Numerical Study of Two-fluid Flowing Equilibria of Helicity-driven Spherical Torus Plasmas

Transcription

1 Numrcal Study of Two-flud Flowng Equlbra of Hlcty-drvn Sphrcal Torus Plasmas T. Kank M. Nagata and T. Uyama Japan Coast Guard Acadmy 5-1 Wakaba Kur Hroshma Japan Dpartmnt of Elctrcal Engnrng and Computr Scncs Unvrsty of Hyogo 167 Shosha Hmj Hyogo Japan Abstract. Two-flud flowng qulbrum confguratons of a hlcty-drvn sphrcal torus (HD-ST) ar numrcally dtrmnd by usng th combnaton of th fnt dffrnc and th boundary lmnt mthods. It s found from th numrcal rsults that lctron fluds nar th cntral conductor ar td to an xtrnal torodal fld and on fluds ar not. Th magntc confguratons chang from th hgh-q HD-ST (q>1) wth paramagntc torodal fld and low- (volum avrag valu <> %) through th hlcty-drvn sphromak and FP to th ultra low-q HD-ST (<q<1) wth damagntc torodal fld and hgh- (<> 18 %) as th xtrnal torodal fld at th nnr dg rgons dcrass and rvrss th sgn. Th two-flud ffcts ar mor sgnfcant n ths qulbrum transton whn th on damagntc drft s domnant n th flowng two-flud. INTODUCTION Durng coaxal hlcty njcton torodal on flow rlatd to a rotatng torodal mod numbr n=1 magntc structur has bn obsrvd n many hlcty-drvn sphrcal torus (HD-ST) xprmnts such as CTX HIST SPHEX SSPX HIT and NSTX. Th n=1 mod structur s consdrd to b playng an ssntal rol n drvng currnt on closd flux surfacs of th HD-ST. On th HIT xprmnts [1] t s found that th n=1 mod s lockd to lctrons and not to ons suggstng a rotatng magntc fld currnt drv. Bcaus of ths fatur th qulbrum computaton of a HD-ST s rqurd to tak nto account two-flud ffcts [-3]. Th two-flud ffcts ar xpctd to xplan th stablty of hgh- ST. Howvr th dtals of how such th flowng two-flud modl affcts th MHD qulbrum confguratons of th HD-ST ar not numrcally nvstgatd. Th purpos of ths study s to numrcally dtrmn th two-flud flowng qulbra of th HD-ST and to nvstgat thr fundamntal proprts. W focus our attnton on contrbuton of th on flow to th magntc confguraton th two-flud ffcts and valus. Th formalsm for flowng two-flud qulbrum s dvlopd by Stnhaur Ishda and co-workrs [4]. It s an xtnson of th MHD qulbrum problm for a non-flowng snglflud whch s govrnd by th Grad-Shafranov quaton. Th axsymmtrc qulbrum of th flowng two-flud s dscrbd by a par of scond-ordr partal dffrntal quatons for th magntc and on flow stram functons and Brnoull quaton for th dnsty [4]. By applyng th two-flud formulaton to th HD-ST qulbrum wth purly torodal on flow w modfy th non-flowng sngl-flud qulbrum cod [5] whch computs th HD-ST qulbrum n th mor ralstc rgon ncludng th sphrcal flux consrvr (FC) and th coaxal hlcty sourc (CHS) of HIST [6]. In ths cod th qulbrum computaton of th HD-ST rflcts th ralstc condton that th bas col flux pntrats th FC wall and th lctrods. In ordr to solv th govrnng quatons of th flowng twoflud qulbrum w mploy th fnt dffrnc and th boundary lmnt mthods as th numrcal approach ncorporatng ths boundary condton.

2 NUMEICAL MODEL For numrcal computaton w modl th mor ralstc rgon ncludng th FC and th CHS of HIST. Accordng to HIST gomtry th sphrcal FC s m n damtr. Th FC adjons th CHS. Th CHS conssts of th outr lctrod (76 m n damtr.35 m n lngth) th nnr lctrod (.18 m n damtr.39 m n lngth) and th outr bas col (.36 m n damtr.3 m n lngth) of rctangular cross scton. Th cntral conductor (.114 m n damtr) s nsrtd along th symmtry axs. Insrton of a torodal fld col currnt I tf along th gomtry axs nsd th cntral conductor producs an xtrnal torodal fld. In Fg. 1 w show th modl of th FC and th CHS whch wll b usd n ths papr. W dvd th rgon n whch th qulbrum s dtrmnd nto thr subrgons 1 and 3. In th HIST xprmnt th bas fld s gnratd long bfor th plasma s njctd nto th FC. Thus th bas fld pntrats th FC wall th lctrods and th cntral conductor and xtnds all ovr th spac whn th qulbrum confguraton s formd. On th othr hands th lftm of th plasma s much shortr than th rsstv pntraton tm of th FC th nnr lctrod and th cntral conductor and t s much longr than that of th outr lctrod. Thrfor w assum that th magntc fld gnratd by th plasma currnt pntrats th outr lctrod and that t dos not pntrat th FC wall th nnr lctrod and th cntral conductor. Lt us us a cylndrcal coordnat systm (r z) n whch th z-axs ls along th symmtry axs of HIST gomtry. Snc th two-flud flowng qulbrum confguraton of th HD-ST plasma s axally symmtrc w can dtrmn t by solvng th coupld par of dffrntal quatons for th gnralzd stram functons and. Th coupld quatons can b wrttn n th form [4] r d d d S ( ) ( ) S r d d r d d dh S ( ) S ( ) r. d d dh d (1) () Hr dnots th Grad-Shafranov oprator and S s dfnd as th rato of th systm sz scal to th on skn dpth. Th flow stram functons and th gnralzd stram functons ar ntroducd to xprss th spcs flow vlocts and th polodal part of th gnralzd vortcty of ach spcs rspctvly. Th total nthalps H and ar arbtrary surfac functons of thr rspctv surfac varabls rspctvly H ( ) p ( ). u / q E (3) (4) Hr p u q and ar th prssur th flow vlocty charg and th scalar potntal rgardng th stady lctrc fld. In ths study w consdr th spcal cas of purly torodal on flow.. ( ). In addton w assum th rmanng arbtrary functons: dh / d ( ) C dh / d C C BT H H C C C 1 H1 H1 C C H3 3 (5) (6) (7) whr C s ar constant paramtrs. Espcally C H1 and C BT ar rlatd to th strngth of on flow and xtrnal torodal fld rspctvly. W chang ths two paramtrs. Nxt lt us consdr th boundary condtons for Eqs. (1) and (). W st = bas on and 5 bcaus th bas flux xtnds all ovr th spac. Hr bas rprsnts th bas flux producd by th bas col currnt I bas. Ampr s law on th surfac c can b wrttn as I bas 1 r n c dl. (8)

3 Hr n dnots th unt vctor whos drcton s outward normal to th boundary. Th boundary condton for on th surfac c s obtand by sttng at an unknown constant. Aftr th lnarzaton of Eqs. (1) and () th problm can b solvd numrcally by mans of th combnaton of th fnt dffrnc and th boundary lmnt mthods [5]. Ths computaton s prformd so that th total torodal currnt I t s constant. FIGUE 1. Modl of th flux consrvr and th coaxal hlcty sourc. NUMEICAL ESULTS W ncras th valu of C H1 rlatd to th strngth of on flow to nvstgat th varaton of th magntc confguraton. As th rsult of tratons th valu of S C BT s dtrmnd as th gnvalu. Paramtrs and computd varous valus such as volum avrag valu <> volum avrag torodal valu < > normalzd valu two-flud ndx f F and volum avrag valu <> ar shown n Tabl I. Hr <> < > f F and <> ar dfnd as p p (9) p p B p T B f F p t / / F F B E B (1) (11) j B. (1) B Hr B j E and F F rprsnt magntc fld currnt dnsty lctrc fld and two-flud corrcton trm n Ohm s law E S u B F F rspctvly. Also Bt s th vacuum torodal fld at th major radus.th avrag s ovr th ntr rgon of 1. If f F 1 ( ff 1) th two-flud ffct s sgnfcant (nglgbl).

4 Th magntc fld profls on th mdplan ar shown n Fg.. Fgur shows th cas of small on flow (C H1 =) and th hgh-q ST wth paramagntc torodal fld B t profl. W ndcat th flow vlocty u and th safty factor q latr. As th ffct of th on flow bcoms mor sgnfcant th xtrnal torodal fld B t. dcrass and furthr rvrss ts sgn. Fgur shows th sphromak confguraton wthout B t.. In Fg. (c) B t at th dg rgons rvrss th sgn whch ndcats th FP-lk confguraton. Evntually B t at th whol rgons rvrss th sgn as shown n Fg. (d). Th magntc confguraton thn changs to th ultra low-q ST wth damagntc B t profl. TABLE I. Paramtrs and computd varous valus of hlcty-drvn sphrcal torus qulbra for C H = C H = C H1 =4. C H3 =- S 1=-.7 S C= and I bas /I t =.. C C H1 S C BT <> < > f F <> [m -1 ] -4.3x1-8.11x x Magntc fld Magntc fld B z - B t B t. B z B t B t. (c) Magntc fld Magntc fld FIGUE. adal profls of magntc fld on th mdplan. hgh-q HD-ST (C H1 =) hlcty-drvn sphromak (C H1 =9.) (c) hlcty-drvn FP (C H1 =15.) and (d) ultra low-q HD-ST (C H1 =8.). Th rd grn and blu lns ndcat th torodal fld th xtrnal torodal fld and th polodal fld rspctvly. 4-4 B t B t. B z B t B t. B z (d) Th flow vlocty profls on th mdplan ar shown n Fg. 3. It s found from Fg. 3 that th torodal currnt s domnantly carrd by th lctron fluds. Th lctron fluds at th nnr dg rgon ar td to B t. whl th on fluds ar not. Fgur 3(c) shows th lctron flow at th nnr dg rgon rvrss th sgn du to th rvrsal of B t. at th nnr dg rgon. As th ffct of th on flow bcoms mor sgnfcant th rvrsd rgon of th torodal lctron flow xtnds as shown n Fg. 3(d).

5 Th torodal currnt dnsty profls on th mdplan ar shown n Fg. 4. As th ffct of th on flow bcoms largr th torodal currnt dnsty changs from th hollow profl to th pakd on. Furthr du to th rvrsal of th torodal lctron flow t rvrss th sgn at th nnr dg rgon. Flow vlocty Flow vlocty u t u t u p u t u t u p Flow vlocty (c) Flow vlocty - u t u t u p u t u t u p (d) FIGUE 3. adal profls of flow vlocty on th mdplan for th sam condton as Fg.. Th rd blu and grn lns ndcat th on torodal flow lctron torodal flow and polodal lctron flow vlocts rspctvly. Torodal currnt dnsty hgh-q ST Sphromak FP low-q ST FIGUE 4. adal profls of torodal currnt dnsty on th mdplan. Th rd blu grn and purpl lns ndcat th hgh-q HD-ST (C H1 =) th hlctydrvn sphromak (C H1 =9.) th hlcty-drvn FP (C H1 =15.) and th ultra low-q HD-ST (C H1 =8.). Safty factor q hgh-q ST Sphromak FP low-q ST axs FIGUE 5. Safty factor q as a functon of th normalzd polodal flux functon / axs for th sam condton as Fg. 4. Hr axs s at th magntc axs. W show th safty factor q as a functon of th normalzd polodal flux functon / axs n Fg. 5. As th ffct of th on flow bcoms largr th q-valu coms down and rvrss th sgn at th nnr dg rgon. Fnally t rvrss th sgn at th whol rgon and bcoms th ultra low-q (<q<1).

6 Th polodal flux contours ar shown n Fg. 6. All ths flux surfacs hav th opn flux pntratng th lctrods and form th hlcty-drvn confguratons. Ths suggsts th possblty of th currnt drv by coaxal hlcty njcton. As th ffct of th on flow bcoms mor sgnfcant th amount of closd flux ncrass. Th HD-STs hav sgnfcantly lowr <> valus than th hlcty-drvn sphromak and FP. Not that th ultra low-q HD-ST wth damagntc B t and hgh- appars n th rgm of <> valu lowr than th lowst gnvalu =9.9 m -1. Thrfor t could b obsrvd n th xprmnt. (c) (d) FIGUE 6. Polodal flux contours for th sam condton as Fg.. W xamn th dpndnc of th maxmum valu of th torodal on flow u tmax on <> and f F n Fg. 7. As u tmax ncrass wth th transton of th hgh-q HD-ST to th hlcty-drvn FP <> ncrass du to th dcras n B t.. On th othr hand <> gradually dcrass as u tmax ncrass from th hlcty-drvn FP to th ultra low-q HD-ST. It s also found from Fg. 7 that xcpt for th rgon of ngatv u tmax all th valus of f F ar largr than unty. In th rgon of slow on flow f F has a sharply pakd valu (u tmax =3) FP low-q ST SPH hgh-q ST u tmax f F u tmax f F=1 ln FIGUE 7. Dpndnc of th valu of u tmax on <> and f F. Hr u tmax rprsnts th maxmum (mnmum) valu of th torodal on flow vlocty u t whn u t s postv (ngatv).

7 Prp. flow x ExB on damagntc drft F F S u xb E FIGUE 8. adal profls of flud drfts F F S u B and E on th mdplan for u tmax =3. E B drft (rd ln) and on damagntc drft (blu ln) and FF (rd ln) S u (blu) and E (grn). B W consdr why f F has th sharply pakd valu. Fgur 8 shows th radal profls of flud drfts F F S u B and E on th mdplan. In th cas of u tmax =3 t s found from Fg. 8 that th E B drft s approxmatly zro and th on damagntc drft s domnant. Du to E th balanc s mantand by u B and F F as shown n Fg. 8. Thrfor f F bcoms sgnfcantly larg. S CONCLUSIONS W hav nvstgatd th two-flud ffcts on th MHD qulbrum confguratons of th HIST HD-ST. Conclusons obtand n ths papr ar summarzd as follows. 1) Equlbrum of th HD-ST basd on th two-flud modl wth flow n th rgon ncludng th FC and CHS ar numrcally dtrmnd by usng th fnt dffrnc and th boundary lmnt mthods. ) Th magntc confguratons chang from th hgh-q HD-ST (q>1) wth paramagntc torodal fld and low- (<> %) through th hlcty-drvn sphromak and FP to th ultra low-q HD-ST (<q<1) wth damagntc torodal fld and hgh- (<> 18 %) as th xtrnal torodal fld at th nnr dg rgons dcrass and rvrss th sgn. 3) In th ultra low-q HD-ST th torodal fld rvrss th sgn but th polodal fld dos not do t. Thus t s dffrnt from th flppd ST obsrvd n th xprmnt. Also th ultra lowq HD-ST appars n th rgm of <> valu (<>= -73 m -1 ) lowr than th lowst gnvalu =9.9 m -1. Thrfor t could b obsrvd n th xprmnt. 4) Th two-flud ffcts ar mor sgnfcant n ths qulbrum transton whn th on damagntc drft s domnant n th flowng two-flud. Th fundamntal proprts of th HD-ST qulbrum basd on th two-flud modl wth flow outlnd hr ar gnrally vry avalabl for prdctng what qulbrum confguraton s formd n th HD-ST xprmnt. Thr ar furthr ssus rlatd to th qulbrum of th ultra low-q HD-ST: 1) Can th gnralzd hlcts consrv durng ths qulbrum transton? ) How do w xprmntally drv a flow of Alfvn Mach numbr M A.7 for producng th ultra low-q HD-ST? 3) Stablty analyss of th flowng two-flud qulbrum of th HD-ST s rqurd. ACKNOLEDGMENTS Ths work was partally supportd by th Elctrc Tchnology sarch Foundaton of Chugoku. EFEENCES [1] K. J. McCollam and T.. Jarbo Plasma Phys. Controlld Fuson (). [] L. C. Stnhaur and A. Ishda Phys. v. Ltt (1997).

8 [3] Z. Yoshda and S. M. Mahajan Phys. v. Ltt (). [4] H. Yamada T. Katano K. Kana A. Ishda and L. C. Stnhaur Phys. Plasmas (). [5] T. Kank M. Nagata T. Uyama S. Ikuno and A. Kamtan J. Phys. Soc. Jpn (1998). [6] M. Nagata T. Kank N. Fukumoto and T. Uyama Phy. Plasmas 1 93 (3).