/[MITgcm]/manual/s_algorithm/text/tracer.tex
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revision 1.17 by jmc, Thu Oct 14 19:53:04 2004 UTC revision 1.22 by jmc, Wed Jun 28 17:01:34 2006 UTC
# Line 3  Line 3 
3    
4  \section{Tracer equations}  \section{Tracer equations}
5  \label{sect: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 16  described here. Line 19  described here.
19    
20  \subsection{Time-stepping of tracers: ABII}  \subsection{Time-stepping of tracers: ABII}
21  \label{sect: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 621  as if in one dimension: Line 627  as if in one dimension:
627  \tau^{n+1/3} & = & \tau^{n}  \tau^{n+1/3} & = & \tau^{n}
628  - \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n})  - \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n})
629             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\
630  \tau^{n+2/3} & = & \tau^{n}  \tau^{n+2/3} & = & \tau^{n+1/3}
631  - \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3})  - \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3})
632             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\
633  \tau^{n+3/3} & = & \tau^{n}  \tau^{n+3/3} & = & \tau^{n+2/3}
634  - \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3})  - \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3})
635             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)
636  \end{eqnarray}  \end{eqnarray}
# Line 660  $W$: {\bf rTrans} (local) Line 666  $W$: {\bf rTrans} (local)
666    
667    
668  \section{Comparison of advection schemes}  \section{Comparison of advection schemes}
669    \label{sect:tracer_advection_schemes}
670    \begin{rawhtml}
671    <!-- CMIREDIR:comparison_of_advection_schemes: -->
672    \end{rawhtml}
673    
674  \begin{table}[htb]  \begin{table}[htb]
675  \centering  \centering
# Line 668  $W$: {\bf rTrans} (local) Line 678  $W$: {\bf rTrans} (local)
678     Advection Scheme & code & use  & use Multi- & Stencil & comments \\     Advection Scheme & code & use  & use Multi- & Stencil & comments \\
679                      &      & A.B. & dimension & (1 dim) & \\                      &      & A.B. & dimension & (1 dim) & \\
680     \hline \hline     \hline \hline
681       $1^{rst}$order upwind  & 1 &  No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
682       \hline
683     centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\     centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
684     \hline     \hline
685     $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/tracer\\     $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/$\tau$\\
686     \hline     \hline
687     centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\     centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
688     \hline \hline     \hline \hline
689  %  Lax-Wendroff       & 10 &  No & Yes & 3 pts & linear/tracer, non-linear/flow\\     $2^{nd}$order DST (Lax-Wendroff)  & 20 &
690  %  \hline                           No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
691     $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/tracer, non-linear/flow\\     \hline
692       $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
693     \hline \hline     \hline \hline
694     $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\     $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
695     \hline     \hline
# Line 686  $W$: {\bf rTrans} (local) Line 699  $W$: {\bf rTrans} (local)
699   \caption{Summary of the different advection schemes available in MITgcm.   \caption{Summary of the different advection schemes available in MITgcm.
700            ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.            ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
701            The code corresponds to the number used to select the corresponding            The code corresponds to the number used to select the corresponding
702            advection scheme in the parameter file (e.g., {\em tempAdvScheme=3} in            advection scheme in the parameter file (e.g., {\bf tempAdvScheme}=3 in
703            file {\em data} selects the $3^{rd}$ order upwind advection scheme            file {\em data} selects the $3^{rd}$ order upwind advection scheme
704            for temperature).            for temperature).
705     }     }
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