Changeset 10354 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_A.tex
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NEMO/trunk/doc/latex/NEMO/subfiles/annex_A.tex
r9414 r10354 19 19 20 20 In order to establish the set of Primitive Equation in curvilinear $s$-coordinates 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 \autoref{subsec: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 25 $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal 26 slope of $s-$surfaces by : 21 ($i.e.$ an orthogonal curvilinear coordinate in the horizontal and 22 an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical), 23 we start from the set of equations established in \autoref{subsec:PE_zco_Eq} for 24 the special case $k = z$ and thus $e_3 = 1$, 25 and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$. 26 Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and 27 the horizontal slope of $s-$surfaces by: 27 28 \begin{equation} \label{apdx:A_s_slope} 28 29 \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s … … 31 32 \end{equation} 32 33 33 The chain rule to establish the model equations in the curvilinear $s-$coordinate 34 system is: 34 The chain rule to establish the model equations in the curvilinear $s-$coordinate system is: 35 35 \begin{equation} \label{apdx:A_s_chain_rule} 36 36 \begin{aligned} … … 52 52 \end{equation} 53 53 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:54 In particular applying the time derivative chain rule to $z$ provides the expression for $w_s$, 55 the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: 56 56 \begin{equation} \label{apdx:A_w_in_s} 57 57 w_s = \left. \frac{\partial z }{\partial t} \right|_s … … 67 67 \label{sec:A_continuity} 68 68 69 Using (\autoref{apdx:A_s_chain_rule}) and the fact that the horizontal scale factors70 $e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of 71 the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows72 in order toobtain its expression in the curvilinear $s-$coordinate system:69 Using (\autoref{apdx:A_s_chain_rule}) and 70 the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, 71 the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to 72 obtain its expression in the curvilinear $s-$coordinate system: 73 73 74 74 \begin{subequations} … … 128 128 \end{subequations} 129 129 130 Here, $w$ is the vertical velocity relative to the $z-$coordinate system. 131 Introducing the dia-surface velocity component, $\omega $, defined as132 the volume flux across the moving $s$-surfaces per unit horizontal area:130 Here, $w$ is the vertical velocity relative to the $z-$coordinate system. 131 Introducing the dia-surface velocity component, 132 $\omega $, defined as the volume flux across the moving $s$-surfaces per unit horizontal area: 133 133 \begin{equation} \label{apdx:A_w_s} 134 134 \omega = w - w_s - \sigma _1 \,u - \sigma _2 \,v \\ 135 135 \end{equation} 136 with $w_s$ given by \autoref{apdx:A_w_in_s}, we obtain the expression for137 the divergence of the velocity in the curvilinear $s-$coordinate system:136 with $w_s$ given by \autoref{apdx:A_w_in_s}, 137 we obtain the expression for the divergence of the velocity in the curvilinear $s-$coordinate system: 138 138 \begin{subequations} 139 139 \begin{align*} {\begin{array}{*{20}l} … … 167 167 \end{subequations} 168 168 169 As a result, the continuity equation \autoref{eq:PE_continuity} in the 170 $s-$coordinates is: 169 As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is: 171 170 \begin{equation} \label{apdx:A_sco_Continuity} 172 171 \frac{1}{e_3 } \frac{\partial e_3}{\partial t} … … 176 175 +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 177 176 \end{equation} 178 A additional term has appeared that take into account the contribution of the time variation179 of the vertical coordinate to the volume budget.177 A additional term has appeared that take into account 178 the contribution of the time variation of the vertical coordinate to the volume budget. 180 179 181 180 … … 186 185 \label{sec:A_momentum} 187 186 188 Here we only consider the first component of the momentum equation, 187 Here we only consider the first component of the momentum equation, 189 188 the generalization to the second one being straightforward. 190 189 … … 193 192 $\bullet$ \textbf{Total derivative in vector invariant form} 194 193 195 Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum 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 194 Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum equation in the vector invariant form. 195 Its total $z-$coordinate time derivative, 196 $\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain 198 197 its expression in the curvilinear $s-$coordinate system: 199 198 … … 258 257 \end{subequations} 259 258 % 260 Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) 261 to $u$ and using (\autoref{apdx:A_w_in_s}) provides the expression of the last term 262 of the right hand side, 259 Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and 260 using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side, 263 261 \begin{equation*} {\begin{array}{*{20}l} 264 262 w_s \;\frac{\partial u}{\partial s} … … 267 265 \end{array} } 268 266 \end{equation*} 269 leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, 267 leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, 270 268 $i.e.$ the total $s-$coordinate time derivative : 271 269 \begin{align} \label{apdx:A_sco_Dt_vect} … … 276 274 + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} 277 275 \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.276 Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in 277 $z-$ and $s-$coordinates. 278 This is not the case for the flux form as shown in next paragraph. 281 279 282 280 $\ $\newline % force a new ligne … … 284 282 $\bullet$ \textbf{Total derivative in flux form} 285 283 286 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 (\autoref{eq:PE_flux_form}), 288 it can be transformed into : 284 Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish. 285 Following the procedure used to establish (\autoref{eq:PE_flux_form}), it can be transformed into : 289 286 %\begin{subequations} 290 287 \begin{align*} {\begin{array}{*{20}l} … … 355 352 \end{subequations} 356 353 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 354 $i.e.$ the total $s-$coordinate time derivative in flux form: 358 355 \begin{flalign}\label{apdx:A_sco_Dt_flux} 359 356 \left. \frac{D u}{D t} \right|_s = \frac{1}{e_3} \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s … … 363 360 \end{flalign} 364 361 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 (\autoref{apdx:A_sco_Continuity}), the continuity equation. 362 It has the same form as in the $z-$coordinate but for 363 the vertical scale factor that has appeared inside the time derivative which 364 comes from the modification of (\autoref{apdx:A_sco_Continuity}), 365 the continuity equation. 368 366 369 367 $\ $\newline % force a new ligne … … 380 378 \end{split} 381 379 \end{equation*} 382 Applying similar manipulation to the second component and replacing383 $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes:380 Applying similar manipulation to the second component and 381 replacing $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes: 384 382 \begin{equation} \label{apdx:A_grad_p_1} 385 383 \begin{split} … … 394 392 \end{equation} 395 393 396 An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for the 397 tilt of $s-$surfaces with respect to geopotential $z-$surfaces. 398 399 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 $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. 394 An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for 395 the tilt of $s-$surfaces with respect to geopotential $z-$surfaces. 396 397 As in $z$-coordinate, 398 the horizontal pressure gradient can be split in two parts following \citet{Marsaleix_al_OM08}. 399 Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$, 400 and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. 402 401 The pressure is then given by: 403 402 \begin{equation*} … … 416 415 \end{equation*} 417 416 418 Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and using the definition of419 the density anomaly it comes the expression in two parts:417 Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and 418 using the definition of the density anomaly it comes the expression in two parts: 420 419 \begin{equation} \label{apdx:A_grad_p_2} 421 420 \begin{split} … … 429 428 \end{split} 430 429 \end{equation} 431 This formulation of the pressure gradient is characterised by the appearance of a term depending on the432 the sea surface height only (last term on the right hand side of expression \autoref{apdx:A_grad_p_2}). 433 This term will be loosely termed \textit{surface pressure gradient} 434 whereas the first term will be termed the 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.439 430 This formulation of the pressure gradient is characterised by the appearance of 431 a term depending on the sea surface height only 432 (last term on the right hand side of expression \autoref{apdx:A_grad_p_2}). 433 This term will be loosely termed \textit{surface pressure gradient} whereas 434 the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to 435 the $z$-coordinate formulation. 436 In fact, the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, 437 and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration. 438 440 439 441 440 $\ $\newline % force a new ligne … … 443 442 $\bullet$ \textbf{The other terms of the momentum equation} 444 443 445 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 \autoref{apdx:B}.444 The coriolis and forcing terms as well as the the vertical physics remain unchanged as 445 they involve neither time nor space derivatives. 446 The form of the lateral physics is discussed in \autoref{apdx:B}. 448 447 449 448 … … 452 451 $\bullet$ \textbf{Full momentum equation} 453 452 454 To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation455 solved by the model has the same mathematical expression as the one in a curvilinear 456 $z-$coordinate, except for the pressure gradient term:453 To sum up, in a curvilinear $s$-coordinate system, 454 the vector invariant momentum equation solved by the model has the same mathematical expression as 455 the one in a curvilinear $z-$coordinate, except for the pressure gradient term: 457 456 \begin{subequations} \label{apdx:A_dyn_vect} 458 457 \begin{multline} \label{apdx:A_PE_dyn_vect_u} … … 475 474 \end{multline} 476 475 \end{subequations} 477 whereas the flux form momentum equation differ from it by the formulation of both478 the time derivative and the pressure gradient term:476 whereas the flux form momentum equation differs from it by 477 the formulation of both the time derivative and the pressure gradient term: 479 478 \begin{subequations} \label{apdx:A_dyn_flux} 480 479 \begin{multline} \label{apdx:A_PE_dyn_flux_u} … … 503 502 \end{equation} 504 503 505 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 507 orthogonal curvilinear set of unit vectors. ($u$,$v$) are always horizontal velocities 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, 510 but the dia-surface velocity component, $i.e.$ the volume flux across the moving 511 $s$-surfaces per unit horizontal area. 504 It is important to realize that the change in coordinate system has only concerned the position on the vertical. 505 It has not affected (\textbf{i},\textbf{j},\textbf{k}), the orthogonal curvilinear set of unit vectors. 506 ($u$,$v$) are always horizontal velocities so that their evolution is driven by \emph{horizontal} forces, 507 in particular the pressure gradient. 508 By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component, 509 $i.e.$ the volume flux across the moving $s$-surfaces per unit horizontal area. 512 510 513 511 … … 518 516 \label{sec:A_tracer} 519 517 520 The tracer equation is obtained using the same calculation as for the continuity 521 equation and thenregrouping the time derivative terms in the left hand side :518 The tracer equation is obtained using the same calculation as for the continuity equation and then 519 regrouping the time derivative terms in the left hand side : 522 520 523 521 \begin{multline} \label{apdx:A_tracer} … … 531 529 532 530 533 The expression for the advection term is a straight consequence of (A.4), the534 expression of the 3D divergence in the $s-$coordinates established above.531 The expression for the advection term is a straight consequence of (A.4), 532 the expression of the 3D divergence in the $s-$coordinates established above. 535 533 536 534 \end{document}
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