1 | \documentclass[../main/NEMO_manual]{subfiles} |
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2 | |
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3 | \begin{document} |
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4 | % ================================================================ |
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5 | % Appendix E : Note on some algorithms |
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6 | % ================================================================ |
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7 | \chapter{Note on some algorithms} |
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8 | \label{apdx:ALGOS} |
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9 | |
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10 | \chaptertoc |
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11 | |
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12 | \newpage |
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13 | |
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14 | This appendix some on going consideration on algorithms used or planned to be used in \NEMO. |
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15 | |
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16 | % ------------------------------------------------------------------------------------------------------------- |
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17 | % UBS scheme |
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18 | % ------------------------------------------------------------------------------------------------------------- |
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19 | \section{Upstream Biased Scheme (UBS) (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})} |
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20 | \label{sec:TRA_adv_ubs} |
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21 | |
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22 | The UBS advection scheme is an upstream biased third order scheme based on |
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23 | an upstream-biased parabolic interpolation. |
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24 | It is also known as Cell Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective Kinematics). |
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25 | For example, in the $i$-direction: |
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26 | \begin{equation} |
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27 | \label{eq:tra_adv_ubs2} |
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28 | \tau_u^{ubs} = \left\{ |
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29 | \begin{aligned} |
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30 | & \tau_u^{cen4} + \frac{1}{12} \,\tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ |
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31 | & \tau_u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 |
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32 | \end{aligned} |
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33 | \right. |
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34 | \end{equation} |
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35 | or equivalently, the advective flux is |
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36 | \begin{equation} |
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37 | \label{eq:tra_adv_ubs2} |
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38 | U_{i+1/2} \ \tau_u^{ubs} |
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39 | =U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2} |
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40 | - \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
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41 | \end{equation} |
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42 | where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and |
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43 | $\tau "_i =\delta_i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$. |
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44 | By choosing this expression for $\tau "$ we consider a fourth order approximation of $\partial_i^2$ with |
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45 | a constant i-grid spacing ($\Delta i=1$). |
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46 | |
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47 | Alternative choice: introduce the scale factors: |
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48 | $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta_{i+1/2}[\tau] \right]$. |
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49 | |
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50 | This results in a dissipatively dominant (\ie\ hyper-diffusive) truncation error |
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51 | \citep{shchepetkin.mcwilliams_OM05}. |
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52 | The overall performance of the advection scheme is similar to that reported in \cite{farrow.stevens_JPO95}. |
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53 | It is a relatively good compromise between accuracy and smoothness. |
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54 | It is not a \emph{positive} scheme meaning false extrema are permitted but |
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55 | the amplitude of such are significantly reduced over the centred second order method. |
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56 | Nevertheless it is not recommended to apply it to a passive tracer that requires positivity. |
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57 | |
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58 | The intrinsic diffusion of UBS makes its use risky in the vertical direction where |
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59 | the control of artificial diapycnal fluxes is of paramount importance. |
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60 | It has therefore been preferred to evaluate the vertical flux using the TVD scheme when |
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61 | \np{ln\_traadv\_ubs}\forcode{ = .true.}. |
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62 | |
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63 | For stability reasons, in \autoref{eq:TRA_adv_ubs}, the first term which corresponds to |
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64 | a second order centred scheme is evaluated using the \textit{now} velocity (centred in time) while |
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65 | the second term which is the diffusive part of the scheme, is evaluated using the \textit{before} velocity |
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66 | (forward in time). |
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67 | This is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the Quick advection scheme. |
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68 | UBS and QUICK schemes only differ by one coefficient. |
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69 | Substituting 1/6 with 1/8 in (\autoref{eq:TRA_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. |
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70 | This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded. |
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71 | Nevertheless it is quite easy to make the substitution in \mdl{traadv\_ubs} module and obtain a QUICK scheme. |
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72 | |
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73 | NB 1: When a high vertical resolution $O(1m)$ is used, the model stability can be controlled by vertical advection |
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74 | (not vertical diffusion which is usually solved using an implicit scheme). |
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75 | Computer time can be saved by using a time-splitting technique on vertical advection. |
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76 | This possibility have been implemented and validated in ORCA05-L301. |
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77 | It is not currently offered in the current reference version. |
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78 | |
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79 | NB 2: In a forthcoming release four options will be proposed for the vertical component used in the UBS scheme. |
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80 | $\tau_w^{ubs}$ will be evaluated using either \textit{(a)} a centered $2^{nd}$ order scheme, |
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81 | or \textit{(b)} a TVD scheme, or \textit{(c)} an interpolation based on conservative parabolic splines following |
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82 | \citet{shchepetkin.mcwilliams_OM05} implementation of UBS in ROMS, or \textit{(d)} an UBS. |
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83 | The $3^{rd}$ case has dispersion properties similar to an eight-order accurate conventional scheme. |
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84 | |
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85 | NB 3: It is straight forward to rewrite \autoref{eq:TRA_adv_ubs} as follows: |
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86 | \begin{equation} |
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87 | \label{eq:tra_adv_ubs2} |
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88 | \tau_u^{ubs} = \left\{ |
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89 | \begin{aligned} |
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90 | & \tau_u^{cen4} + \frac{1}{12} \tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ |
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91 | & \tau_u^{cen4} - \frac{1}{12} \tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 |
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92 | \end{aligned} |
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93 | \right. |
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94 | \end{equation} |
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95 | or equivalently |
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96 | \begin{equation} |
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97 | \label{eq:tra_adv_ubs2} |
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98 | \begin{split} |
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99 | e_{2u} e_{3u}\,u_{i+1/2} \ \tau_u^{ubs} |
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100 | &= e_{2u} e_{3u}\,u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\tau"_i }^{\,i+1/2} \\ |
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101 | & - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
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102 | \end{split} |
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103 | \end{equation} |
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104 | \autoref{eq:TRA_adv_ubs2} has several advantages. |
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105 | First it clearly evidences that the UBS scheme is based on the fourth order scheme to which |
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106 | is added an upstream biased diffusive term. |
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107 | Second, this emphasises that the $4^{th}$ order part have to be evaluated at \emph{now} time step, |
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108 | not only the $2^{th}$ order part as stated above using \autoref{eq:TRA_adv_ubs}. |
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109 | Third, the diffusive term is in fact a biharmonic operator with a eddy coefficient which |
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110 | is simply proportional to the velocity. |
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111 | |
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112 | laplacian diffusion: |
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113 | \begin{equation} |
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114 | \label{eq:tra_ldf_lap} |
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115 | \begin{split} |
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116 | D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\; e_{3T} } &\left[ {\quad \delta_i |
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117 | \left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta_{i+1/2} |
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118 | \left[ T \right]} \right]} \right. \\ |
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119 | &\ \left. {+\; \delta_j \left[ |
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120 | {A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta_{j+1/2} \left[ T |
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121 | \right]} \right)} \right]\quad } \right] |
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122 | \end{split} |
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123 | \end{equation} |
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124 | |
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125 | bilaplacian: |
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126 | \begin{equation} |
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127 | \label{eq:tra_ldf_lap} |
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128 | \begin{split} |
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129 | D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ |
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130 | & \delta_i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2} |
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131 | \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} |
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132 | \delta_i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2} |
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133 | [T] \right] \right] \right] |
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134 | \end{split} |
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135 | \end{equation} |
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136 | with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$, |
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137 | \ie\ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$ |
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138 | it comes: |
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139 | \begin{equation} |
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140 | \label{eq:tra_ldf_lap} |
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141 | \begin{split} |
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142 | D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ |
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143 | & \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2} |
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144 | \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} |
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145 | \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2} |
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146 | [T] \right] \right] \right] |
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147 | \end{split} |
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148 | \end{equation} |
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149 | if the velocity is uniform (\ie\ $|u|=cst$) then the diffusive flux is |
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150 | \begin{equation} |
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151 | \label{eq:tra_ldf_lap} |
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152 | \begin{split} |
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153 | F_u^{lT} = - \frac{1}{12} |
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154 | e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta_{i+1/2} |
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155 | \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} |
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156 | \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta_{i+1/2} |
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157 | [T] \right] \right] |
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158 | \end{split} |
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159 | \end{equation} |
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160 | beurk.... reverte the logic: starting from the diffusive part of the advective flux it comes: |
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161 | |
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162 | \begin{equation} |
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163 | \label{eq:tra_adv_ubs2} |
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164 | \begin{split} |
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165 | F_u^{lT} &= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
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166 | \end{split} |
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167 | \end{equation} |
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168 | if the velocity is uniform (\ie\ $|u|=cst$) and |
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169 | choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right]$ |
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170 | |
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171 | sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$): |
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172 | \begin{equation} |
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173 | \label{eq:tra_adv_ubs2} |
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174 | \begin{split} |
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175 | F_u^{lT} &= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] |
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176 | \end{split} |
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177 | \end{equation} |
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178 | which leads to ${A_T^{lT}}^2 = \frac{1}{12} {e_{1T}}^3\ \overline{|u|}^{\,i+1/2}$ |
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179 | |
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180 | sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$ |
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181 | \begin{equation} |
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182 | \label{eq:tra_adv_ubs2} |
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183 | \begin{split} |
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184 | F_u^{lT} &= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta_i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] \\ |
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185 | &= - \frac{1}{12} e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{1T}\,e_{2T}\,e_{3T}}\,\delta_i \left[ e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u} }{e_{1u}} \delta_{i+1/2}[\tau] \right] \right] |
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186 | \end{split} |
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187 | \end{equation} |
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188 | which leads to ${A_u^{lT}} = \frac{1}{12} {e_{1u}}^3\ |u|$ |
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189 | |
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190 | % ------------------------------------------------------------------------------------------------------------- |
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191 | % Leap-Frog energetic |
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192 | % ------------------------------------------------------------------------------------------------------------- |
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193 | \section{Leapfrog energetic} |
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194 | \label{sec:LF} |
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195 | |
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196 | We adopt the following semi-discrete notation for time derivative. |
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197 | Given the values of a variable $q$ at successive time step, |
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198 | the time derivation and averaging operators at the mid time step are: |
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199 | \[ |
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200 | % \label{eq:dt_mt} |
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201 | \begin{split} |
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202 | \delta_{t+\rdt/2} [q] &= \ \ \, q^{t+\rdt} - q^{t} \\ |
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203 | \overline q^{\,t+\rdt/2} &= \left\{ q^{t+\rdt} + q^{t} \right\} \; / \; 2 |
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204 | \end{split} |
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205 | \] |
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206 | As for space operator, |
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207 | the adjoint of the derivation and averaging time operators are $\delta_t^*=\delta_{t+\rdt/2}$ and |
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208 | $\overline{\cdot}^{\,t\,*}= \overline{\cdot}^{\,t+\Delta/2}$, respectively. |
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209 | |
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210 | The Leap-frog time stepping given by \autoref{eq:DOM_nxt} can be defined as: |
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211 | \[ |
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212 | % \label{eq:LF} |
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213 | \frac{\partial q}{\partial t} |
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214 | \equiv \frac{1}{\rdt} \overline{ \delta_{t+\rdt/2}[q]}^{\,t} |
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215 | = \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} |
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216 | \] |
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217 | Note that \autoref{chap:LF} shows that the leapfrog time step is $\rdt$, |
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218 | not $2\rdt$ as it can be found sometimes in literature. |
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219 | The leap-Frog time stepping is a second order centered scheme. |
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220 | As such it respects the quadratic invariant in integral forms, \ie\ the following continuous property, |
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221 | \[ |
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222 | % \label{eq:Energy} |
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223 | \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} |
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224 | =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} |
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225 | = \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) , |
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226 | \] |
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227 | is satisfied in discrete form. |
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228 | Indeed, |
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229 | \[ |
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230 | \begin{split} |
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231 | \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} |
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232 | &\equiv \sum\limits_{0}^{N} |
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233 | {\frac{1}{\rdt} q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} \ \rdt} |
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234 | \equiv \sum\limits_{0}^{N} { q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} } \\ |
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235 | &\equiv \sum\limits_{0}^{N} { \overline{q}^{\,t+\Delta/2}{ \delta_{t+\rdt/2}[q]}} |
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236 | \equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta_{t+\rdt/2}[q^2] }\\ |
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237 | &\equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta_{t+\rdt/2}[q^2] } |
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238 | \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) |
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239 | \end{split} |
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240 | \] |
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241 | NB here pb of boundary condition when applying the adjoint! |
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242 | In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition |
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243 | (equivalently of the boundary value of the integration by part). |
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244 | In time this boundary condition is not physical and \textbf{add something here!!!} |
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245 | |
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246 | % ================================================================ |
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247 | % Iso-neutral diffusion : |
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248 | % ================================================================ |
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249 | |
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250 | \section{Lateral diffusion operator} |
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251 | |
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252 | % ================================================================ |
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253 | % Griffies' iso-neutral diffusion operator : |
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254 | % ================================================================ |
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255 | \subsection{Griffies iso-neutral diffusion operator} |
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256 | |
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257 | Let try to define a scheme that get its inspiration from the \citet{griffies.gnanadesikan.ea_JPO98} scheme, |
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258 | but is formulated within the \NEMO\ framework |
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259 | (\ie\ using scale factors rather than grid-size and having a position of $T$-points that |
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260 | is not necessary in the middle of vertical velocity points, see \autoref{fig:DOM_zgr_e3}). |
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261 | |
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262 | In the formulation \autoref{eq:TRA_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO, |
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263 | the off-diagonal terms of the small angle diffusion tensor contain several double spatial averages of a gradient, |
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264 | for example $\overline{\overline{\delta_k \cdot}}^{\,i,k}$. |
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265 | It is apparent that the combination of a $k$ average and a $k$ derivative of the tracer allows for |
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266 | the presence of grid point oscillation structures that will be invisible to the operator. |
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267 | These structures are \textit{computational modes}. |
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268 | They will not be damped by the iso-neutral operator, and even possibly amplified by it. |
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269 | In other word, the operator applied to a tracer does not warranties the decrease of its global average variance. |
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270 | To circumvent this, we have introduced a smoothing of the slopes of the iso-neutral surfaces |
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271 | (see \autoref{chap:LDF}). |
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272 | Nevertheless, this technique works fine for $T$ and $S$ as they are active tracers |
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273 | (\ie\ they enter the computation of density), but it does not work for a passive tracer. |
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274 | \citep{griffies.gnanadesikan.ea_JPO98} introduce a different way to discretise the off-diagonal terms that |
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275 | nicely solve the problem. |
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276 | The idea is to get rid of combinations of an averaged in one direction combined with |
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277 | a derivative in the same direction by considering triads. |
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278 | For example in the (\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows: |
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279 | \begin{equation} |
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280 | \label{eq:Gf_triads} |
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281 | _i^k \mathbb{T}_{i_p}^{k_p} (T) |
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282 | = \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \left( |
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283 | \frac{ \delta_{i + i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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284 | -\ {_i^k \mathbb{R}_{i_p}^{k_p}} \ \frac{ \delta_{k+k_p} [T^i] }{ {e_{3w}}_{\,i}^{\,k+k_p} } |
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285 | \right) |
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286 | \end{equation} |
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287 | where the indices $i_p$ and $k_p$ define the four triads and take the following value: |
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288 | $i_p = -1/2$ or $1/2$ and $k_p = -1/2$ or $1/2$, |
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289 | $b_u= e_{1u}\,e_{2u}\,e_{3u}$ is the volume of $u$-cells, |
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290 | $A_i^k$ is the lateral eddy diffusivity coefficient defined at $T$-point, |
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291 | and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad: |
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292 | \begin{equation} |
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293 | \label{eq:Gf_slopes} |
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294 | _i^k \mathbb{R}_{i_p}^{k_p} |
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295 | =\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac |
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296 | {\left(\alpha / \beta \right)_i^k \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] } |
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297 | {\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] } |
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298 | \end{equation} |
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299 | Note that in \autoref{eq:Gf_slopes} we use the ratio $\alpha / \beta$ instead of |
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300 | multiplying the temperature derivative by $\alpha$ and the salinity derivative by $\beta$. |
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301 | This is more efficient as the ratio $\alpha / \beta$ can to be evaluated directly. |
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302 | |
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303 | Note that in \autoref{eq:Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of ${b_{uw}}_{\,i+i_p}^{\,k+k_p}$. |
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304 | This choice has been motivated by the decrease of tracer variance and |
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305 | the presence of partial cell at the ocean bottom (see \autoref{apdx:Gf_operator}). |
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306 | |
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307 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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308 | \begin{figure}[!ht] |
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309 | \begin{center} |
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310 | \includegraphics[width=\textwidth]{Fig_ISO_triad} |
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311 | \caption{ |
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312 | \protect\label{fig:ISO_triad} |
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313 | Triads used in the Griffies's like iso-neutral diffision scheme for |
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314 | $u$-component (upper panel) and $w$-component (lower panel). |
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315 | } |
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316 | \end{center} |
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317 | \end{figure} |
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318 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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319 | |
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320 | The four iso-neutral fluxes associated with the triads are defined at $T$-point. |
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321 | They take the following expression: |
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322 | \begin{flalign*} |
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323 | % \label{eq:Gf_fluxes} |
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324 | \begin{split} |
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325 | {_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) |
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326 | &= \ \; \qquad \quad { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+i_p}^{\,k}} \\ |
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327 | {_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
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328 | &= -\; { _i^k \mathbb{R}_{i_p}^{k_p} } |
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329 | \ \; { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+k_p}} |
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330 | \end{split} |
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331 | \end{flalign*} |
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332 | |
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333 | The resulting iso-neutral fluxes at $u$- and $w$-points are then given by |
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334 | the sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:TRIADS_ISO_triad}): |
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335 | \begin{flalign} |
---|
336 | \label{eq:iso_flux} |
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337 | \textbf{F}_{iso}(T) |
---|
338 | &\equiv \sum_{\substack{i_p,\,k_p}} |
---|
339 | \begin{pmatrix} |
---|
340 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\ |
---|
341 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
---|
342 | \end{pmatrix} |
---|
343 | \notag \\ |
---|
344 | & \notag \\ |
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345 | &\equiv \sum_{\substack{i_p,\,k_p}} |
---|
346 | \begin{pmatrix} |
---|
347 | && { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+1/2}^{\,k} } \\ \\ |
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348 | & -\; { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } |
---|
349 | & {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+1/2} } |
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350 | \end{pmatrix} % \\ |
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351 | % &\\ |
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352 | % &\equiv \sum_{\substack{i_p,\,k_p}} |
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353 | % \begin{pmatrix} |
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354 | % \qquad \qquad \qquad |
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355 | % \frac{1}{ {e_{1u}}_{\,i+1/2}^{\,k} } \ \; |
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356 | % { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T)\\ |
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357 | % \\ |
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358 | % -\frac{1}{ {e_{3w}}_{\,i}^{\,k+1/2} } \ \; |
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359 | % { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; |
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360 | % {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T)\\ |
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361 | % \end{pmatrix} |
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362 | \end{flalign} |
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363 | resulting in a iso-neutral diffusion tendency on temperature given by |
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364 | the divergence of the sum of all the four triad fluxes: |
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365 | \begin{equation} |
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366 | \label{eq:Gf_operator} |
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367 | D_l^T = \frac{1}{b_T} \sum_{\substack{i_p,\,k_p}} \left\{ |
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368 | \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
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369 | + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} |
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370 | \end{equation} |
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371 | where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. |
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372 | |
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373 | This expression of the iso-neutral diffusion has been chosen in order to satisfy the following six properties: |
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374 | \begin{description} |
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375 | \item[$\bullet$ horizontal diffusion] |
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376 | The discretization of the diffusion operator recovers the traditional five-point Laplacian in |
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377 | the limit of flat iso-neutral direction: |
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378 | \[ |
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379 | % \label{eq:Gf_property1a} |
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380 | D_l^T = \frac{1}{b_T} \ \delta_{i} |
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381 | \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] |
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382 | \qquad \text{when} \quad |
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383 | { _i^k \mathbb{R}_{i_p}^{k_p} }=0 |
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384 | \] |
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385 | |
---|
386 | \item[$\bullet$ implicit treatment in the vertical] |
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387 | In the diagonal term associated with the vertical divergence of the iso-neutral fluxes |
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388 | \ie\ the term associated with a second order vertical derivative) |
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389 | appears only tracer values associated with a single water column. |
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390 | This is of paramount importance since it means that |
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391 | the implicit in time algorithm for solving the vertical diffusion equation can be used to evaluate this term. |
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392 | It is a necessity since the vertical eddy diffusivity associated with this term, |
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393 | \[ |
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394 | \sum_{\substack{i_p, \,k_p}} \left\{ |
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395 | A_i^k \; \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
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396 | \right\} |
---|
397 | \] |
---|
398 | can be quite large. |
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399 | |
---|
400 | \item[$\bullet$ pure iso-neutral operator] |
---|
401 | The iso-neutral flux of locally referenced potential density is zero, \ie |
---|
402 | \begin{align*} |
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403 | % \label{eq:Gf_property2} |
---|
404 | \begin{matrix} |
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405 | &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)} |
---|
406 | &= &\alpha_i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) |
---|
407 | &- \ \; \beta _i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (S) & = \ 0 \\ |
---|
408 | &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)} |
---|
409 | &= &\alpha_i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
---|
410 | &- \ \; \beta _i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (S) &= \ 0 |
---|
411 | \end{matrix} |
---|
412 | \end{align*} |
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413 | This result is trivially obtained using the \autoref{eq:Gf_triads} applied to $T$ and $S$ and |
---|
414 | the definition of the triads' slopes \autoref{eq:Gf_slopes}. |
---|
415 | |
---|
416 | \item[$\bullet$ conservation of tracer] |
---|
417 | The iso-neutral diffusion term conserve the total tracer content, \ie |
---|
418 | \[ |
---|
419 | % \label{eq:Gf_property1} |
---|
420 | \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 |
---|
421 | \] |
---|
422 | This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form. |
---|
423 | |
---|
424 | \item[$\bullet$ decrease of tracer variance] |
---|
425 | The iso-neutral diffusion term does not increase the total tracer variance, \ie |
---|
426 | \[ |
---|
427 | % \label{eq:Gf_property1} |
---|
428 | \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 |
---|
429 | \] |
---|
430 | The property is demonstrated in the \autoref{apdx:Gf_operator}. |
---|
431 | It is a key property for a diffusion term. |
---|
432 | It means that the operator is also a dissipation term, |
---|
433 | \ie\ it is a sink term for the square of the quantity on which it is applied. |
---|
434 | It therfore ensures that, when the diffusivity coefficient is large enough, |
---|
435 | the field on which it is applied become free of grid-point noise. |
---|
436 | |
---|
437 | \item[$\bullet$ self-adjoint operator] |
---|
438 | The iso-neutral diffusion operator is self-adjoint, \ie |
---|
439 | \[ |
---|
440 | % \label{eq:Gf_property1} |
---|
441 | \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} |
---|
442 | \] |
---|
443 | In other word, there is no needs to develop a specific routine from the adjoint of this operator. |
---|
444 | We just have to apply the same routine. |
---|
445 | This properties can be demonstrated quite easily in a similar way the "non increase of tracer variance" property |
---|
446 | has been proved (see \autoref{apdx:Gf_operator}). |
---|
447 | \end{description} |
---|
448 | |
---|
449 | % ================================================================ |
---|
450 | % Skew flux formulation for Eddy Induced Velocity : |
---|
451 | % ================================================================ |
---|
452 | \subsection{Eddy induced velocity and skew flux formulation} |
---|
453 | |
---|
454 | When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), |
---|
455 | an additional advection term is added. |
---|
456 | The associated velocity is the so called eddy induced velocity, |
---|
457 | the formulation of which depends on the slopes of iso-neutral surfaces. |
---|
458 | Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, |
---|
459 | \ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinate, |
---|
460 | and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $z^*$ or $s$-coordinates. |
---|
461 | |
---|
462 | The eddy induced velocity is given by: |
---|
463 | \begin{equation} |
---|
464 | \label{eq:eiv_v} |
---|
465 | \begin{split} |
---|
466 | u^* & = - \frac{1}{e_2\,e_{3}} \;\partial_k \left( e_2 \, A_e \; r_i \right) |
---|
467 | = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_i \right) \\ |
---|
468 | v^* & = - \frac{1}{e_1\,e_3}\; \partial_k \left( e_1 \, A_e \; r_j \right) |
---|
469 | = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_j \right) \\ |
---|
470 | w^* & = \frac{1}{e_1\,e_2}\; \left\{ \partial_i \left( e_2 \, A_e \; r_i \right) |
---|
471 | + \partial_j \left( e_1 \, A_e \;r_j \right) \right\} |
---|
472 | \end{split} |
---|
473 | \end{equation} |
---|
474 | where $A_{e}$ is the eddy induced velocity coefficient, |
---|
475 | and $r_i$ and $r_j$ the slopes between the iso-neutral and the geopotential surfaces. |
---|
476 | %%gm wrong: to be modified with 2 2D streamfunctions |
---|
477 | In other words, the eddy induced velocity can be derived from a vector streamfuntion, $\phi$, |
---|
478 | which is given by $\phi = A_e\,\textbf{r}$ as $\textbf{U}^* = \textbf{k} \times \nabla \phi$. |
---|
479 | %%end gm |
---|
480 | |
---|
481 | A traditional way to implement this additional advection is to add it to the eulerian velocity prior to |
---|
482 | compute the tracer advection. |
---|
483 | This allows us to take advantage of all the advection schemes offered for the tracers |
---|
484 | (see \autoref{sec:TRA_adv}) and not just a $2^{nd}$ order advection scheme. |
---|
485 | This is particularly useful for passive tracers where |
---|
486 | \emph{positivity} of the advection scheme is of paramount importance. |
---|
487 | % give here the expression using the triads. It is different from the one given in \autoref{eq:LDF_eiv} |
---|
488 | % see just below a copy of this equation: |
---|
489 | %\begin{equation} \label{eq:ldfeiv} |
---|
490 | %\begin{split} |
---|
491 | % u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ |
---|
492 | % v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\ |
---|
493 | %w^* & = \frac{1}{e_{1w}e_{2w}}\; \left\{ \delta_i \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right] + %\delta_j \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right] \right\} \\ |
---|
494 | %\end{split} |
---|
495 | %\end{equation} |
---|
496 | \[ |
---|
497 | % \label{eq:eiv_vd} |
---|
498 | \textbf{F}_{eiv}^T \equiv \left( |
---|
499 | \begin{aligned} |
---|
500 | \sum_{\substack{i_p,\,k_p}} & |
---|
501 | +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k} |
---|
502 | \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ \\ |
---|
503 | \sum_{\substack{i_p,\,k_p}} & |
---|
504 | - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p} |
---|
505 | \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \delta_{i+i_p}[T^{k+1/2-k_p}] |
---|
506 | \end{aligned} |
---|
507 | \right) |
---|
508 | \] |
---|
509 | |
---|
510 | \citep{griffies_JPO98} introduces another way to implement the eddy induced advection, the so-called skew form. |
---|
511 | It is based on a transformation of the advective fluxes using the non-divergent nature of the eddy induced velocity. |
---|
512 | For example in the (\textbf{i},\textbf{k}) plane, the tracer advective fluxes can be transformed as follows: |
---|
513 | \begin{flalign*} |
---|
514 | \begin{split} |
---|
515 | \textbf{F}_{eiv}^T = |
---|
516 | \begin{pmatrix} |
---|
517 | {e_{2}\,e_{3}\; u^*} \\ |
---|
518 | {e_{1}\,e_{2}\; w^*} |
---|
519 | \end{pmatrix} |
---|
520 | \; T |
---|
521 | &= |
---|
522 | \begin{pmatrix} |
---|
523 | { - \partial_k \left( e_{2} \, A_{e} \; r_i \right) \; T \;} \\ |
---|
524 | {+ \partial_i \left( e_{2} \, A_{e} \; r_i \right) \; T \;} |
---|
525 | \end{pmatrix} |
---|
526 | \\ |
---|
527 | &= |
---|
528 | \begin{pmatrix} |
---|
529 | { - \partial_k \left( e_{2} \, A_{e} \; r_i \; T \right) \;} \\ |
---|
530 | {+ \partial_i \left( e_{2} \, A_{e} \; r_i \; T \right) \;} |
---|
531 | \end{pmatrix} |
---|
532 | + |
---|
533 | \begin{pmatrix} |
---|
534 | {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ |
---|
535 | { - e_{2} \, A_{e} \; r_i \; \partial_i T} |
---|
536 | \end{pmatrix} |
---|
537 | \end{split} |
---|
538 | \end{flalign*} |
---|
539 | and since the eddy induces velocity field is no-divergent, |
---|
540 | we end up with the skew form of the eddy induced advective fluxes: |
---|
541 | \begin{equation} |
---|
542 | \label{eq:eiv_skew_continuous} |
---|
543 | \textbf{F}_{eiv}^T = |
---|
544 | \begin{pmatrix} |
---|
545 | {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ |
---|
546 | { - e_{2} \, A_{e} \; r_i \; \partial_i T} |
---|
547 | \end{pmatrix} |
---|
548 | \end{equation} |
---|
549 | The tendency associated with eddy induced velocity is then simply the divergence of |
---|
550 | the \autoref{eq:eiv_skew_continuous} fluxes. |
---|
551 | It naturally conserves the tracer content, as it is expressed in flux form and, |
---|
552 | as the advective form, it preserves the tracer variance. |
---|
553 | Another interesting property of \autoref{eq:eiv_skew_continuous} form is that when $A=A_e$, |
---|
554 | a simplification occurs in the sum of the iso-neutral diffusion and eddy induced velocity terms: |
---|
555 | \begin{flalign*} |
---|
556 | % \label{eq:eiv_skew+eiv_continuous} |
---|
557 | \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &= |
---|
558 | \begin{pmatrix} |
---|
559 | + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T - e_2 \, A \; r_i \;\partial_k T \\ |
---|
560 | - e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T |
---|
561 | \end{pmatrix} |
---|
562 | + |
---|
563 | \begin{pmatrix} |
---|
564 | {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ |
---|
565 | { - e_{2} \, A_{e} \; r_i \; \partial_i T} |
---|
566 | \end{pmatrix} |
---|
567 | \\ |
---|
568 | &= |
---|
569 | \begin{pmatrix} |
---|
570 | + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T \\ |
---|
571 | - 2\; e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T |
---|
572 | \end{pmatrix} |
---|
573 | \end{flalign*} |
---|
574 | The horizontal component reduces to the one use for an horizontal laplacian operator and |
---|
575 | the vertical one keeps the same complexity, but not more. |
---|
576 | This property has been used to reduce the computational time \citep{griffies_JPO98}, |
---|
577 | but it is not of practical use as usually $A \neq A_e$. |
---|
578 | Nevertheless this property can be used to choose a discret form of \autoref{eq:eiv_skew_continuous} which |
---|
579 | is consistent with the iso-neutral operator \autoref{eq:Gf_operator}. |
---|
580 | Using the slopes \autoref{eq:Gf_slopes} and defining $A_e$ at $T$-point(\ie\ as $A$, |
---|
581 | the eddy diffusivity coefficient), the resulting discret form is given by: |
---|
582 | \begin{equation} |
---|
583 | \label{eq:eiv_skew} |
---|
584 | \textbf{F}_{eiv}^T \equiv \frac{1}{4} \left( |
---|
585 | \begin{aligned} |
---|
586 | \sum_{\substack{i_p,\,k_p}} & |
---|
587 | +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k} |
---|
588 | \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ \\ |
---|
589 | \sum_{\substack{i_p,\,k_p}} & |
---|
590 | - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p} |
---|
591 | \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \delta_{i+i_p}[T^{k+1/2-k_p}] |
---|
592 | \end{aligned} |
---|
593 | \right) |
---|
594 | \end{equation} |
---|
595 | Note that \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells. |
---|
596 | In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces must be added to |
---|
597 | $\mathbb{R}$ for the discret form to be exact. |
---|
598 | |
---|
599 | Such a choice of discretisation is consistent with the iso-neutral operator as |
---|
600 | it uses the same definition for the slopes. |
---|
601 | It also ensures the conservation of the tracer variance (see Appendix \autoref{apdx:eiv_skew}), |
---|
602 | \ie\ it does not include a diffusive component but is a "pure" advection term. |
---|
603 | |
---|
604 | $\ $\newpage %force an empty line |
---|
605 | % ================================================================ |
---|
606 | % Discrete Invariants of the iso-neutral diffrusion |
---|
607 | % ================================================================ |
---|
608 | \subsection{Discrete invariants of the iso-neutral diffrusion} |
---|
609 | \label{subsec:Gf_operator} |
---|
610 | |
---|
611 | Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane. |
---|
612 | |
---|
613 | This part will be moved in an Appendix. |
---|
614 | |
---|
615 | The continuous property to be demonstrated is: |
---|
616 | \[ |
---|
617 | \int_D D_l^T \; T \;dv \leq 0 |
---|
618 | \] |
---|
619 | The discrete form of its left hand side is obtained using \autoref{eq:TRIADS_iso_flux} |
---|
620 | |
---|
621 | \begin{align*} |
---|
622 | &\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\ |
---|
623 | &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
624 | \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
---|
625 | + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\ |
---|
626 | &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
627 | {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] |
---|
628 | + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\ |
---|
629 | &\equiv -\sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
630 | \frac{ _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{1u}}_{\,i+1/2}^{\,k} } \ \delta_{i+1/2} [T] |
---|
631 | - { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; |
---|
632 | \frac{ _i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{3w}}_{\,i}^{\,k+1/2} } \ \delta_{k+1/2} [T] |
---|
633 | \right\} \\ |
---|
634 | % |
---|
635 | \allowdisplaybreaks |
---|
636 | \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} |
---|
637 | % |
---|
638 | &\equiv -\sum_{i,k} |
---|
639 | \begin{Bmatrix} |
---|
640 | &\ \ \Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{-1/2} (T) } |
---|
641 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
642 | & -\ \ {_{i}^{k+1} \mathbb{R}_{-1/2}^{-1/2}} |
---|
643 | & {_{i}^{k+1} \mathbb{T}_{-1/2}^{-1/2} (T) } |
---|
644 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
---|
645 | & \\ |
---|
646 | &+\Bigl( \ \;\; { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } |
---|
647 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
648 | & -\ \ {_i^{k+1} \mathbb{R}_{+1/2}^{-1/2}} |
---|
649 | & { _i^{k+1} \mathbb{T}_{+1/2}^{-1/2} (T) } |
---|
650 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
---|
651 | & \\ |
---|
652 | &+\Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } |
---|
653 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
654 | & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{-1/2}^{+1/2}} |
---|
655 | & \ \;\;{_{i}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } |
---|
656 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
---|
657 | & \\ |
---|
658 | &+\Bigl( \ \;\; { _{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } |
---|
659 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
660 | & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{+1/2}^{+1/2}} |
---|
661 | & \ \;\;{_{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } |
---|
662 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) \\ |
---|
663 | \end{Bmatrix} |
---|
664 | % |
---|
665 | \allowdisplaybreaks |
---|
666 | \intertext{ |
---|
667 | The summation is done over all $i$ and $k$ indices, |
---|
668 | it is therefore possible to introduce a shift of $-1$ either in $i$ or $k$ direction in order to |
---|
669 | regroup all the terms of the summation by triad at a ($i$,$k$) point. |
---|
670 | In other words, we regroup all the terms in the neighbourhood that contain a triad at the same ($i$,$k$) indices. |
---|
671 | It becomes: |
---|
672 | } |
---|
673 | % |
---|
674 | &\equiv -\sum_{i,k} |
---|
675 | \begin{Bmatrix} |
---|
676 | &\ \ \Bigl( {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } |
---|
677 | &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} |
---|
678 | & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} |
---|
679 | & {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } |
---|
680 | &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr) |
---|
681 | & \\ |
---|
682 | &+\Bigl( { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } |
---|
683 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
684 | & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} |
---|
685 | & { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } |
---|
686 | &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr) |
---|
687 | & \\ |
---|
688 | &+\Bigl( {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } |
---|
689 | &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} |
---|
690 | & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} |
---|
691 | & {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } |
---|
692 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) |
---|
693 | & \\ |
---|
694 | &+\Bigl( { _i^k \mathbb{T}_{+1/2}^{+1/2} (T) } |
---|
695 | &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
696 | & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} |
---|
697 | & {_i^k \mathbb{T}_{+1/2}^{+1/2} (T) } |
---|
698 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) \\ |
---|
699 | \end{Bmatrix} \\ |
---|
700 | % |
---|
701 | \allowdisplaybreaks |
---|
702 | \intertext{ |
---|
703 | Then outing in factor the triad in each of the four terms of the summation and |
---|
704 | substituting the triads by their expression given in \autoref{eq:Gf_triads}. |
---|
705 | It becomes: |
---|
706 | } |
---|
707 | % |
---|
708 | &\equiv -\sum_{i,k} |
---|
709 | \begin{Bmatrix} |
---|
710 | &\ \ \Bigl( \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} |
---|
711 | & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} |
---|
712 | &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)^2 |
---|
713 | & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k} \ A_i^k |
---|
714 | & \\ |
---|
715 | &+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
716 | & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} |
---|
717 | &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)^2 |
---|
718 | & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k} \ A_i^k |
---|
719 | & \\ |
---|
720 | &+\Bigl( \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} |
---|
721 | & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} |
---|
722 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)^2 |
---|
723 | & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k} \ A_i^k |
---|
724 | & \\ |
---|
725 | &+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} |
---|
726 | & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} |
---|
727 | &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)^2 |
---|
728 | & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k} \ A_i^k \\ |
---|
729 | \end{Bmatrix} |
---|
730 | \\ |
---|
731 | & \\ |
---|
732 | % |
---|
733 | &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
734 | \begin{matrix} |
---|
735 | &\Bigl( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} |
---|
736 | & -\ \ {_i^k \mathbb{R}_{i_p}^{k_p}} |
---|
737 | &\frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}} \Bigr)^2 |
---|
738 | & \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \ \ |
---|
739 | \end{matrix} |
---|
740 | \right\} |
---|
741 | \quad \leq 0 |
---|
742 | \end{align*} |
---|
743 | The last inequality is obviously obtained as we succeed in obtaining a negative summation of square quantities. |
---|
744 | |
---|
745 | Note that, if instead of multiplying $D_l^T$ by $T$, we were using another tracer field, let say $S$, |
---|
746 | then the previous demonstration would have let to: |
---|
747 | \begin{align*} |
---|
748 | \int_D S \; D_l^T \;dv &\equiv \sum_{i,k} \left\{ S \ D_l^T \ b_T \right\} \\ |
---|
749 | &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
---|
750 | \left( \frac{ \delta_{i +i_p} [S] }{{e_{1u} }_{\,i+i_p}^{\,k}} |
---|
751 | - {_i^k \mathbb{R}_{i_p}^{k_p}} |
---|
752 | \frac{ \delta_{k+k_p} [S] }{{e_{3w}}_{\,i}^{\,k+k_p}} \right) \right. \\ |
---|
753 | & \qquad \qquad \qquad \ \left. |
---|
754 | \left( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} |
---|
755 | - {_i^k \mathbb{R}_{i_p}^{k_p}} |
---|
756 | \frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}} \right) |
---|
757 | \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \ |
---|
758 | \right\} |
---|
759 | % |
---|
760 | \allowdisplaybreaks |
---|
761 | \intertext{ |
---|
762 | which, by applying the same operation as before but in reverse order, leads to: |
---|
763 | } |
---|
764 | % |
---|
765 | &\equiv \sum_{i,k} \left\{ D_l^S \ T \ b_T \right\} |
---|
766 | \end{align*} |
---|
767 | This means that the iso-neutral operator is self-adjoint. |
---|
768 | There is no need to develop a specific to obtain it. |
---|
769 | |
---|
770 | \newpage |
---|
771 | |
---|
772 | % ================================================================ |
---|
773 | % Discrete Invariants of the skew flux formulation |
---|
774 | % ================================================================ |
---|
775 | \subsection{Discrete invariants of the skew flux formulation} |
---|
776 | \label{subsec:eiv_skew} |
---|
777 | |
---|
778 | Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. |
---|
779 | |
---|
780 | This have to be moved in an Appendix. |
---|
781 | |
---|
782 | The continuous property to be demonstrated is: |
---|
783 | \begin{align*} |
---|
784 | \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv \equiv 0 |
---|
785 | \end{align*} |
---|
786 | The discrete form of its left hand side is obtained using \autoref{eq:eiv_skew} |
---|
787 | \begin{align*} |
---|
788 | \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; |
---|
789 | \delta_i &\left[ |
---|
790 | {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} |
---|
791 | \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] |
---|
792 | \right] \; T_i^k \\ |
---|
793 | - \delta_k &\left[ |
---|
794 | {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} |
---|
795 | \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] |
---|
796 | \right] \; T_i^k \ \Biggr\} |
---|
797 | \end{align*} |
---|
798 | apply the adjoint of delta operator, it becomes |
---|
799 | \begin{align*} |
---|
800 | \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; |
---|
801 | &\left( |
---|
802 | {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} |
---|
803 | \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] |
---|
804 | \right) \; \delta_{i+1/2}[T^{k}] \\ |
---|
805 | - &\left( |
---|
806 | {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} |
---|
807 | \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] |
---|
808 | \right) \; \delta_{k+1/2}[T_{i}] \ \Biggr\} |
---|
809 | \end{align*} |
---|
810 | Expending the summation on $i_p$ and $k_p$, it becomes: |
---|
811 | \begin{align*} |
---|
812 | \begin{matrix} |
---|
813 | &\sum\limits_{i,k} \Bigl\{ |
---|
814 | &+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} |
---|
815 | &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{-1/2}} &\delta_{k-1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
816 | &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} |
---|
817 | &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{-1/2}} &\delta_{k-1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
818 | &&+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} |
---|
819 | &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{+1/2}} &\delta_{k+1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
820 | &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} |
---|
821 | &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{k+1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
822 | % |
---|
823 | &&-{e_{2u}}_i^{k+1} &{A_{e}}_i^{k+1} |
---|
824 | &{_i^{k+1} \mathbb{R}_{-1/2}^{- 1/2}} &\delta_{i-1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\ |
---|
825 | &&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} |
---|
826 | &{\ \ \;_i^k \mathbb{R}_{-1/2}^{+1/2}} &\delta_{i-1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] &\\ |
---|
827 | &&-{e_{2u}}_i^{k+1 } &{A_{e}}_i^{k+1} |
---|
828 | &{_i^{k+1} \mathbb{R}_{+1/2}^{- 1/2}} &\delta_{i+1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\ |
---|
829 | &&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} |
---|
830 | &{\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{i+1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] |
---|
831 | &\Bigr\} \\ |
---|
832 | \end{matrix} |
---|
833 | \end{align*} |
---|
834 | The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the same but of opposite signs, |
---|
835 | they cancel out. |
---|
836 | Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{-1/2}}$. |
---|
837 | The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{-1/2}}$ are the same but both of opposite signs and |
---|
838 | shifted by 1 in $k$ direction. |
---|
839 | When summing over $k$ they cancel out with the neighbouring grid points. |
---|
840 | Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{+1/2}}$ in the $i$ direction. |
---|
841 | Therefore the sum over the domain is zero, |
---|
842 | \ie\ the variance of the tracer is preserved by the discretisation of the skew fluxes. |
---|
843 | |
---|
844 | \biblio |
---|
845 | |
---|
846 | \pindex |
---|
847 | |
---|
848 | \end{document} |
---|