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