s_algorithm/text/shap.tex

% $Header: /u/gcmpack/manual/part2/shap.tex,v 1.4 2001/11/06 15:07:32 adcroft Exp $
% $Name:  $

\section{Shapiro Filter} 
\label{sect:shapiro-filter}

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

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

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

In addition, the S2 operator can easily be extended to 
a physical space filter: 
$$
\mathrm{S2g:}\hspace{2cm}
[1 - \frac{\Delta t}{\tau_{shap}} 
\{ \frac{L_{shap}^2}{8} \overline{\nabla}^2 \}^n]
$$

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

\marginpar{Add Response functions and figures}
1	% $Header: /u/gcmpack/manual/part2/shap.tex,v 1.4 2001/11/06 15:07:32 adcroft Exp $
2	% $Name: $
3
4	\section{Shapiro Filter}
5	\label{sect:shapiro-filter}
6
7	The Shapiro filter \cite{Shapiro_70} is a high order horizontal
8	filter that efficiently remove small scale grid noise
9	without affecting the physical structures of a field.
10	It is applied at the end of the time step %(the\_correction\_step),
11	on both velocity and tracer fields.
12
13	Three different space operators are considered here (S1,S2 and S4).
14	They differs essentially by the sequence of derivative in
15	both X and Y directions. Consequently they show different
16	damping response function specially in the diagonal directions
17	X+Y and X-Y.
18
19	Space derivatives can be computed in the real space,
20	taken into account the grid spacing.
21	Alternatively, a pure computational filter can be defined,
22	using pure numerical differences and ignoring
23	grid spacing.
24	This later form is stable whatever the grid is, and therefore
25	specially useful for highly anisotropic grid such as spherical
26	coordinate grid.
27	A damping time-scale parameter $\tau_{shap}$
28	defines the strength of the filter damping.
29
30	The 3 computational filter operators are :
31	$$
32	\mathrm{S1c:}\hspace{2cm}
33	[1 - 1/2 \frac{\Delta t}{\tau_{shap}}
34	\{ (\frac{1}{4}\delta_{ii})^n
35	+ (\frac{1}{4}\delta_{jj})^n \} ]
36	$$
37
38	$$
39	\mathrm{S2c:}\hspace{2cm}
40	[1 - \frac{\Delta t}{\tau_{shap}}
41	\{ \frac{1}{8} (\delta_{ii} + \delta_{jj}) \}^n]
42	$$
43
44	$$
45	\mathrm{S4c:}\hspace{2cm}
46	[1 - \frac{\Delta t}{\tau_{shap}} (\frac{1}{4}\delta_{ii})^n]
47	[1 - \frac{\Delta t}{\tau_{shap}} (\frac{1}{4}\delta_{jj})^n]
48	$$
49
50	In addition, the S2 operator can easily be extended to
51	a physical space filter:
52	$$
53	\mathrm{S2g:}\hspace{2cm}
54	[1 - \frac{\Delta t}{\tau_{shap}}
55	\{ \frac{L_{shap}^2}{8} \overline{\nabla}^2 \}^n]
56	$$
57
58	with the Laplacian operator $\overline{\nabla}^2 $
59	and a length scale parameter $L_{shap}$.
60	The stability of this S2g filter requires
61	$L_{shap} < \mathrm{Min}^{(Global)}(\Delta x,\Delta y)$.
62
63	\marginpar{Add Response functions and figures}