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revision 1.8 by cnh, Thu Oct 25 18:36:53 2001 UTC revision 1.18 by edhill, Sat Oct 16 03:40:13 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    \begin{rawhtml}
7    <!-- CMIREDIR:tracer_equations: -->
8    \end{rawhtml}
9    
10  The basic discretization used for the tracer equations is the second  The basic discretization used for the tracer equations is the second
11  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 18  part of the tracer equations and the var
18  described here.  described here.
19    
20  \subsection{Time-stepping of tracers: ABII}  \subsection{Time-stepping of tracers: ABII}
21  \label{sec:tracer_equations_abII}  \label{sect:tracer_equations_abII}
22    \begin{rawhtml}
23    <!-- CMIREDIR:tracer_equations_abII: -->
24    \end{rawhtml}
25    
26  The default advection scheme is the centered second order method which  The default advection scheme is the centered second order method which
27  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 49  only affects the surface layer since the
49  everywhere else. This term is therefore referred to as the surface  everywhere else. This term is therefore referred to as the surface
50  correction term. Global conservation is not possible using the  correction term. Global conservation is not possible using the
51  flux-form (as here) and a linearized free-surface  flux-form (as here) and a linearized free-surface
52  (\cite{Griffies00,Campin02}).  (\cite{griffies:00,campin:02}).
53    
54  The continuity equation can be recovered by setting  The continuity equation can be recovered by setting
55  $G_{diff}=G_{forc}=0$ and $\tau=1$.  $G_{diff}=G_{forc}=0$ and $\tau=1$.
# Line 123  the forward method. Line 129  the forward method.
129    
130    
131  \section{Linear advection schemes}  \section{Linear advection schemes}
132    \label{sect:tracer-advection}
133    \begin{rawhtml}
134    <!-- CMIREDIR:linear_advection_schemes: -->
135    \end{rawhtml}
136    
137  \begin{figure}  \begin{figure}
138  \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 210  W & = & {\cal A}_c w
210    
211  For non-divergent flow, this discretization can be shown to conserve  For non-divergent flow, this discretization can be shown to conserve
212  the tracer both locally and globally and to globally conserve tracer  the tracer both locally and globally and to globally conserve tracer
213  variance, $\tau^2$. The proof is given in \cite{Adcroft95,Adcroft97}.  variance, $\tau^2$. The proof is given in \cite{adcroft:95,adcroft:97}.
214    
215  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
216  {\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 359  if the limiter is set to zero.
359    
360    
361  \section{Non-linear advection schemes}  \section{Non-linear advection schemes}
362    \begin{rawhtml}
363    <!-- CMIREDIR:non-linear_advection_schemes: -->
364    \end{rawhtml}
365    
366  Non-linear advection schemes invoke non-linear interpolation and are  Non-linear advection schemes invoke non-linear interpolation and are
367  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 400  r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t
400  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
401  \end{eqnarray}  \end{eqnarray}
402  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
403  only provide the Superbee limiter \cite{Roe85}:  only provide the Superbee limiter \cite{roe:85}:
404  \begin{equation}  \begin{equation}
405  \psi(r) = \max[0,\min[1,2r],\min[2,r]]  \psi(r) = \max[0,\min[1,2r],\min[2,r]]
406  \end{equation}  \end{equation}
# Line 449  to centered second order advection in th Line 462  to centered second order advection in th
462    
463  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
464  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
465  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
466  the accuracy increases with the Courant number! For low Courant  the accuracy increases with the Courant number! For low Courant
467  number, DST3 produces very similar results (indistinguishable in  number, DST3 produces very similar results (indistinguishable in
468  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 653  $W$: {\bf rTrans} (local) Line 666  $W$: {\bf rTrans} (local)
666    
667    
668  \section{Comparison of advection schemes}  \section{Comparison of advection schemes}
669    \begin{rawhtml}
670    <!-- CMIREDIR:comparison_of_advection_schemes: -->
671    \end{rawhtml}
672    
673    \begin{table}[htb]
674    \centering
675     \begin{tabular}[htb]{|l|c|c|c|c|l|}
676       \hline
677       Advection Scheme & code & use  & use Multi- & Stencil & comments \\
678                        &      & A.B. & dimension & (1 dim) & \\
679       \hline \hline
680       centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
681       \hline
682       $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/tracer\\
683       \hline
684       centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
685       \hline \hline
686    %  Lax-Wendroff       & 10 &  No & Yes & 3 pts & linear/tracer, non-linear/flow\\
687    %  \hline
688       $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/tracer, non-linear/flow\\
689       \hline \hline
690       $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
691       \hline
692       $3^{nd}$order DST Flux limiter & 33 &  No & Yes & 5 pts & non-linear \\
693       \hline
694     \end{tabular}
695     \caption{Summary of the different advection schemes available in MITgcm.
696              ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
697              The code corresponds to the number used to select the corresponding
698              advection scheme in the parameter file (e.g., {\em tempAdvScheme=3} in
699              file {\em data} selects the $3^{rd}$ order upwind advection scheme
700              for temperature).
701       }
702     \label{tab:advectionShemes_summary}
703    \end{table}
704    
705    
706  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
707  \ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal  \ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal
# Line 675  Figs.~\ref{fig:advect-1d-lo} and \ref{fi Line 724  Figs.~\ref{fig:advect-1d-lo} and \ref{fi
724  phenomenon.  phenomenon.
725    
726  Finally, the bottom left and right panels use the same advection  Finally, the bottom left and right panels use the same advection
727  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
728  Courant number this appears to not matter but for moderate Courant  Courant number this appears to not matter but for moderate Courant
729  number severe distortion of the feature is apparent. Moreover, the  number severe distortion of the feature is apparent. Moreover, the
730  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 753  flux limited scheme is almost essential.
753  non-linear schemes have the most stability (up to Courant number 1).  non-linear schemes have the most stability (up to Courant number 1).
754  \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
755  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.
756  \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
757  strongest argument for using a positive scheme.  strongest argument for using a positive scheme.
758  \end{itemize}  \end{itemize}
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