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2 | % ================================================================ |
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3 | % Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations |
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4 | % ================================================================ |
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5 | \chapter{Appendix A : Curvilinear $s$-Coordinate Equations} |
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6 | \label{Apdx_A} |
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7 | \minitoc |
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8 | |
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9 | In order to establish the set of Primitive Equation in curvilinear |
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10 | $s$-coordinates (i.e. orthogonal curvilinear coordinates in the horizontal and |
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11 | $s$-coordinates in the vertical), we start from the set of equation established |
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12 | in {\S}~I.3 for the special case $k = z$ and thus $e_3 = 1$, and we introduce an arbitrary |
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13 | vertical coordinate $s = s(i,j,z)$. Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z)$) and the horizontal slope of $s$-surfaces by : |
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14 | \begin{equation} \label{Apdx_A_A1} |
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15 | \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s |
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16 | \quad \text{and} |
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17 | \sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s |
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18 | \end{equation} |
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19 | |
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20 | The chain rule to establish the model equations in the curvilinear |
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21 | s-coordinate system is: |
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22 | \begin{equation} \label{Apdx_A_A2} |
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23 | \begin{aligned} |
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24 | &\left. {\frac{\partial \bullet }{\partial i}} \right|_z =\left. |
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25 | {\frac{\partial \bullet }{\partial i}} \right|_s +\frac{\partial \bullet |
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26 | }{\partial s}\;\frac{\partial s}{\partial i}=\left. {\frac{\partial \bullet |
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27 | }{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet |
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28 | }{\partial s} \\ |
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29 | &\left. {\frac{\partial \bullet }{\partial j}} \right|_z =\left. |
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30 | {\frac{\partial \bullet }{\partial j}} \right|_s +\frac{\partial \bullet |
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31 | }{\partial s}\;\frac{\partial s}{\partial j}=\left. {\frac{\partial \bullet |
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32 | }{\partial j}} \right|_s -\frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet |
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33 | }{\partial s} \\ |
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34 | &\;\frac{\partial \bullet }{\partial z} =\frac{1}{e_3 }\frac{\partial \bullet |
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35 | }{\partial s} \\ |
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36 | \end{aligned} |
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37 | \end{equation} |
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38 | |
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39 | Using (\ref{Apdx_A_A2}), the divergence of the velocity is transformed as follows: |
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40 | |
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41 | |
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42 | \begin{equation*} |
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43 | \nabla \cdot {\rm {\bf U}}=\frac{1}{e_1 \,e_2 }\left[ {\left. |
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44 | {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z +\left. {\frac{\partial |
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45 | (e_1 \,v)}{\partial j}} \right|_z } \right]+\frac{\partial w}{\partial z} \\ |
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46 | \end{equation*} |
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47 | |
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48 | %\begin{equation} \label{ } |
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49 | \begin{multline*} |
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50 | =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,u)}{\partial i}} |
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51 | \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (e_2 \,u)}{\partial s}} |
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52 | \right. \\ |
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53 | \shoveright { \left. { +\left. {\frac{\partial (e_1 \,v)}{\partial j}} \right|_s -\frac{e_2 }{e_3 }\sigma _2 \frac{\partial (e_1 v)}{\partial s}} \right]+\frac{\partial w}{\partial s}\frac{\partial s}{\partial z} }\\ |
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54 | \end{multline*} |
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55 | %\end{equation} |
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56 | |
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57 | \begin{equation*} |
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58 | %\begin{multline} |
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59 | =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,u)}{\partial i}} |
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60 | \right|_s +\left. {\frac{\partial (e_1 \,v)}{\partial j}} \right|_s } |
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61 | \right]+\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-\sigma _1 |
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62 | \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} |
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63 | \right] |
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64 | %\end{multline} |
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65 | \end{equation*} |
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66 | |
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67 | %\begin{equation} \label{ } |
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68 | \begin{multline*} |
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69 | =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\frac{\partial (e_2 \,e_3 |
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70 | \,u)}{\partial i}} \right|_s -\left. {e_2 \,u\frac{\partial e_3 }{\partial |
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71 | i}} \right|_s +\left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} |
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72 | \right|_s -\left. {e_1 v\frac{\partial e_3 }{\partial j}} \right|_s } |
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73 | \right] \\ |
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74 | \shoveright{ +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-\sigma _1 |
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75 | \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} |
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76 | \right]} \\ |
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77 | \end{multline*} |
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78 | %\end{equation} |
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79 | |
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80 | |
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81 | Noting that $\frac{1}{e_1 }\left. {\frac{\partial e_3 }{\partial i}} |
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82 | \right|_s =\frac{1}{e_1 }\left. {\frac{\partial ^2z}{\partial i\,\partial |
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83 | s}} \right|_s =\frac{\partial }{\partial s}\left( {\frac{1}{e_1 }\left. |
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84 | {\frac{\partial z}{\partial i}} \right|_s } \right)=\frac{\partial \sigma _1 |
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85 | }{\partial s}$ and $\frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} |
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86 | \right|_s =\frac{\partial \sigma _2 }{\partial s}$, it becomes: |
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87 | |
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88 | \begin{multline*} |
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89 | =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\frac{\partial (e_2 \,e_3 |
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90 | \,u)}{\partial i}} \right|_s +\left. {\frac{\partial (e_1 \,e_3 |
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91 | \,v)}{\partial j}} \right|_s } \right] \\ |
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92 | \shoveright{ +\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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93 | \end{multline*} |
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94 | |
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95 | \begin{multline*} |
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96 | =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\frac{\partial (e_2 \,e_3 |
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97 | \,u)}{\partial i}} \right|_s +\left. {\frac{\partial (e_1 \,e_3 |
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98 | \,v)}{\partial j}} \right|_s } \right] \\ |
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99 | \shoveright{ +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial |
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100 | s}-\frac{\partial (u\;\sigma _1 )}{\partial s}-\frac{\partial (v\;\sigma _2 |
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101 | )}{\partial s}} \right]} \\ |
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102 | \end{multline*} |
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103 | |
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104 | Introducing a "vertical" velocity $\omega $ as the velocity normal to $s$-surfaces: |
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105 | |
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106 | \begin{equation} \label{Apdx_A_A3} |
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107 | \omega =w-\sigma _1 \,u-\sigma _2 \,v |
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108 | \end{equation} |
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109 | |
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110 | the divergence of the velocity is given in curvilinear $s$-coordinates by: |
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111 | \begin{equation} \label{Apdx_A_A4} |
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112 | \nabla \cdot {\rm {\bf U}}=\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. |
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113 | {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s +\left. |
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114 | {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } |
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115 | \right]+\frac{1}{e_3 }\frac{\partial \omega }{\partial s} |
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116 | \end{equation} |
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117 | |
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118 | |
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119 | As a result, the continuity equation (I.1.3) in $s$-coordinates becomes: |
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120 | \begin{equation} \label{Apdx_A_A5} |
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121 | \frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\frac{\partial (e_2 \,e_3 |
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122 | \,u)}{\partial i}} \right|_s +\left. {\frac{\partial (e_1 \,e_3 |
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123 | \,v)}{\partial j}} \right|_s } \right]+\frac{1}{e_3 }\frac{\partial \omega |
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124 | }{\partial s}=0 |
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125 | \end{equation} |
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126 | |
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127 | |
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128 | \textbf{Momentum equation:} |
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129 | |
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130 | As an example let us consider (I.3.10), the first component of the momentum |
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131 | equation. Its non linear term can be transformed as follows: |
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132 | |
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133 | \begin{equation*} |
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134 | \begin{aligned} |
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135 | &+\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial |
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136 | (u^2+v^2)}{\partial i}} \right|_z -w\frac{\partial u}{\partial z} \\ |
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137 | &=\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}} |
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138 | \right|_z -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_z } |
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139 | \right]\;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} |
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140 | \right|_z -w\frac{\partial u}{\partial z} |
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141 | \end{aligned} |
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142 | \end{equation*} |
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143 | \begin{multline*} |
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144 | =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}} |
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145 | \right|_s -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_s } |
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146 | \right. \\ |
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147 | \left. {-\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (e_2 \,v)}{\partial s}+\frac{e_2 }{e_3 }\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right]\;v \\ |
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148 | \shoveright{ -\frac{1}{2e_1 }\left( {\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (u^2+v^2)}{\partial s}} \right)-\frac{w}{e_3 }\frac{\partial u}{\partial s} }\\ |
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149 | \end{multline*} |
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150 | |
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151 | \begin{equation*} |
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152 | =\left. \zeta \right|_s \;v-\frac{1}{2e_1 }\left. {\frac{\partial |
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153 | (u^2+v^2)}{\partial i}} \right|_s -\frac{w}{e_3 }\frac{\partial u}{\partial |
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154 | s}-\left[ {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s}-\frac{\sigma |
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155 | _2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v \\ |
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156 | +\frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} |
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157 | \end{equation*} |
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158 | |
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159 | |
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160 | \begin{multline*} |
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161 | =\left. \zeta \right|_s \;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ |
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162 | \shoveright{ -\frac{1}{e_3 }\left[ {w\frac{\partial u}{\partial s}+\sigma _1 v\frac{\partial v}{\partial s}-\sigma _2 v\frac{\partial u}{\partial s}-\sigma _1 u\frac{\partial u}{\partial s}-\sigma _1 v\frac{\partial v}{\partial s}} \right] }\\ |
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163 | \end{multline*} |
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164 | |
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165 | \begin{equation} \label{Apdx_A_A6} |
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166 | =\left. \zeta \right|_s \;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s -\frac{1}{e_3 }\omega \frac{\partial u}{\partial s} |
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167 | \end{equation} |
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168 | |
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169 | Therefore, the non-linear terms of the momentum equation have the same form |
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170 | in $z- $and $s-$coordinates |
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171 | |
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172 | The pressure gradient term can be transformed as follows: |
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173 | \begin{equation} \label{Apdx_A_A7} |
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174 | \begin{split} |
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175 | -\frac{1}{\rho _o e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z& =-\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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176 | & =-\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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177 | &=-\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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178 | \end{split} |
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179 | \end{equation} |
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180 | |
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181 | An additional term appears in (\ref{Apdx_A_A7}) which accounts for the tilt of model |
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182 | levels. |
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183 | |
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184 | \textbf{Tracer equation:} |
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185 | |
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186 | The tracer equation is obtained using the same calculation as for the |
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187 | continuity equation: |
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188 | |
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189 | %\begin{equation} \label{Eq_ } |
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190 | \begin{multline} \label{Apdx_A_A8} |
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191 | \frac{\partial T}{\partial t} = -\frac{1}{e_1 \,e_2 \,e_3 } \left[ {\frac{\partial }{\partial i}} \left( {e_2 \,e_3 \;Tu} \right) + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \;\right .\\ |
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192 | \shoveright{\left . +\frac{\partial }{\partial k} \left( {e_1 \,e_2 \;T\omega } \right) \right] +D^{lT} +D^{vT} }\\ |
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193 | \end{multline} |
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194 | %\end{equation} |
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195 | |
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196 | |
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197 | The expression of the advection term is a straight consequence of (A.4), the |
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198 | expression of the 3D divergence in $s$-coordinates established above. |
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199 | |
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