} |
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[10414] | 314 | \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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| 315 | = \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} dv |
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[2282] | 316 | - \int\limits_D \frac{p}{e_3}\frac{\partial e_3}{\partial t} \;dv |
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| 317 | \end{flalign*} |
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| 318 | Using this property, we therefore have: |
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| 319 | \begin{flalign*} |
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[10414] | 320 | \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv |
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| 321 | &= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv |
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[2282] | 322 | - \left( \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} dv |
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[10414] | 323 | - \int\limits_D \frac{p}{e_3}\frac{\partial e_3}{\partial t} \;dv \right) &&&\\ |
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| 324 | % |
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| 325 | &=\int\limits_D \frac{p}{e_3} \frac{\partial (e_3\,\rho)}{\partial t}\; \;dv |
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[2282] | 326 | \end{flalign*} |
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| 327 | % end gm comment |
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| 328 | } |
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| 329 | |
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[11597] | 330 | %% ================================================================================================= |
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[9393] | 331 | \section{Discrete total energy conservation: vector invariant form} |
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[11543] | 332 | \label{sec:INVARIANTS_2} |
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[2282] | 333 | |
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[11597] | 334 | %% ================================================================================================= |
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[2282] | 335 | \subsection{Total energy conservation} |
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[11543] | 336 | \label{subsec:INVARIANTS_KE+PE_vect} |
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[2282] | 337 | |
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[11543] | 338 | The discrete form of the total energy conservation, \autoref{eq:INVARIANTS_Tot_Energy}, is given by: |
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[2282] | 339 | \begin{flalign*} |
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[10414] | 340 | \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] | 341 | \end{flalign*} |
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[2282] | 342 | which in vector invariant forms, it leads to: |
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[10414] | 343 | \begin{equation} |
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[11543] | 344 | \label{eq:INVARIANTS_KE+PE_vect_discrete} |
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[10414] | 345 | \begin{split} |
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| 346 | \sum\limits_{i,j,k} \biggl\{ u\, \partial_t u \;b_u |
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| 347 | + v\, \partial_t v \;b_v \biggr\} |
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| 348 | + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{e_{3u}}\partial_t e_{3u} \;b_u |
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| 349 | + \frac{v^2}{e_{3v}}\partial_t e_{3v} \;b_v \biggr\} \\ |
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| 350 | = - \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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| 351 | - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} |
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| 352 | \end{split} |
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| 353 | \end{equation} |
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[707] | 354 | |
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[2282] | 355 | Substituting the discrete expression of the time derivative of the velocity either in vector invariant, |
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[11543] | 356 | leads to the discrete equivalent of the four equations \autoref{eq:INVARIANTS_E_tot_flux}. |
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[2282] | 357 | |
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[11597] | 358 | %% ================================================================================================= |
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[2282] | 359 | \subsection{Vorticity term (coriolis + vorticity part of the advection)} |
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[11543] | 360 | \label{subsec:INVARIANTS_vor} |
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[2282] | 361 | |
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[10354] | 362 | Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$), |
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| 363 | or the planetary ($q=f/e_{3f}$), or the total potential vorticity ($q=(\zeta +f) /e_{3f}$). |
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| 364 | Two discretisation of the vorticity term (ENE and EEN) allows the conservation of the kinetic energy. |
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[11597] | 365 | %% ================================================================================================= |
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[11582] | 366 | \subsubsection{Vorticity term with ENE scheme (\protect\np[=.true.]{ln_dynvor_ene}{ln\_dynvor\_ene})} |
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[11543] | 367 | \label{subsec:INVARIANTS_vorENE} |
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[2282] | 368 | |
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[10354] | 369 | For the ENE scheme, the two components of the vorticity term are given by: |
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[10406] | 370 | \[ |
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[10414] | 371 | - e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h \equiv |
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| 372 | \left( {{ |
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| 373 | \begin{array} {*{20}c} |
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| 374 | + \frac{1} {e_{1u}} \; |
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| 375 | \overline {\, q \ \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} \hfill \\ |
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| 376 | - \frac{1} {e_{2v}} \; |
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| 377 | \overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} \hfill |
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| 378 | \end{array} |
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| 379 | } } \right) |
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[10406] | 380 | \] |
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[707] | 381 | |
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[10354] | 382 | This formulation does not conserve the enstrophy but it does conserve the total kinetic energy. |
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| 383 | Indeed, the kinetic energy tendency associated to the vorticity term and |
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| 384 | averaged over the ocean domain can be transformed as follows: |
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[707] | 385 | \begin{flalign*} |
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[10414] | 386 | &\int\limits_D - \left( e_3 \, q \;\textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv && \\ |
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| 387 | & \qquad \qquad |
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| 388 | { |
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| 389 | \begin{array}{*{20}l} |
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| 390 | &\equiv \sum\limits_{i,j,k} \biggl\{ |
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| 391 | \frac{1} {e_{1u}} \overline { \,q\ \overline{ V }^{\,i+1/2}} ^{\,j} \, u \; b_u |
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| 392 | - \frac{1} {e_{2v}}\overline { \, q\ \overline{ U }^{\,j+1/2}} ^{\,i} \, v \; b_v \; \biggr\} \\ |
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| 393 | &\equiv \sum\limits_{i,j,k} \biggl\{ |
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| 394 | \overline { \,q\ \overline{ V }^{\,i+1/2}}^{\,j} \; U |
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| 395 | - \overline { \,q\ \overline{ U }^{\,j+1/2}}^{\,i} \; V \; \biggr\} \\ |
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| 396 | &\equiv \sum\limits_{i,j,k} q \ \biggl\{ \overline{ V }^{\,i+1/2}\; \overline{ U }^{\,j+1/2} |
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| 397 | - \overline{ U }^{\,j+1/2}\; \overline{ V }^{\,i+1/2} \biggr\} \quad \equiv 0 |
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| 398 | \end{array} |
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[11435] | 399 | } |
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[2282] | 400 | \end{flalign*} |
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| 401 | In other words, the domain averaged kinetic energy does not change due to the vorticity term. |
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| 402 | |
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[11597] | 403 | %% ================================================================================================= |
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[11582] | 404 | \subsubsection{Vorticity term with EEN scheme (\protect\np[=.true.]{ln_dynvor_een}{ln\_dynvor\_een})} |
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[11543] | 405 | \label{subsec:INVARIANTS_vorEEN_vect} |
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[2282] | 406 | |
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[11435] | 407 | With the EEN scheme, the vorticity terms are represented as: |
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[10414] | 408 | \begin{equation} |
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[11558] | 409 | \label{eq:INVARIANTS_dynvor_een1} |
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[10414] | 410 | \left\{ { |
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| 411 | \begin{aligned} |
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| 412 | +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} |
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| 413 | {^{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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| 414 | - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} |
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| 415 | {^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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| 416 | \end{aligned} |
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| 417 | } \right. |
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[11435] | 418 | \end{equation} |
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[10354] | 419 | where the indices $i_p$ and $j_p$ take the following value: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, |
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[11435] | 420 | and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: |
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[10414] | 421 | \begin{equation} |
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[11543] | 422 | \label{eq:INVARIANTS_Q_triads} |
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[10414] | 423 | _i^j \mathbb{Q}^{i_p}_{j_p} |
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| 424 | = \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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[2282] | 425 | \end{equation} |
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| 426 | |
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[10354] | 427 | This formulation does conserve the total kinetic energy. |
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| 428 | Indeed, |
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[2282] | 429 | \begin{flalign*} |
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[10414] | 430 | &\int\limits_D - \textbf{U}_h \cdot \left( \zeta \;\textbf{k} \times \textbf{U}_h \right) \; dv && \\ |
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| 431 | \equiv \sum\limits_{i,j,k} & \biggl\{ |
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| 432 | \left[ \sum_{\substack{i_p,\,k_p}} |
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| 433 | {^{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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| 434 | - \left[ \sum_{\substack{i_p,\,k_p}} |
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| 435 | {^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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| 436 | \equiv \sum\limits_{i,j,k} & \sum_{\substack{i_p,\,k_p}} \biggl\{ \ \ |
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| 437 | {^{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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| 438 | - {^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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| 439 | % |
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| 440 | \allowdisplaybreaks |
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| 441 | \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} |
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| 442 | % |
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| 443 | \equiv \sum\limits_{i,j,k} & \biggl\{ \ \ |
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| 444 | {^{i+1}_j }\mathbb{Q}^{-1/2}_{+1/2} \;V^{i+1}_{j+1/2} \; U^{\,i+1/2}_{j} |
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| 445 | - {^i_{j}\quad}\mathbb{Q}^{-1/2}_{+1/2} \; U^{i-1/2}_{j} \; V^{\,i}_{j+1/2} && \\ |
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| 446 | & + {^{i+1}_j }\mathbb{Q}^{-1/2}_{-1/2} \; V^{i+1}_{j-1/2} \; U^{\,i+1/2}_{j} |
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| 447 | - {^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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| 448 | & + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2} \; U^{\,i+1/2}_{j} |
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| 449 | - {^i_{j}\quad}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j} \; V^{\,i}_{j+1/2} \biggr. && \\ |
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| 450 | & + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{-1/2} \; V^{i}_{j-1/2} \; U^{\,i+1/2}_{j} |
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| 451 | - {^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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| 452 | % |
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| 453 | \allowdisplaybreaks |
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| 454 | \intertext{The summation is done over all $i$ and $j$ indices, it is therefore possible to introduce |
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| 455 | 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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| 456 | first and second lines, second term of the second and fourth lines). By doning so, we can regroup |
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| 457 | 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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| 458 | in the neighbourhood that contain a triad at the same ($i$,$j$) indices. It becomes: } |
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| 459 | \allowdisplaybreaks |
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| 460 | % |
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| 461 | \equiv \sum\limits_{i,j,k} & \biggl\{ \ \ |
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| 462 | {^{i}_j}\mathbb{Q}^{-1/2}_{+1/2} \left[ V^{i}_{j+1/2}\, U^{\,i-1/2}_{j} |
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| 463 | - U^{i-1/2}_{j} \, V^{\,i}_{j+1/2} \right] && \\ |
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| 464 | & + {^{i}_j}\mathbb{Q}^{-1/2}_{-1/2} \left[ V^{i}_{j-1/2} \, U^{\,i-1/2}_{j} |
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| 465 | - U^{i-1/2}_{j} \, V^{\,i}_{j-1/2} \right] \biggr. && \\ |
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| 466 | & + {^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} \left[ V^{i}_{j+1/2} \, U^{\,i+1/2}_{j} |
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| 467 | - U^{i+1/2}_{j} \, V^{\,i}_{j+1/2} \right] \biggr. && \\ |
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| 468 | & + {^{i}_j}\mathbb{Q}^{+1/2}_{-1/2} \left[ V^{i}_{j-1/2} \, U^{\,i+1/2}_{j} |
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| 469 | - U^{i+1/2}_{j-1} \, V^{\,i}_{j-1/2} \right] \ \; \biggr\} \qquad |
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| 470 | \equiv \ 0 && |
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[707] | 471 | \end{flalign*} |
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| 472 | |
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[11597] | 473 | %% ================================================================================================= |
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[9393] | 474 | \subsubsection{Gradient of kinetic energy / Vertical advection} |
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[11543] | 475 | \label{subsec:INVARIANTS_zad} |
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[707] | 476 | |
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[10354] | 477 | The change of Kinetic Energy (KE) due to the vertical advection is exactly balanced by the change of KE due to the horizontal gradient of KE~: |
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[10406] | 478 | \[ |
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[10414] | 479 | \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv |
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| 480 | = - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv |
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| 481 | + \frac{1}{2} \int_D { \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv } |
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[10406] | 482 | \] |
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[11435] | 483 | Indeed, using successively \autoref{eq:DOM_di_adj} (\ie\ the skew symmetry property of the $\delta$ operator) |
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[10354] | 484 | and the continuity equation, then \autoref{eq:DOM_di_adj} again, |
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| 485 | then the commutativity of operators $\overline {\,\cdot \,}$ and $\delta$, and finally \autoref{eq:DOM_mi_adj} |
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[11435] | 486 | (\ie\ the symmetry property of the $\overline {\,\cdot \,}$ operator) |
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[1223] | 487 | applied in the horizontal and vertical directions, it becomes: |
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[707] | 488 | \begin{flalign*} |
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[10414] | 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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| 491 | % |
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| 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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| 495 | % |
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| 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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| 499 | % |
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| 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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| 523 | % |
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| 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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| 529 | % |
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| 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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[11543] | 538 | while the second term corresponds exactly to \autoref{eq:INVARIANTS_KE+PE_vect_discrete}, therefore:} |
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[10414] | 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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[10354] | 545 | There is two main points here. |
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| 546 | First, the satisfaction of this property links the choice of the discrete formulation of the vertical advection and |
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| 547 | of the horizontal gradient of KE. |
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| 548 | Choosing one imposes the other. |
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| 549 | For example KE can also be discretized as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. |
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| 550 | This leads to the following expression for the vertical advection: |
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[10406] | 551 | \[ |
---|
[10414] | 552 | \frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h |
---|
| 553 | \equiv \left( {{ |
---|
| 554 | \begin{array} {*{20}c} |
---|
| 555 | \frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \; \overline{\overline {e_{1t}\,e_{2t} \,\omega\;\delta_{k+1/2} |
---|
| 556 | \left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k} \hfill \\ |
---|
| 557 | \frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \; \overline{\overline {e_{1t}\,e_{2t} \,\omega \;\delta_{k+1/2} |
---|
| 558 | \left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill |
---|
| 559 | \end{array} |
---|
| 560 | } } \right) |
---|
[10406] | 561 | \] |
---|
[11435] | 562 | a formulation that requires an additional horizontal mean in contrast with the one used in \NEMO. |
---|
[10354] | 563 | Nine velocity points have to be used instead of 3. |
---|
[1223] | 564 | This is the reason why it has not been chosen. |
---|
[707] | 565 | |
---|
[10354] | 566 | Second, as soon as the chosen $s$-coordinate depends on time, |
---|
| 567 | an extra constraint arises on the time derivative of the volume at $u$- and $v$-points: |
---|
[2282] | 568 | \begin{flalign*} |
---|
[10414] | 569 | e_{1u}\,e_{2u}\,\partial_t (e_{3u}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,i+1/2} \\ |
---|
| 570 | e_{1v}\,e_{2v}\,\partial_t (e_{3v}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,j+1/2} |
---|
[2282] | 571 | \end{flalign*} |
---|
| 572 | which is (over-)satified by defining the vertical scale factor as follows: |
---|
[10414] | 573 | \begin{flalign*} |
---|
[11543] | 574 | % \label{eq:INVARIANTS_e3u-e3v} |
---|
[10414] | 575 | e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2} \\ |
---|
| 576 | e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2} |
---|
| 577 | \end{flalign*} |
---|
[2282] | 578 | |
---|
| 579 | Blah blah required on the the step representation of bottom topography..... |
---|
| 580 | |
---|
[11597] | 581 | %% ================================================================================================= |
---|
[9393] | 582 | \subsection{Pressure gradient term} |
---|
[11543] | 583 | \label{subsec:INVARIANTS_2.6} |
---|
[2282] | 584 | |
---|
[11693] | 585 | \cmtgm{ |
---|
[10354] | 586 | A pressure gradient has no contribution to the evolution of the vorticity as the curl of a gradient is zero. |
---|
| 587 | In the $z$-coordinate, this property is satisfied locally on a C-grid with 2nd order finite differences |
---|
[11435] | 588 | (property \autoref{eq:DOM_curl_grad}). |
---|
[2282] | 589 | } |
---|
| 590 | |
---|
[10354] | 591 | When the equation of state is linear |
---|
[11435] | 592 | (\ie\ when an advection-diffusion equation for density can be derived from those of temperature and salinity) |
---|
[10354] | 593 | the change of KE due to the work of pressure forces is balanced by |
---|
[11435] | 594 | the change of potential energy due to buoyancy forces: |
---|
[10406] | 595 | \[ |
---|
[10414] | 596 | - \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv |
---|
| 597 | = - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right) \,g\,z \;dv |
---|
[2282] | 598 | + \int_D g\, \rho \; \partial_t (z) \;dv |
---|
[10406] | 599 | \] |
---|
[2282] | 600 | |
---|
[10354] | 601 | This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates. |
---|
| 602 | Indeed, defining the depth of a $T$-point, $z_t$, |
---|
| 603 | as the sum of the vertical scale factors at $w$-points starting from the surface, |
---|
| 604 | the work of pressure forces can be written as: |
---|
[2282] | 605 | \begin{flalign*} |
---|
[10414] | 606 | &- \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv |
---|
| 607 | \equiv \sum\limits_{i,j,k} \biggl\{ \; - \frac{1} {e_{1u}} \Bigl( |
---|
| 608 | \delta_{i+1/2} [p_t] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} [z_t] \Bigr) \; u\;b_u && \\ |
---|
| 609 | & \qquad \qquad \qquad \qquad \qquad \quad \ \, |
---|
| 610 | - \frac{1} {e_{2v}} \Bigl( |
---|
| 611 | \delta_{j+1/2} [p_t] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} [z_t] \Bigr) \; v\;b_v \; \biggr\} && \\ |
---|
| 612 | % |
---|
| 613 | \allowdisplaybreaks |
---|
[11435] | 614 | \intertext{Using successively \autoref{eq:DOM_di_adj}, \ie\ the skew symmetry property of |
---|
[11543] | 615 | the $\delta$ operator, \autoref{eq:DYN_wzv}, the continuity equation, \autoref{eq:DYN_hpg_sco}, |
---|
[10414] | 616 | the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $, |
---|
| 617 | which comes from the definition of $z_t$, it becomes: } |
---|
| 618 | \allowdisplaybreaks |
---|
| 619 | % |
---|
| 620 | \equiv& + \sum\limits_{i,j,k} g \biggl\{ |
---|
| 621 | \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] |
---|
| 622 | + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] |
---|
| 623 | +\Bigl( \delta_i[U] + \delta_j [V] \Bigr)\;\frac{p_t}{g} \biggr\} &&\\ |
---|
| 624 | % |
---|
| 625 | \equiv& + \sum\limits_{i,j,k} g \biggl\{ |
---|
| 626 | \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] |
---|
| 627 | + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] |
---|
| 628 | - \left( \frac{b_t}{e_{3t}} \partial_t (e_{3t}) + \delta_k \left[ W \right] \right) \frac{p_t}{g} \biggr\} &&&\\ |
---|
| 629 | % |
---|
| 630 | \equiv& + \sum\limits_{i,j,k} g \biggl\{ |
---|
| 631 | \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] |
---|
| 632 | + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] |
---|
| 633 | + \frac{W}{g}\;\delta_{k+1/2} [p_t] |
---|
| 634 | - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\ |
---|
| 635 | % |
---|
| 636 | \equiv& + \sum\limits_{i,j,k} g \biggl\{ |
---|
| 637 | \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] |
---|
| 638 | + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] |
---|
| 639 | - W\;e_{3w} \overline \rho^{\,k+1/2} |
---|
| 640 | - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\ |
---|
| 641 | % |
---|
| 642 | \equiv& + \sum\limits_{i,j,k} g \biggl\{ |
---|
| 643 | \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] |
---|
| 644 | + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] |
---|
| 645 | + W\; \overline \rho^{\,k+1/2}\;\delta_{k+1/2} [z_t] |
---|
| 646 | - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\ |
---|
| 647 | % |
---|
| 648 | \allowdisplaybreaks |
---|
| 649 | % |
---|
| 650 | \equiv& - \sum\limits_{i,j,k} g \; z_t \biggl\{ |
---|
| 651 | \delta_i \left[ U\; \overline \rho^{\,i+1/2} \right] |
---|
| 652 | + \delta_j \left[ V\; \overline \rho^{\,j+1/2} \right] |
---|
| 653 | + \delta_k \left[ W\; \overline \rho^{\,k+1/2} \right] \biggr\} |
---|
| 654 | - \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ |
---|
| 655 | % |
---|
| 656 | \equiv& + \sum\limits_{i,j,k} g \; z_t \biggl\{ \partial_t ( e_{3t} \,\rho) \biggr\} \; b_t |
---|
| 657 | - \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ |
---|
| 658 | % |
---|
[2282] | 659 | \end{flalign*} |
---|
[11543] | 660 | The first term is exactly the first term of the right-hand-side of \autoref{eq:INVARIANTS_KE+PE_vect_discrete}. |
---|
[10354] | 661 | It remains to demonstrate that the last term, |
---|
| 662 | which is obviously a discrete analogue of $\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to |
---|
[11543] | 663 | the last term of \autoref{eq:INVARIANTS_KE+PE_vect_discrete}. |
---|
[2282] | 664 | In other words, the following property must be satisfied: |
---|
| 665 | \begin{flalign*} |
---|
[10414] | 666 | \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} |
---|
| 667 | \equiv \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} |
---|
[2282] | 668 | \end{flalign*} |
---|
| 669 | |
---|
[10354] | 670 | Let introduce $p_w$ the pressure at $w$-point such that $\delta_k [p_w] = - \rho \,g\,e_{3t}$. |
---|
[2282] | 671 | The right-hand-side of the above equation can be transformed as follows: |
---|
| 672 | |
---|
| 673 | \begin{flalign*} |
---|
[10414] | 674 | \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\} |
---|
| 675 | &\equiv - \sum\limits_{i,j,k} \biggl\{ \delta_k [p_w]\,\partial_t (z_t) \,e_{1t}\,e_{2t} \biggr\} &&&\\ |
---|
| 676 | % |
---|
| 677 | &\equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \delta_{k+1/2} [\partial_t (z_t)] \,e_{1t}\,e_{2t} \biggr\} |
---|
[2282] | 678 | \equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (e_{3w}) \,e_{1t}\,e_{2t} \biggr\} &&&\\ |
---|
[10414] | 679 | &\equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (b_w) \biggr\} |
---|
| 680 | % |
---|
| 681 | % & \equiv \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w \right) \biggr\} &&&\\ |
---|
| 682 | % & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t \left( \delta_k [z_t] \right) e_{1w}\,e_{2w} \biggr\} &&&\\ |
---|
| 683 | % & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t b_w \biggr\} |
---|
[2282] | 684 | \end{flalign*} |
---|
| 685 | therefore, the balance to be satisfied is: |
---|
| 686 | \begin{flalign*} |
---|
[10414] | 687 | \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\} |
---|
[2282] | 688 | \end{flalign*} |
---|
| 689 | which is a purely vertical balance: |
---|
| 690 | \begin{flalign*} |
---|
[10414] | 691 | \sum\limits_{k} \biggl\{ p_t\;\partial_t (e_{3t}) \biggr\} \equiv \sum\limits_{k} \biggl\{ p_w\, \partial_t (e_{3w}) \biggr\} |
---|
[2282] | 692 | \end{flalign*} |
---|
| 693 | Defining $p_w = \overline{p_t}^{\,k+1/2}$ |
---|
| 694 | |
---|
| 695 | %gm comment |
---|
[11693] | 696 | \cmtgm{ |
---|
[10414] | 697 | \begin{flalign*} |
---|
| 698 | \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\ |
---|
| 699 | % |
---|
| 700 | & \equiv \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w \biggr\} &&&\\ |
---|
| 701 | & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t \left( \delta_{k+1/2} [z_t] \right) e_{1w}\,e_{2w} \biggr\} &&&\\ |
---|
| 702 | & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t b_w \biggr\} |
---|
| 703 | \end{flalign*} |
---|
[2282] | 704 | |
---|
[10414] | 705 | \begin{flalign*} |
---|
| 706 | \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv |
---|
| 707 | \equiv& \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} p \biggr\} \; b_t &&&\\ |
---|
| 708 | \equiv& \sum\limits_{i,j,k} \biggl\{ \biggr\} \; b_t &&&\\ |
---|
| 709 | \end{flalign*} |
---|
[2282] | 710 | |
---|
[10414] | 711 | % |
---|
| 712 | \begin{flalign*} |
---|
| 713 | \equiv& - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv |
---|
| 714 | + \int\limits_D g\, \rho \; \frac{\partial z}{\partial t} \;dv &&& \\ |
---|
| 715 | \end{flalign*} |
---|
| 716 | % |
---|
[2282] | 717 | } |
---|
| 718 | %end gm comment |
---|
| 719 | |
---|
[10354] | 720 | Note that this property strongly constrains the discrete expression of both the depth of $T-$points and |
---|
| 721 | of the term added to the pressure gradient in the $s$-coordinate. |
---|
| 722 | Nevertheless, it is almost never satisfied since a linear equation of state is rarely used. |
---|
[2282] | 723 | |
---|
[11597] | 724 | %% ================================================================================================= |
---|
[9393] | 725 | \section{Discrete total energy conservation: flux form} |
---|
[11543] | 726 | \label{sec:INVARIANTS_3} |
---|
[2282] | 727 | |
---|
[11597] | 728 | %% ================================================================================================= |
---|
[2282] | 729 | \subsection{Total energy conservation} |
---|
[11543] | 730 | \label{subsec:INVARIANTS_KE+PE_flux} |
---|
[2282] | 731 | |
---|
[11543] | 732 | The discrete form of the total energy conservation, \autoref{eq:INVARIANTS_Tot_Energy}, is given by: |
---|
[2282] | 733 | \begin{flalign*} |
---|
[10414] | 734 | \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 \\ |
---|
[2282] | 735 | \end{flalign*} |
---|
| 736 | which in flux form, it leads to: |
---|
| 737 | \begin{flalign*} |
---|
[10414] | 738 | \sum\limits_{i,j,k} \biggl\{ \frac{u }{e_{3u}}\,\frac{\partial (e_{3u}u)}{\partial t} \,b_u |
---|
| 739 | + \frac{v }{e_{3v}}\,\frac{\partial (e_{3v}v)}{\partial t} \,b_v \biggr\} |
---|
| 740 | & - \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{e_{3u}}\frac{\partial e_{3u} }{\partial t} \,b_u |
---|
| 741 | + \frac{v^2}{e_{3v}}\frac{\partial e_{3v} }{\partial t} \,b_v \biggr\} \\ |
---|
| 742 | &= - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_3t}\frac{\partial e_{3t} \rho}{\partial t} \, g \, z_t \,b_t \biggr\} |
---|
| 743 | - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\frac{\partial z_t}{\partial t} \,b_t \biggr\} \\ |
---|
[2282] | 744 | \end{flalign*} |
---|
| 745 | |
---|
[10354] | 746 | Substituting the discrete expression of the time derivative of the velocity either in |
---|
| 747 | vector invariant or in flux form, leads to the discrete equivalent of the ???? |
---|
[2282] | 748 | |
---|
[11597] | 749 | %% ================================================================================================= |
---|
[707] | 750 | \subsection{Coriolis and advection terms: flux form} |
---|
[11543] | 751 | \label{subsec:INVARIANTS_3.2} |
---|
[707] | 752 | |
---|
[11597] | 753 | %% ================================================================================================= |
---|
[9393] | 754 | \subsubsection{Coriolis plus ``metric'' term} |
---|
[11543] | 755 | \label{subsec:INVARIANTS_3.3} |
---|
[707] | 756 | |
---|
[10354] | 757 | In flux from the vorticity term reduces to a Coriolis term in which |
---|
| 758 | the Coriolis parameter has been modified to account for the ``metric'' term. |
---|
| 759 | This altered Coriolis parameter is discretised at an f-point. |
---|
| 760 | It is given by: |
---|
[10406] | 761 | \[ |
---|
[10414] | 762 | f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\; |
---|
| 763 | \equiv \; |
---|
| 764 | f+\frac{1} {e_{1f}\,e_{2f}} \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] |
---|
| 765 | -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u} \right] \right) |
---|
[10406] | 766 | \] |
---|
[707] | 767 | |
---|
[10354] | 768 | Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. |
---|
| 769 | It therefore conserves the total KE. |
---|
[11543] | 770 | The derivation is the same as for the vorticity term in the vector invariant form (\autoref{subsec:INVARIANTS_vor}). |
---|
[707] | 771 | |
---|
[11597] | 772 | %% ================================================================================================= |
---|
[707] | 773 | \subsubsection{Flux form advection} |
---|
[11543] | 774 | \label{subsec:INVARIANTS_3.4} |
---|
[707] | 775 | |
---|
[10354] | 776 | The flux form operator of the momentum advection is evaluated using |
---|
| 777 | a centered second order finite difference scheme. |
---|
| 778 | Because of the flux form, the discrete operator does not contribute to the global budget of linear momentum. |
---|
| 779 | Because of the centered second order scheme, it conserves the horizontal kinetic energy, that is: |
---|
[707] | 780 | |
---|
[10414] | 781 | \begin{equation} |
---|
[11543] | 782 | \label{eq:INVARIANTS_ADV_KE_flux} |
---|
[10414] | 783 | - \int_D \textbf{U}_h \cdot \left( {{ |
---|
| 784 | \begin{array} {*{20}c} |
---|
| 785 | \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ |
---|
| 786 | \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ |
---|
| 787 | \end{array} |
---|
| 788 | } } \right) \;dv |
---|
| 789 | - \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv } =\;0 |
---|
[707] | 790 | \end{equation} |
---|
| 791 | |
---|
[10354] | 792 | Let us first consider the first term of the scalar product |
---|
[11435] | 793 | (\ie\ just the the terms associated with the i-component of the advection): |
---|
[707] | 794 | \begin{flalign*} |
---|
[10414] | 795 | & - \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv \\ |
---|
| 796 | % |
---|
| 797 | \equiv& - \sum\limits_{i,j,k} \biggl\{ \frac{1} {b_u} \biggl( |
---|
| 798 | \delta_{i+1/2} \left[ \overline {U}^{\,i} \;\overline u^{\,i} \right] |
---|
| 799 | + \delta_j \left[ \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] |
---|
| 800 | + \delta_k \left[ \overline {W}^{\,i+1/2}\;\overline u^{\,k+1/2} \right] \biggr) \; \biggr\} \, b_u \;u &&& \\ |
---|
| 801 | % |
---|
| 802 | \equiv& - \sum\limits_{i,j,k} |
---|
| 803 | \biggl\{ |
---|
| 804 | \delta_{i+1/2} \left[ \overline {U}^{\,i}\; \overline u^{\,i} \right] |
---|
| 805 | + \delta_j \left[ \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2} \right] |
---|
| 806 | + \delta_k \left[ \overline {W}^{\,i+12}\;\overline u^{\,k+1/2} \right] |
---|
| 807 | \; \biggr\} \; u \\ |
---|
| 808 | % |
---|
| 809 | \equiv& + \sum\limits_{i,j,k} |
---|
| 810 | \biggl\{ |
---|
| 811 | \overline {U}^{\,i}\; \overline u^{\,i} \delta_i \left[ u \right] |
---|
| 812 | + \overline {V}^{\,i+1/2}\; \overline u^{\,j+1/2} \delta_{j+1/2} \left[ u \right] |
---|
| 813 | + \overline {W}^{\,i+1/2}\; \overline u^{\,k+1/2} \delta_{k+1/2} \left[ u \right] \biggr\} && \\ |
---|
| 814 | % |
---|
| 815 | \equiv& + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ |
---|
| 816 | \overline{U}^{\,i} \delta_i \left[ u^2 \right] |
---|
| 817 | + \overline{V}^{\,i+1/2} \delta_{j+/2} \left[ u^2 \right] |
---|
| 818 | + \overline{W}^{\,i+1/2} \delta_{k+1/2} \left[ u^2 \right] \biggr\} && \\ |
---|
| 819 | % |
---|
| 820 | \equiv& - \sum\limits_{i,j,k} \frac{1}{2} \bigg\{ |
---|
| 821 | U \; \delta_{i+1/2} \left[ \overline {u^2}^{\,i} \right] |
---|
| 822 | + V \; \delta_{j+1/2} \left[ \overline {u^2}^{\,i} \right] |
---|
| 823 | + W \; \delta_{k+1/2} \left[ \overline {u^2}^{\,i} \right] \biggr\} &&& \\ |
---|
| 824 | % |
---|
| 825 | \equiv& - \sum\limits_{i,j,k} \frac{1}{2} \overline {u^2}^{\,i} \biggl\{ |
---|
| 826 | \delta_{i+1/2} \left[ U \right] |
---|
| 827 | + \delta_{j+1/2} \left[ V \right] |
---|
| 828 | + \delta_{k+1/2} \left[ W \right] \biggr\} &&& \\ |
---|
| 829 | % |
---|
| 830 | \equiv& + \sum\limits_{i,j,k} \frac{1}{2} \overline {u^2}^{\,i} |
---|
| 831 | \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} &&& \\ |
---|
[707] | 832 | \end{flalign*} |
---|
[10354] | 833 | Applying similar manipulation applied to the second term of the scalar product leads to: |
---|
[10406] | 834 | \[ |
---|
[10414] | 835 | - \int_D \textbf{U}_h \cdot \left( {{ |
---|
| 836 | \begin{array} {*{20}c} |
---|
| 837 | \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ |
---|
| 838 | \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ |
---|
| 839 | \end{array} |
---|
| 840 | } } \right) \;dv |
---|
| 841 | \equiv + \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) |
---|
| 842 | \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} |
---|
[10406] | 843 | \] |
---|
[10354] | 844 | which is the discrete form of $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv $. |
---|
[11543] | 845 | \autoref{eq:INVARIANTS_ADV_KE_flux} is thus satisfied. |
---|
[707] | 846 | |
---|
[10354] | 847 | When the UBS scheme is used to evaluate the flux form momentum advection, |
---|
| 848 | the discrete operator does not contribute to the global budget of linear momentum (flux form). |
---|
[11435] | 849 | The horizontal kinetic energy is not conserved, but forced to decay (\ie\ the scheme is diffusive). |
---|
[707] | 850 | |
---|
[11597] | 851 | %% ================================================================================================= |
---|
[2282] | 852 | \section{Discrete enstrophy conservation} |
---|
[11543] | 853 | \label{sec:INVARIANTS_4} |
---|
[2282] | 854 | |
---|
[11597] | 855 | %% ================================================================================================= |
---|
[11582] | 856 | \subsubsection{Vorticity term with ENS scheme (\protect\np[=.true.]{ln_dynvor_ens}{ln\_dynvor\_ens})} |
---|
[11543] | 857 | \label{subsec:INVARIANTS_vorENS} |
---|
[707] | 858 | |
---|
[2282] | 859 | In the ENS scheme, the vorticity term is descretized as follows: |
---|
[10414] | 860 | \begin{equation} |
---|
[11543] | 861 | \label{eq:INVARIANTS_dynvor_ens} |
---|
[10414] | 862 | \left\{ |
---|
| 863 | \begin{aligned} |
---|
| 864 | +\frac{1}{e_{1u}} & \overline{q}^{\,i} & {\overline{ \overline{\left( e_{1v}\,e_{3v}\; v \right) } } }^{\,i, j+1/2} \\ |
---|
| 865 | - \frac{1}{e_{2v}} & \overline{q}^{\,j} & {\overline{ \overline{\left( e_{2u}\,e_{3u}\; u \right) } } }^{\,i+1/2, j} |
---|
| 866 | \end{aligned} |
---|
| 867 | \right. |
---|
[11435] | 868 | \end{equation} |
---|
[707] | 869 | |
---|
[10354] | 870 | The scheme does not allow but the conservation of the total kinetic energy but the conservation of $q^2$, |
---|
[11435] | 871 | the potential enstrophy for a horizontally non-divergent flow (\ie\ when $\chi$=$0$). |
---|
[10354] | 872 | Indeed, using the symmetry or skew symmetry properties of the operators |
---|
| 873 | ( \autoref{eq:DOM_mi_adj} and \autoref{eq:DOM_di_adj}), |
---|
| 874 | it can be shown that: |
---|
[10414] | 875 | \begin{equation} |
---|
[11543] | 876 | \label{eq:INVARIANTS_1.1} |
---|
[10414] | 877 | \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 |
---|
[2282] | 878 | \end{equation} |
---|
[10354] | 879 | where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. |
---|
[11543] | 880 | Indeed, using \autoref{eq:DYN_vor_ens}, |
---|
| 881 | the discrete form of the right hand side of \autoref{eq:INVARIANTS_1.1} can be transformed as follow: |
---|
[10414] | 882 | \begin{flalign*} |
---|
| 883 | &\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times |
---|
| 884 | \left( e_3 \, q \; \textbf{k} \times \textbf{U}_h \right)\; dv \\ |
---|
| 885 | % |
---|
| 886 | & \qquad |
---|
| 887 | { |
---|
| 888 | \begin{array}{*{20}l} |
---|
| 889 | &\equiv \sum\limits_{i,j,k} |
---|
| 890 | q \ \left\{ \delta_{i+1/2} \left[ - \,\overline {q}^{\,i}\; \overline{\overline U }^{\,i,j+1/ 2} \right] |
---|
| 891 | - \delta_{j+1/2} \left[ \overline {q}^{\,j}\; \overline{\overline V }^{\,i+1/2, j} \right] \right\} \\ |
---|
| 892 | % |
---|
| 893 | &\equiv \sum\limits_{i,j,k} |
---|
| 894 | \left\{ \delta_i [q] \; \overline{q}^{\,i} \; \overline{ \overline U }^{\,i,j+1/2} |
---|
| 895 | + \delta_j [q] \; \overline{q}^{\,j} \; \overline{\overline V }^{\,i+1/2,j} \right\} && \\ |
---|
| 896 | % |
---|
| 897 | &\equiv \,\frac{1} {2} \sum\limits_{i,j,k} |
---|
| 898 | \left\{ \delta_i \left[ q^2 \right] \; \overline{\overline U }^{\,i,j+1/2} |
---|
| 899 | + \delta_j \left[ q^2 \right] \; \overline{\overline V }^{\,i+1/2,j} \right\} && \\ |
---|
| 900 | % |
---|
| 901 | &\equiv - \frac{1} {2} \sum\limits_{i,j,k} q^2 \; |
---|
| 902 | \left\{ \delta_{i+1/2} \left[ \overline{\overline{ U }}^{\,i,j+1/2} \right] |
---|
| 903 | + \delta_{j+1/2} \left[ \overline{\overline{ V }}^{\,i+1/2,j} \right] \right\} && \\ |
---|
| 904 | \end{array} |
---|
| 905 | } |
---|
| 906 | % |
---|
| 907 | \allowdisplaybreaks |
---|
| 908 | \intertext{ Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2} |
---|
| 909 | \left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$, |
---|
| 910 | and introducing the horizontal divergence $\chi $, it becomes: } |
---|
| 911 | \allowdisplaybreaks |
---|
| 912 | % |
---|
| 913 | & \qquad { |
---|
| 914 | \begin{array}{*{20}l} |
---|
| 915 | &\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} |
---|
| 916 | \quad \equiv 0 && |
---|
| 917 | \end{array} |
---|
| 918 | } |
---|
[707] | 919 | \end{flalign*} |
---|
[11435] | 920 | The later equality is obtain only when the flow is horizontally non-divergent, \ie\ $\chi$=$0$. |
---|
[707] | 921 | |
---|
[11597] | 922 | %% ================================================================================================= |
---|
[11582] | 923 | \subsubsection{Vorticity Term with EEN scheme (\protect\np[=.true.]{ln_dynvor_een}{ln\_dynvor\_een})} |
---|
[11543] | 924 | \label{subsec:INVARIANTS_vorEEN} |
---|
[707] | 925 | |
---|
[11435] | 926 | With the EEN scheme, the vorticity terms are represented as: |
---|
[10414] | 927 | \begin{equation} |
---|
[11558] | 928 | \label{eq:INVARIANTS_dynvor_een2} |
---|
[10414] | 929 | \left\{ { |
---|
| 930 | \begin{aligned} |
---|
| 931 | +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} |
---|
| 932 | {^{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} \\ |
---|
| 933 | - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} |
---|
| 934 | {^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} \\ |
---|
| 935 | \end{aligned} |
---|
| 936 | } \right. |
---|
[11435] | 937 | \end{equation} |
---|
| 938 | where the indices $i_p$ and $k_p$ take the following values: |
---|
[2282] | 939 | $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, |
---|
[11435] | 940 | and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: |
---|
[10414] | 941 | \begin{equation} |
---|
[11543] | 942 | \tag{\ref{eq:INVARIANTS_Q_triads}} |
---|
[10414] | 943 | _i^j \mathbb{Q}^{i_p}_{j_p} |
---|
| 944 | = \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) |
---|
[2282] | 945 | \end{equation} |
---|
[707] | 946 | |
---|
[11435] | 947 | This formulation does conserve the potential enstrophy for a horizontally non-divergent flow (\ie\ $\chi=0$). |
---|
[2282] | 948 | |
---|
[10354] | 949 | Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $, |
---|
| 950 | similar manipulation can be done for the 3 others. |
---|
[11543] | 951 | The discrete form of the right hand side of \autoref{eq:INVARIANTS_1.1} applied to |
---|
[10354] | 952 | this triad only can be transformed as follow: |
---|
[2282] | 953 | |
---|
[10414] | 954 | \begin{flalign*} |
---|
| 955 | &\int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \\ |
---|
| 956 | % |
---|
| 957 | \equiv& \sum\limits_{i,j,k} |
---|
| 958 | {q} \ \biggl\{ \;\; |
---|
| 959 | \delta_{i+1/2} \left[ -\, {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}} \right] |
---|
| 960 | - \delta_{j+1/2} \left[ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \right] |
---|
| 961 | \;\;\biggr\} && \\ |
---|
| 962 | % |
---|
| 963 | \equiv& \sum\limits_{i,j,k} |
---|
| 964 | \biggl\{ \delta_i [q] \ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}} |
---|
| 965 | + \delta_j [q] \ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \biggr\} |
---|
| 966 | && \\ |
---|
| 967 | % |
---|
| 968 | ... & &&\\ |
---|
| 969 | &Demonstation \ to \ be \ done... &&\\ |
---|
| 970 | ... & &&\\ |
---|
| 971 | % |
---|
| 972 | \equiv& \frac{1} {2} \sum\limits_{i,j,k} |
---|
| 973 | \biggl\{ \delta_i \Bigl[ \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \Bigr]\; |
---|
| 974 | \overline{\overline {U}}^{\,i,j+1/2} |
---|
| 975 | + \delta_j \Bigl[ \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \Bigr]\; |
---|
| 976 | \overline{\overline {V}}^{\,i+1/2,j} |
---|
| 977 | \biggr\} |
---|
| 978 | && \\ |
---|
| 979 | % |
---|
| 980 | \equiv& - \frac{1} {2} \sum\limits_{i,j,k} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2\; |
---|
| 981 | \biggl\{ \delta_{i+1/2} |
---|
| 982 | \left[ \overline{\overline {U}}^{\,i,j+1/2} \right] |
---|
| 983 | + \delta_{j+1/2} |
---|
| 984 | \left[ \overline{\overline {V}}^{\,i+1/2,j} \right] |
---|
| 985 | \biggr\} && \\ |
---|
| 986 | % |
---|
| 987 | \equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 |
---|
| 988 | \; \overline{\overline{ b_t^{}\, \chi}}^{\,i+1/2,\,j+1/2} &&\\ |
---|
| 989 | % |
---|
| 990 | \ \ \equiv& \ 0 &&\\ |
---|
[707] | 991 | \end{flalign*} |
---|
| 992 | |
---|
[11597] | 993 | %% ================================================================================================= |
---|
[9393] | 994 | \section{Conservation properties on tracers} |
---|
[11543] | 995 | \label{sec:INVARIANTS_5} |
---|
[707] | 996 | |
---|
[11435] | 997 | All the numerical schemes used in \NEMO\ are written such that the tracer content is conserved by |
---|
[10354] | 998 | the internal dynamics and physics (equations in flux form). |
---|
| 999 | For advection, |
---|
[11435] | 1000 | only the CEN2 scheme (\ie\ $2^{nd}$ order finite different scheme) conserves the global variance of tracer. |
---|
[10354] | 1001 | Nevertheless the other schemes ensure that the global variance decreases |
---|
[11435] | 1002 | (\ie\ they are at least slightly diffusive). |
---|
[10354] | 1003 | For diffusion, all the schemes ensure the decrease of the total tracer variance, except the iso-neutral operator. |
---|
| 1004 | There is generally no strict conservation of mass, |
---|
| 1005 | as the equation of state is non linear with respect to $T$ and $S$. |
---|
[11435] | 1006 | In practice, the mass is conserved to a very high accuracy. |
---|
[11597] | 1007 | %% ================================================================================================= |
---|
[9393] | 1008 | \subsection{Advection term} |
---|
[11543] | 1009 | \label{subsec:INVARIANTS_5.1} |
---|
[707] | 1010 | |
---|
[2282] | 1011 | conservation of a tracer, $T$: |
---|
[10406] | 1012 | \[ |
---|
[10414] | 1013 | \frac{\partial }{\partial t} \left( \int_D {T\;dv} \right) |
---|
| 1014 | = \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=0 |
---|
[10406] | 1015 | \] |
---|
[2282] | 1016 | |
---|
| 1017 | conservation of its variance: |
---|
[10414] | 1018 | \begin{flalign*} |
---|
| 1019 | \frac{\partial }{\partial t} \left( \int_D {\frac{1}{2} T^2\;dv} \right) |
---|
| 1020 | =& \int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv } |
---|
| 1021 | - \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv } |
---|
[2282] | 1022 | \end{flalign*} |
---|
| 1023 | |
---|
[10354] | 1024 | Whatever the advection scheme considered it conserves of the tracer content as |
---|
| 1025 | all the scheme are written in flux form. |
---|
| 1026 | Indeed, let $T$ be the tracer and its $\tau_u$, $\tau_v$, and $\tau_w$ interpolated values at velocity point |
---|
| 1027 | (whatever the interpolation is), |
---|
[11435] | 1028 | the conservation of the tracer content due to the advection tendency is obtained as follows: |
---|
[707] | 1029 | \begin{flalign*} |
---|
[10414] | 1030 | &\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 &&&\\ |
---|
| 1031 | &\equiv - \sum\limits_{i,j,k} \biggl\{ |
---|
| 1032 | \frac{1} {b_t} \left( \delta_i \left[ U \;\tau_u \right] |
---|
| 1033 | + \delta_j \left[ V \;\tau_v \right] \right) |
---|
| 1034 | + \frac{1} {e_{3t}} \delta_k \left[ w\;\tau_w \right] \biggl\} b_t &&&\\ |
---|
| 1035 | % |
---|
| 1036 | &\equiv - \sum\limits_{i,j,k} \left\{ |
---|
| 1037 | \delta_i \left[ U \;\tau_u \right] |
---|
| 1038 | + \delta_j \left[ V \;\tau_v \right] |
---|
[2282] | 1039 | + \delta_k \left[ W \;\tau_w \right] \right\} && \\ |
---|
[10414] | 1040 | &\equiv 0 &&& |
---|
[707] | 1041 | \end{flalign*} |
---|
| 1042 | |
---|
[10354] | 1043 | The conservation of the variance of tracer due to the advection tendency can be achieved only with the CEN2 scheme, |
---|
[11435] | 1044 | \ie\ when $\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$. |
---|
[1223] | 1045 | It can be demonstarted as follows: |
---|
[707] | 1046 | \begin{flalign*} |
---|
[10414] | 1047 | &\int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv } |
---|
| 1048 | = - \int\limits_D \tau\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\ |
---|
| 1049 | \equiv& - \sum\limits_{i,j,k} T\; |
---|
| 1050 | \left\{ |
---|
| 1051 | \delta_i \left[ U \overline T^{\,i+1/2} \right] |
---|
| 1052 | + \delta_j \left[ V \overline T^{\,j+1/2} \right] |
---|
| 1053 | + \delta_k \left[ W \overline T^{\,k+1/2} \right] \right\} && \\ |
---|
| 1054 | \equiv& + \sum\limits_{i,j,k} |
---|
| 1055 | \left\{ U \overline T^{\,i+1/2} \,\delta_{i+1/2} \left[ T \right] |
---|
| 1056 | + V \overline T^{\,j+1/2} \;\delta_{j+1/2} \left[ T \right] |
---|
| 1057 | + W \overline T^{\,k+1/2}\;\delta_{k+1/2} \left[ T \right] \right\} &&\\ |
---|
| 1058 | \equiv& + \frac{1} {2} \sum\limits_{i,j,k} |
---|
| 1059 | \Bigl\{ U \;\delta_{i+1/2} \left[ T^2 \right] |
---|
| 1060 | + V \;\delta_{j+1/2} \left[ T^2 \right] |
---|
| 1061 | + W \;\delta_{k+1/2} \left[ T^2 \right] \Bigr\} && \\ |
---|
| 1062 | \equiv& - \frac{1} {2} \sum\limits_{i,j,k} T^2 |
---|
| 1063 | \Bigl\{ \delta_i \left[ U \right] |
---|
| 1064 | + \delta_j \left[ V \right] |
---|
| 1065 | + \delta_k \left[ W \right] \Bigr\} &&& \\ |
---|
| 1066 | \equiv& + \frac{1} {2} \sum\limits_{i,j,k} T^2 |
---|
| 1067 | \Bigl\{ \frac{1}{e_{3t}} \frac{\partial e_{3t}\,T }{\partial t} \Bigr\} &&& \\ |
---|
[707] | 1068 | \end{flalign*} |
---|
[2282] | 1069 | 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] | 1070 | |
---|
[11597] | 1071 | %% ================================================================================================= |
---|
[9393] | 1072 | \section{Conservation properties on lateral momentum physics} |
---|
[11543] | 1073 | \label{sec:INVARIANTS_dynldf_properties} |
---|
[707] | 1074 | |
---|
[10354] | 1075 | The discrete formulation of the horizontal diffusion of momentum ensures |
---|
| 1076 | the conservation of potential vorticity and the horizontal divergence, |
---|
| 1077 | and the dissipation of the square of these quantities |
---|
[11435] | 1078 | (\ie\ enstrophy and the variance of the horizontal divergence) as well as |
---|
[10354] | 1079 | the dissipation of the horizontal kinetic energy. |
---|
| 1080 | In particular, when the eddy coefficients are horizontally uniform, |
---|
| 1081 | it ensures a complete separation of vorticity and horizontal divergence fields, |
---|
| 1082 | so that diffusion (dissipation) of vorticity (enstrophy) does not generate horizontal divergence |
---|
[11435] | 1083 | (variance of the horizontal divergence) and \textit{vice versa}. |
---|
[707] | 1084 | |
---|
[10354] | 1085 | These properties of the horizontal diffusion operator are a direct consequence of |
---|
| 1086 | properties \autoref{eq:DOM_curl_grad} and \autoref{eq:DOM_div_curl}. |
---|
| 1087 | When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken, |
---|
[11435] | 1088 | the term associated with the horizontal gradient of the divergence is locally zero. |
---|
[707] | 1089 | |
---|
[11597] | 1090 | %% ================================================================================================= |
---|
[9393] | 1091 | \subsection{Conservation of potential vorticity} |
---|
[11543] | 1092 | \label{subsec:INVARIANTS_6.1} |
---|
[707] | 1093 | |
---|
[10354] | 1094 | The lateral momentum diffusion term conserves the potential vorticity: |
---|
[707] | 1095 | \begin{flalign*} |
---|
[10414] | 1096 | &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times |
---|
| 1097 | \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right) |
---|
| 1098 | - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv \\ |
---|
| 1099 | % \end{flalign*} |
---|
| 1100 | %%%%%%%%%% recheck here.... (gm) |
---|
| 1101 | % \begin{flalign*} |
---|
| 1102 | =& \int \limits_D -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times |
---|
| 1103 | \Bigl[ \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv \\ |
---|
| 1104 | % \end{flalign*} |
---|
| 1105 | % \begin{flalign*} |
---|
| 1106 | \equiv& \sum\limits_{i,j} |
---|
| 1107 | \left\{ |
---|
| 1108 | \delta_{i+1/2} \left[ \frac {e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] \right] |
---|
| 1109 | + \delta_{j+1/2} \left[ \frac {e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] \right] |
---|
| 1110 | \right\} \\ |
---|
| 1111 | % |
---|
| 1112 | \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} |
---|
| 1113 | % |
---|
| 1114 | \equiv& \sum\limits_{i,j,k} |
---|
| 1115 | -\,\left\{ |
---|
| 1116 | \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_i \left[ 1\right] |
---|
[6289] | 1117 | + \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_j \left[ 1\right] |
---|
[10414] | 1118 | \right\} \quad \equiv 0 |
---|
| 1119 | \\ |
---|
[707] | 1120 | \end{flalign*} |
---|
| 1121 | |
---|
[11597] | 1122 | %% ================================================================================================= |
---|
[9393] | 1123 | \subsection{Dissipation of horizontal kinetic energy} |
---|
[11543] | 1124 | \label{subsec:INVARIANTS_6.2} |
---|
[707] | 1125 | |
---|
[817] | 1126 | The lateral momentum diffusion term dissipates the horizontal kinetic energy: |
---|
| 1127 | %\begin{flalign*} |
---|
[10406] | 1128 | \[ |
---|
[10414] | 1129 | \begin{split} |
---|
| 1130 | \int_D \textbf{U}_h \cdot |
---|
| 1131 | \left[ \nabla_h \right. & \left. \left( A^{\,lm}\;\chi \right) |
---|
| 1132 | - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right] \; dv \\ |
---|
| 1133 | \\ %%% |
---|
| 1134 | \equiv& \sum\limits_{i,j,k} |
---|
| 1135 | \left\{ |
---|
| 1136 | \frac{1} {e_{1u}} \delta_{i+1/2} \left[ A_T^{\,lm} \chi \right] |
---|
| 1137 | - \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 1138 | \right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u \\ |
---|
| 1139 | &\;\; + \left\{ |
---|
| 1140 | \frac{1} {e_{2u}} \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right] |
---|
| 1141 | + \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 1142 | \right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v \qquad \\ |
---|
| 1143 | \\ %%% |
---|
| 1144 | \equiv& \sum\limits_{i,j,k} |
---|
| 1145 | \Bigl\{ |
---|
| 1146 | e_{2u}\,e_{3u} \;u\; \delta_{i+1/2} \left[ A_T^{\,lm} \chi \right] |
---|
| 1147 | - e_{1u} \;u\; \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 1148 | \Bigl\} |
---|
| 1149 | \\ |
---|
| 1150 | &\;\; + \Bigl\{ |
---|
| 1151 | e_{1v}\,e_{3v} \;v\; \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right] |
---|
| 1152 | + e_{2v} \;v\; \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right] |
---|
| 1153 | \Bigl\} \\ |
---|
| 1154 | \\ %%% |
---|
| 1155 | \equiv& \sum\limits_{i,j,k} |
---|
| 1156 | - \Bigl( |
---|
| 1157 | \delta_i \left[ e_{2u}\,e_{3u} \;u \right] |
---|
| 1158 | + \delta_j \left[ e_{1v}\,e_{3v} \;v \right] |
---|
| 1159 | \Bigr) \; A_T^{\,lm} \chi \\ |
---|
| 1160 | &\;\; - \Bigl( |
---|
| 1161 | \delta_{i+1/2} \left[ e_{2v} \;v \right] |
---|
| 1162 | - \delta_{j+1/2} \left[ e_{1u} \;u \right] |
---|
| 1163 | \Bigr)\; A_f^{\,lm} e_{3f} \zeta \\ |
---|
| 1164 | \\ %%% |
---|
| 1165 | \equiv& \sum\limits_{i,j,k} |
---|
| 1166 | - A_T^{\,lm} \,\chi^2 \;e_{1t}\,e_{2t}\,e_{3t} |
---|
| 1167 | - A_f ^{\,lm} \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f} |
---|
| 1168 | \quad \leq 0 \\ |
---|
| 1169 | \end{split} |
---|
[10406] | 1170 | \] |
---|
[707] | 1171 | |
---|
[11597] | 1172 | %% ================================================================================================= |
---|
[9393] | 1173 | \subsection{Dissipation of enstrophy} |
---|
[11543] | 1174 | \label{subsec:INVARIANTS_6.3} |
---|
[707] | 1175 | |
---|
[10354] | 1176 | The lateral momentum diffusion term dissipates the enstrophy when the eddy coefficients are horizontally uniform: |
---|
[707] | 1177 | \begin{flalign*} |
---|
[10414] | 1178 | &\int\limits_D \zeta \; \textbf{k} \cdot \nabla \times |
---|
| 1179 | \left[ \nabla_h \left( A^{\,lm}\;\chi \right) |
---|
| 1180 | - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \right]\;dv &&&\\ |
---|
| 1181 | &\quad = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times |
---|
| 1182 | \left[ \nabla_h \times \left( \zeta \; \textbf{k} \right) \right]\;dv &&&\\ |
---|
| 1183 | &\quad \equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f} |
---|
| 1184 | \left\{ \delta_{i+1/2} \left[ \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right] |
---|
| 1185 | + \delta_{j+1/2} \left[ \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right] \right\} &&&\\ |
---|
| 1186 | % |
---|
| 1187 | \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} |
---|
| 1188 | % |
---|
| 1189 | &\quad \equiv - A^{\,lm} \sum\limits_{i,j,k} |
---|
| 1190 | \left\{ \left( \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right)^2 b_v |
---|
| 1191 | + \left( \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right)^2 b_u \right\} \quad \leq \;0 &&&\\ |
---|
[707] | 1192 | \end{flalign*} |
---|
| 1193 | |
---|
[11597] | 1194 | %% ================================================================================================= |
---|
[9393] | 1195 | \subsection{Conservation of horizontal divergence} |
---|
[11543] | 1196 | \label{subsec:INVARIANTS_6.4} |
---|
[707] | 1197 | |
---|
[10354] | 1198 | When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken, |
---|
[11435] | 1199 | the term associated with the vertical curl of the vorticity is zero locally, due to \autoref{eq:DOM_div_curl}. |
---|
[10354] | 1200 | The resulting term conserves the $\chi$ and dissipates $\chi^2$ when the eddy coefficients are horizontally uniform. |
---|
[707] | 1201 | \begin{flalign*} |
---|
[10414] | 1202 | & \int\limits_D \nabla_h \cdot |
---|
| 1203 | \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right) |
---|
| 1204 | - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \Bigr] dv |
---|
| 1205 | = \int\limits_D \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi \right) dv \\ |
---|
| 1206 | % |
---|
| 1207 | &\equiv \sum\limits_{i,j,k} |
---|
| 1208 | \left\{ \delta_i \left[ A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] |
---|
| 1209 | + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] \right\} \\ |
---|
| 1210 | % |
---|
| 1211 | \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} |
---|
| 1212 | % |
---|
| 1213 | &\equiv \sum\limits_{i,j,k} |
---|
| 1214 | - \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] |
---|
| 1215 | + \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\} |
---|
| 1216 | \quad \equiv 0 |
---|
[707] | 1217 | \end{flalign*} |
---|
| 1218 | |
---|
[11597] | 1219 | %% ================================================================================================= |
---|
[9393] | 1220 | \subsection{Dissipation of horizontal divergence variance} |
---|
[11543] | 1221 | \label{subsec:INVARIANTS_6.5} |
---|
[707] | 1222 | |
---|
| 1223 | \begin{flalign*} |
---|
[10414] | 1224 | &\int\limits_D \chi \;\nabla_h \cdot |
---|
| 1225 | \left[ \nabla_h \left( A^{\,lm}\;\chi \right) |
---|
| 1226 | - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right]\; dv |
---|
| 1227 | = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\; dv \\ |
---|
| 1228 | % |
---|
| 1229 | &\equiv A^{\,lm} \sum\limits_{i,j,k} \frac{1} {e_{1t}\,e_{2t}\,e_{3t}} \chi |
---|
| 1230 | \left\{ |
---|
| 1231 | \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right] |
---|
| 1232 | + \delta_j \left[ \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] |
---|
| 1233 | \right\} \; e_{1t}\,e_{2t}\,e_{3t} \\ |
---|
| 1234 | % |
---|
| 1235 | \intertext{Using \autoref{eq:DOM_di_adj}, it turns out to be:} |
---|
| 1236 | % |
---|
| 1237 | &\equiv - A^{\,lm} \sum\limits_{i,j,k} |
---|
| 1238 | \left\{ \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right)^2 b_u |
---|
| 1239 | + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 b_v \right\} |
---|
| 1240 | \quad \leq 0 |
---|
[707] | 1241 | \end{flalign*} |
---|
| 1242 | |
---|
[11597] | 1243 | %% ================================================================================================= |
---|
[9393] | 1244 | \section{Conservation properties on vertical momentum physics} |
---|
[11543] | 1245 | \label{sec:INVARIANTS_7} |
---|
[707] | 1246 | |
---|
[10354] | 1247 | As for the lateral momentum physics, |
---|
| 1248 | the continuous form of the vertical diffusion of momentum satisfies several integral constraints. |
---|
| 1249 | The first two are associated with the conservation of momentum and the dissipation of horizontal kinetic energy: |
---|
[817] | 1250 | \begin{align*} |
---|
[10414] | 1251 | \int\limits_D \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1252 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv |
---|
| 1253 | \qquad \quad &= \vec{\textbf{0}} |
---|
| 1254 | % |
---|
| 1255 | \intertext{and} |
---|
| 1256 | % |
---|
| 1257 | \int\limits_D |
---|
| 1258 | \textbf{U}_h \cdot \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1259 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv \quad &\leq 0 |
---|
[817] | 1260 | \end{align*} |
---|
[6289] | 1261 | |
---|
[10354] | 1262 | The first property is obvious. |
---|
| 1263 | The second results from: |
---|
[707] | 1264 | \begin{flalign*} |
---|
[10414] | 1265 | \int\limits_D |
---|
| 1266 | \textbf{U}_h \cdot \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1267 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\;dv &&&\\ |
---|
[707] | 1268 | \end{flalign*} |
---|
| 1269 | \begin{flalign*} |
---|
[10414] | 1270 | &\equiv \sum\limits_{i,j,k} |
---|
| 1271 | \left( |
---|
| 1272 | u\; \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ u \right] \right]\; e_{1u}\,e_{2u} |
---|
| 1273 | + v\; \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right]\; e_{1v}\,e_{2v} \right) &&& |
---|
| 1274 | % |
---|
| 1275 | \intertext{since the horizontal scale factor does not depend on $k$, it follows:} |
---|
| 1276 | % |
---|
| 1277 | &\equiv - \sum\limits_{i,j,k} |
---|
| 1278 | \left( \frac{A_u^{\,vm}} {e_{3uw}} \left( \delta_{k+1/2} \left[ u \right] \right)^2\; e_{1u}\,e_{2u} |
---|
| 1279 | + \frac{A_v^{\,vm}} {e_{3vw}} \left( \delta_{k+1/2} \left[ v \right] \right)^2\; e_{1v}\,e_{2v} \right) |
---|
| 1280 | \quad \leq 0 &&& |
---|
[707] | 1281 | \end{flalign*} |
---|
[817] | 1282 | |
---|
[10354] | 1283 | The vorticity is also conserved. |
---|
| 1284 | Indeed: |
---|
[707] | 1285 | \begin{flalign*} |
---|
[10414] | 1286 | \int \limits_D |
---|
| 1287 | \frac{1} {e_3 } \textbf{k} \cdot \nabla \times |
---|
| 1288 | \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} \left( |
---|
| 1289 | \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k} |
---|
| 1290 | \right) \right)\; dv &&& |
---|
[707] | 1291 | \end{flalign*} |
---|
| 1292 | \begin{flalign*} |
---|
[10414] | 1293 | \equiv \sum\limits_{i,j,k} \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}} |
---|
| 1294 | \bigg\{ \biggr. \quad |
---|
| 1295 | \delta_{i+1/2} |
---|
| 1296 | &\left( \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right] \right) &&\\ |
---|
| 1297 | \biggl. |
---|
| 1298 | - \delta_{j+1/2} |
---|
| 1299 | &\left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{1} {e_{3uw}}\delta_{k+1/2} \left[ u \right] \right] \right) |
---|
| 1300 | \biggr\} \; |
---|
| 1301 | e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0 && |
---|
[707] | 1302 | \end{flalign*} |
---|
[6289] | 1303 | |
---|
[10442] | 1304 | If the vertical diffusion coefficient is uniform over the whole domain, the enstrophy is dissipated, \ie |
---|
[707] | 1305 | \begin{flalign*} |
---|
[10414] | 1306 | \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times |
---|
| 1307 | \left( \frac{1} {e_3}\; \frac{\partial } {\partial k} |
---|
| 1308 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&& |
---|
[707] | 1309 | \end{flalign*} |
---|
[6289] | 1310 | |
---|
[707] | 1311 | This property is only satisfied in $z$-coordinates: |
---|
| 1312 | \begin{flalign*} |
---|
[10414] | 1313 | \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times |
---|
| 1314 | \left( \frac{1} {e_3}\; \frac{\partial } {\partial k} |
---|
| 1315 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv &&& |
---|
[707] | 1316 | \end{flalign*} |
---|
| 1317 | \begin{flalign*} |
---|
[10414] | 1318 | \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \; |
---|
| 1319 | \biggl\{ \biggr. \quad |
---|
| 1320 | \delta_{i+1/2} |
---|
| 1321 | &\left( \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}[v] \right] \right) &&\\ |
---|
| 1322 | - \delta_{j+1/2} |
---|
| 1323 | &\biggl. |
---|
| 1324 | \left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) \biggr\} && |
---|
[707] | 1325 | \end{flalign*} |
---|
| 1326 | \begin{flalign*} |
---|
[10414] | 1327 | \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} |
---|
| 1328 | \biggl\{ \biggr. \quad |
---|
| 1329 | \frac{1} {e_{3v}} \delta_k |
---|
| 1330 | &\left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ \delta_{i+1/2} \left[ e_{2v}\,v \right] \right] \right] &&\\ |
---|
| 1331 | \biggl. |
---|
| 1332 | - \frac{1} {e_{3u}} \delta_k |
---|
| 1333 | &\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] | 1334 | \end{flalign*} |
---|
[10354] | 1335 | Using the fact that the vertical diffusion coefficients are uniform, |
---|
| 1336 | and that in $z$-coordinate, the vertical scale factors do not depend on $i$ and $j$ so that: |
---|
| 1337 | $e_{3f} =e_{3u} =e_{3v} =e_{3t} $ and $e_{3w} =e_{3uw} =e_{3vw} $, it follows: |
---|
[707] | 1338 | \begin{flalign*} |
---|
[10414] | 1339 | \equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k |
---|
| 1340 | \left[ \frac{1} {e_{3w}} \delta_{k+1/2} \Bigl[ \delta_{i+1/2} \left[ e_{2v}\,v \right] |
---|
| 1341 | - \delta_{j+1/ 2} \left[ e_{1u}\,u \right] \Bigr] \right] &&& |
---|
[707] | 1342 | \end{flalign*} |
---|
| 1343 | \begin{flalign*} |
---|
[10414] | 1344 | \equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}} |
---|
| 1345 | \left( \delta_{k+1/2} \left[ \zeta \right] \right)^2 \; e_{1f}\,e_{2f} \; \leq 0 &&& |
---|
[707] | 1346 | \end{flalign*} |
---|
| 1347 | Similarly, the horizontal divergence is obviously conserved: |
---|
| 1348 | |
---|
| 1349 | \begin{flalign*} |
---|
[10414] | 1350 | \int\limits_D \nabla \cdot |
---|
| 1351 | \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1352 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&& |
---|
[707] | 1353 | \end{flalign*} |
---|
[11435] | 1354 | and the square of the horizontal divergence decreases (\ie\ the horizontal divergence is dissipated) if |
---|
[10354] | 1355 | the vertical diffusion coefficient is uniform over the whole domain: |
---|
[707] | 1356 | |
---|
| 1357 | \begin{flalign*} |
---|
[10414] | 1358 | \int\limits_D \chi \;\nabla \cdot |
---|
| 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 = 0 &&& |
---|
[707] | 1361 | \end{flalign*} |
---|
[1223] | 1362 | This property is only satisfied in the $z$-coordinate: |
---|
[707] | 1363 | \begin{flalign*} |
---|
[10414] | 1364 | \int\limits_D \chi \;\nabla \cdot |
---|
| 1365 | \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} |
---|
| 1366 | \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv &&& |
---|
[707] | 1367 | \end{flalign*} |
---|
| 1368 | \begin{flalign*} |
---|
[10414] | 1369 | \equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1t}\,e_{2t}} |
---|
| 1370 | \biggl\{ \Biggr. \quad |
---|
| 1371 | \delta_{i+1/2} |
---|
| 1372 | &\left( \frac{e_{2u}} {e_{3u}} \delta_k |
---|
| 1373 | \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) &&\\ |
---|
| 1374 | \Biggl. |
---|
| 1375 | + \delta_{j+1/2} |
---|
| 1376 | &\left( \frac{e_{1v}} {e_{3v}} \delta_k |
---|
| 1377 | \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} [v] \right] \right) |
---|
| 1378 | \Biggr\} \; e_{1t}\,e_{2t}\,e_{3t} && |
---|
[707] | 1379 | \end{flalign*} |
---|
| 1380 | |
---|
| 1381 | \begin{flalign*} |
---|
[10414] | 1382 | \equiv A^{\,vm} \sum\limits_{i,j,k} \chi \, |
---|
| 1383 | \biggl\{ \biggr. \quad |
---|
| 1384 | \delta_{i+1/2} |
---|
| 1385 | &\left( |
---|
| 1386 | \delta_k \left[ |
---|
| 1387 | \frac{1} {e_{3uw}} \delta_{k+1/2} \left[ e_{2u}\,u \right] \right] \right) && \\ |
---|
| 1388 | \biggl. |
---|
| 1389 | + \delta_{j+1/2} |
---|
| 1390 | &\left( \delta_k \left[ |
---|
| 1391 | \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ e_{1v}\,v \right] \right] \right) \biggr\} && |
---|
[707] | 1392 | \end{flalign*} |
---|
| 1393 | |
---|
| 1394 | \begin{flalign*} |
---|
[10414] | 1395 | \equiv -A^{\,vm} \sum\limits_{i,j,k} |
---|
| 1396 | \frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\; \biggl\{ |
---|
| 1397 | \delta_{k+1/2} \Bigl[ |
---|
| 1398 | \delta_{i+1/2} \left[ e_{2u}\,u \right] |
---|
| 1399 | + \delta_{j+1/2} \left[ e_{1v}\,v \right] \Bigr] \biggr\} &&& |
---|
[707] | 1400 | \end{flalign*} |
---|
| 1401 | |
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| 1402 | \begin{flalign*} |
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[10414] | 1403 | \equiv -A^{\,vm} \sum\limits_{i,j,k} |
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| 1404 | \frac{1} {e_{3w}} \delta_{k+1/2} \left[ \chi \right]\; \delta_{k+1/2} \left[ e_{1t}\,e_{2t} \;\chi \right] &&& |
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[707] | 1405 | \end{flalign*} |
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| 1406 | |
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| 1407 | \begin{flalign*} |
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[10414] | 1408 | \equiv -A^{\,vm} \sum\limits_{i,j,k} |
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| 1409 | \frac{e_{1t}\,e_{2t}} {e_{3w}}\; \left( \delta_{k+1/2} \left[ \chi \right] \right)^2 \quad \equiv 0 &&& |
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[707] | 1410 | \end{flalign*} |
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| 1411 | |
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[11597] | 1412 | %% ================================================================================================= |
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[9393] | 1413 | \section{Conservation properties on tracer physics} |
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[11543] | 1414 | \label{sec:INVARIANTS_8} |
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[707] | 1415 | |
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[10354] | 1416 | The numerical schemes used for tracer subgridscale physics are written such that |
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| 1417 | the heat and salt contents are conserved (equations in flux form). |
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| 1418 | Since a flux form is used to compute the temperature and salinity, |
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[11435] | 1419 | the quadratic form of these quantities (\ie\ their variance) globally tends to diminish. |
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| 1420 | As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear. |
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[707] | 1421 | |
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[11597] | 1422 | %% ================================================================================================= |
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[9393] | 1423 | \subsection{Conservation of tracers} |
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[11543] | 1424 | \label{subsec:INVARIANTS_8.1} |
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[707] | 1425 | |
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| 1426 | constraint of conservation of tracers: |
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| 1427 | \begin{flalign*} |
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[10414] | 1428 | &\int\limits_D \nabla \cdot \left( A\;\nabla T \right)\;dv &&& \\ \\ |
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| 1429 | &\equiv \sum\limits_{i,j,k} |
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| 1430 | \biggl\{ \biggr. |
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| 1431 | \delta_i |
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| 1432 | \left[ |
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| 1433 | A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} |
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| 1434 | \left[ T \right] |
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| 1435 | \right] |
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| 1436 | + \delta_j |
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| 1437 | \left[ |
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| 1438 | A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} |
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| 1439 | \left[ T \right] |
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| 1440 | \right] && \\ |
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| 1441 | & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\; |
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| 1442 | + \delta_k |
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| 1443 | \left[ |
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| 1444 | A_w^{\,vT} \frac{e_{1t}\,e_{2t}} {e_{3t}} \delta_{k+1/2} |
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| 1445 | \left[ T \right] |
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| 1446 | \right] |
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| 1447 | \biggr\} \quad \equiv 0 |
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| 1448 | && |
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[707] | 1449 | \end{flalign*} |
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| 1450 | |
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[11435] | 1451 | In fact, this property simply results from the flux form of the operator. |
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[707] | 1452 | |
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[11597] | 1453 | %% ================================================================================================= |
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[9393] | 1454 | \subsection{Dissipation of tracer variance} |
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[11543] | 1455 | \label{subsec:INVARIANTS_8.2} |
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[707] | 1456 | |
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[1223] | 1457 | constraint on the dissipation of tracer variance: |
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[707] | 1458 | \begin{flalign*} |
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[10414] | 1459 | \int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv &&&\\ |
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| 1460 | &\equiv \sum\limits_{i,j,k} \; T |
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| 1461 | \biggl\{ \biggr. |
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| 1462 | \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right] |
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| 1463 | & + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right] |
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| 1464 | \quad&& \\ |
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| 1465 | \biggl. |
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| 1466 | &&+ \delta_k \left[A_w^{\,vT}\frac{e_{1t}\,e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right] |
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| 1467 | \biggr\} && |
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[707] | 1468 | \end{flalign*} |
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| 1469 | \begin{flalign*} |
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[10414] | 1470 | \equiv - \sum\limits_{i,j,k} |
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| 1471 | \biggl\{ \biggr. \quad |
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| 1472 | & A_u^{\,lT} \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ T \right] \right)^2 e_{1u}\,e_{2u}\,e_{3u} && \\ |
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| 1473 | & + A_v^{\,lT} \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ T \right] \right)^2 e_{1v}\,e_{2v}\,e_{3v} && \\ \biggl. |
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| 1474 | & + A_w^{\,vT} \left( \frac{1} {e_{3w}} \delta_{k+1/2} \left[ T \right] \right)^2 e_{1w}\,e_{2w}\,e_{3w} \biggr\} |
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| 1475 | \quad \leq 0 && |
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[707] | 1476 | \end{flalign*} |
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| 1477 | |
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[817] | 1478 | %%%% end of appendix in gm comment |
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[994] | 1479 | %} |
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[10414] | 1480 | |
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[11693] | 1481 | \subinc{\input{../../global/epilogue}} |
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[10442] | 1482 | |
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[6997] | 1483 | \end{document} |
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