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% $Header$ |
% $Header$ |
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% $Name$ |
% $Name$ |
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%\section{Time Integration} |
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\subsection{Non-linear free surface} |
\subsection{Non-linear free surface} |
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\label{sect:nonlinear-freesurface} |
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Recently, 2 options have added to the model |
Recently, new options have been added to the model |
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(and therefore, have not yet been extensively tested) |
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that concern the free surface formulation. |
that concern the free surface formulation. |
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%------------------------------------------ |
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\subsubsection{Non-uniform linear-relation for the surface potential} |
\subsubsection{Non-uniform linear-relation for the surface potential} |
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The linear relation between |
The linear relation between surface pressure/geo-potential |
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surface pressure / geo- potential ($\Phi_{surf}$) |
($\Phi_{surf}$) and surface displacement ($\eta$) could be considered |
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and surface displacement ($\eta$) |
to be a constant ($b_s=$ constant) |
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has been considered as uniform ($b_s =$ Constant) |
\marginpar{add a reference to part.1 here} |
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but is in fact |
but is in fact dependent on the position ($x,y,r$) |
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dependent on the position ($x,y,r$) |
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since we linearize: |
since we linearize: |
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$$\Phi_{surf}=\int_{R_o}^{R_o+\eta} b dr \simeq b_s \eta |
$$\Phi_{surf}=\int_{R_o}^{R_o+\eta} b dr \simeq b_s \eta |
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~\mathrm{with}~ b_s = b(\theta,S,r)_{r=R_o} |
~\mathrm{with}~ b_s = b(\theta,S,r)_{r=R_o} |
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\simeq b_s(\theta_{ref}(R_o),S_{ref}(R_o),R_o)$$ |
\simeq b_s(\theta_{ref}(R_o),S_{ref}(R_o),R_o)$$ |
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Note that, for convinience, the effect of the local |
Note that, for convenience, the effect on $b_s$ of the local surface |
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surface $\theta,S$ on $b_s$ |
$\theta,S$ are not considered here, but are incorporated in to |
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are not considered here, but incorporated in $\Phi'_{hyd}$. |
$\Phi'_{hyd}$. |
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For the ocean, $b_s = g \rho_{surf} / \rho_o \simeq g$ |
For the ocean, $b_s = g \rho_{surf} / \rho_o \simeq g$ is a very good |
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is a fairly good approximation since the relative difference |
approximation since the relative difference in surface density are |
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in surface density are usually small and only due to |
usually small and only due to local $\theta,S$ gradients (because the |
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local $\theta,S$ gradient (because $R_o = 0$); |
upper surface, $R_o = 0$, is essentially flat). Therefore, they can |
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Therefore, they can easely be incorporated in $\Phi'_{hyd}$. |
easily be incorporated in $\Phi'_{hyd}$. |
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For the atmosphere, because of topographic effects, |
For the atmosphere, however, because of topographic effects, the |
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the reference surface pressure $R_o$ has large spacial differences |
reference surface pressure $R_o$ has large spatial variations that |
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that are responsible for significant $b_s$ variations |
are responsible for significant $b_s$ variations (from 0.8 to 1.2 |
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(from 0.8 to 1.2 $[m^3/kg]$). For this reason, |
$[m^3/kg]$). For this reason, we use a non-uniform linear coefficient |
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we use a non-uniform linear coefficient $b_s$. |
$b_s$. |
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Practically, in an oceanic configuration or |
In practice, in an oceanic configuration or when the default value |
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when the default value (TRUE) of the parameter |
(TRUE) of the parameter {\bf uniformLin\_PhiSurf} is used, then $b_s$ |
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{\bf uniformLin\_PhiSurf} is used |
is simply set to $g$ for the ocean and $1.$ for the atmosphere. |
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then $b_s$ is simply set to $g$ for the ocean |
Turning {\bf uniformLin\_PhiSurf} to "FALSE", tells the code to |
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and $1.$ for the atmosphere.\\ |
evaluate $b_s$ from the reference vertical profile $\theta_{ref}$ |
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Turning {\bf uniformLin\_PhiSurf} to "FALSE", allows to |
({\it S/R INI\_LINEAR\_PHISURF}) according to the reference surface |
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evaluate $b_s$ from the reference vertical profile $\theta_{ref}$ |
pressure $P_o$ ($=R_o$): $b_s = c_p \kappa (P_o / Pc)^{(\kappa - 1)} |
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({\it S/R INI\_LINEAR\_PHISURF}) |
\theta_{ref}(P_o)$ |
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according to the reference surface pressure $P_o$ ($=R_o$): |
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$b_s = c_p \kappa (P_o / Pc)^{(\kappa - 1)} \theta_{ref}(P_o)$ |
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%------------------------------------------ |
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\subsubsection{Free surface effect on column total thickness |
\subsubsection{Free surface effect on column total thickness |
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(Non-linear free surface)} |
(Non-linear free surface)} |
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The total thickness of the fluid column is |
The total thickness of the fluid column is $r_{surf} - R_{fixed} = |
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$r_{surf} - R_{min} = \eta + R_o - R_{min}$ |
\eta + R_o - R_{fixed}$. In most applications, the free surface |
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In the linear free surface approximation |
displacements are small compared to the total thickness |
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(detailed before), only the fixed part of |
$\eta << H_o = R_o - R_{fixed}$. |
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it ($R_o - R_{min})$ is considered when we integrate the |
In the previous sections and in older version of the model, |
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continuity equation or compute tracer and momentum advection term. |
the linearized free-surface approximation was made, assuming |
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$r_{surf} - R_{fixed} \simeq H_o$ when the horizontal transport is |
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This approximation is dropped when using |
computed, either in the continuity equation or in tracer and momentum |
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the non-linear free surface formulation. |
advection terms. |
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Details follow here after for the barotropic part |
This approximation is dropped when using the non-linear free surface |
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and in the 2 next subsections for the baroclinic |
formulation and the total thickness, including the time varying part |
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part. |
$\eta$, is consisdered when computing horizontal transport. |
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Implications for the barotropic part are presented hereafter. |
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%------------------------------------------ |
In sections \ref{sect:freesurf-tracer-advection} and |
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% Non-Linear Barotropic part |
\ref{sect:freesurf-momentum-advection}, consequences for tracer |
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and momentum are brifly discussed. a more detailed description |
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The continuous form of the model equations remains |
is available in \cite{campin:02}. |
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unchanged, except for the 2D continuity equation |
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(\ref{eq-tCsC-eta}) that is now integrated |
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from $R_{min}(x,y)$ up to $r_{surf}=R_o+\eta$ : |
In the non-linear formulation, the continuous form of the model equations |
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remains unchanged, except for the 2D continuity equation |
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(\ref{eq:discrete-time-backward-free-surface}) which is now |
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integrated from $R_{fixed}(x,y)$ up to $r_{surf}=R_o+\eta$ : |
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\begin{displaymath} |
\begin{displaymath} |
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\epsilon_{fs} \partial_t \eta = |
\epsilon_{fs} \partial_t \eta = |
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\left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) = |
\left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) = |
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- {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o+\eta} \vec{\bf v} dr |
- {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o+\eta} \vec{\bf v} dr |
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+ \epsilon_{fw} (P-E) |
+ \epsilon_{fw} (P-E) |
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\end{displaymath} |
\end{displaymath} |
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Since $\eta$ has a direct effect on the horizontal |
Since $\eta$ has a direct effect on the horizontal velocity (through |
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velocity (through $\nabla_h \Phi_{surf}$), this |
$\nabla_h \Phi_{surf}$), this adds a non-linear term to the free |
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adds a non-linear term to the free surface equation. |
surface equation. Several options for the time discretization of this |
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non-linear part can be considered, as detailed below. |
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Regarding the time discretization of this non-linear part, |
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several options are now being tested: |
If the column thickness is evaluated at time step $n$, and with |
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implicit treatment of the surface potential gradient, equations |
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With the column thickness evaluated at time step $n$, |
(\ref{eq-solve2D} and \ref{eq-solve2D_rhs}) becomes: |
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and the surface potential gradient still implicit, |
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equation (\ref{eq-solve2D} \& \ref{eq-solve2D_rhs}) |
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become: |
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\begin{eqnarray*} |
\begin{eqnarray*} |
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\epsilon_{fs} {\eta}^{n+1} - |
\epsilon_{fs} {\eta}^{n+1} - |
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{\bf \nabla}_h \cdot \Delta t^2 (\eta^{n}+R_o-R_{min}) |
{\bf \nabla}_h \cdot \Delta t^2 (\eta^{n}+R_o-R_{fixed}) |
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{\bf \nabla}_h b_s {\eta}^{n+1} |
{\bf \nabla}_h b_s {\eta}^{n+1} |
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= {\eta}^* |
= {\eta}^* |
|
%\label{solve_2d} |
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\end{eqnarray*} |
\end{eqnarray*} |
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where |
where |
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\begin{eqnarray*} |
\begin{eqnarray*} |
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{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
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\Delta t {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o+\eta^n} \vec{\bf v}^* dr |
\Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o+\eta^n} \vec{\bf v}^* dr |
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\: + \: \epsilon_{fw} \Delta_t (P-E)^{n} |
\: + \: \epsilon_{fw} \Delta_t (P-E)^{n} |
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%\label{solve_2d_rhs} |
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\end{eqnarray*} |
\end{eqnarray*} |
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This method requires to update the solver matrix at each time step. |
This method requires us to update the solver matrix at each time step. |
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Alternatively, the non-linear contribution can be evaluated fully |
Alternatively, the non-linear contribution can be evaluated fully |
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explicitly: |
explicitly: |
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\begin{eqnarray*} |
\begin{eqnarray*} |
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\epsilon_{fs} {\eta}^{n+1} - |
\epsilon_{fs} {\eta}^{n+1} - |
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{\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{min}) |
{\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{fixed}) |
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{\bf \nabla}_h b_s {\eta}^{n+1} |
{\bf \nabla}_h b_s {\eta}^{n+1} |
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= {\eta}^* |
= {\eta}^* |
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+{\bf \nabla}_h \cdot \Delta t^2 (\eta^{n}) |
+{\bf \nabla}_h \cdot \Delta t^2 (\eta^{n}) |
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{\bf \nabla}_h b_s {\eta}^{n} |
{\bf \nabla}_h b_s {\eta}^{n} |
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\end{eqnarray*} |
\end{eqnarray*} |
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This formulation allows to keep the initial solver matrix |
This formulation allows one to keep the initial solver matrix |
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since the non-linear free surface only affects the RHS. |
unchanged though throughout the integration, since the non-linear free |
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surface only affects the RHS. |
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Finally, another option is a "linearized" formulation where the total |
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column thickness appears only in the integral term of the RHS |
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(\ref{eq-solve2D_rhs}) but not directly in the equation |
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(\ref{eq-solve2D}). |
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Those different options (see tab.?? for the one still available) |
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have been tested and show litle differences. However, we recommand |
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the use of the most precise method (the 1rst one) since the |
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computation cost involved in the solver matrix update are negligeable. |
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\begin{center} |
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\begin{tabular}[htb]{|l|c|l|} |
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\hline |
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parameter & value & description \\ |
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\hline |
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& -1 & linear free-surface, restart from a pickup file \\ |
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& & produced with \#undef EXACT\_CONSERV code\\ |
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\cline{2-3} |
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& 0 & Linear free-Surface \\ |
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\cline{2-3} |
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nonlinFreeSurf & 4 & Non-linear free-surface \\ |
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\cline{2-3} |
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& 3 & same as 4 but neglecting |
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$\int_{R_o}^{R_o+\eta} b' dr $ in $\Phi'_{hyd}$ \\ |
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\cline{2-3} |
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& 2 & same as 3 but do not update cg2d solver matrix \\ |
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\cline{2-3} |
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& 1 & same as 2 but treat momentum as in Linear FS \\ |
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\hline |
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& 0 & do not use $r*$ vertical coordinate (= default)\\ |
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\cline{2-3} |
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select\_rStar & 2 & use $r^*$ vertical coordinate \\ |
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\cline{2-3} |
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& 1 & same as 2 but without the contribution of the\\ |
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& & slope of the coordinate in $\nabla \Phi$ \\ |
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\hline |
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\end{tabular} |
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\end{center} |
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An other option is a "linearized" formulation where the |
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total column thickness appears only in the integral term of |
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the RHS (\ref{eq-solve2D_rhs}) but not directly in |
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the equation (\ref{eq-solve2D}). |
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%------------------------------------------ |
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\subsubsection{Free surface effect on the surface level thickness |
\subsubsection{Free surface effect on the surface level thickness |
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(Non-linear free surface): Tracer advection} |
(Non-linear free surface): Tracer advection} |
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\label{sect:freesurf-tracer-advection} |
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To ensure a global tracer conservation (i.e., the total amount) |
To ensure global tracer conservation (i.e., the total amount) as well |
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as well as the local one (see tracer section for more details), |
as local conservation, the change in the surface level thickness must |
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the change in the surface level thickness must be consistent with |
be consistent with the way the continuity equation is integrated, both |
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the way the continuity equation is integrated, both in |
in the barotropic part (to find $\eta$) and baroclinic part (to find |
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in the barotropic part (to find $\eta$) and baroclinic part |
$w = \dot{r}$). |
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(to find $w = \dot{r}$). |
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To illustrate this, let's consider the shallow water model, |
To illustrate this, consider the shallow water model, with uniform |
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with uniform cartesian horizontal grid: |
Cartesian horizontal grid: |
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$$ |
$$ |
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\partial_t h + \nabla \cdot h \vec{\bf v} = 0 |
\partial_t h + \nabla \cdot h \vec{\bf v} = 0 |
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$$ |
$$ |
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$$ |
$$ |
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\partial_t (h \theta) + \nabla \cdot ( h \theta \vec{\bf v})= 0 |
\partial_t (h \theta) + \nabla \cdot ( h \theta \vec{\bf v})= 0 |
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$$ |
$$ |
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Using the implicit (non-linear) free surface described before, we have: |
Using the implicit (non-linear) free surface described above (section |
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\ref{sect:pressure-method-linear-backward}) we have: |
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\begin{eqnarray*} |
\begin{eqnarray*} |
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h^{n+1} = h^{n} - \Delta_t \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) \\ |
h^{n+1} = h^{n} - \Delta_t \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) \\ |
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\end{eqnarray*} |
\end{eqnarray*} |
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The discretized form of the tracer equation must use the same |
The discretized form of the tracer equation must adopt the same |
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"geometry" to compute the tracer fluxes, that is, the same value of |
``form'' in the computation of tracer fluxes, that is, the same value |
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h, as the continuity equation did: |
of $h$, as used in the continuity equation: |
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\begin{eqnarray*} |
\begin{eqnarray*} |
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h^{n+1} \, \theta^{n+1} = h^n \, \theta^n |
h^{n+1} \, \theta^{n+1} = h^n \, \theta^n |
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- \Delta_t \nabla \cdot (h^n \, \theta^n \, \vec{\bf v}^{n+1}) |
- \Delta_t \nabla \cdot (h^n \, \theta^n \, \vec{\bf v}^{n+1}) |
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\end{eqnarray*} |
\end{eqnarray*} |
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In order to deal with the Adams-Bashforth time stepping, |
The use of a 3 time-levels timestepping scheme such as the Adams-Bashforth |
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we implement this scheme slightly differently, in two step: |
make the conservation less straitforward. |
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the variation of the water column thickness (from |
The current implementation with the Adams-Bashforth time-stepping |
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$h^n$ to $h^{n+1}$) |
provides an exact local conservation and prevents any drift in |
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is not incorporated directly to the tracer equation. |
the global tracer content (\cite{campin:02}). |
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Instead, |
Compared to the linear free-surface method, an additional step is required: |
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the model continues to evaluate the $G_\theta$ term (first step) |
the variation of the water column thickness (from $h^n$ to $h^{n+1}$) is |
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as it use to do with the Linear free surface formulation |
not incorporated directly into the tracer equation. Instead, the |
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(with the "{\it surface correction}" turned "on", see tracer section): |
model uses the $G_\theta$ terms (first step) as in the linear free |
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surface formulation (with the "{\it surface correction}" turned "on", |
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see tracer section): |
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$$ |
$$ |
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G_\theta^n = \left(- \nabla \cdot (h^n \, \theta^n \, \vec{\bf v}^{n+1}) |
G_\theta^n = \left(- \nabla \cdot (h^n \, \theta^n \, \vec{\bf v}^{n+1}) |
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- \dot{r}_{surf}^{n+1} \theta^n \right) / h^n |
- \dot{r}_{surf}^{n+1} \theta^n \right) / h^n |
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$$ |
$$ |
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Then in a second step, |
Then, in a second step, the thickness variation (expansion/reduction) |
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thickness variation (expansion/reduction) is taken into account : |
is taken into account: |
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$$ |
$$ |
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\theta^{n+1} = \theta^n + \Delta_t \frac{h^n}{h^{n+1}} G_\theta^{(n+1/2)} |
\theta^{n+1} = \theta^n + \Delta_t \frac{h^n}{h^{n+1}} G_\theta^{(n+1/2)} |
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$$ |
$$ |
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= - \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) = (h^{n+1} - h^{n}) |
= - \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) = (h^{n+1} - h^{n}) |
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/ \Delta_t |
/ \Delta_t |
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$ |
$ |
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those 2 formulations are equivalent. |
these two formulations are equivalent. |
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Implementation in the MITgcm is as follows. The model ``geometry'' |
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(here the {\bf hFacC,W,S}) is updated just before entering {\it |
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SOLVE\_FOR\_PRESSURE}, using the current $\eta$ field. Then, at the |
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end of the time step, the variables are advanced in time, so that |
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$\eta^n$ replaces $\eta^{n-1}$. At the next time step, the tracer |
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tendency ($G_\theta$) is computed using the same geometry, which is |
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now consistent with $\eta^{n-1}$. Finally, in S/R {\it |
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TIMESTEP\_TRACER}, the expansion/reduction ratio is applied to the |
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surface level to compute the new tracer field. |
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The implementation in the MITgcm follows this scheme. |
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The model "geometry" (here the {\bf hFacC,W,S}) is updated |
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just before entering {\it SOLVE\_FOR\_PRESSURE}, |
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according to the current $\eta$ field. |
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Then, at the end of the time step, the variables are |
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advanced in time, so that $\eta^n$ becomes $\eta^{n-1}$. |
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At the next time step, the tracer tendency ($G_\theta$) is computed |
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using the same geometry, that is now consistent with |
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$\eta^{n-1}$. |
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Finally, in S/R {\it TIMESTEP\_TRACER}, the expansion/reduction |
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ratio is applied to the surface level to compute the new tracer field. |
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%------------------------------------------ |
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\subsubsection{Free surface effect on the surface level thickness |
\subsubsection{Free surface effect on the surface level thickness |
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(Non-linear free surface): Momentum advection} |
(Non-linear free surface): Momentum advection} |
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\label{sect:freesurf-momentum-advection} |
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With the flux form formulation, advection of momentum |
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can be treated exactly as the tracer advection is. |
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Here the expansion/reduction factors ($hFacW^{n+1}/hFacW^n$ for $u$ |
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and $hFacS^{n+1}/hFacS^n$ for $v$) are simply applied in the |
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subroutine {\it TIMESTEP}. |
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Regarding momentum advection, |
Regarding momentum advection, |
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the vector invariant formulation is similar to the |
the vector invariant formulation is similar to the |
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Updating the {\bf hFacC,W,S} and the {\bf recip\_hFac}(s) |
Updating the {\bf hFacC,W,S} and the {\bf recip\_hFac}(s) |
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at one given place (like describe before) is sufficient. |
at one given place (like describe before) is sufficient. |
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With the flux form formulation, advection of momentum |
\subsubsection{Non-linear free surface and vertical resolution} |
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can be treated exactly as the tracer advection is. |
\label{sect:nonlin-freesurf-dz_surf} |
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Here the expansion/reduction factors ($hFacW^{n+1}/hFacW^n$ for $u$ |
|
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and $hFacS^{n+1}/hFacS^n$ for $v$) are simply applied in the |
When the amplitude of the free-surface variations becomes |
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subroutine {\it TIMESTEP}. |
as large as the vertical resolution near the surface, |
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the surface layer thickness can decrease to nearly zero or |
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can even vanishe completly. |
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This later possibility has not been implemented, and a |
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minimum relative thickness is imposed ({\bf hFacInf}, |
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parameter file {\em data}, namelist {\em PARM01}) to prevent |
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numerical instabilities caused by very thin surface level. |
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A better atlternative to the vanishing level problem has been |
| 264 |
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found and implemented recently, and rely on a different |
| 265 |
|
vertical coordinate $r^*$~: |
| 266 |
|
The time variation ot the total column thickness becomes |
| 267 |
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part of the r* coordinate motion, as in a $\sigma_{z},\sigma_{p}$ |
| 268 |
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model, but the fixed part related to topography is treated |
| 269 |
|
as in a height or pressure coordinate model. |
| 270 |
|
A complete description is given in \cite{adcroft:04}. |
| 271 |
|
|
| 272 |
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The time-stepping implementation of the $r^*$ coordinate is |
| 273 |
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identical to the non-linear free-surface in $r$ coordinate, |
| 274 |
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and differences appear only in the spacial discretisation. |
| 275 |
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\marginpar{needs a subsection ref. here} |
| 276 |
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