Geography Reference
In-Depth Information
Equation (3.30) is often referred to as the
thermal wind
equation. However, it
is actually a relationship for the vertical wind
shear
(i.e., the rate of change of the
geostrophic wind with respect to ln
p
). Strictly speaking, the term
thermal wind
refers to the vector difference between geostrophic winds at two levels. Designating
the thermal wind vector by
V
T
, we may integrate (3.30) from pressure level p
0
to
level p
1
(p
1
<p
0
) to get
p
1
k
×
∇
p
T
d ln p
R
f
V
T
≡
V
g
(p
1
)
−
V
g
(p
0
)
=−
(3.31)
p
0
denote the mean temperature in the layer between pressure p
0
and
p
1
, the
x
and
y
components of the thermal wind are thus given by
Letting
T
∂
ln
p
0
p
1
∂
ln
p
0
p
1
R
f
T
R
f
T
u
T
=−
;
v
T
=
(3.32)
∂y
∂x
p
p
Alternatively, we may express the thermal wind for a given layer in terms of the
horizontal gradient of the geopotential difference between the top and the bottom
of the layer:
1
f
∂
∂y
(
1
−
1
f
∂
∂x
(
1
−
u
T
=−
0
)
;
v
T
=
0
)
(3.33)
The equivalence of (3.32) and (3.33) can be verified readily by integrating the
hydrostatic equation (3.27) vertically from p
0
to p
1
after replacing T by the mean
T
. The result is the hypsometric equation (1.22):
ln
p
0
p
1
1
−
0
≡
=
gZ
T
R
T
(3.34)
The quantity Z
T
is the
thickness
of the layer between p
0
and p
1
measured in units
of geopotential meters. From (3.34) we see that the thickness is proportional to the
mean temperature in the layer. Hence, lines of equal Z
T
(isolines of thickness) are
equivalent to the isotherms of mean temperature in the layer.
The thermal wind equation is an extremely useful diagnostic tool, which is often
used to check analyses of the observed wind and temperature fields for consistency.
It can also be used to estimate the mean horizontal temperature advection in a layer
as shown in Fig. 3.9. It is clear from the vector form of the thermal wind relation:
ln
p
0
p
1
1
f
k
g
f
k
R
f
k
V
T
=
×
∇
(
1
−
0
)
=
×
∇
Z
T
=
×
∇
T
(3.35)
that the thermal wind blows parallel to the isotherms (lines of constant thickness)
with the warm air to the right facing downstream in the Northern Hemisphere.
