[10414] | 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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[2282] | 5 | \chapter{Curvilinear $s-$Coordinate Equations} |
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[11543] | 6 | \label{apdx:SCOORD} |
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[10414] | 7 | |
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[11435] | 8 | \chaptertoc |
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[707] | 9 | |
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[11337] | 10 | \vfill |
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| 11 | \begin{figure}[b] |
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| 12 | \subsubsection*{Changes record} |
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| 13 | \begin{tabular}{l||l|m{0.65\linewidth}} |
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| 14 | Release & Author & Modifications \\ |
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| 15 | {\em 4.0} & {\em Mike Bell} & {\em review} \\ |
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| 16 | {\em 3.x} & {\em Gurvan Madec} & {\em original} \\ |
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| 17 | \end{tabular} |
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| 18 | \end{figure} |
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| 19 | |
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[9393] | 20 | \section{Chain rule for $s-$coordinates} |
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[11543] | 21 | \label{sec:SCOORD_chain} |
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[2282] | 22 | |
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[3294] | 23 | In order to establish the set of Primitive Equation in curvilinear $s$-coordinates |
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[11435] | 24 | (\ie\ an orthogonal curvilinear coordinate in the horizontal and |
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[10354] | 25 | an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical), |
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[11543] | 26 | we start from the set of equations established in \autoref{subsec:MB_zco_Eq} for |
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[10354] | 27 | the special case $k = z$ and thus $e_3 = 1$, |
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| 28 | and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$. |
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| 29 | 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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| 30 | the horizontal slope of $s-$surfaces by: |
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[10414] | 31 | \begin{equation} |
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[11543] | 32 | \label{eq:SCOORD_s_slope} |
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[11335] | 33 | \sigma_1 =\frac{1}{e_1 } \; \left. {\frac{\partial z}{\partial i}} \right|_s |
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[10414] | 34 | \quad \text{and} \quad |
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[11335] | 35 | \sigma_2 =\frac{1}{e_2 } \; \left. {\frac{\partial z}{\partial j}} \right|_s . |
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[707] | 36 | \end{equation} |
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| 37 | |
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[11335] | 38 | The model fields (e.g. pressure $p$) can be viewed as functions of $(i,j,z,t)$ (e.g. $p(i,j,z,t)$) or as |
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[11543] | 39 | functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of |
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| 40 | these fields. Any ``infinitesimal'' change in $\bullet$ can be written in two forms: |
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[10414] | 41 | \begin{equation} |
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[11543] | 42 | \label{eq:SCOORD_s_infin_changes} |
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[11335] | 43 | \begin{aligned} |
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[11543] | 44 | & \delta \bullet = \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t} |
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| 45 | + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t} |
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| 46 | + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t} |
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[11335] | 47 | + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,s} , \\ |
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[11543] | 48 | & \delta \bullet = \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t} |
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| 49 | + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t} |
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| 50 | + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t} |
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[11335] | 51 | + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,z} . |
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| 52 | \end{aligned} |
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| 53 | \end{equation} |
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| 54 | Using the first form and considering a change $\delta i$ with $j, z$ and $t$ held constant, shows that |
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| 55 | \begin{equation} |
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[11558] | 56 | \label{eq:SCOORD_s_chain_rule1} |
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[11335] | 57 | \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,z,t} = |
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| 58 | \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,s,t} |
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[11543] | 59 | + \left. {\frac{\partial s }{\partial i}} \right|_{j,z,t} \; |
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| 60 | \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} . |
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[11335] | 61 | \end{equation} |
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[11543] | 62 | The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces, |
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| 63 | (\autoref{eq:SCOORD_s_slope}), by applying the second of (\autoref{eq:SCOORD_s_infin_changes}) with $\bullet$ set to |
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[11335] | 64 | $s$ and $j, t$ held constant |
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| 65 | \begin{equation} |
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[11543] | 66 | \label{eq:SCOORD_delta_s} |
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| 67 | \delta s|_{j,t} = |
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| 68 | \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} |
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[11335] | 69 | + \delta z \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} . |
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| 70 | \end{equation} |
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| 71 | Choosing to look at a direction in the $(i,z)$ plane in which $\delta s = 0$ and using |
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[11543] | 72 | (\autoref{eq:SCOORD_s_slope}) we obtain |
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[11335] | 73 | \begin{equation} |
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[11543] | 74 | \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} = |
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[11335] | 75 | - \left. \frac{ \partial z }{\partial i} \right|_{j,s,t} \; |
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| 76 | \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} |
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| 77 | = - \frac{e_1 }{e_3 }\sigma_1 . |
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[11543] | 78 | \label{eq:SCOORD_ds_di_z} |
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[11335] | 79 | \end{equation} |
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[11543] | 80 | Another identity, similar in form to (\autoref{eq:SCOORD_ds_di_z}), can be derived |
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| 81 | by choosing $\bullet$ to be $s$ and using the second form of (\autoref{eq:SCOORD_s_infin_changes}) to consider |
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[11335] | 82 | changes in which $i , j$ and $s$ are constant. This shows that |
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| 83 | \begin{equation} |
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[11543] | 84 | \label{eq:SCOORD_w_in_s} |
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| 85 | w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} = |
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[11335] | 86 | - \left. \frac{ \partial z }{\partial s} \right|_{i,j,t} |
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[11543] | 87 | \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} |
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| 88 | = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} . |
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[11335] | 89 | \end{equation} |
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| 90 | |
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[11543] | 91 | In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is |
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| 92 | usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish |
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[11335] | 93 | the model equations in the curvilinear $s-$coordinate system are: |
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| 94 | \begin{equation} |
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[11558] | 95 | \label{eq:SCOORD_s_chain_rule2} |
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[10414] | 96 | \begin{aligned} |
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| 97 | &\left. {\frac{\partial \bullet }{\partial t}} \right|_z = |
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[11543] | 98 | \left. {\frac{\partial \bullet }{\partial t}} \right|_s |
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[11335] | 99 | + \frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial t} , \\ |
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[10414] | 100 | &\left. {\frac{\partial \bullet }{\partial i}} \right|_z = |
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| 101 | \left. {\frac{\partial \bullet }{\partial i}} \right|_s |
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[11335] | 102 | +\frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial i}= |
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[11543] | 103 | \left. {\frac{\partial \bullet }{\partial i}} \right|_s |
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[11335] | 104 | -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} , \\ |
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[10414] | 105 | &\left. {\frac{\partial \bullet }{\partial j}} \right|_z = |
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[11543] | 106 | \left. {\frac{\partial \bullet }{\partial j}} \right|_s |
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[11335] | 107 | + \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= |
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[11543] | 108 | \left. {\frac{\partial \bullet }{\partial j}} \right|_s |
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[11335] | 109 | - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} , \\ |
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| 110 | &\;\frac{\partial \bullet }{\partial z} \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} . |
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[10414] | 111 | \end{aligned} |
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[707] | 112 | \end{equation} |
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| 113 | |
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[9393] | 114 | \section{Continuity equation in $s-$coordinates} |
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[11543] | 115 | \label{sec:SCOORD_continuity} |
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[707] | 116 | |
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[11558] | 117 | Using (\autoref{eq:SCOORD_s_chain_rule1}) and |
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[10354] | 118 | the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, |
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| 119 | the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to |
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| 120 | obtain its expression in the curvilinear $s-$coordinate system: |
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[707] | 121 | |
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[10414] | 122 | \begin{subequations} |
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| 123 | \begin{align*} |
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| 124 | { |
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| 125 | \begin{array}{*{20}l} |
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[11151] | 126 | \nabla \cdot {\mathrm {\mathbf U}} |
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[10414] | 127 | &= \frac{1}{e_1 \,e_2 } \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z |
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| 128 | +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z \right] |
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| 129 | + \frac{\partial w}{\partial z} \\ \\ |
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| 130 | & = \frac{1}{e_1 \,e_2 } \left[ |
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| 131 | \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s |
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| 132 | - \frac{e_1 }{e_3 } \sigma_1 \frac{\partial (e_2 \,u)}{\partial s} |
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| 133 | + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s |
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| 134 | - \frac{e_2 }{e_3 } \sigma_2 \frac{\partial (e_1 \,v)}{\partial s} \right] |
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| 135 | + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\ \\ |
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| 136 | & = \frac{1}{e_1 \,e_2 } \left[ |
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| 137 | \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s |
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| 138 | + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s \right] |
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| 139 | + \frac{1}{e_3 }\left[ \frac{\partial w}{\partial s} |
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| 140 | - \sigma_1 \frac{\partial u}{\partial s} |
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| 141 | - \sigma_2 \frac{\partial v}{\partial s} \right] \\ \\ |
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| 142 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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| 143 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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| 144 | -\left. e_2 \,u \frac{\partial e_3 }{\partial i} \right|_s |
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| 145 | + \left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s |
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| 146 | - \left. e_1 v \frac{\partial e_3 }{\partial j} \right|_s \right] \\ |
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| 147 | & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad |
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| 148 | + \frac{1}{e_3 } \left[ \frac{\partial w}{\partial s} |
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| 149 | - \sigma_1 \frac{\partial u}{\partial s} |
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| 150 | - \sigma_2 \frac{\partial v}{\partial s} \right] \\ |
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| 151 | % |
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| 152 | \intertext{Noting that $ |
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| 153 | \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s |
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| 154 | =\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s |
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| 155 | =\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right) |
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| 156 | =\frac{\partial \sigma_1}{\partial s} |
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| 157 | $ and $ |
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| 158 | \frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s |
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| 159 | =\frac{\partial \sigma_2}{\partial s} |
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| 160 | $, it becomes:} |
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| 161 | % |
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[11151] | 162 | \nabla \cdot {\mathrm {\mathbf U}} |
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[10414] | 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 | & \qquad \qquad \qquad \qquad \quad |
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| 167 | +\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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| 168 | \\ |
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| 169 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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| 170 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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| 171 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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| 172 | + \frac{1}{e_3 } \; \frac{\partial}{\partial s} \left[ w - u\;\sigma_1 - v\;\sigma_2 \right] |
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| 173 | \end{array} |
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| 174 | } |
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| 175 | \end{align*} |
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[2282] | 176 | \end{subequations} |
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| 177 | |
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[11543] | 178 | Here, $w$ is the vertical velocity relative to the $z-$coordinate system. |
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| 179 | Using the first form of (\autoref{eq:SCOORD_s_infin_changes}) |
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| 180 | and the definitions (\autoref{eq:SCOORD_s_slope}) and (\autoref{eq:SCOORD_w_in_s}) for $\sigma_1$, $\sigma_2$ and $w_s$, |
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[11335] | 181 | one can show that the vertical velocity, $w_p$ of a point |
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[11543] | 182 | moving with the horizontal velocity of the fluid along an $s$ surface is given by |
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[10414] | 183 | \begin{equation} |
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[11543] | 184 | \label{eq:SCOORD_w_p} |
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[11335] | 185 | \begin{split} |
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| 186 | w_p = & \left. \frac{ \partial z }{\partial t} \right|_s |
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[11543] | 187 | + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s |
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[11335] | 188 | + \frac{v}{e_2} \left. \frac{ \partial z }{\partial j} \right|_s \\ |
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| 189 | = & w_s + u \sigma_1 + v \sigma_2 . |
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[11543] | 190 | \end{split} |
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[11335] | 191 | \end{equation} |
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| 192 | The vertical velocity across this surface is denoted by |
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| 193 | \begin{equation} |
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[11543] | 194 | \label{eq:SCOORD_w_s} |
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| 195 | \omega = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v ) . |
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[707] | 196 | \end{equation} |
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[11543] | 197 | Hence |
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[11335] | 198 | \begin{equation} |
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[11543] | 199 | \frac{1}{e_3 } \frac{\partial}{\partial s} \left[ w - u\;\sigma_1 - v\;\sigma_2 \right] = |
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| 200 | \frac{1}{e_3 } \frac{\partial}{\partial s} \left[ \omega + w_s \right] = |
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| 201 | \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s} |
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| 202 | + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] = |
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| 203 | \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s |
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[11335] | 204 | \end{equation} |
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| 205 | |
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[11543] | 206 | Using (\autoref{eq:SCOORD_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain |
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[11335] | 207 | our final expression for the divergence of the velocity in the curvilinear $s-$coordinate system: |
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| 208 | \begin{equation} |
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| 209 | \nabla \cdot {\mathrm {\mathbf U}} = |
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| 210 | \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
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[10414] | 211 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
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| 212 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
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| 213 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
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[11335] | 214 | + \frac{1}{e_3 } \left. \frac{\partial e_3}{\partial t} \right|_s . |
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| 215 | \end{equation} |
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[707] | 216 | |
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[11543] | 217 | As a result, the continuity equation \autoref{eq:MB_PE_continuity} in the $s-$coordinates is: |
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[10414] | 218 | \begin{equation} |
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[11543] | 219 | \label{eq:SCOORD_sco_Continuity} |
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[10414] | 220 | \frac{1}{e_3 } \frac{\partial e_3}{\partial t} |
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| 221 | + \frac{1}{e_1 \,e_2 \,e_3 }\left[ |
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| 222 | {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s |
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| 223 | + \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right] |
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[11335] | 224 | +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 . |
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[707] | 225 | \end{equation} |
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[11335] | 226 | An additional term has appeared that takes into account |
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[10354] | 227 | the contribution of the time variation of the vertical coordinate to the volume budget. |
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[707] | 228 | |
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[9393] | 229 | \section{Momentum equation in $s-$coordinate} |
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[11543] | 230 | \label{sec:SCOORD_momentum} |
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[707] | 231 | |
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[10354] | 232 | Here we only consider the first component of the momentum equation, |
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[2282] | 233 | the generalization to the second one being straightforward. |
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| 234 | |
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| 235 | $\bullet$ \textbf{Total derivative in vector invariant form} |
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| 236 | |
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[11543] | 237 | Let us consider \autoref{eq:MB_dyn_vect}, the first component of the momentum equation in the vector invariant form. |
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[10354] | 238 | Its total $z-$coordinate time derivative, |
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| 239 | $\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain |
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[2282] | 240 | its expression in the curvilinear $s-$coordinate system: |
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[707] | 241 | |
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[10414] | 242 | \begin{subequations} |
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| 243 | \begin{align*} |
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| 244 | { |
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| 245 | \begin{array}{*{20}l} |
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| 246 | \left. \frac{D u}{D t} \right|_z |
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| 247 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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| 248 | - \left. \zeta \right|_z v |
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| 249 | + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z |
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| 250 | + w \;\frac{\partial u}{\partial z} \\ \\ |
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| 251 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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[11335] | 252 | - \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z |
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[10414] | 253 | -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v |
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| 254 | + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z |
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| 255 | + w \;\frac{\partial u}{\partial z} \\ |
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| 256 | % |
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[11558] | 257 | \intertext{introducing the chain rule (\autoref{eq:SCOORD_s_chain_rule1}) } |
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[10414] | 258 | % |
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| 259 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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| 260 | - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s |
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| 261 | -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right. |
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| 262 | \left. {-\frac{e_1}{e_3}\sigma_1 \frac{\partial (e_2 \,v)}{\partial s} |
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| 263 | +\frac{e_2}{e_3}\sigma_2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v \\ |
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| 264 | & \qquad \qquad \qquad \qquad |
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| 265 | { |
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| 266 | + \frac{1}{2e_1} \left( \left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s |
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| 267 | - \frac{e_1}{e_3}\sigma_1 \frac{\partial (u^2+v^2)}{\partial s} \right) |
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| 268 | + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} |
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| 269 | } \\ \\ |
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| 270 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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[11335] | 271 | - \left. \zeta \right|_s \;v |
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[10414] | 272 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ |
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| 273 | &\qquad \qquad \qquad \quad |
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| 274 | + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} |
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[11335] | 275 | + \left[ {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s} |
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[10414] | 276 | - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v |
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| 277 | - \frac{\sigma_1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ \\ |
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| 278 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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[11335] | 279 | - \left. \zeta \right|_s \;v |
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[10414] | 280 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ |
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| 281 | &\qquad \qquad \qquad \quad |
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| 282 | + \frac{1}{e_3} \left[ {w\frac{\partial u}{\partial s} |
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| 283 | +\sigma_1 v\frac{\partial v}{\partial s} - \sigma_2 v\frac{\partial u}{\partial s} |
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| 284 | - \sigma_1 u\frac{\partial u}{\partial s} - \sigma_1 v\frac{\partial v}{\partial s}} \right] \\ \\ |
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| 285 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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[11335] | 286 | - \left. \zeta \right|_s \;v |
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[10414] | 287 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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| 288 | + \frac{1}{e_3} \left[ w - \sigma_2 v - \sigma_1 u \right] |
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[11335] | 289 | \; \frac{\partial u}{\partial s} . \\ |
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[10414] | 290 | % |
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[11543] | 291 | \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{eq:SCOORD_w_s}) } |
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[10414] | 292 | % |
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| 293 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
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[11335] | 294 | - \left. \zeta \right|_s \;v |
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[10414] | 295 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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[11335] | 296 | + \frac{1}{e_3 } \left( \omega + w_s \right) \frac{\partial u}{\partial s} \\ |
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[10414] | 297 | \end{array} |
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| 298 | } |
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| 299 | \end{align*} |
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[2282] | 300 | \end{subequations} |
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| 301 | % |
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[11558] | 302 | Applying the time derivative chain rule (first equation of (\autoref{eq:SCOORD_s_chain_rule1})) to $u$ and |
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[11543] | 303 | using (\autoref{eq:SCOORD_w_in_s}) provides the expression of the last term of the right hand side, |
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[10414] | 304 | \[ |
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| 305 | { |
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| 306 | \begin{array}{*{20}l} |
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[11335] | 307 | \frac{w_s}{e_3} \;\frac{\partial u}{\partial s} |
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| 308 | = - \left. \frac{\partial s}{\partial t} \right|_z \; \frac{\partial u }{\partial s} |
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| 309 | = \left. {\frac{\partial u }{\partial t}} \right|_s - \left. {\frac{\partial u }{\partial t}} \right|_z \ . |
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[10414] | 310 | \end{array} |
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| 311 | } |
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[10406] | 312 | \] |
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[11335] | 313 | This leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, |
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[11435] | 314 | \ie\ the total $s-$coordinate time derivative : |
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[10414] | 315 | \begin{align} |
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[11543] | 316 | \label{eq:SCOORD_sco_Dt_vect} |
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[10414] | 317 | \left. \frac{D u}{D t} \right|_s |
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| 318 | = \left. {\frac{\partial u }{\partial t}} \right|_s |
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[11335] | 319 | - \left. \zeta \right|_s \;v |
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[10414] | 320 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
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[11543] | 321 | + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} . |
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[2282] | 322 | \end{align} |
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[10354] | 323 | Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in |
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| 324 | $z-$ and $s-$coordinates. |
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| 325 | This is not the case for the flux form as shown in next paragraph. |
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[2282] | 326 | |
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| 327 | $\bullet$ \textbf{Total derivative in flux form} |
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| 328 | |
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[10354] | 329 | Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish. |
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[11543] | 330 | Following the procedure used to establish (\autoref{eq:MB_flux_form}), it can be transformed into : |
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[10414] | 331 | % \begin{subequations} |
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| 332 | \begin{align*} |
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| 333 | { |
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| 334 | \begin{array}{*{20}l} |
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| 335 | \left. \frac{D u}{D t} \right|_s &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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| 336 | & - \zeta \;v |
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| 337 | + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} |
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| 338 | + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s} \\ \\ |
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| 339 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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| 340 | &+\frac{1}{e_1\;e_2} \left( \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i} |
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| 341 | + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j} \right) |
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| 342 | + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\ |
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| 343 | &&- \,u \left[ \frac{1}{e_1 e_2 } \left( \frac{\partial(e_2 u)}{\partial i} |
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| 344 | + \frac{\partial(e_1 v)}{\partial j} \right) |
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| 345 | + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ \\ |
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| 346 | &&- \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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[11335] | 347 | -u \;\frac{\partial e_1 }{\partial j} \right) . \\ |
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[10414] | 348 | \end{array} |
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| 349 | } |
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[817] | 350 | \end{align*} |
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[2282] | 351 | % |
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[11543] | 352 | Introducing the vertical scale factor inside the horizontal derivative of the first two terms |
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[11435] | 353 | (\ie\ the horizontal divergence), it becomes : |
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[10414] | 354 | \begin{align*} |
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| 355 | { |
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| 356 | \begin{array}{*{20}l} |
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| 357 | % \begin{align*} {\begin{array}{*{20}l} |
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[11543] | 358 | % {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s |
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[10414] | 359 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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| 360 | &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u^2 )}{\partial i} |
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| 361 | + \frac{\partial( e_1 e_3 \,u v )}{\partial j} |
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| 362 | - e_2 u u \frac{\partial e_3}{\partial i} |
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| 363 | - e_1 u v \frac{\partial e_3 }{\partial j} \right) |
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| 364 | + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\ |
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| 365 | && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} |
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| 366 | + \frac{\partial(e_1 e_3 \, v)}{\partial j} |
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| 367 | - e_2 u \;\frac{\partial e_3 }{\partial i} |
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| 368 | - e_1 v \;\frac{\partial e_3 }{\partial j} \right) |
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[11335] | 369 | + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ \\ |
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[10414] | 370 | && - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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| 371 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ \\ |
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| 372 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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| 373 | &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u\,u )}{\partial i} |
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| 374 | + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j} \right) |
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| 375 | + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\ |
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| 376 | && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} |
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| 377 | + \frac{\partial(e_1 e_3 \, v)}{\partial j} \right) |
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[11335] | 378 | + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] |
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[10414] | 379 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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[11335] | 380 | -u \;\frac{\partial e_1 }{\partial j} \right) . \\ |
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[10414] | 381 | % |
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| 382 | \intertext {Introducing a more compact form for the divergence of the momentum fluxes, |
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[11543] | 383 | and using (\autoref{eq:SCOORD_sco_Continuity}), the $s-$coordinate continuity equation, |
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[10414] | 384 | it becomes : } |
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| 385 | % |
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| 386 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
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[11151] | 387 | &+ \left. \nabla \cdot \left( {{\mathrm {\mathbf U}}\,u} \right) \right|_s |
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[10414] | 388 | + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t} |
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[2282] | 389 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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[10414] | 390 | -u \;\frac{\partial e_1 }{\partial j} \right) |
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| 391 | \\ |
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| 392 | \end{array} |
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| 393 | } |
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[2282] | 394 | \end{align*} |
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[11543] | 395 | which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, |
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[11435] | 396 | \ie\ the total $s-$coordinate time derivative in flux form: |
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[10414] | 397 | \begin{flalign} |
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[11543] | 398 | \label{eq:SCOORD_sco_Dt_flux} |
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[10414] | 399 | \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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[11151] | 400 | + \left. \nabla \cdot \left( {{\mathrm {\mathbf U}}\,u} \right) \right|_s |
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[10414] | 401 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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[11335] | 402 | -u \;\frac{\partial e_1 }{\partial j} \right). |
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[2282] | 403 | \end{flalign} |
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| 404 | which is the total time derivative expressed in the curvilinear $s-$coordinate system. |
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[10354] | 405 | It has the same form as in the $z-$coordinate but for |
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| 406 | the vertical scale factor that has appeared inside the time derivative which |
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[11543] | 407 | comes from the modification of (\autoref{eq:SCOORD_sco_Continuity}), |
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[10354] | 408 | the continuity equation. |
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[707] | 409 | |
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[2282] | 410 | $\bullet$ \textbf{horizontal pressure gradient} |
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| 411 | |
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| 412 | The horizontal pressure gradient term can be transformed as follows: |
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[10406] | 413 | \[ |
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[10414] | 414 | \begin{split} |
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| 415 | -\frac{1}{\rho_o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z |
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| 416 | & =-\frac{1}{\rho_o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial p}{\partial s}} \right] \\ |
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| 417 | & =-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma_1 }{\rho_o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\ |
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[11335] | 418 | &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1 . |
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[10414] | 419 | \end{split} |
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[10406] | 420 | \] |
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[10354] | 421 | Applying similar manipulation to the second component and |
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[11543] | 422 | replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{eq:SCOORD_s_slope}, it becomes: |
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[10414] | 423 | \begin{equation} |
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[11543] | 424 | \label{eq:SCOORD_grad_p_1} |
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[10414] | 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}{\rho_o \,e_1 } \left( \left. {\frac{\partial p}{\partial i}} \right|_s |
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| 428 | + g\;\rho \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) \\ |
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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}{\rho_o \,e_2 } \left( \left. {\frac{\partial p}{\partial j}} \right|_s |
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[11335] | 432 | + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) . \\ |
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[10414] | 433 | \end{split} |
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[707] | 434 | \end{equation} |
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| 435 | |
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[11543] | 436 | An additional term appears in (\autoref{eq:SCOORD_grad_p_1}) which accounts for |
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[10354] | 437 | the tilt of $s-$surfaces with respect to geopotential $z-$surfaces. |
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[707] | 438 | |
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[10354] | 439 | As in $z$-coordinate, |
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[11123] | 440 | the horizontal pressure gradient can be split in two parts following \citet{marsaleix.auclair.ea_OM08}. |
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[10354] | 441 | Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$, |
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| 442 | and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. |
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[2282] | 443 | The pressure is then given by: |
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[10414] | 444 | \[ |
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| 445 | \begin{split} |
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[11335] | 446 | p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \rho_o \left( d + 1 \right) \; e_3 \; dk \\ |
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| 447 | &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + \rho_o g \, \int_z^\eta e_3 \; dk . |
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[10414] | 448 | \end{split} |
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[10406] | 449 | \] |
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[2282] | 450 | Therefore, $p$ and $p_h'$ are linked through: |
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[10414] | 451 | \begin{equation} |
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[11543] | 452 | \label{eq:SCOORD_pressure} |
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[11335] | 453 | p = \rho_o \; p_h' + \rho_o \, g \, ( \eta - z ) |
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[2282] | 454 | \end{equation} |
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| 455 | and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is: |
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[10414] | 456 | \[ |
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[11335] | 457 | \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 . |
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[10406] | 458 | \] |
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[2282] | 459 | |
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[11543] | 460 | Substituing \autoref{eq:SCOORD_pressure} in \autoref{eq:SCOORD_grad_p_1} and |
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[11335] | 461 | using the definition of the density anomaly it becomes an expression in two parts: |
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[10414] | 462 | \begin{equation} |
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[11543] | 463 | \label{eq:SCOORD_grad_p_2} |
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[10414] | 464 | \begin{split} |
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| 465 | -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z |
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| 466 | &=-\frac{1}{e_1 } \left( \left. {\frac{\partial p_h'}{\partial i}} \right|_s |
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[11335] | 467 | + g\; d \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} , \\ |
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[10414] | 468 | % |
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| 469 | -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z |
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| 470 | &=-\frac{1}{e_2 } \left( \left. {\frac{\partial p_h'}{\partial j}} \right|_s |
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[11335] | 471 | + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} . \\ |
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[10414] | 472 | \end{split} |
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[2282] | 473 | \end{equation} |
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[10354] | 474 | This formulation of the pressure gradient is characterised by the appearance of |
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| 475 | a term depending on the sea surface height only |
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[11543] | 476 | (last term on the right hand side of expression \autoref{eq:SCOORD_grad_p_2}). |
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[10354] | 477 | This term will be loosely termed \textit{surface pressure gradient} whereas |
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| 478 | the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to |
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| 479 | the $z$-coordinate formulation. |
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| 480 | In fact, the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, |
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| 481 | and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration. |
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[2282] | 482 | |
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| 483 | $\bullet$ \textbf{The other terms of the momentum equation} |
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| 484 | |
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[10354] | 485 | The coriolis and forcing terms as well as the the vertical physics remain unchanged as |
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| 486 | they involve neither time nor space derivatives. |
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[11543] | 487 | The form of the lateral physics is discussed in \autoref{apdx:DIFFOPERS}. |
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[2282] | 488 | |
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| 489 | $\bullet$ \textbf{Full momentum equation} |
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| 490 | |
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[10354] | 491 | To sum up, in a curvilinear $s$-coordinate system, |
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| 492 | the vector invariant momentum equation solved by the model has the same mathematical expression as |
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| 493 | the one in a curvilinear $z-$coordinate, except for the pressure gradient term: |
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[10414] | 494 | \begin{subequations} |
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[11543] | 495 | \label{eq:SCOORD_dyn_vect} |
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[10414] | 496 | \begin{multline} |
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[11543] | 497 | \label{eq:SCOORD_PE_dyn_vect_u} |
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[10414] | 498 | \frac{\partial u}{\partial t}= |
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| 499 | + \left( {\zeta +f} \right)\,v |
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| 500 | - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right) |
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| 501 | - \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\ |
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| 502 | - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) |
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| 503 | - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} |
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[11335] | 504 | + D_u^{\vect{U}} + F_u^{\vect{U}} , |
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[10414] | 505 | \end{multline} |
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| 506 | \begin{multline} |
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[11543] | 507 | \label{eq:SCOORD_dyn_vect_v} |
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[10414] | 508 | \frac{\partial v}{\partial t}= |
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| 509 | - \left( {\zeta +f} \right)\,u |
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| 510 | - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right) |
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| 511 | - \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\ |
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| 512 | - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) |
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| 513 | - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} |
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[11335] | 514 | + D_v^{\vect{U}} + F_v^{\vect{U}} . |
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[10414] | 515 | \end{multline} |
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[817] | 516 | \end{subequations} |
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[10354] | 517 | whereas the flux form momentum equation differs from it by |
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| 518 | the formulation of both the time derivative and the pressure gradient term: |
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[10414] | 519 | \begin{subequations} |
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[11543] | 520 | \label{eq:SCOORD_dyn_flux} |
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[10414] | 521 | \begin{multline} |
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[11543] | 522 | \label{eq:SCOORD_PE_dyn_flux_u} |
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[10414] | 523 | \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t} = |
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[11335] | 524 | - \nabla \cdot \left( {{\mathrm {\mathbf U}}\,u} \right) |
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[10414] | 525 | + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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| 526 | -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,v \\ |
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| 527 | - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) |
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| 528 | - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} |
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[11335] | 529 | + D_u^{\vect{U}} + F_u^{\vect{U}} , |
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[10414] | 530 | \end{multline} |
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| 531 | \begin{multline} |
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[11543] | 532 | \label{eq:SCOORD_dyn_flux_v} |
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[10414] | 533 | \frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}= |
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[11151] | 534 | - \nabla \cdot \left( {{\mathrm {\mathbf U}}\,v} \right) |
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[11335] | 535 | - \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
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[10414] | 536 | -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,u \\ |
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| 537 | - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) |
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| 538 | - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} |
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[11543] | 539 | + D_v^{\vect{U}} + F_v^{\vect{U}} . |
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[10414] | 540 | \end{multline} |
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[2282] | 541 | \end{subequations} |
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| 542 | Both formulation share the same hydrostatic pressure balance expressed in terms of |
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[3294] | 543 | hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: |
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[10414] | 544 | \begin{equation} |
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[11543] | 545 | \label{eq:SCOORD_dyn_zph} |
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[11335] | 546 | \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 . |
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[2282] | 547 | \end{equation} |
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[707] | 548 | |
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[10354] | 549 | It is important to realize that the change in coordinate system has only concerned the position on the vertical. |
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| 550 | It has not affected (\textbf{i},\textbf{j},\textbf{k}), the orthogonal curvilinear set of unit vectors. |
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| 551 | ($u$,$v$) are always horizontal velocities so that their evolution is driven by \emph{horizontal} forces, |
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| 552 | in particular the pressure gradient. |
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| 553 | By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component, |
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[11543] | 554 | \ie\ the volume flux across the moving $s$-surfaces per unit horizontal area. |
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[817] | 555 | |
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[9393] | 556 | \section{Tracer equation} |
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[11543] | 557 | \label{sec:SCOORD_tracer} |
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[817] | 558 | |
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[10354] | 559 | The tracer equation is obtained using the same calculation as for the continuity equation and then |
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| 560 | regrouping the time derivative terms in the left hand side : |
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[707] | 561 | |
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[10414] | 562 | \begin{multline} |
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[11543] | 563 | \label{eq:SCOORD_tracer} |
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[10414] | 564 | \frac{1}{e_3} \frac{\partial \left( e_3 T \right)}{\partial t} |
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| 565 | = -\frac{1}{e_1 \,e_2 \,e_3} |
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| 566 | \left[ \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right) |
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| 567 | + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] \\ |
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[11335] | 568 | - \frac{1}{e_3} \frac{\partial }{\partial k} \left( Tw \right) |
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[10414] | 569 | + D^{T} +F^{T} |
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[707] | 570 | \end{multline} |
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| 571 | |
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[11543] | 572 | The expression for the advection term is a straight consequence of (\autoref{eq:SCOORD_sco_Continuity}), |
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| 573 | the expression of the 3D divergence in the $s-$coordinates established above. |
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[707] | 574 | |
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[11584] | 575 | \onlyinsubfile{\input{../../global/epilogue}} |
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[10414] | 576 | |
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[6997] | 577 | \end{document} |
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