Geography Reference
In-Depth Information
where as in the previous subsection (z
∗
) is a complex amplitude and c a complex
phase speed, and substituting from (8.64) into (8.62) we find that the vertical
structure is given by the solution of the standard second-order differential equation
d
2
dz
∗
2
α
2
−
=
0
(8.65)
=
k
2
l
2
/ε. A similar substitution into (8.63) yields the boundary
where α
2
+
conditions
z
∗
−
c
d/dz
∗
−
at z
∗
=
=
0,
0,H
(8.66)
valid for rigid horizontal boundaries (w
∗
0) at the surface (z
∗
=
=
0) and the
tropopause (z
∗
=
H ).
The general solution of (8.65) can be written in the form
z
∗
=
A sinh αz
∗
+
B cosh αz
∗
(8.67)
Substituting from (8.67) into the boundary conditions (8.66) for z
∗
0 and H
yields a set of two linear homogeneous equations in the amplitude coefficients A
and B:
=
−
cαA
−
B
=
0
α (H
−
c)(A cosh αH
+
B sinh αH)
−
(A sinh αH
+
B cosh αH)
=
0
As in the two-layer model a nontrivial solution exists only if the determinant of
the coefficients of A and B vanishes. Again, this leads to a quadratic equation in
the phase speed c. The solution (see Problem 8.12) has the form
1
1/2
H
2
H
2
4 cosh αH
αH sinh αH
+
4
α
2
H
2
c
=
±
−
(8.68)
Thus
4 cosh αH
αH sinh αH
+
4
α
2
H
2
c
i
=
0 f1
−
< 0
and the flow is then baroclinically unstable. When the quantity in square brackets
in (8.68) is equal to zero, the flow is said to be
neutrally stable
. This condition
occurs for α
=
α
c
where
α
c
H
(
tanh α
c
H
)
−
1
α
c
H
2
/4
−
+
1
=
0
(8.69)
Using the identity
2 tanh
α
c
H
2
1
tanh
2
α
c
H
2
tanh α
c
H
=
+