Geoscience Reference
In-Depth Information
elevation into velocity. A brief theoretical description
of inertial and Rossby waves is given in Section 5.4.
Examples of different flows studied with the altimetric
imaging velocimetry (AIV) technique are described in
Section 5.5.
where
H
0
is the depth of the layer in the absence of
rotation,
is the rotation rate of the tank, and
g
is
the gravitational acceleration. The dynamical equivalence
of the varying depth to the quadratically varying Corio-
lis parameter in the polar
β
-plane approximation results
from the conservation of the potential vorticity (PV),
defined as
q
=
(ζ
+2
)/h
.Here
ζ
is the vertical component
of the relative vorticity and
h
is given by (5.3). Assum-
ing that the percentage change in
h
is small and that
ζ
is
much smaller than the background vorticity 2
(i.e., the
Rossby number Ro =
5.2. POLAR
β
-PLANE
In GFD, an ocean or the atmosphere is often consid-
ered as thin layers of fluid if one is concerned about
large-scale motions. In order to avoid using a spherical
coordinate system, a local Cartesian system (
x
,
y)
tan-
gent to the surface of the planet at some latitude
ϕ
0
is
often employed together with the
β
-plane approximation
to describe the variation of the Coriolis parameter in the
meridional direction. The Coriolis parameter is defined as
f
=2
sin
ϕ
,where
is the rotation rate of the planet
and
ϕ
the latitude. The
β
-plane approximation gives a lin-
ear dependence on
y
of the form
f
=
f
0
+
βy
, where the
distance
y
=
a(ϕ
|
|
1
)
, we obtain the following
approximate expression for PV:
ζ
/
2
ζ
+2
1
r
2
.
2
2
gH
0
R
2
2
1
H
0
q
≈
−
−
(5.4)
Comparing the term in curly brackets in the above for-
mula with (5.2), we define the laboratory polar
β
-plane
with parameter
γ
given by
3
gH
0
γ
=
.
(5.5)
ϕ
0
)
increases toward the north. Here
a
is the radius of the planet. This linear dependence is the
result of the expansion of the Coriolis parameter about a
reference latitude
ϕ
0
,
−
Within this laboratory framework we can also introduce
a local
β
-plane if desired. Choosing
r
0
to be the reference
distance from the pole (“midlatitudes” of the tank), we
obtain an approximate expression for the depth as follows:
2
sin
ϕ
0
+
2
cos
ϕ
0
a
f
≈
y
,
(5.1)
(r
0
−
h(r)
=
H
0
+
2
2
g
R
2
2
y)
2
−
such that
f
0
=2
sin
ϕ
0
and
β
=2
cos
ϕ
0
/a
. The valid-
ity of this approximation has been extensively discussed in
theliterature [
Phillips
1963, 1966;
Veronis
, 1963]. However,
the
β
-plane cannot be centered at the pole of the planet.
A different approximation can be used instead for the
domains which include the pole. The Coriolis parameter
can be written as
f
=2
cos
φ
,where
φ
is the colatitude.
Expanding near the pole,
φ
= 0, we obtain
r
0
−
=
h(r
0
)
H
0
+
2
2
g
R
2
2
2
r
0
g
≈
2
r
0
y
−
−
y
.
Here the
y
axis of the local Cartesian system with the ori-
gin at
r
0
is directed toward the pole. The PV can then be
written in the form
ζ
+2
1+
2
r
0
gh(r
0
)
y
2
1
=2
1
h(r
0
)
q
≈
(5.6)
φ
2
2
a
2
r
2
=
f
0
−
γ r
2
,
f
≈
−
−
(5.2)
such that the
β
-parameter is defined as
β
=
2
3
r
0
where
r
is the radial distance from the pole and
f
0
=2
.
We have now adopted the polar coordinate system (
r
,
θ
)
and introduced the parameter
γ
=
/a
2
. This approxima-
tion is called the polar
β
-plane or
γ
-plane approximation.
Note that the Coriolis parameter
f
varies quadratically
with distance.
In the laboratory, we do not reproduce the variation of
the Coriolis parameter with distance. Instead, the depth
of the fluid is varied. The depth of a rotating fluid with a
free surface varies quadratically with the distance
r
from
the axis of rotation. In a cylindrical tank of radius
R
the
depth
h
is given by
gh(r
0
)
.
(5.7)
5.3. ALTIMETRIC IMAGING VELOCIMETRY
One can think of a paraboloidal surface of a liquid
when in a solid-body rotation as an undisturbed reference
surface similar to a geoid in satellite altimetry. A flow gen-
erated in the rotating layer creates perturbations of the
reference surface. Let
η
be the elevation of the surface
measured with respect to the reference paraboloidal sur-
face. Unlike satellite altimetry where the distance between
the satellite and the sea surface is measured, laboratory
altimetry measures the angles, namely the gradient of the
surface elevation,
r
2
,
h(r)
=
H
0
+
2
2
g
R
2
2
−
(5.3)
∇
η
=
(∂η/∂x
,
∂η/∂y)
, in the horizontal
