Geography Reference
In-Depth Information
7.12.
Determine the perturbation horizontal and vertical velocity fields for sta-
tionary gravity waves forced by flow over sinusoidally varying topography
given the following conditions: the height of the grou
nd
is h
=
h
0
cos kx
10
−
2
s
−
1
; u
5ms
−
1
where h
0
=
50 m is a constant; N
=
2
×
=
; and
10
−
3
m
−
1
.
Hint
: For small amplitude topography (h
0
k
k
=
3
×
1) we
can approximate the lower boundary condition by
u∂h
∂x
w
=
Dh/Dt
=
at z
=
0.
7.13.
For the topographic gravity wave pr
ob
lem discussed in Section 7.4.2, with
vertical velocity given by (7.48) and uk <
N , fin
d the zonal wind perturba-
tion. Compute the v
er
tical momentum flux u
w
and show that this flux is
N
2
2. Determine the slope of the phase lines in
a maximum when k
2
u
2
=
the x, z plane for this case.
7.14.
Verify the group velocity relationship for inertia-gravity waves given in
(7.67).
7.15.
Show that when u
=
0 the wave number vector
κ
for an internal gravity
wave is perpendicular to the group velocity vector.
7.16.
Using the linearized form of the vorticity equation (6.18) and the β-plane
approximation, derive the Rossby wave speed for a homogeneous incom-
pressible ocean of depth h. Assume a motionless basic state and small per-
turbations that depend only on x and t ,
u
(x, t) ,v
v
(x, t) ,h
h
(x, t)
=
=
=
+
u
H
where H is the mean depth of the ocean. With the aid of the continuity equa-
tion for a homogeneous layer (7.21) and the geostrophic wind relationship
v
gf
−
0
∂h
/∂x, show that the perturbation potential vorticity equation
can be written in the form
=
∂
2
∂x
2
−
h
+
f
0
gH
β
∂h
∂
∂t
∂x
=
0
and that h
=
h
0
e
ik(x
−
ct)
is a solution provided that
β
k
2
f
0
/gH
−
1
c
=−
+
If the ocean is 4 km deep, what is the Rossby wave speed at latitude 45˚ for
a wave of 10,000 km zonal wavelength?
