Geography Reference
In-Depth Information
way to cool the atmosphere as required by the thickness tendency is by adiabatic
cooling through the vertical motion field. Thus, the vertical motion maintains a
hydrostatic temperature field (i.e., a field in which temperature and thickness are
proportional) in the presence of differential vorticity advection. Without this com-
pensating vertical motion, either the vorticity changes at 500 hPa could not remain
geostrophic or the temperature changes in the 500- to 1000-hPa layer could not
remain hydrostatic.
To summarize, we have shown as a result of scaling arguments that for synoptic-
scale motions where vorticity is constrained to be geostrophic and temperature is
constrained to be hydrostatic, the vertical motion field is determined uniquely by
the geopotential field. Further, we have shown that this vertical motion field is just
that required to ensure that changes in vorticity will be geostrophic and changes in
temperature will be hydrostatic. These constraints, whose importance can hardly
be overemphasized, are elaborated in the next subsection.
6.4.2
The Q Vector
In order to better appreciate the essential role of the divergent ageostrophic motion
in quasi-geostrophic flow, it is useful to examine separately the rates of change,
following the geostrophic wind, of the vertical shear of the geostrophic wind and
of the horizontal temperature gradient.
On the midlatitude β-plane the quasi-geostrophic prediction equations may be
expressed simply as
D
g
u
g
Dt
−
f
0
v
a
−
βyv
g
=
0
(6.38)
D
g
v
g
Dt
+
f
0
u
a
+
βyu
g
=
0
(6.39)
D
g
T
Dt
−
σp
R
J
c
p
ω
=
(6.40)
These are coupled by the thermal wind relationship
f
0
∂u
g
R
p
∂T
∂y
,
f
0
∂v
g
R
p
∂T
∂x
∂p
=
∂p
=−
(6.41a,b)
or in vector form:
f
0
k
∂
V
g
∂p
R
p
∇
×
=
T
(6.42)
Equations for the evolution of the thermal wind components are obtained by taking
partial derivatives with respect to p in (6.38) and (6.39), multiplying through by
f
0
, and applying the chain rule of differentiation in the advective part of the total
derivative to obtain
