Geography Reference
In-Depth Information
However, from (8.51)
i
=
L so that the first integral in (8.60)
vanishes. Furthermore, if remains finite as z
∗
r
=
0aty
=±
the contribution to the
second integral of (8.60) at the upper boundary vanishes. If we then use (8.57) to
eliminate the vertical derivatives in this term at the lower boundary, (8.60) can be
expressed as
→∞
z
∗
=
0
+
L
∞
+
L
2
2
∂q
∂y
ρ
0
|
|
ε
∂u
∂z
∗
ρ
0
|
|
2
dydz
∗
−
=
c
i
dy
0
(8.61)
2
|
u
−
c
|
|
u
−
c
|
−
L
0
−
L
i
is the disturbance amplitude squared.
Equation (8.61) has important implications for the stability of quasi-geostrophic
perturbations. For unstable modes, c
i
must be nonzero,
a
nd thus the quantity in
square brackets in (8.61) must v
a
nish. Because
2
r
where
|
|
=
+
2
is
no
nnegative,
instability is possible only when ∂u/∂z
∗
at the lower boundary and (∂q/∂y) in the
whole domain satisfy certain constraints:
2
/
|
|
|
u
−
c
|
(a) If ∂u/∂z
∗
at z
∗
=
0 (which by thermal wind balance implies that the merid-
ional temperature gradient vanishes at the boundary), the second integral
in (8.61) vanishes. Thus, the first i
nt
egral must also vanish for instability
to occ
u
r. This can occur only if ∂q/∂y changes sign within the domain
(i.e., ∂q/∂y
0 somewhere). This is referred to as the
Rayleigh necessary
condition
and is another dem
o
nstration of the fundamental role played by
potential vorticity. Because ∂q/∂y is normally positive, it is clear that in the
absence of temperature gradients at the lower boundary, a region of neg-
ative meridional potential vorticity gradients must exist in the interior for
inst
ab
ility to be possible.
(b) If ∂q/∂y
=
0 everywhere, then it is necessary that ∂u/∂z
∗
> 0 somewhere
at t
he
lower boundary for c
i
>0.
(c) If ∂u/∂z
∗
≥
< 0 everywhere at z
∗
=
0, then it is necessary that ∂q/∂y < 0
somewhere for instability to occur. Thus, there is an asymmetry between
westerly and easterly shear at the lower boundary, with the former more
favorable for baroclinic instability.
The basic state potential vorticity gradient in (8.49) can be written in the form
∂
2
u
∂y
2
ε
∂
2
u
∂z
∗
∂q
∂y
=
ε
H
∂u
∂z
∗
−
∂ε
∂z
∗
∂u
∂z
∗
β
−
+
−
2
Because β is positive everywhere, if ε is constant a negative basic state potential
vorticity gradient can occur only for strong positive mean flow curvature (i.e.,
∂
2
u/∂y
2
2
or ∂
2
u/∂z
∗
>> 0 ) or strong negative vertical shear (∂u/∂z
∗
<< 0).
