Geography Reference
In-Depth Information
The geostrophic wind defined in (6.7) is nondivergent. Thus,
∂u
a
∂x
+
∂v
a
∂y
∇·
V
=
∇·
V
a
=
and the continuity equation (6.3) can be rewritten as
∂u
a
∂x
+
∂v
a
∂y
+
∂ω
∂p
=
0
(6.12)
which shows that if the geostrophic wind is defined by (6.7), ω is determined only
by the ageostrophic part of the wind field.
In the thermodynamic energy equation (6.4), the horizontal advection can be
approximated by its geostrophic value. However, the vertical advection is not
neglected, but forms part of the adiabatic heating and cooling term. This term
must be retained because the static stability is usually large enough on the synoptic
scale so that the adiabatic heating or cooling due to vertical motion is of the same
order as the horizontal temperature advection, despite the smallness of the vertical
velocity. The adiabatic heating and cooling term can be somewhat simplified by
dividing the total temperature field, T
tot
, into a basic state (standard atmosphere)
portion that depends only on pressure, T
0
(p), plus a deviation from the basic state,
T (
x, y, p, t
), as was done for potential temperature in Section 2.7.4. Thus,
T
tot
(x, y, p, t)
=
T
0
(p)
+
T (x, y, p, t)
, only the basic state portion of the temper-
ature field need be included in the static stability term, and the quasi-geostrophic
thermodynamic energy equation may be expressed in the form
∂
∂t
+
Now because
|
dT
0
/dp
||
∂T /∂p
|
T
σp
R
ω
J
c
p
V
g
·∇
−
=
(6.13a)
RT
0
p
−
1
dlnθ
0
/dp and θ
0
is the potential temperature corresponding
to the basic state temperature T
0
[
where σ
≡−
10
−
6
m
2
Pa
−
2
s
−
2
in the midtropo-
sphere]. From (6.2) the thermodynamic energy equation can be expressed in terms
of the geopotential field. The result is
∂
∂t
+
σ
≈
2.5
×
−
∂
∂p
κJ
p
V
g
·∇
−
=
σω
(6.13b)
R
c
p
.
Equations (6.7), (6.11), (6.12), and (6.13b) constitute the quasi-geostrophic
equations. These form a complete set in the dependent variables ,
V
g
,
V
a
, and ω
(provided that the diabatic heating rate is known). This form of the equations is not,
however, suitable as a prediction system. As shown in the next section, it proves
where κ
≡
