Geography Reference
In-Depth Information
Fig. 7.8
Parcel oscillation path (heavy arrow) for pure gravity waves with phase lines tilted at an
angle α to the vertical.
depends only on the static stability (measured by the buoyancy frequency N ) and
the angle of the phase lines to the vertical.
The above heuristic derivation can be verified by considering the linearized equa-
tions for two-dimensional internal gravity waves. For simplicity, we employ the
Boussinesq approximation
, in which density is treated as a constant except where
it is coupled with gravity in the buoyancy term of the vertical momentum equation.
Thus, in this approximation the atmosphere is considered to be incompressible,
and local density variations are assumed to be small perturbations of the constant
basic state density field. Because the vertical variation of the basic state density
is neglected except where coupled with gravity, the Boussinesq approximation is
only valid for motions in which the vertical scale is less than the atmospheric scale
height H(
8km).
Neglecting effects of rotation, the basic equations for two-dimensional motion
of an incompressible atmosphere may be written as
≈
∂u
∂t
+
u
∂u
w
∂u
1
ρ
∂p
∂x
=
∂x
+
∂z
+
0
(7.25)
∂w
∂t
+
u
∂w
w
∂w
1
ρ
∂p
∂z
+
∂x
+
∂z
+
g
=
0
(7.26)
∂u
∂x
+
∂w
∂z
=
0
(7.27)
∂θ
∂t
+
u
∂θ
w
∂θ
∂x
+
∂z
=
0
(7.28)
where the potential temperature θ is related to pressure and density by
p
s
p
κ
p
ρR
θ
=
which after taking logarithms on both sides yields
γ
−
1
ln p
ln θ
=
−
ln ρ
+
constant
(7.29)
