Geography Reference
In-Depth Information
Fig. 7.10
Streamlines in steady flow over an infinite series of sinusoidal ridges for the narrow ridge
case (a) and broad ridge case (b). The dashed line in (b) shows the phase of maximum
upward displacement. (After Durran, 1990.)
this case there are solutions in the form of waves that are stationary relative to the
ground [i.e, ν
0 in (7.43)]. For such stationary waves, w
depends only on (x, z)
and (7.42) simplifies to
=
∂
2
w
∂x
2
∂
2
w
∂z
2
N
2
u
2
w
=
+
+
0
(7.46)
Substituting from (7.43) into (7.46) then yields the dispersion relationship
N
2
u
2
m
2
k
2
=
−
(7.47)
Fo
r given values of N , k, and u, (7.47) determines the vertical structure. Then
<N/k, (7.47) shows that m
2
> 0 (i.e., m must be real) and solutions of
(7.46) have the form of vertically propagating waves:
w
=ˆ
if
|
u
|
w exp [i (kx
+
mz)]
