/[MITgcm]/manual/s_algorithm/text/tracer.tex

Diff of /manual/s_algorithm/text/tracer.tex

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-revision 1.17 by jmc,
Thu Oct 14 19:53:04 2004 UTC
+revision 1.27 by heimbach,
Wed Oct 26 17:30:23 2016 UTC
 Line 2
  % $Name$
  \section{Tracer equations}
- \label{sect:tracer_equations}
+ \label{sec:tracer_equations}
+ \begin{rawhtml}
+ <!-- CMIREDIR:tracer_equations: -->
+ \end{rawhtml}
  The basic discretization used for the tracer equations is the second
  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
  described here.
  \subsection{Time-stepping of tracers: ABII}
- \label{sect:tracer_equations_abII}
+ \label{sec:tracer_equations_abII}
+ \begin{rawhtml}
+ <!-- CMIREDIR:tracer_equations_abII: -->
+ \end{rawhtml}
  The default advection scheme is the centered second order method which
  requires a second order or quasi-second order time-stepping scheme to
-Line 123 
 the forward method.
+Line 129 
 the forward method.
  \section{Linear advection schemes}
- \label{sect:tracer-advection}
+ \label{sec:tracer-advection}
  \begin{rawhtml}
  <!-- CMIREDIR:linear_advection_schemes: -->
  \end{rawhtml}
  \begin{figure}
- \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}
+ \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-lo.eps}}
  \caption{
  Comparison of 1-D advection schemes. Courant number is 0.05 with 60
  points and solutions are shown for T=1 (one complete period).
-Line 147 
 $\mu=c/(1-c)$.
+Line 153 
 $\mu=c/(1-c)$.
  \end{figure}
  \begin{figure}
- \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-hi.eps}}
+ \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-1d-hi.eps}}
  \caption{
  Comparison of 1-D advection schemes. Courant number is 0.89 with 60
  points and solutions are shown for T=1 (one complete period).
-Line 353 
 if the limiter is set to zero.
+Line 359 
 if the limiter is set to zero.
  \section{Non-linear advection schemes}
+ \label{sec:non-linear_advection_schemes}
  \begin{rawhtml}
  <!-- CMIREDIR:non-linear_advection_schemes: -->
  \end{rawhtml}
-Line 543 
 $\tau$: {\bf tracer} (argument)
+Line 550 
 $\tau$: {\bf tracer} (argument)
  \subsection{Multi-dimensional advection}
  \begin{figure}
- \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-lo-diag.eps}}
+ \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-lo-diag.eps}}
  \caption{
  Comparison of advection schemes in two dimensions; diagonal advection
  of a resolved Gaussian feature. Courant number is 0.01 with
-Line 564 
 lines).
+Line 571 
 lines).
  \end{figure}
  \begin{figure}
- \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-mid-diag.eps}}
+ \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-mid-diag.eps}}
  \caption{
  Comparison of advection schemes in two dimensions; diagonal advection
  of a resolved Gaussian feature. Courant number is 0.27 with
-Line 585 
 lines).
+Line 592 
 lines).
  \end{figure}
  \begin{figure}
- \resizebox{5.5in}{!}{\includegraphics{part2/advect-2d-hi-diag.eps}}
+ \resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/advect-2d-hi-diag.eps}}
  \caption{
  Comparison of advection schemes in two dimensions; diagonal advection
  of a resolved Gaussian feature. Courant number is 0.47 with
-Line 621 
 as if in one dimension:
+Line 628 
 as if in one dimension:
  \tau^{n+1/3} & = & \tau^{n}
  - \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n})
             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\
- \tau^{n+2/3} & = & \tau^{n}
+ \tau^{n+2/3} & = & \tau^{n+1/3}
  - \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3})
             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\
- \tau^{n+3/3} & = & \tau^{n}
+ \tau^{n+3/3} & = & \tau^{n+2/3}
  - \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3})
             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)
  \end{eqnarray}
-Line 658 
 $W$: {\bf rTrans} (local)
+Line 665 
 $W$: {\bf rTrans} (local)
  \end{minipage} }
+ \begin{figure}
+ \resizebox{3.5in}{!}{\includegraphics{s_algorithm/figs/multiDim_CS.eps}}
+ \caption{Muti-dimensional advection time-stepping with Cubed-Sphere topology
+ \label{fig:advect-multidim_cs}
+ }
+ \end{figure}
  \section{Comparison of advection schemes}
+ \label{sec:tracer_advection_schemes}
+ \begin{rawhtml}
+ <!-- CMIREDIR:comparison_of_advection_schemes: -->
+ \end{rawhtml}
  \begin{table}[htb]
  \centering
+ {\small
   \begin{tabular}[htb]{|l|c|c|c|c|l|}
     \hline
     Advection Scheme & code & use  & use Multi- & Stencil & comments \\
                      &      & A.B. & dimension & (1 dim) & \\
     \hline \hline
+    $1^{rst}$order upwind  & 1 &  No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
+    \hline
     centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
     \hline
-    $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/tracer\\
+    $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/$\tau$\\
     \hline
     centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
     \hline \hline
- %  Lax-Wendroff       & 10 &  No & Yes & 3 pts & linear/tracer, non-linear/flow\\
+    $2^{nd}$order DST (Lax-Wendroff)  & 20 &
- %  \hline
+                          No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
-    $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/tracer, non-linear/flow\\
+    \hline
+    $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
+    \hline
+    $2^{nd}$order-moment Prather & 80 & No & Yes & ~ & ~ \\
     \hline \hline
     $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
     \hline
     $3^{nd}$order DST Flux limiter & 33 &  No & Yes & 5 pts & non-linear \\
     \hline
+    $2^{nd}$order-moment Prather w. limiter & 81 & No & Yes & ~ & ~ \\
+    \hline
+    piecewise parabolic w. ``null'' limiter & 40 & No & Yes & ~ & ~ \\
+    \hline
+    piecewise parabolic w. ``mono'' limiter & 41 & No & Yes & ~ & ~ \\
+    \hline
+    piecewise quartic w. ``null'' limiter & 50 & No & Yes & ~ & ~ \\
+    \hline
+    piecewise quartic w. ``mono'' limiter & 51 & No & Yes & ~ & ~ \\
+    \hline
+    piecewise quartic w. ``weno'' limiter & 52 & No & Yes & ~ & ~ \\
+    \hline
+    $7^{nd}$order one-step method & 7 & No & Yes & ~ & ~ \\
+    with Monotonicity Preserving Limiter & ~ & ~ & ~ & ~ & ~ \\
+    \hline
   \end{tabular}
+  }
   \caption{Summary of the different advection schemes available in MITgcm.
            ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
            The code corresponds to the number used to select the corresponding
-           advection scheme in the parameter file (e.g., {\em tempAdvScheme=3} in
+           advection scheme in the parameter file (e.g., {\bf tempAdvScheme}=3 in
            file {\em data} selects the $3^{rd}$ order upwind advection scheme
            for temperature).
     }

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