Geoscience Reference
In-Depth Information
where
n
D
1
6
C
1
3n
:
(8.50)
Taking (
8.46
)and(
8.47
) into account and replacing
by
D
, the desired ansatz for
the first-order approximation (LH1) of the diagonal elements is obtained:
d
g
D exp
d
g
D
2
n
(8.51)
The relationship (
8.51
)wasderivedby
Purser et al.
(
2003
) for the one-dimensional
case (
) and tested by
Mirouze and Weaver
(
2010
), who reported a
significant (2-4 times) improvement of the accuracy in 1d simulations.
An estimate similar to (
8.51
) can also be obtained for
B
m
, possibly with a
different coefficient
1
D
0:5
Q
n
. It is assumed, however, that
Q
n
may not differ too much
from
n
given similarity in the shapes (Fig.
8.1
) of the correlation functions (
8.42
)
and (
8.43
). Furthermore, because of the approximate nature of (
8.51
), the best
representation of
d
n
that ts significantly different from the one given by (
8.50
). For this reason, a more
general form of (
8.51
) was adopted in the numerical experiments, assuming
.
x
/
in realistic applications may be achieved with a value of
d
g
.
d
g
.
d
2
.
=4
2
d
2
.
x
/
exp
Œ
D
=2
x
/
I
x
/
Œ
I
D
x
/
(8.52)
m
D
2
for the Gaussian model and its second-order (
) spline approximation (
8.10
).
The following experiments investigate the dependence of the respective approx-
imation errors h
"
g;2
i on the free parameter
.
8.3.3
Numerical Results
To assess the efficiency of the methods outlined in Sects.
8.3.1
and
8.3.2
,two
series of numerical experiments with realistically inhomogeneous BEC models are
performed. In the first series the methods were tested in the 2d case with the state
vector having a dimension of several thousand. In the second series, the LH0 and
LH1 techniques are examined in a realistic 3d setting with a state space dimension
of
N
10
6
.
8.3.3.1
Experimental Setting in 2d
The state space is described by scalar functions defined on the orthogonal curvilinear
grid of the Navy Coastal Ocean Model (NCOM) (
Martin et al. 2008
)setupin
the Monterrey Bay (Fig.
8.4
). The number
N
of grid points (dimension of the
state space) was 3,438. A vector field
u
.
x
/
was used
t
o generate the diffusion
2
of
p
tensor as follows. The smaller principal axis
is set to be orthogonal
Search WWH ::
Custom Search