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% $Header: /u/gcmpack/mitgcmdoc/part2/nonlin_frsurf.tex,v 1.3 2001/09/24 19:30:40 jmc Exp $ |
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% $Name: $ |
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\subsection{Non-linear free surface} |
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Recently, two options have been added to the model (and have not yet |
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been extensively tested) that concern the free surface formulation. |
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\subsubsection{Non-uniform linear-relation for the surface potential} |
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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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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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For the atmosphere, however, because of topographic effects, the |
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reference surface pressure $R_o$ has large spacial 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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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 the linear free surface approximation |
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(detailed before), only the fixed part of it ($R_o - R_{fixed})$ is |
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considered when we integrate the continuity equation or compute tracer |
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and momentum advection term. |
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|
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This approximation is dropped when using the non-linear free surface |
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formulation. Here we discuss sections the barotropic part. In |
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sections \ref{sect:freesurf-tracer-advection} and |
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\ref{sect:freesurf-momentum-advection} we consider the baroclinic |
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component. |
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|
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The continuous form of the model equations remains unchanged, except |
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for the 2D continuity equation (\ref{eq-tCsC-eta}) 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 have been tested. |
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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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|
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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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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:??}, 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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For Adams-Bashforth time-stepping, we implement this scheme slightly |
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differently from the linear free-surface method, using two steps: the |
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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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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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\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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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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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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