J. C. Sprott OHMIC HEATING RATE IN A TOROIDAL OCTUPOLE. August 1975 PLP 643. Plasma Studies. University of Wisconsin

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Reprts are infrmal and preliminary and as such may cntain errrs nt yet

1 OHMC HEATNG RATE N A TORODAL OCTUPOLE J. C. Sprtt August 1975 PLP 643 Plasma Studies University f Wiscnsin These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated. Tbey are fr private circulatin nly and are nt t be further transmitted withut cnsent f the authrs and majr prfessr.

2 Cnsider a tridal ctuple field B p with superimpsed rthgnal tridal field B T and bth fields varying in time. We wish t estimate the pwer P delivered t the plasma as a functin f time via hmic heating assuming the vacuu m fields are unperturbed by any currents in the plasma. As a starting pint P = J (j E dv where (j is the dc cnductivity Ell is the cmpnent f E parallel t B (= 8p + 8 T ) and the integratin is ver the vlume f the plasma. Nw Ell cnsists f tw parts: The tridal electric field is given in terms f the plidal flux by E T = 111 d :1f dt (/J - /JHOOP) where R is the majr radius. The plidal electric field can be apprximated by where d is the length f a ( clsed) plidal field line and the area integral is evaluated in the plidal plane between the hps and the flux surface n which E p is t be calculated. A frtunate prperty f an ctuple is that in plidal flux space mst f the plasma is lcated very near the separatrix and s it is reasnable t expect that electric fields calculated n the separatrix wuld represent an apprpriate average fr determining the pwer absrbed by the plasma. We further make a large aspect rati apprximatin s that

3 -2- the variatin f the tridal field ver the vlu me f the plasma is negligible. Then and The cnstants C 1 and C 2 can be determined with a bit f wrk directly fr0m the flux plt. Fr the small ctuple C meters and C meter Nw since we have used an apprximatin in which B T E T and E p are cnstant in space the vlume integral invlves nly the spatial dependence f B p which we take t be B = (. ) 1 B p Therefre the pwer can be written as p a. t is reasnable t assume that the cnductivity is cnstant in space and t remve it frm the integral. The remaining integral is just barely duble with an integral table and a very ambitius graduate student. Lacking the latter a satisfactry apprximatin can be btained by evaluating the integral in the limits a (= B p /B T ) «1 and a» 1 and fitting the intermediate case with a smth analytic functin t get the result:

4 -3- Finally we wish t btain an expressin fr the cnductivity which we assume t arise frm bth electrn-in and electrn-neutral (H 2 ) cllisins. The electrn-in (Spitzer) resistivity is x10 l eff n W T 3 /2 e 6.24x10 l eff = T 3/2 e hm-meters where T e is in ev and l eff is any anmaly factr that ma y be present. The electrn-neutral resistivity is -6 n N = 6.29x10 n ln e hm-meters and the ttal cnductivity is a = = {l ff /1600 T x10-6 n n ) -1 n sp n N e e 0 e Putting it all tgether gives Nte that the middle (crss) term implies that the heating is enhanced.. whenever ne field is rising while the ther is falling (B p B T < 0). The abve expressin was incrprated as an electrn heating term in prgram SMULT (see PLP ) in an attempt t mdel the experiments being dne by Etzweiler n the small ctuple (see PLP ). On that experiment the plidal field pulse is a 5 msec half sine wave and the trid al field pulse is a 2 msec damped half sine wave given apprximately by

5 -4- B T = 1.77 B TMAX sin( 1571 t ) exp( -654 t )( t > 0 ) which can be initiated at an arbitrary time relative t the plidal field pulse. The simulatin was run with B T = 1.25 kg ( crrespnding t 2.0 kv n the B T bank) B p = 4.8 kg ( crrespnding t 2.0 kv n the B p bank ) p = 1x 10-5 trr ( gauge) and l eff = 1. The 500 watt cw 2.45 GHz micrwaves were turned ff and the tridal field pulse turned n at 3.5 msec int the 5mmsec plidal field pulse. The electrn temperature rises abruptly t - 4 kev when the tridal field cmes n while the density rises mre slwly 12-3 t - 1.5x10 cm. The neutrals are fairly well burned ut (!l'/ e /1f - 10 ). The cnductivity is limited by electrn-neutral cllisins and a cmputer simulatin with l eff = 10 lks al mst identical t the l eff = 1 case. The in temperature always remains less than 1 ev and in fact drps t ev during the hmic heating because f prer cupling t the electrns and enhanced lsses. The dminant lss mechanism is thermal flw t 2 bstacles ( taken as 90 cm ). The neutral density rises abut a factr f tw as a result f electrn impact desrptin which Grebner estimates t be The results f the simulatin are reasnable except fr the 4 kev electrn temperature which is prbably much t high. t des suggest the imprtance f imprving ur sft x-ray diagnstics hwever. There are a number f effects that culd reduce the temperature in the experiment: impurities turbulent enhancement f the resistivity pr penetratin f the electric fields instabilities anamlus lsses electrn runaway etc. n particular tr 4 kev and 1.5x10 12 cm -3 the Dreicer critical runaway electric field is nly 0.04 vlts/meter which is surely exceeded in the experiment. Lencini (PLP 276) has experimentally measured the plasma resistivity in

6 -5- this runaway regime and he finds ( hms/meter) where T e is in ev and n is in cm -3 Wf we!add this resistivity t nsp and nn' t imu[ati6h ReBiets a much mre credible val ue f T e 100 ev and n 7x10 11 cm -3. The detailed results are shwn in Figure 1. The simulatins thrughut the rest f this paper include the Lencini term t accunt fr electrn runaway. t is f curse desirable t have sme cmparisns f the simulatin with experimental results. The ne quantity that is easy t measure experimentally is the vlume-averaged in saturatin current t a Langmuir prbe in the bridge ( see Strait PLP 566). The in saturatin current can nt be interpreted in the usual way hwever because we anticipate that kt e /e may cnsiderably exceed the usual prbe bias vltage f 45 vlts. Hwever if plasma lsses are gverned by thermal fl w t grunded bstacles in the plasma ( i. e. hp supprts) we expect the fl ating ptential V f t be near zer. n fact experimental measurements shw V f l 10 vlts fr the case cnsidered here. Fr V f = 0 the current t a prbe biased t -V B is given by = 10. ( 1 - exp ( -ev B /kt e )) 1 s that in the limit kt e /e» V B ' the prbe current beys the rather unusual relatin a n/ prvided T > T.. Prgram SMULT was mdified t predict the current density e 1 t a Langmuir prbe biased t - 45 vlts fr arbitrary T e assu ming V f = O. The predicted current is cmpared with the bserved current in Fig. 2. The

7 -6- agreement is pretty gd cnsidering all the apprximatins that are invlved. Recall als that the secndary electrn emissin cefficient fr a prbe may exceed unity at these electrn temperatures and s detailed agreement is even less likely. T further test the simulatin the peak in current was measured as a functin f the fur easily varied parameters: B T B p ' T (tridal field turn-n time) and (filling pressure). Th e experimental results are cmpared with the c mputer simulatin in Figs The agreement is nt t impressive and shws a tendency fr the simulatin t verestimate the heating especially at lw values f ma gnetic field. Finally as an aside the change in in gauge pressure reading frm befre t immediately after the pulse was recrded as a functin f initial pressure and the result is shwn in Fig. 7. Nte that ver the intermediate range f pressures the hmic discharge acts as a vac uum pump and prduces as much as a 40% 'reductin in backgrund pressure. The simulatin always predicts a psitive p because it is assumed that every in that strikes a surface recmbines and returns as a neutral. This is clearly an errneus assumptin and needs t be replaced with smething mre realistic. n cnclusin we have derived an expressin fr the vlume-averaged pwer absrbed by the plasma in an ctuple due t hmic heating when an arbitrarily time varying plidal and tridal field are present. Cmputer simulatins shw an unreasnably high electrn temperature (4 kev) if the resistivity is classical. Hwever since the electric field exceeds the critical field fr electrn runaway it is mre reasnable t use a resistivity such as that measured experimentally by Lencini in which case the simulatin give a mre believable value f 100 ev. The calculatin is available as a functin subprgra m (OHMC) which can be called fr m SMULT. in. ent is t cntinually refine the. r.

8 - 7 - The intent is t cntinually refine the subprgram as mre detailed infrmatin n the cnductivity becmes available frm the experiment.

l d------------------------------- /a 5 1 M U L..A1"" 0 N 3..-. E u v l-.- (j) z U J O" 10 '0 ---... f.!. J r r... '...... -... --. -.- n.. -..--... - - - - - -.

9 l d /a 5 1 M U L..A1"" 0 N E u v l-.- (j) z U J O" 10 ' f.!. J r r... ' n a_ --.tt '1'fl/!lf.". t t... f ) i ) t -. t r." J ' '< OZ #. 10 '.J -Q} t.. l 2 ECRH OFF BT ON! 3 TME.:' (rns) 4-5 FG

10 .. 10 e S M).""TlN '). 10 r /'.; " ;' - -- CRH OFF s.. ON 2 TME ems) s FG 2

11 .3 10 r / SMULATON 000 O JO Bp'= f -::: X f) -5 T 1: == 35 ms TRDAL 2 3 8At-lK VOL-rAGE. (<V) F t;..3

12 a 0 O'Z. (0 BT = c kv f == X 10-5 T 1: = 35 rv S s

f03 --- SMULATON)...... n... _. 1 02 00 000000 - a. E ct.. '-...J.

13 f SMULATON) n... _ a. E ct.. '-...J... H 10 Sp ::: 2 k V BT = 2 kv p::: X la -.5 T L- J 2 7:: (ms) :; 5 FG S

14 .) S MUL.p.. ON N <t f ""'-i - < D 0 Bp = 2 kv BT = 2 kv 7: =.s rns FG- 6

15 "' J) <:).;.. C a J ::t--.. () FG 7

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

OTHER USES OF THE ICRH COUPL ING CO IL J. C. Sprtt Nvember 1975 -I,," PLP 663 Plasma Studies University f Wiscnsin These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated.