| 21 |
using pure numerical differences and ignoring |
using pure numerical differences and ignoring |
| 22 |
grid spacing. |
grid spacing. |
| 23 |
This later form is stable whatever the grid is, and therefore |
This later form is stable whatever the grid is, and therefore |
| 24 |
specially useful for highly anisotrope grid such as spherical |
specially useful for highly anisotropic grid such as spherical |
| 25 |
coordinate grid. |
coordinate grid. |
| 26 |
A damping time-scale parameter $\tau_{shap}$ |
A damping time-scale parameter $\tau_{shap}$ |
| 27 |
defines the strength of the filter damping. |
defines the strength of the filter damping. |
| 46 |
[1 - \frac{\Delta t}{\tau_{shap}} (\frac{1}{4}\delta_{jj})^n] |
[1 - \frac{\Delta t}{\tau_{shap}} (\frac{1}{4}\delta_{jj})^n] |
| 47 |
$$ |
$$ |
| 48 |
|
|
| 49 |
In addition, the S2 operator can easly be extended to |
In addition, the S2 operator can easily be extended to |
| 50 |
a physical space filter: |
a physical space filter: |
| 51 |
$$ |
$$ |
| 52 |
\mathrm{S2g:}\hspace{2cm} |
\mathrm{S2g:}\hspace{2cm} |
| 54 |
\{ \frac{L_{shap}^2}{8} \overline{\nabla}^2 \}^n] |
\{ \frac{L_{shap}^2}{8} \overline{\nabla}^2 \}^n] |
| 55 |
$$ |
$$ |
| 56 |
|
|
| 57 |
with the Laplacien operator $\overline{\nabla}^2 $ |
with the Laplacian operator $\overline{\nabla}^2 $ |
| 58 |
and a length scale parameter $L_{shap}$. |
and a length scale parameter $L_{shap}$. |
| 59 |
The stability of this S2g filter requires |
The stability of this S2g filter requires |
| 60 |
$L_{shap} < \mathrm{Min}^{(Global)}(\Delta x,\Delta y)$. |
$L_{shap} < \mathrm{Min}^{(Global)}(\Delta x,\Delta y)$. |
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