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 | \newpage |
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10 | $\ $\newline % force a new ligne |
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11 | |
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12 | % ================================================================ |
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13 | % Chain rule |
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14 | % ================================================================ |
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15 | \section{Chain rule of $s-$coordinate} |
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16 | \label{Apdx_A_continuity} |
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17 | |
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18 | In order to establish the set of Primitive Equation in curvilinear $s-$coordinates |
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19 | ($i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian |
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20 | Eulerian (ALE) coordinate in the vertical), we start from the set of equations established |
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21 | in \S\ref{PE_zco_Eq} for the special case $k = z$ and thus $e_3 = 1$, and we introduce |
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22 | an arbitrary vertical coordinate $a = a(i,j,z,t)$. Let us define a new vertical scale factor by |
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23 | $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal |
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24 | slope of $s-$surfaces by : |
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25 | \begin{equation} \label{Apdx_A_s_slope} |
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26 | \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s |
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27 | \quad \text{and} \quad |
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28 | \sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s |
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29 | \end{equation} |
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30 | |
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31 | The chain rule to establish the model equations in the curvilinear $s-$coordinate |
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32 | system is: |
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33 | \begin{equation} \label{Apdx_A_s_chain_rule} |
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34 | \begin{aligned} |
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35 | &\left. {\frac{\partial \bullet }{\partial t}} \right|_z = |
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36 | \left. {\frac{\partial \bullet }{\partial t}} \right|_s |
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37 | -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\ |
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38 | &\left. {\frac{\partial \bullet }{\partial i}} \right|_z = |
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39 | \left. {\frac{\partial \bullet }{\partial i}} \right|_s |
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40 | -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}= |
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41 | \left. {\frac{\partial \bullet }{\partial i}} \right|_s |
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42 | -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet }{\partial s} \\ |
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43 | &\left. {\frac{\partial \bullet }{\partial j}} \right|_z = |
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44 | \left. {\frac{\partial \bullet }{\partial j}} \right|_s |
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45 | - \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= |
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46 | \left. {\frac{\partial \bullet }{\partial j}} \right|_s |
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47 | - \frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet }{\partial s} \\ |
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48 | &\;\frac{\partial \bullet }{\partial z} \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} \\ |
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49 | \end{aligned} |
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50 | \end{equation} |
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51 | |
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52 | In particular applying the time derivative chain rule to $z$ provides the expression |
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53 | for $w_s$, the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: |
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54 | \begin{equation} \label{Apdx_A_w_in_s} |
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55 | w_s = \left. \frac{\partial z }{\partial t} \right|_s |
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56 | = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t} |
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57 | = e_3 \, \frac{\partial s}{\partial t} |
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58 | \end{equation} |
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59 | |
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60 | |
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61 | % ================================================================ |
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62 | % continuity equation |
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63 | % ================================================================ |
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64 | \section{Continuity Equation in $s-$coordinate} |
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65 | \label{Apdx_A_continuity} |
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66 | |
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67 | Using (\ref{Apdx_A_s_chain_rule}) and the fact that the horizontal scale factors |
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68 | $e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of |
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69 | the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows |
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70 | in order to obtain its expression in the curvilinear $s-$coordinate system: |
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71 | |
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72 | \begin{subequations} |
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73 | \begin{align*} {\begin{array}{*{20}l} |
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74 | \nabla \cdot {\rm {\bf U}} |
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75 | &= \frac{1}{e_1 \,e_2 } \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z |
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76 | +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z \right] |
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77 | + \frac{\partial w}{\partial z} \\ |
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78 | \\ |
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79 | & = \frac{1}{e_1 \,e_2 } \left[ |
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80 | \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s |
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81 | - \frac{e_1 }{e_3 } \sigma _1 \frac{\partial (e_2 \,u)}{\partial s} |
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82 | + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s |
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83 | - \frac{e_2 }{e_3 } \sigma _2 \frac{\partial (e_1 \,v)}{\partial s} \right] |
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84 | + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\ |
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85 | \\ |
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86 | & = \frac{1}{e_1 \,e_2 } \left[ |
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87 | \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s |
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88 | + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s \right] |
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89 | + \frac{1}{e_3 }\left[ \frac{\partial w}{\partial s} |
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90 | - \sigma _1 \frac{\partial u}{\partial s} |
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91 | - \sigma _2 \frac{\partial v}{\partial s} \right] \\ |
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92 | \\ |
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93 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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94 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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95 | -\left. e_2 \,u \frac{\partial e_3 }{\partial i} \right|_s |
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96 | + \left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s |
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97 | - \left. e_1 v \frac{\partial e_3 }{\partial j} \right|_s \right] \\ |
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98 | & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad |
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99 | + \frac{1}{e_3 } \left[ \frac{\partial w}{\partial s} |
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100 | - \sigma _1 \frac{\partial u}{\partial s} |
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101 | - \sigma _2 \frac{\partial v}{\partial s} \right] \\ |
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102 | % |
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103 | \intertext{Noting that $ |
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104 | \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s |
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105 | =\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s |
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106 | =\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right) |
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107 | =\frac{\partial \sigma _1}{\partial s} |
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108 | $ and $ |
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109 | \frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s |
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110 | =\frac{\partial \sigma _2}{\partial s} |
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111 | $, it becomes:} |
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112 | % |
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113 | \nabla \cdot {\rm {\bf U}} |
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114 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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115 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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116 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] \\ |
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117 | & \qquad \qquad \qquad \qquad \quad |
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118 | +\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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119 | \\ |
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120 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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121 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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122 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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123 | + \frac{1}{e_3 } \; \frac{\partial}{\partial s} \left[ w - u\;\sigma _1 - v\;\sigma _2 \right] |
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124 | \end{array} } |
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125 | \end{align*} |
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126 | \end{subequations} |
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127 | |
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128 | Here, $w$ is the vertical velocity relative to the $z-$coordinate system. |
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129 | Introducing the dia-surface velocity component, $\omega $, defined as |
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130 | the velocity relative to the moving $s-$surfaces and normal to them: |
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131 | \begin{equation} \label{Apdx_A_w_s} |
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132 | \omega = w - w_s - \sigma _1 \,u - \sigma _2 \,v \\ |
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133 | \end{equation} |
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134 | with $w_s$ given by \eqref{Apdx_A_w_in_s}, we obtain the expression for |
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135 | the divergence of the velocity in the curvilinear $s-$coordinate system: |
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136 | \begin{subequations} |
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137 | \begin{align*} {\begin{array}{*{20}l} |
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138 | \nabla \cdot {\rm {\bf U}} |
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139 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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140 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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141 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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142 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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143 | + \frac{1}{e_3 } \frac{\partial w_s }{\partial s} \\ |
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144 | \\ |
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145 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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146 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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147 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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148 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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149 | + \frac{1}{e_3 } \frac{\partial}{\partial s} \left( e_3 \; \frac{\partial s}{\partial t} \right) \\ |
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150 | \\ |
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151 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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152 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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153 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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154 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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155 | + \frac{\partial}{\partial s} \frac{\partial s}{\partial t} |
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156 | + \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s} \\ |
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157 | \\ |
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158 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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159 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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160 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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161 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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162 | + \frac{1}{e_3 } \frac{\partial e_3}{\partial t} \\ |
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163 | \end{array} } |
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164 | \end{align*} |
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165 | \end{subequations} |
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166 | |
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167 | As a result, the continuity equation \eqref{Eq_PE_continuity} in the |
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168 | $s-$coordinates is: |
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169 | \begin{equation} \label{Apdx_A_sco_Continuity} |
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170 | \frac{1}{e_3 } \frac{\partial e_3}{\partial t} |
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171 | + \frac{1}{e_1 \,e_2 \,e_3 }\left[ |
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172 | {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s |
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173 | + \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right] |
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174 | +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 |
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175 | \end{equation} |
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176 | A additional term has appeared that take into account the contribution of the time variation |
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177 | of the vertical coordinate to the volume budget. |
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178 | |
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179 | |
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180 | % ================================================================ |
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181 | % momentum equation |
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182 | % ================================================================ |
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183 | \section{Momentum Equation in $s-$coordinate} |
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184 | \label{Apdx_A_momentum} |
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185 | |
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186 | Here we only consider the first component of the momentum equation, |
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187 | the generalization to the second one being straightforward. |
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188 | |
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189 | $\ $\newline % force a new ligne |
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190 | |
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191 | $\bullet$ \textbf{Total derivative in vector invariant form} |
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192 | |
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193 | Let us consider \eqref{Eq_PE_dyn_vect}, the first component of the momentum |
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194 | equation in the vector invariant form. Its total $z-$coordinate time derivative, |
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195 | $\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain |
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196 | its expression in the curvilinear $s-$coordinate system: |
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197 | |
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198 | \begin{subequations} |
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199 | \begin{align*} {\begin{array}{*{20}l} |
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200 | \left. \frac{D u}{D t} \right|_z |
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201 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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202 | - \left. \zeta \right|_z v |
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203 | + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z |
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204 | + w \;\frac{\partial u}{\partial z} \\ |
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205 | \\ |
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206 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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207 | - \left. \zeta \right|_z v |
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208 | + \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z |
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209 | -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v |
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210 | + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z |
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211 | + w \;\frac{\partial u}{\partial z} \\ |
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212 | % |
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213 | \intertext{introducing the chain rule (\ref{Apdx_A_s_chain_rule}) } |
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214 | % |
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215 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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216 | - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s |
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217 | -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right. |
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218 | \left. {-\frac{e_1}{e_3}\sigma _1 \frac{\partial (e_2 \,v)}{\partial s} |
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219 | +\frac{e_2}{e_3}\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v \\ |
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220 | & \qquad \qquad \qquad \qquad |
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221 | { + \frac{1}{2e_1} \left( \left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s |
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222 | - \frac{e_1}{e_3}\sigma _1 \frac{\partial (u^2+v^2)}{\partial s} \right) |
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223 | + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} } \\ |
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224 | \\ |
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225 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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226 | + \left. \zeta \right|_s \;v |
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227 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ |
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228 | &\qquad \qquad \qquad \quad |
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229 | + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} |
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230 | - \left[ {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s} |
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231 | - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v |
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232 | - \frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ |
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233 | \\ |
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234 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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235 | + \left. \zeta \right|_s \;v |
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236 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ |
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237 | &\qquad \qquad \qquad \quad |
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238 | + \frac{1}{e_3} \left[ {w\frac{\partial u}{\partial s} |
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239 | +\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s} |
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240 | - \sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\ |
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241 | \\ |
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242 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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243 | + \left. \zeta \right|_s \;v |
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244 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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245 | + \frac{1}{e_3} \left[ w - \sigma _2 v - \sigma _1 u \right] |
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246 | \; \frac{\partial u}{\partial s} \\ |
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247 | % |
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248 | \intertext{Introducing $\omega$, the dia-a-surface velocity given by (\ref{Apdx_A_w_s}) } |
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249 | % |
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250 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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251 | + \left. \zeta \right|_s \;v |
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252 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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253 | + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s} \\ |
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254 | \end{array} } |
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255 | \end{align*} |
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256 | \end{subequations} |
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257 | % |
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258 | Applying the time derivative chain rule (first equation of (\ref{Apdx_A_s_chain_rule})) |
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259 | to $u$ and using (\ref{Apdx_A_w_in_s}) provides the expression of the last term |
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260 | of the right hand side, |
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261 | \begin{equation*} {\begin{array}{*{20}l} |
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262 | w_s \;\frac{\partial u}{\partial s} |
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263 | = \frac{\partial s}{\partial t} \; \frac{\partial u }{\partial s} |
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264 | = \left. {\frac{\partial u }{\partial t}} \right|_s - \left. {\frac{\partial u }{\partial t}} \right|_z \quad , |
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265 | \end{array} } |
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266 | \end{equation*} |
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267 | leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, |
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268 | $i.e.$ the total $s-$coordinate time derivative : |
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269 | \begin{align} \label{Apdx_A_sco_Dt_vect} |
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270 | \left. \frac{D u}{D t} \right|_s |
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271 | = \left. {\frac{\partial u }{\partial t}} \right|_s |
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272 | + \left. \zeta \right|_s \;v |
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273 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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274 | + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} |
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275 | \end{align} |
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276 | Therefore, the vector invariant form of the total time derivative has exactly the same |
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277 | mathematical form in $z-$ and $s-$coordinates. This is not the case for the flux form |
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278 | as shown in next paragraph. |
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279 | |
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280 | $\ $\newline % force a new ligne |
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281 | |
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282 | $\bullet$ \textbf{Total derivative in flux form} |
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283 | |
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284 | Let us start from the total time derivative in the curvilinear $s-$coordinate system |
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285 | we have just establish. Following the procedure used to establish (\ref{Eq_PE_flux_form}), |
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286 | it can be transformed into : |
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287 | %\begin{subequations} |
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288 | \begin{align*} {\begin{array}{*{20}l} |
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289 | \left. \frac{D u}{D t} \right|_s &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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290 | & - \zeta \;v |
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291 | + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} |
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292 | + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s} \\ |
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293 | \\ |
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294 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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295 | &+\frac{1}{e_1\;e_2} \left( \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i} |
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296 | + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j} \right) |
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297 | + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ |
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298 | \\ |
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299 | &&- \,u \left[ \frac{1}{e_1 e_2 } \left( \frac{\partial(e_2 u)}{\partial i} |
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300 | + \frac{\partial(e_1 v)}{\partial j} \right) |
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301 | + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ |
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302 | \\ |
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303 | &&- \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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304 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ |
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305 | \end{array} } |
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306 | \end{align*} |
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307 | % |
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308 | Introducing the vertical scale factor inside the horizontal derivative of the first two terms |
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309 | ($i.e.$ the horizontal divergence), it becomes : |
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310 | \begin{subequations} |
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311 | \begin{align*} {\begin{array}{*{20}l} |
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312 | %\begin{align*} {\begin{array}{*{20}l} |
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313 | %{\begin{array}{*{20}l} |
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314 | \left. \frac{D u}{D t} \right|_s |
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315 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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316 | &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u^2 )}{\partial i} |
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317 | + \frac{\partial( e_1 e_3 \,u v )}{\partial j} |
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318 | - e_2 u u \frac{\partial e_3}{\partial i} |
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319 | - e_1 u v \frac{\partial e_3 }{\partial j} \right) |
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320 | + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ |
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321 | \\ |
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322 | && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} |
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323 | + \frac{\partial(e_1 e_3 \, v)}{\partial j} |
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324 | - e_2 u \;\frac{\partial e_3 }{\partial i} |
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325 | - e_1 v \;\frac{\partial e_3 }{\partial j} \right) |
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326 | -\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ |
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327 | \\ |
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328 | && - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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329 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ |
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330 | \\ |
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331 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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332 | &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u\,u )}{\partial i} |
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333 | + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j} \right) |
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334 | + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ |
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335 | \\ |
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336 | && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} |
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337 | + \frac{\partial(e_1 e_3 \, v)}{\partial j} \right) |
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338 | -\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] |
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339 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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340 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ |
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341 | % |
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342 | \intertext {Introducing a more compact form for the divergence of the momentum fluxes, |
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343 | and using (\ref{Apdx_A_sco_Continuity}), the $s-$coordinate continuity equation, |
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344 | it becomes : } |
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345 | % |
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346 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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347 | &+ \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s |
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348 | + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t} |
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349 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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350 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ |
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351 | \end{array} } |
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352 | \end{align*} |
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353 | \end{subequations} |
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354 | which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, |
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355 | $i.e.$ the total $s-$coordinate time derivative in flux form : |
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356 | \begin{flalign}\label{Apdx_A_sco_Dt_flux} |
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357 | \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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358 | + \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s |
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359 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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360 | -u \;\frac{\partial e_1 }{\partial j} \right) |
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361 | \end{flalign} |
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362 | which is the total time derivative expressed in the curvilinear $s-$coordinate system. |
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363 | It has the same form as in the $z-$coordinate but for the vertical scale factor |
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364 | that has appeared inside the time derivative which comes from the modification |
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365 | of (\ref{Apdx_A_sco_Continuity}), the continuity equation. |
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366 | |
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367 | $\ $\newline % force a new ligne |
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368 | |
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369 | $\bullet$ \textbf{horizontal pressure gradient} |
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370 | |
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371 | The horizontal pressure gradient term can be transformed as follows: |
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372 | \begin{equation*} |
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373 | \begin{split} |
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374 | -\frac{1}{\rho _o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z |
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375 | & =-\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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376 | & =-\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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377 | &=-\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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378 | \end{split} |
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379 | \end{equation*} |
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380 | Applying similar manipulation to the second component and replacing |
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381 | $\sigma _1$ and $\sigma _2$ by their expression \eqref{Apdx_A_s_slope}, it comes: |
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382 | \begin{equation} \label{Apdx_A_grad_p} |
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383 | \begin{split} |
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384 | -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z |
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385 | &=-\frac{1}{\rho _o \,e_1 } \left( \left. {\frac{\partial p}{\partial i}} \right|_s |
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386 | + g\;\rho \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) \\ |
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387 | % |
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388 | -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z |
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389 | &=-\frac{1}{\rho _o \,e_2 } \left( \left. {\frac{\partial p}{\partial j}} \right|_s |
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390 | + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) \\ |
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391 | \end{split} |
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392 | \end{equation} |
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393 | |
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394 | An additional term appears in (\ref{Apdx_A_grad_p}) which accounts for the |
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395 | tilt of $s-$surfaces with respect to geopotential $z-$surfaces. |
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396 | |
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397 | As in $z$-coordinate, the horizontal pressure gradient can be split in two parts |
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398 | following \citet{Marsaleix_al_OM08}. Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$, |
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399 | and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. |
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400 | The pressure is then given by: |
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401 | \begin{equation*} |
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402 | \begin{split} |
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403 | p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left( \rho_o \, d + 1 \right) \; e_3 \; dk \\ |
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404 | &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk |
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405 | \end{split} |
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406 | \end{equation*} |
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407 | Therefore, $p$ and $p_h'$ are linked through: |
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408 | \begin{equation} \label{Apdx_A_pressure} |
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409 | p = \rho_o \; p_h' + g \, ( z + \eta ) |
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410 | \end{equation} |
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411 | and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is: |
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412 | \begin{equation*} |
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413 | \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 |
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414 | \end{equation*} |
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415 | |
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416 | Substituing \eqref{Apdx_A_pressure} in \eqref{Apdx_A_grad_p} and using the definition of |
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417 | the density anomaly it comes the expression in two parts: |
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418 | \begin{equation} \label{Apdx_A_grad_p} |
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419 | \begin{split} |
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420 | -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z |
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421 | &=-\frac{1}{e_1 } \left( \left. {\frac{\partial p_h'}{\partial i}} \right|_s |
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422 | + g\; d \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\ |
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423 | % |
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424 | -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z |
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425 | &=-\frac{1}{e_2 } \left( \left. {\frac{\partial p_h'}{\partial j}} \right|_s |
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426 | + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\ |
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427 | \end{split} |
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428 | \end{equation} |
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429 | This formulation of the pressure gradient is characterised by the appearance of a term depending on the |
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430 | the sea surface height only (last term on the right hand side of expression \eqref{Apdx_A_grad_p}). |
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431 | This term will be abusively named \textit{surface pressure gradient} whereas the first term will be named |
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432 | \textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation. |
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433 | In fact, the the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and |
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434 | $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of |
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435 | the vertical integration. |
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436 | |
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437 | |
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438 | $\ $\newline % force a new ligne |
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439 | |
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440 | $\bullet$ \textbf{The other terms of the momentum equation} |
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441 | |
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442 | The coriolis and forcing terms as well as the the vertical physics remain unchanged |
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443 | as they involve neither time nor space derivatives. The form of the lateral physics is |
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444 | discussed in appendix~\ref{Apdx_B}. |
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445 | |
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446 | |
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447 | $\ $\newline % force a new ligne |
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448 | |
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449 | $\bullet$ \textbf{Full momentum equation} |
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450 | |
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451 | To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation |
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452 | solved by the model has the same mathematical expression as the one in a curvilinear |
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453 | $z-$coordinate, but the pressure gradient term : |
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454 | \begin{subequations} \label{Apdx_A_dyn_vect} |
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455 | \begin{multline} \label{Apdx_A_PE_dyn_vect_u} |
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456 | \frac{\partial u}{\partial t}= |
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457 | + \left( {\zeta +f} \right)\,v |
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458 | - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right) |
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459 | - \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\ |
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460 | - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) |
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461 | - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} |
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462 | + D_u^{\vect{U}} + F_u^{\vect{U}} |
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463 | \end{multline} |
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464 | \begin{multline} \label{Apdx_A_dyn_vect_v} |
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465 | \frac{\partial v}{\partial t}= |
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466 | - \left( {\zeta +f} \right)\,u |
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467 | - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right) |
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468 | - \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\ |
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469 | - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) |
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470 | - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} |
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471 | + D_v^{\vect{U}} + F_v^{\vect{U}} |
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472 | \end{multline} |
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473 | \end{subequations} |
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474 | whereas the flux form momentum equation differ from it by the formulation of both |
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475 | the time derivative and the pressure gradient term : |
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476 | \begin{subequations} \label{Apdx_A_dyn_flux} |
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477 | \begin{multline} \label{Apdx_A_PE_dyn_flux_u} |
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478 | \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t} = |
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479 | \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) |
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480 | + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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481 | -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,v \\ |
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482 | - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) |
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483 | - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} |
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484 | + D_u^{\vect{U}} + F_u^{\vect{U}} |
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485 | \end{multline} |
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486 | \begin{multline} \label{Apdx_A_dyn_flux_v} |
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487 | \frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}= |
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488 | - \nabla \cdot \left( {{\rm {\bf U}}\,v} \right) |
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489 | + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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490 | -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,u \\ |
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491 | - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) |
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492 | - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} |
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493 | + D_v^{\vect{U}} + F_v^{\vect{U}} |
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494 | \end{multline} |
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495 | \end{subequations} |
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496 | Both formulation share the same hydrostatic pressure balance expressed in terms of |
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497 | hydrostatic pressure and density anmalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: |
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498 | \begin{equation} \label{Apdx_A_dyn_zph} |
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499 | \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 |
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500 | \end{equation} |
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501 | |
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502 | It is important to realize that the change in coordinate system has only concerned |
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503 | the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the |
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504 | orthogonal curvilinear set of unit vector. ($u$,$v$) are always horizontal velocities |
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505 | so that their evolution is driven by \emph{horizontal} forces, in particular |
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506 | the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity, |
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507 | but the dia-surface velocity component, $i.e.$ the velocity relative to the moving |
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508 | $s-$surfaces and normal to them. |
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509 | |
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510 | |
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511 | % ================================================================ |
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512 | % Tracer equation |
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513 | % ================================================================ |
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514 | \section{Tracer Equation} |
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515 | \label{Apdx_A_tracer} |
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516 | |
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517 | The tracer equation is obtained using the same calculation as for the continuity |
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518 | equation and then regrouping the time derivative terms in the left hand side : |
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519 | |
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520 | \begin{multline} \label{Apdx_A_tracer} |
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521 | \frac{1}{e_3} \frac{\partial \left( e_3 T \right)}{\partial t} |
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522 | = -\frac{1}{e_1 \,e_2 \,e_3} |
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523 | \left[ \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right) |
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524 | + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] \\ |
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525 | + \frac{1}{e_3} \frac{\partial }{\partial k} \left( Tw \right) |
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526 | + D^{T} +F^{T} |
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527 | \end{multline} |
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528 | |
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529 | |
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530 | The expression for the advection term is a straight consequence of (A.4), the |
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531 | expression of the 3D divergence in the $s-$coordinates established above. |
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532 | |
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