/[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 1.17 by jmc, Thu Oct 14 19:53:04 2004 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    \begin{rawhtml}
128    <!-- CMIREDIR:linear_advection_schemes: -->
129    \end{rawhtml}
130    
131  \begin{figure}  \begin{figure}
132  \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 204  W & = & {\cal A}_c w
204    
205  For non-divergent flow, this discretization can be shown to conserve  For non-divergent flow, this discretization can be shown to conserve
206  the tracer both locally and globally and to globally conserve tracer  the tracer both locally and globally and to globally conserve tracer
207  variance, $\tau^2$. The proof is given in \cite{Adcroft95,Adcroft97}.  variance, $\tau^2$. The proof is given in \cite{adcroft:95,adcroft:97}.
208    
209  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
210  {\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 349  if the limiter is set to zero. Line 353  if the limiter is set to zero.
353    
354    
355  \section{Non-linear advection schemes}  \section{Non-linear advection schemes}
356    \begin{rawhtml}
357    <!-- CMIREDIR:non-linear_advection_schemes: -->
358    \end{rawhtml}
359    
360  Non-linear advection schemes invoke non-linear interpolation and are  Non-linear advection schemes invoke non-linear interpolation and are
361  widely used in computational fluid dynamics (non-linear does not refer  widely used in computational fluid dynamics (non-linear does not refer
# Line 387  r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t Line 394  r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t
394  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
395  \end{eqnarray}  \end{eqnarray}
396  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
397  only provide the Superbee limiter \cite{Roe85}:  only provide the Superbee limiter \cite{roe:85}:
398  \begin{equation}  \begin{equation}
399  \psi(r) = \max[0,\min[1,2r],\min[2,r]]  \psi(r) = \max[0,\min[1,2r],\min[2,r]]
400  \end{equation}  \end{equation}
# Line 449  to centered second order advection in th Line 456  to centered second order advection in th
456    
457  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
458  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
459  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
460  the accuracy increases with the Courant number! For low Courant  the accuracy increases with the Courant number! For low Courant
461  number, DST3 produces very similar results (indistinguishable in  number, DST3 produces very similar results (indistinguishable in
462  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 654  $W$: {\bf rTrans} (local) Line 661  $W$: {\bf rTrans} (local)
661    
662  \section{Comparison of advection schemes}  \section{Comparison of advection schemes}
663    
664    \begin{table}[htb]
665    \centering
666     \begin{tabular}[htb]{|l|c|c|c|c|l|}
667       \hline
668       Advection Scheme & code & use  & use Multi- & Stencil & comments \\
669                        &      & A.B. & dimension & (1 dim) & \\
670       \hline \hline
671       centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
672       \hline
673       $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/tracer\\
674       \hline
675       centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
676       \hline \hline
677    %  Lax-Wendroff       & 10 &  No & Yes & 3 pts & linear/tracer, non-linear/flow\\
678    %  \hline
679       $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/tracer, non-linear/flow\\
680       \hline \hline
681       $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
682       \hline
683       $3^{nd}$order DST Flux limiter & 33 &  No & Yes & 5 pts & non-linear \\
684       \hline
685     \end{tabular}
686     \caption{Summary of the different advection schemes available in MITgcm.
687              ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
688              The code corresponds to the number used to select the corresponding
689              advection scheme in the parameter file (e.g., {\em tempAdvScheme=3} in
690              file {\em data} selects the $3^{rd}$ order upwind advection scheme
691              for temperature).
692       }
693     \label{tab:advectionShemes_summary}
694    \end{table}
695    
696    
697  Figs.~\ref{fig:advect-2d-lo-diag}, \ref{fig:advect-2d-mid-diag} and  Figs.~\ref{fig:advect-2d-lo-diag}, \ref{fig:advect-2d-mid-diag} and
698  \ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal  \ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal
699  advection problem using a selection of schemes for low, moderate and  advection problem using a selection of schemes for low, moderate and
# Line 675  Figs.~\ref{fig:advect-1d-lo} and \ref{fi Line 715  Figs.~\ref{fig:advect-1d-lo} and \ref{fi
715  phenomenon.  phenomenon.
716    
717  Finally, the bottom left and right panels use the same advection  Finally, the bottom left and right panels use the same advection
718  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
719  Courant number this appears to not matter but for moderate Courant  Courant number this appears to not matter but for moderate Courant
720  number severe distortion of the feature is apparent. Moreover, the  number severe distortion of the feature is apparent. Moreover, the
721  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 744  flux limited scheme is almost essential.
744  non-linear schemes have the most stability (up to Courant number 1).  non-linear schemes have the most stability (up to Courant number 1).
745  \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
746  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.
747  \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
748  strongest argument for using a positive scheme.  strongest argument for using a positive scheme.
749  \end{itemize}  \end{itemize}
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