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(2.50) as€vx ¼Let us defineω2cexiω Evx ωc B!ð2:61Þ382evp exEevE BSingle-Particle Motionsexiω Eωc Bð2:62Þwhere the tilde has been added merely to emphasize that the drift is oscillating. Theupper (lower) sign, as usual, denotes positive (negative) q. Now Eqs. (2.50) and(2.51) become€vx ¼ ω2c vx evp€v y ¼ ω2c v y evEð2:63ÞBy analogy with Eq. (2.12), we try a solution which is the sum of a drift and agyratory motion:vx ¼ v⊥ eiωc t þ evpv y ¼ iv⊥ eiωc t þ evEð2:64ÞIf we now differentiate twice with respect to time, we find€vx ¼ ω2c vx þ ω2c ω2 evp222€v y ¼ ωc v y þ ωc ω evEð2:65ÞThis is not the same as Eq.
(2.63) unless ω2 ω2c . If we now make the assumptionthat E varies slowly, so that ω2 ω2c , then Eq. (2.64) is the approximate solution toEq. (2.63).Equation (2.64) tells us that the guiding center motion has two components. They component, perpendicular to B and E, is the usual E B drift, except that vE nowoscillates slowly at the frequency ω. The x component, a new drift along thedirection of E, is called the polarization drift. By replacing iω by ∂/∂t, we cangeneralize Eq.
(2.62) and define the polarization drift asvp ¼ 1 dEωc B dtð2:66ÞSince vp is in opposite directions for ions and electrons, there is a polarizationcurrent; for Z ¼ 1, this isnedEρ dE¼ 2j p ¼ ne vi p ve p ¼ 2 ðM þ mÞdteBB dtwhere ρ is the mass density.ð2:67Þ2.6 Time-Varying B Field39Fig. 2.12 The polarizationdriftThe physical reason for the polarization current is simple (Fig. 2.12). Consideran ion at rest in a magnetic field. If a field E is suddenly applied, the first thing theion does is to move in the direction of E. Only after picking up a velocity v does theion feel a Lorentz force ev B and begin to move downward in Fig. (2.12).
IfE is now kept constant, there is no further vp drift but only a vE drift. However,if E is reversed, there is again a momentary drift, this time to the left. Thus vp is astartup drift due to inertia and occurs only in the first half-cycle of each gyrationduring which E changes. Consequently, vp goes to zero with ω/ωc.The polarization effect in a plasma is similar to that in a solid dielectric, whereD ¼ ε0 E þ P. The dipoles in a plasma are ions and electrons separated by a distancerL. But since ions and electrons can move around to preserve quasineutrality, theapplication of a steady E field does not result in a polarization field P.
However, ifE oscillates, an oscillating current jp results from the lag due to the ion inertia.2.6Time-Varying B FieldFinally, we allow the magnetic field to vary in time. Since the Lorentz force isalways perpendicular to v, a magnetic field itself cannot impart energy to a chargedparticle. However, associated with B is an electric field given by∇E¼ B_ð2:68Þand this can accelerate the particles. We can no longer assume the fields to becompletely uniform. Let v⊥ ¼ dl/dt be the transverse velocity, l being the element ofpath along a particle trajectory (with vk neglected).
Taking the scalar product of theequation of motion (2.8) with v⊥, we haved 1 2 dlmv ¼ qE v⊥ ¼ qE dt 2 ⊥dtð2:69Þ402Single-Particle MotionsThe change in one gyration is obtained by integrating over one period:δ12 ð 2π=ωcdlqE dtmv2⊥ ¼dt0If the field changes slowly, we can replace the time integral by a line integral overthe unperturbed orbit:ð þ1δ mv2⊥ ¼ qE dl ¼ q ð∇EÞ dS2sðð2:70Þ¼ q B_ dSSHere S is the surface enclosed by the Larmor orbit and has a direction given by theright-hand rule when the fingers point in the direction of v.
Since the plasma isdiamagnetic, we have B · dS < 0 for ions and >0 for electrons. Then Eq. (2.70)becomes221_1_ 2 ¼ qπB_ v⊥ m ¼ 2 mv⊥ 2πBδ mv2⊥ ¼ qBπrL2ωcωc qBBð2:71Þ_ c ¼ B=_ f c is just the change δB during one period of gyration.The quantity 2πB=ωThus1δ mv2⊥ ¼ μ δBð2:72Þ2Since the left-hand side is δ(μB), we have the desired resultδμ ¼ 0ð2:73ÞThe magnetic moment is invariant in slowly varying magnetic fields.As the B field varies in strength, the Larmor orbits expand and contract, and theparticles lose and gain transverse energy. This exchange of energy between theparticles and the field is described very simply by Eq. (2.73). The invariance of μallows us to prove easily the following well-known theorem:The magnetic flux through a Larmor orbit is constant.The flux Ф is given by BS, with S ¼ πr 2L ThusΦ ¼ Bπv2⊥v2 m2 2πm 1mv22πm¼ Bπ ⊥2 2 ¼ 2 2 ⊥ ¼ 2 μ2qqωcBqBð2:74ÞTherefore, Ф is constant if μ is constant.This property is used in a method of plasma heating known as adiabaticcompression.
Figure 2.13 shows a schematic of how this is done. A plasma isinjected into the region between the mirrors A and B. Coils A and B are then pulsed2.7 Summary of Guiding Center Drifts41Fig. 2.13 Two-stage adiabatic compression of a plasmato increase B and hence v2⊥ . The heated plasma can then be transferred to the regionC–D by a further pulse in A, increasing the mirror ratio there. The coils C and D arethen pulsed to further compress and heat the plasma. Early magnetic mirror fusiondevices employed this type of heating.
Adiabatic compression has also been usedsuccessfully on toroidal plasmas and is an essential element of laser-driven fusionschemes using either magnetic or inertial confinement.2.7Summary of Guiding Center DriftsGeneral force F :vf ¼1F Bq B2ð2:17ÞElectric field :vE ¼EBB2ð2:15Þm gBq B2ð2:18ÞGravitational field :Nonuniform E :vg ¼ EB1vE ¼ 1 þ r 2L ∇ 24B2ð2:59ÞB ∇BB2ð2:24ÞNonuniform B fieldGrad B drift :Curvature drift :12v∇B ¼ v⊥ r LvR ¼mv2k Rc Bq R2c B2ð2:26Þ422Curved vacuum field :vR þ v∇B ¼Polarization drift :2.8Single-Particle Motionsm 2 1 2 Rc Bv þ vq k 2 ⊥ R2c B2ð2:30Þ1 dEωc B dtð2:66Þvp ¼ Adiabatic InvariantsIt is well known in classicalþ mechanics that whenever a system has a periodicmotion, the action integral pdq taken over a period is a constant of the motion.Here p and q are the generalized momentum and coordinate which repeat themselves in the motion.
If a slow change is made in the system, so that the motion isnot quite periodic, the constant of the motion does not change and is then called anadiabatic invariant. By slowþ here we mean slow compared with the period ofmotion, so that the integralpdq is well defined even though it is strictly no longeran integral over a closed path. Adiabatic invariants play an important role in plasmaphysics; they allow us to obtain simple answers in many instances involvingcomplicated motions. There are three adiabatic invariants, each corresponding toa different type of periodic motion.2.8.1The First Adiabatic Invariant, μWe have already met the quantityμ ¼ mv2⊥ =2Band have proved its invariance in spatially and temporally varying B fields. Theperiodic motion involved, of course, is the Larmor gyration.
If we take p to beangular momentum mv⊥r and dq to be the coordinate dθ, the action integralbecomesþþp dq ¼ mv⊥ r L dθ ¼ 2πr L mv⊥ ¼ 2πmv2⊥m¼ 4πμωcjqjð2:75ÞThus μ is a constant of the motion as long as q/m is not changed. We have provedthe invariance of μ only with the implicit assumption ω/ωc 1, where ω is afrequency characterizing the rate of change of B as seen by the particle.
A proofexists, however, that μ is invariant even when ω ωc. In theorists’ language, μ isinvariant “to all orders in an expansion in ω/ωc.” What this means in practice is thatμ remains much more nearly constant than B does during one period of gyration.2.8 Adiabatic Invariants43It is just as important to know when an adiabatic invariant does not exist as toknow when it does.
Adiabatic invariance of μ is violated when ω is not smallcompared with ωc. We give three examples of this.(A) Magnetic Pumping. If the strength of B in a mirror confinement system isvaried sinusoidally, the particles’ v⊥ would oscillate; but there would be nogain of energy in the long run. However, if the particles make collisions, theinvariance of μ is violated, and the plasma can be heated. In particular, aparticle making a collision during the compression phase can transfer part of itsgyration energy into vk energy, and this is not taken out again in the expansionphase.(B) Cyclotron Heating.
Now imagine that the B field is oscillated at the frequencyωc. The induced electric field will then rotate in phase with some of theparticles and accelerate their Larmor motion continuously. The conditionω ωc is violated, μ is not conserved, and the plasma can be heated.(C) Magnetic Cusps. If the current in one of the coils in a simple magnetic mirrorsystem is reversed, a magnetic cusp is formed (Fig. 2.14). This configurationhas, in addition to the usual mirrors, a spindle-cusp mirror extending over 360in azimuth.
A plasma confined in a cusp device is supposed to have betterstability properties than that in an ordinary mirror. Unfortunately, the loss-conelosses are larger because of the additional loss region; and the particle motion isnonadiabatic. Since the B field vanishes at the center of symmetry, ωc is zerothere; and μ is not preserved. The local Larmor radius near the center is largerthan the device. Because of this, the adiabatic invariant μ does not guaranteethat particles outside a loss cone will stay outside after passing through thenonadiabatic region. Fortunately, there is in this case another invariant: thecanonical angular momentum pθ ¼ mrvθ erAθ.
This ensures that there will bea population of particles trapped indefinitely until they make a collision.Fig. 2.14 Plasma confinement in a cusped magnetic field442.8.22Single-Particle MotionsThe Second Adiabatic Invariant, JConsider a particle trapped between two magnetic mirrors: It bounces betweenthem and therefore has a periodicmotion at the “bounce frequency.” A constant ofþthis motion is given bymvk ds, where ds is an element of path length (of theguiding center) along a field line. However, since the guiding center drifts acrossfield lines, the motion is not exactly periodic, and the constant of the motionbecomes an adiabatic invariant. This is called the longitudinal invariant J and isdefined for a half-cycle between the two turning points (Fig.
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