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