Geography Reference
In-Depth Information
where µ is defined below (8.71). The corresponding group velocity is then
U
T
µ
1
4k
2
λ
2
k
4
∂ν
∂k
=
c
gx
=
U
m
±
+
(8.77)
4λ
4
−
Comparison of (8.77 ) with (8.71) shows that relative to the mean zonal flow
the group velocity exceeds the phase velocity by the factor in parentheses on the
right side of (8.77 ) for both
ea
stward and westward directed neutral modes. For
example, if k
2
√
2
, the group velocity equals twice the phase speed,
which is the situation shown schematically in Fig. 7.4.
2λ
2
1
=
+
PROBLEMS
8.1.
Show using (8.25) that the maximum growth rate for baroclinic instability
when β
=
0 occurs for
2λ
2
√
2
1
k
2
=
−
How long does it take the most rapidly growing wave to amplify by a factor
of e
1
20 m s
−
1
?
8.2.
Solve for ψ
3
and ω
2
in terms of ψ
1
for a baroclinic Rossby wave whose
phase speed satisfies (8.23). Explain the phase relationship among ψ
1
, ψ
3
,
ω
2
if λ
2
10
−
12
m
−
1
=
2
×
and U
T
=
and in terms of the quasi-geostrophic theory. (Note that U
T
=
0 in this
case.)
8.3.
For the case U
1
=−
U
3
and k
2
λ
2
solve for ψ
3
and ω
2
in terms of ψ
1
for
=
=
marginally stable waves [i.e., δ
0 in (8.21)].
U
T
solve for ψ
3
and ω
2
in terms of
ψ
1
. Explain the phase relationships among ω
2
, ψ
1
, and ψ
3
in terms of the
energetics of quasi-geostrophic waves for the amplifying wave.
0,k
2
λ
2
, and U
m
=
8.4.
For the case β
=
=
8.5.
Suppose that a baroclinic fluid is confined between two rigid horizontal lids
in a rotating tank in which β
0 but friction is present in the form of
linear drag proportional to the velocity (i.e.,
Fr
=
µ
V
). Show that the
two-level model perturbation vorticity equations in Cartesian coordinates
can be written as
=−
∂
∂t
+
µ
∂
2
ψ
1
∂x
2
∂
∂x
+
f
δp
ω
2
=
U
1
−
0
∂
∂t
+
µ
∂
2
ψ
3
∂x
2
∂
∂x
+
f
δp
ω
2
=
U
3
+
0
