Geoscience Reference
In-Depth Information
Substituting (5.13) and (5.14) into (5.23) and then lineariz-
ing and changing to polar coordinates (
r
,
θ
), we obtain a
single equation for surface elevation,
was fixed to the bottom of the tank rotating initially with
“null” rate
0
=2.29s
−
1
(Figure 5.1). The mountain was
located at “midlatitudes” of the tank. The height of the
mountain was
h
m
=0.1
H
0
= 1cm and its diameter was
d
= 0.11
R
= 6cm. Here
R
= 55cm is the radius of the
tank. To generate a flow above the mountain, the rota-
tion rate of the table was changed abruptly to a slightly
lower value,
= 0.98
0
, which imposed a solid-body
cyclonic (eastward) flow. The tank rotates counterclock-
wise, modeling the Northern Hemisphere with the North
Pole in the center.
This mean zonal flow, which is initially of rotation rate
0
−
η
+2
γ
∂η
∂
∂t
1
R
b
2
η
∇
−
∂θ
= 0,
(5.24)
where
R
b
=
√
gH
0
/f
0
is the barotropic radius of deforma-
tion. The solution of (5.24) can be found in the form of
wave modes,
iωt) J
m
α
mn
r
R
η
=
η
0
exp
(imθ
−
(5.25)
with dispersion relation
, the
n g
radually relaxes on the Ekman time scale
T
E
=
H
0
/
√
ν
=65s,where
ν
is the kinematic viscosity
of water. The perturbation resulting from the interaction
of the mean zonal flow with the bottom topography is a
lee wave pattern downstream of the mountain shown in
Figure 5.3. The velocity vector field calculated from the
measured gradient of the surface elevation using equa-
tion (5.22) is superimposed on the altimetry image in
Figure 5.3a. Note that the color altimetry image is con-
verted to gray scale and only every 40th velocity vector
in each direction is displayed here to avoid overcrowd-
ing. Figure 5.3b shows the isolines of the surface elevation
η
calculated by integrating
2
mγ R
b
−
ω
mn
=
α
mn
R
b
/R
2
+1
.
(5.26)
Here
α
mn
is the
n
th root of the Bessel function of the
m
th
order and
R
is the radius of the domain (the radius of
the tank). Figure 5.2b shows frequency given by (5.26) for
different values of the wave numbers
n
and
m
.
5.5. OBSERVATIONS
Here we give several examples of experiments where
flows were observed using AIV. Consider first flows over
bottom topography. These flows are rich in phenomena,
including inertial and Rossby waves, vortices, and jets.
In our experiment a mountain with a flat top (plateau)
η
over
x
and
y
. The scale
of
η
indicates that the amplitude of the wave is about
75
μ
m. To gain some physical insight into the interaction
of the incoming low with the mountain, consider a simple
∇
(a)
(b)
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
Figure 5.3.
A lee wave behind a small mountain in eastward flow (counterclockwise) at
t
= 25 s. The mountain (white circle) is
positioned at “6 o'clock.” (a) Velocity vectors are superimposed on the altimetric image of the surface slope which is converted
from color to gray scale. Only every 40
th
velocity vector in each direction is displayed here. (b) Isolines show surface topography
in centimeters. A stationary Rossby wave is clearly seen as a pattern of hills and valleys above and downstream of the mountain.
The amplitude of the first wave crest in the lee of the mountain is about 75
μ
m.
