/[MITgcm]/manual/s_algorithm/text/shap.tex
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revision 1.1 by adcroft, Wed Aug 8 16:15:22 2001 UTC revision 1.5 by jmc, Mon Aug 8 17:57:06 2005 UTC
# Line 1  Line 1 
1  % $Header$  % $Header$
2  % $Name$  % $Name$
3    
4  \subsection{Shapiro Filter}  \section{Shapiro Filter}
5    \label{sect:shapiro-filter}
6    
7  The Shapiro filter (Shapiro 1970, 1975) is a high order horizontal  The Shapiro filter \cite{Shapiro_70} is a high order horizontal
8  filter that efficiently remove small scale grid noise  filter that efficiently remove small scale grid noise
9  without affecting the physical structures of a field.  without affecting the physical structures of a field.
10  It is applied at the end of the time step %(the\_correction\_step),  It is applied at the end of the time step %(the\_correction\_step),
# Line 21  Alternatively, a pure computational filt Line 22  Alternatively, a pure computational filt
22  using pure numerical differences and ignoring  using pure numerical differences and ignoring
23  grid spacing.  grid spacing.
24  This later form is stable whatever the grid is, and therefore  This later form is stable whatever the grid is, and therefore
25  specially useful for highly anisotrope grid such as spherical  specially useful for highly anisotropic grid such as spherical
26  coordinate grid.  coordinate grid.
27  A damping time-scale parameter $\tau_{shap}$  A damping time-scale parameter $\tau_{shap}$
28  defines the strength of the filter damping.  defines the strength of the filter damping.
# Line 46  $$ Line 47  $$
47  [1 - \frac{\Delta t}{\tau_{shap}} (\frac{1}{4}\delta_{jj})^n]  [1 - \frac{\Delta t}{\tau_{shap}} (\frac{1}{4}\delta_{jj})^n]
48  $$  $$
49    
50  In addition, the S2 operator can easly be extended to  In addition, the S2 operator can easily be extended to
51  a physical space filter:  a physical space filter:
52  $$  $$
53  \mathrm{S2g:}\hspace{2cm}  \mathrm{S2g:}\hspace{2cm}
# Line 54  $$ Line 55  $$
55  \{ \frac{L_{shap}^2}{8} \overline{\nabla}^2 \}^n]  \{ \frac{L_{shap}^2}{8} \overline{\nabla}^2 \}^n]
56  $$  $$
57    
58  with the Laplacien operator $\overline{\nabla}^2 $  with the Laplacian operator $\overline{\nabla}^2 $
59  and a length scale parameter $L_{shap}$.  and a length scale parameter $L_{shap}$.
60  The stability of this S2g filter requires  The stability of this S2g filter requires
61  $L_{shap} < \mathrm{Min}^{(Global)}(\Delta x,\Delta y)$.  $L_{shap} < \mathrm{Min}^{(Global)}(\Delta x,\Delta y)$.
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