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These fields can be measured by Faraday rotation of frequency-doubled light (λ0 ¼ 0.53 μm) derived from thesame laser. If B ¼ 100 T, n ¼ 1027 m3, and the path length in the plasma is30 μm, what is the Faraday rotation angle in degrees? (Assume k║B.)4.25. A microwave interferometer employing the ordinary wave cannot be usedabove the cutoff density nc. To measure higher densities, one can use theextraordinary wave.(a) Write an expression for the cutoff density ncx for the X wave.(b) On a v2ϕ /c2 vs. ω diagram, show the branch of the X-wave dispersionrelation on which such an interferometer would work.4.18Hydromagnetic WavesThe last part of our survey of fundamental plasma waves concerns low-frequencyion oscillations in the presence of a magnetic field. Of the many modes possible, weshall treat only two: the hydromagnetic wave along B0, or Alfvén wave, and themagnetosonic wave.
The Alfvén wave in plane geometry has k along B0; E1 and j1perpendicular to B0; and B1 and v1 perpendicular to both B0 and E1 (Fig. 4.45).From Maxwell’s equation we have, as usual (Eq. (4.80)),∇ ∇ E1 ¼ kðk E1 Þ þ k2 E1 ¼ω2iωE1 þj2ε0 c2 1cð4:118Þ4.18Hydromagnetic Waves127Fig. 4.45 Geometry of anAlfvén wave propagatingalong B0^ by assumption, only the x component of this equation isSince k ¼ k^z and E1 ¼ E1 xnontrivial. The current j1 now has contributions from both ions and electrons, sincewe are considering low frequencies.
The x component of Eq. (4.118) becomesε0 ω2 c2 k2 E1 ¼ iωn0 eðvix vex Þð4:119Þ1ieΩ2c1 2E1vix ¼Mωω1e ΩcΩ21 2cE1viy ¼Mω ωωð4:120ÞThermal motions are not important for this wave; we may therefore use the solutionof the ion equation of motion with Ti ¼ 0 obtained previously in Eq. (4.63). Forcompleteness, we include here the component viy, which was not written explicitlybefore:The corresponding solution to the electron equation of motion is found by lettingM ! m, e ! e, Ωc ! ωc and then taking the limit ω2c ω2 :ie ω2E1 ! 0mω ω2ce ωc ω2E1¼E1 ¼ 22m ω ωcB0vex ¼veyð4:121Þ1284 Waves in PlasmasIn this limit, the Larmor gyrations of the electrons are neglected, and the electronshave simply an E B drift in the y direction.
Inserting these solutions intoEq. (4.119), we obtainε01ieΩ2c1 2E1ω c k E1 ¼ iωn0 eMωω22 2ð4:122ÞThe y components of v1 are needed only for the physical picture to be given later.Using the definition of the ion plasma frequency Ωp (Eq. (4.49)), we have1Ω2ω2 c2 k2 ¼ Ω2p 1 2cωð4:123ÞWe must now make the further assumption ω2 Ω2c ; hydromagnetic waves havefrequencies well below ion cyclotron resonance. In this limit, Eq. (4.123) becomesω2 c2 k2 ¼ ω2Ω2pΩ2c¼ ω2n0 e 2 M 2ρ¼ ω2ε0 M e2 B20ε0 B20ω2c2c2¼¼1 þ ρμ0 =B20 c2k2 1 þ ρ=ε0 B20ð4:124Þwhere ρ is the mass density n0M. This answer is no surprise, since the denominatorcan be recognized as the relative dielectric constant for low-frequency perpendicular motions (Eq. (3.28)).
Equation (4.124) simply gives the phase velocity for anelectromagnetic wave in a dielectric medium:ωcc¼¼k ðεR μR Þ1=2 ε1=2Rfor μR ¼ 1As we have seen previously, ε is much larger than unity for most laboratoryplasmas, and Eq. (4.124) can be written approximately asωB0¼ vϕ ¼kðμ0 ρÞ1=2ð4:125ÞThese hydromagnetic waves travel along B0 at a constant velocity vA, called theAlfvén velocity:vA B=ðμ0 ρÞ1=2ð4:126ÞThis is a characteristic velocity at which perturbations of the lines of force travel.The dielectric constant of Eq.
(3.28) can now be written4.18Hydromagnetic Waves129ER ¼ E=E0 ¼ 1 þ c2 =v2Að4:127ÞNote that vA is small for well-developed plasmas with appreciable density, andtherefore ER is large.To understand what happens physically in an Alfvén wave, recall that this is anelectromagnetic wave with a fluctuating magnetic field B1 given by∇ E1 ¼ B_ 1Ex ¼ ðω=kÞB yð4:128ÞThe small component By, when added to B0, gives the lines of force a sinusoidalripple, shown exaggerated in Fig. 4.46.
At the point shown, By is in the positivey direction, so, according to Eq. (4.128), Ex is in the positive x direction if ω/k is inthe z direction. The electric field Ex gives the plasma an E1 B0 drift in the negativey direction. Since we have taken the limit ω2 Ω2c ; both ions and electrons willhave the same drift vy, according to Eqs. (4.120) and (4.121).
Thus, the fluid movesup and down in the y direction, as previously indicated in Fig. 4.45. The magnitudeof this velocity is |Ex/B0|. Since the ripple in the field is moving by at the phasevelocity ω/k, the line of force is also moving downward at the point indicated inFig. 4.46. The downward velocity of the line of force is (ω/k)|By/B0|, which,according to Eq. (4.128), is just equal to the fluid velocity |Ex/B0|. Thus, the fluidand the field lines oscillate together as if the particles were stuck to the lines.
Thelines of force act as if they were mass-loaded strings under tension, and an Alfvénwave can be regarded as the propagating disturbance occurring when the strings areplucked. This concept of plasma frozen to lines of force and moving with them is auseful one for understanding many low-frequency plasma phenomena. It can beshown that this notion is an accurate one as long as there is no electric field along B.It remains for us to see what sustains the electric field Ex which we presupposedwas there. As E1 fluctuates, the ions’ inertia causes them to lag behind the electrons,and there is a polarization drift vp in the direction of E1.
This drift vix is given byEq. (4.120) and causes a current j1 to flow in the x direction. The resulting j1 B0force on the fluid is in the y direction and is 90 out of phase with the velocity v1.Fig. 4.46 Relation among the oscillating quantities in an Alfvén wave and the (exaggerated)distortion of the lines of force1304 Waves in PlasmasFig. 4.47 Geometry of ashear Alfvén wave in acylindrical columnFig.
4.48 Schematic of an experiment to detect Alfvén waves. [From J. M. Wilcox, F. I. Boley,and A. W. DeSilva, Phys. Fluids 3, 15 (1960).]This force perpetuates the oscillation in the same way as in any oscillator where theforce is out of phase with the velocity. It is, of course, always the ion inertia thatcauses an overshoot and a sustained oscillation, but in a plasma the momentum istransferred in a complicated way via the electromagnetic forces.In a more realistic geometry for experiments, E1 would be in the radial directionand v1 in the azimuthal direction (Fig. 4.47).
The motion of the plasma is thenincompressible. This is the reason the ∇p term in the equation of motion could beneglected. This mode is called the torsional Alfvén wave or shear Alfvén wave. Itwas first produced in liquid mercury by B. Lehnert.Alfvén waves in a plasma were first generated and detected by Allen, Baker,Pyle, and Wilcox at Berkeley, California, and by Jephcott in England in 1959. Thework was done in a hydrogen plasma created in a “slow pinch” discharge betweentwo electrodes aligned along a magnetic field (Fig. 4.48).
Discharge of a slowcapacitor bank A created the plasma. The fast capacitor B, connected to the metalwall, was then fired to create an electric field E1 perpendicular to B0. The ringing ofthe capacitor generated a wave, which was detected, with an appropriate time delay,4.19Magnetosonic Waves131Fig. 4.49 Measured phase velocity of Alfvén waves vs.
magnetic field. [From Wilcox, Boley, andDeSilva, loc. cit.]by probes P. Figure 4.49 shows measurements of phase velocity vs. magnetic field,demonstrating the linear dependence predicted by Eq. (4.126).This experiment was a difficult one, because a large magnetic field of 1 T wasneeded to overcome damping. With large B0, vA (and hence the wavelength),become uncomfortably large unless the density is high. In the experiment of Wilcoxet al., a density of 6 1021 m3 was used to achieve a low Alfvén speed of2.8 105 m/s. Note that it is not possible to increase ρ by using a heavier atom.The frequency ω ¼ kvA is proportional to M1/2, while the cyclotron frequency Ωc isproportional to M1.
Therefore, the ratio ω/Ωc is proportional to M1/2. With heavieratoms it is not possible to satisfy the condition ω2 Ω2c :4.19Magnetosonic WavesFinally, we consider low-frequency electromagnetic waves propagating across B0.Again we may take B0 ¼ B0^z and E1 ¼ E1 ^x ; but we now let k ¼ k^y (Fig. 4.50).Now we see that the E1 B0 drifts lie along k, so that the plasma will becompressed and released in the course of the oscillation. It is necessary, therefore,to keep the ∇p term in the equation of motion.
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