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% $Header: /u/gcmpack/manual/s_algorithm/text/nonlin_frsurf.tex,v 1.14 2010/08/30 23:09:18 jmc Exp $ |
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% $Name: $ |
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\subsection{Non-linear free-surface} |
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\label{sec:nonlinear-freesurface} |
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|
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Recently, new options have been added to the model |
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that concern the free surface formulation. |
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|
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|
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\subsubsection{pressure/geo-potential and free surface} |
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\label{sec:phi-freesurface} |
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|
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For the atmosphere, since $\phi = \phi_{topo} - \int^p_{p_s} \alpha dp$, |
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subtracting the reference state defined in section |
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\ref{sec:hpe-p-geo-potential-split}~:\\ |
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$$ |
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\phi_o = \phi_{topo} - \int^p_{p_o} \alpha_o dp |
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\hspace{5mm}\mathrm{with}\hspace{3mm} \phi_o(p_o)=\phi_{topo} |
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$$ |
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we get: |
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$$ |
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\phi' = \phi - \phi_o = \int^{p_s}_p \alpha dp - \int^{p_o}_p \alpha_o dp |
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$$ |
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For the ocean, the reference state is simpler since $\rho_c$ does not dependent |
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on $z$ ($b_o=g$) and the surface reference position is uniformly $z=0$ ($R_o=0$), |
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and the same subtraction leads to a similar relation. |
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For both fluid, using the isomorphic notations, we can write: |
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$$ |
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\phi' = \int^{r_{surf}}_r b~ dr - \int^{R_o}_r b_o dr |
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$$ |
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and re-write as: |
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\begin{equation} |
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\phi' = \int^{r_{surf}}_{R_o} b~ dr + \int^{R_o}_r (b - b_o) dr |
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\label{eq:split-phi-Ro} |
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\end{equation} |
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or: |
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\begin{equation} |
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\phi' = \int^{r_{surf}}_{R_o} b_o dr + \int^{r_{surf}}_r (b - b_o) dr |
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\label{eq:split-phi-bo} |
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\end{equation} |
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|
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In section \ref{sec:finding_the_pressure_field}, following eq.\ref{eq:split-phi-Ro}, |
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the pressure/geo-potential $\phi'$ has been separated into surface ($\phi_s$), |
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and hydrostatic anomaly ($\phi'_{hyd}$). |
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In this section, the split between $\phi_s$ and $\phi'_{hyd}$ is |
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made according to equation \ref{eq:split-phi-bo}. This slightly |
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different definition reflects the actual implementation in the code |
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and is valid for both linear and non-linear |
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free-surface formulation, in both r-coordinate and r*-coordinate. |
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|
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Because the linear free-surface approximation ignore the tracer content |
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of the fluid parcel between $R_o$ and $r_{surf}=R_o+\eta$, |
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for consistency reasons, this part is also neglected in $\phi'_{hyd}$~: |
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$$ |
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\phi'_{hyd} = \int^{r_{surf}}_r (b - b_o) dr \simeq \int^{R_o}_r (b - b_o) dr |
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$$ |
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Note that in this case, the two definitions of $\phi_s$ and $\phi'_{hyd}$ |
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from equation \ref{eq:split-phi-Ro} and \ref{eq:split-phi-bo} converge toward |
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the same (approximated) expressions: $\phi_s = \int^{r_{surf}}_{R_o} b_o dr$ |
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and $\phi'_{hyd}=\int^{R_o}_r b' dr$.\\ |
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On the contrary, the unapproximated formulation ("non-linear free-surface", |
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see the next section) retains the full expression: |
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$\phi'_{hyd} = \int^{r_{surf}}_r (b - b_o) dr $~. |
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This is obtained by selecting {\bf nonlinFreeSurf}=4 in parameter |
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file {\em data}.\\ |
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|
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Regarding the surface potential: |
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$$\phi_s = \int_{R_o}^{R_o+\eta} b_o dr = b_s \eta |
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\hspace{5mm}\mathrm{with}\hspace{5mm} |
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b_s = \frac{1}{\eta} \int_{R_o}^{R_o+\eta} b_o dr $$ |
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$b_s \simeq b_o(R_o)$ is an excellent approximation (better than |
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the usual numerical truncation, since generally $|\eta|$ is smaller |
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than the vertical grid increment). |
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|
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For the ocean, $\phi_s = g \eta$ and $b_s = g$ is uniform. |
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For the atmosphere, however, because of topographic effects, the |
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reference surface pressure $R_o=p_o$ has large spatial variations that |
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are responsible for significant $b_s$ variations (from 0.8 to 1.2 |
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$[m^3/kg]$). For this reason, when {\bf uniformLin\_PhiSurf} {\em=.FALSE.} |
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(parameter file {\em data}, namelist {\em PARAM01}) |
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a non-uniform linear coefficient $b_s$ is used and computed |
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({\it S/R INI\_LINEAR\_PHISURF}) according to the reference surface |
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pressure $p_o$: |
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$b_s = b_o(R_o) = c_p \kappa (p_o / P^o_{SL})^{(\kappa - 1)} \theta_{ref}(p_o)$. |
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with $P^o_{SL}$ the mean sea-level pressure. |
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|
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|
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\subsubsection{Free surface effect on column total thickness |
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(Non-linear free-surface)} |
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|
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The total thickness of the fluid column is $r_{surf} - R_{fixed} = |
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\eta + R_o - R_{fixed}$. In most applications, the free surface |
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displacements are small compared to the total thickness |
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$\eta \ll H_o = R_o - R_{fixed}$. |
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In the previous sections and in older version of the model, |
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the linearized free-surface approximation was made, assuming |
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$r_{surf} - R_{fixed} \simeq H_o$ when computing horizontal transports, |
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either in the continuity equation or in tracer and momentum |
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advection terms. |
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This approximation is dropped when using the non-linear free-surface |
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formulation and the total thickness, including the time varying part |
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$\eta$, is considered when computing horizontal transports. |
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Implications for the barotropic part are presented hereafter. |
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In section \ref{sec:freesurf-tracer-advection} consequences for |
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tracer conservation is briefly discussed (more details can be |
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found in \cite{campin:02})~; the general time-stepping is presented |
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in section \ref{sec:nonlin-freesurf-timestepping} with some |
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limitations regarding the vertical resolution in section |
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\ref{sec:nonlin-freesurf-dz_surf}. |
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|
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In the non-linear formulation, the continuous form of the model |
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equations 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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|
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\begin{displaymath} |
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\epsilon_{fs} \partial_t \eta = |
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\left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) = |
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- {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o+\eta} \vec{\bf v} dr |
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+ \epsilon_{fw} (P-E) |
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\end{displaymath} |
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|
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Since $\eta$ has a direct effect on the horizontal velocity (through |
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$\nabla_h \Phi_{surf}$), this adds a non-linear term to the free |
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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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|
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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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(\ref{eq-solve2D} and \ref{eq-solve2D_rhs}) becomes: |
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\begin{eqnarray*} |
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\epsilon_{fs} {\eta}^{n+1} - |
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{\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} |
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= {\eta}^* |
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\end{eqnarray*} |
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where |
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\begin{eqnarray*} |
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{\eta}^* = \epsilon_{fs} \: {\eta}^{n} - |
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\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} |
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\end{eqnarray*} |
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This method requires us to update the solver matrix at each time step. |
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|
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Alternatively, the non-linear contribution can be evaluated fully |
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explicitly: |
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\begin{eqnarray*} |
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\epsilon_{fs} {\eta}^{n+1} - |
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{\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{fixed}) |
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{\bf \nabla}_h b_s {\eta}^{n+1} |
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= {\eta}^* |
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+{\bf \nabla}_h \cdot \Delta t^2 (\eta^{n}) |
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{\bf \nabla}_h b_s {\eta}^{n} |
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\end{eqnarray*} |
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This formulation allows one to keep the initial solver matrix |
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unchanged though throughout the integration, since the non-linear free |
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surface only affects the RHS. |
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|
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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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|
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Those different options (see Table \ref{tab:nonLinFreeSurf_flags}) |
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have been tested and show little differences. However, we recommend |
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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 is negligible. |
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|
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\begin{table}[htb] |
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%\begin{center} |
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\centering |
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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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\caption{Non-linear free-surface flags} |
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\label{tab:nonLinFreeSurf_flags} |
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%\end{center} |
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\end{table} |
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|
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|
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\subsubsection{Tracer conservation with non-linear free-surface} |
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\label{sec:freesurf-tracer-advection} |
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|
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To ensure global tracer conservation (i.e., the total amount) as well |
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as local conservation, the change in the surface level thickness must |
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be consistent with the way the continuity equation is integrated, both |
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in the barotropic part (to find $\eta$) and baroclinic part (to find |
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$w = \dot{r}$). |
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|
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To illustrate this, consider the shallow water model, with a source |
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of fresh water (P): |
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$$ |
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\partial_t h + \nabla \cdot h \vec{\bf v} = P |
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$$ |
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where $h$ is the total thickness of the water column. |
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To conserve the tracer $\theta$ we have to discretize: |
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$$ |
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\partial_t (h \theta) + \nabla \cdot ( h \theta \vec{\bf v}) |
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= P \theta_{\mathrm{rain}} |
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$$ |
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Using the implicit (non-linear) free surface described above (section |
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\ref{sec:pressure-method-linear-backward}) we have: |
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\begin{eqnarray*} |
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h^{n+1} = h^{n} - \Delta t \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) + \Delta t P \\ |
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\end{eqnarray*} |
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The discretized form of the tracer equation must adopt the same |
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``form'' in the computation of tracer fluxes, that is, the same value |
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of $h$, as used in the continuity equation: |
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\begin{eqnarray*} |
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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}) |
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+ \Delta t P \theta_{rain} |
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\end{eqnarray*} |
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|
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The use of a 3 time-levels time-stepping scheme such as the Adams-Bashforth |
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make the conservation sightly tricky. |
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The current implementation with the Adams-Bashforth time-stepping |
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provides an exact local conservation and prevents any drift in |
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the global tracer content (\cite{campin:02}). |
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Compared to the linear free-surface method, an additional step is required: |
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the variation of the water column thickness (from $h^n$ to $h^{n+1}$) is |
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not incorporated directly into the tracer equation. Instead, the |
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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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$$ |
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G_\theta^n = \left(- \nabla \cdot (h^n \, \theta^n \, \vec{\bf v}^{n+1}) |
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- \dot{r}_{surf}^{n+1} \theta^n \right) / h^n |
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$$ |
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Then, in a second step, the thickness variation (expansion/reduction) |
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is taken into account: |
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$$ |
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\theta^{n+1} = \theta^n + \Delta t \frac{h^n}{h^{n+1}} |
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\left( G_\theta^{(n+1/2)} + P (\theta_{\mathrm{rain}} - \theta^n )/h^n \right) |
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%\theta^{n+1} = \theta^n + \frac{\Delta t}{h^{n+1}} |
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% \left( h^n G_\theta^{(n+1/2)} + P (\theta_{\mathrm{rain}} - \theta^n ) \right) |
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$$ |
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Note that with a simple forward time step (no Adams-Bashforth), |
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these two formulations are equivalent, |
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since |
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$ |
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(h^{n+1} - h^{n})/ \Delta t = |
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P - \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) = P + \dot{r}_{surf}^{n+1} |
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$ |
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|
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\subsubsection{Time stepping implementation of the |
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non-linear free-surface} |
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\label{sec:nonlin-freesurf-timestepping} |
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|
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The grid cell thickness was hold constant with the linear |
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free-surface~; with the non-linear free-surface, it is now varying |
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in time, at least at the surface level. |
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This implies some modifications of the general algorithm described |
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earlier in sections \ref{sec:adams-bashforth-sync} and |
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\ref{sec:adams-bashforth-staggered}. |
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|
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A simplified version of the staggered in time, non-linear |
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free-surface algorithm is detailed hereafter, and can be compared |
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to the equivalent linear free-surface case (eq. \ref{eq:Gv-n-staggered} |
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to \ref{eq:t-n+1-staggered}) and can also be easily transposed |
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to the synchronous time-stepping case. |
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Among the simplifications, salinity equation, implicit operator |
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and detailed elliptic equation are omitted. Surface forcing is |
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explicitly written as fluxes of temperature, fresh water and |
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momentum, $Q^{n+1/2}, P^{n+1/2}, F_{\bf v}^n$ respectively. |
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$h^n$ and $dh^n$ are the column and grid box thickness in r-coordinate. |
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%------------------------------------------------------------- |
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\begin{eqnarray} |
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\phi^{n}_{hyd} & = & \int b(\theta^{n},S^{n},r) dr |
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\label{eq:phi-hyd-nlfs} \\ |
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\vec{\bf G}_{\vec{\bf v}}^{n-1/2}\hspace{-2mm} & = & |
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\vec{\bf G}_{\vec{\bf v}} (dh^{n-1},\vec{\bf v}^{n-1/2}) |
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\hspace{+2mm};\hspace{+2mm} |
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\vec{\bf G}_{\vec{\bf v}}^{(n)} = |
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\frac{3}{2} \vec{\bf G}_{\vec{\bf v}}^{n-1/2} |
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- \frac{1}{2} \vec{\bf G}_{\vec{\bf v}}^{n-3/2} |
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\label{eq:Gv-n-nlfs} \\ |
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%\vec{\bf G}_{\vec{\bf v}}^{(n)} & = & |
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% \frac{3}{2} \vec{\bf G}_{\vec{\bf v}}^{n-1/2} |
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%- \frac{1}{2} \vec{\bf G}_{\vec{\bf v}}^{n-3/2} |
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%\label{eq:Gv-n+5-nlfs} \\ |
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%\vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \frac{\Delta t}{dh^{n}} \left( |
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%dh^{n-1}\vec{\bf G}_{\vec{\bf v}}^{(n)} + F_{\vec{\bf v}}^{n} \right) |
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\vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \frac{dh^{n-1}}{dh^{n}} \left( |
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\vec{\bf G}_{\vec{\bf v}}^{(n)} + F_{\vec{\bf v}}^{n}/dh^{n-1} \right) |
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- \Delta t \nabla \phi_{hyd}^{n} |
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\label{eq:vstar-nlfs} |
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\end{eqnarray} |
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\hspace{3cm}$\longrightarrow$~~{\it update model~geometry~:~}${\bf hFac}(dh^n)$\\ |
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\begin{eqnarray} |
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\eta^{n+1/2} \hspace{-2mm} & = & |
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\eta^{n-1/2} + \Delta t P^{n+1/2} - \Delta t |
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\nabla \cdot \int \vec{\bf v}^{n+1/2} dh^{n} \nonumber \\ |
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& = & \eta^{n-1/2} + \Delta t P^{n+1/2} - \Delta t |
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\nabla \cdot \int \!\!\! \left( \vec{\bf v}^* - g \Delta t \nabla \eta^{n+1/2} \right) dh^{n} |
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\label{eq:nstar-nlfs} \\ |
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\vec{\bf v}^{n+1/2}\hspace{-2mm} & = & |
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\vec{\bf v}^{*} - g \Delta t \nabla \eta^{n+1/2} |
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\label{eq:v-n+1-nlfs} \\ |
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h^{n+1} & = & h^{n} + \Delta t P^{n+1/2} - \Delta t |
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\nabla \cdot \int \vec{\bf v}^{n+1/2} dh^{n} |
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\label{eq:h-n+1-nlfs} \\ |
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G_{\theta}^{n} & = & G_{\theta} ( dh^{n}, u^{n+1/2}, \theta^{n} ) |
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\hspace{+2mm};\hspace{+2mm} |
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G_{\theta}^{(n+1/2)} = \frac{3}{2} G_{\theta}^{n} - \frac{1}{2} G_{\theta}^{n-1} |
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\label{eq:Gt-n-nlfs} \\ |
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%\theta^{n+1} & = &\theta^{n} + \frac{\Delta t}{dh^{n+1}} \left( dh^n |
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%G_{\theta}^{(n+1/2)} + Q^{n+1/2} + P^{n+1/2} (\theta_{\mathrm{rain}}-\theta^n) \right) |
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\theta^{n+1} & = &\theta^{n} + \Delta t \frac{dh^n}{dh^{n+1}} \left( |
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G_{\theta}^{(n+1/2)} |
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+( P^{n+1/2} (\theta_{\mathrm{rain}}-\theta^n) + Q^{n+1/2})/dh^n \right) |
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\nonumber \\ |
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& & \label{eq:t-n+1-nlfs} |
| 340 |
\end{eqnarray} |
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%------------------------------------------------------------- |
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Two steps have been added to linear free-surface algorithm |
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(eq. \ref{eq:Gv-n-staggered} to \ref{eq:t-n+1-staggered}): |
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Firstly, 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 $dh^{n}$ field. |
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Secondly, the vertically integrated continuity equation |
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(eq.\ref{eq:h-n+1-nlfs}) has been added ({\bf exactConserv}{\em =TRUE}, |
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in parameter file {\em data}, namelist {\em PARM01}) |
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just before computing the vertical velocity, in subroutine |
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{\em INTEGR\_CONTINUITY}. |
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%This ensures that tracer and continuity equation discretization a |
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Although this equation might appear redundant with eq.\ref{eq:nstar-nlfs}, |
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the integrated column thickness $h^{n+1}$ will be different from |
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$\eta^{n+1/2} + H$~ in the following cases: |
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\begin{itemize} |
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\item when Crank-Nickelson time-stepping is used (see section |
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\ref{sec:freesurf-CrankNick}). |
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\item when filters are applied to the flow field, after |
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(\ref{eq:v-n+1-nlfs}) and alter the divergence of the flow. |
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\item when the solver does not iterate until convergence~; |
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for example, because a too large residual target was set |
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({\bf cg2dTargetResidual}, parameter file {\em data}, namelist |
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{\em PARM02}). |
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\end{itemize}\noindent |
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In this staggered time-stepping algorithm, the momentum tendencies |
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are computed using $dh^{n-1}$ geometry factors. |
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(eq.\ref{eq:Gv-n-nlfs}) and then rescaled in subroutine {\it TIMESTEP}, |
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(eq.\ref{eq:vstar-nlfs}), similarly to tracer tendencies (see section |
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\ref{sec:freesurf-tracer-advection}). |
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The tracers are stepped forward later, using the recently updated |
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flow field ${\bf v}^{n+1/2}$ and the corresponding model geometry |
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$dh^{n}$ to compute the tendencies (eq.\ref{eq:Gt-n-nlfs}); |
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Then the tendencies are rescaled by $dh^n/dh^{n+1}$ to derive |
| 375 |
the new tracers values $(\theta,S)^{n+1}$ (eq.\ref{eq:t-n+1-nlfs}, |
| 376 |
in subroutine {\em CALC\_GT, CALC\_GS}). |
| 377 |
|
| 378 |
Note that the fresh-water input is added in a consistent way in the |
| 379 |
continuity equation and in the tracer equation, taking into account |
| 380 |
the fresh-water temperature $\theta_{\mathrm{rain}}$. |
| 381 |
|
| 382 |
Regarding the restart procedure, |
| 383 |
two 2.D fields $h^{n-1}$ and $(h^n-h^{n-1})/\Delta t$ |
| 384 |
in addition to the standard |
| 385 |
state variables and tendencies ($\eta^{n-1/2}$, ${\bf v}^{n-1/2}$, |
| 386 |
$\theta^n$, $S^n$, ${\bf G}_{\bf v}^{n-3/2}$, $G_{\theta,S}^{n-1}$) |
| 387 |
are stored in a "{\em pickup}" file. |
| 388 |
The model restarts reading this "{\em pickup}" file, |
| 389 |
then update the model geometry according to $h^{n-1}$, |
| 390 |
and compute $h^n$ and the vertical velocity |
| 391 |
%$h^n=h^{n-1} + \Delta t [(h^n-h^{n-1})/\Delta t]$, |
| 392 |
before starting the main calling sequence (eq.\ref{eq:phi-hyd-nlfs} |
| 393 |
to \ref{eq:t-n+1-nlfs}, {\em S/R FORWARD\_STEP}). |
| 394 |
\\ |
| 395 |
|
| 396 |
\fbox{ \begin{minipage}{4.75in} |
| 397 |
{\em INTEGR\_CONTINUITY} ({\em integr\_continuity.F}) |
| 398 |
|
| 399 |
$h^{n+1} -H_o$: {\bf etaH} ({\em DYNVARS.h}) |
| 400 |
|
| 401 |
$h^{n} -H_o$: {\bf etaHnm1} ({\em SURFACE.h}) |
| 402 |
|
| 403 |
$(h^{n+1}-h^{n})/\Delta t$: {\bf dEtaHdt} ({\em SURFACE.h}) |
| 404 |
|
| 405 |
\end{minipage} } |
| 406 |
|
| 407 |
\subsubsection{Non-linear free-surface and vertical resolution} |
| 408 |
\label{sec:nonlin-freesurf-dz_surf} |
| 409 |
|
| 410 |
When the amplitude of the free-surface variations becomes |
| 411 |
as large as the vertical resolution near the surface, |
| 412 |
the surface layer thickness can decrease to nearly zero or |
| 413 |
can even vanish completely. |
| 414 |
This later possibility has not been implemented, and a |
| 415 |
minimum relative thickness is imposed ({\bf hFacInf}, |
| 416 |
parameter file {\em data}, namelist {\em PARM01}) to prevent |
| 417 |
numerical instabilities caused by very thin surface level. |
| 418 |
|
| 419 |
A better alternative to the vanishing level problem has been |
| 420 |
found and implemented recently, and rely on a different |
| 421 |
vertical coordinate $r^*$~: |
| 422 |
The time variation ot the total column thickness becomes |
| 423 |
part of the r* coordinate motion, as in a $\sigma_{z},\sigma_{p}$ |
| 424 |
model, but the fixed part related to topography is treated |
| 425 |
as in a height or pressure coordinate model. |
| 426 |
A complete description is given in \cite{adcroft:04a}. |
| 427 |
|
| 428 |
The time-stepping implementation of the $r^*$ coordinate is |
| 429 |
identical to the non-linear free-surface in $r$ coordinate, |
| 430 |
and differences appear only in the spacial discretization. |
| 431 |
\marginpar{needs a subsection ref. here} |
| 432 |
|
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