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( ) with i = 1,K,n; j = 1,K,m;
ψ i, j
=ψ r i
,λ j
r i
∈ [ 0,ar]; λ j
∈ [ 0, 2π[., then the convolution formula
is written as follows:
ψ
=
r , λ
( r , λ )
i
∑
∑
=
∑
k l
r λ
d i − k ≤dmax
d j − l ≤d
max
where
j
∑
k
r
d i − k ≤dmax
ψ
r r
( r , λ ) ⋅ w( d ) ⋅ s( d )
r r
( ) ⋅ s( d )
r
i − k max
λ
r r λ
( r , λ ) ⋅ w( d ) ⋅ w( d ) ⋅ s( d ) ⋅ s( d )
∑
k l
r λ
d i − k ≤dmax
d j − l ≤d
max
r
d
i-k and
d
k
ψ
λ
j-l
d
λ
l
k
∑
k
≤d
λ
r r λ
( ) ⋅ w( d ) ⋅ s( d ) ⋅ s( d )
w d
j−l
between two points r i
,λ j
λ
( )
d j l
j
w d
i−k
j−l
i−k
i−k
i−k
i−k
i−k
i−k
i−k
j−l
j−l
are the radial and azimuthally distances
r
( ) and ( rk , λ
l
), and ( d i k
)
s
− are the surface areas around the ( )
s
− and
rk , λ
l point.
The convolution is calculated up to a user prescribed
distance d max between the application point and all other
points necessary for the convolution.
When the convolution is applied for the components of a
vector (e.g. the horizontal winds), we must consider the
representation of the vector components relative to the same
referential centred on the application point.
The polar grid was used in this project as an intermediate
step to the application of the filter on a spherical latitudelongitude
stretched grid. In both cases, at the origin of a
polar coordinate system or at the north and south poles of
the sphere, the lines of constant polar angle or constant
longitude converge in a single point. This convergence has
consequence a severe time-stepping limit (the so-called
«pole problem»). One way to stabilize the solution of a
numerical differential equation near the pole is to damp the
waves that are unstable for a chosen timestep at a rate that
exceeds the exponential growth (Williamson and Laprise
2000).
4. Results
We present in this section two sets of tests performed with
this smoothing operator. In both cases we used a large-scale
cosine signal represented into a polar domain. An artificial
noise was added to represent the small-scale noise to be
removed.
We first verified the efficiency of the convolution operator
to act as polar filter using a uniform polar grid. In figure 1 is
presented the total signal (a) and the filtered signal (b). We
find that the amplitude of the large-scale scale signal is
unchanged and that the short-scale noise is removed.
In the second case we used a stretched polar grid and the
filter was applied outside the uniform high-resolution area to
remove the anisotropy. When the filter was built, we had
decreased the resolution with 8% in the radial direction and
with 3.8% in azimuthally direction for every grid point in
the stretching zones. The total stretching factor is almost 6 in
both directions. The large-scale signal was represented on
the entire domain and the noise was added in the stretching
zones. The initial signal is represented in figure 2a and the
filtered signal is represented in figure 2b. One thing to
remark is that the filtering operator conserves the quantities
and keeps unchanged the amplitude of the long waves.
Figure 1: The initial signal composed from a large-scale
signal and a small-scale noise in (a) and the filtered signal
in b).
Figure 2: The initial signal composed of a long wave and
the noise added in the stretching zones in a) and the
filtered signal in b).
References
Fox-Rabinovitz, M., J. Côté, B. Dugas, M. Déqué and J.L.
McGregor, Variable resolution general circulation
models: Stretched-grid model intercomparison project
(SGMIP), J. Geophys. Res., 111, D16104,
doi:10.1029/2005JD006520, 2006.
M. Fox-Rabinovitz, J. Côté, B. Dugas, M. Déqué, J.L.
McGregor and A. Belochitski, Stretched-grid Model
Intercomparison Project: decadal regional climate
simulations with enhanced variable and uniformresolution
GCMs, Meteorology and Atmospheric
Physics, Volume 100, Numbers 1-4, pp. 159-178,
2008.
R. Laprise, Regional climate modelling, J. Comput. Phys.,
doi:10.1016/j.jcp.2006.10.024, 2006.
Surcel, D., Filtres universels pour les modèles numériques
à résolution variable, Mémoire de maîtrise en Sciences
de l'atmosphère, Université du Québec à Montréal,
104 pp., 2005.
Williamson, D., and R. Laprise: Numerical
approximations for global atmospheric General
Circulation Model, P. Mote and A. O’Neil (eds.)
Numerical modeling of global atmosphere in the
climate system, Kluwer Academic Publishers, 127-
219, 2000.