Fundamentals of Plasma Physics

6.5 High frequency waves: Altar-Appleton-Hartree dispersion relation 197

can be accessed for any given plasma density and magnetic field. Plasmaswithω^2 pe>ω^2 ce
are often labelled ‘overdense’ and plasmas withω^2 pe< ω^2 ceare correspondingly labeled
‘underdense’.For overdense plasmas, the mode line passes to the rightof the intersection
of theP = 0,R=∞bounding surfaces while for underdense plasmas the mode line
passes to the left of this intersection. Two different plasmas will beself-similar if they have
similar mode lines. For example if a lab plasma has the same mode line as a space plasma
it will support the same kind of modes, but do so in a scaled fashion. Becausethe CMA
diagram is log-log the bounding surface curves extend infinitely to the left and right of the
figure and also infinitely above and below it;however no new regions exist outside of what
is sketched in Fig.6.2.
The weakly-magnetized case corresponds to the lower parts of regions 1-5,while the
low-density case corresponds to the left parts of regions 1, 2, 3, 6, 9, 10, and 12. The
CMA diagram provides a visual way for categorizing a great deal of useful information. In
particular, it allows identification of isomorphisms between modes indifferent regions of
parameter space so that understanding developed about the behavior for one kind of mode
can be readily adapted to explain the behavior of a different, but isomorphicmode located
in another region of parameter space.

6.5 High frequency waves: Altar-Appleton-Hartree dispersion relation

Examination of the dielectric tensor elementsS,P,andDshows that while both ion and
electron terms are of importance for low frequency waves, for high frequencywaves (ω>>
ωci,ωpi)the ion terms are unimportant and may be dropped. Thus, for high frequency
waves the dielectric tensor elements simplify to

S=1−

ω^2 pe

ω^2 −ω^2 ce

P=1−

ω^2 pe

ω^2

D=

ωce

ω

ω^2 pe

(ω^2 −ω^2 ce)

(6.64)

and the correspondingRandLterms are

R=1−

ω^2 pe

ω(ω+ωce)

L=1−

ω^2 pe

ω(ω−ωce)

.

(6.65)

The development of long distance short-wave radio communication in the 1930’s motivated
investigations into how radio waves bounce from the ionosphere. Because the bouncing
involves aP=0cutoff and because the ionosphere hasω^2 peof orderω^2 cebut usually larger,
the relevant frequencies must be of the order of the electron plasma frequency and so are
much higher than both the ion cyclotron and ion plasma frequencies. Thus, ion effects are
unimportant and so all ion terms may be dropped in order to simplify the analysis.