--- manual/s_algorithm/text/tracer.tex 2001/10/25 18:36:53 1.8 +++ manual/s_algorithm/text/tracer.tex 2001/11/13 20:13:54 1.12 @@ -1,8 +1,8 @@ -% $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 1.12 2001/11/13 20:13:54 adcroft Exp $ % $Name: $ \section{Tracer equations} -\label{sec:tracer_equations} +\label{sect:tracer_equations} The basic discretization used for the tracer equations is the second order piece-wise constant finite volume form of the forced @@ -15,7 +15,7 @@ described here. \subsection{Time-stepping of tracers: ABII} -\label{sec:tracer_equations_abII} +\label{sect:tracer_equations_abII} 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 +43,7 @@ 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 +123,7 @@ \section{Linear advection schemes} +\label{sect:tracer-advection} \begin{figure} \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}} @@ -200,7 +201,7 @@ 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}) @@ -387,7 +388,7 @@ 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} @@ -675,7 +676,7 @@ 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 +705,6 @@ 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}