/[MITgcm]/manual/s_algorithm/text/tracer.tex
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revision 1.8 by cnh, Thu Oct 25 18:36:53 2001 UTC revision 1.13 by adcroft, Mon May 6 19:19:31 2002 UTC
# Line 2  Line 2 
2  % $Name$  % $Name$
3    
4  \section{Tracer equations}  \section{Tracer equations}
5  \label{sec:tracer_equations}  \label{sect:tracer_equations}
6    
7  The basic discretization used for the tracer equations is the second  The basic discretization used for the tracer equations is the second
8  order piece-wise constant finite volume form of the forced  order piece-wise constant finite volume form of the forced
# Line 15  part of the tracer equations and the var Line 15  part of the tracer equations and the var
15  described here.  described here.
16    
17  \subsection{Time-stepping of tracers: ABII}  \subsection{Time-stepping of tracers: ABII}
18  \label{sec:tracer_equations_abII}  \label{sect:tracer_equations_abII}
19    
20  The default advection scheme is the centered second order method which  The default advection scheme is the centered second order method which
21  requires a second order or quasi-second order time-stepping scheme to  requires a second order or quasi-second order time-stepping scheme to
# Line 43  only affects the surface layer since the Line 43  only affects the surface layer since the
43  everywhere else. This term is therefore referred to as the surface  everywhere else. This term is therefore referred to as the surface
44  correction term. Global conservation is not possible using the  correction term. Global conservation is not possible using the
45  flux-form (as here) and a linearized free-surface  flux-form (as here) and a linearized free-surface
46  (\cite{Griffies00,Campin02}).  (\cite{griffies:00,campin:02}).
47    
48  The continuity equation can be recovered by setting  The continuity equation can be recovered by setting
49  $G_{diff}=G_{forc}=0$ and $\tau=1$.  $G_{diff}=G_{forc}=0$ and $\tau=1$.
# Line 123  the forward method. Line 123  the forward method.
123    
124    
125  \section{Linear advection schemes}  \section{Linear advection schemes}
126    \label{sect:tracer-advection}
127    
128  \begin{figure}  \begin{figure}
129  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}
# Line 200  W & = & {\cal A}_c w Line 201  W & = & {\cal A}_c w
201    
202  For non-divergent flow, this discretization can be shown to conserve  For non-divergent flow, this discretization can be shown to conserve
203  the tracer both locally and globally and to globally conserve tracer  the tracer both locally and globally and to globally conserve tracer
204  variance, $\tau^2$. The proof is given in \cite{Adcroft95,Adcroft97}.  variance, $\tau^2$. The proof is given in \cite{adcroft:95,adcroft:97}.
205    
206  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
207  {\em S/R GAD\_C2\_ADV\_X} ({\em gad\_c2\_adv\_x.F})  {\em S/R GAD\_C2\_ADV\_X} ({\em gad\_c2\_adv\_x.F})
# Line 387  r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t Line 388  r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t
388  r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} } & \forall & u < 0  r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} } & \forall & u < 0
389  \end{eqnarray}  \end{eqnarray}
390  as it's argument. There are many choices of limiter function but we  as it's argument. There are many choices of limiter function but we
391  only provide the Superbee limiter \cite{Roe85}:  only provide the Superbee limiter \cite{roe:85}:
392  \begin{equation}  \begin{equation}
393  \psi(r) = \max[0,\min[1,2r],\min[2,r]]  \psi(r) = \max[0,\min[1,2r],\min[2,r]]
394  \end{equation}  \end{equation}
# Line 449  to centered second order advection in th Line 450  to centered second order advection in th
450    
451  The DST3 method described above must be used in a forward-in-time  The DST3 method described above must be used in a forward-in-time
452  manner and is stable for $0 \le |c| \le 1$. Although the scheme  manner and is stable for $0 \le |c| \le 1$. Although the scheme
453  appears to be forward-in-time, it is in fact second order in time and  appears to be forward-in-time, it is in fact third order in time and
454  the accuracy increases with the Courant number! For low Courant  the accuracy increases with the Courant number! For low Courant
455  number, DST3 produces very similar results (indistinguishable in  number, DST3 produces very similar results (indistinguishable in
456  Fig.~\ref{fig:advect-1d-lo}) to the linear third order method but for  Fig.~\ref{fig:advect-1d-lo}) to the linear third order method but for
# Line 675  Figs.~\ref{fig:advect-1d-lo} and \ref{fi Line 676  Figs.~\ref{fig:advect-1d-lo} and \ref{fi
676  phenomenon.  phenomenon.
677    
678  Finally, the bottom left and right panels use the same advection  Finally, the bottom left and right panels use the same advection
679  scheme but the right does not use the mutli-dimensional method. At low  scheme but the right does not use the multi-dimensional method. At low
680  Courant number this appears to not matter but for moderate Courant  Courant number this appears to not matter but for moderate Courant
681  number severe distortion of the feature is apparent. Moreover, the  number severe distortion of the feature is apparent. Moreover, the
682  stability of the multi-dimensional scheme is determined by the maximum  stability of the multi-dimensional scheme is determined by the maximum
# Line 704  flux limited scheme is almost essential. Line 705  flux limited scheme is almost essential.
705  non-linear schemes have the most stability (up to Courant number 1).  non-linear schemes have the most stability (up to Courant number 1).
706  \item If you need to know how much diffusion/dissipation has occurred you  \item If you need to know how much diffusion/dissipation has occurred you
707  will have a lot of trouble figuring it out with a non-linear method.  will have a lot of trouble figuring it out with a non-linear method.
708  \item The presence of false extrema is unphysical and this alone is the  \item The presence of false extrema is non-physical and this alone is the
709  strongest argument for using a positive scheme.  strongest argument for using a positive scheme.
710  \end{itemize}  \end{itemize}
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