--- manual/s_algorithm/text/shap.tex 2001/08/09 19:48:39 1.2 +++ manual/s_algorithm/text/shap.tex 2005/08/08 17:57:06 1.5 @@ -1,9 +1,10 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/shap.tex,v 1.2 2001/08/09 19:48:39 adcroft Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/shap.tex,v 1.5 2005/08/08 17:57:06 jmc Exp $ % $Name: $ \section{Shapiro Filter} +\label{sect:shapiro-filter} -The Shapiro filter (Shapiro 1970, 1975) is a high order horizontal +The Shapiro filter \cite{Shapiro_70} 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 %(the\_correction\_step), @@ -21,7 +22,7 @@ using pure numerical differences and ignoring grid spacing. This later form is stable whatever the grid is, and therefore -specially useful for highly anisotrope grid such as spherical +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. @@ -46,7 +47,7 @@ [1 - \frac{\Delta t}{\tau_{shap}} (\frac{1}{4}\delta_{jj})^n] $$ -In addition, the S2 operator can easly be extended to +In addition, the S2 operator can easily be extended to a physical space filter: $$ \mathrm{S2g:}\hspace{2cm} @@ -54,7 +55,7 @@ \{ \frac{L_{shap}^2}{8} \overline{\nabla}^2 \}^n] $$ -with the Laplacien operator $\overline{\nabla}^2 $ +with the Laplacian operator $\overline{\nabla}^2 $ and a length scale parameter $L_{shap}$. The stability of this S2g filter requires $L_{shap} < \mathrm{Min}^{(Global)}(\Delta x,\Delta y)$.