Geography Reference
In-Depth Information
in conservation form we simplify the right-hand side by using the chain rule of
differentiation to write term C as
f
0
σ
f
0
σ
∂
∂p
∂
∂p
∂
V
g
∂p
·∇
∂
∂p
−
V
g
·∇
−
However, from the thermal wind relation, f
0
∂
V
g
/∂p
=
k
×
∇
(∂/∂p),so
that ∂
V
g
/∂p is perpendicular to
(∂/∂p), and the second term in the above
expression vanishes. The first term (which is proportional to the geostrophic advec-
tion of the normalized static stability anomaly) can be combined with term B in
(6.23). Recalling the definition of χ and dividing through by f
0
, (6.23) can then
be expressed simply in the form of a conservation equation:
∂
∂t
+
∇
q
D
g
q
Dt
=
V
g
·∇
=
0
(6.24)
where q is the quasi-geostrophic potential vorticity defined by
1
f
0
∇
f
0
σ
∂
∂p
∂
∂p
2
q
≡
+
f
+
(6.25)
The three parts of (6.25), reading from left to right, are the geostrophic
relative
vorticity, the
planetary
vorticity, and the
stretching
vorticity. As a parcel moves
about in the atmosphere, the geostrophic relative vorticity, the planetary vorticity,
and the stretching vorticity terms may each change. However, according to (6.24)
their sum is conserved following the geostrophic motion. This conservation is
consistent with (6.21) and (6.22), which show that for adiabatic flow, changes in
the absolute vorticity are balanced by opposite changes in the stretching vorticity.
The scalar q is proportional to a linearized form of the Ertel potential vorticity,
P , discussed in Section 4.3. However, unlike Ertel's potential vorticity, q has units
of vorticity and is determined solely by the isobaric geopotential distribution. Note
that whereas Ertel's potential vorticity is conserved following the adiabatic motion
on an isentropic surface, q is conserved under adiabatic conditions following the
geostrophic motion on an isobaric surface. For this reason some authors prefer
the term
pseudo-potential vorticity
. However, we will generally refer to q as the
quasi-geostrophic potential vorticity.
According to (6.25) quasi-geostrophic potential vorticity tends to be proportional
to minus the geopotential. A local increase in q is associated with trough develop-
ment, whereas a decrease in q is associated with ridge development. Because q is
a conserved quantity following the geostrophic motion, we can diagnose the ten-
dency purely from the geostrophic advection of q. Furthermore, (6.24) shows that
the tendency will be zero (i.e., the flow will be steady) provided that the geostrophic
wind is everywhere parallel to lines of constant q (i.e., that
V
g
·∇
q
=
0).
