Stability of zonal flows
225
(r =0), there is no bottom topography (η b =0), S is constant and we put β =0
(f -plane approximation). Consider the steady state ¯
ψ(y, z)= − y(z +1),such
that the velocity field is given by U (z)=¯ u(z)=z +1,V =¯ v =0. Hence there
is only vertical shear in the basic flow and from the geostrophic and hydrostatic
balances
∂ψ
∂x
= v ; −
∂ψ
∂y
= u ;
∂ψ
∂z
= −ρ,
(10.25)
the thermal wind balance is (section 8.3)
∂u
∂z
=
∂ρ
∂y
;
∂v
∂z
= −
∂ρ
∂x
,
(10.26)
The density field of the zonal flow is hence given by
¯
ρ(y, z)=y.
(10.27)
The equations which describe the evolution of small perturbations on this steady
flow are (with q = ∇ 2 φ + S −1 φ zz )
∂q
∂t
+(z +1)
∂q
∂x
=0,
(10.28a)
z =0:−(
∂
∂t
+
∂
∂x
)
∂φ
∂z
+
∂φ
∂x
=0,
(10.28b)
z = −1:−
∂
∂t
∂φ
∂z
+
∂φ
∂x
=0,
(10.28c)
y = ±1:
∂φ
∂x
=0.
(10.28d)
These contain solutions
φ(x, y, z, t)=Φ(y, z)e
ik(x−ct) ,
(10.29)
and substitution gives
(z +1− c)(
1
S
∂ 2 Φ
∂z 2 +
∂ 2 Φ
∂y 2 − k
2 Φ) = 0,
(10.30a)
z = −1:c
∂Φ
∂z
+Φ=0,
(10.30b)
z =0:(1− c)
∂Φ
∂z
− Φ=0,
(10.30c)
y = ±1:Φ=0.
(10.30d)
For z +1− c =0we find solutions of (10.30a) of the form
Φ(y, z)=A(z)cos(n +
1
2
)πy,
(10.31)