Geography Reference
In-Depth Information
where
f
f
∂b
∂z
,F
2
∂u
g
∂y
f
∂M
∂y
∂b
∂y
N
s
N
2
,S
2
≡
+
≡
−
=
≡−
(9.16)
where M is the absolute momentum defined below (7.53).
Equation (9.15) can be compared with the quasi-geostrophic version obtained
by neglecting advection by the ageostrophic circulation in (9.7) and (9.8). This has
the form
N
2
∂
2
ψ
M
∂y
2
f
2
∂
2
ψ
M
∂z
2
+
=
2Q
2
(9.17)
Thus, in the quasi-geostrophic case the coefficients in the differential operator
on the left depend only on the standard atmosphere static stability, N , and the
planetary vorticity, f , whereas in the semigeostrophic case they depend on the
deviation of potential temperature from its standard profile through the N
s
and S
terms and the absolute vorticity through the F term.
An equation of the form (9.17), in which the coefficients of the derivatives on
the left are positive, is referred to as an
elliptic boundary value problem
. It has
a solution ψ
M
that is uniquely determined by Q
2
plus the boundary conditions.
For a situation such as that of Fig. 9.1b, with both ∂v
g
/∂y and ∂b/∂y negative,
the forcing term Q
2
is negative in the frontal region. The streamfunction in that
case describes a circulation symmetric about the y axis, with rising on the warm
side and sinking on the cold side. The semigeostrophic case (9.15) is also an
elliptic boundary value problem provided that N
s
F
2
S
4
> 0. It can be shown
(see Problem 9.1) that this condition requires that the Ertel potential vorticity be
positive in the Northern Hemisphere and negative in the Southern Hemisphere,
which is nearly always the case in the extratropics for an unsaturated atmosphere.
The spatial variation of the coefficients in (9.16), and the presence of the cross-
derivative term, produce a distortion of the secondary circulation, as shown in
Fig. 9.5. The frontal zone slopes toward the cold air side with height; there is
an intensification of the cross-frontal flow near the surface in the region of large
absolute vorticity on the warm air side of the front, and a tilting of the circulation
with height.
The influence of the ageostrophic circulation on the time scale for frontogenesis
can be illustrated by comparing the processes included in quasi-geostrophic and
semigeostrophic frontogenesis. For semigeostrophic advection there is a positive
feedback that reduces the time scale of frontogenesis greatly compared to that in
the quasi-geostrophic case. As the temperature contrast increases, Q
2
increases,
and the secondary circulation must also increase so that the amplification
rate
of
|
−
rather than remaining constant as in the quasi-
geostrophic case. Because of this feedback, in the absence of frictional effects, the
semigeostrophic model can produce an infinite temperature gradient at the surface
in less than half a day.
∂T /∂y
|
increases with
|
∂T /∂y
|
