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2.20 Shapiro Filter

The Shapiro filter Shapiro [1970] is a high order horizontal filter that efficiently remove small scale grid noise without affecting the physical structures of a field. It is applied at the end of the time step on both velocity and tracer fields.

Three different space operators are considered here (S1,S2 and S4). They differs essentially by the sequence of derivative in both X and Y directions. Consequently they show different damping response function specially in the diagonal directions X+Y and X-Y.

Space derivatives can be computed in the real space, taken into account the grid spacing. Alternatively, a pure computational filter can be defined, using pure numerical differences and ignoring grid spacing. This later form is stable whatever the grid is, and therefore specially useful for highly anisotropic grid such as spherical coordinate grid. A damping time-scale parameter $ \tau_{shap}$ defines the strength of the filter damping.

The 3 computational filter operators are :

$\displaystyle \mathrm{S1c:}\hspace{2cm}
[1 - 1/2 \frac{\Delta t}{\tau_{shap}}
\{ (\frac{1}{4}\delta_{ii})^n
+ (\frac{1}{4}\delta_{jj})^n \} ]
$

$\displaystyle \mathrm{S2c:}\hspace{2cm}
[1 - \frac{\Delta t}{\tau_{shap}}
\{ \frac{1}{8} (\delta_{ii} + \delta_{jj}) \}^n]
$

$\displaystyle \mathrm{S4c:}\hspace{2cm}
[1 - \frac{\Delta t}{\tau_{shap}} (\fra...
...}\delta_{ii})^n]
[1 - \frac{\Delta t}{\tau_{shap}} (\frac{1}{4}\delta_{jj})^n]
$

In addition, the S2 operator can easily be extended to a physical space filter:

$\displaystyle \mathrm{S2g:}\hspace{2cm}
[1 - \frac{\Delta t}{\tau_{shap}}
\{ \frac{L_{shap}^2}{8} \overline{\nabla}^2 \}^n]
$

with the Laplacian operator $ \overline{\nabla}^2 $ and a length scale parameter $ L_{shap}$ . The stability of this S2g filter requires $ L_{shap} < \mathrm{Min}^{(Global)}(\Delta x,\Delta y)$ .



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Next: 2.20.1 SHAP Diagnostics Up: 2. Discretization and Algorithm Previous: 2.19 Comparison of advection   Contents
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