Geography Reference
In-Depth Information
If the basic state flow is assumed to be a zonally directed geostrophic wind u
g
,
and it is assumed that the parcel displacement does not perturb the pressure field,
the approximate equations of motion become
Du
Dt
=
f
Dy
Dt
=
fv
(7.49)
f
u
g
−
u
Dv
Dt
=
(7.50)
We consider a parcel that is moving with the geostrophic basic state motion at a
position y
y
0
. If the parcel is displaced across stream by a distance δy,wecan
obtain its new zonal velocity from the integrated form of (7.49):
=
u (y
0
+
δy)
=
u
g
(y
0
)
+
fδy
(7.51)
The geostrophic wind at y
0
+
δy can be approximated as
∂u
g
∂y
u
g
(y
0
+
δy)
=
u
g
(y
0
)
+
δy
(7.52)
Using (7.51) and (7.52) to evaluate (7.50) at y
0
+
δy yields
f
f
δy
D
2
δy
Dt
2
Dv
Dt
=
∂u
g
∂y
f
∂M
∂y
=−
−
=−
δy
(7.53)
where we have defined the
absolute momentum
, M
u
g
.
This equation is mathematically of the same form as (2.52), the equation for the
motion of a vertically displaced particle in a stratified atmosphere. Depending on
the sign of the coefficient on the right-hand side in (7.53), the parcel will either be
forced to return to its original position or will accelerate further from that position.
This coefficient thus determines the condition for
inertial instability
:
≡
fy
−
f
f
> 0
stable
f
∂M
∂u
g
∂y
∂y
=
−
=
0
neutral
(7.54)
< 0
unstable
Viewed in an inertial reference frame, instability results from an imbalance
between the pressure gradient and inertial forces for a parcel displaced radially in
an axisymmetric vortex. In the Northern Hemisphere, where f is positive, the flow
is inertially stable provided that the absolute vorticity of the basic flow, ∂M
∂y,is
positive. In the Southern Hemisphere, however, inertial stability requires that the
absolute vorticity be negative. Observations show that for extratropical synoptic-
scale systems the flow is always inertially stable, although near neutrality often
