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DYNAMICAL OCEANOGRAPHY
(10.4) Barotropic and baroclinic instability
Consider a zonal channel with y ∈ [−1, 1] and a background stratification
ρ b = ρ b (z). Assess the stability (and the type of instability mechanism
responsible) of the following (dimensionless) flows:
a. U (y)=6sinπy, V = W =0.
b. ψ(y, z)=−y 4 (z +1).
c.
U =
⎧
⎨
⎩
−
¯
U
2
: −1 1
2
¯
Uy : −
1
2 1
2
¯
U
2
:
1
2 where ¯
U is constant.
(10.5) Baroclinic instability in the Eady model
The relation between the condition of the motion of fluid particles with respect
to isopycnals and the instability condition in the Eady model (μ<μ c ) is not
immediately transparent.
a. Show that
∂ρ∗
∂y∗
∂ρ∗
∂z∗
= −
Dǫ
SL
∂ ¯
ρ
∂y
+ O(
D
L
ǫF )
b. Derive for this case that the acceleration a ∗ is
a ∗ = −g.P
ζ ∗
ρ ∗
∂ρ ∗
∂z ∗
(1 −
Dǫ
SL
∂ ¯
ρ
∂y
η ∗
ζ ∗
)
c. Subsequently show that η ∗ and ζ ∗ are related to the perturbation velocities
through
η ∗
ζ ∗
=
˜
v ∗
˜
w ∗
=
L
Dǫ
˜
v 0
˜
w 1
where ˜
v 0 and ˜
w 1 are the dimensionless velocities of the perturbations; these
are determined from the eigensolutions with positive growth factor.