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monumental proportions in the E and F regions. Eventually, either these waves
break down nonlinearly due to their large amplitude or due to the fact that the
kinematic viscosity (
ρ
0
decreases) that viscous dissipation
balances growth and eventually destroys the wave. Wave breaking is discussed
in Chapter 7.
Equation (6.4) can also be written in the form:
ν/ρ
0
) gets so large (as
2
C
0
k
y
+
k
z
4
2
b
C
0
k
y
−
ω
2
2
ω
−
ω
+
ω
a
ω
=
0
(6.6)
2
b
g
2
C
0
a
where
ω
=
(γ
−
1
)
/
is the square of the Brunt-Väisälä frequency,
ω
=
C
0
/
4
H
2
is the acoustic frequency, and where
k
y
and
k
z
are real. At a given value
of
k
y
,if
ω>ω
a
) the first and
second terms dominate and we recover the sound wave dispersion relation found
previously for
g
ω
is large enough (the high-frequency branch with
=
0. The low-frequency branch corresponds to gravity waves
that propagate only for
ω<ω
b
. Physically, the Brunt-Väisälä frequency is the
frequency at which a parcel of air oscillates about its equilibrium position when
it is initially displaced from that position. For a nonisothermal atmosphere, one
in which
C
0
varies with height, it can be shown (Holton, 1979) that
C
0
+
g
C
0
dC
0
/
dz
2
g
2
ω
b
=
(γ
−
1
)
/
/
(6.7a)
and also
g
θ
d
θ
dz
2
ω
b
=
(6.7b)
where
is the potential temperature (the temperature that a parcel of air would
attain if adiabatically brought to the ground).
Both branches of this dispersion relation (and one more!) are shown in Fig. 6.4.
For
θ
ω>ω
a
we have normal sound waves. There is a forbidden band where
m
(
k
z
)
is imaginary (dotted region) and then for
ω<ω
b
we have inertio-internal gravity
waves, which become modified near the inertial period,
f
, where
f
=
2
sin
θ
(
being the latitude).
Inertial waves are discussed following. Notice that IGWs travel much slower
than the sound speed (for the same
k
being the radian frequency corresponding to a day and
θ
is much less), which allows them to
interact with winds. There is somewhat of a similarity here between the role
of the plasma frequency in electrodynamics and the Brunt-Väisälä frequency in
neutral dynamics. For a plasma to have propagation, you must have
,ω
ω>ω
p
, the
plasma frequency, while for IGWs
ω<ω
b
is required for propagation. Both
ω
p
and
ω
b
represent natural oscillations of the medium. Representative values for
the buoyancy period
T
b
=
(
derived from (6.7a,b) are plotted in Fig. 6.5 as
a function of height. Clearly, the 1-3 h oscillations in the ionospheric parameters
just discussed fall in the gravity wave branch
2
π/ω
b
)
τ
=
(
2
π/ω) >
T
b
.
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