/[MITgcm]/manual/s_algorithm/text/nonlin_frsurf.tex
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revision 1.13 by edhill, Tue Jun 27 22:31:23 2006 UTC revision 1.14 by jmc, Mon Aug 30 23:09:18 2010 UTC
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6  \subsection{Non-linear free-surface}  \subsection{Non-linear free-surface}
7  \label{sect:nonlinear-freesurface}  \label{sec:nonlinear-freesurface}
8    
9  Recently, new options have been added to the model  Recently, new options have been added to the model
10  that concern the free surface formulation.  that concern the free surface formulation.
11    
12    
13  \subsubsection{pressure/geo-potential and free surface}  \subsubsection{pressure/geo-potential and free surface}
14  \label{sect:phi-freesurface}  \label{sec:phi-freesurface}
15    
16  For the atmosphere, since $\phi = \phi_{topo} - \int^p_{p_s} \alpha dp$,  For the atmosphere, since $\phi = \phi_{topo} - \int^p_{p_s} \alpha dp$,
17  subtracting the reference state defined in section  subtracting the reference state defined in section
# Line 104  This approximation is dropped when using Line 104  This approximation is dropped when using
104  formulation and the total thickness, including the time varying part  formulation and the total thickness, including the time varying part
105  $\eta$, is considered when computing horizontal transports.  $\eta$, is considered when computing horizontal transports.
106  Implications for the barotropic part are presented hereafter.  Implications for the barotropic part are presented hereafter.
107  In section \ref{sect:freesurf-tracer-advection} consequences for  In section \ref{sec:freesurf-tracer-advection} consequences for
108  tracer conservation is briefly discussed (more details can be  tracer conservation is briefly discussed (more details can be
109  found in \cite{campin:02})~; the general time-stepping is presented  found in \cite{campin:02})~; the general time-stepping is presented
110  in section \ref{sect:nonlin-freesurf-timestepping} with some  in section \ref{sec:nonlin-freesurf-timestepping} with some
111  limitations regarding the vertical resolution in section  limitations regarding the vertical resolution in section
112  \ref{sect:nonlin-freesurf-dz_surf}.  \ref{sec:nonlin-freesurf-dz_surf}.
113    
114  In the non-linear formulation, the continuous form of the model  In the non-linear formulation, the continuous form of the model
115  equations remains unchanged, except for the 2D continuity equation  equations remains unchanged, except for the 2D continuity equation
# Line 205  computation cost involved in the solver Line 205  computation cost involved in the solver
205    
206    
207  \subsubsection{Tracer conservation with non-linear free-surface}  \subsubsection{Tracer conservation with non-linear free-surface}
208  \label{sect:freesurf-tracer-advection}  \label{sec:freesurf-tracer-advection}
209    
210  To ensure global tracer conservation (i.e., the total amount) as well  To ensure global tracer conservation (i.e., the total amount) as well
211  as local conservation, the change in the surface level thickness must  as local conservation, the change in the surface level thickness must
# Line 225  $$ Line 225  $$
225    = P \theta_{\mathrm{rain}}    = P \theta_{\mathrm{rain}}
226  $$  $$
227  Using the implicit (non-linear) free surface described above (section  Using the implicit (non-linear) free surface described above (section
228  \ref{sect:pressure-method-linear-backward}) we have:  \ref{sec:pressure-method-linear-backward}) we have:
229  \begin{eqnarray*}  \begin{eqnarray*}
230  h^{n+1} = h^{n} - \Delta t \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) + \Delta t P \\  h^{n+1} = h^{n} - \Delta t \nabla \cdot (h^n \, \vec{\bf v}^{n+1} ) + \Delta t P \\
231  \end{eqnarray*}  \end{eqnarray*}
# Line 271  $ Line 271  $
271    
272  \subsubsection{Time stepping implementation of the  \subsubsection{Time stepping implementation of the
273  non-linear free-surface}      non-linear free-surface}    
274  \label{sect:nonlin-freesurf-timestepping}  \label{sec:nonlin-freesurf-timestepping}
275    
276  The grid cell thickness was hold constant with the linear  The grid cell thickness was hold constant with the linear
277  free-surface~; with the non-linear free-surface, it is now varying  free-surface~; with the non-linear free-surface, it is now varying
278  in time, at least at the surface level.  in time, at least at the surface level.
279  This implies some modifications of the general algorithm described  This implies some modifications of the general algorithm described
280  earlier in sections \ref{sect:adams-bashforth-sync} and  earlier in sections \ref{sec:adams-bashforth-sync} and
281  \ref{sect:adams-bashforth-staggered}.  \ref{sec:adams-bashforth-staggered}.
282    
283  A simplified version of the staggered in time, non-linear  A simplified version of the staggered in time, non-linear
284  free-surface algorithm is detailed hereafter, and can be compared  free-surface algorithm is detailed hereafter, and can be compared
# Line 354  redundant with eq.\ref{eq:nstar-nlfs}, t Line 354  redundant with eq.\ref{eq:nstar-nlfs}, t
354  thickness $h^{n+1}$ can be different from $\eta^{n+1/2} + H$~:  thickness $h^{n+1}$ can be different from $\eta^{n+1/2} + H$~:
355  \begin{itemize}  \begin{itemize}
356  \item when Crank-Nickelson time-stepping is used (see section  \item when Crank-Nickelson time-stepping is used (see section
357  \ref{sect:freesurf-CrankNick}).  \ref{sec:freesurf-CrankNick}).
358  \item when filters are applied to the flow field, after  \item when filters are applied to the flow field, after
359  (\ref{eq:v-n+1-nlfs}) and alter the divergence of the flow.  (\ref{eq:v-n+1-nlfs}) and alter the divergence of the flow.
360  \item when the solver does not iterate until convergence~;  \item when the solver does not iterate until convergence~;
# Line 366  In this staggered time-stepping algorith Line 366  In this staggered time-stepping algorith
366  are computed using $dh^{n-1}$ geometry factors.  are computed using $dh^{n-1}$ geometry factors.
367  (eq.\ref{eq:Gv-n-nlfs}) and then rescaled in subroutine {\it TIMESTEP},  (eq.\ref{eq:Gv-n-nlfs}) and then rescaled in subroutine {\it TIMESTEP},
368  (eq.\ref{eq:vstar-nlfs}), similarly to tracer tendencies (see section  (eq.\ref{eq:vstar-nlfs}), similarly to tracer tendencies (see section
369  \ref{sect:freesurf-tracer-advection}).  \ref{sec:freesurf-tracer-advection}).
370  The tracers are stepped forward later, using the recently updated  The tracers are stepped forward later, using the recently updated
371  flow field ${\bf v}^{n+1/2}$ and the corresponding model geometry  flow field ${\bf v}^{n+1/2}$ and the corresponding model geometry
372  $dh^{n}$ to compute the tendencies (eq.\ref{eq:Gt-n-nlfs});  $dh^{n}$ to compute the tendencies (eq.\ref{eq:Gt-n-nlfs});
# Line 404  $h^{n+1}-h^{n}/\Delta t$: {\bf dEtaHdt} Line 404  $h^{n+1}-h^{n}/\Delta t$: {\bf dEtaHdt}
404  \end{minipage} }  \end{minipage} }
405    
406  \subsubsection{Non-linear free-surface and vertical resolution}  \subsubsection{Non-linear free-surface and vertical resolution}
407  \label{sect:nonlin-freesurf-dz_surf}  \label{sec:nonlin-freesurf-dz_surf}
408    
409  When the amplitude of the free-surface variations becomes  When the amplitude of the free-surface variations becomes
410  as large as the vertical resolution near the surface,  as large as the vertical resolution near the surface,
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