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(7.64). The damping is then given byπ ω3p 2ω p 1ω2ImðωÞ ¼ 2 pffiffiffi 3 exp 2 22 k k π vthk vth!! ω2ppffiffiffiωp 33¼ πω pexp 2 2 exp2kvthk vthωImωppffiffiffi ω p 31exp¼ 0:22 πkvth2k2 λ2Dð7:69Þð7:70ÞSince Im(ω) is negative, there is a collisionless damping of plasma waves; thisis called Landau damping. As is evident from Eq. (7.70), this damping isextremely small for small kλD, but becomes important for kλD ¼ O(1).
This effectis connected with f1, the distortion of the distribution function caused bythe wave.2307.57 Kinetic TheoryThe Meaning of Landau DampingThe theoretical discovery of wave damping without energy dissipation by collisionsis perhaps the most astounding result of plasma physics research. That this is areal effect has been demonstrated in the laboratory. Although a simple physicalexplanation for this damping is now available, it is a triumph of applied mathematicsthat this unexpected effect was first discovered purely mathematically in the courseof a careful analysis of a contour integral. Landau damping is a characteristicof collisionless plasmas, but it may also have application in other fields. For instance,in the kinetic treatment of galaxy formation, stars can be considered as atomsof a plasma interacting via gravitational rather than electromagnetic forces.Instabilities of the gas of stars can cause spiral arms to form, but this process islimited by Landau damping.To see what is responsible for Landau damping, we first notice that Im(ω) arisesfrom the pole at v ¼ vϕ.
Consequently, the effect is connected with those particles inthe distribution that have a velocity nearly equal to the phase velocity—the “resonant particles.” These particles travel along with the wave and do not see a rapidlyfluctuating electric field: They can, therefore, exchange energy with the waveeffectively. The easiest way to understand this exchange of energy is to picture asurfer trying to catch an ocean wave (Fig. 7.17). (Warning: this picture is only fordirecting our thinking along the right lines; it does not correctly explain Eq. (7.70).)If the surfboard is not moving, it merely bobs up and down as the wave goes by anddoes not gain any energy on the average. Similarly, a boat propelled much fasterthan the wave cannot exchange much energy with the wave.
However, if thesurfboard has almost the same velocity as the wave, it can be caught and pushedalong by the wave; this is, after all, the main purpose of the exercise. In that case,the surfboard gains energy, and therefore the wave must lose energy and is damped.On the other hand, if the surfboard should be moving slightly faster than the wave, itwould push on the wave as it moves uphill; then the wave could gain energy. In aFig. 7.17 Customary physical picture of Landau damping7.5 The Meaning of Landau Damping231Fig.
7.18 Distortion of aMaxwellian distributionin the region v ’ vϕ causedby Landau dampingFig. 7.19 A doublehumped distribution andthe region whereinstabilities will developplasma, there are electrons both faster and slower than the wave. A Maxwelliandistribution, however, has more slow electrons than fast ones (Fig. 7.18). Consequently, there are more particles taking energy from the wave than vice versa, andthe wave is damped. As particles with v vϕ are trapped in the wave, f(v) isflattened near the phase velocity.
This distortion is f1(v) which we calculated. Asseen in Fig. 7.18, the perturbed distribution function contains the same number ofparticles but has gained total energy (at the expense of the wave).From this discussion, one can surmise that if f0(v) contained more fast particlesthan slow particles, a wave can be excited. Indeed, from Eq. (7.67), it is apparentthat Im(ω) is positive if ∂ ^f 0 =∂v is positive at v ¼ vϕ. Such a distribution is shown inFig. 7.19. Waves with vϕ in the region of positive slope will be unstable, gainingenergy at the expense of the particles. This is just the finite-temperature analogy ofthe two-stream instability.
When there are two cold (KT ¼ 0) electron streams inmotion, f0(v) consists of two δ-functions. This is clearly unstable because ∂f0/∂v isinfinite; and, indeed, we found the instability from fluid theory. When the streamshave finite temperature, kinetic theory tells us that the relative densities andtemperatures of the two streams must be such as to have a region of positive ∂f0/∂v between them; more precisely, the total distribution function must have aminimum for instability.2327 Kinetic TheoryThe physical picture of a surfer catching waves is very appealing, but it isnot precise enough to give us a real understanding of Landau damping.
Thereare actually two kinds of Landau damping: linear Landau damping, andnonlinear Landau damping. Both kinds are independent of dissipative collisionalmechanisms. If a particle is caught in the potential well of a wave, thephenomenon is called “trapping.” As in the case of the surfer, particles canindeed gain or lose energy in trapping. However, trapping does not lie withinthe purview of the linear theory. That this is true can be seen from the equationof motionm d2 x=dt2 ¼ qEðxÞð7:71ÞIf one evaluates E(x) by inserting the exact value of x, the equation would benonlinear, since E(x) is something like sin kx.
What is done in linear theory is to usefor x the unperturbed orbit; i.e., x ¼ x0 + v0t. Then Eq. (7.71) is linear. This approximation, however, is no longer valid when a particle is trapped. When it encountersa potential hill large enough to reflect it, its velocity and position are, of course,greatly affected by the wave and are not close to their unperturbed values. In fluidtheory, the equation of motion is∂vmþ ðv ∇ Þv ¼ qEðxÞ∂tð7:72ÞHere, E(x) is to be evaluated in the laboratory frame, which is easy; but to make upfor it, there is the (v · ∇)v term. The neglect of (v1 · ∇)v1 in linear theory amounts tothe same thing as using unperturbed orbits.
In kinetic theory, the nonlinear term thatis neglected is, from Eq. (7.45),q ∂f1E1m∂vð7:73ÞWhen particles are trapped, they reverse their direction of travel relative to thewave, so the distribution function f(v) is greatly disturbed near v ¼ ω/k. This meansthat ∂f1/∂v is comparable to ∂f0/∂v, and the term (7.73) is not negligible. Hence,trapping is not in the linear theory.When a wave grows to a large amplitude, collisionless damping with trappingdoes occur.
One then finds that the wave does not decay monotonically; rather, theamplitude fluctuates during the decay as the trapped particles bounce back and forthin the potential wells. This is nonlinear Landau damping. Since the result ofEq. (7.67) was derived from a linear theory, it must arise from a different physicaleffect. The question is: Can untrapped electrons moving close to the phase velocityof the wave exchange energy with the wave? Before giving the answer, let usexamine the energy of such electrons.7.5 The Meaning of Landau Damping7.5.1233The Kinetic Energy of a Beam of ElectronsWe may divide the electron distribution f0(v) into a large number of monoenergeticbeams (Fig.
7.20). Consider one of these beams: It has unperturbed velocity u anddensity nu. The velocity u may lie near vϕ, so that this beam may consist of resonantelectrons. We now turn on a plasma oscillation E1(x, t) and consider the kineticenergy of the beam as it moves through the crests and troughs of the wave.The wave is caused by a self-consistent motion of all the beams together. If nu issmall enough (the number of beams large enough), the beam being examinedhas a negligible effect on the wave and may be considered as moving in a givenfield E(x, t). LetE ¼ E1 sin ðkx ωtÞ ¼ dϕ=dxð7:74Þϕ ¼ ðE1 =kÞ cos ðkx ωtÞð7:75ÞThe linearized fluid equation for the beam is∂v1∂v1mþu¼ eE1 sin ðkx ωtÞ∂t∂xð7:76ÞA possible solution isv1 ¼ eE1 cos ðkx ωtÞω kumð7:77ÞThis is the velocity modulation caused by the wave as the beam electrons movepast.
To conserve particle flux, there is a corresponding oscillation in density, givenby the linearized continuity equation:∂n1∂n1∂v1þu¼ nu∂t∂x∂xFig. 7.20 Dissection of adistribution f0(v) into a largenumber of monoenergeticbeams with velocityu and density nuð7:78Þ2347 Kinetic TheoryFig. 7.21 Phase relations of velocity and density for electrons moving in an electrostatic waveSince v1 is proportional to cos(kx ωt), we can try n1 ¼ n1 cos(kx ωt). Substitution of this into Eq. (7.78) yieldsn1 ¼ nueE1 k cos ðkx ωtÞmðω kuÞ2ð7:79ÞFigure 7.21 shows what Eqs. (7.77) and (7.79) mean.
The first two curves showone wavelength of E and of the potential eϕ seen by the beam electrons. The thirdcurve is a plot of Eq. (7.77) for the case ω ku < 0, or u > vϕ. This is easilyunderstood: When the electron a has climbed the potential hill, its velocity is7.5 The Meaning of Landau Damping235small, and vice versa. The fourth curve is v1 for the case u < vϕ and it is seen that thesign is reversed. This is because the electron b, moving to the left in the frame of thewave, is decelerated going up to the top of the potential barrier; but since it ismoving the opposite way, its velocity v1 in the positive x direction is maximumthere. The moving potential hill accelerates electron b to the right, so by the time itreaches the top, it has the maximum v1.
The final curve on Fig. 7.21 shows thedensity n1, as given by Eq. (7.79). This does not change sign with u vϕ, because inthe frame of the wave, both electron a and electron b are slowest at the top of thepotential hill, and therefore the density is highest there. The point is that the relativephase between n1 and v1 changes sign with u vϕ.We may now compute the kinetic energy Wk of the beam:1W k ¼ m ð nu þ n1 Þ ð u þ v 1 Þ 221 ¼ m nu u2 þ nu v21 þ 2un1 v1 þ n1 u2 þ 2nu uv1 þ n1 v212ð7:80ÞThe last three terms contain odd powers of oscillating quantities, so they will vanishwhen we average over a wavelength.
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