Waves and Instabilities at Midlatitudes - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 268

dynamics are discussed in terms of propagating diurnal and semidiurnal atmo-
spheric tides. This distinction is partly historical but is also related to the origin
of the atmospheric forcing. The upper thermospheric winds are driven in situ by
solar heating, Joule heating, and momentum transfer with the plasma, whereas
the lower E-region winds are usually ascribed to upward-propagating tides gen-
erated at tropospheric and/or stratospheric heights. The oscillation periods of
the semidiurnal and diurnal propagating tides are, of course, 12 and 24 h, respec-
tively. Higher-order tides also occur, but as the oscillation period nears several
hours, the motions are usually referred to as gravity waves. Upper thermospheric
forcing via solar UV heating has a strong diurnal component and drives an in
situ diurnal tide.
The upper atmosphere is continuously bombarded with gravity waves from
a number of sources. These include tropospheric weather fronts, tornadoes and
thunderstorms (Kelley, 1997; Hung et al., 1978), impulsive auroral zone momen-
tum injection and heating events (Richmond andMatsushita, 1975; Nicolls et al.,
2004), and even earthquakes (Kelley et al., 1985). The famous monograph enti-
tled The Upper Atmosphere in Motion by Hines (1974) is an excellent annotated
collection of gravity wave studies published by Hines and coworkers over about
a 10-year period. The reader is referred to that work for details about gravity
waves and tidal oscillations, as well as to the excellent review of tidal theory by
Chapman and Lindzen (1970) mentioned earlier. Here our approach is much
more modest in scope, aiming at physical intuition rather than detailed analysis.
We study gravity waves first, in effect finding the normal modes of a flat
nonrotating inviscid atmosphere. These results will be valid as long as the periods
do not approach the tidal range and the wavelengths are not long enough that the
curvature of the earth matters. We assume an isothermal, inviscid atmosphere
initially in hydrostatic equilibrium, so that if

ρ 0 and p 0 are the zero-order mass

density and pressure, the relation

ρ 0 g =−∇ p 0

ρ 0 and p 0 , which vary only in the
vertical direction in this model, are of the form

applies. In addition, it can be shown that

e − z / H ρ 0 , p 0 ∝

where H is the scale height of the atmosphere—that is, 1 .

Here we again choose our coordinates using the meteorological convention and
take x eastward, y northward, and z vertically upward. We assume there are
no neutral winds in the unperturbed atmosphere. The equations governing the
behavior of the atmosphere are the mass continuity equation (2.2), the equation
of motion (2.20), and the adiabatic condition (see Yeh and Liu, 1974). In the
equation of motion, only terms due to gravity, pressure gradients, and inertia
are retained.

/ H =− ( 1 /ρ 0 )( d ρ 0 / dz )