Geography Reference
In-Depth Information
9.4
MOUNTAIN WAVES
Section 7.5.2 showed that stably stratified air forced to flow over sinusoidally vary-
ing surface topography creates oscillations, which can be either vertically propa-
gating or vertically decaying, depending on whether the intrinsic wave frequency
relative to the mean flow is less than or greater than the buoyancy frequency. Most
topographic features on the surface of the earth do not, however, consist of regu-
larly repeating lines of ridges. In general the distance between large topographic
barriers is large compared to the horizontal scales of the barriers. Also, the static
stability and the basic state flow are not usually constants, as was assumed in
Section 7.5.2, but may vary strongly with height. Furthermore, nonlinear modi-
fications of mountain waves are sometimes associated with strong surface winds
along the lee slopes of ridges. Thus, mountain waves are significant features of
mesoscale meteorology.
9.4.1
Flow over Isolated Ridges
Just as flow over a periodic series of sinusoidal ridges can be represented by a
single sinusoidal function, that is, by a single Fourier harmonic, flow over an
isolated ridge can be approximated by the sum of a number of Fourier components
(see Section 7.2.1). Thus, any zonally varying topography can be represented by
a Fourier series of the form
∞
Re
h
s
exp (ik
s
x)
h
M
(x)
=
(9.29)
s
=
1
where h
s
is the amplitude of the sth Fourier component of the topography. We
can then express the solution to the wave equation (7.46) as the sum of Fourier
components:
Re
W
s
exp
i (k
s
x
m
s
z)
∞
=
+
w (x, z)
(9.30)
s
=
1
k
s
.
Individual Fourier modes will yield vertically propagating or vertically decay-
ing contributions to the total solution (9.30) depending on whether m
s
is real or
imaginary. This in turn depends on whether k
s
is less than or greater than N
2
/
ik
s
uh
s
, and m
s
N
2
/u
2
where W
s
=
=
−
u
2
.
Thus, each Fourier mode behaves just as the solution (7.48) for periodic sinusoidal
topography. For a narrow ridge, Fourier components with wave numbers greater
than N/
¯
u dominate in (9.29), and the resulting disturbance decays with height. For
a broad ridge, components with wave numbers less than N/
¯
u dominate and the
disturbance propagates vertically. In the wide mountain limit where m
s
¯
N
2
/
u
2
,
≈
¯
the flow is periodic in the vertical with a vertical wavelength of 2πm
−
1
s
, and phase
lines tilt upstream with height as shown in Fig. 9.7.
