Waves and Instabilities at Midlatitudes - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 269

Now consider atmospheric oscillations in the presence of gravity. We assume
there are small perturbations in the mass density, pressure, and wind veloc-
ity denoted by

. Without the Coriolis or viscous
forces there is no coupling between oscillations in the y - z plane and those in
the x direction, so we can ignore the x component of velocity, making the
problem two-dimensional. For meridional propagation we define a column
vector F by

δρ , δ p , and U = ( u , v , w )

δρ/ρ 0
δ p / p 0

F =

v
w

and assume that atmospheric perturbations can be described by plane waves of
the form

( ω t − k y y − k z z )

e i F ∝ (6.1)

into the equations
describing the atmosphere (see Chapter 2) and retaining terms up to first order
in

Substituting ρ = ρ 0 + δρ , p = p 0 + δ p , U = ( 0 , v , w )

p , and U gives the linearized forms of the mass continuity, motion, and
adiabatic state equations—that is,

δρ , δ

∂(δρ)/∂ t + U ·∇ ρ 0 + ρ 0 ∇· U = 0 (6.2a)

ρ 0 ∂ v /∂ t + ∂(δ p )/∂ y = 0 (6.2b)

ρ 0 ∂ w /∂ t + ∂(δ p )/∂ z + δρ g = 0 (6.2c)

C 0 ∂(δρ)/∂ C 0 U ∂(δ p )/∂ t + U ·∇ p 0 − t − ·∇ ρ 0 = 0 (6.2d)

We have taken the atmosphere to be isothermal with temperature T . In (6.2d),
C 0 is the speed of sound, given by

C 0 = γ p 0 /ρ 0 = γ gH

where γ is the ratio of specific heats at constant pressure and constant volume

and H Mg is the scale height. Viscosity has been ignored (inviscid fluid).

Using (6.1) and the condition for hydrostatic equilibrium, (6.2) can be rewritten
as a matrix equation:

= k B T /

i ω 0 − ik y − 1 / H − ik z

ik y C 0 /γ 0 − i ω 0

C 0 1 ik z /γ

· F = 0 (6.2e)

g − / H + 0 i ω

C 0 C 0 /γ

− i ω i ω 0 (γ − 1 ) g