[6997] | 1 | \documentclass[NEMO_book]{subfiles} |
---|
| 2 | \begin{document} |
---|
[707] | 3 | |
---|
| 4 | % ================================================================ |
---|
| 5 | % Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations |
---|
| 6 | % ================================================================ |
---|
[2282] | 7 | \chapter{Curvilinear $s-$Coordinate Equations} |
---|
[707] | 8 | \label{Apdx_A} |
---|
| 9 | \minitoc |
---|
| 10 | |
---|
[2282] | 11 | \newpage |
---|
| 12 | $\ $\newline % force a new ligne |
---|
[996] | 13 | |
---|
[2282] | 14 | % ================================================================ |
---|
| 15 | % Chain rule |
---|
| 16 | % ================================================================ |
---|
[3294] | 17 | \section{The chain rule for $s-$coordinates} |
---|
[2282] | 18 | \label{Apdx_A_continuity} |
---|
| 19 | |
---|
[3294] | 20 | In order to establish the set of Primitive Equation in curvilinear $s$-coordinates |
---|
[2282] | 21 | ($i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian |
---|
| 22 | Eulerian (ALE) coordinate in the vertical), we start from the set of equations established |
---|
| 23 | in \S\ref{PE_zco_Eq} for the special case $k = z$ and thus $e_3 = 1$, and we introduce |
---|
| 24 | an arbitrary vertical coordinate $a = a(i,j,z,t)$. Let us define a new vertical scale factor by |
---|
[1223] | 25 | $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal |
---|
[2282] | 26 | slope of $s-$surfaces by : |
---|
[817] | 27 | \begin{equation} \label{Apdx_A_s_slope} |
---|
[707] | 28 | \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s |
---|
[817] | 29 | \quad \text{and} \quad |
---|
[707] | 30 | \sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s |
---|
| 31 | \end{equation} |
---|
| 32 | |
---|
[2282] | 33 | The chain rule to establish the model equations in the curvilinear $s-$coordinate |
---|
[1223] | 34 | system is: |
---|
[817] | 35 | \begin{equation} \label{Apdx_A_s_chain_rule} |
---|
[707] | 36 | \begin{aligned} |
---|
[817] | 37 | &\left. {\frac{\partial \bullet }{\partial t}} \right|_z = |
---|
| 38 | \left. {\frac{\partial \bullet }{\partial t}} \right|_s |
---|
| 39 | -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\ |
---|
| 40 | &\left. {\frac{\partial \bullet }{\partial i}} \right|_z = |
---|
| 41 | \left. {\frac{\partial \bullet }{\partial i}} \right|_s |
---|
| 42 | -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}= |
---|
| 43 | \left. {\frac{\partial \bullet }{\partial i}} \right|_s |
---|
| 44 | -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet }{\partial s} \\ |
---|
| 45 | &\left. {\frac{\partial \bullet }{\partial j}} \right|_z = |
---|
| 46 | \left. {\frac{\partial \bullet }{\partial j}} \right|_s |
---|
| 47 | - \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= |
---|
| 48 | \left. {\frac{\partial \bullet }{\partial j}} \right|_s |
---|
| 49 | - \frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet }{\partial s} \\ |
---|
| 50 | &\;\frac{\partial \bullet }{\partial z} \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} \\ |
---|
[707] | 51 | \end{aligned} |
---|
| 52 | \end{equation} |
---|
| 53 | |
---|
[2282] | 54 | In particular applying the time derivative chain rule to $z$ provides the expression |
---|
| 55 | for $w_s$, the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: |
---|
[817] | 56 | \begin{equation} \label{Apdx_A_w_in_s} |
---|
| 57 | w_s = \left. \frac{\partial z }{\partial t} \right|_s |
---|
| 58 | = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t} |
---|
| 59 | = e_3 \, \frac{\partial s}{\partial t} |
---|
| 60 | \end{equation} |
---|
[707] | 61 | |
---|
[2282] | 62 | |
---|
[817] | 63 | % ================================================================ |
---|
| 64 | % continuity equation |
---|
| 65 | % ================================================================ |
---|
[3294] | 66 | \section{Continuity Equation in $s-$coordinates} |
---|
[2282] | 67 | \label{Apdx_A_continuity} |
---|
[707] | 68 | |
---|
[1223] | 69 | Using (\ref{Apdx_A_s_chain_rule}) and the fact that the horizontal scale factors |
---|
| 70 | $e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of |
---|
[2282] | 71 | the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows |
---|
| 72 | in order to obtain its expression in the curvilinear $s-$coordinate system: |
---|
[707] | 73 | |
---|
[2282] | 74 | \begin{subequations} |
---|
| 75 | \begin{align*} {\begin{array}{*{20}l} |
---|
[817] | 76 | \nabla \cdot {\rm {\bf U}} |
---|
| 77 | &= \frac{1}{e_1 \,e_2 } \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z |
---|
| 78 | +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z \right] |
---|
| 79 | + \frac{\partial w}{\partial z} \\ |
---|
| 80 | \\ |
---|
| 81 | & = \frac{1}{e_1 \,e_2 } \left[ |
---|
| 82 | \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s |
---|
| 83 | - \frac{e_1 }{e_3 } \sigma _1 \frac{\partial (e_2 \,u)}{\partial s} |
---|
| 84 | + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s |
---|
| 85 | - \frac{e_2 }{e_3 } \sigma _2 \frac{\partial (e_1 \,v)}{\partial s} \right] |
---|
| 86 | + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\ |
---|
| 87 | \\ |
---|
| 88 | & = \frac{1}{e_1 \,e_2 } \left[ |
---|
| 89 | \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s |
---|
| 90 | + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s \right] |
---|
| 91 | + \frac{1}{e_3 }\left[ \frac{\partial w}{\partial s} |
---|
| 92 | - \sigma _1 \frac{\partial u}{\partial s} |
---|
| 93 | - \sigma _2 \frac{\partial v}{\partial s} \right] \\ |
---|
| 94 | \\ |
---|
| 95 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
---|
| 96 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
---|
| 97 | -\left. e_2 \,u \frac{\partial e_3 }{\partial i} \right|_s |
---|
| 98 | + \left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s |
---|
| 99 | - \left. e_1 v \frac{\partial e_3 }{\partial j} \right|_s \right] \\ |
---|
| 100 | & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad |
---|
| 101 | + \frac{1}{e_3 } \left[ \frac{\partial w}{\partial s} |
---|
| 102 | - \sigma _1 \frac{\partial u}{\partial s} |
---|
| 103 | - \sigma _2 \frac{\partial v}{\partial s} \right] \\ |
---|
[2282] | 104 | % |
---|
| 105 | \intertext{Noting that $ |
---|
| 106 | \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s |
---|
| 107 | =\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s |
---|
| 108 | =\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right) |
---|
| 109 | =\frac{\partial \sigma _1}{\partial s} |
---|
| 110 | $ and $ |
---|
| 111 | \frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s |
---|
| 112 | =\frac{\partial \sigma _2}{\partial s} |
---|
| 113 | $, it becomes:} |
---|
| 114 | % |
---|
[817] | 115 | \nabla \cdot {\rm {\bf U}} |
---|
| 116 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
---|
| 117 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
---|
| 118 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] \\ |
---|
[2282] | 119 | & \qquad \qquad \qquad \qquad \quad |
---|
[817] | 120 | +\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] \\ |
---|
| 121 | \\ |
---|
| 122 | & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
---|
| 123 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
---|
| 124 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
---|
| 125 | + \frac{1}{e_3 } \; \frac{\partial}{\partial s} \left[ w - u\;\sigma _1 - v\;\sigma _2 \right] |
---|
[2282] | 126 | \end{array} } |
---|
| 127 | \end{align*} |
---|
| 128 | \end{subequations} |
---|
| 129 | |
---|
[1223] | 130 | Here, $w$ is the vertical velocity relative to the $z-$coordinate system. |
---|
| 131 | Introducing the dia-surface velocity component, $\omega $, defined as |
---|
[3294] | 132 | the volume flux across the moving $s$-surfaces per unit horizontal area: |
---|
[817] | 133 | \begin{equation} \label{Apdx_A_w_s} |
---|
| 134 | \omega = w - w_s - \sigma _1 \,u - \sigma _2 \,v \\ |
---|
[707] | 135 | \end{equation} |
---|
[1223] | 136 | with $w_s$ given by \eqref{Apdx_A_w_in_s}, we obtain the expression for |
---|
[2282] | 137 | the divergence of the velocity in the curvilinear $s-$coordinate system: |
---|
| 138 | \begin{subequations} |
---|
| 139 | \begin{align*} {\begin{array}{*{20}l} |
---|
[817] | 140 | \nabla \cdot {\rm {\bf U}} |
---|
| 141 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
---|
| 142 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
---|
| 143 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
---|
| 144 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
---|
| 145 | + \frac{1}{e_3 } \frac{\partial w_s }{\partial s} \\ |
---|
| 146 | \\ |
---|
| 147 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
---|
| 148 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
---|
| 149 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
---|
| 150 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
---|
| 151 | + \frac{1}{e_3 } \frac{\partial}{\partial s} \left( e_3 \; \frac{\partial s}{\partial t} \right) \\ |
---|
| 152 | \\ |
---|
| 153 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
---|
| 154 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
---|
| 155 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
---|
| 156 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
---|
| 157 | + \frac{\partial}{\partial s} \frac{\partial s}{\partial t} |
---|
| 158 | + \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s} \\ |
---|
| 159 | \\ |
---|
| 160 | &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ |
---|
| 161 | \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s |
---|
| 162 | +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] |
---|
| 163 | + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} |
---|
| 164 | + \frac{1}{e_3 } \frac{\partial e_3}{\partial t} \\ |
---|
[2282] | 165 | \end{array} } |
---|
[817] | 166 | \end{align*} |
---|
[2282] | 167 | \end{subequations} |
---|
[707] | 168 | |
---|
[1223] | 169 | As a result, the continuity equation \eqref{Eq_PE_continuity} in the |
---|
[2282] | 170 | $s-$coordinates is: |
---|
| 171 | \begin{equation} \label{Apdx_A_sco_Continuity} |
---|
[817] | 172 | \frac{1}{e_3 } \frac{\partial e_3}{\partial t} |
---|
| 173 | + \frac{1}{e_1 \,e_2 \,e_3 }\left[ |
---|
| 174 | {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s |
---|
| 175 | + \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right] |
---|
| 176 | +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 |
---|
[707] | 177 | \end{equation} |
---|
[2282] | 178 | A additional term has appeared that take into account the contribution of the time variation |
---|
| 179 | of the vertical coordinate to the volume budget. |
---|
[707] | 180 | |
---|
[2282] | 181 | |
---|
[817] | 182 | % ================================================================ |
---|
| 183 | % momentum equation |
---|
| 184 | % ================================================================ |
---|
[2282] | 185 | \section{Momentum Equation in $s-$coordinate} |
---|
| 186 | \label{Apdx_A_momentum} |
---|
[707] | 187 | |
---|
[2282] | 188 | Here we only consider the first component of the momentum equation, |
---|
| 189 | the generalization to the second one being straightforward. |
---|
| 190 | |
---|
| 191 | $\ $\newline % force a new ligne |
---|
| 192 | |
---|
| 193 | $\bullet$ \textbf{Total derivative in vector invariant form} |
---|
| 194 | |
---|
[1223] | 195 | Let us consider \eqref{Eq_PE_dyn_vect}, the first component of the momentum |
---|
[2282] | 196 | equation in the vector invariant form. Its total $z-$coordinate time derivative, |
---|
| 197 | $\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain |
---|
| 198 | its expression in the curvilinear $s-$coordinate system: |
---|
[707] | 199 | |
---|
[2282] | 200 | \begin{subequations} |
---|
| 201 | \begin{align*} {\begin{array}{*{20}l} |
---|
| 202 | \left. \frac{D u}{D t} \right|_z |
---|
| 203 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
---|
| 204 | - \left. \zeta \right|_z v |
---|
| 205 | + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z |
---|
| 206 | + w \;\frac{\partial u}{\partial z} \\ |
---|
[817] | 207 | \\ |
---|
[2282] | 208 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
---|
| 209 | - \left. \zeta \right|_z v |
---|
| 210 | + \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z |
---|
| 211 | -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v |
---|
| 212 | + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z |
---|
| 213 | + w \;\frac{\partial u}{\partial z} \\ |
---|
| 214 | % |
---|
| 215 | \intertext{introducing the chain rule (\ref{Apdx_A_s_chain_rule}) } |
---|
| 216 | % |
---|
| 217 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
---|
| 218 | - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s |
---|
| 219 | -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right. |
---|
| 220 | \left. {-\frac{e_1}{e_3}\sigma _1 \frac{\partial (e_2 \,v)}{\partial s} |
---|
| 221 | +\frac{e_2}{e_3}\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v \\ |
---|
| 222 | & \qquad \qquad \qquad \qquad |
---|
| 223 | { + \frac{1}{2e_1} \left( \left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s |
---|
| 224 | - \frac{e_1}{e_3}\sigma _1 \frac{\partial (u^2+v^2)}{\partial s} \right) |
---|
| 225 | + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} } \\ |
---|
[817] | 226 | \\ |
---|
[2282] | 227 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
---|
| 228 | + \left. \zeta \right|_s \;v |
---|
| 229 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ |
---|
| 230 | &\qquad \qquad \qquad \quad |
---|
| 231 | + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} |
---|
| 232 | - \left[ {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s} |
---|
| 233 | - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v |
---|
| 234 | - \frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ |
---|
[817] | 235 | \\ |
---|
[2282] | 236 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
---|
| 237 | + \left. \zeta \right|_s \;v |
---|
| 238 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ |
---|
| 239 | &\qquad \qquad \qquad \quad |
---|
| 240 | + \frac{1}{e_3} \left[ {w\frac{\partial u}{\partial s} |
---|
| 241 | +\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s} |
---|
| 242 | - \sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\ |
---|
| 243 | \\ |
---|
| 244 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
---|
| 245 | + \left. \zeta \right|_s \;v |
---|
| 246 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
---|
| 247 | + \frac{1}{e_3} \left[ w - \sigma _2 v - \sigma _1 u \right] |
---|
[817] | 248 | \; \frac{\partial u}{\partial s} \\ |
---|
[2282] | 249 | % |
---|
| 250 | \intertext{Introducing $\omega$, the dia-a-surface velocity given by (\ref{Apdx_A_w_s}) } |
---|
| 251 | % |
---|
| 252 | &= \left. {\frac{\partial u }{\partial t}} \right|_z |
---|
| 253 | + \left. \zeta \right|_s \;v |
---|
| 254 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
---|
| 255 | + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s} \\ |
---|
| 256 | \end{array} } |
---|
| 257 | \end{align*} |
---|
| 258 | \end{subequations} |
---|
| 259 | % |
---|
| 260 | Applying the time derivative chain rule (first equation of (\ref{Apdx_A_s_chain_rule})) |
---|
| 261 | to $u$ and using (\ref{Apdx_A_w_in_s}) provides the expression of the last term |
---|
| 262 | of the right hand side, |
---|
| 263 | \begin{equation*} {\begin{array}{*{20}l} |
---|
| 264 | w_s \;\frac{\partial u}{\partial s} |
---|
| 265 | = \frac{\partial s}{\partial t} \; \frac{\partial u }{\partial s} |
---|
| 266 | = \left. {\frac{\partial u }{\partial t}} \right|_s - \left. {\frac{\partial u }{\partial t}} \right|_z \quad , |
---|
| 267 | \end{array} } |
---|
| 268 | \end{equation*} |
---|
| 269 | leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, |
---|
| 270 | $i.e.$ the total $s-$coordinate time derivative : |
---|
| 271 | \begin{align} \label{Apdx_A_sco_Dt_vect} |
---|
| 272 | \left. \frac{D u}{D t} \right|_s |
---|
| 273 | = \left. {\frac{\partial u }{\partial t}} \right|_s |
---|
| 274 | + \left. \zeta \right|_s \;v |
---|
| 275 | + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s |
---|
| 276 | + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} |
---|
| 277 | \end{align} |
---|
| 278 | Therefore, the vector invariant form of the total time derivative has exactly the same |
---|
| 279 | mathematical form in $z-$ and $s-$coordinates. This is not the case for the flux form |
---|
| 280 | as shown in next paragraph. |
---|
| 281 | |
---|
| 282 | $\ $\newline % force a new ligne |
---|
| 283 | |
---|
| 284 | $\bullet$ \textbf{Total derivative in flux form} |
---|
| 285 | |
---|
| 286 | Let us start from the total time derivative in the curvilinear $s-$coordinate system |
---|
| 287 | we have just establish. Following the procedure used to establish (\ref{Eq_PE_flux_form}), |
---|
| 288 | it can be transformed into : |
---|
| 289 | %\begin{subequations} |
---|
| 290 | \begin{align*} {\begin{array}{*{20}l} |
---|
| 291 | \left. \frac{D u}{D t} \right|_s &= \left. {\frac{\partial u }{\partial t}} \right|_s |
---|
| 292 | & - \zeta \;v |
---|
| 293 | + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} |
---|
| 294 | + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s} \\ |
---|
[817] | 295 | \\ |
---|
[2282] | 296 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
---|
| 297 | &+\frac{1}{e_1\;e_2} \left( \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i} |
---|
| 298 | + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j} \right) |
---|
| 299 | + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ |
---|
| 300 | \\ |
---|
| 301 | &&- \,u \left[ \frac{1}{e_1 e_2 } \left( \frac{\partial(e_2 u)}{\partial i} |
---|
| 302 | + \frac{\partial(e_1 v)}{\partial j} \right) |
---|
| 303 | + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ |
---|
| 304 | \\ |
---|
| 305 | &&- \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
---|
| 306 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ |
---|
| 307 | \end{array} } |
---|
[817] | 308 | \end{align*} |
---|
[2282] | 309 | % |
---|
| 310 | Introducing the vertical scale factor inside the horizontal derivative of the first two terms |
---|
| 311 | ($i.e.$ the horizontal divergence), it becomes : |
---|
| 312 | \begin{subequations} |
---|
| 313 | \begin{align*} {\begin{array}{*{20}l} |
---|
| 314 | %\begin{align*} {\begin{array}{*{20}l} |
---|
| 315 | %{\begin{array}{*{20}l} |
---|
| 316 | \left. \frac{D u}{D t} \right|_s |
---|
| 317 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
---|
| 318 | &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u^2 )}{\partial i} |
---|
| 319 | + \frac{\partial( e_1 e_3 \,u v )}{\partial j} |
---|
| 320 | - e_2 u u \frac{\partial e_3}{\partial i} |
---|
| 321 | - e_1 u v \frac{\partial e_3 }{\partial j} \right) |
---|
| 322 | + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ |
---|
| 323 | \\ |
---|
| 324 | && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} |
---|
| 325 | + \frac{\partial(e_1 e_3 \, v)}{\partial j} |
---|
| 326 | - e_2 u \;\frac{\partial e_3 }{\partial i} |
---|
| 327 | - e_1 v \;\frac{\partial e_3 }{\partial j} \right) |
---|
| 328 | -\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ |
---|
| 329 | \\ |
---|
| 330 | && - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
---|
| 331 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ |
---|
| 332 | \\ |
---|
| 333 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
---|
| 334 | &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u\,u )}{\partial i} |
---|
| 335 | + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j} \right) |
---|
| 336 | + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ |
---|
| 337 | \\ |
---|
| 338 | && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} |
---|
| 339 | + \frac{\partial(e_1 e_3 \, v)}{\partial j} \right) |
---|
| 340 | -\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] |
---|
| 341 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
---|
| 342 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ |
---|
| 343 | % |
---|
| 344 | \intertext {Introducing a more compact form for the divergence of the momentum fluxes, |
---|
| 345 | and using (\ref{Apdx_A_sco_Continuity}), the $s-$coordinate continuity equation, |
---|
| 346 | it becomes : } |
---|
| 347 | % |
---|
| 348 | &= \left. {\frac{\partial u }{\partial t}} \right|_s |
---|
| 349 | &+ \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s |
---|
| 350 | + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t} |
---|
| 351 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
---|
| 352 | -u \;\frac{\partial e_1 }{\partial j} \right) \\ |
---|
| 353 | \end{array} } |
---|
| 354 | \end{align*} |
---|
| 355 | \end{subequations} |
---|
| 356 | which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, |
---|
| 357 | $i.e.$ the total $s-$coordinate time derivative in flux form : |
---|
| 358 | \begin{flalign}\label{Apdx_A_sco_Dt_flux} |
---|
| 359 | \left. \frac{D u}{D t} \right|_s = \frac{1}{e_3} \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s |
---|
| 360 | + \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s |
---|
| 361 | - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
---|
| 362 | -u \;\frac{\partial e_1 }{\partial j} \right) |
---|
| 363 | \end{flalign} |
---|
| 364 | which is the total time derivative expressed in the curvilinear $s-$coordinate system. |
---|
| 365 | It has the same form as in the $z-$coordinate but for the vertical scale factor |
---|
| 366 | that has appeared inside the time derivative which comes from the modification |
---|
| 367 | of (\ref{Apdx_A_sco_Continuity}), the continuity equation. |
---|
[707] | 368 | |
---|
[2282] | 369 | $\ $\newline % force a new ligne |
---|
[707] | 370 | |
---|
[2282] | 371 | $\bullet$ \textbf{horizontal pressure gradient} |
---|
| 372 | |
---|
| 373 | The horizontal pressure gradient term can be transformed as follows: |
---|
| 374 | \begin{equation*} |
---|
[707] | 375 | \begin{split} |
---|
[2282] | 376 | -\frac{1}{\rho _o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z |
---|
| 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] \\ |
---|
[707] | 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) \\ |
---|
| 379 | &=-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho _o }\sigma _1 |
---|
| 380 | \end{split} |
---|
[2282] | 381 | \end{equation*} |
---|
| 382 | Applying similar manipulation to the second component and replacing |
---|
| 383 | $\sigma _1$ and $\sigma _2$ by their expression \eqref{Apdx_A_s_slope}, it comes: |
---|
| 384 | \begin{equation} \label{Apdx_A_grad_p} |
---|
| 385 | \begin{split} |
---|
| 386 | -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z |
---|
| 387 | &=-\frac{1}{\rho _o \,e_1 } \left( \left. {\frac{\partial p}{\partial i}} \right|_s |
---|
| 388 | + g\;\rho \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) \\ |
---|
| 389 | % |
---|
| 390 | -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z |
---|
| 391 | &=-\frac{1}{\rho _o \,e_2 } \left( \left. {\frac{\partial p}{\partial j}} \right|_s |
---|
| 392 | + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) \\ |
---|
| 393 | \end{split} |
---|
[707] | 394 | \end{equation} |
---|
| 395 | |
---|
[1223] | 396 | An additional term appears in (\ref{Apdx_A_grad_p}) which accounts for the |
---|
[2282] | 397 | tilt of $s-$surfaces with respect to geopotential $z-$surfaces. |
---|
[707] | 398 | |
---|
[2282] | 399 | As in $z$-coordinate, the horizontal pressure gradient can be split in two parts |
---|
| 400 | following \citet{Marsaleix_al_OM08}. Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$, |
---|
| 401 | and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. |
---|
| 402 | The pressure is then given by: |
---|
| 403 | \begin{equation*} |
---|
| 404 | \begin{split} |
---|
| 405 | p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left( \rho_o \, d + 1 \right) \; e_3 \; dk \\ |
---|
| 406 | &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk |
---|
| 407 | \end{split} |
---|
| 408 | \end{equation*} |
---|
| 409 | Therefore, $p$ and $p_h'$ are linked through: |
---|
| 410 | \begin{equation} \label{Apdx_A_pressure} |
---|
| 411 | p = \rho_o \; p_h' + g \, ( z + \eta ) |
---|
| 412 | \end{equation} |
---|
| 413 | and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is: |
---|
| 414 | \begin{equation*} |
---|
| 415 | \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 |
---|
| 416 | \end{equation*} |
---|
| 417 | |
---|
| 418 | Substituing \eqref{Apdx_A_pressure} in \eqref{Apdx_A_grad_p} and using the definition of |
---|
| 419 | the density anomaly it comes the expression in two parts: |
---|
| 420 | \begin{equation} \label{Apdx_A_grad_p} |
---|
| 421 | \begin{split} |
---|
| 422 | -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z |
---|
| 423 | &=-\frac{1}{e_1 } \left( \left. {\frac{\partial p_h'}{\partial i}} \right|_s |
---|
| 424 | + g\; d \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\ |
---|
| 425 | % |
---|
| 426 | -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z |
---|
| 427 | &=-\frac{1}{e_2 } \left( \left. {\frac{\partial p_h'}{\partial j}} \right|_s |
---|
| 428 | + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\ |
---|
| 429 | \end{split} |
---|
| 430 | \end{equation} |
---|
| 431 | This formulation of the pressure gradient is characterised by the appearance of a term depending on the |
---|
| 432 | the sea surface height only (last term on the right hand side of expression \eqref{Apdx_A_grad_p}). |
---|
[3294] | 433 | This term will be loosely termed \textit{surface pressure gradient} |
---|
| 434 | whereas the first term will be termed the |
---|
[2282] | 435 | \textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation. |
---|
| 436 | In fact, the the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and |
---|
| 437 | $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of |
---|
| 438 | the vertical integration. |
---|
| 439 | |
---|
| 440 | |
---|
| 441 | $\ $\newline % force a new ligne |
---|
| 442 | |
---|
| 443 | $\bullet$ \textbf{The other terms of the momentum equation} |
---|
| 444 | |
---|
| 445 | The coriolis and forcing terms as well as the the vertical physics remain unchanged |
---|
| 446 | as they involve neither time nor space derivatives. The form of the lateral physics is |
---|
| 447 | discussed in appendix~\ref{Apdx_B}. |
---|
| 448 | |
---|
| 449 | |
---|
| 450 | $\ $\newline % force a new ligne |
---|
| 451 | |
---|
| 452 | $\bullet$ \textbf{Full momentum equation} |
---|
| 453 | |
---|
| 454 | To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation |
---|
| 455 | solved by the model has the same mathematical expression as the one in a curvilinear |
---|
[3294] | 456 | $z-$coordinate, except for the pressure gradient term : |
---|
[817] | 457 | \begin{subequations} \label{Apdx_A_dyn_vect} |
---|
| 458 | \begin{multline} \label{Apdx_A_PE_dyn_vect_u} |
---|
[2282] | 459 | \frac{\partial u}{\partial t}= |
---|
[817] | 460 | + \left( {\zeta +f} \right)\,v |
---|
| 461 | - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right) |
---|
| 462 | - \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\ |
---|
[2282] | 463 | - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) |
---|
| 464 | - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} |
---|
[817] | 465 | + D_u^{\vect{U}} + F_u^{\vect{U}} |
---|
| 466 | \end{multline} |
---|
| 467 | \begin{multline} \label{Apdx_A_dyn_vect_v} |
---|
[2282] | 468 | \frac{\partial v}{\partial t}= |
---|
[817] | 469 | - \left( {\zeta +f} \right)\,u |
---|
| 470 | - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right) |
---|
| 471 | - \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\ |
---|
[2282] | 472 | - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) |
---|
| 473 | - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} |
---|
[817] | 474 | + D_v^{\vect{U}} + F_v^{\vect{U}} |
---|
| 475 | \end{multline} |
---|
| 476 | \end{subequations} |
---|
[2282] | 477 | whereas the flux form momentum equation differ from it by the formulation of both |
---|
| 478 | the time derivative and the pressure gradient term : |
---|
| 479 | \begin{subequations} \label{Apdx_A_dyn_flux} |
---|
| 480 | \begin{multline} \label{Apdx_A_PE_dyn_flux_u} |
---|
| 481 | \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t} = |
---|
| 482 | \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) |
---|
| 483 | + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
---|
| 484 | -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,v \\ |
---|
| 485 | - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) |
---|
| 486 | - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} |
---|
| 487 | + D_u^{\vect{U}} + F_u^{\vect{U}} |
---|
| 488 | \end{multline} |
---|
| 489 | \begin{multline} \label{Apdx_A_dyn_flux_v} |
---|
| 490 | \frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}= |
---|
| 491 | - \nabla \cdot \left( {{\rm {\bf U}}\,v} \right) |
---|
| 492 | + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} |
---|
| 493 | -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,u \\ |
---|
| 494 | - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) |
---|
| 495 | - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} |
---|
| 496 | + D_v^{\vect{U}} + F_v^{\vect{U}} |
---|
| 497 | \end{multline} |
---|
| 498 | \end{subequations} |
---|
| 499 | Both formulation share the same hydrostatic pressure balance expressed in terms of |
---|
[3294] | 500 | hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: |
---|
[2282] | 501 | \begin{equation} \label{Apdx_A_dyn_zph} |
---|
| 502 | \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 |
---|
| 503 | \end{equation} |
---|
[707] | 504 | |
---|
[2282] | 505 | It is important to realize that the change in coordinate system has only concerned |
---|
| 506 | the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the |
---|
[3294] | 507 | orthogonal curvilinear set of unit vectors. ($u$,$v$) are always horizontal velocities |
---|
[2282] | 508 | so that their evolution is driven by \emph{horizontal} forces, in particular |
---|
| 509 | the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity, |
---|
[3294] | 510 | but the dia-surface velocity component, $i.e.$ the volume flux across the moving |
---|
| 511 | $s$-surfaces per unit horizontal area. |
---|
[817] | 512 | |
---|
[2282] | 513 | |
---|
[817] | 514 | % ================================================================ |
---|
| 515 | % Tracer equation |
---|
| 516 | % ================================================================ |
---|
| 517 | \section{Tracer Equation} |
---|
[2282] | 518 | \label{Apdx_A_tracer} |
---|
[817] | 519 | |
---|
[1223] | 520 | The tracer equation is obtained using the same calculation as for the continuity |
---|
| 521 | equation and then regrouping the time derivative terms in the left hand side : |
---|
[707] | 522 | |
---|
[817] | 523 | \begin{multline} \label{Apdx_A_tracer} |
---|
| 524 | \frac{1}{e_3} \frac{\partial \left( e_3 T \right)}{\partial t} |
---|
[2282] | 525 | = -\frac{1}{e_1 \,e_2 \,e_3} |
---|
| 526 | \left[ \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right) |
---|
| 527 | + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] \\ |
---|
| 528 | + \frac{1}{e_3} \frac{\partial }{\partial k} \left( Tw \right) |
---|
| 529 | + D^{T} +F^{T} |
---|
[707] | 530 | \end{multline} |
---|
| 531 | |
---|
| 532 | |
---|
[1223] | 533 | The expression for the advection term is a straight consequence of (A.4), the |
---|
[2282] | 534 | expression of the 3D divergence in the $s-$coordinates established above. |
---|
[707] | 535 | |
---|
[6997] | 536 | \end{document} |
---|