Geography Reference
In-Depth Information
downward and negatively buoyant when displaced upward so that it will tend to
return to its equilibrium level and the atmosphere is said to be statically stable or
stably stratified
.
Adiabatic oscillations of a fluid parcel about its equilibrium level in a stably
stratified atmosphere are referred to as
buoyancy oscillations
. The characteristic
frequency of such oscillations can be derived by considering a parcel that is dis-
placed vertically a small distance δz without disturbing its environment. If the
environment is in hydrostatic balance, ρ
0
g
dp
0
/dz, where p
0
and ρ
0
are the
pressure and density of the environment. The vertical acceleration of the parcel is
=−
D
2
Dt
2
(δz)
Dw
Dt
=
1
ρ
∂p
∂z
=−
g
−
(2.50)
where p and ρ are the pressure and density of the parcel. In the parcel method it is
assumed that the pressure of the parcel adjusts instantaneously to the environmental
pressure during the displacement: p
p
0
. This condition must be true if the
parcel is to leave the environment undisturbed. Thus with the aid of the hydrostatic
relationship, pressure can be eliminated in (2.50) to give
=
g
ρ
0
−
D
2
Dt
2
(δz)
ρ
g
θ
θ
0
=
=
(2.51)
ρ
where (2.44) and the ideal gas law have been used to express the buoyancy force
in terms of potential temperature. Here θ designates the deviation of the potential
temperature of the parcel from its basic state (environmental) value θ
0
(z). If the
parcel is initially at level z
0 where the potential temperature is θ
0
(0), then for a
small displacement δz we can represent the environmental potential temperature as
=
+
dθ
0
dz
δz
θ
0
(δz)
≈
θ
0
(0)
If the parcel displacement is adiabatic, the potential temperature of the parcel is
conserved. Thus, θ(δz)
=
θ
0
(0)
−
θ
0
(δz)
=−
(dθ
0
/dz)δz, and (2.51) becomes
D
2
Dt
2
(δz)
N
2
δz
=−
(2.52)
where
g
d ln θ
0
dz
is a measure of the static stability of the environment. Equation (2.52) has a general
solution of the form δz
N
2
=
A exp(
iNt
). Therefore, if N
2
> 0, the parcel will oscillate
about its initial level with a period τ
=
=
2π/N. The corresponding frequency N is
