OTHER USES OF THE ICRH COUPL ING CO IL. November 1975

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Randolf Arthur Malone
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1 OTHER USES OF THE ICRH COUPL ING CO IL J. C. Sprtt Nvember I,," PLP 663 Plasma Studies University f Wiscnsin These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated. They are fr private cirulatin nly and are nt t be further transmitted withut cnsent f the authrs and majr prfessr.

2 It has been suggested that ther nn-sinusidal wavefrms culd be fed acrss the terminals f the ICRH cupling cil in the small ctuple t prduce different types f plasma heating. This nte examines several such pssibilities. An experimentally easy wavefrm t prduce wul d be a vltage that rse abruptly t sme value V and then decayed expnentially with a lng time cnstant, T. Such a wavefrm culd be prduced by switching a charged capacitr acrss the cil terminals by means f an ignitin. Fr a sufficiently large capacitr the circuit wuld be verdamped and wuld prduce the desired wavefrm. Such a wavefrm wuld cntain a furier spectrum f frequencies frm w lit up t infinity. Thus fr T large, bth electrn and in cycltrn heating shuld ccur thrughut the vlume f nn-unifrm field in the ctuple. In rder t cmpare the vlume-averaged heating rates fr such a pulse with, fr example, the ICRH rate using sine waves, we assume that the cupling cil impses a maximum zer-t-peak vltage that is allwed acrss its terminals, and we ask hw a wavefrm f V = V sin wt cmpares with ne that has V = v e -t / T (t; A}, (;}e furthet assume that T is very large, s that the pulse is essentially a step functin. { V = v V t < 0 t > 0. Fr sine waves, it is well knwn that the in (r electrn) energy is given by (1)

3 -2- where G is a gemetric factr $ 0. 1 in the ctuple, m is the particle (electrn r in) mass, E2 is the spatially averaged mean square electric field, B is the field at cycltrn resnance, and w is the rf frequency. Nte that the energy rises linearly in time as expected fr a stchastic prcess. Fr the step functin case, we calculate the heating by nting that a particle will accelerate fr abut ne cycltrn radian befre it begins t gyrate, and s the heating is strngly peaked near the zer field axis. Anther way t say this is that -panticles near the zer field axis are nn-adiabatic and hence are mst effected by the fast rising electric field. The vlume-averaged heating rate is determined frm But t = = c,- 3 eb r (wt) 1/3 s (2) Cmparing this t Eq. (1) shws that the cefficient and the time dependence imply a greater heating fr the step functin than fr the sine wave prvided the E fields (i.e., cil vltage) are the same. It is nly necessary that the step functin remain n fr several cycltrn radians (wt > 1). In this case, w eb /m, where B is the field near the wall = in the ctuple. Nte als that since W a m -1/3, mst (91%) f the energy ges int the electrns rather than the ins.

-3- If a tridal field were added t the ctuple, we wuld expect a runaway cnditin with average energies given by r (3) - -1 This prduces an even larger heating fr wt > 1, but since Wa rn, nearly all

4 -3- If a tridal field were added t the ctuple, we wuld expect a runaway cnditin with average energies given by r (3) - -1 This prduces an even larger heating fr wt > 1, but since Wa rn, nearly all (99.95%) f the energy ges int the electrns. In rder t test these predictins, a cmputer cde (SPEMS2) was used t fllw the trajectries f 100 nn-interacting charged particles in am ideal linear ctuple field with filimentary currents, and a spatially cnstant E field. The average energy f the 100 particles as a functin f time was determined fr the three cases discussed abve, and the results are shwn in the figure. The scaling f energy with time and the apprximate magnitude f the energy agree well with the predictins (Eqn. 1, 2, and 3). In the ctuple, the E field is nt spatially unifrm and the field n axis abut abut 1/20 f the field at the surface f the cil (which is V /2.7 vlts/meter), r abut 185 vlts/meter fr 10 kv acrss the cil. This cnsiderably exceeds the vltages nrmally used fr tri-dal hmic heating experiments. The cupling cil can be represented apprximately by an 0.7 llh\r by"ililductance.'l ifl2.serne,$1 wittb: 9"ecsti-st cef:f.<id5qg, & fjr 10 kv 60 llf capacitr culd be used t dump int the cil and it wuld prduce a slightly underdamped (Q = 1.8) wave with a 40 II sec perid, and a peak current f abut 63 ka. Using the impulse apprximatin and assuming that the cil weighs 10 k it wuld nly jump 0.01 cm ff the flr as a result f image currents in the wall.

5 -4- The apprximate maximum energy gained by a particle is given by WMAX = m (f E dt ) 2 ;; ev fr a particle n axis. Nte that since the maximum energy is [jedt] 2 ex LCV 2, the particle energy is determined slely by the energy stred in the capacitr and is apprximately 10 ev/kjule. This rather small value, tgether with the fact that mst f the energy ges int a few electrns n axis has led us t be rather pessimistic abut the future f this type f experiment. T say it anther way, in rder t get 100% cupling t the plasma wuld require a density f 2 x cm -3 and at thse densities there wu ij be difficulty getting the field t penetrate the plasma. On the ther hand it may happen that instabilities r parametric effects cme int play at lw densities and give rise t ther interesting effects such as anmalus heating.

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GENERAL FORMULAS FOR FLAT-TOPPED WAVEFORMS. J.e. Sprott. Plasma Studies. University of Wisconsin

GENERAL FORMULAS FOR FLAT-TOPPED WAVEFORMS J.e. Sprtt PLP 924 September 1984 Plasma Studies University f Wiscnsin These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated.