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branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_A.tex
r1223 r2282 3 3 % Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations 4 4 % ================================================================ 5 \chapter{Curvilinear $s $-Coordinate Equations}5 \chapter{Curvilinear $s-$Coordinate Equations} 6 6 \label{Apdx_A} 7 7 \minitoc 8 8 9 10 In order to establish the set of Primitive Equation in curvilinear $s$-coordinates 11 ($i.e.$ an orthogonal curvilinear coordinate in the horizontal and $s$-coordinate 12 in the vertical), we start from the set of equations established in \S\ref{PE_zco_Eq} 13 for the special case $k = z$ and thus $e_3 = 1$, and we introduce an arbitrary 14 vertical coordinate $s = s(i,j,z,t)$. Let us define a new vertical scale factor by 9 \newpage 10 $\ $\newline % force a new ligne 11 12 % ================================================================ 13 % Chain rule 14 % ================================================================ 15 \section{Chain rule of $s-$coordinate} 16 \label{Apdx_A_continuity} 17 18 In order to establish the set of Primitive Equation in curvilinear $s-$coordinates 19 ($i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian 20 Eulerian (ALE) coordinate in the vertical), we start from the set of equations established 21 in \S\ref{PE_zco_Eq} for the special case $k = z$ and thus $e_3 = 1$, and we introduce 22 an arbitrary vertical coordinate $a = a(i,j,z,t)$. Let us define a new vertical scale factor by 15 23 $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal 16 slope of $s $-surfaces by :24 slope of $s-$surfaces by : 17 25 \begin{equation} \label{Apdx_A_s_slope} 18 26 \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s … … 21 29 \end{equation} 22 30 23 The chain rule to establish the model equations in the curvilinear $s $-coordinate31 The chain rule to establish the model equations in the curvilinear $s-$coordinate 24 32 system is: 25 33 \begin{equation} \label{Apdx_A_s_chain_rule} … … 42 50 \end{equation} 43 51 44 In particular applying the time derivative chain rule to $z$ provides the 45 expression for $w_s$, the vertical velocity of the $s-$surfaces:52 In particular applying the time derivative chain rule to $z$ provides the expression 53 for $w_s$, the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: 46 54 \begin{equation} \label{Apdx_A_w_in_s} 47 55 w_s = \left. \frac{\partial z }{\partial t} \right|_s … … 50 58 \end{equation} 51 59 60 52 61 % ================================================================ 53 62 % continuity equation 54 63 % ================================================================ 55 \section{Continuity Equation }56 \label{Apdx_ B_continuity}64 \section{Continuity Equation in $s-$coordinate} 65 \label{Apdx_A_continuity} 57 66 58 67 Using (\ref{Apdx_A_s_chain_rule}) and the fact that the horizontal scale factors 59 68 $e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of 60 the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows: 61 62 \begin{align*} 69 the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows 70 in order to obtain its expression in the curvilinear $s-$coordinate system: 71 72 \begin{subequations} 73 \begin{align*} {\begin{array}{*{20}l} 63 74 \nabla \cdot {\rm {\bf U}} 64 75 &= \frac{1}{e_1 \,e_2 } \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z … … 89 100 - \sigma _1 \frac{\partial u}{\partial s} 90 101 - \sigma _2 \frac{\partial v}{\partial s} \right] \\ 91 \\ 92 \ end{align*}93 94 Noting that $\frac{1}{e_1 }\left. {\frac{\partial e_3 }{\partial i}}95 \right|_s =\frac{1}{e_1 }\left. {\frac{\partial ^2z}{\partial i\,\partial 96 s}} \right|_s =\frac{\partial }{\partial s}\left( {\frac{1}{e_1 }\left. 97 {\frac{\partial z}{\partial i}} \right|_s } \right)=\frac{\partial \sigma _1 98 }{\partial s}$ and $\frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}}99 \right|_s =\frac{\partial \sigma _2 }{\partial s}$, it becomes: 100 101 \begin{align*} 102 % 103 \intertext{Noting that $ 104 \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s 105 =\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s 106 =\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right) 107 =\frac{\partial \sigma _1}{\partial s} 108 $ and $ 109 \frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s 110 =\frac{\partial \sigma _2}{\partial s} 111 $, it becomes:} 112 % 102 113 \nabla \cdot {\rm {\bf U}} 103 114 & = \frac{1}{e_1 \,e_2 \,e_3 } \left[ 104 115 \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 105 116 +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] \\ 106 & \qquad \qquad \qquad \qquad \q quad \quad117 & \qquad \qquad \qquad \qquad \quad 107 118 +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma _1 }{\partial s}-v\frac{\partial \sigma _2 }{\partial s}-\sigma _1 \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} \right] \\ 108 119 \\ … … 111 122 +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] 112 123 + \frac{1}{e_3 } \; \frac{\partial}{\partial s} \left[ w - u\;\sigma _1 - v\;\sigma _2 \right] 113 \end{align*} 114 124 \end{array} } 125 \end{align*} 126 \end{subequations} 127 115 128 Here, $w$ is the vertical velocity relative to the $z-$coordinate system. 116 129 Introducing the dia-surface velocity component, $\omega $, defined as 117 the velocity relative to the moving $s $-surfaces and normal to them:130 the velocity relative to the moving $s-$surfaces and normal to them: 118 131 \begin{equation} \label{Apdx_A_w_s} 119 132 \omega = w - w_s - \sigma _1 \,u - \sigma _2 \,v \\ 120 133 \end{equation} 121 134 with $w_s$ given by \eqref{Apdx_A_w_in_s}, we obtain the expression for 122 the divergence of the velocity in the curvilinear $s$-coordinate system: 123 \begin{align*} \label{Apdx_A_A4} 135 the divergence of the velocity in the curvilinear $s-$coordinate system: 136 \begin{subequations} 137 \begin{align*} {\begin{array}{*{20}l} 124 138 \nabla \cdot {\rm {\bf U}} 125 139 &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ … … 147 161 + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 148 162 + \frac{1}{e_3 } \frac{\partial e_3}{\partial t} \\ 163 \end{array} } 149 164 \end{align*} 165 \end{subequations} 150 166 151 167 As a result, the continuity equation \eqref{Eq_PE_continuity} in the 152 $s $-coordinates becomes:153 \begin{equation} \label{Apdx_A_ A5}168 $s-$coordinates is: 169 \begin{equation} \label{Apdx_A_sco_Continuity} 154 170 \frac{1}{e_3 } \frac{\partial e_3}{\partial t} 155 171 + \frac{1}{e_1 \,e_2 \,e_3 }\left[ … … 158 174 +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 159 175 \end{equation} 176 A additional term has appeared that take into account the contribution of the time variation 177 of the vertical coordinate to the volume budget. 178 160 179 161 180 % ================================================================ 162 181 % momentum equation 163 182 % ================================================================ 164 \section{Momentum Equation} 165 \label{Apdx_B_momentum} 183 \section{Momentum Equation in $s-$coordinate} 184 \label{Apdx_A_momentum} 185 186 Here we only consider the first component of the momentum equation, 187 the generalization to the second one being straightforward. 188 189 $\ $\newline % force a new ligne 190 191 $\bullet$ \textbf{Total derivative in vector invariant form} 166 192 167 193 Let us consider \eqref{Eq_PE_dyn_vect}, the first component of the momentum 168 equation in the vector invariant form (similar manipulations can be performed 169 on the second component). Its non-linear term can be transformed as follows: 170 171 \begin{align*} 172 &+\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_z 173 - w \;\frac{\partial u}{\partial z} \\ 174 \\ 175 &\qquad=\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}} 176 \right|_z -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_z } 177 \right]\;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} 178 \right|_z -w\frac{\partial u}{\partial z} \\ 179 \\ 180 &\qquad =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}} 181 \right|_s -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_s } \right. 182 \left. {-\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (e_2 \,v)}{\partial s}+\frac{e_2 }{e_3 }\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right]\;v \\ 183 &\qquad \qquad \qquad \qquad \qquad 184 { -\frac{1}{2e_1 }\left( {\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (u^2+v^2)}{\partial s}} \right) 185 -\frac{w}{e_3 }\frac{\partial u}{\partial s} } \\ 194 equation in the vector invariant form. Its total $z-$coordinate time derivative, 195 $\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain 196 its expression in the curvilinear $s-$coordinate system: 197 198 \begin{subequations} 199 \begin{align*} {\begin{array}{*{20}l} 200 \left. \frac{D u}{D t} \right|_z 201 &= \left. {\frac{\partial u }{\partial t}} \right|_z 202 - \left. \zeta \right|_z v 203 + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z 204 + w \;\frac{\partial u}{\partial z} \\ 205 \\ 206 &= \left. {\frac{\partial u }{\partial t}} \right|_z 207 - \left. \zeta \right|_z v 208 + \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z 209 -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v 210 + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z 211 + w \;\frac{\partial u}{\partial z} \\ 212 % 213 \intertext{introducing the chain rule (\ref{Apdx_A_s_chain_rule}) } 214 % 215 &= \left. {\frac{\partial u }{\partial t}} \right|_z 216 - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s 217 -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right. 218 \left. {-\frac{e_1}{e_3}\sigma _1 \frac{\partial (e_2 \,v)}{\partial s} 219 +\frac{e_2}{e_3}\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v \\ 220 & \qquad \qquad \qquad \qquad 221 { + \frac{1}{2e_1} \left( \left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s 222 - \frac{e_1}{e_3}\sigma _1 \frac{\partial (u^2+v^2)}{\partial s} \right) 223 + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} } \\ 224 \\ 225 &= \left. {\frac{\partial u }{\partial t}} \right|_z 226 + \left. \zeta \right|_s \;v 227 + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ 228 &\qquad \qquad \qquad \quad 229 + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} 230 - \left[ {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s} 231 - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v 232 - \frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ 233 \\ 234 &= \left. {\frac{\partial u }{\partial t}} \right|_z 235 + \left. \zeta \right|_s \;v 236 + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ 237 &\qquad \qquad \qquad \quad 238 + \frac{1}{e_3} \left[ {w\frac{\partial u}{\partial s} 239 +\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s} 240 - \sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\ 241 \\ 242 &= \left. {\frac{\partial u }{\partial t}} \right|_z 243 + \left. \zeta \right|_s \;v 244 + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 245 + \frac{1}{e_3} \left[ w - \sigma _2 v - \sigma _1 u \right] 246 \; \frac{\partial u}{\partial s} \\ 247 % 248 \intertext{Introducing $\omega$, the dia-a-surface velocity given by (\ref{Apdx_A_w_s}) } 249 % 250 &= \left. {\frac{\partial u }{\partial t}} \right|_z 251 + \left. \zeta \right|_s \;v 252 + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 253 + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s} \\ 254 \end{array} } 186 255 \end{align*} 187 \begin{align*} 188 \qquad &= \left. \zeta \right|_s \;v 189 - \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 190 - \frac{w}{e_3 }\frac{\partial u}{\partial s} 191 - \left[ {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s} 192 - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v \\ 193 \qquad&\qquad \qquad \qquad \qquad \qquad \qquad 194 \qquad \qquad \qquad \qquad \quad 195 +\frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ 196 %\\ 197 \qquad &= \left. \zeta \right|_s \;v 198 - \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ 199 \qquad&\qquad \qquad \qquad 200 -\frac{1}{e_3} \left[ {w\frac{\partial u}{\partial s} 201 +\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s} 202 -\sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\ 203 \\ 204 \qquad &= \left. \zeta \right|_s \;v 205 - \frac{1}{2e_1 }\left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s 206 - \frac{1}{e_3} \left[ w - \sigma _2 v - \sigma _1 u \right] 207 \; \frac{\partial u}{\partial s} \\ 208 \\ 209 \qquad &= \left. \zeta \right|_s \;v 210 - \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 211 - \frac{1}{e_3 }\omega \frac{\partial u}{\partial s} 212 - \frac{\partial s}{\partial t} \frac{\partial u}{\partial s} 256 \end{subequations} 257 % 258 Applying the time derivative chain rule (first equation of (\ref{Apdx_A_s_chain_rule})) 259 to $u$ and using (\ref{Apdx_A_w_in_s}) provides the expression of the last term 260 of the right hand side, 261 \begin{equation*} {\begin{array}{*{20}l} 262 w_s \;\frac{\partial u}{\partial s} 263 = \frac{\partial s}{\partial t} \; \frac{\partial u }{\partial s} 264 = \left. {\frac{\partial u }{\partial t}} \right|_s - \left. {\frac{\partial u }{\partial t}} \right|_z \quad , 265 \end{array} } 266 \end{equation*} 267 leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, 268 $i.e.$ the total $s-$coordinate time derivative : 269 \begin{align} \label{Apdx_A_sco_Dt_vect} 270 \left. \frac{D u}{D t} \right|_s 271 = \left. {\frac{\partial u }{\partial t}} \right|_s 272 + \left. \zeta \right|_s \;v 273 + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 274 + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} 275 \end{align} 276 Therefore, the vector invariant form of the total time derivative has exactly the same 277 mathematical form in $z-$ and $s-$coordinates. This is not the case for the flux form 278 as shown in next paragraph. 279 280 $\ $\newline % force a new ligne 281 282 $\bullet$ \textbf{Total derivative in flux form} 283 284 Let us start from the total time derivative in the curvilinear $s-$coordinate system 285 we have just establish. Following the procedure used to establish (\ref{Eq_PE_flux_form}), 286 it can be transformed into : 287 %\begin{subequations} 288 \begin{align*} {\begin{array}{*{20}l} 289 \left. \frac{D u}{D t} \right|_s &= \left. {\frac{\partial u }{\partial t}} \right|_s 290 & - \zeta \;v 291 + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} 292 + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s} \\ 293 \\ 294 &= \left. {\frac{\partial u }{\partial t}} \right|_s 295 &+\frac{1}{e_1\;e_2} \left( \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i} 296 + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j} \right) 297 + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ 298 \\ 299 &&- \,u \left[ \frac{1}{e_1 e_2 } \left( \frac{\partial(e_2 u)}{\partial i} 300 + \frac{\partial(e_1 v)}{\partial j} \right) 301 + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ 302 \\ 303 &&- \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 304 -u \;\frac{\partial e_1 }{\partial j} \right) \\ 305 \end{array} } 213 306 \end{align*} 214 215 Therefore, the non-linear terms of the momentum equation have the same 216 form in $z-$ and $s-$coordinates but with the addition of the time derivative 217 of the velocity: 218 \begin{multline} \label{Apdx_A_momentum_NL} 219 +\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_z 220 - w \;\frac{\partial u}{\partial z} \\ 221 = - \frac{\partial u}{\partial t} + \left. \zeta \right|_s \;v 222 - \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 223 - \frac{1}{e_3 }\omega \frac{\partial u}{\partial s} 224 \end{multline} 225 226 The pressure gradient term can be transformed as follows: 227 \begin{equation} \label{Apdx_A_grad_p} 307 % 308 Introducing the vertical scale factor inside the horizontal derivative of the first two terms 309 ($i.e.$ the horizontal divergence), it becomes : 310 \begin{subequations} 311 \begin{align*} {\begin{array}{*{20}l} 312 %\begin{align*} {\begin{array}{*{20}l} 313 %{\begin{array}{*{20}l} 314 \left. \frac{D u}{D t} \right|_s 315 &= \left. {\frac{\partial u }{\partial t}} \right|_s 316 &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u^2 )}{\partial i} 317 + \frac{\partial( e_1 e_3 \,u v )}{\partial j} 318 - e_2 u u \frac{\partial e_3}{\partial i} 319 - e_1 u v \frac{\partial e_3 }{\partial j} \right) 320 + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ 321 \\ 322 && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} 323 + \frac{\partial(e_1 e_3 \, v)}{\partial j} 324 - e_2 u \;\frac{\partial e_3 }{\partial i} 325 - e_1 v \;\frac{\partial e_3 }{\partial j} \right) 326 -\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ 327 \\ 328 && - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 329 -u \;\frac{\partial e_1 }{\partial j} \right) \\ 330 \\ 331 &= \left. {\frac{\partial u }{\partial t}} \right|_s 332 &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u\,u )}{\partial i} 333 + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j} \right) 334 + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ 335 \\ 336 && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} 337 + \frac{\partial(e_1 e_3 \, v)}{\partial j} \right) 338 -\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] 339 - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 340 -u \;\frac{\partial e_1 }{\partial j} \right) \\ 341 % 342 \intertext {Introducing a more compact form for the divergence of the momentum fluxes, 343 and using (\ref{Apdx_A_sco_Continuity}), the $s-$coordinate continuity equation, 344 it becomes : } 345 % 346 &= \left. {\frac{\partial u }{\partial t}} \right|_s 347 &+ \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s 348 + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t} 349 - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 350 -u \;\frac{\partial e_1 }{\partial j} \right) \\ 351 \end{array} } 352 \end{align*} 353 \end{subequations} 354 which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, 355 $i.e.$ the total $s-$coordinate time derivative in flux form : 356 \begin{flalign}\label{Apdx_A_sco_Dt_flux} 357 \left. \frac{D u}{D t} \right|_s = \frac{1}{e_3} \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s 358 + \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s 359 - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 360 -u \;\frac{\partial e_1 }{\partial j} \right) 361 \end{flalign} 362 which is the total time derivative expressed in the curvilinear $s-$coordinate system. 363 It has the same form as in the $z-$coordinate but for the vertical scale factor 364 that has appeared inside the time derivative which comes from the modification 365 of (\ref{Apdx_A_sco_Continuity}), the continuity equation. 366 367 $\ $\newline % force a new ligne 368 369 $\bullet$ \textbf{horizontal pressure gradient} 370 371 The horizontal pressure gradient term can be transformed as follows: 372 \begin{equation*} 228 373 \begin{split} 229 -\frac{1}{\rho _o e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z& =-\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] \\ 374 -\frac{1}{\rho _o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z 375 & =-\frac{1}{\rho _o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial p}{\partial s}} \right] \\ 230 376 & =-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma _1 }{\rho _o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\ 231 377 &=-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho _o }\sigma _1 232 378 \end{split} 379 \end{equation*} 380 Applying similar manipulation to the second component and replacing 381 $\sigma _1$ and $\sigma _2$ by their expression \eqref{Apdx_A_s_slope}, it comes: 382 \begin{equation} \label{Apdx_A_grad_p} 383 \begin{split} 384 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 385 &=-\frac{1}{\rho _o \,e_1 } \left( \left. {\frac{\partial p}{\partial i}} \right|_s 386 + g\;\rho \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) \\ 387 % 388 -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 389 &=-\frac{1}{\rho _o \,e_2 } \left( \left. {\frac{\partial p}{\partial j}} \right|_s 390 + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) \\ 391 \end{split} 233 392 \end{equation} 234 393 235 394 An additional term appears in (\ref{Apdx_A_grad_p}) which accounts for the 236 tilt of model levels. 237 238 Introducing \eqref{Apdx_A_momentum_NL} and \eqref{Apdx_A_grad_p} in \eqref{Eq_PE_dyn_vect} and regrouping the time derivative terms in the left 239 hand side, and performing the same manipulation on the second component, 240 we obtain the vector invariant form of the momentum equations in the 241 $s-$coordinate : 395 tilt of $s-$surfaces with respect to geopotential $z-$surfaces. 396 397 As in $z$-coordinate, the horizontal pressure gradient can be split in two parts 398 following \citet{Marsaleix_al_OM08}. Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$, 399 and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. 400 The pressure is then given by: 401 \begin{equation*} 402 \begin{split} 403 p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left( \rho_o \, d + 1 \right) \; e_3 \; dk \\ 404 &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk 405 \end{split} 406 \end{equation*} 407 Therefore, $p$ and $p_h'$ are linked through: 408 \begin{equation} \label{Apdx_A_pressure} 409 p = \rho_o \; p_h' + g \, ( z + \eta ) 410 \end{equation} 411 and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is: 412 \begin{equation*} 413 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 414 \end{equation*} 415 416 Substituing \eqref{Apdx_A_pressure} in \eqref{Apdx_A_grad_p} and using the definition of 417 the density anomaly it comes the expression in two parts: 418 \begin{equation} \label{Apdx_A_grad_p} 419 \begin{split} 420 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 421 &=-\frac{1}{e_1 } \left( \left. {\frac{\partial p_h'}{\partial i}} \right|_s 422 + g\; d \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\ 423 % 424 -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 425 &=-\frac{1}{e_2 } \left( \left. {\frac{\partial p_h'}{\partial j}} \right|_s 426 + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\ 427 \end{split} 428 \end{equation} 429 This formulation of the pressure gradient is characterised by the appearance of a term depending on the 430 the sea surface height only (last term on the right hand side of expression \eqref{Apdx_A_grad_p}). 431 This term will be abusively named \textit{surface pressure gradient} whereas the first term will be named 432 \textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation. 433 In fact, the the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and 434 $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of 435 the vertical integration. 436 437 438 $\ $\newline % force a new ligne 439 440 $\bullet$ \textbf{The other terms of the momentum equation} 441 442 The coriolis and forcing terms as well as the the vertical physics remain unchanged 443 as they involve neither time nor space derivatives. The form of the lateral physics is 444 discussed in appendix~\ref{Apdx_B}. 445 446 447 $\ $\newline % force a new ligne 448 449 $\bullet$ \textbf{Full momentum equation} 450 451 To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation 452 solved by the model has the same mathematical expression as the one in a curvilinear 453 $z-$coordinate, but the pressure gradient term : 242 454 \begin{subequations} \label{Apdx_A_dyn_vect} 243 455 \begin{multline} \label{Apdx_A_PE_dyn_vect_u} 244 \frac{ 1}{e_3} \frac{\partial \left( e_3\,u \right)}{\partial t}=456 \frac{\partial u}{\partial t}= 245 457 + \left( {\zeta +f} \right)\,v 246 458 - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right) 247 459 - \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\ 248 - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o} \right)249 + g\frac{\rho }{\rho _o}\sigma _1 460 - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) 461 - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} 250 462 + D_u^{\vect{U}} + F_u^{\vect{U}} 251 463 \end{multline} 252 464 \begin{multline} \label{Apdx_A_dyn_vect_v} 253 \frac{1}{e_3}\frac{\partial \left( e_3\,v \right)}{\partial t}=465 \frac{\partial v}{\partial t}= 254 466 - \left( {\zeta +f} \right)\,u 255 467 - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right) 256 468 - \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\ 257 - \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o} \right)258 + g\frac{\rho }{\rho _o }\sigma _2 469 - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) 470 - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 259 471 + D_v^{\vect{U}} + F_v^{\vect{U}} 260 472 \end{multline} 261 473 \end{subequations} 262 263 It has the same form as in the $z-$coordinate but for the vertical scale factor 264 that has appeared inside the time derivative. The form of the vertical physics 265 and forcing terms remains unchanged. The form of the lateral physics is 266 discussed in appendix~\ref{Apdx_B}. 474 whereas the flux form momentum equation differ from it by the formulation of both 475 the time derivative and the pressure gradient term : 476 \begin{subequations} \label{Apdx_A_dyn_flux} 477 \begin{multline} \label{Apdx_A_PE_dyn_flux_u} 478 \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t} = 479 \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) 480 + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 481 -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,v \\ 482 - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) 483 - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} 484 + D_u^{\vect{U}} + F_u^{\vect{U}} 485 \end{multline} 486 \begin{multline} \label{Apdx_A_dyn_flux_v} 487 \frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}= 488 - \nabla \cdot \left( {{\rm {\bf U}}\,v} \right) 489 + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 490 -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,u \\ 491 - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) 492 - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 493 + D_v^{\vect{U}} + F_v^{\vect{U}} 494 \end{multline} 495 \end{subequations} 496 Both formulation share the same hydrostatic pressure balance expressed in terms of 497 hydrostatic pressure and density anmalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: 498 \begin{equation} \label{Apdx_A_dyn_zph} 499 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 500 \end{equation} 501 502 It is important to realize that the change in coordinate system has only concerned 503 the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the 504 orthogonal curvilinear set of unit vector. ($u$,$v$) are always horizontal velocities 505 so that their evolution is driven by \emph{horizontal} forces, in particular 506 the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity, 507 but the dia-surface velocity component, $i.e.$ the velocity relative to the moving 508 $s-$surfaces and normal to them. 509 267 510 268 511 % ================================================================ … … 270 513 % ================================================================ 271 514 \section{Tracer Equation} 272 \label{Apdx_ B_tracer}515 \label{Apdx_A_tracer} 273 516 274 517 The tracer equation is obtained using the same calculation as for the continuity … … 277 520 \begin{multline} \label{Apdx_A_tracer} 278 521 \frac{1}{e_3} \frac{\partial \left( e_3 T \right)}{\partial t} 279 = -\frac{1}{e_1 \,e_2 \,e_3 } 280 \left[ {\frac{\partial }{\partial i}} \left( {e_2 \,e_3 \;Tu} \right) \right . 281 + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \\ 282 + \left. \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] +D^{T} +F^{T} \; \; 522 = -\frac{1}{e_1 \,e_2 \,e_3} 523 \left[ \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right) 524 + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] \\ 525 + \frac{1}{e_3} \frac{\partial }{\partial k} \left( Tw \right) 526 + D^{T} +F^{T} 283 527 \end{multline} 284 528 285 529 286 530 The expression for the advection term is a straight consequence of (A.4), the 287 expression of the 3D divergence in the $s $-coordinates established above.288 531 expression of the 3D divergence in the $s-$coordinates established above. 532
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