Changeset 10368 for NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_A.tex

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

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 r9414 In order to establish the set of Primitive Equation in curvilinear $s$-coordinates ($i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical), we start from the set of equations established in \autoref{subsec:PE_zco_Eq} for the special case $k = z$ and thus $e_3 = 1$, and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$. Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal slope of $s-$surfaces by : ($i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical), we start from the set of equations established in \autoref{subsec:PE_zco_Eq} for the special case $k = z$ and thus $e_3 = 1$, and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$. Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal slope of $s-$surfaces by: \label{apdx:A_s_slope} \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s The chain rule to establish the model equations in the curvilinear $s-$coordinate system is: The chain rule to establish the model equations in the curvilinear $s-$coordinate system is: \label{apdx:A_s_chain_rule} \begin{aligned} In particular applying the time derivative chain rule to $z$ provides the expression for $w_s$,  the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: In particular applying the time derivative chain rule to $z$ provides the expression for $w_s$, the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: \label{apdx:A_w_in_s} w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s \label{sec:A_continuity} Using (\autoref{apdx:A_s_chain_rule}) and the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to obtain its expression in the curvilinear $s-$coordinate system: Using (\autoref{apdx:A_s_chain_rule}) and the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to obtain its expression in the curvilinear $s-$coordinate system: \begin{subequations} \end{subequations} Here, $w$ is the vertical velocity relative to the $z-$coordinate system. Introducing the dia-surface velocity component, $\omega$, defined as the volume flux across the moving $s$-surfaces per unit horizontal area: Here, $w$ is the vertical velocity relative to the $z-$coordinate system. Introducing the dia-surface velocity component, $\omega$, defined as the volume flux across the moving $s$-surfaces per unit horizontal area: \label{apdx:A_w_s} \omega  = w - w_s - \sigma _1 \,u - \sigma _2 \,v    \\ with $w_s$ given by \autoref{apdx:A_w_in_s}, we obtain the expression for the divergence of the velocity in the curvilinear $s-$coordinate system: with $w_s$ given by \autoref{apdx:A_w_in_s}, we obtain the expression for the divergence of the velocity in the curvilinear $s-$coordinate system: \begin{subequations} \begin{align*} {\begin{array}{*{20}l} \end{subequations} As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is: As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is: \label{apdx:A_sco_Continuity} \frac{1}{e_3 } \frac{\partial e_3}{\partial t} +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 A additional term has appeared that take into account the contribution of the time variation of the vertical coordinate to the volume budget. A additional term has appeared that take into account the contribution of the time variation of the vertical coordinate to the volume budget. \label{sec:A_momentum} Here we only consider the first component of the momentum equation, Here we only consider the first component of the momentum equation, the generalization to the second one being straightforward. $\bullet$ \textbf{Total derivative in vector invariant form} Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum equation in the vector invariant form. Its total $z-$coordinate time derivative, $\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum equation in the vector invariant form. Its total $z-$coordinate time derivative, $\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain its expression in the curvilinear $s-$coordinate system: \end{subequations} % Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side, Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side, \begin{equation*} {\begin{array}{*{20}l} w_s  \;\frac{\partial u}{\partial s} \end{array} } \end{equation*} leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, $i.e.$ the total $s-$coordinate time derivative : \begin{align} \label{apdx:A_sco_Dt_vect} + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} \end{align} Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in $z-$ and $s-$coordinates. This is not the case for the flux form as shown in next paragraph. Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in $z-$ and $s-$coordinates. This is not the case for the flux form as shown in next paragraph. $\$\newline    % force a new ligne $\bullet$ \textbf{Total derivative in flux form} Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish. Following the procedure used to establish (\autoref{eq:PE_flux_form}), it can be transformed into : Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish. Following the procedure used to establish (\autoref{eq:PE_flux_form}), it can be transformed into : %\begin{subequations} \begin{align*} {\begin{array}{*{20}l} \end{subequations} which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, $i.e.$ the total $s-$coordinate time derivative in flux form : $i.e.$ the total $s-$coordinate time derivative in flux form: \begin{flalign}\label{apdx:A_sco_Dt_flux} \left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s \end{flalign} which is the total time derivative expressed in the curvilinear $s-$coordinate system. It has the same form as in the $z-$coordinate but for the vertical scale factor that has appeared inside the time derivative which comes from the modification of (\autoref{apdx:A_sco_Continuity}), the continuity equation. It has the same form as in the $z-$coordinate but for the vertical scale factor that has appeared inside the time derivative which comes from the modification of (\autoref{apdx:A_sco_Continuity}), the continuity equation. $\$\newline    % force a new ligne \end{split} \end{equation*} Applying similar manipulation to the second component and replacing $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes: Applying similar manipulation to the second component and replacing $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes: \label{apdx:A_grad_p_1} \begin{split} An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for the tilt of $s-$surfaces with respect to geopotential $z-$surfaces. As in $z$-coordinate, the horizontal pressure gradient can be split in two parts following \citet{Marsaleix_al_OM08}. Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$, and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for the tilt of $s-$surfaces with respect to geopotential $z-$surfaces. As in $z$-coordinate, the horizontal pressure gradient can be split in two parts following \citet{Marsaleix_al_OM08}. Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$, and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. The pressure is then given by: \begin{equation*} \end{equation*} Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and using the definition of the density anomaly it comes the expression in two parts: Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and using the definition of the density anomaly it comes the expression in two parts: \label{apdx:A_grad_p_2} \begin{split} \end{split} This formulation of the pressure gradient is characterised by the appearance of a term depending on the the sea surface height only (last term on the right hand side of expression \autoref{apdx:A_grad_p_2}). This term will be loosely termed \textit{surface pressure gradient} whereas the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation. In fact, the the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration. This formulation of the pressure gradient is characterised by the appearance of a term depending on the sea surface height only (last term on the right hand side of expression \autoref{apdx:A_grad_p_2}). This term will be loosely termed \textit{surface pressure gradient} whereas the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation. In fact, the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration. $\$\newline    % force a new ligne $\bullet$ \textbf{The other terms of the momentum equation} The coriolis and forcing terms as well as the the vertical physics remain unchanged as they involve neither time nor space derivatives. The form of the lateral physics is discussed in \autoref{apdx:B}. The coriolis and forcing terms as well as the the vertical physics remain unchanged as they involve neither time nor space derivatives. The form of the lateral physics is discussed in \autoref{apdx:B}. $\bullet$ \textbf{Full momentum equation} To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation solved by the model has the same mathematical expression as the one in a curvilinear $z-$coordinate, except for the pressure gradient term : To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation solved by the model has the same mathematical expression as the one in a curvilinear $z-$coordinate, except for the pressure gradient term: \begin{subequations} \label{apdx:A_dyn_vect} \begin{multline} \label{apdx:A_PE_dyn_vect_u} \end{multline} \end{subequations} whereas the flux form momentum equation differ from it by the formulation of both the time derivative and the pressure gradient term  : whereas the flux form momentum equation differs from it by the formulation of both the time derivative and the pressure gradient term: \begin{subequations} \label{apdx:A_dyn_flux} \begin{multline} \label{apdx:A_PE_dyn_flux_u} It is important to realize that the change in coordinate system has only concerned the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the orthogonal curvilinear set of unit vectors. ($u$,$v$) are always horizontal velocities so that their evolution is driven by \emph{horizontal} forces, in particular the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component, $i.e.$ the volume flux across the moving $s$-surfaces per unit horizontal area. It is important to realize that the change in coordinate system has only concerned the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the orthogonal curvilinear set of unit vectors. ($u$,$v$) are always horizontal velocities so that their evolution is driven by \emph{horizontal} forces, in particular the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component, $i.e.$ the volume flux across the moving $s$-surfaces per unit horizontal area. \label{sec:A_tracer} The tracer equation is obtained using the same calculation as for the continuity equation and then regrouping the time derivative terms in the left hand side : The tracer equation is obtained using the same calculation as for the continuity equation and then regrouping the time derivative terms in the left hand side : \begin{multline} \label{apdx:A_tracer} The expression for the advection term is a straight consequence of (A.4), the expression of the 3D divergence in the $s-$coordinates established above. The expression for the advection term is a straight consequence of (A.4), the expression of the 3D divergence in the $s-$coordinates established above. \end{document}