VELOCITY SPACE DIFFUSION FROM WEAK PLASMA TURBULENCE IN A STRONG MAGNETIC FIELD

Transcription

1 REFERENCE IC/66/17 INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS VELOCITY SPACE DIFFUSION FROM WEAK PLASMA TURBULENCE IN A STRONG MAGNETIC FIELD C. F. KENNEL 1966 PIAZZA OBERDAN TRIESTE

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3 IC/66/17 INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS VELOCITY SPACE DIFFUSION FROM WEAK PLASMA TURBULENCE IN A STRONG MAGNETIC FIELD 1 " CF. KENNEL* TRIESTE March 1966 t Submitted to "Physics of Fluids" * On leave of absence from AVCO-Everett Research Laboratory, Everett, Mass., USA

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5 * ABSTRACT We consider quasi-linear velocity space" diffusion for waves of any oscillation branch propagating at an arbitrary angle to a uniform magnetic field in a spatially uniform plasma. We neglect non-linear mode coupling. In the limit that the distribution function changes slowly compared to a gyroperiod, this diffusion takes a simple form. An H-like theorem suggests that a steady state exists which is marginally stable to all modes. Particles with velocities much larger than typical phase velocities in the excited spectrum'are scattered primarily ih pitch angle, while particles are efficiently energized by turbulent diffusion only to energies corresponding to typical wave phase velocities.

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7 VELOCITY SPACE DIFFUSION FROM WEAK PLASMA TURBULENCE IN A STRONG MAGNETIC FIELD 1. INTRODUCTION The theory of weak plasma turbulence has recently undergone a considerable development (1-13). Much of this extensive development clarified the formal nature of the theory. The present paper makes no contribution to the formal understanding of quasi-linear theory, but attempts to draw general conclusions assuming the correctness of the present theoretical approach with the aim of possibly being useful in the interpretation of experimental phenomena. Therefore, we remove some restrictions which were unimportant in investigations of the formal structure of weak turbulence theory but which greatly simplified the algebraic manipulations. These restrictions involve mainly the nature of the waves considered, and are not basic mathematical restrictions inherent in the weak turbulence approximation. One such limitation was the usual but by no means universal consideration of turbulence only in electrostatic wave modes. Because of the very low particle energy densities (relative to magnetic) commonly encountered in laboratory devices, electrostatic waves may play the dominant role in the laboratory., However, the investigations of the Earth's immediate envirpnment in space using artificial Earth satellites have created a new class of plasma observations in which electromagnetic waves such as whistlers and Ion cyclotron waves most certainly play a role. Thus we consider electromagnetic as well as electrostatic wave turbulence. In addition, special propagation directions, usually parallel to the magnetic field, were usually chosen for analytic simplicity. Clearly, waves propagating at an angle to the magnetic field will be present in actual experiments. The removal of the above two restrictions are the basic effects we investigate. -1-

8 The basic plasma is assumed non-relativistic, collisionless, infinite and spatially uniform in the absence of wave excitation, and imrnersed in a strong static magnetic field of constant magnitude with no curvature. We do not restrict the excitation spectrum, except to require that wave amplitudes be small. However, we investigate only the form of the lowest order turbulent velocity diffusion, neglecting the direct nonlinear coupling between waves. Previous work {7,13) indicates that the non-linear coupling is formally the same order as the diffusion considered here. The present equation will adequately describe the evolution of the distribution function when velocity space gradients are moderately large, so that the particle distribution changes more rapidly then the wave distribution, or when resonant mode r couplings are forbidden, and forms a part of a more complete picture when these other effects cannot be neglected. Interestingly enough, this apparently undiscriminating approach leads to several interesting conclusions, which are, albeit of a qualitative nature, quite general because both the diffusion operator and the diffusion coefficient, when expressed in terms of the electric field polarization amplitudes, have a simple form. In particular, we investigate the conditions under which an arbitrary turbulent wave spectrum scatters particles in pitch angle relative to the magnetic field and when it scatters particles in energy. Particles with velocities much larger than typical phase velocities in the excited spectrum are scattered primarily in pitch angles, whereas only particles with velocities the order of wave phase velocities or less can be efficiently energized. Many investigations have considered the so-called "self-consistent" initial-value quasi-linear problem, where an initially unstable particle distribution evolves by the unstable growth of waves and subsequent velocity space diffusion to a final steady state. Experimental situations are likely to have sources and sinks of particles and waves which determine the observable physical processes. Clearly, detailed understanding of these effects is necessary for each particular experimental realization of -2-

9 the theory. Nevertheless, we discuss the problem of "plateau formation 11 in the three-dimensional velocity distribution and the approach, via turbulent diffusion, to a final steady state of marginal stability to all waves. Naturally, here we are limited td infinite plasmas without sinks and sources. An H-like theorem suggests that steady,marginally stable, states can exist in the presence of a finite wave amplitude, ignoring mode coupling. However, the approach to the steady state could well be slow. In Section 2 we derive formally the quasi-linear diffusion equation. Here no new paths are broken. Essential to the success of the present derivation is the assumption that the distribution function changes slowly on time scales of the typical gyro-periods, so that the coupling between the inhomogeneities of the distribution function in the Larmor phase variable due to the turbulent wave spectrum can be broken. Thus diffusion changes the distribution slowly compared with typical Larmor periods, and the distribution function may be taken independent of the Larmor phase variable. This is a strong magnetic field assumption. The diffusion in "weak" magnetic fields may well be considerably different. In Section 3, we derive the specific form of the diffusion equation, and discuss the broad consequences of this form in Section FORMAL DERIVATION OF QUASI-LINEAR DIFFUSION EQUATION It is sufficient for our purposes to calculate the diffusion equation and coefficients to the order of the square of the wave amplitude only. The method of derivation of the second order quasi-linear diffusion equation is by now quite standard, and will be only outlined in brief. plasma: I We start with the Vlasov equations for a collision-free two-component (2.1) where f± (x, y, t) is the one-particle distribution function for ions or electrons, denoted by ± respectively. E and B are the electric and mag- -3-

10 netic fields respectively. Maxwell's equations using moments of the distribution functions as source terms complete the set. The plasma is assumed to consist of an average steady state which is infinite and spatially homogeneous on which is superposed a spectrum of randomly fluctuating fields whose amplitudes are considered small, though non-linear effects are not totally negligible. The unidirectional static magnetic field B o points along the z-axis for simplicity and there is no uniform static electric field in equilibrium. Denoting fluctuating components of the distribution functions by 6f, we write: where we have Fourier-analyzed the fluctuating components in space. Transforming to cylindrical velocity space co-ordinates aligned along the static field v x =Vx cos0 v x (2. 3) V z = in equation (2.1) leads to 9F o ± /3 0 = 0. The base distribution functions p ± are therefore independent of the phase 0 of particle gyration about the magnetic field. Upon Fourier-analysis of (2.1) and transformation to cylindrical co-ordinates, the Vlasov equation for a fluctuating component Sf^1, (k \ 0) becomes (2.4) -4-

11 + it where Lji and P denote the operators ay (2. 6) and fl =ieb( /M ± c the Larmor frequency for each species. Note that 1+ contains the sign of the charge. Wherever the sense is clear, we shall drop the + notation distinguishing between species. The above equations apply only to the finite wavelength components with non-zero k. We treat k. = 0 terms separately. The spatially uniform component of the distribution function evolves slowly from nonlinear interactions between waves and particles. This will introduce formally divergent terms in an expansion in powers of the wave amplitude. However, Galeev, et al. (13) have shown that summation of these divergent terms leads to a renormalization of the base spatially homogeneous distribution function. It is sufficient to replace F^ by g* = F^ + 5f(f (t), where 6f 0 is the k = 0 component of the perturbed distribution function. Putting k = 0 in (2. 4) we find A. where we have taken I _ o = 0. Thus the evolution of g- is formally second order in the linearization amplitude. 6fu, with k'-k 0, has time variations from two sources: rapid oscillations from the high frequency wave distributions, and slow changes from the Sf^t) variation and other non-linear effects. Neglecting the slow changes, 7rr 6f k may be replaced in lowest order by -iwef^, where w is the wave frequency. Then (2.4) reduces to -5-

12 f 2 8 ) The terms of 0(A^) represent non-linear mode coupling and will not be considered further. (2.8) can be solved for 6f7, which then can be substituted into (2. 7) to estimate the slow evolution of gi. An extremely rapid time variation, on the order of the "fast" wave 5 ' frequency itself, is implied by the term -Q t 9/9 # in equation (2. 7). If g 1 had appreciable 4> inhomogeneities, changes in g ± in times of the order of l/fi ± would violate the physically reasonable assumption that particle distributions are sensibly constant over wave and gyro-periods. These considerations lead us to look for an expansion of g ± in powers of 1/Q t ; Assuming that the RHS of (2. 7) leads only to a slow time variation in g*, of order yg*, where y is a typical wave growth rate, then we have 7 and so on. The physically necessary requirement that g( n )± be periodic in 0 so that \ <L<$> *$ L ± a C leads to the following equation for g(01± -6-

13 v* «J*^ (2>u) We now justify this procedure by rough order of magnitude estimates, has the rough magnitude where we have estimated 9/9V;~1/VTH * a typical (thermal) particle velocity, and 6 k is a typical wave field; 0> kw )» to be evaluated in Section 3, is roughly since the perturbed distributions 6i^ become largs at a series of resonances given by w - k» v,, - nfi+ =,0. Combining terms we find p. 14, where ^(k) ari6g k l 2 / B o We take c 2 /v ±2^l for estimation purposes. There are two contributions to the integral (2.14). The first is from by far the largest part of the wave distribution which is non-resonant, where u - k,i v,i - nf2 ± ^0+. This contribution is of order J t (2.15) where j^,, the wave beta, by analogy with the usual definition of the ratio of particle to magnetic pressure = 87rp/B 2, is the ratio of the total wave energy to the static magnetic energy. -7-

14 Waves In resonance contribute a large term of order 1/7 where 7 is a typical wave growth rate. On the other hand, resonance is relatively narrow, Ak,)~> * ~^ k,,, and thus this term is "of order (2.16, When 0 W «1, the estimates (2.15) and (2.16) therefore suggest that it is consistent to take Od (2.17) and that the expansion (2.10) is consistent for small growth rates and wave amplitudes. There are physical situations for which these estimates may be violated. For instance, plasma oscillations have a growth rate roughly propdrtional to the electron plasma frequency Wp (14). If Wp >> n ± the assumption that 7/^+ «1 may be violated. Similarly, other modes involving mostly electronic motion and high frequencies may have growth rates much larger than the ion gyrofrequency (15). These cases will not be considered further. 3. DERIVATION OF QUASI-LINEAR DIFFUSION EQUATION In the last section, we arrived formally at the following form for (0) the diffusion equation where we now drop the superscript notation: (3.1) -8-

15 Even though the basic plasma has only two basic directions, those of the static magnetic field and the wave vectors, we retairi a threedimensional wave distribution for generality. The cylindrical co-ordinate system in k space aligned along the magnetic field direction is convenient, and defined by: Kx K, 2>W > T (3.2) Since the plasma is spatially isotropic, the dispersion relation for a single wave mode can depend only upon (kj., k n ) and must be independent Of Tp. Since wave frequencies may be comparable with gyrofrequencies, it is also useful to express the wave fields perpendicular to the static magnetic field in terms of complex polarization vectors denoting right and left-handed rotation about the magnetic field, and plane components aligned parallel to the field '* (3.3) Using Faraday's law to express wave magnetic fields in terms of wave electric fields, and the cylindrical velocity apace co-ordinates (2. 3), we arrive at the following form for P +k -9-

16 * (3.4) f where we have abbreviated the following operators; (3.5) The basic eigenvector E for a wave (k^., k, u if/) has the form (3.6) Only these components appear in (3.4), as is necessary. From Section 2, the operator L k is defined as (3,7) When L^ operates on the fluctuating wave term P +k t the time dependence is approximately harmonic, proportional to e" iwt. -**. ^ Anticipating this result, we write L k, for kf 0, using the definition (3. 2) and the cylindrical velocity co-ordinate system as -10- ihs~ * &;. -J :- Li y a

17 > 8) and therefore, by inspection, the inverse operator becomes»'e -n* (3.9) X has a small imaginary part to insure convergence of L k and therefore, causal solutions of the linearized Vlasov equation. Bessel function identity The V- «±g Si*<*~v) m ^ j ; (H^S «O-iK) (3.10) further simplifies L kw to L J5,rt Henceforth the arguments of the Bessel functions will not be explicitly written. Which particle species is being referred to will usually be clear from the context. Now we may solve for the 6fJ, to lowest order. Since g is independent of 0, all tf-derivatives drop out and (3.12) -11-

18 Here we have defined To complete the derivation of (3.1) we need ;in expression for In cylindrical co-ordinates the vector -k is simply (k tj -k, t, ip + w). Since the wave fields are real quantities, E(-k) = E*(+k) and g(-kj = B*(+k), where * denotes complex conjugate. Thus, In addition, we require wf+k) = -u*f-k), 7^+k) = y(-k) where w and 7 are the real and imaginary parts respectively of the eigenfrequency. Substitution of these relations in (3.4) then expresses E^ in terms of +k quantities. R k must eventually be integrated over O-2TT in <$t. The terms i3/3< in P_ k may then be integrated by parts to yield the following relations between operators before and after integration: -12-

19 e «. (3.14) if** The modified operator PJ k which results from making the substitution (3.14) arising from integration by parts is L V Wj p, ^ ^Vj. (3.15) * We substitute (3.15), together with (3.12), into (3.1) and perform the integration. The relationship T^-m)* d* = $, m then reduces the double sum of Bessel functions to a single sum: Where U n < (l) -13-

20 ^ 0 ṙv. 1± which then is non-resonant. Assuming that the distribution of waves in k space is an even function of k, t, the diffusion equation reduces to a form involving G 1 alone. From the relations u(+ k,,l) = = 7(- k,, ), 81 (+. k,,l ) = + 8 and E nj 2 n = 0, the non-resonant terms vanish, so that (3.16) reduces to 5-4Z (3.17) where (3.18) This completes the derivation of the quasi-linear diffusion equation. We have tacitly assumed for this derivation that only one branch of oscillations Go can be expressed in terms of G, and a term proportional to O

21 is excited. However, since waves are uncorrelated in lowest order (16), the diffusion coefficient for a plasma with many different excited branches will be a sum of terms, each of the above form. To find specific information about diffusion rates the above form for the diffusion equation must be coupled with the wave dispersion relation = o in order to relate the various polarizations e r e 1^, e^e" 1^1, and E,, to each other, and to the net energy in wave motion. The solution of the dispersion relation, [D = 0, is often difficult since each element of ID is in general a transcendental function (24). The above form for A n in which the polarizations appear explicitly as a positive definite term, enables some conclusions to be drawn without solving the dispersion relation. Furthermore, (3.1 7) is a convenient form for insertion of approximate solutions for the wave polarizations. 4. GENERAL DISCUSSION Parallel Wave Distribution and Principal Resonances Suppose that k x (or v^) = 0. Then all A n are zero but the following three, which we call principal resonances: v,, E f -15-

22 Thus, diffusion at the two principal cyclotron resonances, n = ±1, occurs only if pure left and right-hand circularly polarized waves are present. e { should be identified with the ion cyclotron wave, e r with the'whistler mode and E,, with plasma oscillations (25). The diffusion equation with A +1 alone agrees with that found by Chang and Pearlstein (17) for the ion cyclotron wave, while A.j agrees with the whistler results of Engel (18) and Sudan (19) Quasi-linear H-Theorem and Final Diffusion State An H-theorem may be constructed which sheds light on the final steady diffusion state at asymptotically long times, t * w. Suppose we construct the positive definite functional: (4.2) Then using (3.17) and integrating by parts Notice that H is not the entropy. In the limit that resonant particle interactions and turbulent wavewave interactions change the wave distribution on time scales much longer than the periods of typical excited waves, so that the imaginary part, y, of the wave frequency may be considered small, 1y/u «1, the resonant part of A n may be simplified using the well-known Dirac relation;. \,TT V u - ^ (4.4) -16-

23 P denotes the Cauchy principal part. The principal part describes nonresonant wave-particle scattering, whereas'ithe 6-function picks out resonant scattering. The sign of the 6-function term corresponds to a causal solution (Y > 0) (14). By referring to equations (3.1 3), it may be seen that so that the principal part terms vanish upon integration over k M and summation over n, leaving only (4.6) This agrees with the result of Sudan (20) for whistler waves (e fi = E,, = 0, e r X 0) propagating along the field. Since V n > 0 for all n and both species, dh/dt is negative definite Furthermore, dh/dt = 0, implying a stationary plasma, only when f -17-

24 in all those regions of velocity space where V n is non-zero. This result holds for all resonances and both particle species. The generalization to many particle species is obvious. The state with dh/dt = 0 is a state of marginal stability with respect to all eigenmodes. We calculate the rate of change of wave energy w k in one mode, k- to. Mj *f) (4. 9) where j is the perturbed current Using equations (3.12) and (3.13) and assuming that d 3 v g*^, dvit = 0 the above reduces to A Thus when G 2 g± = 0, dw k /dt = 0 for all waves, and the state with stationary particle distribution corresponds to a stationary wave distribution in the absence of non-linear mode coupling. Since V n ^ 0 for all k lf G^ Mlc,., k,,)) g 1 = 0 must be satisfied independently for all k x. In general, g 1 must be constant along contours in (v x, Vir) velocity space, which differ for each value of k^. Another more formal statement of this follows. Suppose we consider waves with a given kj. individually, and then require that the H-theorem be satisfied for each k x separately. Then for the given k A, the distribution functions g 1 must be constant along the characteristics of the operator Gj,which are now fully defined since k, t may be related to w -18- \K\

25 through the dispersion relation, w = wfk^, k,,), and since w/k n is related to V, through the resonance condition v,, These characteristics are the curves in the v t, v M plane: (VJ^+VJ, =Co^+^t (4 13) Since g is independent of <f>, diffusion occurs on the surfaces generated by rotating the above characteristic lines about the v M axis. Now we ask whether a steady diffusion state can exist. First let us postulate reasonable properties for the wave distributions. A wave distribution with finite total energy will be restricted to a finite volume of k-space. We assume that all the parallel phase velocity components lie between minimum and maximum values, (^/knimin^ w/k t t <^ f w /ku) max. Then the diffusion coefficient will be non-zero in "bands" in velocity surrounding each resonance for -oo^ n ^+«3 and 0 ^ y,. ^oo. For the sake of discussion and clarity we shall assume that a given particle has only one resonance with the excited wave spectrum. In practice, however, the diffusion coefficients at different resonances may overlap in velocity space. Here, just as when there: are excited waves with a distribution in kj., a particle has a choice of characteristic line to follow at the intersection point of characteristics. In practice, one diffusion coefficient will probably dominate all others, and diffusion will effectively occur on the characteristic line belonging to the strongest resonance. In any case, these effects do not alter the discussion to follow. Within the regions where V n *? 0, the steady distribution function must be constant along all characteristic lines. Where these lines intersect in a wide variety of angles in the (v^, v M ) plane, the steady distribution -19-

26 will have effectively to be a constant function of both Vj. and v M in that region of velocity space. However, g± does not necessarily contain an infinite energy when It is constant on characteristic surfaces. Since J n ( ^ *) +0 as Vi *oo, V n also approaches zero, and g* (v M> y^-^oo) are undetermined by the above equation. Similarly, since there is a maximum phase velocity, particles with very large v M must resonate with very high cyclotron harmonics; thus, since <k»0 as inl» «, g(vj., v, t -*oo) is similarly zero. We conclude that the diffusion coefficient is nearly zero for particles of very large energy. Furthermore, none of the characteristic lines intersect points at infinity. Consider the case where both v t, >> ( w /k,,) ma ax v L» (tj/k,,^^. Then the characteristic lines are given by Vu + v?= const., or surfaces of constant energy. Since the characteristics do not cross surfaces of constant energy at infinity, we conclude that stationary distributions with finite particle energy can exist with D + 0. Similarly, when v n >> (w/kn) max and v A, the characteristic lines are v, 2, = constant, and when v t >> (w/kn) max > v t,, the characteristics are v t == const. In all cases, for wave distributions with a maximum u/k t,, the energy on a characteristic line is finite everywhere on the line if it is finite at one point. Thus, these energy considerations do not exclude a steady quasilinear diffusion state which has a finite excitation amplitude. This conclusion differs from that drawn by Bernstein and Engelmann (21), who concluded that no such state exists for a three-dimensional distribution of electrostatic plasma waves with no external magnetic field. However, their characteristic lines intersect the point at infinity, and, in addition, diffusion coefficient for that problem does not vanish for large values of the component of velocity perpendicular to the wave vector. This result also suggests that diffusion in a weak magnetic field may differ from the strong field case treated here. Even though steady quasi-linear diffusion state may exist in principle, it may take an extraordinarily long time to reach it, since the -20-

27 diffusion coefficient is small for large velocities. If the high-energy particle distribution is unstable overall, the wave intensity may still grow, albeit slowly, at Ion;? times after the initial period of rapid unstable growth Diffusion Surfaces A simple physical argument suggests that the characteristic surfaces are actually those on which particles diffuse (22). Diffusion occurs because particles interact with fluctuation quanta in the turbulent background. If the particles gain energy AE from a wave, the wave loses energy-tiw, so that AE = -tiw. Similarly, the gain in parallel energy ^En^Mj v n Av,,* -tik,, v,,. Taking the ratio,we find AE/AE,, = u/k n v M, or noting that AE = M + (v x Av^ + v M Av M ) we find Integrating this just leads to v 2 A + (v M -u/k,,) 2 = const. Thus particles diffuse on surfaces of constant energy as measured in the frame of the component of wave phase velocity parallel to the magnetic field. If resonances do not overlap and if the k A spectrum is narrow, particles would be effectively restricted to one such, surface Pitch Angle Diffusion and Turbulent Viscosity From the nature of the diffusion surface, we may draw inferences on the general nature of quasi-linear diffusion. First of all, when the particle velocities are much larger than (w/k,,^^, diffusion takes place essentially on surfaces of constant energy, independent of the wave spectrum. Diffusion at very high cyclotron harmonic resonances is almost certainly in pitch angle. Wave-particle scattering is thus elastic, with dif fusion primarily in pitch angle and little change in energy. The turbulent dissipation coefficients constructed from the wave distribution will correspond to viscosity in this region of velocity space. -21-

28 4. 5 Energization of Particles Turbulent plasmas often develop high-energy tails to their velocity distribution. We investigate here to what extent these can arise as a result of quasi-linear diffusion processes. For diffusion to create a net energization of a certain subset of particles in the velocity distribution, the distribution of these particles must fall off with increasing particle energy and also the collisions of particles and waves must be inelastic. In other words, a wave and a particle must exchange energy in an elementary collision. Strongly inelastic collisions can occur only for those particles with v u ^ ("/kitjm^, and thus energy exchanges comparable with pitch angle changes only occur here. Furthermore, there are two different cases. At Landau resonances, v, t = u/k M, and diffusion occurs on the surfaces v, 2 = constant. Thus, Landau resonance diffusion may only increase the parallel energy of particles. On the other hand, if the v L component is observed to increase without iruch change in Vn, this must result from waves with frequencies close to a cyclotron harmonic, since then the diffusion surfaces are (u/k n ) 2 + v? & constant, and vj 2 can change by the spread in (u/k n ) in the excited spectrum. Of course, some change in the particle energy occurs in cyclotron resonance diffusion even when u/nst ± J «1, but it is slow compared to the rate of change of pitch angle. If pitch angle scattering drives particles out of the system, as in magnetic mirror configurations, they will not be appreciably accelerated or decelerated before they are lost from the system. If tana = v^/vn, where a = the pitch angle, then the pitch angle diffusion coefficient, E^, is formally (4.15) where (Aa) is the mean square step taken in a in a time At. When -22-

29 n n t \ «1, AE is roughly t)slt I ' (4.16) so that the energy diffusion coefficient Dp is approximately Comparing the time to diffuse a radian, T a 1/D a» ant * the time to random walk an energy E, T E - E 2 /!^, we find (4.17) Thus efficient turbulent acceleration of particles primarily occurs for those particles with v,, ^ (u/k,,) max The maximum particle energy achievable by quasi-linear turbulence is then approximately where (w/k,i) max is the maximum parallel phase velocity in the turbulent spectrum. If the system is spatially inhomogeneous, (w/k,,) max is the largest parallel phase velocity attained anywhere in the experimental region. From the above formula, we may draw some rather general conclusions. Fcjr instance, protons can in general achieve higher energies than electrons in turbulent diffusion. If the weak turbulence under consideration is hydromagnetic, with jw/n + «; 1, the phase velocities are quite generally the order of the Alfve'n velocity, so that the maximum -23-

30 energy attainable by protons in weak hydromagnetic turbulence is the order of B /8TTN, the magnetic energy per particle, whereas electrons reach only (MJ M+) B 2 /8JTN. Clearly, it is perilous to proceed further without specific information about the detailed nature of the wave spectrum, and accurate calculation of diffusion rates. However, these general consid arations suggest that the turbulent acceleration of particles to velocities well above typical phase velocities is probably an exceptional circumstance. 5. DISCUSSION AND SUMMARY In the limit that wave-particle interactions change the velocity distribution slowly compared to typical particle gyroperiods, a simple form for the diffusion equation may be found which allows some conclusions to be drawn independent of the nature of the wave excitation, and which is suitable for application to specific problems upon insertion of information about wave polarizations. There exists a three-dimensional "plateau" in the velocity distribution, marginally stable to all waves, towards which diffusion drives the distribution function in the absence of other effects. High harmonic resonances are effective pitch angle scatterers while those resonances involving velocities comparable with typical phase velocities scatter particles in energy as efficiently as in pitch angle. Effects not included in the present treatment may alter the threedimensional plateau. Particle-particle collisions tend to drive the velocity distribution towards a Maxwellian. The small (^-dependent part of the distribution function g^ may remove the plateau, since the Larmor orbiting then changes the distribution function. These effects should be more pronounced as growth rates approach the gyrofrequencies. Similarly, when mode-couplings are allowed,' the waves must reach a stationary state among themselves. It should further be noted that when mode couplings are included, product waves with very large phase velocities can be constructed which can extend the acceleration regime to higher particle velocities. However, this effect will usually be somewhat slower than the primary diffusion (23), The present diffusion equation -24-

31 may well be an improvement over simple stochastic methods even for stronger turbulence regimes, because it contains information on the spectral distribution of wave energy,on polarizations,and on the resonance nature of the wave-particle interaction. Waves could preserve some memory of their linear polarizations and resonances into the strongly nonlinear regime, provided that each wave is permitted several full oscillations before scattering with other waves. ACKNOWLEDGMENTS The author is indebted to Professors R. N. Sudan and M. N. Rosenbluth and Dr. H. V. Wong for useful discussions. Academician R. Z. Sagdeev read and commented upon the manuscript. It is a pleasure to acknowledge the hospitality of Professor Abdus Salam and the IAEA at the International Centre for Theoretical Physics, Trieste, and the financial support of the U. S. National Science Foundation. -25-

32 REFERENCES (1) P.A. Sturrock, Proc. Roy. Soc. (London) A 2A2_, (1957). (2) M. M. Litvak, Avco-Everett Research Laboratory, Everett, Mass., Research Report 92, 1960, unpublished. (3) F.J. Pishman, A.R. Kantrowitz and H. E. Petschek, Rev. Mod. Phys. J32, (1960). (4) M. Camac, A.R. Kantrowitz, M. M. Litvak, R. M. Patrick and H. E. Petschek, Nucl. Fusion Suppl., part 2, (1962). (5) W. E. Drummond and D. Pines, Nucl. Fusion Suppl., part 3, (1962). (6) A. A. Vedenov, E. P. Velikhov and R. Z. Sagdeev, Nucl. Fusion Suppl., part 2, (1962). (7) A. A. Galeev and V. I. Karpman, Sov. Phys. -JETP r?, 2, (1963). (8) B. B. Kadomtsev and V. I. Petviashvili, Sov. Phys. -JETP lji, 6, (1963). (9) R.E. Aamodt and W.E. Drummond, Phys. Fluids J, 11, (1964). (10) L. M. Al'tshuJ and V. I. Karpman, Sov. Phys.-JETP 20, 4, (1965). (11) A.A. Galeev, V.I. Karpman and R. Z. Sagdeov, Nucl. Fusion _5, 1, (1965). (12) B. B. Kadomtsev, Plasma Turbulence, Academic Press, London, New York, (1965). -26-

33 (13) A.A, Galeev, V,I, Karpraan and R. Z. Sagdeev, op. cit,, (14) L.D. Landau, J. Phys. USSR 10, 25 (1946). (15) M. N. Rosenbluth and R. F. Post, Phys. Fluids, 8, (1965). (16) B. B. Kadomtsev, op, cit., (17) D. B. Chang and L.D. Pearlstein, J. Geophys. Res. JQ, 13, (1965). (18) R.D. Engel, Phys. Fluids 8., 5, 939 (1965). (19) R.N. Sudan, private communication. (20) R. N. Sudan, private communication. (21) I. B. Bernstein and F. Engelmann, Lecture Notes for NUFFIC. summer course on plasma physics, Breukelen, Netherlands, (1965). (22) C.F. Kennel and H. E. Petschek, J. Geophys. Res. 71, 1, 1-28 (1966). (23) R.Z. Sagdeev, private communication. (24) T. H. Stix, The Theory of Plasma Waves, McGraw Hill, New York, (25) T. H. Stix, op. cit., a -27-

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