Geography Reference
In-Depth Information
the
buoyancy frequency
.
3
For average tropospheric conditions, N
10
−
1
s
−
1
≈
1.2
×
so that the period of a buoyancy oscillation is about 8 min.
In the case of N
0, examination of (2.52) indicates that no accelerating force
will exist and the parcel will be in neutral equilibrium at its new level. However,
if N
2
< 0 (potential temperature decreasing with height) the displacement will
increase exponentially in time. We thus arrive at the familiar gravitational or static
stability criteria for dry air:
=
d
θ
0
/dz
> 0
statically stable,
d
θ
0
/dz
=
0
statically neutral,
d
θ
0
/dz
< 0
statically unstable.
On the synoptic scale the atmosphere is always stably stratified because any
unstable regions that develop are stabilized quickly by convective overturning.
For a moist atmosphere, the situation is more complicated and discussion of that
situation will be deferred until Chapter 11.
2.7.4
Scale Analysis of the Thermodynamic Energy Equation
If potential temperature is divided into a basic state θ
0
(z) and a deviation θ (x, y, z, t)
so that the total potential temperature at any point is given by θ
tot
+
θ (x, y, z, t), the first law of thermodynamics (2.46) can be written approximately
for synoptic scaling as
=
θ
0
(z)
∂θ
∂t
+
1
θ
0
u
∂θ
v
∂θ
∂y
w
d ln θ
0
dz
J
c
p
T
∂x
+
+
=
(2.53)
1,
dθ
dz
dθ
0
dz, and
where we have used the facts that for
|
θ/θ
0
|
ln
θ
0
1
θ
θ
0
≈
ln θ
tot
=
+
ln θ
0
+
θ/θ
0
Outside regions of active precipitation, diabatic heating is due primarily to net
radiative heating. In the troposphere, radiative heating is quite weak so that typi-
cally J
c
p
≤
1˚C d
−
1
(except near cloud tops, where substantially larger cooling
can occur due to thermal emission by the cloud particles). The typical amplitude
of horizontal potential temperature fluctuations in a midlatitude synoptic system
(above the boundary layer) is θ
∼
4˚C. Thus,
∂θ
∂t
+
T
θ
0
u
∂θ
v
∂θ
∂y
θU
L
∼
4
◦
Cd
-1
∂x
+
∼
3
N
is often referred to as the Brunt-Vaisala frequency.
