Stability of zonal flows
231
where
q n = ∇
2 φ n − F n (−1)
n (φ 2 − φ 1 ),
(10.48a)
¯
Π n (y)=βy + ¯
ψ
′′ (y) − F n (−1)
n ( ¯
ψ
′
2 (y) − ¯
ψ
′
1 (y)),
(10.48b)
with boundary conditions
y = ±1:
∂φ n
∂x
=0.
(10.49)
We consider the special case that
¯
ψ n (y)=−U n y.
(10.50)
Here, the U n are the constant zonal velocities in both layers and this flow is a
solution of (10.45) and satisfies the boundary conditions (Fig. 10.9).
H 1
H 2
z = - h
z = 0
z = -1
U 1
U 2
Figure 10.9. Sketch of the steady flow and perturbations in the Phillips model.
We search for separable solutions of (10.47) of the form
φ n (x, y, t)=Φ n (y)e
ik(x−ct) .
(10.51)
This gives two coupled equations for the Φ n as
(U 1 − c)(Φ
′′
1 − k
2 Φ 1 − F 1 (Φ 1 − Φ 2 ))
+Φ 1 (β + F 1 (U 1 − U 2 )) = 0,
(10.52a)
(U 2 − c)(Φ
′′
2 − k
2 Φ 2 − F 2 (Φ 2 − Φ 1 ))
+Φ 2 (β + F 2 (U 2 − U 1 )) = 0,
(10.52b)
with
Φ 1 (−1) = Φ 1 (1) = Φ 2 (−1) = Φ 2 (1) = 0.
(10.53)