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% $Header: /u/gcmpack/manual/part2/nonlin_frsurf.tex,v 1.7 2001/11/13 20:27:54 adcroft Exp $ |
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% $Name: $ |
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\subsection{Non-linear free surface} |
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\label{sect: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{Non-uniform linear-relation for the surface potential} |
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|
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The linear relation between surface pressure/geo-potential |
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($\Phi_{surf}$) and surface displacement ($\eta$) could be considered |
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to be a constant ($b_s=$ constant) |
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\marginpar{add a reference to part.1 here} |
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but is in fact dependent on the position ($x,y,r$) |
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since we linearize: |
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$$\Phi_{surf}=\int_{R_o}^{R_o+\eta} b dr \simeq b_s \eta |
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~\mathrm{with}~ b_s = b(\theta,S,r)_{r=R_o} |
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\simeq b_s(\theta_{ref}(R_o),S_{ref}(R_o),R_o)$$ |
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Note that, for convenience, the effect on $b_s$ of the local surface |
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$\theta,S$ are not considered here, but are incorporated in to |
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$\Phi'_{hyd}$. |
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|
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For the ocean, $b_s = g \rho_{surf} / \rho_o \simeq g$ is a very good |
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approximation since the relative difference in surface density are |
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usually small and only due to local $\theta,S$ gradients (because the |
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upper surface, $R_o = 0$, is essentially flat). Therefore, they can |
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easily be incorporated in $\Phi'_{hyd}$. |
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|
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For the atmosphere, however, because of topographic effects, the |
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reference surface pressure $R_o$ 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, we use a non-uniform linear coefficient |
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$b_s$. |
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|
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In practice, in an oceanic configuration or when the default value |
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(TRUE) of the parameter {\bf uniformLin\_PhiSurf} is used, then $b_s$ |
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is simply set to $g$ for the ocean and $1.$ for the atmosphere. |
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Turning {\bf uniformLin\_PhiSurf} to "FALSE", tells the code to |
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evaluate $b_s$ from the reference vertical profile $\theta_{ref}$ |
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({\it S/R INI\_LINEAR\_PHISURF}) according to the reference surface |
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pressure $P_o$ ($=R_o$): $b_s = c_p \kappa (P_o / Pc)^{(\kappa - 1)} |
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\theta_{ref}(P_o)$ |
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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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$\eta << H_o = R_o - R_{fixed}$. |
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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 the horizontal transport is |
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computed, 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 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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\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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is available in \cite{campin:02}. |
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|
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|
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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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|
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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 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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|
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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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|
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|
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\subsubsection{Free surface effect on the surface level thickness |
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(Non-linear free surface): Tracer advection} |
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\label{sect: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 uniform |
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Cartesian horizontal grid: |
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$$ |
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\partial_t h + \nabla \cdot h \vec{\bf v} = 0 |
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$$ |
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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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$$ |
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\partial_t (h \theta) + \nabla \cdot ( h \theta \vec{\bf v})= 0 |
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$$ |
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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*} |
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h^{n+1} = h^{n} - \Delta_t \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) \\ |
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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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\end{eqnarray*} |
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|
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The use of a 3 time-levels timestepping scheme such as the Adams-Bashforth |
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make the conservation less straitforward. |
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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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$$ |
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\theta^{n+1} = \theta^n + \Delta_t \frac{h^n}{h^{n+1}} G_\theta^{(n+1/2)} |
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$$ |
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Note that with a simple forward time step (no Adams-Bashforth), |
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since |
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$ |
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\dot{r}_{surf}^{n+1} |
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= - \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) = (h^{n+1} - h^{n}) |
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/ \Delta_t |
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$ |
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these two formulations are equivalent. |
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|
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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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|
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|
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\subsubsection{Free surface effect on the surface level thickness |
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(Non-linear free surface): Momentum advection} |
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\label{sect:freesurf-momentum-advection} |
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|
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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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|
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Regarding momentum advection, |
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the vector invariant formulation is similar to the |
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advective form (as opposed to the flux form) and therefore |
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does not need specific modification to include the |
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free surface effect on the surface level thickness. |
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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. |
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|
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\subsubsection{Non-linear free surface and vertical resolution} |
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\label{sect:nonlin-freesurf-dz_surf} |
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|
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When the amplitude of the free-surface variations becomes |
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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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|
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A better atlternative to the vanishing level problem has been |
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found and implemented recently, and rely on a different |
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vertical coordinate $r^*$~: |
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The time variation ot the total column thickness becomes |
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part of the r* coordinate motion, as in a $\sigma_{z},\sigma_{p}$ |
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model, but the fixed part related to topography is treated |
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as in a height or pressure coordinate model. |
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A complete description is given in \cite{adcroft:04}. |
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|
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The time-stepping implementation of the $r^*$ coordinate is |
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identical to the non-linear free-surface in $r$ coordinate, |
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and differences appear only in the spacial discretisation. |
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\marginpar{needs a subsection ref. here} |
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