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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
=
+
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