--- manual/s_algorithm/text/tracer.tex	2004/04/20 23:32:59	1.16
+++ manual/s_algorithm/text/tracer.tex	2010/08/30 23:09:19	1.26
@@ -1,8 +1,11 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/tracer.tex,v 1.16 2004/04/20 23:32:59 edhill Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/tracer.tex,v 1.26 2010/08/30 23:09:19 jmc Exp $
 % $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
@@ -15,7 +18,10 @@
 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
@@ -123,13 +129,13 @@
 
 
 \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).
@@ -147,7 +153,7 @@
 \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).
@@ -353,6 +359,7 @@
 
 
 \section{Non-linear advection schemes}
+\label{sec:non-linear_advection_schemes}
 \begin{rawhtml}
 <!-- CMIREDIR:non-linear_advection_schemes: -->
 \end{rawhtml}
@@ -543,7 +550,7 @@
 \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
@@ -564,7 +571,7 @@
 \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
@@ -585,7 +592,7 @@
 \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
@@ -621,10 +628,10 @@
 \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}
@@ -658,8 +665,54 @@
 
 \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
+ \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/$\tau$\\
+   \hline
+   centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
+   \hline \hline
+   $2^{nd}$order DST (Lax-Wendroff)  & 20 &
+                         No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
+   \hline
+   $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
+   \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
+ \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., {\bf tempAdvScheme}=3 in
+          file {\em data} selects the $3^{rd}$ order upwind advection scheme 
+          for temperature).
+   }
+ \label{tab:advectionShemes_summary}
+\end{table}
+
 
 Figs.~\ref{fig:advect-2d-lo-diag}, \ref{fig:advect-2d-mid-diag} and
 \ref{fig:advect-2d-hi-diag} show solutions to a simple diagonal
