Stability of zonal flows
233
In the case β =0, the eigenvalues become
c = U 2 +
U s K 2 (K 2 +2F 2 )
2K 2 (K 2 + F 2 + F 1 )
±
[U 2
s (K 4 − 4F 1 F 2 )]
1
2
(K 2 + F 2 + F 1 )
,
(10.57)
and hence there is instability (c i > 0) when
K
2 = k
2 +(j +
1
2
)
2 π
2 < 2(F 1 F 2 )
1/2 .
(10.58)
If (j +
1
2 ) 2 π 2 < 2(F 1 F 2 ) 1/2 , then there is an interval of wavenumbers [0,k 0 ] for
which the basic flow is unstable. For a given F 1 and F 2 the largest wavenumber
k 0m (for j =0) perturbation that can grow is given by
k
2
0m =2(F 1 F 2 )
1/2 −
π 2
4
.
(10.59)
This condition is an approximation (i.e., in the two-layer model) of the condition
μ<μ c in the Eady model. Wavenumbers larger than (4F 1 F 2 ) 1/4 and hence
perturbations with wavelength λ ∗ with
λ ∗ >
2πL
k 0m
=2πL(4F 1 F 2 )
−1/4 ,
(10.60)
will be damped.
In the case where F 1 = F 2 = F , the eigenvalues are given by
c =
1
2
(U 1 + U 2 ) −
β(K 2 + F )
K 2 (K 2 +2F )
±
[4β 2 F 2 − K 4 U 2
s (4F 2 − K 4 )] 1/2
2K 2 (K 2 +2F )
, (10.61)
and hence if K 2 > 2F , then the basic state is stable. For K 2 < 2F , the flow only
becomes unstable if U s is large enough, i.e.
U
2
s >U
2
sc =
4β 2 F 2
K 4 (4F 2 − K 4 )
.
(10.62)
This critical value of the shear stress depends on β but is independent of the sign
of U s .T h eβ-effect is hence stabilizing in the baroclinic instability mechanism.
In Fig. 10.10a, U sc > 0 is plotted as a function of K. The minimum of the curve
(∂U s /∂K =0)is given by U sm = β/F. For U s >U sm , there exists an interval of
perturbations (characterized by K) that will grow exponentially. For U s =2U sm ,
the growth factors c i are plotted in Fig. 10.10b.