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2.12 Continuity and horizontal pressure gradient terms

The core algorithm is based on the ``C grid'' discretization of the continuity equation which can be summarized as:

$\displaystyle \partial_t u + \frac{1}{\Delta x_c} \delta_i \left. \frac{ \parti...
... r}\right\vert _{s} \eta + \frac{\epsilon_{nh}}{\Delta x_c} \delta_i \Phi_{nh}'$ $\displaystyle =$ $\displaystyle G_u - \frac{1}{\Delta x_c} \delta_i \Phi_h'$ (2.87)
$\displaystyle \partial_t v + \frac{1}{\Delta y_c} \delta_j \left. \frac{ \parti...
... r}\right\vert _{s} \eta + \frac{\epsilon_{nh}}{\Delta y_c} \delta_j \Phi_{nh}'$ $\displaystyle =$ $\displaystyle G_v - \frac{1}{\Delta y_c} \delta_j \Phi_h'$ (2.88)
$\displaystyle \epsilon_{nh} \left( \partial_t w + \frac{1}{\Delta r_c} \delta_k \Phi_{nh}' \right)$ $\displaystyle =$ $\displaystyle \epsilon_{nh} G_w + \overline{b}^k - \frac{1}{\Delta r_c} \delta_k \Phi_{h}'$ (2.89)
$\displaystyle \delta_i \Delta y_g \Delta r_f h_w u +
\delta_j \Delta x_g \Delta r_f h_s v +
\delta_k {\cal A}_c w$ $\displaystyle =$ $\displaystyle {\cal A}_c \delta_k (P-E)_{r=0}$ (2.90)

where the continuity equation has been most naturally discretized by staggering the three components of velocity as shown in Fig. 2.9. The grid lengths $ \Delta x_c$ and $ \Delta y_c$ are the lengths between tracer points (cell centers). The grid lengths $ \Delta x_g$ , $ \Delta y_g$ are the grid lengths between cell corners. $ \Delta r_f$ and $ \Delta r_c$ are the distance (in units of $ r$ ) between level interfaces (w-level) and level centers (tracer level). The surface area presented in the vertical is denoted $ {\cal
A}_c$ . The factors $ h_w$ and $ h_s$ are non-dimensional fractions (between 0 and 1) that represent the fraction cell depth that is ``open'' for fluid flow.

The last equation, the discrete continuity equation, can be summed in the vertical to yield the free-surface equation:

$\displaystyle {\cal A}_c \partial_t \eta + \delta_i \sum_k \Delta y_g \Delta r_f h_w u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v = {\cal A}_c(P-E)_{r=0}$ (2.91)

The source term $ P-E$ on the rhs of continuity accounts for the local addition of volume due to excess precipitation and run-off over evaporation and only enters the top-level of the ocean model.


next up previous contents
Next: 2.13 Hydrostatic balance Up: 2. Discretization and Algorithm Previous: 2.11.6 Topography: partially filled   Contents
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