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Model vorticity budget on a B-grid


Using Equations 2.21 and 2.22, the model vorticity equation is obtained at the center of the cell (also the $\eta$ location):

\frac{\partial \zeta}{\partial t} &
+ D^+_x \ov...
...erline{F_y }^y
- D^+_y \overline{F_x }^x \mbox{~.}
\end{split}\end{displaymath} (8.1)

Study of this equation shows that all terms cancel out in the interior. However, close to corners, they are not necessarily zero. The pressure term, hereafter IP, for instance gives after summation over the domain and emphasizing the terms close to the northeast corner (assuming $\Delta x = \Delta y = \Delta$):

I_P & = \sum_i \sum_j \Delta x \Delta y \left(
...}^* \right) +
\cdots +
\mbox{other corner terms}
\end{split}\end{displaymath} (8.2)

where $\eta^*$ is the extrapolated value of the elevation along the wall and (i,j) are the indices taken at the velocity point directly southwest of the corner. When using a linear extrapolating law, $\eta_{i+1,j}^* = 3/2 \eta_{ij}- 1/2
\eta_{i-1,j}$, IP becomes

\begin{displaymath}I_P = \frac{g}{4}\left( \eta_{i,j-1} - \eta_{i-1,j} \right) +
\cdots +
\mbox{other corner terms} \mbox{~.}
\end{displaymath} (8.3)

Hence, the pressure term in the vorticity equation does not cancel out. However, for the advective terms, hereafter Iadv, we have

\begin{displaymath}I_{adv}= \sum_i \sum_j \Delta x \Delta y \left(
D^+_x \overl...
- D^+_y \overline{u \Delta_x u + v \Delta_y u }^x \right)
\end{displaymath} (8.4)

which simplifies first to a circulation integral of the form

\begin{displaymath}I_{adv}= \sum
\overline{ \mbox{\bf u}\mbox{\boldmath$\nabla$ }\mbox{\bf u}} \cdot \mbox{\boldmath\bf$\Delta$ l}\mbox{~.}
\end{displaymath} (8.5)

One can prove that this summation cancels out. Hence, Iadv=0. The same occurs for the Coriolis terms because the summation can be recast as a summation of flux normal to the wall at velocity points along the wall, which are zero due to the impermeability condition. Therefore, the model vorticity budget reduces in the case of the B-grid to

\begin{displaymath}\sum_i \sum_j \frac{\partial \zeta}{\partial t} \Delta x \Delta y = \mathcal{F}_{i}+
\mathcal{F}_{vis}- I_P \mbox{~.}
\end{displaymath} (8.6)

Hence IP is present in the model budget whereas it should ideally be zero. Note that a zero order extrapolation ( $\eta_{i+1,j}^* =\eta_{ij}$) results in the cancellation of IP. However, this extrapolation leads to countercurrents along the boundaries for the Munk problem on a beta plane, as shown in Section 4.2.
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Frederic Dupont