Changeset 11335 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_A.tex
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NEMO/trunk/doc/latex/NEMO/subfiles/annex_A.tex
r11151 r11335 29 29 \begin{equation} 30 30 \label{apdx:A_s_slope} 31 \sigma_1 =\frac{1}{e_1 } \;\left. {\frac{\partial z}{\partial i}} \right|_s31 \sigma_1 =\frac{1}{e_1 } \; \left. {\frac{\partial z}{\partial i}} \right|_s 32 32 \quad \text{and} \quad 33 \sigma_2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s 34 \end{equation} 35 36 The chain rule to establish the model equations in the curvilinear $s-$coordinate system is: 33 \sigma_2 =\frac{1}{e_2 } \; \left. {\frac{\partial z}{\partial j}} \right|_s . 34 \end{equation} 35 36 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 37 functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of 38 these fields. Any ``infinitesimal'' change in $\bullet$ can be written in two forms: 39 \begin{equation} 40 \label{apdx:A_s_infin_changes} 41 \begin{aligned} 42 & \delta \bullet = \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t} 43 + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t} 44 + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t} 45 + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,s} , \\ 46 & \delta \bullet = \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t} 47 + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t} 48 + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t} 49 + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,z} . 50 \end{aligned} 51 \end{equation} 52 Using the first form and considering a change $\delta i$ with $j, z$ and $t$ held constant, shows that 53 \begin{equation} 54 \label{apdx:A_s_chain_rule} 55 \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,z,t} = 56 \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,s,t} 57 + \left. {\frac{\partial s }{\partial i}} \right|_{j,z,t} \; 58 \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} . 59 \end{equation} 60 The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces, 61 (\autoref{apdx:A_s_slope}), by applying the second of (\autoref{apdx:A_s_infin_changes}) with $\bullet$ set to 62 $s$ and $j, t$ held constant 63 \begin{equation} 64 \label{apdx:a_delta_s} 65 \delta s|_{j,t} = 66 \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} 67 + \delta z \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} . 68 \end{equation} 69 Choosing to look at a direction in the $(i,z)$ plane in which $\delta s = 0$ and using 70 (\autoref{apdx:A_s_slope}) we obtain 71 \begin{equation} 72 \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} = 73 - \left. \frac{ \partial z }{\partial i} \right|_{j,s,t} \; 74 \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} 75 = - \frac{e_1 }{e_3 }\sigma_1 . 76 \label{apdx:a_ds_di_z} 77 \end{equation} 78 Another identity, similar in form to (\autoref{apdx:a_ds_di_z}), can be derived 79 by choosing $\bullet$ to be $s$ and using the second form of (\autoref{apdx:A_s_infin_changes}) to consider 80 changes in which $i , j$ and $s$ are constant. This shows that 81 \begin{equation} 82 \label{apdx:A_w_in_s} 83 w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} = 84 - \left. \frac{ \partial z }{\partial s} \right|_{i,j,t} 85 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} 86 = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} . 87 \end{equation} 88 89 In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is 90 usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish 91 the model equations in the curvilinear $s-$coordinate system are: 37 92 \begin{equation} 38 93 \label{apdx:A_s_chain_rule} 39 94 \begin{aligned} 40 95 &\left. {\frac{\partial \bullet }{\partial t}} \right|_z = 41 \left. {\frac{\partial \bullet }{\partial t}} \right|_s 42 -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t}\\96 \left. {\frac{\partial \bullet }{\partial t}} \right|_s 97 + \frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial t} , \\ 43 98 &\left. {\frac{\partial \bullet }{\partial i}} \right|_z = 44 99 \left. {\frac{\partial \bullet }{\partial i}} \right|_s 45 -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}=46 \left. {\frac{\partial \bullet }{\partial i}} \right|_s 47 -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} \\100 +\frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial i}= 101 \left. {\frac{\partial \bullet }{\partial i}} \right|_s 102 -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} , \\ 48 103 &\left. {\frac{\partial \bullet }{\partial j}} \right|_z = 49 \left. {\frac{\partial \bullet }{\partial j}} \right|_s 50 -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}=51 \left. {\frac{\partial \bullet }{\partial j}} \right|_s 52 - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} \\53 &\;\frac{\partial \bullet }{\partial z} \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} 104 \left. {\frac{\partial \bullet }{\partial j}} \right|_s 105 + \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= 106 \left. {\frac{\partial \bullet }{\partial j}} \right|_s 107 - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} , \\ 108 &\;\frac{\partial \bullet }{\partial z} \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} . 54 109 \end{aligned} 55 110 \end{equation} 56 111 57 In particular applying the time derivative chain rule to $z$ provides the expression for $w_s$,58 the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate:59 \begin{equation}60 \label{apdx:A_w_in_s}61 w_s = \left. \frac{\partial z }{\partial t} \right|_s62 = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t}63 = e_3 \, \frac{\partial s}{\partial t}64 \end{equation}65 112 66 113 % ================================================================ … … 131 178 \end{subequations} 132 179 133 Here, $w$ is the vertical velocity relative to the $z-$coordinate system. 134 Introducing the dia-surface velocity component, 135 $\omega $, defined as the volume flux across the moving $s$-surfaces per unit horizontal area: 180 Here, $w$ is the vertical velocity relative to the $z-$coordinate system. 181 Using the first form of (\autoref{apdx:A_s_infin_changes}) 182 and the definitions (\autoref{apdx:A_s_slope}) and (\autoref{apdx:A_w_in_s}) for $\sigma_1$, $\sigma_2$ and $w_s$, 183 one can show that the vertical velocity, $w_p$ of a point 184 moving with the horizontal velocity of the fluid along an $s$ surface is given by 185 \begin{equation} 186 \label{apdx:A_w_p} 187 \begin{split} 188 w_p = & \left. \frac{ \partial z }{\partial t} \right|_s 189 + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s 190 + \frac{v}{e_2} \left. \frac{ \partial z }{\partial j} \right|_s \\ 191 = & w_s + u \sigma_1 + v \sigma_2 . 192 \end{split} 193 \end{equation} 194 The vertical velocity across this surface is denoted by 136 195 \begin{equation} 137 196 \label{apdx:A_w_s} 138 \omega = w - w_s - \sigma_1 \,u - \sigma_2 \,v \\ 139 \end{equation} 140 with $w_s$ given by \autoref{apdx:A_w_in_s}, 141 we obtain the expression for the divergence of the velocity in the curvilinear $s-$coordinate system: 142 \begin{subequations} 143 \begin{align*} 144 { 145 \begin{array}{*{20}l} 146 \nabla \cdot {\mathrm {\mathbf U}} 147 &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ 197 \omega = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v ) . 198 \end{equation} 199 Hence 200 \begin{equation} 201 \frac{1}{e_3 } \frac{\partial}{\partial s} \left[ w - u\;\sigma_1 - v\;\sigma_2 \right] = 202 \frac{1}{e_3 } \frac{\partial}{\partial s} \left[ \omega + w_s \right] = 203 \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s} 204 + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] = 205 \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s 206 \end{equation} 207 208 Using (\autoref{apdx:A_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain 209 our final expression for the divergence of the velocity in the curvilinear $s-$coordinate system: 210 \begin{equation} 211 \nabla \cdot {\mathrm {\mathbf U}} = 212 \frac{1}{e_1 \,e_2 \,e_3 } \left[ 148 213 \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 149 214 +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] 150 215 + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 151 + \frac{1}{e_3 } \frac{\partial w_s }{\partial s} \\ \\ 152 &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ 153 \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 154 +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] 155 + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 156 + \frac{1}{e_3 } \frac{\partial}{\partial s} \left( e_3 \; \frac{\partial s}{\partial t} \right) \\ \\ 157 &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ 158 \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 159 +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] 160 + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 161 + \frac{\partial}{\partial s} \frac{\partial s}{\partial t} 162 + \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s} \\ \\ 163 &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ 164 \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 165 +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] 166 + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 167 + \frac{1}{e_3 } \frac{\partial e_3}{\partial t} 168 \end{array} 169 } 170 \end{align*} 171 \end{subequations} 216 + \frac{1}{e_3 } \left. \frac{\partial e_3}{\partial t} \right|_s . 217 \end{equation} 172 218 173 219 As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is: … … 178 224 {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s 179 225 + \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right] 180 +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 181 \end{equation} 182 A additional term has appeared that takeinto account226 +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 . 227 \end{equation} 228 An additional term has appeared that takes into account 183 229 the contribution of the time variation of the vertical coordinate to the volume budget. 184 230 … … 210 256 + w \;\frac{\partial u}{\partial z} \\ \\ 211 257 &= \left. {\frac{\partial u }{\partial t}} \right|_z 212 - \left. \zeta \right|_z v 213 + \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z 258 - \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z 214 259 -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v 215 260 + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z … … 230 275 } \\ \\ 231 276 &= \left. {\frac{\partial u }{\partial t}} \right|_z 232 +\left. \zeta \right|_s \;v277 - \left. \zeta \right|_s \;v 233 278 + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ 234 279 &\qquad \qquad \qquad \quad 235 280 + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} 236 -\left[ {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s}281 + \left[ {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s} 237 282 - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v 238 283 - \frac{\sigma_1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ \\ 239 284 &= \left. {\frac{\partial u }{\partial t}} \right|_z 240 +\left. \zeta \right|_s \;v285 - \left. \zeta \right|_s \;v 241 286 + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ 242 287 &\qquad \qquad \qquad \quad … … 245 290 - \sigma_1 u\frac{\partial u}{\partial s} - \sigma_1 v\frac{\partial v}{\partial s}} \right] \\ \\ 246 291 &= \left. {\frac{\partial u }{\partial t}} \right|_z 247 +\left. \zeta \right|_s \;v292 - \left. \zeta \right|_s \;v 248 293 + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 249 294 + \frac{1}{e_3} \left[ w - \sigma_2 v - \sigma_1 u \right] 250 \; \frac{\partial u}{\partial s} \\295 \; \frac{\partial u}{\partial s} . \\ 251 296 % 252 \intertext{Introducing $\omega$, the dia- a-surface velocity given by (\autoref{apdx:A_w_s}) }297 \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{apdx:A_w_s}) } 253 298 % 254 299 &= \left. {\frac{\partial u }{\partial t}} \right|_z 255 +\left. \zeta \right|_s \;v300 - \left. \zeta \right|_s \;v 256 301 + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 257 + \frac{1}{e_3 } \left( \omega -w_s \right) \frac{\partial u}{\partial s} \\302 + \frac{1}{e_3 } \left( \omega + w_s \right) \frac{\partial u}{\partial s} \\ 258 303 \end{array} 259 304 } … … 266 311 { 267 312 \begin{array}{*{20}l} 268 w_s\;\frac{\partial u}{\partial s}269 = \frac{\partial s}{\partial t}\; \frac{\partial u }{\partial s}270 = \left. {\frac{\partial u }{\partial t}} \right|_s - \left. {\frac{\partial u }{\partial t}} \right|_z \ quad ,313 \frac{w_s}{e_3} \;\frac{\partial u}{\partial s} 314 = - \left. \frac{\partial s}{\partial t} \right|_z \; \frac{\partial u }{\partial s} 315 = \left. {\frac{\partial u }{\partial t}} \right|_s - \left. {\frac{\partial u }{\partial t}} \right|_z \ . 271 316 \end{array} 272 317 } 273 318 \] 274 leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative,319 This leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, 275 320 \ie the total $s-$coordinate time derivative : 276 321 \begin{align} … … 278 323 \left. \frac{D u}{D t} \right|_s 279 324 = \left. {\frac{\partial u }{\partial t}} \right|_s 280 +\left. \zeta \right|_s \;v325 - \left. \zeta \right|_s \;v 281 326 + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 282 + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} 327 + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} . 283 328 \end{align} 284 329 Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in … … 306 351 + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ \\ 307 352 &&- \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 308 -u \;\frac{\partial e_1 }{\partial j} \right) \\353 -u \;\frac{\partial e_1 }{\partial j} \right) . \\ 309 354 \end{array} 310 355 } … … 328 373 - e_2 u \;\frac{\partial e_3 }{\partial i} 329 374 - e_1 v \;\frac{\partial e_3 }{\partial j} \right) 330 -\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ \\375 + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\ \\ 331 376 && - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 332 377 -u \;\frac{\partial e_1 }{\partial j} \right) \\ \\ … … 337 382 && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i} 338 383 + \frac{\partial(e_1 e_3 \, v)}{\partial j} \right) 339 -\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right]384 + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] 340 385 - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 341 -u \;\frac{\partial e_1 }{\partial j} \right) \\386 -u \;\frac{\partial e_1 }{\partial j} \right) . \\ 342 387 % 343 388 \intertext {Introducing a more compact form for the divergence of the momentum fluxes, … … 361 406 + \left. \nabla \cdot \left( {{\mathrm {\mathbf U}}\,u} \right) \right|_s 362 407 - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 363 -u \;\frac{\partial e_1 }{\partial j} \right) 408 -u \;\frac{\partial e_1 }{\partial j} \right). 364 409 \end{flalign} 365 410 which is the total time derivative expressed in the curvilinear $s-$coordinate system. … … 377 422 & =-\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] \\ 378 423 & =-\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_1 424 &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1 . 380 425 \end{split} 381 426 \] 382 427 Applying similar manipulation to the second component and 383 replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it comes:428 replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it becomes: 384 429 \begin{equation} 385 430 \label{apdx:A_grad_p_1} … … 391 436 -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 392 437 &=-\frac{1}{\rho_o \,e_2 } \left( \left. {\frac{\partial p}{\partial j}} \right|_s 393 + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) \\438 + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) . \\ 394 439 \end{split} 395 440 \end{equation} … … 405 450 \[ 406 451 \begin{split} 407 p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \ left( \rho_o \,d + 1 \right) \; e_3 \; dk \\408 &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk452 p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \rho_o \left( d + 1 \right) \; e_3 \; dk \\ 453 &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + \rho_o g \, \int_z^\eta e_3 \; dk . 409 454 \end{split} 410 455 \] … … 412 457 \begin{equation} 413 458 \label{apdx:A_pressure} 414 p = \rho_o \; p_h' + g \, ( z + \eta)459 p = \rho_o \; p_h' + \rho_o \, g \, ( \eta - z ) 415 460 \end{equation} 416 461 and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is: 417 462 \[ 418 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 463 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 . 419 464 \] 420 465 421 466 Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and 422 using the definition of the density anomaly it comes theexpression in two parts:467 using the definition of the density anomaly it becomes an expression in two parts: 423 468 \begin{equation} 424 469 \label{apdx:A_grad_p_2} … … 426 471 -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 427 472 &=-\frac{1}{e_1 } \left( \left. {\frac{\partial p_h'}{\partial i}} \right|_s 428 + g\; d \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\473 + g\; d \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} , \\ 429 474 % 430 475 -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 431 476 &=-\frac{1}{e_2 } \left( \left. {\frac{\partial p_h'}{\partial j}} \right|_s 432 + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} \\477 + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} . \\ 433 478 \end{split} 434 479 \end{equation} … … 463 508 - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) 464 509 - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} 465 + D_u^{\vect{U}} + F_u^{\vect{U}} 510 + D_u^{\vect{U}} + F_u^{\vect{U}} , 466 511 \end{multline} 467 512 \begin{multline} … … 473 518 - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) 474 519 - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 475 + D_v^{\vect{U}} + F_v^{\vect{U}} 520 + D_v^{\vect{U}} + F_v^{\vect{U}} . 476 521 \end{multline} 477 522 \end{subequations} … … 483 528 \label{apdx:A_PE_dyn_flux_u} 484 529 \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t} = 485 \nabla \cdot \left( {{\mathrm {\mathbf U}}\,u} \right)530 - \nabla \cdot \left( {{\mathrm {\mathbf U}}\,u} \right) 486 531 + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 487 532 -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,v \\ 488 533 - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right) 489 534 - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} 490 + D_u^{\vect{U}} + F_u^{\vect{U}} 535 + D_u^{\vect{U}} + F_u^{\vect{U}} , 491 536 \end{multline} 492 537 \begin{multline} … … 494 539 \frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}= 495 540 - \nabla \cdot \left( {{\mathrm {\mathbf U}}\,v} \right) 496 +\left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i}541 - \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i} 497 542 -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,u \\ 498 543 - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right) 499 544 - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 500 + D_v^{\vect{U}} + F_v^{\vect{U}} 545 + D_v^{\vect{U}} + F_v^{\vect{U}} . 501 546 \end{multline} 502 547 \end{subequations} … … 505 550 \begin{equation} 506 551 \label{apdx:A_dyn_zph} 507 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 552 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 . 508 553 \end{equation} 509 554 … … 531 576 \left[ \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right) 532 577 + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] \\ 533 +\frac{1}{e_3} \frac{\partial }{\partial k} \left( Tw \right)578 - \frac{1}{e_3} \frac{\partial }{\partial k} \left( Tw \right) 534 579 + D^{T} +F^{T} 535 580 \end{multline} 536 581 537 The expression for the advection term is a straight consequence of ( A.4),582 The expression for the advection term is a straight consequence of (\autoref{apdx:A_sco_Continuity}), 538 583 the expression of the 3D divergence in the $s-$coordinates established above. 539 584
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