[10419] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
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| 2 | |
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[6997] | 3 | \begin{document} |
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[707] | 4 | |
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| 5 | % ================================================================ |
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[10419] | 6 | % Chapter Appendix A : Curvilinear s-Coordinate Equations |
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[707] | 7 | % ================================================================ |
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[2282] | 8 | \chapter{Curvilinear $s-$Coordinate Equations} |
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[9407] | 9 | \label{apdx:A} |
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[10419] | 10 | |
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[707] | 11 | \minitoc |
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| 12 | |
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[2282] | 13 | \newpage |
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[996] | 14 | |
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[2282] | 15 | % ================================================================ |
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| 16 | % Chain rule |
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| 17 | % ================================================================ |
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[9393] | 18 | \section{Chain rule for $s-$coordinates} |
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[9414] | 19 | \label{sec:A_chain} |
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[2282] | 20 | |
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[3294] | 21 | In order to establish the set of Primitive Equation in curvilinear $s$-coordinates |
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[10368] | 22 | ($i.e.$ an orthogonal curvilinear coordinate in the horizontal and |
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| 23 | an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical), |
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| 24 | we start from the set of equations established in \autoref{subsec:PE_zco_Eq} for |
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| 25 | the special case $k = z$ and thus $e_3 = 1$, |
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| 26 | and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$. |
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| 27 | Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and |
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| 28 | the horizontal slope of $s-$surfaces by: |
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[10419] | 29 | \begin{equation} |
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| 30 | \label{apdx:A_s_slope} |
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| 31 | \sigma_1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s |
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| 32 | \quad \text{and} \quad |
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| 33 | \sigma_2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s |
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[707] | 34 | \end{equation} |
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| 35 | |
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[10368] | 36 | The chain rule to establish the model equations in the curvilinear $s-$coordinate system is: |
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[10419] | 37 | \begin{equation} |
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| 38 | \label{apdx:A_s_chain_rule} |
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| 39 | \begin{aligned} |
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| 40 | &\left. {\frac{\partial \bullet }{\partial t}} \right|_z = |
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| 41 | \left. {\frac{\partial \bullet }{\partial t}} \right|_s |
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[817] | 42 | -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\ |
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[10419] | 43 | &\left. {\frac{\partial \bullet }{\partial i}} \right|_z = |
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| 44 | \left. {\frac{\partial \bullet }{\partial i}} \right|_s |
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| 45 | -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}= |
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| 46 | \left. {\frac{\partial \bullet }{\partial i}} \right|_s |
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| 47 | -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} \\ |
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| 48 | &\left. {\frac{\partial \bullet }{\partial j}} \right|_z = |
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| 49 | \left. {\frac{\partial \bullet }{\partial j}} \right|_s |
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| 50 | - \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= |
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| 51 | \left. {\frac{\partial \bullet }{\partial j}} \right|_s |
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| 52 | - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} \\ |
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| 53 | &\;\frac{\partial \bullet }{\partial z} \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} |
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| 54 | \end{aligned} |
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[707] | 55 | \end{equation} |
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| 56 | |
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[10368] | 57 | In particular applying the time derivative chain rule to $z$ provides the expression for $w_s$, |
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| 58 | the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: |
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[10419] | 59 | \begin{equation} |
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| 60 | \label{apdx:A_w_in_s} |
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| 61 | w_s = \left. \frac{\partial z }{\partial t} \right|_s |
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| 62 | = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t} |
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| 63 | = e_3 \, \frac{\partial s}{\partial t} |
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[817] | 64 | \end{equation} |
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[707] | 65 | |
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[817] | 66 | % ================================================================ |
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| 67 | % continuity equation |
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| 68 | % ================================================================ |
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[9393] | 69 | \section{Continuity equation in $s-$coordinates} |
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[9407] | 70 | \label{sec:A_continuity} |
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[707] | 71 | |
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[10368] | 72 | Using (\autoref{apdx:A_s_chain_rule}) and |
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| 73 | the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, |
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| 74 | the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to |
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| 75 | obtain its expression in the curvilinear $s-$coordinate system: |
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[707] | 76 | |
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[10419] | 77 | \begin{subequations} |
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| 78 | \begin{align*} |
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| 79 | { |
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| 80 | \begin{array}{*{20}l} |
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| 81 | \nabla \cdot {\rm {\bf U}} |
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| 82 | &= \frac{1}{e_1 \,e_2 } \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z |
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| 83 | +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z \right] |
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| 84 | + \frac{\partial w}{\partial z} \\ \\ |
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| 85 | & = \frac{1}{e_1 \,e_2 } \left[ |
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| 86 | \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s |
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| 87 | - \frac{e_1 }{e_3 } \sigma_1 \frac{\partial (e_2 \,u)}{\partial s} |
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| 88 | + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s |
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| 89 | - \frac{e_2 }{e_3 } \sigma_2 \frac{\partial (e_1 \,v)}{\partial s} \right] |
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| 90 | + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\ \\ |
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| 91 | & = \frac{1}{e_1 \,e_2 } \left[ |
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| 92 | \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s |
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| 93 | + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s \right] |
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| 94 | + \frac{1}{e_3 }\left[ \frac{\partial w}{\partial s} |
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| 95 | - \sigma_1 \frac{\partial u}{\partial s} |
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| 96 | - \sigma_2 \frac{\partial v}{\partial s} \right] \\ \\ |
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| 97 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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| 98 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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| 99 | -\left. e_2 \,u \frac{\partial e_3 }{\partial i} \right|_s |
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| 100 | + \left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s |
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| 101 | - \left. e_1 v \frac{\partial e_3 }{\partial j} \right|_s \right] \\ |
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| 102 | & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad |
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| 103 | + \frac{1}{e_3 } \left[ \frac{\partial w}{\partial s} |
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| 104 | - \sigma_1 \frac{\partial u}{\partial s} |
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| 105 | - \sigma_2 \frac{\partial v}{\partial s} \right] \\ |
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| 106 | % |
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| 107 | \intertext{Noting that $ |
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| 108 | \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s |
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| 109 | =\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s |
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| 110 | =\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right) |
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| 111 | =\frac{\partial \sigma_1}{\partial s} |
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| 112 | $ and $ |
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| 113 | \frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s |
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| 114 | =\frac{\partial \sigma_2}{\partial s} |
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| 115 | $, it becomes:} |
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| 116 | % |
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| 117 | \nabla \cdot {\rm {\bf U}} |
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| 118 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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| 119 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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| 120 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] \\ |
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| 121 | & \qquad \qquad \qquad \qquad \quad |
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| 122 | +\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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| 123 | \\ |
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| 124 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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| 125 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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| 126 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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| 127 | + \frac{1}{e_3 } \; \frac{\partial}{\partial s} \left[ w - u\;\sigma_1 - v\;\sigma_2 \right] |
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| 128 | \end{array} |
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| 129 | } |
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| 130 | \end{align*} |
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[2282] | 131 | \end{subequations} |
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| 132 | |
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[10368] | 133 | Here, $w$ is the vertical velocity relative to the $z-$coordinate system. |
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| 134 | Introducing the dia-surface velocity component, |
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| 135 | $\omega $, defined as the volume flux across the moving $s$-surfaces per unit horizontal area: |
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[10419] | 136 | \begin{equation} |
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| 137 | \label{apdx:A_w_s} |
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| 138 | \omega = w - w_s - \sigma_1 \,u - \sigma_2 \,v \\ |
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[707] | 139 | \end{equation} |
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[10368] | 140 | with $w_s$ given by \autoref{apdx:A_w_in_s}, |
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| 141 | we obtain the expression for the divergence of the velocity in the curvilinear $s-$coordinate system: |
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[10419] | 142 | \begin{subequations} |
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| 143 | \begin{align*} |
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| 144 | { |
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| 145 | \begin{array}{*{20}l} |
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| 146 | \nabla \cdot {\rm {\bf U}} |
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| 147 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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| 148 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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| 149 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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| 150 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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| 151 | + \frac{1}{e_3 } \frac{\partial w_s }{\partial s} \\ \\ |
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| 152 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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| 153 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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| 154 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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| 155 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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| 156 | + \frac{1}{e_3 } \frac{\partial}{\partial s} \left( e_3 \; \frac{\partial s}{\partial t} \right) \\ \\ |
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| 157 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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| 158 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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| 159 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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| 160 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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| 161 | + \frac{\partial}{\partial s} \frac{\partial s}{\partial t} |
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| 162 | + \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s} \\ \\ |
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| 163 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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| 164 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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| 165 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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| 166 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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| 167 | + \frac{1}{e_3 } \frac{\partial e_3}{\partial t} |
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| 168 | \end{array} |
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| 169 | } |
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| 170 | \end{align*} |
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[2282] | 171 | \end{subequations} |
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[707] | 172 | |
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[10368] | 173 | As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is: |
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[10419] | 174 | \begin{equation} |
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| 175 | \label{apdx:A_sco_Continuity} |
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| 176 | \frac{1}{e_3 } \frac{\partial e_3}{\partial t} |
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| 177 | + \frac{1}{e_1 \,e_2 \,e_3 }\left[ |
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| 178 | {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s |
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| 179 | + \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right] |
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| 180 | +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 |
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[707] | 181 | \end{equation} |
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[10368] | 182 | A additional term has appeared that take into account |
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| 183 | the contribution of the time variation of the vertical coordinate to the volume budget. |
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[707] | 184 | |
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[2282] | 185 | |
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[817] | 186 | % ================================================================ |
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| 187 | % momentum equation |
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| 188 | % ================================================================ |
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[9393] | 189 | \section{Momentum equation in $s-$coordinate} |
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[9407] | 190 | \label{sec:A_momentum} |
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[707] | 191 | |
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[10368] | 192 | Here we only consider the first component of the momentum equation, |
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[2282] | 193 | the generalization to the second one being straightforward. |
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| 194 | |
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| 195 | $\bullet$ \textbf{Total derivative in vector invariant form} |
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| 196 | |
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[10368] | 197 | Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum equation in the vector invariant form. |
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| 198 | Its total $z-$coordinate time derivative, |
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| 199 | $\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain |
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[2282] | 200 | its expression in the curvilinear $s-$coordinate system: |
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[707] | 201 | |
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[10419] | 202 | \begin{subequations} |
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| 203 | \begin{align*} |
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| 204 | { |
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| 205 | \begin{array}{*{20}l} |
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| 206 | \left. \frac{D u}{D t} \right|_z |
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| 207 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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| 208 | - \left. \zeta \right|_z v |
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| 209 | + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z |
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| 210 | + w \;\frac{\partial u}{\partial z} \\ \\ |
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| 211 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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| 212 | - \left. \zeta \right|_z v |
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| 213 | + \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z |
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| 214 | -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v |
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| 215 | + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z |
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| 216 | + w \;\frac{\partial u}{\partial z} \\ |
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| 217 | % |
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| 218 | \intertext{introducing the chain rule (\autoref{apdx:A_s_chain_rule}) } |
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| 219 | % |
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| 220 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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| 221 | - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s |
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| 222 | -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right. |
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| 223 | \left. {-\frac{e_1}{e_3}\sigma_1 \frac{\partial (e_2 \,v)}{\partial s} |
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| 224 | +\frac{e_2}{e_3}\sigma_2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v \\ |
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| 225 | & \qquad \qquad \qquad \qquad |
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| 226 | { |
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| 227 | + \frac{1}{2e_1} \left( \left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s |
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| 228 | - \frac{e_1}{e_3}\sigma_1 \frac{\partial (u^2+v^2)}{\partial s} \right) |
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| 229 | + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} |
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| 230 | } \\ \\ |
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| 231 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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| 232 | + \left. \zeta \right|_s \;v |
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| 233 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ |
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| 234 | &\qquad \qquad \qquad \quad |
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| 235 | + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} |
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| 236 | - \left[ {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s} |
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| 237 | - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v |
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| 238 | - \frac{\sigma_1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ \\ |
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| 239 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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| 240 | + \left. \zeta \right|_s \;v |
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| 241 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ |
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| 242 | &\qquad \qquad \qquad \quad |
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| 243 | + \frac{1}{e_3} \left[ {w\frac{\partial u}{\partial s} |
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| 244 | +\sigma_1 v\frac{\partial v}{\partial s} - \sigma_2 v\frac{\partial u}{\partial s} |
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| 245 | - \sigma_1 u\frac{\partial u}{\partial s} - \sigma_1 v\frac{\partial v}{\partial s}} \right] \\ \\ |
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| 246 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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| 247 | + \left. \zeta \right|_s \;v |
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| 248 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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| 249 | + \frac{1}{e_3} \left[ w - \sigma_2 v - \sigma_1 u \right] |
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| 250 | \; \frac{\partial u}{\partial s} \\ |
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| 251 | % |
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| 252 | \intertext{Introducing $\omega$, the dia-a-surface velocity given by (\autoref{apdx:A_w_s}) } |
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| 253 | % |
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| 254 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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| 255 | + \left. \zeta \right|_s \;v |
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| 256 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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| 257 | + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s} \\ |
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| 258 | \end{array} |
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| 259 | } |
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| 260 | \end{align*} |
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[2282] | 261 | \end{subequations} |
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| 262 | % |
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[10368] | 263 | Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and |
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| 264 | using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side, |
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[10419] | 265 | \[ |
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| 266 | { |
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| 267 | \begin{array}{*{20}l} |
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| 268 | w_s \;\frac{\partial u}{\partial s} |
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| 269 | = \frac{\partial s}{\partial t} \; \frac{\partial u }{\partial s} |
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| 270 | = \left. {\frac{\partial u }{\partial t}} \right|_s - \left. {\frac{\partial u }{\partial t}} \right|_z \quad , |
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| 271 | \end{array} |
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| 272 | } |
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| 273 | \] |
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[10368] | 274 | leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, |
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[2282] | 275 | $i.e.$ the total $s-$coordinate time derivative : |
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[10419] | 276 | \begin{align} |
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| 277 | \label{apdx:A_sco_Dt_vect} |
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| 278 | \left. \frac{D u}{D t} \right|_s |
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| 279 | = \left. {\frac{\partial u }{\partial t}} \right|_s |
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[2282] | 280 | + \left. \zeta \right|_s \;v |
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[10419] | 281 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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| 282 | + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} |
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[2282] | 283 | \end{align} |
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[10368] | 284 | Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in |
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| 285 | $z-$ and $s-$coordinates. |
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| 286 | This is not the case for the flux form as shown in next paragraph. |
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[2282] | 287 | |
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| 288 | $\bullet$ \textbf{Total derivative in flux form} |
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| 289 | |
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[10368] | 290 | Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish. |
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| 291 | Following the procedure used to establish (\autoref{eq:PE_flux_form}), it can be transformed into : |
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[10419] | 292 | % \begin{subequations} |
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| 293 | \begin{align*} |
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| 294 | { |
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| 295 | \begin{array}{*{20}l} |
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| 296 | \left. \frac{D u}{D t} \right|_s &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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| 297 | & - \zeta \;v |
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| 298 | + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} |
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| 299 | + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s} \\ \\ |
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| 300 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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| 301 | &+\frac{1}{e_1\;e_2} \left( \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i} |
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| 302 | + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j} \right) |
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| 303 | + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\ |
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| 304 | &&- \,u \left[ \frac{1}{e_1 e_2 } \left( \frac{\partial(e_2 u)}{\partial i} |
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| 305 | + \frac{\partial(e_1 v)}{\partial j} \right) |
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| 306 | + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ \\ |
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| 307 | &&- \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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| 308 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ |
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| 309 | \end{array} |
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| 310 | } |
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[817] | 311 | \end{align*} |
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[2282] | 312 | % |
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| 313 | Introducing the vertical scale factor inside the horizontal derivative of the first two terms |
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| 314 | ($i.e.$ the horizontal divergence), it becomes : |
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[10419] | 315 | \begin{align*} |
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| 316 | { |
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| 317 | \begin{array}{*{20}l} |
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| 318 | % \begin{align*} {\begin{array}{*{20}l} |
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| 319 | % {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s |
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| 320 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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| 321 | &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u^2 )}{\partial i} |
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| 322 | + \frac{\partial( e_1 e_3 \,u v )}{\partial j} |
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| 323 | - e_2 u u \frac{\partial e_3}{\partial i} |
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| 324 | - e_1 u v \frac{\partial e_3 }{\partial j} \right) |
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| 325 | + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\ |
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| 326 | && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} |
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| 327 | + \frac{\partial(e_1 e_3 \, v)}{\partial j} |
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| 328 | - e_2 u \;\frac{\partial e_3 }{\partial i} |
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| 329 | - e_1 v \;\frac{\partial e_3 }{\partial j} \right) |
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| 330 | -\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ \\ |
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| 331 | && - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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| 332 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ \\ |
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| 333 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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| 334 | &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u\,u )}{\partial i} |
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| 335 | + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j} \right) |
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| 336 | + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\ |
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| 337 | && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} |
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| 338 | + \frac{\partial(e_1 e_3 \, v)}{\partial j} \right) |
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| 339 | -\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] |
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| 340 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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| 341 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ |
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| 342 | % |
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| 343 | \intertext {Introducing a more compact form for the divergence of the momentum fluxes, |
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| 344 | and using (\autoref{apdx:A_sco_Continuity}), the $s-$coordinate continuity equation, |
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| 345 | it becomes : } |
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| 346 | % |
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| 347 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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| 348 | &+ \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s |
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| 349 | + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t} |
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[2282] | 350 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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[10419] | 351 | -u \;\frac{\partial e_1 }{\partial j} \right) |
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| 352 | \\ |
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| 353 | \end{array} |
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| 354 | } |
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[2282] | 355 | \end{align*} |
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| 356 | which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, |
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[10368] | 357 | $i.e.$ the total $s-$coordinate time derivative in flux form: |
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[10419] | 358 | \begin{flalign} |
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| 359 | \label{apdx:A_sco_Dt_flux} |
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| 360 | \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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| 361 | + \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s |
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| 362 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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| 363 | -u \;\frac{\partial e_1 }{\partial j} \right) |
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[2282] | 364 | \end{flalign} |
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| 365 | which is the total time derivative expressed in the curvilinear $s-$coordinate system. |
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[10368] | 366 | It has the same form as in the $z-$coordinate but for |
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| 367 | the vertical scale factor that has appeared inside the time derivative which |
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| 368 | comes from the modification of (\autoref{apdx:A_sco_Continuity}), |
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| 369 | the continuity equation. |
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[707] | 370 | |
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[2282] | 371 | $\bullet$ \textbf{horizontal pressure gradient} |
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| 372 | |
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| 373 | The horizontal pressure gradient term can be transformed as follows: |
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[10419] | 374 | \[ |
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| 375 | \begin{split} |
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| 376 | -\frac{1}{\rho_o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z |
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| 377 | & =-\frac{1}{\rho_o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial p}{\partial s}} \right] \\ |
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| 378 | & =-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma_1 }{\rho_o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\ |
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| 379 | &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1 |
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| 380 | \end{split} |
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| 381 | \] |
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[10368] | 382 | Applying similar manipulation to the second component and |
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[10419] | 383 | replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it comes: |
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| 384 | \begin{equation} |
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| 385 | \label{apdx:A_grad_p_1} |
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| 386 | \begin{split} |
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| 387 | -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z |
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| 388 | &=-\frac{1}{\rho_o \,e_1 } \left( \left. {\frac{\partial p}{\partial i}} \right|_s |
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| 389 | + g\;\rho \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) \\ |
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| 390 | % |
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| 391 | -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z |
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| 392 | &=-\frac{1}{\rho_o \,e_2 } \left( \left. {\frac{\partial p}{\partial j}} \right|_s |
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| 393 | + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) \\ |
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| 394 | \end{split} |
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[707] | 395 | \end{equation} |
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| 396 | |
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[10368] | 397 | An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for |
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| 398 | the tilt of $s-$surfaces with respect to geopotential $z-$surfaces. |
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[707] | 399 | |
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[10368] | 400 | As in $z$-coordinate, |
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| 401 | the horizontal pressure gradient can be split in two parts following \citet{Marsaleix_al_OM08}. |
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| 402 | Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$, |
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| 403 | and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. |
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[2282] | 404 | The pressure is then given by: |
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[10419] | 405 | \[ |
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| 406 | \begin{split} |
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| 407 | p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left( \rho_o \, d + 1 \right) \; e_3 \; dk \\ |
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| 408 | &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk |
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| 409 | \end{split} |
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| 410 | \] |
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[2282] | 411 | Therefore, $p$ and $p_h'$ are linked through: |
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[10419] | 412 | \begin{equation} |
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| 413 | \label{apdx:A_pressure} |
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| 414 | p = \rho_o \; p_h' + g \, ( z + \eta ) |
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[2282] | 415 | \end{equation} |
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| 416 | and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is: |
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[10419] | 417 | \[ |
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| 418 | \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 |
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| 419 | \] |
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[2282] | 420 | |
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[10368] | 421 | Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and |
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| 422 | using the definition of the density anomaly it comes the expression in two parts: |
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[10419] | 423 | \begin{equation} |
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| 424 | \label{apdx:A_grad_p_2} |
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| 425 | \begin{split} |
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| 426 | -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z |
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| 427 | &=-\frac{1}{e_1 } \left( \left. {\frac{\partial p_h'}{\partial i}} \right|_s |
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| 428 | + g\; d \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\ |
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| 429 | % |
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| 430 | -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z |
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| 431 | &=-\frac{1}{e_2 } \left( \left. {\frac{\partial p_h'}{\partial j}} \right|_s |
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| 432 | + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\ |
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| 433 | \end{split} |
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[2282] | 434 | \end{equation} |
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[10368] | 435 | This formulation of the pressure gradient is characterised by the appearance of |
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| 436 | a term depending on the sea surface height only |
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| 437 | (last term on the right hand side of expression \autoref{apdx:A_grad_p_2}). |
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| 438 | This term will be loosely termed \textit{surface pressure gradient} whereas |
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| 439 | the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to |
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| 440 | the $z$-coordinate formulation. |
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| 441 | In fact, the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, |
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| 442 | and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration. |
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[2282] | 443 | |
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| 444 | $\bullet$ \textbf{The other terms of the momentum equation} |
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| 445 | |
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[10368] | 446 | The coriolis and forcing terms as well as the the vertical physics remain unchanged as |
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| 447 | they involve neither time nor space derivatives. |
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| 448 | The form of the lateral physics is discussed in \autoref{apdx:B}. |
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[2282] | 449 | |
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| 450 | $\bullet$ \textbf{Full momentum equation} |
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| 451 | |
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[10368] | 452 | To sum up, in a curvilinear $s$-coordinate system, |
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| 453 | the vector invariant momentum equation solved by the model has the same mathematical expression as |
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| 454 | the one in a curvilinear $z-$coordinate, except for the pressure gradient term: |
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[10419] | 455 | \begin{subequations} |
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| 456 | \label{apdx:A_dyn_vect} |
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| 457 | \begin{multline} |
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| 458 | \label{apdx:A_PE_dyn_vect_u} |
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| 459 | \frac{\partial u}{\partial t}= |
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| 460 | + \left( {\zeta +f} \right)\,v |
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| 461 | - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right) |
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| 462 | - \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\ |
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| 463 | - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) |
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| 464 | - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} |
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| 465 | + D_u^{\vect{U}} + F_u^{\vect{U}} |
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| 466 | \end{multline} |
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| 467 | \begin{multline} |
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| 468 | \label{apdx:A_dyn_vect_v} |
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| 469 | \frac{\partial v}{\partial t}= |
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| 470 | - \left( {\zeta +f} \right)\,u |
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| 471 | - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right) |
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| 472 | - \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\ |
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| 473 | - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) |
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| 474 | - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} |
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| 475 | + D_v^{\vect{U}} + F_v^{\vect{U}} |
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| 476 | \end{multline} |
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[817] | 477 | \end{subequations} |
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[10368] | 478 | whereas the flux form momentum equation differs from it by |
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| 479 | the formulation of both the time derivative and the pressure gradient term: |
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[10419] | 480 | \begin{subequations} |
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| 481 | \label{apdx:A_dyn_flux} |
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| 482 | \begin{multline} |
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| 483 | \label{apdx:A_PE_dyn_flux_u} |
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| 484 | \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t} = |
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| 485 | \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) |
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| 486 | + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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| 487 | -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,v \\ |
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| 488 | - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) |
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| 489 | - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} |
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| 490 | + D_u^{\vect{U}} + F_u^{\vect{U}} |
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| 491 | \end{multline} |
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| 492 | \begin{multline} |
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| 493 | \label{apdx:A_dyn_flux_v} |
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| 494 | \frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}= |
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| 495 | - \nabla \cdot \left( {{\rm {\bf U}}\,v} \right) |
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| 496 | + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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| 497 | -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,u \\ |
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| 498 | - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) |
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| 499 | - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} |
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| 500 | + D_v^{\vect{U}} + F_v^{\vect{U}} |
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| 501 | \end{multline} |
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[2282] | 502 | \end{subequations} |
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| 503 | Both formulation share the same hydrostatic pressure balance expressed in terms of |
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[3294] | 504 | hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: |
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[10419] | 505 | \begin{equation} |
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| 506 | \label{apdx:A_dyn_zph} |
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| 507 | \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 |
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[2282] | 508 | \end{equation} |
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[707] | 509 | |
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[10368] | 510 | It is important to realize that the change in coordinate system has only concerned the position on the vertical. |
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| 511 | It has not affected (\textbf{i},\textbf{j},\textbf{k}), the orthogonal curvilinear set of unit vectors. |
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| 512 | ($u$,$v$) are always horizontal velocities so that their evolution is driven by \emph{horizontal} forces, |
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| 513 | in particular the pressure gradient. |
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| 514 | By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component, |
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| 515 | $i.e.$ the volume flux across the moving $s$-surfaces per unit horizontal area. |
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[817] | 516 | |
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[2282] | 517 | |
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[817] | 518 | % ================================================================ |
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| 519 | % Tracer equation |
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| 520 | % ================================================================ |
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[9393] | 521 | \section{Tracer equation} |
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[9407] | 522 | \label{sec:A_tracer} |
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[817] | 523 | |
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[10368] | 524 | The tracer equation is obtained using the same calculation as for the continuity equation and then |
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| 525 | regrouping the time derivative terms in the left hand side : |
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[707] | 526 | |
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[10419] | 527 | \begin{multline} |
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| 528 | \label{apdx:A_tracer} |
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| 529 | \frac{1}{e_3} \frac{\partial \left( e_3 T \right)}{\partial t} |
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| 530 | = -\frac{1}{e_1 \,e_2 \,e_3} |
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| 531 | \left[ \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right) |
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| 532 | + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] \\ |
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| 533 | + \frac{1}{e_3} \frac{\partial }{\partial k} \left( Tw \right) |
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| 534 | + D^{T} +F^{T} |
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[707] | 535 | \end{multline} |
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| 536 | |
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[10368] | 537 | The expression for the advection term is a straight consequence of (A.4), |
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| 538 | the expression of the 3D divergence in the $s-$coordinates established above. |
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[707] | 539 | |
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[10419] | 540 | \biblio |
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| 541 | |
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[6997] | 542 | \end{document} |
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