Stability of zonal flows
221
where the prime indicates differentiation to y. The boundary conditions are
z = −1, 0:(
∂
∂t
− ¯
ψ
′ ∂
∂x
)
∂φ
∂z
=0,
(10.13a)
y = −1, 1:
∂φ
∂x
=0.
(10.13b)
This system of equations has solutions of the form
φ(x, y, z, t)= ˆ
φ(y, z)e
ik(x−ct) ,
(10.14)
where c = c r + ic i is the complex growth factor. If c i > 0, then perturbations will
grow and the zonal flow will be unstable. The problem for ˆ
φ(y, z) becomes
−( ¯
ψ
′ + c)
∂ 2 ˆ
φ
∂y 2 − k
2 ˆ
φ +
∂
∂z
(
1
S
∂ ˆ
φ
∂z
)
+( ¯
ψ
′′′ + β) ˆ
φ =0,
(10.15)
with boundary conditions
z = −1, 0:
∂ ˆ
φ
∂z
=0,
(10.16a)
y = −1, 1: ˆ
φ =0.
(10.16b)
If (10.18a) is satisfied then (10.13a) will certainly be satisfied; we will restrict
ourselves here to perturbations that satisfy (10.16a).
Just as with the structure of the stratified Rossby waves in chapter 7, there again
exist separable solutions
ˆ
φ(y, z)=Φ(z)A(y),
(10.17)
if Φ satisfies the equation
(S
−1 Φ
′ )
′ = −χΦ,
(10.18a)
Φ
′ (0) = Φ
′ (−1) = 0.
(10.18b)
With this splitting, the eigenvalue problem (10.15) (with eigenvalue c) becomes
A
′′ − (χ + k
2 )A − A
U ′′ − β
U − c
=0
(10.19a)
A(−1) = A(1) = 0.
(10.19b)
The growth of the barotropic (χ =0 ) as well as of the baroclinic (χ>0) modes
is described by a similar eigenvalue problem as in section 8.3.1.
If we multiply (10.19a) by A ∗ , the complex conjugate of A,wefind
A
∗ A
′′ − (χ + k
2 )|A|
2 −|A |
2 U ′′ − β
U − c
=0,
(10.20)