Geography Reference
In-Depth Information
7.17.
Section 4.3 showed that for a homogeneous incompressible fluid a decrease
in depth with latitude has the same dynamic effect as a latitudinal depen-
dence of the Coriolis parameter. Thus, Rossby-type waves can be produced
in a rotating cylindrical vessel if the depth of the fluid is dependent on the
radial coordinate. To determine the Rossby wave speed formula for this
equivalent
β-
effect
, we assume that the flow is confined between rigid lids
in an annular region whose distance from the axis or rotation is large enough
so that the curvature terms in the equations can be neglected. We then can
refer the motion to Cartesian coordinates with x directed azimuthally and
y directed toward the axis of rotation. If the system is rotating at angular
velocity and the depth is linearly dependent on y,
=
H
0
−
H (y)
γy
show that the perturbation continuity equation can be written as
H
0
∂u
∂v
∂y
γv
=
∂x
+
−
0
and that the perturbation quasi-geostrophic vorticity equation is thus
β
∂ψ
∂
∂t
∇
2
ψ
+
∂x
=
0
where ψ
is the perturbation geostrophic streamfunction and β
2γ /H
0
.
What is the Rossby wave speed in this situation for waves of wavelength 100
cm in both the x and y directions if
=
1s
−
1
, H
0
=
=
=
0.05?
Hint
: Assume that the velocity field is geostrophic except in the divergence
term.
7.18.
Show by scaling arguments that if the horizontal wavelength is much greater
than the depth of the fluid, two-dimensional surface gravity waves will be
hydrostatic so that the “shallow water” approximation applies.
7.19.
The linearized form of the quasi-geostrophic vorticity equation (6.18) can
be written as
20 cm, and γ
∂
∂t
+
β
∂ψ
∂
∂x
2
ψ
+
∇
∂x
=−
f
0
∇·
u
V
Suppose that the horizontal divergence field is given by
∇·
V
=
A cos [k (x
−
ct)]
where A is a constant. Find a solution for the corresponding relative vorticity
field. What is the phase relationship between vorticity and divergence? For
what value of c does the vorticity become infinite?
