Geography Reference
In-Depth Information
ν
2
f
2
−
1
v
ˆ
=
−
(νl
−
ikf )
p
ˆ
(7.64)
νm
N
2
w
ˆ
=−
p
ˆ
(7.65)
which with the aid of (7.60) yields the dispersion relation for hydrostatic waves.
N
2
k
2
l
2
m
−
2
ν
2
f
2
=
+
+
(7.66)
Because hydrostatic waves must have (k
2
l
2
)/m
2
1, (7.66) indicates that
for vertical propagation to be possible (m real) the frequency must satisfy the
inequality
+
|
f
|
<
|
ν
|
N . Equation (7.66) is just the limit of (7.56) when we let
k
2
l
2
m
2
sin
2
α
1,cos
2
α
→
=
+
which is consistent with the hydrostatic approximation.
If axes are chosen to make l
0, it may be shown (see Problem 7.14) that the
ratio of the vertical to horizontal components of group velocity is given by
c
gz
/c
gx
=
|
=
ν
2
f
2
1
/
2
|
=
k/m
−
/N
(7.67)
Thus, for fixed v, inertia-gravity waves propagate more closely to the horizontal
than pure internal gravity waves. However, as in the latter case the group velocity
vector is again parallel to lines of constant phase.
Eliminating
p between (7.63) and (7.64) for the case l
ˆ
=
0 yields the relation-
ship
v
ˆ
=−
if
u/ν, from which it is easily verified that if
ˆ
u is real, the perturbation
ˆ
horizontal motions satisfy the relations
u
f
ν
sin (kx
u
=ˆ
νt) ,v
=ˆ
u cos (kx
+
mz
−
+
mz
−
νt)
(7.68)
so that the horizontal velocity vector rotates anticyclonically (that is, clockwise in
the Northern Hemisphere) with time. As a result, parcels follow elliptical trajecto-
ries in a plane orthogonal to the wavenumber vector. Equations (7.68) also show
that the horizontal velocity vector turns anticyclonically with height for waves
with upward energy propagation (e. g., waves with m<0 and ν<0). These
characteristics are illustrated by the vertical cross section shown in Fig. 7.12. The
anticyclonic turning of the horizontal wind with height and time is a primary
method for identifying inertia-gravity oscillations in meteorological data.
7.6
ADJUSTMENT TO GEOSTROPHIC BALANCE
Chapter 6 showed that synoptic-scale motions in midlatitudes are in approxi-
mate geostrophic balance. Departures from this balance can lead to the exci-
tation of inertia-gravity waves, which act to adjust the mass and momentum
