Geography Reference
In-Depth Information
ν/m respectively.
6
The components of
the group velocity, c
gx
and c
gz
, however, are given by
mean flow) given by c
x
=ˆ
ν/k and , c
z
=ˆ
Nm
2
∂ν
∂k
=
c
gx
=
u
±
(7.45a)
k
2
m
2
3
/
2
+
∂ν
∂m
=±
(
−
Nkm)
c
gz
=
(7.45b)
k
2
m
2
3
/
2
+
where the upper or lower signs are chosen in the same way as in (7.44). Thus,
the vertical component of group velocity has a sign opposite to that of the verti-
cal phase speed relative to the mean flow (downward phase propagation implies
upward energy propagation). Furthermore, it is easily shown from (7.45) that the
group velocity vector is parallel to lines of constant phase. Internal gravity waves
thus have the remarkable property that group velocity is perpendicular to the direc-
tion of phase propagation. Because energy propagates at the group velocity this
implies that energy propagates parallel to the wave crests and troughs, rather than
perpendicular to them as in acoustic waves or shallow water gravity waves. In the
atmosphere, internal gravity waves generated in the troposphere by cumulus con-
vection, by flow over topography, and by other processes may propagate upward
many scale heights into the middle atmosphere, even though individual fluid parcel
oscillations may be confined to vertical distances much less than a kilometer.
Referring again to Fig. 7.9 it is evident that the angle of the phase lines to the
local vertical is given by
L
z
)
1
/
2
m
2
)
1
/
2
L
z
/(L
x
+
k/(k
2
cos α
=
=±
+
=±
k/
|
κ
|
N cos α (i.e., gravity wave frequencies must be less than the buoy-
ancy frequency) in agreement with the heuristic parcel oscillation model (7.24).
The tilt of phase lines for internal gravity waves depends only on the ratio of
the intrinsic wave frequency to the buoyancy frequency, and is independent of
wavelength.
Thus,
ν
ˆ
=±
7.4.2 Topographic Waves
When air with mean wind speed u is forced to flow over a sinusoidal pattern of
ridges under statically stable conditions, individual air parcels are alternately dis-
placed upward and downward from their equilibrium levels and will thus undergo
buoyancy oscillations as they move across the ridges as shown in Fig. 7.10. In
6
Note that phase speed is not a vector. The phase speed in the direction perpendicular to constant
phase lines (i.e., the blunt arrows in Fig. 7.9) is given by ν/(k
2
+
m
2
)
1/2
, which is not equal to
(c
x
+
c
z
)
1/2
.
