--- manual/s_algorithm/text/tracer.tex	2001/10/25 18:36:53	1.8
+++ manual/s_algorithm/text/tracer.tex	2004/10/16 03:40:13	1.18
@@ -1,8 +1,11 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/tracer.tex,v 1.8 2001/10/25 18:36:53 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/tracer.tex,v 1.18 2004/10/16 03:40:13 edhill Exp $
 % $Name:  $
 
 \section{Tracer equations}
-\label{sec:tracer_equations}
+\label{sect: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{sec:tracer_equations_abII}
+\label{sect: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
@@ -43,7 +49,7 @@
 everywhere else. This term is therefore referred to as the surface
 correction term. Global conservation is not possible using the
 flux-form (as here) and a linearized free-surface
-(\cite{Griffies00,Campin02}).
+(\cite{griffies:00,campin:02}).
 
 The continuity equation can be recovered by setting
 $G_{diff}=G_{forc}=0$ and $\tau=1$.
@@ -123,6 +129,10 @@
 
 
 \section{Linear advection schemes}
+\label{sect:tracer-advection}
+\begin{rawhtml}
+<!-- CMIREDIR:linear_advection_schemes: -->
+\end{rawhtml}
 
 \begin{figure}
 \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}
@@ -200,7 +210,7 @@
 
 For non-divergent flow, this discretization can be shown to conserve
 the tracer both locally and globally and to globally conserve tracer
-variance, $\tau^2$. The proof is given in \cite{Adcroft95,Adcroft97}.
+variance, $\tau^2$. The proof is given in \cite{adcroft:95,adcroft:97}.
 
 \fbox{ \begin{minipage}{4.75in}
 {\em S/R GAD\_C2\_ADV\_X} ({\em gad\_c2\_adv\_x.F})
@@ -349,6 +359,9 @@
 
 
 \section{Non-linear advection schemes}
+\begin{rawhtml}
+<!-- CMIREDIR:non-linear_advection_schemes: -->
+\end{rawhtml}
 
 Non-linear advection schemes invoke non-linear interpolation and are
 widely used in computational fluid dynamics (non-linear does not refer
@@ -387,7 +400,7 @@
 r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} } & \forall & u < 0
 \end{eqnarray}
 as it's argument. There are many choices of limiter function but we
-only provide the Superbee limiter \cite{Roe85}:
+only provide the Superbee limiter \cite{roe:85}:
 \begin{equation}
 \psi(r) = \max[0,\min[1,2r],\min[2,r]]
 \end{equation}
@@ -449,7 +462,7 @@
 
 The DST3 method described above must be used in a forward-in-time
 manner and is stable for $0 \le |c| \le 1$. Although the scheme
-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
 the accuracy increases with the Courant number! For low Courant
 number, DST3 produces very similar results (indistinguishable in
 Fig.~\ref{fig:advect-1d-lo}) to the linear third order method but for
@@ -653,6 +666,42 @@
 
 
 \section{Comparison of 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
+   centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
+   \hline
+   $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/tracer\\
+   \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\\
+%  \hline
+   $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/tracer, non-linear/flow\\
+   \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., {\em 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
@@ -675,7 +724,7 @@
 phenomenon.
 
 Finally, the bottom left and right panels use the same advection
-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
 Courant number this appears to not matter but for moderate Courant
 number severe distortion of the feature is apparent. Moreover, the
 stability of the multi-dimensional scheme is determined by the maximum
@@ -704,6 +753,6 @@
 non-linear schemes have the most stability (up to Courant number 1).
 \item If you need to know how much diffusion/dissipation has occurred you
 will have a lot of trouble figuring it out with a non-linear method.
-\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
 strongest argument for using a positive scheme.
 \end{itemize}