Bers, A.

Hizanidis, K.

Ram, A.K.

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PFC/JA-85-27TRAPPED ELECTRON STOCHASTICITYBY FREQUENCY-MODULATED WAVES*A. K. Ram, K. Hizanidis, and A. Bers**August 1985* This work was supported in part by National Science FoundationGrant ECS 82-13430 and in part by Department of Energy ContractNo. DE-AC02-78ET-51013**Plasma Fusion Center and Research Laboratory of Electronics,Massachusetts Institute of Technology, Cambridge, MA 02139.TRAPPED ELECTRON STOCHASTICITYBY FREQUENCY-MODULATED WAVESA. K. Ram, K. Hizanidis, and A. BersPlasma Fusion CenterandResearch Laboratory of ElectronicsMassachusetts Institute of TechnologyCambridge, Massachusetts 02139AbstractIt is shown that for electrons trapped in a static potential a finite amplitudefrequency-modulated wave induces stochastic motions (and diffusion) in electron orbitsover a significantly larger area of phase-space than a single frequency wave of the sameamplitude. The wavelength of the FM fields is unimportant if it is greater than twicethe width of the static potential. It is proposed that this be used for pumping of mirrortrapped electrons to enhance the confining potential in a tandem mirror cell.The purpose of this letter is to show that the distribution function of electronstrapped in a sinusoidal potential well can be significantly modified by using anelectro-tatic wave (ESW) that is frequency-modulated (FM). Previous studies with asingle frequency ESW as a perturbation have shown that the only electrons that exhibitstochastic motion are those that are near the separatrix of the confining potential orthose that have velocities which are in a narrow range of the phase-velocity of theESN\.' 2 In particular, to induce stochasticity in trapped electrons would require a verylarge amplitude ESW. In contrast, we will show that an FM wave with appropriatelychosen parameters can lead to stochastic motion over an appreciable portion of phasespace of the trapped electrons for a relatively small, finite amplitude wave. Theappropriately normalized one-dimensional equation of motion for such an electron is:d'z AW--g= - sin(z) - E sin ikz - wt - - sin(Wmt) (1)dt2 I Wmwhere c is the strength of the ESW with wavenumber, k, frequency, w, and modulationalfrequency, waM. The distances have been normalized to the wavelength of the staticpotential and the time to the bounce period of an electron at the bottom of the staticpotential well; E measures the strength of the ESW relative to the maximum of thestatic potential; Aw is the frequency bandwidth of the FM wave about the carrierfrequency, w. The Hamiltonian corresponding to (1) is:AWH = Ho - cos kz - wt - - sin(wmt) (2)k Wmwhere H1o = (A/2) - cos z is the unperturbed Hamiltonian corresponding to the motion= dz/dt) of an electron in the static field. Here we will be concerned only with thetrapped electrons so that -1 < Ho < 1. The lower limit corresponds to an electron atthe bottom and the upper limit to an electron at the separatrix of the potential well.We transform H into the action-angle (I - 0) representation of Ho yielding:7(I, ) Ho(I) - - E JI(-) E V,(k, I) cos(nO - flt) (3)k t=_00 Wm n=-owhere JI is a Bessel function of order e, I = (8/7r)[E(i) - (1 - K2)K(K)], K and E arethe complete elliptic integrals of the first and second kind 3, respectively, r 2 = 1(1+Ho)and:IV,,(k, I) = -f docos 2ksin-' {Ksn(, )- ndj (4)with Q = 7r/2K(K) being the unperturbed non-linear frequency of the trapped electrons(0 < Q < 1) and sn is a Jacobian elliptic function 3. The condition for the onset ofstochastic instability is given by the Chirikov resonance overlap criterion 4:(A IT ?)i, , + (AITRh(s = ()Itn' - Ifnwhere h , satisfies the resonance condition:nP(Ia) = II = w - twn (6)(', n') is the nearest resonance to (i, n) and(AI T )t = 2 [- (A) ( -- 11/] (7)k WM (d2/dI) 11tis the resonance width. We want to find the conditions on w, k, Wm and Aw such thatstochasticity can occur for the smallest possible c. In an experiment this would implythat, for a given (practical) bandwidth, we want to keep the power requirements forthe FM-ESW to a minimum. An analysis of (4) shows that VI is the largest coefficientand IVl 0.6 for integer Inj ;> 1 (V_ = (-1)"V.) and for any value of k. Then,from (7) we see that k should be less than 1. Assuming that only the n = 1 term isimportant and that k g 0.5, we find, f7 K(V/1 - r-9V1(k,I) ~ 2k / cosh{ 2 K(r) (8)and (7) is independent of k. From (6) the separation between resonances is proportionalto win. Since 0 < n < 1 for the trapped electrons, we require that Wm < 1 so thatseveral resonances can occur in the region of interest. This is supported by the exactnumerical analysis of (1) as shown in figures la and lb where the only parameter thatis changed between the two plots is w. 5  For Wm = 0.2 islands are clearly visible( n = 1 islands at I ~ 1.35, 2.18 and n = 2 islands at I ~ 0.74 ) and there is nostochasticity except near the separatrix ( I ~~ 2.55 ) where it is expected. However,when wm = 0.05 there appears a wide region of stochasticity in the trapped-electron2region for 0.1 ( I < 1.2. The remanants of n = 1 islands at I ; 0.38, 0.74 and 1.06are still discernable even though their width in 4-space is considerably diminished. ForWm = 0.01 there is no clear island structure left and the whole region for I < 1.2 isstochastic.These conditions on k and Wm help determine the range in which w should lie.Since Aw < 1 (a usual experimental constraint) and Wm < 1, and, since the onset ofstochasticity will occur near the maximum value of JI(Aw/wm) in (7), we concludefrom the properties of Bessel functions and (6) that 0 < w < 1 + Aw. With theserestrictions on w, k and Win, the threshold for onset of stochasticity is found to beq7,1 a WM for wm < Aw. The constant of proportionality basically depends on wand Aw. For instance, when w = 0.9, Aw = 0.1, E7,1 -z 0.27wm n. This dependenceof ETJI on wm is supported by the numerical results. As the amplitude is increasedabove threshold the region of stochasticity widens in I-space. If we choose w = 0.9,Wi = 0.01, Aw = 0.1, k = 0.2, E = 4 X 10-3 connected stochasticity appears over theregion 0 < I < 1.65. This is in excellent agreement with the results obtained from theexact numerical integration of (1). For E > 0.03, the approximate analysis with justn = 1 breaks down. The exact analysis indicates that the region of stochasticity nowencompasses all trapped electrons while the n = 1 analysis shows that there still existsa region of coherent motion for the trapped electrons. This implies that the higherorder (n > 1) contributions have become important. The bandwidth, Aw, plays a rolein determining the width of the region in I-space that goes stochastic. For a fixed C,increasing Aw increases the width of the stochastic regime. However, we fix Aw = 0.1as that is about the maximum bandwidth that would be experimentally reasonable.Once there is a region of connected stochasticity, we can determine the quasilineardiffusion coefficient for electrons in that region either from the Vlasov theory for theelectron distribution function or from the two-point correlation function of dI/dt. Thetwo approaches give the same diffusion coefficient:DQL = 1 7TE2 V2(k, Iln)I,? Idu - (9)EJ Wk r, (m |M)d0/dIJ&nwhere Al is the width in I-space of the stochastic region and the sum is over those Iand n such that I& is in the stochastic region. This is compared with the numerically3evaiuatrd diffusion coefficient, DNUM = <(61)2 > /(26t) where <> indicates anaverage over the number of initial conditions (usually taken to be 1000) which liewithin the stochastic region, and 61 is the change in I in a time 6t r. = 27r/wm; Tmis the time scale over which the electrons have sampled the entire bandwidth of the FMwave. In figure 2 we have plotted DN1Nf and DQL versus E for two different values ofwOm. For values of E beginning near the threshold for onset of stochasticity and going upto about two orders of magnitude above threshold, the diffusion is almost quasilinear.However, for c > 4 X 10-3 significant deviations of DNUM from DQL take place withD NU A< DQL. The change in the slope of DQL for wm = iO- around , 10-4is due to the increase in Al as c is increased. For 4 X 10-3 < K 2.4 X 10-2 thewidth Al remains almost constant. (So as to avoid numerical difficulties encounteredfor electrons very close to the separatrix of the static well, the highest value of C welook at is just below the value for which the whole trapped region becomes stochastic).Hence, over a large range of c, DN'M does not behave like DQL. Further, as Wm isdecreased DNUMI is found to decrease; 6 e.g. for E > 5 X 10-, DNUM for Wm = 10-2is greater than DNUM for wm = 10~3 as shown in figure 2. As Wm -+ 0 we approachthe single frequency ESW limit when there is no diffusion in the region of interest forthe range of F considered here.Apart from the change in diffusion as Wm is decreased, there is a marked differencein < J 2 > as a function of time for the two values of win. In figure 3 we plot < 12 >versus time for w7m = 10-2 and Wm = 10-3 while keeping E = 8 X 10-3, k = 0.2,w = 0.9, and Aw = 0.1. For wm = 0.01, < J 2 > increases until it saturates. Theincrease implies diffusion while the saturation implies that the distribution function hasbeen flattened in the stochastic region. The time required to flatten the distributionfunction is between 2 rm and 4 rm. But, for Wm = 10-3, < J 2 > oscillates with time(with a period rm) over the entire I-space that satisfies the Chirikov resonance overlap.The small changes in < 2 > for every rm time step are reflected by the significantlylower diffusion coefficient in figure 2 for this case relative to the case of Wm = 10-2.For the lower value of Wm the distribution function takes a longer time to flatten out.It is intersting to note that for wm = 10-2, < 12 > never has the large oscillatorybehavior as in the case of W, = 10-3 even though the width in I-space over whichresonance overlap occurs is approximately the same in the two cases. We have observed4the same trend in < 12 > for all ( > 4 X 10- 3 for the two values of wi.In conclusion, we find that an FM-ESW can induce stochastic motion in electronstrapped in a static well for very small amplitudes by an appropriate choice of wavefrequency and modulational frequency. The stochastic motion occurs in phase-spacefar away from the phase velocity of the FM wave and becomes independent of thewavelength of the ESW if it is longer than twice the dimensions of the static well.The frequency of the ESW has to be close (approximately within the bandwidth of theFM wave) to the bounce frequency of the electrons whose motion is to be perturbed.The modulational frequency has to be about one percent of the frequency. For largermodulational frequencies it is difficult to produce stochastic motion for reasonableamplitudes of the FM-ESW while for smaller modulational frequencies there is verylittle diffusion and it takes a very long time to flatten the distribution function. Thediffusion is most effective (essentially quasilinear) for amplitudes about one to twoorders of magnitude above threshold. Larger amplitudes lead to a decrease in theeffective diffusion constant, as compared to the quasilinar diffusion.Finally, we suggest that the FM-ESW can be used for enhancing the confiningelectrostatic potential in the plug region of a tandem mirror plasma by stochasticallymodifying the electron distribution function in that region7 . The time required toflatten the electron distribution function is smaller than the electron-electron collisionaltime. Since the bounce frequency of the electrons in the plug region is in the rangeof ion-cyclotron frequencies, a regular antenna used for exciting waves in this rangeof frequencies can also excite an electric field along the d.c. magnetic field which canpenetrate into the plasma 8 . A single antenna usually launches a large portion of theinput power near small wavenumbers which is ideal for affecting the trapped electronsas our calculations indicate. The passing electrons from the central cell would be hardlyaffected as far as their motion along the d.c. magnetic field is concerned.The authors acknowledge helpful discussions with Drs. M. Mauel and D. Smithand with Profs. A. Lichtenberg and M. Lieberman. This work was supported in partby National Science Foundation Grant ECS 82-13430 and in part by Department ofEnergy Contract No. DE-AC02-78ET-51013.5References1. G. M. Zaslavskii and N. N. Filonenko, Sov. Phys. JETP 27, 851 (1968).2. A. B. Rechester and T. H. Stix, Phys. Rev. A 12, 1656 (1979).3. "Handbook of Mathematical Functions," edited by M. Abramowitz and I. A.Stegun (Dover, New York 1965).4. B. V. Chirikov, Phys. Reports 52, 263 (1979).5. This has also been observed for the standard mapping with a FM perturbationby A. J. Lichtenberg, M. A. Lieberman, "Regular and Stochastic Motion,"(Springer Verlag, New York, 1983).6. This behavior of the diffusion with respect to the modulational frequency forthe standard mapping with FM perturbation has been discussed by B. V.Chirikov, D. L. Shepelyanskii, Sov. Phys. Tech. Phys. 27, 156 (1982). Howeverthe variation with E is significantly different in our case than in this reference.There is also some discussion of diffusion with multiple frequency waves inG. M. Zaslavskii and B. V. Chirikov, Sov. Phys. Uspekhi M&5, 549 (1972),but a detailed treatment of FM perturbation is not given; their quasilineardiffusion calculation is only valid for n = 1.7. R. C. Cohen, Phys. Fluids 2, 2774 (1983); in this paper the quasilinearmodification to the electron distribution function is due to electron cyclotronresonance heating.8. D. K. Smith, B. D. McVey, Bull. Am. Phys. Soc. 7, 915 (1982).6Figure CaptionsFigure la Surface of section plot in I- 0 space for E = 8 X 10-3, w = 1, k = 0.2, Aw = 0.1and .;,, = 0.2.Figure lb Same as figure la except that wm = 0.05.Figure 2 Diffusion coefficients, DQL (Eq. 9) and DNUM, versus e for w = 0.9, k = 0.2,AW = 0.1, and for (a) Wm = 10-2. (the solid line being DQL and the pointsdenoted by + being DN'UiA1); (b) Wm = 10-3 (the dashed line being DQL and thepoints denoted by ® being DNU Al).Figure 3 < I2 > versus time for c = 8 X 10-3, 0.9, k = 0.2, Aw = 0.1 and(a) w,,, = 0.01; (b) w,, = 0.001.70SlI IIi"I,.* IiI~li~1-Iil~i~i~jij*~ ~a1* c I jjiI!* *-~ ~IL-. ~tFdI~flull IIn'iiU.iiiiiiii11111;''I 'II'Jil/'II II I(0 N~ co600II8a . . . no~loll 1.Li~iir~ J lie II~f~4:3qRI/I/ 4 IIJJ~ijIU ~I .NbfeeAt0I1Jii- p * * - -* * II40* . .14g,~1IttL.* ... iI~A.1~alit100jilli'~ A' U~I ~. *~'~'hil1141J''I I _I t _ ~~I. ! 1bI ,* 2*'I,I I I1 f4atf'II I(0II9L d~ V 4* nP*561%a~qvXtzm ,rIIC\j OCa 60 H0-rc,1J S)++ 00*1-+0+000 NEE30o Ez00E3n0E1000II -E3I  I I I If~j.10 ' 0(0'0 10-II-111111 I1 0fr)00'o1 00'o0)0Ci ' NOISficdIO-*-10re)1'011111! 1 1 1 11111i I I 1 11111111 1 11111111 1I I I i I IN 9 O- - O0 wd0ci110)00NrzdHYwz-0Nd--4 Lo-a - <. I

Trapped electron stochasticity by frequency-modulated waves

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Trapped electron stochasticity by frequency-modulated waves

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