Geoscience Reference
In-Depth Information
a
b
37.2
37.2
36.8
36.8
0.05
36.4
1
36.4
0.04
0.8
0.6
0.03
36.0
36.0
0.4
0.02
0.2
0.01
0
35.6
35.6
−123.2
−122.8
−122.4
−122.0
−121.6
−123.2
−122.8
−122.4
−122.0
−121.6
LONGITUDE
LONGITUDE
Fig. 8.4
Left
: A composite map of five columns of the
B
g
operator.
White circles
denote locations
of the diagonal elements of the corresponding correlation matrices.
Right panel
shows the map of
the non-normalized diagonal elements of
B
g
. Depth contours are in meters
to
u
with the corresponding “background” length scale
2
D
3ı
,where
ı.
x
/
is
the spatia
lly va
rying grid step. The length of the larger axis
1
is set to be equal to
.1;
p
j
u
j
=
max
/
2
,where
u
is a prescribed threshold value of j
u
j.If
u
is a velocity
field, then a structure like this simulates enhanced diffusive transport of model errors
in the regions of strong currents on the background of isotropic error diffusion with
the decorrelation scale
u
2
.
In the 2d experiments, the vector field
u
is generated by treating bottom
topography
(Fig.
8.4
) as a stream function. The threshold value
v
was taken
to be one-fifth of the rms variation of jr
h
j over the domain.
All the experiments described in the next two sections are performed using the
BEC models (
8.42
)and(
8.43
) with the parameters
h.
x
/
. A composite map of
five columns of
B
g
is shown in Fig.
8.4
a. The diffusion operator (
8.1
) is constrained
to have zero normal derivative at the open and rigid boundaries of the domain in
both 2d and 3d experiments.
Numerically, the action of
B
g
on a state vector
y
n
D
m
D
2
was evaluated by explicitly
0
integrating the corresponding diffusion equation
y
t
D
D
=2
y
for the virtual “time
period” defined by
. The minimum number
of “time steps” required for the scheme's stability in such a setting was 5,256. The
action of
B
2
was computed by solving the system of equations
, starting from the “initial condition”
y
0
=4/
2
y
.
I
D
D
y
0
with a conjugate gradient method. The number of iterations, required for obtaining
a solution, varied within 2,000-2,500. To make the shapes of the
B
g
and
B
2
compatible (Fig.
8.1
), the diffusion tensor in
B
2
was multiplied by
8=
(see
Tab le
8.1
).
The exact values
d
of the diagonal elements are shown in Fig.
8.4
b. Their
magnitude appears to be lower in the regions of “strong currents” (large
u
), as
the corresponding
.
x
/
ı
-functions are dispersed over larger areas by diffusion.
d
.
x
/
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