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2018-12-03T12:45:01+01:00 (23 months ago)
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smasson
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dev_r10164_HPC09_ESIWACE_PREP_MERGE: merge with trunk@10365, see #2133

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  • NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_A.tex

    r9414 r10368  
    1919 
    2020In 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 
     22an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical), 
     23we start from the set of equations established in \autoref{subsec:PE_zco_Eq} for 
     24the special case $k = z$ and thus $e_3 = 1$, 
     25and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$. 
     26Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and 
     27the horizontal slope of $s-$surfaces by: 
    2728\begin{equation} \label{apdx:A_s_slope} 
    2829\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s  
     
    3132\end{equation} 
    3233 
    33 The chain rule to establish the model equations in the curvilinear $s-$coordinate  
    34 system is: 
     34The chain rule to establish the model equations in the curvilinear $s-$coordinate system is: 
    3535\begin{equation} \label{apdx:A_s_chain_rule} 
    3636\begin{aligned} 
     
    5252\end{equation} 
    5353 
    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: 
     54In particular applying the time derivative chain rule to $z$ provides the expression for $w_s$, 
     55the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: 
    5656\begin{equation} \label{apdx:A_w_in_s} 
    5757w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s  
     
    6767\label{sec:A_continuity} 
    6868 
    69 Using (\autoref{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  
    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: 
     69Using (\autoref{apdx:A_s_chain_rule}) and 
     70the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, 
     71the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to 
     72obtain its expression in the curvilinear $s-$coordinate system: 
    7373 
    7474\begin{subequations}  
     
    128128\end{subequations} 
    129129 
    130 Here, $w$ is the vertical velocity relative to the $z-$coordinate system.  
    131 Introducing the dia-surface velocity component, $\omega $, defined as  
    132 the volume flux across the moving $s$-surfaces per unit horizontal area: 
     130Here, $w$ is the vertical velocity relative to the $z-$coordinate system. 
     131Introducing the dia-surface velocity component, 
     132$\omega $, defined as the volume flux across the moving $s$-surfaces per unit horizontal area: 
    133133\begin{equation} \label{apdx:A_w_s} 
    134134\omega  = w - w_s - \sigma _1 \,u - \sigma _2 \,v    \\ 
    135135\end{equation} 
    136 with $w_s$ given by \autoref{apdx:A_w_in_s}, we obtain the expression for  
    137 the divergence of the velocity in the curvilinear $s-$coordinate system: 
     136with $w_s$ given by \autoref{apdx:A_w_in_s}, 
     137we obtain the expression for the divergence of the velocity in the curvilinear $s-$coordinate system: 
    138138\begin{subequations}  
    139139\begin{align*} {\begin{array}{*{20}l}  
     
    167167\end{subequations} 
    168168 
    169 As a result, the continuity equation \autoref{eq:PE_continuity} in the  
    170 $s-$coordinates is: 
     169As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is: 
    171170\begin{equation} \label{apdx:A_sco_Continuity} 
    172171\frac{1}{e_3 } \frac{\partial e_3}{\partial t}  
     
    176175 +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0    
    177176\end{equation} 
    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. 
     177A additional term has appeared that take into account 
     178the contribution of the time variation of the vertical coordinate to the volume budget. 
    180179 
    181180 
     
    186185\label{sec:A_momentum} 
    187186 
    188 Here we only consider the first component of the momentum equation,  
     187Here we only consider the first component of the momentum equation, 
    189188the generalization to the second one being straightforward. 
    190189 
     
    193192$\bullet$ \textbf{Total derivative in vector invariant form} 
    194193 
    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  
     194Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum equation in the vector invariant form. 
     195Its total $z-$coordinate time derivative, 
     196$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain 
    198197its expression in the curvilinear $s-$coordinate system: 
    199198 
     
    258257\end{subequations} 
    259258% 
    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, 
     259Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and 
     260using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side, 
    263261\begin{equation*} {\begin{array}{*{20}l}  
    264262w_s  \;\frac{\partial u}{\partial s}  
     
    267265\end{array} }      
    268266\end{equation*} 
    269 leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative,  
     267leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, 
    270268$i.e.$ the total $s-$coordinate time derivative : 
    271269\begin{align} \label{apdx:A_sco_Dt_vect} 
     
    276274  + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s}    
    277275\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. 
     276Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in 
     277$z-$ and $s-$coordinates. 
     278This is not the case for the flux form as shown in next paragraph. 
    281279 
    282280$\ $\newline    % force a new ligne 
     
    284282$\bullet$ \textbf{Total derivative in flux form} 
    285283 
    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 : 
     284Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish. 
     285Following the procedure used to establish (\autoref{eq:PE_flux_form}), it can be transformed into : 
    289286%\begin{subequations}  
    290287\begin{align*} {\begin{array}{*{20}l}  
     
    355352\end{subequations} 
    356353which 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: 
    358355\begin{flalign}\label{apdx:A_sco_Dt_flux} 
    359356\left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s   
     
    363360\end{flalign} 
    364361which 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. 
     362It has the same form as in the $z-$coordinate but for 
     363the vertical scale factor that has appeared inside the time derivative which 
     364comes from the modification of (\autoref{apdx:A_sco_Continuity}), 
     365the continuity equation. 
    368366 
    369367$\ $\newline    % force a new ligne 
     
    380378\end{split} 
    381379\end{equation*} 
    382 Applying similar manipulation to the second component and replacing  
    383 $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes: 
     380Applying similar manipulation to the second component and 
     381replacing $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes: 
    384382\begin{equation} \label{apdx:A_grad_p_1} 
    385383\begin{split} 
     
    394392\end{equation} 
    395393 
    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 $p_h'= g \; \int_z^\eta d \; e_3 \; dk$.  
     394An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for 
     395the tilt of $s-$surfaces with respect to geopotential $z-$surfaces. 
     396 
     397As in $z$-coordinate, 
     398the horizontal pressure gradient can be split in two parts following \citet{Marsaleix_al_OM08}. 
     399Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$, 
     400and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. 
    402401The pressure is then given by: 
    403402\begin{equation*}  
     
    416415\end{equation*} 
    417416 
    418 Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and using the definition of  
    419 the density anomaly it comes the expression in two parts: 
     417Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and 
     418using the definition of the density anomaly it comes the expression in two parts: 
    420419\begin{equation} \label{apdx:A_grad_p_2} 
    421420\begin{split} 
     
    429428\end{split} 
    430429\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 \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   
     430This formulation of the pressure gradient is characterised by the appearance of 
     431a term depending on the sea surface height only 
     432(last term on the right hand side of expression \autoref{apdx:A_grad_p_2}). 
     433This term will be loosely termed \textit{surface pressure gradient} whereas 
     434the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to 
     435the $z$-coordinate formulation. 
     436In fact, the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, 
     437and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration. 
     438 
    440439 
    441440$\ $\newline    % force a new ligne 
     
    443442$\bullet$ \textbf{The other terms of the momentum equation} 
    444443 
    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}. 
     444The coriolis and forcing terms as well as the the vertical physics remain unchanged as 
     445they involve neither time nor space derivatives. 
     446The form of the lateral physics is discussed in \autoref{apdx:B}. 
    448447 
    449448 
     
    452451$\bullet$ \textbf{Full momentum equation} 
    453452 
    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  
    456 $z-$coordinate, except for the pressure gradient term : 
     453To sum up, in a curvilinear $s$-coordinate system, 
     454the vector invariant momentum equation solved by the model has the same mathematical expression as 
     455the one in a curvilinear $z-$coordinate, except for the pressure gradient term: 
    457456\begin{subequations} \label{apdx:A_dyn_vect} 
    458457\begin{multline} \label{apdx:A_PE_dyn_vect_u} 
     
    475474\end{multline} 
    476475\end{subequations} 
    477 whereas the flux form momentum equation differ from it by the formulation of both 
    478 the time derivative and the pressure gradient term  : 
     476whereas the flux form momentum equation differs from it by 
     477the formulation of both the time derivative and the pressure gradient term: 
    479478\begin{subequations} \label{apdx:A_dyn_flux} 
    480479\begin{multline} \label{apdx:A_PE_dyn_flux_u} 
     
    503502\end{equation} 
    504503 
    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.  
     504It is important to realize that the change in coordinate system has only concerned the position on the vertical. 
     505It 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, 
     507in particular the pressure gradient. 
     508By 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.  
    512510 
    513511 
     
    518516\label{sec:A_tracer} 
    519517 
    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 : 
     518The tracer equation is obtained using the same calculation as for the continuity equation and then 
     519regrouping the time derivative terms in the left hand side : 
    522520 
    523521\begin{multline} \label{apdx:A_tracer} 
     
    531529 
    532530 
    533 The expression for the advection term is a straight consequence of (A.4), the  
    534 expression of the 3D divergence in the $s-$coordinates established above.  
     531The expression for the advection term is a straight consequence of (A.4), 
     532the expression of the 3D divergence in the $s-$coordinates established above.  
    535533 
    536534\end{document} 
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