1 | |
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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{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 | |
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10 | In order to establish the set of Primitive Equation in curvilinear $s$-coordinates |
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11 | ($i.e.$ an orthogonal curvilinear coordinate in the horizontal and $s$-coordinate |
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12 | in the vertical), we start from the set of equations established in \S\ref{PE_zco_Eq} |
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13 | for the special case $k = z$ and thus $e_3 = 1$, and we introduce an arbitrary |
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14 | vertical coordinate $s = s(i,j,z,t)$. Let us define a new vertical scale factor by |
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15 | $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal |
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16 | slope of $s$-surfaces by : |
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17 | \begin{equation} \label{Apdx_A_s_slope} |
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18 | \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s |
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19 | \quad \text{and} \quad |
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20 | \sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s |
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21 | \end{equation} |
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22 | |
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23 | The chain rule to establish the model equations in the curvilinear $s$-coordinate |
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24 | system is: |
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25 | \begin{equation} \label{Apdx_A_s_chain_rule} |
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26 | \begin{aligned} |
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27 | &\left. {\frac{\partial \bullet }{\partial t}} \right|_z = |
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28 | \left. {\frac{\partial \bullet }{\partial t}} \right|_s |
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29 | -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\ |
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30 | &\left. {\frac{\partial \bullet }{\partial i}} \right|_z = |
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31 | \left. {\frac{\partial \bullet }{\partial i}} \right|_s |
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32 | -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}= |
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33 | \left. {\frac{\partial \bullet }{\partial i}} \right|_s |
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34 | -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet }{\partial s} \\ |
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35 | &\left. {\frac{\partial \bullet }{\partial j}} \right|_z = |
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36 | \left. {\frac{\partial \bullet }{\partial j}} \right|_s |
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37 | - \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= |
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38 | \left. {\frac{\partial \bullet }{\partial j}} \right|_s |
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39 | - \frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet }{\partial s} \\ |
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40 | &\;\frac{\partial \bullet }{\partial z} \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} \\ |
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41 | \end{aligned} |
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42 | \end{equation} |
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43 | |
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44 | In particular applying the time derivative chain rule to $z$ provides the |
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45 | expression for $w_s$, the vertical velocity of the $s-$surfaces: |
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46 | \begin{equation} \label{Apdx_A_w_in_s} |
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47 | w_s = \left. \frac{\partial z }{\partial t} \right|_s |
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48 | = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t} |
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49 | = e_3 \, \frac{\partial s}{\partial t} |
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50 | \end{equation} |
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51 | |
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52 | % ================================================================ |
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53 | % continuity equation |
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54 | % ================================================================ |
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55 | \section{Continuity Equation} |
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56 | \label{Apdx_B_continuity} |
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57 | |
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58 | Using (\ref{Apdx_A_s_chain_rule}) and the fact that the horizontal scale factors |
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59 | $e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of |
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60 | the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows: |
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61 | |
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62 | \begin{align*} |
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63 | \nabla \cdot {\rm {\bf U}} |
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64 | &= \frac{1}{e_1 \,e_2 } \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z |
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65 | +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z \right] |
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66 | + \frac{\partial w}{\partial z} \\ |
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67 | \\ |
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68 | & = \frac{1}{e_1 \,e_2 } \left[ |
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69 | \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s |
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70 | - \frac{e_1 }{e_3 } \sigma _1 \frac{\partial (e_2 \,u)}{\partial s} |
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71 | + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s |
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72 | - \frac{e_2 }{e_3 } \sigma _2 \frac{\partial (e_1 \,v)}{\partial s} \right] |
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73 | + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\ |
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74 | \\ |
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75 | & = \frac{1}{e_1 \,e_2 } \left[ |
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76 | \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s |
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77 | + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s \right] |
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78 | + \frac{1}{e_3 }\left[ \frac{\partial w}{\partial s} |
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79 | - \sigma _1 \frac{\partial u}{\partial s} |
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80 | - \sigma _2 \frac{\partial v}{\partial s} \right] \\ |
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81 | \\ |
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82 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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83 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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84 | -\left. e_2 \,u \frac{\partial e_3 }{\partial i} \right|_s |
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85 | + \left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s |
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86 | - \left. e_1 v \frac{\partial e_3 }{\partial j} \right|_s \right] \\ |
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87 | & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad |
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88 | + \frac{1}{e_3 } \left[ \frac{\partial w}{\partial s} |
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89 | - \sigma _1 \frac{\partial u}{\partial s} |
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90 | - \sigma _2 \frac{\partial v}{\partial s} \right] \\ |
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91 | \\ |
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92 | \end{align*} |
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93 | |
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94 | Noting that $\frac{1}{e_1 }\left. {\frac{\partial e_3 }{\partial i}} |
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95 | \right|_s =\frac{1}{e_1 }\left. {\frac{\partial ^2z}{\partial i\,\partial |
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96 | s}} \right|_s =\frac{\partial }{\partial s}\left( {\frac{1}{e_1 }\left. |
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97 | {\frac{\partial z}{\partial i}} \right|_s } \right)=\frac{\partial \sigma _1 |
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98 | }{\partial s}$ and $\frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} |
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99 | \right|_s =\frac{\partial \sigma _2 }{\partial s}$, it becomes: |
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100 | |
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101 | \begin{align*} |
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102 | \nabla \cdot {\rm {\bf U}} |
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103 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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104 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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105 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] \\ |
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106 | & \qquad \qquad \qquad \qquad \qquad \quad |
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107 | +\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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108 | \\ |
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109 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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110 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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111 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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112 | + \frac{1}{e_3 } \; \frac{\partial}{\partial s} \left[ w - u\;\sigma _1 - v\;\sigma _2 \right] |
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113 | \end{align*} |
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114 | |
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115 | Here, $w$ is the vertical velocity relative to the $z-$coordinate system. |
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116 | Introducing the dia-surface velocity component, $\omega $, defined as |
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117 | the velocity relative to the moving $s$-surfaces and normal to them: |
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118 | \begin{equation} \label{Apdx_A_w_s} |
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119 | \omega = w - w_s - \sigma _1 \,u - \sigma _2 \,v \\ |
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120 | \end{equation} |
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121 | with $w_s$ given by \eqref{Apdx_A_w_in_s}, we obtain the expression for |
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122 | the divergence of the velocity in the curvilinear $s$-coordinate system: |
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123 | \begin{align*} \label{Apdx_A_A4} |
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124 | \nabla \cdot {\rm {\bf U}} |
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125 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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126 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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127 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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128 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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129 | + \frac{1}{e_3 } \frac{\partial w_s }{\partial s} \\ |
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130 | \\ |
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131 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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132 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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133 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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134 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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135 | + \frac{1}{e_3 } \frac{\partial}{\partial s} \left( e_3 \; \frac{\partial s}{\partial t} \right) \\ |
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136 | \\ |
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137 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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138 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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139 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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140 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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141 | + \frac{\partial}{\partial s} \frac{\partial s}{\partial t} |
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142 | + \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s} \\ |
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143 | \\ |
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144 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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145 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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146 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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147 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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148 | + \frac{1}{e_3 } \frac{\partial e_3}{\partial t} \\ |
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149 | \end{align*} |
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150 | |
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151 | As a result, the continuity equation \eqref{Eq_PE_continuity} in the |
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152 | $s$-coordinates becomes: |
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153 | \begin{equation} \label{Apdx_A_A5} |
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154 | \frac{1}{e_3 } \frac{\partial e_3}{\partial t} |
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155 | + \frac{1}{e_1 \,e_2 \,e_3 }\left[ |
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156 | {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s |
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157 | + \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right] |
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158 | +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 |
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159 | \end{equation} |
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160 | |
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161 | % ================================================================ |
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162 | % momentum equation |
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163 | % ================================================================ |
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164 | \section{Momentum Equation} |
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165 | \label{Apdx_B_momentum} |
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166 | |
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167 | Let us consider \eqref{Eq_PE_dyn_vect}, the first component of the momentum |
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168 | equation in the vector invariant form (similar manipulations can be performed |
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169 | on the second component). Its non-linear term can be transformed as follows: |
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170 | |
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171 | \begin{align*} |
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172 | &+\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_z |
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173 | - w \;\frac{\partial u}{\partial z} \\ |
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174 | \\ |
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175 | &\qquad=\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}} |
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176 | \right|_z -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_z } |
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177 | \right]\;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} |
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178 | \right|_z -w\frac{\partial u}{\partial z} \\ |
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179 | \\ |
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180 | &\qquad =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}} |
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181 | \right|_s -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_s } \right. |
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182 | \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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183 | &\qquad \qquad \qquad \qquad \qquad |
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184 | { -\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) |
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185 | -\frac{w}{e_3 }\frac{\partial u}{\partial s} } \\ |
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186 | \end{align*} |
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187 | \begin{align*} |
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188 | \qquad &= \left. \zeta \right|_s \;v |
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189 | - \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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190 | - \frac{w}{e_3 }\frac{\partial u}{\partial s} |
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191 | - \left[ {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s} |
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192 | - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v \\ |
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193 | \qquad&\qquad \qquad \qquad \qquad \qquad \qquad |
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194 | \qquad \qquad \qquad \qquad \quad |
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195 | +\frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ |
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196 | %\\ |
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197 | \qquad &= \left. \zeta \right|_s \;v |
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198 | - \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ |
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199 | \qquad&\qquad \qquad \qquad |
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200 | -\frac{1}{e_3} \left[ {w\frac{\partial u}{\partial s} |
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201 | +\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s} |
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202 | -\sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\ |
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203 | \\ |
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204 | \qquad &= \left. \zeta \right|_s \;v |
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205 | - \frac{1}{2e_1 }\left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s |
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206 | - \frac{1}{e_3} \left[ w - \sigma _2 v - \sigma _1 u \right] |
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207 | \; \frac{\partial u}{\partial s} \\ |
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208 | \\ |
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209 | \qquad &= \left. \zeta \right|_s \;v |
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210 | - \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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211 | - \frac{1}{e_3 }\omega \frac{\partial u}{\partial s} |
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212 | - \frac{\partial s}{\partial t} \frac{\partial u}{\partial s} |
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213 | \end{align*} |
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214 | |
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215 | Therefore, the non-linear terms of the momentum equation have the same |
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216 | form in $z-$ and $s-$coordinates but with the addition of the time derivative |
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217 | of the velocity: |
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218 | \begin{multline} \label{Apdx_A_momentum_NL} |
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219 | +\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_z |
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220 | - w \;\frac{\partial u}{\partial z} \\ |
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221 | = - \frac{\partial u}{\partial t} + \left. \zeta \right|_s \;v |
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222 | - \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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223 | - \frac{1}{e_3 }\omega \frac{\partial u}{\partial s} |
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224 | \end{multline} |
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225 | |
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226 | The pressure gradient term can be transformed as follows: |
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227 | \begin{equation} \label{Apdx_A_grad_p} |
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228 | \begin{split} |
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229 | -\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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230 | & =-\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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231 | &=-\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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232 | \end{split} |
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233 | \end{equation} |
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234 | |
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235 | An additional term appears in (\ref{Apdx_A_grad_p}) which accounts for the |
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236 | tilt of model levels. |
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237 | |
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238 | Introducing \eqref{Apdx_A_momentum_NL} and \eqref{Apdx_A_grad_p} in \eqref{Eq_PE_dyn_vect} and regrouping the time derivative terms in the left |
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239 | hand side, and performing the same manipulation on the second component, |
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240 | we obtain the vector invariant form of the momentum equations in the |
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241 | $s-$coordinate : |
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242 | \begin{subequations} \label{Apdx_A_dyn_vect} |
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243 | \begin{multline} \label{Apdx_A_PE_dyn_vect_u} |
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244 | \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t}= |
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245 | + \left( {\zeta +f} \right)\,v |
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246 | - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right) |
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247 | - \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\ |
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248 | - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o} \right) |
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249 | + g\frac{\rho }{\rho _o}\sigma _1 |
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250 | + D_u^{\vect{U}} + F_u^{\vect{U}} |
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251 | \end{multline} |
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252 | \begin{multline} \label{Apdx_A_dyn_vect_v} |
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253 | \frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}= |
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254 | - \left( {\zeta +f} \right)\,u |
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255 | - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right) |
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256 | - \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\ |
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257 | - \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o} \right) |
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258 | + g\frac{\rho }{\rho _o }\sigma _2 |
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259 | + D_v^{\vect{U}} + F_v^{\vect{U}} |
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260 | \end{multline} |
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261 | \end{subequations} |
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262 | |
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263 | It has the same form as in the $z-$coordinate but for the vertical scale factor |
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264 | that has appeared inside the time derivative. The form of the vertical physics |
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265 | and forcing terms remains unchanged. The form of the lateral physics is |
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266 | discussed in appendix~\ref{Apdx_B}. |
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267 | |
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268 | % ================================================================ |
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269 | % Tracer equation |
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270 | % ================================================================ |
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271 | \section{Tracer Equation} |
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272 | \label{Apdx_B_tracer} |
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273 | |
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274 | The tracer equation is obtained using the same calculation as for the continuity |
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275 | equation and then regrouping the time derivative terms in the left hand side : |
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276 | |
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277 | \begin{multline} \label{Apdx_A_tracer} |
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278 | \frac{1}{e_3} \frac{\partial \left( e_3 T \right)}{\partial t} |
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279 | = -\frac{1}{e_1 \,e_2 \,e_3 } |
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280 | \left[ {\frac{\partial }{\partial i}} \left( {e_2 \,e_3 \;Tu} \right) \right . |
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281 | + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \\ |
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282 | + \left. \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] +D^{T} +F^{T} \; \; |
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283 | \end{multline} |
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284 | |
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285 | |
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286 | The expression for the advection term is a straight consequence of (A.4), the |
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287 | expression of the 3D divergence in the $s$-coordinates established above. |
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288 | |
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