222
DYNAMICAL OCEANOGRAPHY
where | A | is the norm of the complex number A. Integration over the interval
[−1, 1] and use of the boundary conditions (10.19b), which also must be satisfied
by A ∗ ,gives
1
−1
A
′ |
2 − (λ + k
2 )|A|
2
dy −
1
−1
|A|
2 U ′′ − β
U − c
dy =0.
(10.21)
By multiplying the numerator and denominator of the integrand in the second
-0.5
0
0.5
1
1.5
2
2.5
-1
-0.5
0
0.5
1
y
a = 2
a = 1
a = 0.5
U(y)
Figure 10.2. Plot of the zonal velocity U (y)=
a
2
(1 + cos πy) for a =0.5, 1.0 and a =2.0.
integral of (10.21) by U − (c r − ic i ), the integral can be written as
1
−1
| A |
2 (U ′′ − β)(U − (c r − ic i ))
| U − c | 2
dy.
(10.22)
The equality (10.21) must hold for both real and imaginary part. The first integral
is a real number and for the second (10.22) gives
c i
1
−1
|A|
2 U ′′ − β
| U − c | 2 dy =0.
(10.23)
Ex. 10.1
For instability, the inequality c i > 0 must hold and hence a necessary condition
for instability is that the function U ′′ − β must change sign on the interval [−1, 1].
If this does not occur, then c i =0must hold and the flow U (y) cannot be unstable.
For β>0, this condition is called the Kuo criterium; for β =0it is called the
Rayleigh criterium. Be careful with this criterium, because it does not provide a
sufficient condition for instability; a flow which satisfies this condition can still