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The differencein potential means there is an electric field E1 between A and B. Just as in the case ofthe flute instability, E1 causes a drift v1 ¼ E1 B0 =B20 in the x direction. As thewave passes by, traveling in the y direction, an observer at point A will see n1 and ϕ1oscillating in time. The drift v1 will also oscillate in time, and in fact it is v1 whichcauses the density to oscillate.
Since there is a gradient ∇n0 in the x direction, thedrift v1 will bring plasma of different density to a fixed observer A. A drift wave,therefore, has a motion such that the fluid moves back and forth in the x directionalthough the wave travels in the y direction.To be more quantitative, the magnitude of v1x is given byv1x ¼ E y =B0 ¼ ik y ϕ1 =B0ð6:59ÞWe shall assume v1x does not vary with x and that kz is much less than ky; that is, thefluid oscillates incompressibly in the x direction.
Consider now the number ofguiding centers brought into 1 m3 at a fixed point A; it is obviously∂n1 =∂t ¼ v1x ∂n0 =∂xð6:60ÞThis is just the equation of continuity for guiding centers, which, of course, donot have a fluid drift vD. The term n0 ∇ · v1 vanishes because of our previousassumption. The difference between the density of guiding centers and the densityof particles n1 gives a correction to Eq. (6.60) which is higher order and may beneglected here.
Using Eqs. (6.59) and (6.58), we can write Eq. (6.60) asiωn1 ¼ik y ϕ1 0eϕn ¼ iω 1 n0KT eB0 0ð6:61ÞThus we have0ωKT e n0¼¼ vDekyeB0 n0ð6:62ÞThese waves, therefore, travel with the electron diamagnetic drift velocity and arecalled drift waves. This is the velocity in the y, or azimuthal, direction. In addition,there is a component of k in the z direction. For reasons not given here, thiscomponent must satisfy the conditionskz << k yvthi << ω=kz << vtheð6:63Þ6.8 Resistive Drift Waves209To see why drift waves are unstable, one must realize that v1x is not quite Ey/B0 forthe ions. There are corrections due to the polarization drift, Eq. (2.66), and thenonuniform E drift, Eq.
(2.59). The result of these drifts is always to make thepotential distribution ϕ1 lag behind the density distribution n1 (Problem 4.1). Thisphase shift causes v1 to be outward where the plasma has already been shiftedoutward, and vice versa; hence the perturbation grows. In the absence of the phaseshift, n1 and ϕ1 would be 90 out of phase, as shown in Fig.
6.14, and drift waveswould be purely oscillatory.The role of resistivity comes in because the field E1 must not be short-circuitedby electron flow along B0. Electron–ion collisions, together with a long distance1λz2between crest and trough of the wave, make it possible to have a resistivepotential drop and a finite value of E1. The dispersion relation for resistive driftwaves is approximatelywhereω2 þ iσ jj ω ω* ¼ 0ð6:64Þω* k y vDeð6:65Þandσ jj k2zΩc ðωc τei Þk2yð6:66ÞIf σk is large compared with ω, Eq. (6.64) can be satisfied only if ω ω*.
In thatcase, we may replace ω by ω* in the first term. Solving for ω, we then obtainω ω* þ iω2* =σ jjð6:67ÞThis shows that Im(ω) is always positive and is proportional to the resistivity η.Drift waves are, therefore, unstable and will eventually occur in any plasma with adensity gradient. Fortunately, the growth rate is rather small, and there are ways tostop it altogether by making B0 nonuniform.Note that Eq. (6.52) for the flute instability and Eq. (6.64) for the drift instabilityhave different structures. In the former, the coefficients are real, and ω is complexwhen the discriminant of the quadratic is negative; this is typical of a reactiveinstability. In the latter, the coefficients are complex, so ω is always complex; this istypical of a dissipative instability.Problem6.10 A toroidal hydrogen plasma with circular cross section has major radiusR ¼ 50 cm, minor radius a ¼ 2 cm, B ¼ 1 T, KTe ¼ 10 eV, KTi ¼ 1 eV, andn0 ¼ 1019m3. Taking n0/n00 ’ a/2 and g ’ (KTe + KTi)/MR, estimate the2106 Equilibrium and Stabilitygrowth rates of the m ¼ 1 resistive drift wave and the m ¼ 1 gravitational flutemode.
(One can usually apply the slab-geometry formulas to cylindricalgeometry by replacing ky by m/r, where m is the azimuthal mode number.)6.9The Weibel Instability1As an example of an instability driven by anisotropy of the distribution function, wegive a physical picture (due to B. D. Fried) of the Weibel instability, in which amagnetic perturbation is made to grow. This will also serve as an example of anelectromagnetic instability. Let the ions be fixed, and let the electrons be hotter inthe y direction than in the x or z directions. There is then a preponderance of fastelectrons in the y directions (Fig.
6.15), but equal numbers flow up and down,so that there is no net current. Suppose a field B ¼ Bz z^ cos kx spontaneously arisesfrom noise. The Lorentz force ev B then bends the e1ectron trajectories asshown by the dashed curves, with the result that downward-moving electronscongregate at A and upward-moving ones at B. The resulting current sheetsj ¼ en0ve are phased exactly right to generate a B field of the shape assumed,and the perturbation grows. Though the general case requires a kinetic treatment,the simple case vy ¼ v0, vx ¼ vz ¼ 0 has been solved by Fried from this physicalpicture, yielding a growth rate γ ωpv0/c.Fig. 6.15 Physical mechanism of the Weibel instability1A salute to a good friend. Erich Weibel (1925–1983).Chapter 7Kinetic Theory7.1The Meaning of f(v)The fluid theory we have been using so far is the simplest description of a plasma; itis indeed fortunate that this approximation is sufficiently accurate to describe themajority of observed phenomena.
There are some phenomena, however, for whicha fluid treatment is inadequate. For these, we need to consider the velocity distribution function f(v) for each species; this treatment is called kinetic theory. In fluidtheory, the dependent variables are functions of only four independent variables: x,y, z, and t.
This is possible because the velocity distribution of each species isassumed to be Maxwellian everywhere and can therefore be uniquely specified byonly one number, the temperature T. Since collisions can be rare in hightemperature plasmas, deviations from thermal equilibrium can be maintained forrelatively long times.
As an example, consider two velocity distributions f1(vx) andf2(vx) in a one-dimensional system (Fig. 7.1). These two distributions will haveentirely different behaviors, but as long as the areas under the curves are the same,fluid theory does not distinguish between them.The density is a function of four scalar variables: n ¼ n(r, t). When weconsider velocity distributions, we have seven independent variables: f ¼ f(r, v, t).By f(r, v, t), we mean that the number of particles per m3 at position r and time t withvelocity components between vx and vx + dvx, vy and vy + dvy, and vz and vz + dvz isf x; y; z; vx ; v y ; vz ; t dvx dv y dvzThe integral of this is written in several equivalent ways:nðr; tÞ ¼ð11dvxð11dv yð11dvz f ðr; v; tÞ ¼¼ð11ð11© Springer International Publishing Switzerland 2016F.F.
Chen, Introduction to Plasma Physics and Controlled Fusion,DOI 10.1007/978-3-319-22309-4_7f ðr; v; tÞd3 vf ðr; v; tÞdvð7:1Þ2112127 Kinetic TheoryFig. 7.1 Examples of non-Maxwellian distribution functionsNote that dv is not a vector; it stands for a three-dimensional volume element invelocity space. If f is normalized so thatð11^f ðr; v; tÞdv ¼ 1ð7:2Þit is a probability, which we denote by ^f . Thusf ðr; v; tÞ ¼ nðr; tÞ ^f ðr; v; tÞð7:3ÞNote that ^f is still a function of seven variables, since the shape of the distribution,as well as the density, can change with space and time.
From Eq. (7.2), it is clearthat ^f has the dimensions (m/s)3; and consequently, from Eq. (7.3), f has thedimensions s3-m6.A particularly important distribution function is the Maxwellian:where^f m ¼ ðm=2πKT Þ3=2 exp v2 =v2th1=2v v2x þ v2y þ v2zandvth ð2KT=mÞ1=2ð7:4Þð7:5ÞBy using the definite integralð11pffiffiffiexp x2 dx ¼ πone easily verifies that the integral of ^f m over dvx dvy dvz is unity.ð7:6Þ7.1 The Meaning of f(v)213Fig.
7.2 Threedimensional velocity spaceThere are several average velocities of a Maxwellian distribution that arecommonly used. In Sect. 1.3, we saw that the root-mean-square velocity is given by 1=2¼ ð3KT=mÞ1=2v2ð7:7ÞThe average magnitude of the velocity jvj, or simply v, is found as follows:v¼ð11v ^f ðvÞd3 vð7:8ÞSince ^f m is isotropic, the integral is most easily done in spherical coordinates in vspace (Fig. 7.2). Since the volume element of each spherical shell is 4πv2 dv,we havev ¼ ðm=2πKT Þ3=2ð103=2¼ πv2th4πv4th v exp v2 =v2th 4πv2 dvð10 2 3exp y y dyð7:9Þð7:10ÞThe definite integral has a value 12, found by integration by parts.
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