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 | \chapter{Curvilinear $s-$Coordinate Equations} |
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6 | \label{apdx:SCOORD} |
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7 | |
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8 | % {\em 4.0} & {\em Mike Bell} & {\em review} \\ |
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9 | % {\em 3.x} & {\em Gurvan Madec} & {\em original} \\ |
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10 | |
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11 | \thispagestyle{plain} |
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12 | |
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13 | \chaptertoc |
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14 | |
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15 | \paragraph{Changes record} ~\\ |
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16 | |
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17 | {\footnotesize |
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18 | \begin{tabularx}{\textwidth}{l||X|X} |
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19 | Release & Author(s) & Modifications \\ |
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20 | \hline |
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21 | {\em 4.0} & {\em ...} & {\em ...} \\ |
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22 | {\em 3.6} & {\em ...} & {\em ...} \\ |
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23 | {\em 3.4} & {\em ...} & {\em ...} \\ |
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24 | {\em <=3.4} & {\em ...} & {\em ...} |
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25 | \end{tabularx} |
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26 | } |
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27 | |
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28 | \clearpage |
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29 | |
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30 | \section{Chain rule for $s-$coordinates} |
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31 | \label{sec:SCOORD_chain} |
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32 | |
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33 | In order to establish the set of Primitive Equation in curvilinear $s$-coordinates |
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34 | (\ie\ an orthogonal curvilinear coordinate in the horizontal and |
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35 | an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical), |
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36 | we start from the set of equations established in \autoref{subsec:MB_zco_Eq} for |
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37 | the special case $k = z$ and thus $e_3 = 1$, |
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38 | and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$. |
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39 | Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and |
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40 | the horizontal slope of $s-$surfaces by: |
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41 | \begin{equation} |
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42 | \label{eq:SCOORD_s_slope} |
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43 | \sigma_1 =\frac{1}{e_1 } \; \left. {\frac{\partial z}{\partial i}} \right|_s |
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44 | \quad \text{and} \quad |
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45 | \sigma_2 =\frac{1}{e_2 } \; \left. {\frac{\partial z}{\partial j}} \right|_s . |
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46 | \end{equation} |
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47 | |
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48 | The model fields (e.g. pressure $p$) can be viewed as functions of $(i,j,z,t)$ (e.g. $p(i,j,z,t)$) or as |
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49 | functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of |
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50 | these fields. Any ``infinitesimal'' change in $\bullet$ can be written in two forms: |
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51 | \begin{equation} |
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52 | \label{eq:SCOORD_s_infin_changes} |
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53 | \begin{aligned} |
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54 | & \delta \bullet = \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t} |
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55 | + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t} |
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56 | + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t} |
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57 | + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,s} , \\ |
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58 | & \delta \bullet = \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t} |
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59 | + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t} |
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60 | + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t} |
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61 | + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,z} . |
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62 | \end{aligned} |
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63 | \end{equation} |
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64 | Using the first form and considering a change $\delta i$ with $j, z$ and $t$ held constant, shows that |
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65 | \begin{equation} |
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66 | \label{eq:SCOORD_s_chain_rule1} |
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67 | \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,z,t} = |
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68 | \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,s,t} |
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69 | + \left. {\frac{\partial s }{\partial i}} \right|_{j,z,t} \; |
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70 | \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} . |
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71 | \end{equation} |
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72 | The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces, |
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73 | (\autoref{eq:SCOORD_s_slope}), by applying the second of (\autoref{eq:SCOORD_s_infin_changes}) with $\bullet$ set to |
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74 | $s$ and $j, t$ held constant |
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75 | \begin{equation} |
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76 | \label{eq:SCOORD_delta_s} |
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77 | \delta s|_{j,t} = |
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78 | \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} |
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79 | + \delta z \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} . |
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80 | \end{equation} |
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81 | Choosing to look at a direction in the $(i,z)$ plane in which $\delta s = 0$ and using |
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82 | (\autoref{eq:SCOORD_s_slope}) we obtain |
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83 | \begin{equation} |
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84 | \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} = |
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85 | - \left. \frac{ \partial z }{\partial i} \right|_{j,s,t} \; |
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86 | \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} |
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87 | = - \frac{e_1 }{e_3 }\sigma_1 . |
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88 | \label{eq:SCOORD_ds_di_z} |
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89 | \end{equation} |
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90 | Another identity, similar in form to (\autoref{eq:SCOORD_ds_di_z}), can be derived |
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91 | by choosing $\bullet$ to be $s$ and using the second form of (\autoref{eq:SCOORD_s_infin_changes}) to consider |
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92 | changes in which $i , j$ and $s$ are constant. This shows that |
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93 | \begin{equation} |
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94 | \label{eq:SCOORD_w_in_s} |
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95 | w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} = |
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96 | - \left. \frac{ \partial z }{\partial s} \right|_{i,j,t} |
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97 | \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} |
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98 | = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} . |
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99 | \end{equation} |
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100 | |
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101 | In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is |
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102 | usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish |
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103 | the model equations in the curvilinear $s-$coordinate system are: |
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104 | \begin{equation} |
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105 | \label{eq:SCOORD_s_chain_rule2} |
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106 | \begin{aligned} |
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107 | &\left. {\frac{\partial \bullet }{\partial t}} \right|_z = |
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108 | \left. {\frac{\partial \bullet }{\partial t}} \right|_s |
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109 | + \frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial t} , \\ |
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110 | &\left. {\frac{\partial \bullet }{\partial i}} \right|_z = |
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111 | \left. {\frac{\partial \bullet }{\partial i}} \right|_s |
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112 | +\frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial i}= |
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113 | \left. {\frac{\partial \bullet }{\partial i}} \right|_s |
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114 | -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} , \\ |
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115 | &\left. {\frac{\partial \bullet }{\partial j}} \right|_z = |
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116 | \left. {\frac{\partial \bullet }{\partial j}} \right|_s |
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117 | + \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= |
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118 | \left. {\frac{\partial \bullet }{\partial j}} \right|_s |
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119 | - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} , \\ |
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120 | &\;\frac{\partial \bullet }{\partial z} \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} . |
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121 | \end{aligned} |
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122 | \end{equation} |
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123 | |
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124 | %% ================================================================================================= |
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125 | \section{Continuity equation in $s-$coordinates} |
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126 | \label{sec:SCOORD_continuity} |
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127 | |
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128 | Using (\autoref{eq:SCOORD_s_chain_rule1}) and |
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129 | the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, |
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130 | the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to |
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131 | obtain its expression in the curvilinear $s-$coordinate system: |
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132 | |
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133 | \begin{subequations} |
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134 | \begin{align*} |
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135 | { |
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136 | \begin{array}{*{20}l} |
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137 | \nabla \cdot {\mathrm {\mathbf U}} |
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138 | &= \frac{1}{e_1 \,e_2 } \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z |
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139 | +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z \right] |
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140 | + \frac{\partial w}{\partial z} \\ \\ |
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141 | & = \frac{1}{e_1 \,e_2 } \left[ |
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142 | \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s |
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143 | - \frac{e_1 }{e_3 } \sigma_1 \frac{\partial (e_2 \,u)}{\partial s} |
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144 | + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s |
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145 | - \frac{e_2 }{e_3 } \sigma_2 \frac{\partial (e_1 \,v)}{\partial s} \right] |
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146 | + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\ \\ |
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147 | & = \frac{1}{e_1 \,e_2 } \left[ |
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148 | \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s |
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149 | + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s \right] |
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150 | + \frac{1}{e_3 }\left[ \frac{\partial w}{\partial s} |
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151 | - \sigma_1 \frac{\partial u}{\partial s} |
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152 | - \sigma_2 \frac{\partial v}{\partial s} \right] \\ \\ |
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153 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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154 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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155 | -\left. e_2 \,u \frac{\partial e_3 }{\partial i} \right|_s |
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156 | + \left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s |
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157 | - \left. e_1 v \frac{\partial e_3 }{\partial j} \right|_s \right] \\ |
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158 | & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad |
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159 | + \frac{1}{e_3 } \left[ \frac{\partial w}{\partial s} |
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160 | - \sigma_1 \frac{\partial u}{\partial s} |
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161 | - \sigma_2 \frac{\partial v}{\partial s} \right] \\ |
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162 | % |
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163 | \intertext{Noting that $ |
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164 | \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s |
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165 | =\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s |
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166 | =\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right) |
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167 | =\frac{\partial \sigma_1}{\partial s} |
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168 | $ and $ |
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169 | \frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s |
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170 | =\frac{\partial \sigma_2}{\partial s} |
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171 | $, it becomes:} |
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172 | % |
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173 | \nabla \cdot {\mathrm {\mathbf U}} |
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174 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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175 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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176 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] \\ |
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177 | & \qquad \qquad \qquad \qquad \quad |
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178 | +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma_1 }{\partial s}-v\frac{\partial \sigma_2 }{\partial s}-\sigma_1 \frac{\partial u}{\partial s}-\sigma_2 \frac{\partial v}{\partial s}} \right] \\ |
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179 | \\ |
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180 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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181 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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182 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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183 | + \frac{1}{e_3 } \; \frac{\partial}{\partial s} \left[ w - u\;\sigma_1 - v\;\sigma_2 \right] |
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184 | \end{array} |
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185 | } |
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186 | \end{align*} |
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187 | \end{subequations} |
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188 | |
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189 | Here, $w$ is the vertical velocity relative to the $z-$coordinate system. |
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190 | Using the first form of (\autoref{eq:SCOORD_s_infin_changes}) |
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191 | and the definitions (\autoref{eq:SCOORD_s_slope}) and (\autoref{eq:SCOORD_w_in_s}) for $\sigma_1$, $\sigma_2$ and $w_s$, |
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192 | one can show that the vertical velocity, $w_p$ of a point |
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193 | moving with the horizontal velocity of the fluid along an $s$ surface is given by |
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194 | \begin{equation} |
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195 | \label{eq:SCOORD_w_p} |
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196 | \begin{split} |
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197 | w_p = & \left. \frac{ \partial z }{\partial t} \right|_s |
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198 | + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s |
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199 | + \frac{v}{e_2} \left. \frac{ \partial z }{\partial j} \right|_s \\ |
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200 | = & w_s + u \sigma_1 + v \sigma_2 . |
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201 | \end{split} |
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202 | \end{equation} |
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203 | The vertical velocity across this surface is denoted by |
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204 | \begin{equation} |
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205 | \label{eq:SCOORD_w_s} |
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206 | \omega = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v ) . |
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207 | \end{equation} |
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208 | Hence |
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209 | \begin{equation} |
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210 | \frac{1}{e_3 } \frac{\partial}{\partial s} \left[ w - u\;\sigma_1 - v\;\sigma_2 \right] = |
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211 | \frac{1}{e_3 } \frac{\partial}{\partial s} \left[ \omega + w_s \right] = |
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212 | \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s} |
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213 | + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] = |
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214 | \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s |
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215 | \end{equation} |
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216 | |
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217 | Using (\autoref{eq:SCOORD_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain |
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218 | our final expression for the divergence of the velocity in the curvilinear $s-$coordinate system: |
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219 | \begin{equation} |
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220 | \nabla \cdot {\mathrm {\mathbf U}} = |
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221 | \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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222 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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223 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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224 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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225 | + \frac{1}{e_3 } \left. \frac{\partial e_3}{\partial t} \right|_s . |
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226 | \end{equation} |
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227 | |
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228 | As a result, the continuity equation \autoref{eq:MB_PE_continuity} in the $s-$coordinates is: |
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229 | \begin{equation} |
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230 | \label{eq:SCOORD_sco_Continuity} |
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231 | \frac{1}{e_3 } \frac{\partial e_3}{\partial t} |
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232 | + \frac{1}{e_1 \,e_2 \,e_3 }\left[ |
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233 | {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s |
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234 | + \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right] |
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235 | +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 . |
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236 | \end{equation} |
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237 | An additional term has appeared that takes into account |
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238 | the contribution of the time variation of the vertical coordinate to the volume budget. |
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239 | |
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240 | %% ================================================================================================= |
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241 | \section{Momentum equation in $s-$coordinate} |
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242 | \label{sec:SCOORD_momentum} |
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243 | |
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244 | Here we only consider the first component of the momentum equation, |
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245 | the generalization to the second one being straightforward. |
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246 | |
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247 | $\bullet$ \textbf{Total derivative in vector invariant form} |
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248 | |
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249 | Let us consider \autoref{eq:MB_dyn_vect}, the first component of the momentum equation in the vector invariant form. |
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250 | Its total $z-$coordinate time derivative, |
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251 | $\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain |
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252 | its expression in the curvilinear $s-$coordinate system: |
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253 | |
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254 | \begin{subequations} |
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255 | \begin{align*} |
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256 | { |
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257 | \begin{array}{*{20}l} |
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258 | \left. \frac{D u}{D t} \right|_z |
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259 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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260 | - \left. \zeta \right|_z v |
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261 | + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z |
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262 | + w \;\frac{\partial u}{\partial z} \\ \\ |
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263 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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264 | - \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z |
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265 | -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v |
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266 | + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z |
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267 | + w \;\frac{\partial u}{\partial z} \\ |
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268 | % |
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269 | \intertext{introducing the chain rule (\autoref{eq:SCOORD_s_chain_rule1}) } |
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270 | % |
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271 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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272 | - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s |
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273 | -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right. |
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274 | \left. {-\frac{e_1}{e_3}\sigma_1 \frac{\partial (e_2 \,v)}{\partial s} |
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275 | +\frac{e_2}{e_3}\sigma_2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v \\ |
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276 | & \qquad \qquad \qquad \qquad |
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277 | { |
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278 | + \frac{1}{2e_1} \left( \left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s |
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279 | - \frac{e_1}{e_3}\sigma_1 \frac{\partial (u^2+v^2)}{\partial s} \right) |
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280 | + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} |
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281 | } \\ \\ |
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282 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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283 | - \left. \zeta \right|_s \;v |
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284 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ |
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285 | &\qquad \qquad \qquad \quad |
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286 | + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} |
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287 | + \left[ {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s} |
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288 | - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v |
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289 | - \frac{\sigma_1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ \\ |
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290 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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291 | - \left. \zeta \right|_s \;v |
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292 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ |
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293 | &\qquad \qquad \qquad \quad |
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294 | + \frac{1}{e_3} \left[ {w\frac{\partial u}{\partial s} |
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295 | +\sigma_1 v\frac{\partial v}{\partial s} - \sigma_2 v\frac{\partial u}{\partial s} |
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296 | - \sigma_1 u\frac{\partial u}{\partial s} - \sigma_1 v\frac{\partial v}{\partial s}} \right] \\ \\ |
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297 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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298 | - \left. \zeta \right|_s \;v |
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299 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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300 | + \frac{1}{e_3} \left[ w - \sigma_2 v - \sigma_1 u \right] |
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301 | \; \frac{\partial u}{\partial s} . \\ |
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302 | % |
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303 | \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{eq:SCOORD_w_s}) } |
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304 | % |
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305 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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306 | - \left. \zeta \right|_s \;v |
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307 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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308 | + \frac{1}{e_3 } \left( \omega + w_s \right) \frac{\partial u}{\partial s} \\ |
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309 | \end{array} |
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310 | } |
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311 | \end{align*} |
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312 | \end{subequations} |
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313 | Applying the time derivative chain rule (first equation of (\autoref{eq:SCOORD_s_chain_rule1})) to $u$ and |
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314 | using (\autoref{eq:SCOORD_w_in_s}) provides the expression of the last term of the right hand side, |
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315 | \[ |
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316 | { |
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317 | \begin{array}{*{20}l} |
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318 | \frac{w_s}{e_3} \;\frac{\partial u}{\partial s} |
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319 | = - \left. \frac{\partial s}{\partial t} \right|_z \; \frac{\partial u }{\partial s} |
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320 | = \left. {\frac{\partial u }{\partial t}} \right|_s - \left. {\frac{\partial u }{\partial t}} \right|_z \ . |
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321 | \end{array} |
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322 | } |
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323 | \] |
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324 | This leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, |
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325 | \ie\ the total $s-$coordinate time derivative : |
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326 | \begin{align} |
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327 | \label{eq:SCOORD_sco_Dt_vect} |
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328 | \left. \frac{D u}{D t} \right|_s |
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329 | = \left. {\frac{\partial u }{\partial t}} \right|_s |
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330 | - \left. \zeta \right|_s \;v |
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331 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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332 | + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} . |
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333 | \end{align} |
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334 | Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in |
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335 | $z-$ and $s-$coordinates. |
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336 | This is not the case for the flux form as shown in next paragraph. |
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337 | |
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338 | $\bullet$ \textbf{Total derivative in flux form} |
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339 | |
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340 | Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish. |
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341 | Following the procedure used to establish (\autoref{eq:MB_flux_form}), it can be transformed into : |
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342 | % \begin{subequations} |
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343 | \begin{align*} |
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344 | { |
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345 | \begin{array}{*{20}l} |
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346 | \left. \frac{D u}{D t} \right|_s &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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347 | & - \zeta \;v |
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348 | + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} |
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349 | + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s} \\ \\ |
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350 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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351 | &+\frac{1}{e_1\;e_2} \left( \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i} |
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352 | + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j} \right) |
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353 | + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\ |
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354 | &&- \,u \left[ \frac{1}{e_1 e_2 } \left( \frac{\partial(e_2 u)}{\partial i} |
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355 | + \frac{\partial(e_1 v)}{\partial j} \right) |
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356 | + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ \\ |
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357 | &&- \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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358 | -u \;\frac{\partial e_1 }{\partial j} \right) . \\ |
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359 | \end{array} |
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360 | } |
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361 | \end{align*} |
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362 | Introducing the vertical scale factor inside the horizontal derivative of the first two terms |
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363 | (\ie\ the horizontal divergence), it becomes : |
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364 | \begin{align*} |
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365 | { |
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366 | \begin{array}{*{20}l} |
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367 | % \begin{align*} {\begin{array}{*{20}l} |
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368 | % {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s |
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369 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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370 | &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u^2 )}{\partial i} |
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371 | + \frac{\partial( e_1 e_3 \,u v )}{\partial j} |
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372 | - e_2 u u \frac{\partial e_3}{\partial i} |
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373 | - e_1 u v \frac{\partial e_3 }{\partial j} \right) |
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374 | + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\ |
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375 | && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} |
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376 | + \frac{\partial(e_1 e_3 \, v)}{\partial j} |
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377 | - e_2 u \;\frac{\partial e_3 }{\partial i} |
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378 | - e_1 v \;\frac{\partial e_3 }{\partial j} \right) |
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379 | + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ \\ |
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380 | && - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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381 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ \\ |
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382 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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383 | &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u\,u )}{\partial i} |
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384 | + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j} \right) |
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385 | + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\ |
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386 | && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} |
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387 | + \frac{\partial(e_1 e_3 \, v)}{\partial j} \right) |
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388 | + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] |
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389 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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390 | -u \;\frac{\partial e_1 }{\partial j} \right) . \\ |
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391 | % |
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392 | \intertext {Introducing a more compact form for the divergence of the momentum fluxes, |
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393 | and using (\autoref{eq:SCOORD_sco_Continuity}), the $s-$coordinate continuity equation, |
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394 | it becomes : } |
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395 | % |
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396 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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397 | &+ \left. \nabla \cdot \left( {{\mathrm {\mathbf U}}\,u} \right) \right|_s |
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398 | + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t} |
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399 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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400 | -u \;\frac{\partial e_1 }{\partial j} \right) |
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401 | \\ |
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402 | \end{array} |
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403 | } |
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404 | \end{align*} |
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405 | which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, |
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406 | \ie\ the total $s-$coordinate time derivative in flux form: |
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407 | \begin{flalign} |
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408 | \label{eq:SCOORD_sco_Dt_flux} |
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409 | \left. \frac{D u}{D t} \right|_s = \frac{1}{e_3} \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s |
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410 | + \left. \nabla \cdot \left( {{\mathrm {\mathbf U}}\,u} \right) \right|_s |
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411 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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412 | -u \;\frac{\partial e_1 }{\partial j} \right). |
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413 | \end{flalign} |
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414 | which is the total time derivative expressed in the curvilinear $s-$coordinate system. |
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415 | It has the same form as in the $z-$coordinate but for |
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416 | the vertical scale factor that has appeared inside the time derivative which |
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417 | comes from the modification of (\autoref{eq:SCOORD_sco_Continuity}), |
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418 | the continuity equation. |
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419 | |
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420 | $\bullet$ \textbf{horizontal pressure gradient} |
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421 | |
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422 | The horizontal pressure gradient term can be transformed as follows: |
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423 | \[ |
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424 | \begin{split} |
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425 | -\frac{1}{\rho_o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z |
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426 | & =-\frac{1}{\rho_o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial p}{\partial s}} \right] \\ |
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427 | & =-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma_1 }{\rho_o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\ |
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428 | &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1 . |
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429 | \end{split} |
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430 | \] |
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431 | Applying similar manipulation to the second component and |
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432 | replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{eq:SCOORD_s_slope}, it becomes: |
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433 | \begin{equation} |
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434 | \label{eq:SCOORD_grad_p_1} |
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435 | \begin{split} |
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436 | -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z |
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437 | &=-\frac{1}{\rho_o \,e_1 } \left( \left. {\frac{\partial p}{\partial i}} \right|_s |
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438 | + g\;\rho \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) \\ |
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439 | % |
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440 | -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z |
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441 | &=-\frac{1}{\rho_o \,e_2 } \left( \left. {\frac{\partial p}{\partial j}} \right|_s |
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442 | + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) . \\ |
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443 | \end{split} |
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444 | \end{equation} |
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445 | |
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446 | An additional term appears in (\autoref{eq:SCOORD_grad_p_1}) which accounts for |
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447 | the tilt of $s-$surfaces with respect to geopotential $z-$surfaces. |
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448 | |
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449 | As in $z$-coordinate, |
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450 | the horizontal pressure gradient can be split in two parts following \citet{marsaleix.auclair.ea_OM08}. |
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451 | Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$, |
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452 | and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. |
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453 | The pressure is then given by: |
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454 | \[ |
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455 | \begin{split} |
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456 | p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \rho_o \left( d + 1 \right) \; e_3 \; dk \\ |
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457 | &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + \rho_o g \, \int_z^\eta e_3 \; dk . |
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458 | \end{split} |
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459 | \] |
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460 | Therefore, $p$ and $p_h'$ are linked through: |
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461 | \begin{equation} |
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462 | \label{eq:SCOORD_pressure} |
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463 | p = \rho_o \; p_h' + \rho_o \, g \, ( \eta - z ) |
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464 | \end{equation} |
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465 | and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is: |
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466 | \[ |
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467 | \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 . |
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468 | \] |
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469 | |
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470 | Substituing \autoref{eq:SCOORD_pressure} in \autoref{eq:SCOORD_grad_p_1} and |
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471 | using the definition of the density anomaly it becomes an expression in two parts: |
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472 | \begin{equation} |
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473 | \label{eq:SCOORD_grad_p_2} |
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474 | \begin{split} |
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475 | -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z |
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476 | &=-\frac{1}{e_1 } \left( \left. {\frac{\partial p_h'}{\partial i}} \right|_s |
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477 | + g\; d \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} , \\ |
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478 | % |
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479 | -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z |
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480 | &=-\frac{1}{e_2 } \left( \left. {\frac{\partial p_h'}{\partial j}} \right|_s |
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481 | + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} . \\ |
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482 | \end{split} |
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483 | \end{equation} |
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484 | This formulation of the pressure gradient is characterised by the appearance of |
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485 | a term depending on the sea surface height only |
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486 | (last term on the right hand side of expression \autoref{eq:SCOORD_grad_p_2}). |
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487 | This term will be loosely termed \textit{surface pressure gradient} whereas |
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488 | the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to |
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489 | the $z$-coordinate formulation. |
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490 | In fact, the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, |
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491 | and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration. |
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492 | |
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493 | $\bullet$ \textbf{The other terms of the momentum equation} |
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494 | |
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495 | The coriolis and forcing terms as well as the the vertical physics remain unchanged as |
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496 | they involve neither time nor space derivatives. |
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497 | The form of the lateral physics is discussed in \autoref{apdx:DIFFOPERS}. |
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498 | |
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499 | $\bullet$ \textbf{Full momentum equation} |
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500 | |
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501 | To sum up, in a curvilinear $s$-coordinate system, |
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502 | the vector invariant momentum equation solved by the model has the same mathematical expression as |
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503 | the one in a curvilinear $z-$coordinate, except for the pressure gradient term: |
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504 | \begin{subequations} |
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505 | \label{eq:SCOORD_dyn_vect} |
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506 | \begin{multline} |
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507 | \label{eq:SCOORD_PE_dyn_vect_u} |
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508 | \frac{\partial u}{\partial t}= |
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509 | + \left( {\zeta +f} \right)\,v |
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510 | - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right) |
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511 | - \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\ |
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512 | - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) |
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513 | - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} |
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514 | + D_u^{\vect{U}} + F_u^{\vect{U}} , |
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515 | \end{multline} |
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516 | \begin{multline} |
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517 | \label{eq:SCOORD_dyn_vect_v} |
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518 | \frac{\partial v}{\partial t}= |
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519 | - \left( {\zeta +f} \right)\,u |
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520 | - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right) |
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521 | - \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\ |
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522 | - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) |
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523 | - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} |
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524 | + D_v^{\vect{U}} + F_v^{\vect{U}} . |
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525 | \end{multline} |
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526 | \end{subequations} |
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527 | whereas the flux form momentum equation differs from it by |
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528 | the formulation of both the time derivative and the pressure gradient term: |
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529 | \begin{subequations} |
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530 | \label{eq:SCOORD_dyn_flux} |
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531 | \begin{multline} |
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532 | \label{eq:SCOORD_PE_dyn_flux_u} |
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533 | \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t} = |
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534 | - \nabla \cdot \left( {{\mathrm {\mathbf U}}\,u} \right) |
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535 | + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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536 | -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,v \\ |
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537 | - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) |
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538 | - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} |
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539 | + D_u^{\vect{U}} + F_u^{\vect{U}} , |
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540 | \end{multline} |
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541 | \begin{multline} |
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542 | \label{eq:SCOORD_dyn_flux_v} |
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543 | \frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}= |
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544 | - \nabla \cdot \left( {{\mathrm {\mathbf U}}\,v} \right) |
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545 | - \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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546 | -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,u \\ |
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547 | - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) |
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548 | - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} |
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549 | + D_v^{\vect{U}} + F_v^{\vect{U}} . |
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550 | \end{multline} |
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551 | \end{subequations} |
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552 | Both formulation share the same hydrostatic pressure balance expressed in terms of |
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553 | hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: |
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554 | \begin{equation} |
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555 | \label{eq:SCOORD_dyn_zph} |
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556 | \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 . |
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557 | \end{equation} |
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558 | |
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559 | It is important to realize that the change in coordinate system has only concerned the position on the vertical. |
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560 | It has not affected (\textbf{i},\textbf{j},\textbf{k}), the orthogonal curvilinear set of unit vectors. |
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561 | ($u$,$v$) are always horizontal velocities so that their evolution is driven by \emph{horizontal} forces, |
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562 | in particular the pressure gradient. |
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563 | By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component, |
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564 | \ie\ the volume flux across the moving $s$-surfaces per unit horizontal area. |
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565 | |
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566 | %% ================================================================================================= |
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567 | \section{Tracer equation} |
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568 | \label{sec:SCOORD_tracer} |
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569 | |
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570 | The tracer equation is obtained using the same calculation as for the continuity equation and then |
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571 | regrouping the time derivative terms in the left hand side : |
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572 | |
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573 | \begin{multline} |
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574 | \label{eq:SCOORD_tracer} |
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575 | \frac{1}{e_3} \frac{\partial \left( e_3 T \right)}{\partial t} |
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576 | = -\frac{1}{e_1 \,e_2 \,e_3} |
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577 | \left[ \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right) |
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578 | + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] \\ |
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579 | - \frac{1}{e_3} \frac{\partial }{\partial k} \left( Tw \right) |
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580 | + D^{T} +F^{T} |
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581 | \end{multline} |
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582 | |
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583 | The expression for the advection term is a straight consequence of (\autoref{eq:SCOORD_sco_Continuity}), |
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584 | the expression of the 3D divergence in the $s-$coordinates established above. |
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585 | |
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586 | \subinc{\input{../../global/epilogue}} |
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587 | |
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588 | \end{document} |
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