| 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 $\tau=1$. |
| 200 |
|
|
| 201 |
For non-divergent flow, this discretization can be shown to conserve |
For non-divergent flow, this discretization can be shown to conserve |
| 202 |
the tracer both locally and globally and to globally conserve tracer |
the tracer both locally and globally and to globally conserve tracer |
| 203 |
variance, $\tau^2$. The proof is given in \cite{Adcroft95,Adcroft97}. |
variance, $\tau^2$. The proof is given in \cite{adcroft:95,adcroft:97}. |
| 204 |
|
|
| 205 |
\fbox{ \begin{minipage}{4.75in} |
\fbox{ \begin{minipage}{4.75in} |
| 206 |
{\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}) |
| 387 |
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 |
| 388 |
\end{eqnarray} |
\end{eqnarray} |
| 389 |
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 |
| 390 |
only provide the Superbee limiter \cite{Roe85}: |
only provide the Superbee limiter \cite{roe:85}: |
| 391 |
\begin{equation} |
\begin{equation} |
| 392 |
\psi(r) = \max[0,\min[1,2r],\min[2,r]] |
\psi(r) = \max[0,\min[1,2r],\min[2,r]] |
| 393 |
\end{equation} |
\end{equation} |
| 675 |
phenomenon. |
phenomenon. |
| 676 |
|
|
| 677 |
Finally, the bottom left and right panels use the same advection |
Finally, the bottom left and right panels use the same advection |
| 678 |
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 |
| 679 |
Courant number this appears to not matter but for moderate Courant |
Courant number this appears to not matter but for moderate Courant |
| 680 |
number severe distortion of the feature is apparent. Moreover, the |
number severe distortion of the feature is apparent. Moreover, the |
| 681 |
stability of the multi-dimensional scheme is determined by the maximum |
stability of the multi-dimensional scheme is determined by the maximum |
| 704 |
non-linear schemes have the most stability (up to Courant number 1). |
non-linear schemes have the most stability (up to Courant number 1). |
| 705 |
\item If you need to know how much diffusion/dissipation has occurred you |
\item If you need to know how much diffusion/dissipation has occurred you |
| 706 |
will have a lot of trouble figuring it out with a non-linear method. |
will have a lot of trouble figuring it out with a non-linear method. |
| 707 |
\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 |
| 708 |
strongest argument for using a positive scheme. |
strongest argument for using a positive scheme. |
| 709 |
\end{itemize} |
\end{itemize} |
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