Contains reports on one research project.United States Atomic Energy Commission under Contract AT(30-1)-328

Lidsky, Lawrence M.

Murakami, M.

Contains report on one research project.United States Atomic Energy Commission under Contract AT(30-1)-328

Fancher, R. E.

DSpace@MIT

XII. INTERACTION OF LASER RADIATION WITH PLASMAS AND NONADIABATICMOTION OF PARTICLES IN MAGNETIC FIELDSAcademic and Research StaffProf. T. H. DupreeProf. L. M. LidskyProf. N. L. OlesonGraduate StudentsK. R. S. Chen R. E. Fancher P. MargosianH. Ching J. N. Hamawi M. MurakamiA. EFFECTS OF HIGH ENERGY ABSORPTION ON AFREELY EXPANDING FLUID FLOWA fluid freely expanding through a high-pressure ratio nozzle has a large densitygradient which suggests possible use as a gas target. For example, by using high-energy tritium ions and a deuterium-free expansion jet, it is hoped that neutrons can beproduced at higher flux densities than was possible before. For interestingly high tri-tium ion currents, large amounts of power are delivered to the deuterium while it isundergoing its free expansion. The effect of this power addition to the fluid flow is ofinterest because if the flow rate of the fluid is decreased or if the density gradient isreduced, the attractiveness of this neutron source is diminished. Colombant has com-puted the effect of the energy absorption at high power levels on the deuterium jet.The purpose of this work is to determine experimentally the effects of high-powerabsorption on a free expanding jet and therefore give some experimental basis to judgethe validity of the theoretical calculation. Because the expanding fluid' s mass flow rateis of particular interest, the possibility of thermal choking has been investigated.To simulate a 1-A, 250-keV tritium ion beam impinging on an MW deuterium-freeexpansion gas target, a 35-W CO laser is focused on an ethylene-free expansion jetwith a 0. 9 mm nozzle orifice. The CO 2 laser is used as the means of adding power,because just as with the tritium ion beam, the power absorbed is dependent on density.The ratio of absorption length to nozzle orifice diameter is 4. 2, as compared with 0. 5for the tritium-deuterium system.Figure XII-1 shows a highly schematic diagram of the experiment; there are 3 sub-systems. The first creates the free jet expansion by expanding ethylene through a largepressure ratio. For the flow rates used (150 cc-atm/sec) and pumped volumes indicatedin the figure, the useful operating time was approximately 2. 5 sec. The second majorsubsystem is the 10. 6 . CO 2 laser and associated focusing optics. The laser beam isThis work was supported by the United States Atomic Energy Commission underContract AT(30-1)-3285.QPR No. 91 187105-mm LENSHe-Ne LASER6-mm LENS----- 10- PINHOLE/I\I N>S I "-.PRESSUREGAUGEHEATER FSOLENOIDVALVEFig. XII- 1.CO2 LASER BEAM20-cm FOCAL LENGTHSPHERICAL MIRRORVALVETOVACUUM PUMPHighly schematized configuration of the experiment.QPR No. 91 188(a)(b)Fig. XII-2. Schlieren photographs sensitive to density gradients along the flowaxis. The nozzle is at the left, and the CO 2 laser beam entersthrough the crystal at the right. (a) Without heating. (b) Withheating.QPR No. 91 189(XII. INTERACTION OF LASER RADIATION WITH PLASMAS)focused at the nozzle orifice with a 20-cm focal length spherical mirror used slightlyoff-axis. We find that approximately 80% of the laser power can be passed through the0.9 mm diameter nozzle. The optical diagnostics is based on a 1-mW He-Ne laser and"spatial filter." The spatial filtration is necessitated by the large optical magnificationratio (~20X).The change in pressure versus time measurement was used to determine whetherthermal choking occurred. Large changes in the characteristics of the flow wereobserved using a Schlieren system (see Fig. XII-2).Approximately 25 W of the laser's power was delivered to the ethylene expansion jet.The resultant heating increased the ethylene's internal energy 25%, and the followingeffects were observed: (i) 5% reduction in the mass flow rate; (ii) 10o% increase in theexpansion plume's maximum diameter; and (iii) a larger density gradient close to thesonic line.The Schlieren photographs show that the flow experiences very large changes in thedensity gradient as a result of laser heating and, at the same time, the pressure mea-surement shows that the flow rate is only slightly reduced. These two results stronglysuggest that the sonic line bows slightly outward into the expansion plume. This two-dimensional effect increases the sonic cross-section area and helps eliminate thechoking effect. With two dimensions for expansion available, the flow can spread outover the expansion cone in a much shorter distance, thereby keeping the density gradientcloser to the nozzle.The data on thermal choking will have to be checked by Colombant to see how itcompares to what is expected theoretically in a one-dimensional flow. Increasing thestagnation pressure overcomes the effect of thermal choking, and thus the attractivenessof a gas target remains. The fractional change in enthalpy is not quite as large aswould be encountered in interesting neutron targets. We plan to repeat these measure-ments with a more powerful laser.R. E. Fancher, L. M. LidskyReferences1. L. M. Lidsky and D. Colombant, IEEE Trans. on Nuclear Science 14, 945 (1967).QPR No. 91 190

Interaction of Laser Radiation with Plasmas and Nonadiabatic Motion of Particles in Magnetic Fields

XIV. INTERACTION OF LASER RADIATION WITH PLASMAS AND NONADIABATICMOTION OF PARTICLES IN MAGNETIC FIELDSAcademic and Research StaffProf. T. H. DupreeProf. L. M. LidskyProf. N. L. OlesonGraduate StudentsK. R. S. Chen J. N. Hamawi M. MurakamiH. Ching P. Margosian A. A. OffenbergerS. I. Friedman L. C. PittengerA. RESONANT AND NEAR-RESONANT WAVE-PARTICLE INTER-ACTIONS IN A TRANSVERSE PLASMA WAVEOur original motivation of this study is related to the problem of the "corkscrew,"1which is a device to inject charged particles "resonantly" into thermonuclear fusiondevices. Wingerson's original idea was the following: If an external helical magneticfield perturbation is applied in such a way that the pitch of the perturbation is tuned con-tinuously to the orbit pitch of the injected particle, then the particle can be "resonantly"accelerated perpendicularly to the main field, and trapped inside the mirror field. Thistuning, or "resonance" condition is expressed in the form2Trv -=P (1)V corc - kv =0, (2)where v 11 is an instantaneous velocity component parallel to the main magnetic field, ecis the cyclotron frequency in the main magnetic field, P is the pitch of the applied per-turbation, and k = 27r/P. The particles that have been trapped can inversely resonatewith the perturbation field, reverse their orbit, and escape from the confinementregion.2 To estimate the trapping efficiency, we have to know the particle's scatteringprocesses in velocity space, which are due to the perturbation; therefore, we have tostudy the particle motiondvm - = q v x (B +B (3)in the vicinity of the resonance expressed by Eq. 1.This work was supported by the United States Atomic Energy Commission underContract AT(30-1)-3285QPR No. 89 183(XIV. INTERACTION OF LASER RADIATION WITH PLASMAS)A very similar situation takes place in a transverse electromagnetic wave propagatingalong an external magnetic field. In the laboratory frame, charged particles areaccelerated by both the oscillatory electric field E and the oscillatory magnetic field-IB , which are related throughaBVX E t (4)S at•In the wave frame, moving with the constant wave phase velocity (under the assumptionthat () = 0), the electric field vanishes, however, and the equation of motion in theframe is given by Eq. 3. One of the natural modes of the transverse electromagneticwaves is the so-called whistler mode whose frequency range is wci 0 W < Wce, where Wciand oce are the ion's and electron's cyclotron frequency, respectively. This mode isinteresting from the viewpoint of communication noise in the ionosphere and of charged-particle precipitation from the magnetosphere to the atmosphere. Recently, plasmaphysicists have become interested in these matters from the viewpoint of testing plasmakinetic theories. 3 Quasi-linear theory takes into account the wave-particle interactionin the linear sense; if the Doppler-shifted frequency of the whistler mode is equal tothe electron-cyclotron frequency, that is,o-k v =o (5)- -L cor in the wave framek.v = c (6)-W c(notice its similarity to the corkscrew resonance condition, Eq. 2), then the particlesexchange their energy with the wave, whose growth or damping depends on the slope ofthe particle's distribution function. The limitation of the quasi-linear theory may beattributed to the fact that straight orbit calculations are used. In fact, for the longitudi-nal case, O'Neil 4 has shown that the linear damping rate breaks down when the particlesare trapped in the potential trough of the electrostatic wave. Dupree5 constructed theso-called strong turbulence theory in which he introduced a diffusive term into the orbitof particles interacting with the electrostatic wave. A very similar situation is expectedfor the transverse-wave case.We shall discuss the two problems on the same basis; we shall try to solve Eq. 3for the resonant and near-resonant particle cases. Actually, in a corkscrew the per-turbation amplitude B] and the pitch P have spatial dependences, and in whistler-modewaves there is a finite spectral width instead of a single wave. Here we shall treat thecase of a perturbation of spatially independent amplitude and pitch in order to understandits basic features.In our last progress report, 6 we showed that there are two kinds of particles,QPR No. 89 184(XIV. INTERACTION OF LASER RADIATION WITH PLASMAS)untrapped and trapped in the sense similar to the electrostatic case, and that for theuntrapped particles straight orbit calculation does give right answers.In this report we shall discuss some relationships with the quasi-linear theory forthe transverse mode, and also discuss the limitation of the theory, where the theorybreaks down because of particle trapping. In order to cope with the inability of thestraightforward orbit calculation to treat the turning points properly, we shall introducea new method, singular perturbation technique.? The results will be explained on aphysical basis, and their relationship to Dupree's strong turbulence theory will bepointed out.1. Limitation of "Straight-Line Orbit" ApproximationUsing the conservation of the particle's kinetic energy (v = vi + v), we reducethe equation of motion, Eq. 3, into a set of two simpler normalized equations:du= p cos x (7)dX 1 2r (8)d, u11  p 'where ull and u are the velocity components normalized by the total velocity vo, is thespatial variable along the main magnetic field normalized by the Larmor radius, r o , ofthe totally wound-up particles, P is the small-perturbation magnetic field amplitude nor-malized by the main magnetic amplitude, p is the perturbation pitch normalized by theLarmor radius ro, and X is the angle between the direction of the perpendicular velocityand that of the perturbation magnetic field.We have shown previously 6 that the untrapped particle is a particle whose initialcondition satisfiesy2 >> A, (9)whereY 1 2r (10)(values with the subscript o are the initial value at S = 0), anduAs Co (11)UIloWe also showed that the straightforward perturbation solution converges with the exactsolution. Condition (9) can be rewritten in more familiar notationQPR No. 89 185(XIV. INTERACTION OF LASER RADIATION WITH PLASMAS)2 2( c -vIwc YLo (2- vvo Bo (12)The condition is satisfied when the particle's orbit pitch is far from the perturbation'sfield pitch and/or the perturbation field strength is weak enough. In plasma kinetictheory this state is called "weak turbulence," which can be described by the quasi-lineartheory. In the quasi-linear theory, the diffusion coefficient for the right-hand polarizedwave is given by2q2 Bk V YkDQL 2 (13)m (w-k ' v+c )cwhere yk is the linear damping rate for the k wave. In our case we have a single setof w and k. If we average over the termination position, z, then the term cos yz inthe expression Au = p2(1-cos yz)/y2 will drop out. Then2 2 2Au = u Au = . (14)2 - 2 2" (14)llo U o110If we take the diffusion coefficient in the formD = (15)where T is a characteristic time during which the perturbation changes significantly,then we get2 2 I1q 2Bi yoD - 2 (16)m 2 (W c-kvll 0)This is the diffusion coefficient inthe wave frame. Therefore, in the laboratory framev should be replaced by v - . Then the diffusion coefficient in the laboratory frameisq 2 B2 vo2(D)lb 2 2 (17)lab m (w-kvl1 + )2Noticing that T can be replaced by the reciprocal of the growth rate, we realize thatEq. 17 agrees with the quasi-linear diffusion coefficient DQL.QPR No. 89 186(XIV. INTERACTION OF LASER RADIATION WITH PLASMAS)When the trapping phenomena begin to take place, the straightforward orbit calcula-6tion does not converge near the turning points. Our previous calculation and our latercalculation tell us that the trapping occurs when the particle's initial condition lies inthe rangey2 zAP (18)or, in more familiar notation,(- ) ovLo _I (19)\k IIo k vl o BoThe condition is satisfied when the particle's orbit pitch is close enough to the perturba-tion's field pitch and/or the perturbation's field strength is strong enough. We call theformer "resonance" and the latter "strong perturbation." Equation 2 can be consideredto be a similarity law between "resonance" and "strong perturbation" for the trappedparticles. According to Dupree's strong-turbulence theory, we denote wo/k - V11 fortrapped particles by the "resonance width" w. Thusw vLo Bw - 0 1 (20)The terminology "strong turbulence" does not necessarily mean the state in which waveamplitude is large, but it may also mean the state in which the "resonance" effect isvery important even if wave amplitude is small.2. Trapped-Particle CaseFor the trapped particles, the straight-line orbit approximation on which the quasi-linear theory is based cannot describe the motion because turning points appear on theirorbits. The exact solution with elliptic integrals does not seem tractable (or at leastit is difficult to grasp its gross features). 8 Here we call for a new method, the singularperturbation technique.? Since the original set of equations (7) and (8) is an autonomousset, it is integrated with the initial condition ul = ulo at = 0 in the form-1 -1 21rsin u- sin - - ( ) = in X - sinXo). (21)Therefore, Eqs. 7 and 8 with Eq. 21 give us the following exact differential equation:2 u +sin- u -1 2+p 1S+ sin- - sin u_ - ul-ulo) + p sin Xo X - = 0. (22)Using the Taylor expansion around ulo (= u -Au ), we obtainQPR No. 89 187(XIV. INTERACTION OF LASER RADIATION WITH PLASMAS)2+ Py sin X + (y 2 +Ap sinXo)AUl + Ay + C sin Xo Aud5+ (LyF + A + -L C + - Ap sin Xo Au + O Au4 = 0, (23)whereSloA- C= F=U1lo Ullo Ulloand all A, C, and F are of the order of unity.In passing, for the case y2 >> Ap, which is the untrapped particle case, the positive-definite third term of Eq. 23 shows the sinusoidal change of Au . In fact, from Eq. 23,we can show thatAul) = O(p 3 ) (24)Aul = ( - cos yz) + O(P4), (25)Ywhich agree with the previous results of the untrapped particle case.6Returning to the problem of the trapped particles, we recall that the parameter y isof the order of IJT, or y = - r, where F is a constant of the order of unity. Herewe use the following transformation:au± =P v (26)= z/fp. (27)Then Eq. 23 is reduced toP3/2 d v 3/2 sinX +P3/2( 2A sin X) 3/2 FA C sinX)v2+ 3/2 A+J rc + - pF) v3 + O(p) = 0. (28)Therefore the most dominant terms are22d + sin3 Asin 3 2 2+ A3+ O()= 0. (29)__2 + +Av + 3rAv2 -  + O('1-) = 0. (Z9)dz + in s n)v - A - vdz we get an ordinary differentialNeglecting the small correction terms of order , we get an ordinary differentialQPR No. 89 188(XIV. INTERACTION OF LASER RADIATION WITH PLASMAS)equation with variables of the order of unity.More formally, we could have used the following singular expansion in terms of -,Aul = - V(o) (1) 3/2 (2) + ... (30)S i z(o) + z(1) + 4-P z(2) + ... (31)and introduced it in Eq. 23; then the smallest order equation would have been the sameequation as Eq. 29.The energy integral of Eq. 29 with the initial conditions v = 0, dv/dz = cos Xo atz = 0 is of the form2 2cdz os Xo + 21 sinXo+ ( +A sinXo + Av + v = 0. (32)From this equation we obtain an exact solution in terms of elliptic integrals which turnsout to be too complicated to demonstrate the important physical situation. Singular pointanalysis will do for the purpose of grasping gross features of the solution. Equation 14can be re-formed into the following form:d (dv v+ ((v+ 2 -2A sinx )dv dz (33)dz2The solution of Eq. 33 brings out two cases: one is the case F - 2A sin Xo 0, theother is 2 - 2A sin Xo < 0. In the former case there are three singular points, whilein the latter case only one singular point exists. Expanding Eq. 33 around those singularpoints, we realize that for the case 2 - 2A sin Xo 0, (v = 0, dv - ) is a saddleand (0, -- + Z - A sin Xo are centers, and for r 2 - 2A sin X < 0, (0, -is a saddle. Notice that, since the numerator on the right-hand side is a function ofF d irFv + A and d \A = 0, the phase plane plot is symmetric with this point. We shall showthat this point does correspond to the difference between the particle's initial velocityand the phase velocity of the perturbation.From the information above we can draw the outline of the phase plane plot, whichis shown in Fig. XIV-1. The initial condition is v = 0 and dv/dz = 0; therefore, initiallythe particles are distributed on the dv/dz axis. For the case 2 - 2A sinXo 0, there aretwo kinds of motion: the particles inside the separatrix (which is shown by a dark line)rotate around v = - + - 2A sin Xo , while the particles outside the separatrixrotate in larger cycles around v = - . The broken line shows the outermost particle'sQPR No. 89 189(XIV. INTERACTION OF LASER RADIATION WITH PLASMAS)- - -2F4 3F2+1- 2AV422F+1 - 3 FA2 2AVAv- r 2 -2Asinx v --- + 2 -2AsinXoA A2  iF 4+ (4A2 +F42Ap2)AF - 2AsinXo>0 2 - 2AsinXo<0Fig. XIV-1. Phase plane plot for the trapped particles.orbit. There is a forbidden region that particles starting on the dv/dz axis cannot reach.For the case r2 - 2A sin Xo < 0, all particles rotate around v = -We have mentioned that both diagrams are symmetrical with respect to v = - . Andwe can say that over-all most particles rotate around this point after a long time; there-fore, if we call this the ultimate value of v, then the ultimate change in velocity is.V o k o )AV = - v v  (34)1 Il yo 0 k oor~ k / o oAv, =v 0 ,,o o v1 1  (35)lb bOQPR No. 89 190(XIV. INTERACTION OF LASER RADIATION WITH PLASMAS)If the ultimate parallel velocity is defined by vll10 + 11' then(v ) + (36)11 (Vll)ultimate - Vllo + -- knamely, any trapped particle's parallel velocity tends to coincide with the phase velocityof the perturbation. The rotation in the phase plane plot corresponds to particle oscil-lation in a trough of the pseudo-potential, 9 centered on the phase velocity. In that sense,the terminology "trapping" is justified.The standard deviation is estimated from the width of the rotation on the v axis, andis (F/A) 2 times some factor a, where a is of the order of unity but larger than unity;therefore, the standard deviation of AVll isAv2 = ao aw 2 (37)where w is the "resonance width," and, from Eq. 20, we can evaluate2h J2 (w o )lo BAV Z a (38)II k v BIio oLet us consider the damping rate of the transverse wave. Similarly to the longitudi-nal case, if there are many particles streaming slower than the wave phase velocity andsomewhat fewer particles streaming faster than the wave phase velocity, then the waveenergy is absorbed by the particles. The damping rate can be calculated by balancingbetween the increase of the particle energy and the decrease of the wave energy. Theinitial damping rate is proportional to the slope of the distribution function, as Landaudamping for the longitudinal wave case. Once the particle trapping takes place, however,the linear damping rate breaks down. After the trapping time, the phase mixing takesplace as shown in Fig. XIV-1, and the growth rate approaches zero, as in O'Neil's cal-culation in the longitudinal wave.4The trapping time can be calculated for our case. In the set of original equations wehave used spatial coordinate z instead of time variable t; however, these variables arerelated with each other throught =-. (39)Therefore, if we can calculate the "trapping distance," ztrap , which is the distance theparticle travels before being trapped, we can compute the trapping time ttrap. We haveshown that the trapping takes place in the distance inversely proportional to qFI. Althoughwe can compute it more explicitly in the trapping range, here we give a more intuitiveQPR No. 89 191(XIV. INTERACTION OF LASER RADIATION WITH PLASMAS)derivation.10 The particles are accelerated along the main magnetic field by the pertur-bation roughly in the ratemall = qvB. (40)If we define the trapping time by the time during which the particle travels one wave-length, then1 1 2k 2 al ttrap* (41)The combination of Eqs. 40 and 41 gives us the trapping timet = . (42)trap q kv BPreliminary consideration of the extension of Dupree's strong-turbulence theory to thetransverse wave shows the consistency of the trapping time and the resonance width w.M. Murakami, L. M. LidskyReferences1. R. V. Wingerson, Phys. Rev. Letters 6, 446 (1961).2. R. V. Wingerson, T. H. Dupree, and D. J. Rose, Phys. Fluids 7, 1475 (1964).3. C. F. Kennel, and H. E. Petschek, J. Geophys. Res. 71, 1 (1966).4. T. O'Neil, Phys. Fluids 8, 2255 (1965).5. T. H. Dupree, Phys. Fluids 9, 1773 (1966).6. M. Murakami and L. M. Lidsky, Quarterly Progress Report No. 88, ResearchLaboratory of Electronics, M. I. T. , January 15, 1968, p. 220.7. M. Van Dyke, Perturbation Methods in Fluid Mechanics (Academic Press,New York and London, 1964).8. R. F. Lutomirski and R. N. Sudan, Phys. Rev. 147, 156 (1966).9. C. A. Robert and S. J. Buchsbaum, Phys. Rev. 135, A381 (1964).10. L. M. Lidsky, "Symposium on Fluctuations and Diffusion in Plasmas," PrincetonUniversity, June 26-30, 1967.QPR No. 89 192

XV. INTERACTION OF LASER RADIATION WITH PLASMAS AND NONADIABATICMOTION OF PARTICLES IN MAGNETIC FIELDSAcademic and Research StaffProf. T. H. DupreeProf. L. M. LidskyProf. N. L. OlesonGraduate StudentsK. R. S. Chen J. N. Hamawi M. MurakamiH. Ching P. Margosian A. A. OffenbergerA. STRONG TURBULENCE THEORY FOR A TRANSVERSEELECTROMAGNETIC WAVE - COMPARISON WITH THESINGLE-PARTICLE CALCULATIONIn previous reports1, 2 we have discussed single charged-particle motion in a con-stant pitch and constant amplitude magnetic perturbation with relation to "corkscrew"injection and wave-particle interaction in magnetized plasma. We have worked out themotion of untrapped 1 and trapped 2 particles.Here we apply Dupree's strong turbulence theory 3 to wave-particle interaction ina transverse electromagnetic wave propagating parallel to an external magnetic field.By using a transformation of the velocity coordinate,4 which is essentially transforma-tion to the wave frame used in studies of the single-particle motion,2 we reduce theVlasov equation to a one-dimensional diffusion-type equation. Then manipulation sim-ilar to that of the longitudinal wave case by Dupree3 leads to nonlinear wave-particleinteraction including "trapped" particle effects.We compare the results in the strong field limit, where the trapping effect is impor-tant, with the results of the single-particle model. The results agree except for numer-ical factors. The single-particle calculation gives a proof of the correctness of thestrong turbulence theory, which includes some crucial assumptions and provides someclearer physical intuitions.1. Strong Turbulence TheoryWe start with the Vlasov equation:8f 8 q 8+ v -f +- (E+vX B) f = 0. (1)t r m avWe consider a transverse electromagnetic wave propagating parallel to an externalThis work was supported by the United States Atomic Energy Commission underContract AT(30-1)-3285.QPR No. 90 145(XV. INTERACTION OF LASER RADIATION WITH PLASMAS)magnetic field. And for simplicity, we consider a spatially homogeneous plasma in theabsence of zero-order electric field. We shall follow the discussion of Dupree's longi-tudinal wave case.To solve Eq. 1, we expand the field in Fourier seriesE(r,t) = Ek (t ) eik rk IB(r, t) = B 0 z + (t) ei--rkIntroducing (2) into Eq. 1, we obtain1a a q _a q+v- +-vXB f+-- ar m- -o v m (E + v XB )  f = 0.kWe divide the distribution function f into two parts:f (r, v,t) = f(v, ,vi t) + k(vt)e , (4)kwhere (f(viI, v1 , t)) is the distribution function averaged over a time much longer thana Larmor period so that it is independent of the gyration angle 0, which is shown inFig. XV-1 with the definitions of vll, v1 , Bo, and k. In the time averaging we assumeZ9011 Fig. XV-1. Velocity coordinate system.that fluctuation phases are uncorrelated; then we can use ensemble averaging forthe correlation terms instead of time averaging. ThusQPR No. 90 146(XV. INTERACTION OF LASER RADIATION WITH PLASMAS)+ v - f (v',v, t) +- - (Ek+VXB = 0, (5)kq awhere the - vX B term drops out, because of the 0 independence of <f>.m- -o 8vThe fluctuation distribution function fk is given correctly to first order in Bk (or Ek)by [The proof is parallel to the electrostatic case by Dupree 3.]fk(t) e- m = -- dr (Ek(T) +v(T) XBk(T)) U(t, T)oik -r a /t ik(rke- vr - U(T, to) f(to) + (U(t, to) fk(t=O) e 'r  (6)where the propagation operator U(t, t ) is the solution ofa q qa+ v + -- vXB U + - +vXB) - 0 (7)- _r m- -o 8v m k k - vkand the angular brackets indicate an ensemble average over wave phases.Substituting Eq. 6 in Eq. 5, we obtain2(+t )( ( , VI, t) = q k - -kk oik .r(t) ik (T) a(E_ (T)+V(T) X B k) e r(t) <U(t, T) > e - av >,(8)where we have neglected the initial value term, since we are not interested in it.The integrand of Eq. 8 is nonvanishing in only a small interval around t = T, and ifthe time dependence of it is sufficiently slow during this time interval, then we canreplace <f(T) > with <f(t) >. Thus, (7) is reduced to a diffusion equation:a v - D(v) - , t) = 0, (9)where2D(v) - 2 d (k(t) (t) XBk(t)) (Ek -k(T))m k oik - r(t) < (t, T) -ik. r(T) (10)Se - < U(t, 7)>eQPR No. 90 147(XV. INTERACTION OF LASER RADIATION WITH PLASMAS)If the acceleration forces represent stationary time series, then the orbit does too;that is, <U(t, T) > = <U(t-T) >, and therefore2q 2 -tD(v) m 2 d (Ekt t XBk k(t-T) +V(t-T)XBk(tT))Seik - r(t) < (T) > e-ik -r(t-T).Continu e- <U(T)> e observe thatContinuing to follow Dupree, we observe that(11)<U(T)> e- ik ' r(t- T ) (U(T) e-ik - r(t-T)= (e-ik r' c (-T)where r (-T) is a trajectory at time -T with initial values r(t) and v(t).large t - t o (compared with the nonlinear correlation time which willEq. 25), we can replace the upper limit of the integral in Eq. 11 by oo.(12)For sufficientlybe defined inThenD(Y) - 2 dT (_Ek(t) +v(t) X Bk(t)) (E_k (t-7) + v(t-T) XBk(t-T))kik - r (t) e-ick ) (13)The quantity (e-k" rc(-t))the same equation as <f>,is a function only of the orbit; therefore, it must satisfyS- ar av av0 (14)We must solve Eqs. 13 and 14 simultaneously. To facilitate this solution, we usethe following coordinate transform, 4u = V L -2 k k dvl (15)v = V1This transform is a transform in which one of the coordinates is along the diffusion pathof the particles (see sec. 2). Ek and B k are related through the Maxwell equations. Fora transverse wave propagating parallel to the external magnetic field, the fluctuatingelectric field is0kEk - kXBkk 2 -k (16)Therefore, noting again that j ( v.) does not include a 0 component, we obtainQPR No. 90 148(XV. INTERACTION OF LASER RADIATION WITH PLASMAS)(E +vXB (OB ) VIl _ (V1_k - -- k av (- B )  vi( B ) a (17)a- D (v) - m- " (Ek(t) +y(t) X Bk(t)) e i k z (t )  c ( - Tke avm k ke ei(t( ) (-ikz (-- 8vk ikz ( -t)D(v, u) 0. (208)whereD(v, u) 20 is very d (ficul(t) to so(t))lve. I(t-)f we ignoreBk(t-)) v dependence(t) of D, how(t-T)ikz. eikz(t) ikzikvt . D(u)(19)Therefore Eq. 13 is converted to the following one-dimensional equationS8 Dvu) _. ) (e-ikz ( - t )+ v D(v, = 0. (20)In general, the diffusion coefficient is a function of v and u. If the dependence on v isretained, Eq. 20 is very difficult to solve. If we ignore the v dependence of D, how-ever, the solution is simple-ikz e(-t) -ikz + ikvt - 1 k 2D(u)t 3e =e3 (21)We shall assume that this result is also approximately true for the diffusion coefficientD(v, u). This is one of the crucial approximations in this theory. The agreement ofthe results in this theory with those of the single-particle model (which will be describedQPR No. 90 149(XV. INTERACTION OF LASER RADIATION WITH PLASMAS)later) implies that the assumption does not hide any important gross features of thetheory.To evaluate the diffusion coefficient, Eq. 19, we substitute the unperturbed orbitfor v 1 (t), namelyiW TA A Cv (t-T) 0(t-T) = v (t) 0(t) e (22)(which will be correct to "first order"), where c is the cyclotron frequency includingthe sign of the charge; that is,S- -B .c m 0oAlso, the complex conjugate of the magnetic field is(23)k = (t) k T__-k(t-T) = __k (t) e= Bk(t) eSubstituting these in Eq. 19, we obtain an integral equation for D(v, u),2D(v, u) -2 1m ki(kv-w k+Wc )T-k T D(v, u)dT e (25)2 2k -L (v, u) SqNoticing that -m-BkL is an acceleration that produces the random process, we define5nonlinear correlation time TC,NLTC, NL1232 i(kv-k+) T - k T D(v, u)SiBk2 f0 dT e3kS 1k12k(26)If we neglect the diffusion coefficient in the integrand, TC, NLautocorrelation time TACTACreduces to the normal20 i (kv-wk + c)TE ok 0 d ekk(27)The autocorrelation time is approximately evaluated from the difference in phase veloc-ity of the wave packet 01 k2k1 k2 'QPR No. 90(24)150(XV. INTERACTION OF LASER RADIATION WITH PLASMAS)01 27T k 1 (28)AC o k1 k2where k is the average wave number of the spectrum.o 3As shown by Dupree, the resonant function123o00 i(kv-wk+oc)T 3 k D(v, u)R[kV-k c , D(v, u)] = Real dT e (29)for a real frequency wk has a maximum at v = ( k-Wc)/k, at which R = (1/3)!2 -1/3 ~ 2 1/3(1/3k D) / 3 , and R goes to zero for Ikv- k+ l > (1/3k D) , and the area under Ris 7T. For simplicity, we replace this bell shape by a square shape with the same areaand approximately the same height and width. For the complex frequency, Wk + iYk'the nonresonant interaction attributable to the small imaginary part, yk' is ineffectivefor the resonant region, while it is only the remaining contribution in the nonresonantregion. Therefore R is approximated by the following function:2kw I kv-w +Wc < kw2kw kR[kv-wk+wc ,D(v, u)] (30)k )2 Ikv-wkw c> kw(k1-wk - cwhere w = (D/3k)1 / 3 is k - 1 times the half-width of the resonance function R[kv- k +WcD(v, u)]. The resonance function for the average wave number ko is one important timescale, the trapping time, TTR:_ 1 _ (31)/ 3TTR 2 kw 2k2D (31)Now we consider the two limiting cases.1. TTR >> TAC This case corresponds to the case in which the spectrum of thewaves is so broad and/or the wave amplitude is so small that the trapping effect is notimportant. Because of the resonance width w << k k , the resonant part of the reso-nance function becomes delta function. Therefore, the diffusion coefficient in this isgiven by2 2q 12 2 ^V, 2) q 2 'YkD(v, u) 12 1(v, u) 6(kv-wkwc)+ 2 Bk 1 (kv, ) c) 2 .m k m k(32)QPR No. 90 151(XV. INTERACTION OF LASER RADIATION WITH PLASMAS)This is the diffusion coefficient predicted by the conventional quasi-linear theory. Thefirst term represents the resonant interaction of wave and particle, and the second rep-resents the additional "long-range" term, which describes the nonresonant or adiabaticinteraction of wave and particle.2. TTR << TAC. This case corresponds to the case in which the spectrum of the waveis so sharp around the resonance condition and/or the wave amplitude is so large thattrapping phenomena are important. The nonlinear correlation time TC, NL is equal tothe trapping time TTR.Using the simple resonance function with Eq. 25, we obtain an approximate diffusioncoefficient for a finite bandwidth of the spectrum 1 2 I:2 /3D(v, u) = 2  Bk 2 2(v, u) 3km (v, u) k(33)2q 2 2 kD(v, u)= 2 jB1 k 2 v (v, u) 2 otherwise. (34)m k (kvck+c)Note the 3/2 power dependence of the resonant diffusion coefficient on the amplitude ofthe wave, and also that the nonresonant diffusion coefficient has the same form as in theTTR >> TAC case. We shall now compare these results with those of the single-particlemodel.2. Comparisons of the Strong Turbulence Theory with theSingle-Particle ModelWe have already solved the problem of determining particle orbits in resonant andnonresonant constant-pitch, constant-amplitude, magnetic perturbation., 2 Here wecompare the results of the single-particle model (SPM) with the strong turbulence the-ory (STT) derived above. We shall show that the two solutions are identical within theaccuracy of the assumptions made in their derivations. Thus, in the future, we shallbe able to use the strong physical insight into the dynamics of the interaction gained inthe SPM to elucidate and justify results gained by use of the mathematically more power-ful STT.We first note some differences between them. In the STT, an infinitely large numberQPR No. 90 152(XV. INTERACTION OF LASER RADIATION WITH PLASMAS)of incoherent waves allows us to ensemble-average with respect to wave phases, whilein the SPM we discuss the particle motion in a single coherent wave. In the very laststage of the SPM, however, we disregard the fine structure of the particle orbit toobtain the diffusion coefficients. In effect, we study diffusion, resonant or nonresonant,resulting from the sequential interaction of the particle with many uncorrelated waves.Thus, in the SPM we perform a time average, and in the STT an ensemble aver-age. As expected, the two models give the same results if the corresponding corre-lation times are properly defined.In the STT, we have used the transform, Eq. 15. Here we discuss the physicalmeaning of this coordinate transform. For simplicity, we consider a case of singlewave w and k. Then, (15) can be reduced tou = v + v - VI W= +v ). (35)Since the diffusion equation (18) does not include derivatives with respect to u, particlesdiffuse on the constant u-surface, as described above. We shall explain this fact on thebasis of energy and momentum conservation in the wave-particle system. If we denotewave total energy (including wave electromagnetic energy and nonresonant particle kin-etic energy) by I k , it can be shown 6 that the total wave momentum density is klk/w.(Notice the analogy with quantum mechanics.) Therefore the conservations of the energydensity and momentum density parallel to the propagation (and B o ) in the wave-particlesystem aredn imv2+v + d Ik = 0 (36)d (nmv) + d  Ik ) 0, (37)where n is particle density. (In passing, the perpendicular momentum is automaticallyconserved in Larmor motion.) These equations lead us to the following relationship:(--) dvl + vLdv = 0 (38)orv - + v = constant. (39)This shows that particles diffuse on a constant u-surface upon which the energy andmomentum densities of the wave-particle system are conserved.Now we return to comparison of the results of both theories. Let us first considerQPR No. 90 153(XV. INTERACTION OF LASER RADIATION WITH PLASMAS)the resonant interaction. In the STT, TTR < TAC, the diffusion of the particle broadensthe 6-function resonance interaction. The SPM shows more clearly what is going onwith each particle. In the electrostatic (longitudinal) wave O'Neil 7 shows that the lineartheory breaks down when the resonant particle becomes trapped inside the trough of thewave electrostatic potential. A very similar situation occurs in the transverse-wavecase, namely, the particle becomes trapped inside the trough of some "pseudopotential." Once the particle becomes trapped, the simple perturbation theory fails,Also, since this situation is more complicated than the electrostatic case, the exactorbit calculation,8 does not help in understanding the gross structure. Instead we intro-duce a singular perturbation technique that treats the trapping phenomena (including2reflection points) properly. The important consequences are the following: In theexpanded scale with a scaling factor 1/ J (p is the ratio of the wave magnetic fieldamplitude to the external magnetic field amplitude), there exist two kinds of "pseudopotentials," which are always symmetric with respect to (cwk-W c)/k (see Fig. XV-2).I PSEUDOPOTENTIAL -- PSEUDOPOTENTIALV VK - c  K ck kINITIAL VALUE INITIAL VALUEFig. XV-2. Two kinds of "pseudo potential" inside which the particle is trapped.The initial condition determines the detailed structures (forbidden region) of the particlemotion. But we need use only the gross features to compare with the STT whichassumes random wave phases. As shown in our last report, 2 the trapping time, whichis evaluated from the spatial oscillation period, is given bym 1,(TTR SPM = cl qk (40)where c 1 is a numerical factor of the order of unity. In the STT, the nonlinear correla-tion time TC, NL is evaluated asCNL k? )-1/3 2IT m (41)TC, NL k q kvBk'QPR No. 90 154(XV. INTERACTION OF LASER RADIATION WITH PLASMAS)therefore, this correlation time can be properly interpreted as the trapping time.From the symmetrical point of the "pseudo potentials," we observe that the averagevelocity of the trapped particle is (wk-W c)/k, which is the phase velocity seen by agyrating particle.The diffusion coefficient is estimated by the width of the phase plane trajectoriesand the trapping time in the SPM:(B)3/2(DTR SPM =2k mk , (42)where c 2 is a numerical factor of the order of unity. In the STT, Eq. 33 shows(D k /2(43)TR STT = kmk 3 (43)which agrees with (42), except for a numerical factor.In the STT, we define the resonance width w, which is the maximum differencebetween particle velocity and wave-phase velocity for which a particle can still be dif-fused by the wave and the value is(w)STT /3 1/4 k /2 (44)On the other hand, in the SPM, the resonance width is decided by the boundary betweentrapping and nontrapping, which givesB1/2(w) k (45)SPM mkAgain both are in agreement.A particle whose velocity is beyond the resonance width around (wk- wc)/k interactsnonresonantly (or adiabatically) with the waves. It oscillates in the electromagneticfield of the wave with an amplitude proportional to the field strength. The diffusioncoefficient of the untrapped particles is given both in the SPM and STT in the form2q 22 _ kUTR STT UTR SPM=- B2 kV 2 (46)m (kv-wk+wc)We may conclude that the results of the strong turbulence theory agree withthose of the single-particle model. The strong turbulence theory includes someQPR No. 90 155(XV. INTERACTION OF LASER RADIATION WITH PLASMAS)crucial assumptions, but the agreement implies that none of them hides physicallyimportant gross features. Therefore they give a more rigid basis to the strongturbulence theory. Furthermore, the single-particle model provides clearer insightto what is going on physically.M. Murakami, L. M. LidskyReferences1. M. Murakami and L. M. Lidsky, Quarterly Progress Report No. 88, Research Lab-oratory of Electronics, M. I. T., January 15, 1968, p. 220.2. M. Murakami and L. M. Lidsky, Quarterly Progress Report No. 89, ResearchLaboratory of Electronics, M.I. T., April 15, 1968, p. 183.3. T. H. Dupree, Phys. Fluids 9, 1773 (1966).4. J. Rowlands, V. D. Shapiro, and V. I. Shevchenko, Soviet Phys. - JETP 23, 651(1966).5. T. H. Dupree, Private communication (1968).6. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley Press, Inc., Reading, Mass., 1960).7. T. O'Neil, Phys. Fluids 8, 2255 (1965).8. R. F. Lutomirski and R. N. Sudan, Phys. Rev. 147, 156 (1966).QPR No. 90 156

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Interaction of Laser Radiation with Plasmas and Nonadiabatic Motion of Particles in Magnetic Fields

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