/[MITgcm]/manual/s_algorithm/text/tracer.tex
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revision 1.9 by adcroft, Tue Nov 13 15:32:28 2001 UTC revision 1.12 by adcroft, Tue Nov 13 20:13:54 2001 UTC
# Line 2  Line 2 
2  % $Name$  % $Name$
3    
4  \section{Tracer equations}  \section{Tracer equations}
5  \label{sec:tracer_equations}  \label{sect:tracer_equations}
6    
7  The basic discretization used for the tracer equations is the second  The basic discretization used for the tracer equations is the second
8  order piece-wise constant finite volume form of the forced  order piece-wise constant finite volume form of the forced
# Line 15  part of the tracer equations and the var Line 15  part of the tracer equations and the var
15  described here.  described here.
16    
17  \subsection{Time-stepping of tracers: ABII}  \subsection{Time-stepping of tracers: ABII}
18  \label{sec:tracer_equations_abII}  \label{sect:tracer_equations_abII}
19    
20  The default advection scheme is the centered second order method which  The default advection scheme is the centered second order method which
21  requires a second order or quasi-second order time-stepping scheme to  requires a second order or quasi-second order time-stepping scheme to
# Line 43  only affects the surface layer since the Line 43  only affects the surface layer since the
43  everywhere else. This term is therefore referred to as the surface  everywhere else. This term is therefore referred to as the surface
44  correction term. Global conservation is not possible using the  correction term. Global conservation is not possible using the
45  flux-form (as here) and a linearized free-surface  flux-form (as here) and a linearized free-surface
46  (\cite{Griffies00,Campin02}).  (\cite{griffies:00,campin:02}).
47    
48  The continuity equation can be recovered by setting  The continuity equation can be recovered by setting
49  $G_{diff}=G_{forc}=0$ and $\tau=1$.  $G_{diff}=G_{forc}=0$ and $\tau=1$.
# Line 123  the forward method. Line 123  the forward method.
123    
124    
125  \section{Linear advection schemes}  \section{Linear advection schemes}
126    \label{sect:tracer-advection}
127    
128  \begin{figure}  \begin{figure}
129  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}  \resizebox{5.5in}{!}{\includegraphics{part2/advect-1d-lo.eps}}
# Line 200  W & = & {\cal A}_c w Line 201  W & = & {\cal A}_c w
201    
202  For non-divergent flow, this discretization can be shown to conserve  For non-divergent flow, this discretization can be shown to conserve
203  the tracer both locally and globally and to globally conserve tracer  the tracer both locally and globally and to globally conserve tracer
204  variance, $\tau^2$. The proof is given in \cite{Adcroft95,Adcroft97}.  variance, $\tau^2$. The proof is given in \cite{adcroft:95,adcroft:97}.
205    
206  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
207  {\em S/R GAD\_C2\_ADV\_X} ({\em gad\_c2\_adv\_x.F})  {\em S/R GAD\_C2\_ADV\_X} ({\em gad\_c2\_adv\_x.F})
# Line 387  r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t Line 388  r = \frac{ \tau_{i-1} - \tau_{i-2} }{ \t
388  r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} } & \forall & u < 0  r = \frac{ \tau_{i+1} - \tau_{i} }{ \tau_{i} - \tau_{i-1} } & \forall & u < 0
389  \end{eqnarray}  \end{eqnarray}
390  as it's argument. There are many choices of limiter function but we  as it's argument. There are many choices of limiter function but we
391  only provide the Superbee limiter \cite{Roe85}:  only provide the Superbee limiter \cite{roe:85}:
392  \begin{equation}  \begin{equation}
393  \psi(r) = \max[0,\min[1,2r],\min[2,r]]  \psi(r) = \max[0,\min[1,2r],\min[2,r]]
394  \end{equation}  \end{equation}
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