6.3 Dispersion relation expressed as a relation betweenn^2 xandn^2 z 193taining two normal modes corresponding to two qualitatively distinct propagating waves.
However, since two modes do not exist in all bounded volumes, the actual number of modes
is smaller than twenty-six. As an example of a bounded volume with two waves, the wave
normal surfaces of the fast and slow Alfvén waves are in the upper right hand corner of the
CMA diagram since this bounded volume corresponds toω<<ωciandω<<ωpe,i.e.,
aboveL=0and to the right ofP=0.
6.2.8 Use of the CMA diagram
The CMA diagram can be used in several ways. For example it can be used to(i) identify
all allowed cold plasma modes in a given plasma for various values ofωor (ii) investigate
how a given mode evolves as it propagates through a spatially inhomogeneous plasmaand
possibly intersects resonances or cutoffs due to spatial variation of density or magnetic
field.
Let us consider the first example. Suppose the plasma is uniform and has a prescribed
density and magnetic field. Sinceln
[(
ω^2 pe+ω^2 pi)
/ω^2]
andln[
ω^2 ce/ω^2]
are the coordi-
nates of the CMA diagram, varyingωcorresponds to tracing out a line having a slope
of 45^0 and an offset determined by the prescribed density and magnetic field. Highfre-
quencies correspond to the lower left portion of this line and low frequencies to the upper
right. Since the allowed modes lie along this line, if the line does not pass through a given
bounded volume, then modes inside that bounded volume do not exist in the specified
plasma.
Now consider the second example. Suppose the plasma is spatially non-uniform in such
a way that both density and magnetic field are a function of position. To be specific, suppose
that density increases as one moves in thexdirection while magnetic field increases as one
moves in theydirection. Thus, the CMA diagram becomes a map of the actual plasma. A
wave with prescribed frequencyωis launched at some positionx,yand then propagates
along some trajectory in parameter space as determined by its local dispersion relation.
The wave will continuously change its character as determined by the local wave normal
surface. Thus a wave which is injected with a downward velocity as a fast Alfvén mode in
the upper-right bounded volume will pass through theL=∞bounding surface and will
undergo only a quantitative deformation. In contrast, a wave which is launched as a slow
Alfvén mode (dumbbell shape) from the same position will disappear when it reaches the
L=∞bounding surface, because the slow mode does not exist on the lower side (high
frequency side) of theL=∞bounding surface. The slow Alfvén wave undergoes ion
cyclotron resonance at theL=∞bounding surface and will be absorbed there.
6.3 Dispersion relation expressed as a relation betweenn^2 xandn^2 z
The CMA diagram is very useful for classifying waves, but is often notso useful in practical
situations where it is not obvious how to specify the angleθ. In a practical situation a wave
is typically excited by an antenna that lies in a plane and the geometry ofthe antenna
imposes the component of the wavevector in the antenna plane. The transmitterfrequency
determinesω.