Geography Reference
In-Depth Information
however, be complex, in which case m
r
describes sinusoidal variation in z and m
i
describes exponential decay or growth in z depending on whether m
i
is positive
or negative. When m is real, the total wave number may be regarded as a vector
κ
≡
(k, m), directed perpendicular to lines of constant phase, and in the direction
of phase increase, whose components, k
2π/L
z
, are inversely
proportional to the horizontal and vertical wavelengths, respectively. Substitution
of the assumed solution into (7.42) yields the dispersion relationship
=
2π/L
x
and m
=
uk)
2
k
2
m
2
N
2
k
2
(ν
−
+
−
=
0
so that
Nk/
k
2
m
2
1
/
2
ν
ˆ
≡
ν
−
uk
=±
+
=±
Nk/
|
κ
|
(7.44)
where
ν, the
intrinsic frequency
, is the frequency relative to the mean wind. Here,
the plus sign is to be taken for eastward phase propagation and the minus sign for
westward phase propagation, relative to the mean wind.
If we let k>0 and m<0, then lines of constant phase tilt eastward with
increasing height as shown in Fig. 7.9 (i.e., for φ
ˆ
mz to remain constant as x
increases, z must also increase when k>0 and m<0). The choice of the positive
root in (7.44) then corresponds to eastward and downward phase propagation
relative to the mean flow with horizontal and vertical phase speeds (relative to the
=
kx
+
Horizontal distance
Fig. 7.9
Idealized cross section showing phases of pressure, temperature, and velocity perturbations
for an internal gravity wave. Thin arrows indicate the perturbation velocity field, blunt solid
arrows the phase velocity. Shading shows regions of upward motion.
