[707] | 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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[2282] | 5 | \chapter{Curvilinear $s-$Coordinate Equations} |
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[707] | 6 | \label{Apdx_A} |
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| 7 | \minitoc |
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| 8 | |
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[2282] | 9 | \newpage |
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| 10 | $\ $\newline % force a new ligne |
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[996] | 11 | |
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[2282] | 12 | % ================================================================ |
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| 13 | % Chain rule |
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| 14 | % ================================================================ |
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[3294] | 15 | \section{The chain rule for $s-$coordinates} |
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[2282] | 16 | \label{Apdx_A_continuity} |
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| 17 | |
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[3294] | 18 | In order to establish the set of Primitive Equation in curvilinear $s$-coordinates |
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[2282] | 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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[1223] | 23 | $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal |
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[2282] | 24 | slope of $s-$surfaces by : |
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[817] | 25 | \begin{equation} \label{Apdx_A_s_slope} |
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[707] | 26 | \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s |
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[817] | 27 | \quad \text{and} \quad |
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[707] | 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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[2282] | 31 | The chain rule to establish the model equations in the curvilinear $s-$coordinate |
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[1223] | 32 | system is: |
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[817] | 33 | \begin{equation} \label{Apdx_A_s_chain_rule} |
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[707] | 34 | \begin{aligned} |
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[817] | 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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[707] | 49 | \end{aligned} |
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| 50 | \end{equation} |
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| 51 | |
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[2282] | 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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[817] | 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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[707] | 59 | |
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[2282] | 60 | |
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[817] | 61 | % ================================================================ |
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| 62 | % continuity equation |
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| 63 | % ================================================================ |
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[3294] | 64 | \section{Continuity Equation in $s-$coordinates} |
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[2282] | 65 | \label{Apdx_A_continuity} |
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[707] | 66 | |
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[1223] | 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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[2282] | 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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[707] | 71 | |
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[2282] | 72 | \begin{subequations} |
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| 73 | \begin{align*} {\begin{array}{*{20}l} |
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[817] | 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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[2282] | 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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[817] | 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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[2282] | 117 | & \qquad \qquad \qquad \qquad \quad |
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[817] | 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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[2282] | 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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[1223] | 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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[3294] | 130 | the volume flux across the moving $s$-surfaces per unit horizontal area: |
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[817] | 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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[707] | 133 | \end{equation} |
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[1223] | 134 | with $w_s$ given by \eqref{Apdx_A_w_in_s}, we obtain the expression for |
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[2282] | 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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[817] | 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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[2282] | 163 | \end{array} } |
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[817] | 164 | \end{align*} |
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[2282] | 165 | \end{subequations} |
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[707] | 166 | |
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[1223] | 167 | As a result, the continuity equation \eqref{Eq_PE_continuity} in the |
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[2282] | 168 | $s-$coordinates is: |
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| 169 | \begin{equation} \label{Apdx_A_sco_Continuity} |
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[817] | 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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[707] | 175 | \end{equation} |
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[2282] | 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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[707] | 178 | |
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[2282] | 179 | |
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[817] | 180 | % ================================================================ |
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| 181 | % momentum equation |
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| 182 | % ================================================================ |
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[2282] | 183 | \section{Momentum Equation in $s-$coordinate} |
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| 184 | \label{Apdx_A_momentum} |
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[707] | 185 | |
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[2282] | 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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[1223] | 193 | Let us consider \eqref{Eq_PE_dyn_vect}, the first component of the momentum |
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[2282] | 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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[707] | 197 | |
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[2282] | 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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[817] | 205 | \\ |
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[2282] | 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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[817] | 224 | \\ |
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[2282] | 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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[817] | 233 | \\ |
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[2282] | 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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[817] | 246 | \; \frac{\partial u}{\partial s} \\ |
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[2282] | 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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[817] | 293 | \\ |
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[2282] | 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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[817] | 306 | \end{align*} |
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[2282] | 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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[707] | 366 | |
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[2282] | 367 | $\ $\newline % force a new ligne |
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[707] | 368 | |
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[2282] | 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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[707] | 373 | \begin{split} |
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[2282] | 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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[707] | 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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[2282] | 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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[707] | 392 | \end{equation} |
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| 393 | |
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[1223] | 394 | An additional term appears in (\ref{Apdx_A_grad_p}) which accounts for the |
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[2282] | 395 | tilt of $s-$surfaces with respect to geopotential $z-$surfaces. |
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[707] | 396 | |
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[2282] | 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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[3294] | 431 | This term will be loosely termed \textit{surface pressure gradient} |
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| 432 | whereas the first term will be termed the |
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[2282] | 433 | \textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation. |
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| 434 | In fact, the the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and |
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| 435 | $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of |
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| 436 | the vertical integration. |
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| 437 | |
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| 438 | |
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| 439 | $\ $\newline % force a new ligne |
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| 440 | |
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| 441 | $\bullet$ \textbf{The other terms of the momentum equation} |
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| 442 | |
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| 443 | The coriolis and forcing terms as well as the the vertical physics remain unchanged |
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| 444 | as they involve neither time nor space derivatives. The form of the lateral physics is |
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| 445 | discussed in appendix~\ref{Apdx_B}. |
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| 446 | |
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| 447 | |
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| 448 | $\ $\newline % force a new ligne |
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| 449 | |
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| 450 | $\bullet$ \textbf{Full momentum equation} |
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| 451 | |
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| 452 | To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation |
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| 453 | solved by the model has the same mathematical expression as the one in a curvilinear |
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[3294] | 454 | $z-$coordinate, except for the pressure gradient term : |
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[817] | 455 | \begin{subequations} \label{Apdx_A_dyn_vect} |
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| 456 | \begin{multline} \label{Apdx_A_PE_dyn_vect_u} |
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[2282] | 457 | \frac{\partial u}{\partial t}= |
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[817] | 458 | + \left( {\zeta +f} \right)\,v |
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| 459 | - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right) |
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| 460 | - \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\ |
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[2282] | 461 | - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) |
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| 462 | - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} |
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[817] | 463 | + D_u^{\vect{U}} + F_u^{\vect{U}} |
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| 464 | \end{multline} |
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| 465 | \begin{multline} \label{Apdx_A_dyn_vect_v} |
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[2282] | 466 | \frac{\partial v}{\partial t}= |
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[817] | 467 | - \left( {\zeta +f} \right)\,u |
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| 468 | - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right) |
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| 469 | - \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\ |
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[2282] | 470 | - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) |
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| 471 | - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} |
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[817] | 472 | + D_v^{\vect{U}} + F_v^{\vect{U}} |
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| 473 | \end{multline} |
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| 474 | \end{subequations} |
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[2282] | 475 | whereas the flux form momentum equation differ from it by the formulation of both |
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| 476 | the time derivative and the pressure gradient term : |
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| 477 | \begin{subequations} \label{Apdx_A_dyn_flux} |
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| 478 | \begin{multline} \label{Apdx_A_PE_dyn_flux_u} |
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| 479 | \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t} = |
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| 480 | \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) |
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| 481 | + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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| 482 | -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,v \\ |
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| 483 | - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) |
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| 484 | - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} |
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| 485 | + D_u^{\vect{U}} + F_u^{\vect{U}} |
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| 486 | \end{multline} |
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| 487 | \begin{multline} \label{Apdx_A_dyn_flux_v} |
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| 488 | \frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}= |
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| 489 | - \nabla \cdot \left( {{\rm {\bf U}}\,v} \right) |
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| 490 | + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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| 491 | -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,u \\ |
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| 492 | - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) |
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| 493 | - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} |
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| 494 | + D_v^{\vect{U}} + F_v^{\vect{U}} |
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| 495 | \end{multline} |
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| 496 | \end{subequations} |
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| 497 | Both formulation share the same hydrostatic pressure balance expressed in terms of |
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[3294] | 498 | hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: |
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[2282] | 499 | \begin{equation} \label{Apdx_A_dyn_zph} |
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| 500 | \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 |
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| 501 | \end{equation} |
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[707] | 502 | |
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[2282] | 503 | It is important to realize that the change in coordinate system has only concerned |
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| 504 | the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the |
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[3294] | 505 | orthogonal curvilinear set of unit vectors. ($u$,$v$) are always horizontal velocities |
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[2282] | 506 | so that their evolution is driven by \emph{horizontal} forces, in particular |
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| 507 | the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity, |
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[3294] | 508 | but the dia-surface velocity component, $i.e.$ the volume flux across the moving |
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| 509 | $s$-surfaces per unit horizontal area. |
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[817] | 510 | |
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[2282] | 511 | |
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[817] | 512 | % ================================================================ |
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| 513 | % Tracer equation |
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| 514 | % ================================================================ |
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| 515 | \section{Tracer Equation} |
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[2282] | 516 | \label{Apdx_A_tracer} |
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[817] | 517 | |
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[1223] | 518 | The tracer equation is obtained using the same calculation as for the continuity |
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| 519 | equation and then regrouping the time derivative terms in the left hand side : |
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[707] | 520 | |
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[817] | 521 | \begin{multline} \label{Apdx_A_tracer} |
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| 522 | \frac{1}{e_3} \frac{\partial \left( e_3 T \right)}{\partial t} |
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[2282] | 523 | = -\frac{1}{e_1 \,e_2 \,e_3} |
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| 524 | \left[ \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right) |
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| 525 | + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] \\ |
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| 526 | + \frac{1}{e_3} \frac{\partial }{\partial k} \left( Tw \right) |
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| 527 | + D^{T} +F^{T} |
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[707] | 528 | \end{multline} |
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| 529 | |
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| 530 | |
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[1223] | 531 | The expression for the advection term is a straight consequence of (A.4), the |
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[2282] | 532 | expression of the 3D divergence in the $s-$coordinates established above. |
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[707] | 533 | |
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