Geography Reference
In-Depth Information
It turns out that vertically propagating equatorial waves share a number of physical
properties with ordinary gravity modes. Section 7.5 discussed vertically propagat-
ing gravity waves in the presence of rotation for a simple situation in which the
Coriolis parameter was assumed to be constant and the waves were assumed to be
sinusoidal in both x and y. We found that such inertia-gravity waves can propagate
vertically only when the wave frequency satisfies the inequality f<ν<N. Thus,
at middle latitudes, waves with periods in the range of several days are generally
vertically trapped (i.e., they are not able to propagate significantly into the strato-
sphere). As the equator is approached, however, the decreasing Coriolis frequency
should allow vertical propagation to occur for lower frequency waves. Thus, in
the equatorial region there is the possibility for existence of long-period vertically
propagating internal gravity waves.
As in Section 11.4 we consider linearized perturbations on an equatorial βplane.
The linearized equations of motion, continuity equation, and first law of thermo-
dynamics can then be expressed in log-pressure coordinates as
∂u /∂t
βyv =−
/∂x
(12.29)
∂v /∂t
βyu =−
/∂y
+
(12.30)
ρ 0 w /∂z
∂u /∂x
∂v /∂y
ρ 1
0
+
+
=
0
(12.31)
2 /∂t∂z
w N 2
0 (12.32)
We again assume that the perturbations are zonally propagating waves, but we
now assume that they also propagate vertically with vertical wave number m. Due
to the basic state density stratification, there will also be an amplitude growth in
height proportional to ρ 1/2
0
+
=
. Thus, the x, y, z, and t dependencies can be separated
as
=
u
v w
u (y)
ˆ
ˆ
v (y)
ˆ
e z/2H
+
exp [i (kx
mz
νt)]
(12.33)
w (y)
ˆ
(y)
Substituting from (12.33) into (12.29)-(12.32) yields a set of ordinary differen-
tial equations for the meridional structure:
ik ˆ
u
ˆ
βy
v
ˆ
=−
(12.34)
ˆ
ˆ
+
ˆ
=−
v
βy
u
/∂y
(12.35)
ik
v/∂y +
u
ˆ
+
ˆ
i (m
+
i/2H )
w
ˆ
=
0
(12.36)
i/2H ) ˆ
wN 2
ν (m
=
0
(12.37)
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