Geography Reference
In-Depth Information
rotating horizontal planes so that the gravity and rotation vectors are everywhere
parallel. In such a situation
U
T
k
2
1/2
2λ
2
−
c
=
U
m
±
(8.24)
k
2
2λ
2
+
For waves with zonal wave numbers satisfying k
2
< 2λ
2
, (8.24
)
has an imaginary
part. Thus, all waves longer than the critical wavelength L
c
=
√
2π/λ will amplify.
From the definition of λ we can write
δpπ (2σ )
1/2
/f
0
L
c
=
10
−
3
N
−
1
m
3
s
−
1
. Thus, with
For typical tropospheric conditions (2σ)
1/2
≈
2
×
10
−
4
s
−
1
δp
3000 km. It is also clear
from this formula that the critical wavelength for baroclinic instability increases
with static stability. The role of static stability in stabilizing the shorter waves can
be understood qualitatively as follows: For a sinusoidal perturbation, the relative
vorticity, and hence the differential vorticity advection, increases with the square of
the wave number. However, as shown in Chapter 6, a secondary vertical circulation
is required to maintain hydrostatic temperature changes and geostrophic vorticity
changes in the presence of differential vorticity advection. Thus, for a geopotential
perturbation of fixed amplitude the relative strength of the accompanying vertical
circulation must increase as the wavelength of the disturbance decreases. Because
static stability tends to resist vertical displacements, the shortest wavelengths will
thus be stabilized.
It is also of interest that with β
=
500 hPa and f
0
=
we find that L
c
≈
0 the critical wavelength for instability does
not depend on the magnitude of the basic state thermal wind U
T
. The growth rate,
however, does depend on U
T
. According to (8.17) the time dependence of the dis-
turbance solution has the form exp(-
ikct
). Thus, the exponential growth rate is α
=
=
kc
i
, where c
i
designates the imaginary part of the phase speed. In the present case
kU
T
2λ
2
1/2
k
2
−
=
α
(8.25)
2λ
2
k
2
+
so that the growth rate increases linearly with the mean thermal wind.
Returning to the general case where all terms are retained in (8.21), the stability
criterion is understood most easily by computing the
neutral curve
, which connects
all values of U
T
and k for which δ
=
0 so that the flow is
marginally stable
. From
(8.21), the condition δ
=
0 implies that
U
T
2λ
2
k
2
β
2
λ
4
k
4
2λ
2
k
2
=
−
(8.26)
+