190 Chapter 6. Cold plasma waves in a magnetized plasma
modes. Parameter space is divided up into a finite number of regions, called bounded vol-
umes, separated by curves in parameter space, called bounding surfaces,across which the
modes changequalitatively. Thus, within a bounded volume, modes change quantitatively
but not qualitatively. For example, if Alfvén waves exist at one point in aparticular bounded
volume, they must exist everywhere in that bounded volume, although the dispersion may
not be quantitatively the same at different locations in the volume.
The appropriate choice of bounding surfaces consists of:
- Theprinciple resonanceswhich are the curves in parameter space wheren^2 has a
resonance at eitherθ=0orθ=π/ 2 .Thus, the principle resonances are the curves
R=∞(i.e., electron cyclotron resonance),L=∞(i.e., ion cyclotron resonance),
andS=0(i.e., the upper and lower hybrid resonances). - The cutoffsR=0,L=0, andP=0.
The behavior of wave normal surfaces inside a bounded volume and when crossing a
bounded surface can be summarized in five theorems (Stix 1962), each a simpleconse-
quence of the results derived so far: - Inside a bounded volumencannot vanish. Proof: nvanishes only whenPRL=0,
butP=0,R=0andL=0have been defined to be bounding surfaces. - Ifn^2 has a resonance (i.e., goes to infinity) atanypoint in a bounded volume, then
foreveryother point in the same bounded volume, there exists a resonance at some
unique angleθresand its associated mirror angles, namely−θres,π−θres, and
−(π−θres)but atnoother angles. Proof: Ifn^2 →∞thenA→ 0 in which case
tan^2 θres=−P/S determines the uniqueθres.Nowtan^2 (π−θres) = tan^2 θres
so there is also a resonance at the supplementθ=π−θres.Also, since the square
of the tangent is involved, bothθresandπ−θresmay be replaced by their negatives.
NeitherP norScan change sign inside a bounded volume and both are single valued
functions of their location in parameter space. Thus,−P/Scan only change sign at
a bounding surface. In summary, if a resonance occurs at any point in a bounding
surface, then a resonance exists at some unique angleθresand its associated mirror
angles at every point in the bounding surface. Resonances only occur whenPand
Shave opposite signs. Since 1 /ngoes to zero at a resonance, the radius of a wave
normal surface goes to zero at a resonance. - At any point in parameter space, for a given interval inθ in whichnis finite,n is
either pure real or pure imaginary throughout that interval. Proof:n^2 is always real
and is a continuous function ofθ.The only situation wherencan change from being
pure real to being pure imaginary is whenn^2 changes sign. This occurs whenn^2
passes through zero, but because of the definition for bounding surfaces,n^2 does not
vanish inside a bounded volume. Althoughn^2 may change sign when going through
infinity,this situation is not relevant because the theorem was restrictedto finiten. - nis symmetric about θ=0andθ=π/ 2 .Proof:n is a function ofsin^2 θand of
cos^2 θ, both of which are symmetric aboutθ=0andθ=π/ 2. - Except for the special case where the surfacesPD=0andRL=PSintersect, the
two modes may coincide only atθ=0 or atθ=π/ 2 .Proof: For 0 <θ<π/ 2 the