/[MITgcm]/manual/s_algorithm/text/tracer.tex
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revision 1.22 by jmc, Wed Jun 28 17:01:34 2006 UTC revision 1.27 by heimbach, Wed Oct 26 17:30:23 2016 UTC
# Line 2  Line 2 
2  % $Name$  % $Name$
3    
4  \section{Tracer equations}  \section{Tracer equations}
5  \label{sect:tracer_equations}  \label{sec:tracer_equations}
6  \begin{rawhtml}  \begin{rawhtml}
7  <!-- CMIREDIR:tracer_equations: -->  <!-- CMIREDIR:tracer_equations: -->
8  \end{rawhtml}  \end{rawhtml}
# Line 18  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{sect:tracer_equations_abII}  \label{sec:tracer_equations_abII}
22  \begin{rawhtml}  \begin{rawhtml}
23  <!-- CMIREDIR:tracer_equations_abII: -->  <!-- CMIREDIR:tracer_equations_abII: -->
24  \end{rawhtml}  \end{rawhtml}
# Line 129  the forward method. Line 129  the forward method.
129    
130    
131  \section{Linear advection schemes}  \section{Linear advection schemes}
132  \label{sect:tracer-advection}  \label{sec:tracer-advection}
133  \begin{rawhtml}  \begin{rawhtml}
134  <!-- CMIREDIR:linear_advection_schemes: -->  <!-- CMIREDIR:linear_advection_schemes: -->
135  \end{rawhtml}  \end{rawhtml}
136    
137  \begin{figure}  \begin{figure}
138  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-lo.eps}}
139  \caption{  \caption{
140  Comparison of 1-D advection schemes. Courant number is 0.05 with 60  Comparison of 1-D advection schemes. Courant number is 0.05 with 60
141  points and solutions are shown for T=1 (one complete period).  points and solutions are shown for T=1 (one complete period).
# Line 153  $\mu=c/(1-c)$. Line 153  $\mu=c/(1-c)$.
153  \end{figure}  \end{figure}
154    
155  \begin{figure}  \begin{figure}
156  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-hi.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-hi.eps}}
157  \caption{  \caption{
158  Comparison of 1-D advection schemes. Courant number is 0.89 with 60  Comparison of 1-D advection schemes. Courant number is 0.89 with 60
159  points and solutions are shown for T=1 (one complete period).  points and solutions are shown for T=1 (one complete period).
# Line 359  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    \label{sec:non-linear_advection_schemes}
363  \begin{rawhtml}  \begin{rawhtml}
364  <!-- CMIREDIR:non-linear_advection_schemes: -->  <!-- CMIREDIR:non-linear_advection_schemes: -->
365  \end{rawhtml}  \end{rawhtml}
# Line 549  $\tau$: {\bf tracer} (argument) Line 550  $\tau$: {\bf tracer} (argument)
550  \subsection{Multi-dimensional advection}  \subsection{Multi-dimensional advection}
551    
552  \begin{figure}  \begin{figure}
553  \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-lo-diag.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-lo-diag.eps}}
554  \caption{  \caption{
555  Comparison of advection schemes in two dimensions; diagonal advection  Comparison of advection schemes in two dimensions; diagonal advection
556  of a resolved Gaussian feature. Courant number is 0.01 with  of a resolved Gaussian feature. Courant number is 0.01 with
# Line 570  lines). Line 571  lines).
571  \end{figure}  \end{figure}
572    
573  \begin{figure}  \begin{figure}
574  \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-mid-diag.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-mid-diag.eps}}
575  \caption{  \caption{
576  Comparison of advection schemes in two dimensions; diagonal advection  Comparison of advection schemes in two dimensions; diagonal advection
577  of a resolved Gaussian feature. Courant number is 0.27 with  of a resolved Gaussian feature. Courant number is 0.27 with
# Line 591  lines). Line 592  lines).
592  \end{figure}  \end{figure}
593    
594  \begin{figure}  \begin{figure}
595  \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-hi-diag.eps}}  \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-hi-diag.eps}}
596  \caption{  \caption{
597  Comparison of advection schemes in two dimensions; diagonal advection  Comparison of advection schemes in two dimensions; diagonal advection
598  of a resolved Gaussian feature. Courant number is 0.47 with  of a resolved Gaussian feature. Courant number is 0.47 with
# Line 664  $W$: {\bf rTrans} (local) Line 665  $W$: {\bf rTrans} (local)
665    
666  \end{minipage} }  \end{minipage} }
667    
668    \begin{figure}
669    \resizebox{3.5in}{!}{\includegraphics{s_algorithm/figs/multiDim_CS.eps}}
670    \caption{Muti-dimensional advection time-stepping with Cubed-Sphere topology
671    \label{fig:advect-multidim_cs}
672    }
673    \end{figure}
674    
675  \section{Comparison of advection schemes}  \section{Comparison of advection schemes}
676  \label{sect:tracer_advection_schemes}  \label{sec:tracer_advection_schemes}
677  \begin{rawhtml}  \begin{rawhtml}
678  <!-- CMIREDIR:comparison_of_advection_schemes: -->  <!-- CMIREDIR:comparison_of_advection_schemes: -->
679  \end{rawhtml}  \end{rawhtml}
680    
681  \begin{table}[htb]  \begin{table}[htb]
682  \centering  \centering
683    {\small
684   \begin{tabular}[htb]{|l|c|c|c|c|l|}   \begin{tabular}[htb]{|l|c|c|c|c|l|}
685     \hline     \hline
686     Advection Scheme & code & use  & use Multi- & Stencil & comments \\     Advection Scheme & code & use  & use Multi- & Stencil & comments \\
# Line 690  $W$: {\bf rTrans} (local) Line 698  $W$: {\bf rTrans} (local)
698                           No & Yes & 3 pts & linear/$\tau$, non-linear/v\\                           No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
699     \hline     \hline
700     $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\     $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
701       \hline
702       $2^{nd}$order-moment Prather & 80 & No & Yes & ~ & ~ \\
703     \hline \hline     \hline \hline
704     $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\     $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
705     \hline     \hline
706     $3^{nd}$order DST Flux limiter & 33 &  No & Yes & 5 pts & non-linear \\     $3^{nd}$order DST Flux limiter & 33 &  No & Yes & 5 pts & non-linear \\
707     \hline     \hline
708       $2^{nd}$order-moment Prather w. limiter & 81 & No & Yes & ~ & ~ \\
709       \hline
710       piecewise parabolic w. ``null'' limiter & 40 & No & Yes & ~ & ~ \\
711       \hline
712       piecewise parabolic w. ``mono'' limiter & 41 & No & Yes & ~ & ~ \\
713       \hline
714       piecewise quartic w. ``null'' limiter & 50 & No & Yes & ~ & ~ \\
715       \hline
716       piecewise quartic w. ``mono'' limiter & 51 & No & Yes & ~ & ~ \\
717       \hline
718       piecewise quartic w. ``weno'' limiter & 52 & No & Yes & ~ & ~ \\
719       \hline
720       $7^{nd}$order one-step method & 7 & No & Yes & ~ & ~ \\
721       with Monotonicity Preserving Limiter & ~ & ~ & ~ & ~ & ~ \\
722       \hline
723      
724   \end{tabular}   \end{tabular}
725     }
726   \caption{Summary of the different advection schemes available in MITgcm.   \caption{Summary of the different advection schemes available in MITgcm.
727            ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.            ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
728            The code corresponds to the number used to select the corresponding            The code corresponds to the number used to select the corresponding
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