Waves and Instabilities at Midlatitudes - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 270

In deriving (6.2e) we used the fact from (6.1) that δ p ∝ p 0 exp i (ω t − k y y − k z z )

and that the zero-order pressure varies only vertically. This leads to ∂(δ p )/∂ z =

δ and eventually to the corresponding entry in row 3.

Setting the determinant of the 4

p [ ( 1 / p 0 )( dp 0 / dz ) − ik z ]

4 matrix equal to zero yields the dispersion
relation for linear modes of a nonrotating neutral atmosphere on a flat earth,

×

2 C 0 k y +

k z

4

g 2 k y +

2 k z =

ω − ω + (γ − 1 ) i γ g ω 0 (6.3)

A variety of possible wave modes are buried in this dispersion relation. Suppose
we take the limit that g

= 0. Then (6.3) reduces to

C 0 k y + k z

2

ω =

which is the dispersion relation for sound waves propagating without attenua-
tion, growth (pure real constant).

We now turn to the gravity wave case. If there are no sources of energy or
dissipation (viscosity was ignored), waves will not growor decay in time at a fixed
point in space, so we can assume

ω and k ), or dispersion ( ω/ k =

ω

is real. If we are including gravity, however, it
can be shown that there are no solutions of (6.3) with both k y and k z purely real.
Anticipating the final result, let us assume k y is purely real and investigate k z . This
corresponds to a wave propagating in an unattenuated fashion with a component
in the horizontal direction. Then we can write (6.3) as

4 2 C 0 k y + (γ −

g 2 k y =−

2 k z + ω 2 C 0 k z

ω − ω 1 ) i γ g ω (6.4)

where the left-hand side is purely real. Now if we let k z be a complex number,

k z + ik z k z =

it is straightforward to show that the right-hand side of (6.4) is purely real if and
only if

k z = ( 1 / 2 H )

Dropping the superscript (prime) notation, we can now see that the solutions
for the quantities in the column vector F are of the form

( ω t − k y y − k z z ) e z / 2 H

e i (6.5)

In (6.5) , k y , and k z are real.

Atmospheric waves that propagate in the manner described by (6.5) are termed
internal gravity waves (IGW). Some of the complexity of the wind patterns that
arise in the 90-120 km height range due to such waves can be gauged from the
photograph of a trimethyl aluminum (TMA) vapor trail deployed by a sounding
rocket, which is shown in Fig. 6.1. This photograph yields only one perspective

ω