ELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA. December 4, PLP No. 322

Transcription

1 ELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA by J. C. SPROTT December 4, 1969 PLP N. 3 These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated. They are fr private circulatin nly and are nt t be further transmitted withut cnsent f the authr(s) r majr prfessr. Plasma Studies University f Wiscnsin

2 Electrn Cycltrn Heating f an Anistrpic Plasma by J. C. Sprtt Departments f Physics and Electrical Engineering University f Wiscnsin In this PLP, the heating mdel f PLP 8 will be extended t plasmas in which the electrn velcity distributin is anistrpic (r, equivalently, plasmas in which the particle density is nt cnstant n a magnetic field line: V" n 0). Magnetic mirrr cnfined plasmas are necessarily anistrpic because f the lss cnes, and multiple cnfined plasmas als exhibit a degree f anistrpy at high energies when electrn cycltrn 1 resnance heating is applied. The calculated heating rate fr nn-relativistic electrns is cmpletely general in the sense that, fr a given density distributin net), the heating rate can be determined exactly fr an arbitrary magnetic field. It will be shwn that the calculatins f Guest and the nn-relativistic limit f Grawe's 3 results are a special case f this mre general thery in which the magnetic field alng the axis is parablic in psitin and in which a special distributin functin is assumed. It was shwn in PLP 8 that the pwer absrbed by the plasma in a flux shell dw is given by dp 1Tn ee.l d1jj B 1\7 B I ' 110 (1) where E is the rms perpendicular rf electric field, and the subscript 0 refers t the value f a quantity at the resnance zne. The number f electrns in d1jj is given by s that the average heating rate f electrns in dw is

3 dp / dn _ _1T...,.n...,,..._ E.J.,-",-- dw dt d$ d$ B Iv BI B (t) " 0 n(t)dt Fr n(t) cnst, equatin () reduces t _.J. d\,j 1TeE dt - B VI I V, B I '0 () (3) which is the special case previusly used t calculate heating rates. The nly apprximatins used t btain equatin () are the fllwing: 1) v «c ) v «wb /Iv BI ) v B ) Iv nl «n v B Cnditins 1), ), and 3) are easily satisfied by nn - relativistic plasmas in mst experimental devices. Cnditin 4) is satisfied except fr a very special distributin functin that will be described later. In rder t cmpare equatin () with ther calculatins in the literature, we will cnsider the special case f a magnetic field given by B(Z) B(O)(l+,Z ). Fr n(t) cnst, V' - fa> - _ 1_ fa> dz _...J1T,,--_ - B(Z) - B(O) 0 1+8Z I8B(O) and lv, BI B(O) BZ 10 0 The heating rate is then ee E dw J. e dt B 1Sz B 0 Ir-l 0 (4)

4 3 where r B /B (O). As a secnd example, we will cnsider a distributin functin in which all the particles mirrr at the same distance frm the midplane, Z M ' In the absence f parallel electric fields, the density distributin fr this [ case is given by a. B(Z)/v (Z) fr Z$Z l,, H n(z) fr Z>Z ' M where v II (Z) 'i (0) 18 ';z _z. Fr this n(z), we btain Z J 11 n(z)dz B(Z) and where Substituting the abve results int equatin () gives a heating rate f ee dw J. r dt.([' r-l ' B (5) In a recent paper n stchastic heating, Grawe 3 treats the same prblem by slving the equatin f mtin f a relativistic particle that executes sinusidal scillatins alng the magnetic field: He makes the fllwing apprximatins:

5 4 1) hmgeneus rf electric field ) randm phases 3) B(Z) B (O) (1+BZ ) 4) E ll 0 5) n cllisins r ther scattering prcesses 6) II W J.. «W.1. 7 ) we «W c 8) W r-l a ---» W 4r s l. Grawe's result in the nn-relativistic limit is dw 1TeE W dt ( ;- )J a(l-) [a(l+»), S (6) where J is a Bessel functin f rder a(l-). In rder t cmpare the tw results, cnsider the case 01, fr which the Bessel functin can be apprximated by W S r W r-l '" cs [--- 1T W r-l W r s Since the argument f the cs term is large as required by cnditin 8) abve, the slutin scillates rapidly, and we can take the average value: cs r-l _!..] 1 [ W r 4. s Substituting int equatin (6) gives dw ee r dt 'i3" r-l ' which is identical t equatin (5) with 01. The equivalence f the tw slutins is a satisfying result since the appraches are cnsiderably different. The advantage f the present treatment ver mst f thse cnsidered in the literature is that the dynamics

6 5 f the particle mtin is hidden in the cnductivity used t derive equatin (1), and s the technique is nt restricted t gemetries fr which the particle trajectry can be analytically evaluated. Furthermre, fr nnrelativistic electrns, the assumptins required t derive equatin (5) are less restrictive than thse required fr (6), and the algebra required t prduce the answer is simpler. The same result can als be btained by a simple phenmenlgical argument. The average energy gained by an electrn during ne transit thrugh the resnance can be written as e E LO AW " m A 'i m where T is the time during which the electrn stays in phase with the electric field, r the time required fr the electrn t crss the resnance regin. Kawamuxa and Terashima4 used the abve expressin t calculate the heating rate in the TP-M machine at Nagya University (Japan). Lichtenberg5 et al. have prpsed a similar expressin, differing nly by a numerical factr f rder unity, by slving the Fkker-Planck equatin fr a Markvian prcess. The transit time f the particle thrugh the resnance can be apprximated by as suggested by varius authrs, 6.7 I:::. tv '!TeE --_,:.1..0"-::-0 v ll lv" BI The resulting energy gain is Kuckes7 arrived at an identical result by slving explicitly the equatin f mtin f an electrn that mves thrugh the resnance regin with cnstant parallel velcity in a field with a cnstant gradient parallel t B. The heating rate f an electrn in a mirrr field f the frm cnsidered by Guest

7 6 and by Grawe can be calculated frm the abve result as fllws: ee..lo _ w dw Q - ll' -I-' dt 1T 18B r r-l. The result is identical t equatin (5). The dependence f the heating rate n 6 is reasnable since 6 appraches zer as the mirrr pint f the particles mves clser t the resnance regin. Fr a distributin f particles that all mirrr at the resnance (60), the density ris es rapidly as the resnance is apprached, and becmes infinite at the resnance pint. Equivalently, particles that mirrr at resnance spend an infinitely lnger time in a regin dz at ZZ than in a M regin dz at Z<Z M ' and s the particles stay in resnance fr a much lnger time than wuld be the case if they traversed the resnance with a nn-zer Nevertheless, the results f Guest and Grawe 3 indicate a finite heating rate at 60. This apparent cntradictin cmes frm the fact that cnditin 4) n page is vilated fr this special distributin functin. The heating mdel can be easily extended t this case, if we replace the density at n with an average density within the resnance regin: A mre precise derivatin wuld start with the cnductivity used t derive equatin (1), and allw the density t vary near the resnance at 0 accrding t The integral is evaluated in ee v dp (J E d.1.0 dl/j J...LO B w I a straightfrward manner: f. (w +w c )n(t)dt ( ) +4 - w -w w v c

8 7 The result is the same as wuld have been btained if the pre viusly estimated average density had been substituted int equatin (1). The average heating rate is calculated by dividing by the number f particles in d, dn 5 nd d Ct 1 I s 0 -{ZZ_Z Z 0 t get the result: dw dt The width f the resnance z is calculated by a methd suggested by Guest: I Z/ d (w-w )-- - c v. TI ' r /3 Using this value fr Z. the heating rate is dt-l - dt ee J. w fib S 0 1/3 r [ 3 1TW ] [ r -l ] /3 0.1 e E 1/3 J. B [ ] W 0 s r [ r - l ] /3 (7) which is identical t the nn-relativistic limit f Grawe's result (equatin (6» with 00, except that his cefficient is Guest als btained a similar result with a cefficient f The heating efficiency f particles that mirrr at the resnance zne is a factr (w/w ) 1/3 greater than fr S particles that mirrr well beynd resnance (0)0). Fr typical cases, this enhancement is less than an rder f magnitude. The nn-relativistic mdel predicts n heating fr electrns that mirrr befre the resnance (0<0). 3 Grawe has shwn, that at relativistic energies, heating des ccur fr these nn-resnant particles.

9 8 In many experimental situatins, the distributin required t prduce the heating rate given by equatin (7) prbably cannt be attained. Fr example, in mu1tip1es where mst f the particles pass thrugh a regin where B is very small, the magnetic mment is nt well cnserved, and the 8 mirrr pints f energetic electrns fluctuate widely. At lw energies, culmb cllisins cause the electrns t diffuse rapidly in velcity space. Parallel electric fields, either frm the micrwaves r frm space charge effects, casue a bradening f the density distributin near resnance. Magnetic field errrs may als smear ut the particles. In these situatins, it is prbably mre realistic t assume a distributin functin in the midplane f the frm W i (0) W (0) " f Ae kt.&. e Using the cnservatin f energy and magnetic mment, we can write W.,L (O) B(O) W B J. and B(O) w" (0) W,. + W l. (1 - - B -), s that the distributin functin ff the midplane is given by B(O) W l. W,. +W.L. (l-b (O)/B) f(w J.,W..,B) Ae BkT J. e kt II The density distributin is determined by a simple integratin: n(r,) This distributin, incidently, suggests a simple means fr estimating the anistrpy f a plasma by measuring the density gradient, since

10 9 If we assume a parablic mirrr field, we btain n.::.. rn ;;.;... (;:..;; O 1+8Z and dn d I CC> ndz ltrn(o) B 0 B ;sa Substituting int equatin () gives the heating rate: dw ee.&. Nte that fr 61, n( ) n(o) cnst, as required fr an istrpic distributin. re- dt B(0) (1+8Z 6) 1Ir:r (8) Fr an istrpic plasma (61), the heating rate reduces t the case in equatin (4). The effect f the anistrpy is t enhance the heating if the resnance well away is near the midplane (Z 0) frm the midplane (8Z >1). and t reduce the heating if it is This result is reasnable since a large anistrpy causes the electrns t remain near the bttm f the magnetic well. Fr a real magnetic mirrr with a finite mirrr rati, the assumed distributin functin wuld have t be replaced by a lss cne distributin, and the prblem wuld have t be slved using a mre realistic B(Z). In rder t arrive at equatin (8) using the usual analytic evaluatin f particle trajectries, it wuld be necessary t integrate Grawe's Bessel functin slutin (equatin (6» ver a weighted distributin f values f 6. Such a prcedure wuld almst certainly require numerical methds.

11 10 References 1. J. C. Sprtt, Univ. f Wis. Ph.D. Thesis (1969).. G. E. Guest, in Oak Ridge Natinal Labratry reprt ORNL-4150 (1967), page H. Grawe, Plasma Physics 11, 151 (1969). 4. T. Kawamura and Y. Terashima, in Institute f Plasma Physics, Nagya University Annual review (1968), page A. J. Lichtenberg, M. J. Schwartz, and D. T. Tuma, Plasma Physics 11, 101 (1969). 6. W. B. Ard, in Oak Ridge Natinal Labratry reprt ORNL-4401 (1968), page A. F. Kuckes, Plasma Physics 10, 367 (1967). 8. J. E. Hward, Univ. f Wis. Ph.D. Thesis (1969).