Minimum Energy Stte of Plsms with n Internl Trnsport Brrier T. Tmno ), I. Ktnum ), Y. Skmoto ) ) Formerly, Plsm Reserch Center, University of Tsukub, Tsukub, Ibrki, Jpn ) Plsm Reserch Center, University of Tsukub, Tsukub, Ibrki, Jpn ) Jpn Atomic Energy Reserch Institute, Nk, Ibrki, Jpn e-mil contct of min uthor: tmno@mil.ccsnet.ne.jp Abstrct. The condition for the minimum energy stte of tokmk plsms is emined under the constrint of the totl ngulr momentum conservtion. This problem cn be treted s vrition problem nd the solution gives specific condition between density nd rottion profiles. This condition hs been tested ginst ctul DIII-D nd JT-U eperimentl dt with creful evlution of second order vrition vlues. It is concluded tht the stedy stte tokmk plsms with n internl trnsport brrier (ITB) re in the minimum energy stte, but the trnsition sttes re not. These plsms with ITB re likely self-orgnized, which eplins stiff profiles of ITB plsms seen in mny tokmk devices.. Introduction It ws reported t the Sorrento IAEA Energy Conference tht the profiles of the tokmk plsms with n internl trnsport brrier (ITB) re eplined by the constrint due to the totl ngulr momentum conservtion []. However, the physicl mechnism presented t tht time ws not very solid. Since then, studies on second order vritions nd further nlysis of DIII- D nd JT-U eperimentl dt hve reveled tht stedy stte ITB plsms correspond to the minimum energy stte under the constrint of the totl ngulr momentum conservtion []. In this pper we present the theoreticl derivtion of the condition for the minimum energy stte nd the results of its tests ginst the ctul tokmk plsms with ITB.. Bsic Theory In toroidlly symmetric system such s tokmk, the totl toroidl ngulr momentum P φ is conserved. Tking n verge over the flu surfces, one my write Pφ = π R nmuφrdr, () where R is the mjor rdius of the is, n is the plsm density, m is the sum of the ion nd electron msses, nd u φ is the toroidl component of rottion velocity. The totl number of prticles N is conserved: N = π R nrdr. () The totl energy U, the sum of the totl kinetic energy K nd the totl potentil energy W, my lso be conserved, i.e., U = K + W = const. () The totl kinetic energy verged over flu surfces my be written s K = π R nmu rdr, ()
where u = uφ + uθ, nd u θ is the poloidl component of the rottion velocity. The totl potentil energy verged over flu surfces my be written s B W = R nk Ti + Te + E + rdr π { ( ) ε }. (5) µ Since the totl energy is conserved, the mimum totl kinetic energy corresponds to the minimum totl potentil energy. Accordingly, the condition for the minimum (potentil) energy stte is obtined by solving the vrition problem of finding the mimum kinetic energy stte by use of the Lgrngin method δ( K λn λp φ ) =, () where λ nd λ re constnt. Eqution () is eplicitly written s n ( u λ λuφ ) + ( uφ λ) n =. (7) uφ The solution for Eq. (7) gives specific condition between the density nd the toroidl rottion: n + α + β =, () n uφ u + + u α β u where n nd u re the density nd the toroidl rottion velocity t r =, nd α nd β re djustble constnts. This solution generlly mens either the mimum or the minimum. The second order vrition δ K must be evluted to distinguish the two: δ π d n dn δ φ φ φ φ K = R m ( u + du du u + n )( u ) rdr. (9) This cn numericlly be evluted for Eq. () s δ π K = R nm Fd( δu φ ), () where ( + α + β)( αβw βw + αw+ ) F =. () ( + αw+ βw ) Here, = r/ nd w= u φ / u. The minimum energy stte corresponds to negtive δ K.. Emintion ginst Eperimentl Dt The conditions described by Eq. () nd Eq () for the minimum energy stte hve been emined ginst ctul DIII-D nd JT-U plsms with n internl trnsport brrier. Figure shows the density profile of the DIII-D dischrge #59 t.975 s in Quiescent Double Brrier (QDB) Mode []. The blue circle dt points with error brs mesured by Thomson scttering systems re plotted ginst the normlized rdius. Figure shows the mesured rottion velocity t.975 s. Then the density n is re-plotted ginst the rottion velocity u s shown in Fig. nd the best fit stisfying Eq. () is lso shown. Note tht u is nerly the sme s u φ becuse of smll u θ. The χ vlue for the best fit is.. The density corresponding to
the best fit (red curve) is drwn in Figure. The fit seems ecellent. The sign of the second order vrition for the best fit is numericlly evluted by use of Eq. () nd it is indeed negtive since the clculted G vlue is.9, where G is defined s G = Fd. () The sensitivity of the G vlue sign ginst choices of the fitted prmeters n, α, nd β re crefully emined. For emple, the n vlue must be higher by lmost % for the G vlue slightly positive. This over-evluted cse is shown in Figure nd the fit is clerly poor s indicted by the higher χ vlue of.. Therefore, it is concluded tht the DIII-D QDB is in the minimum energy stte under the constrint of the totl ngulr momentum conservtion. n ( 9 m - ) u ( 5 m/s) QDB u_fit.... Fig. : QDB density profile nd the best fit..... Fig. : QDB rottion profile. n ( 9 m - ) n ( 9 m - ) + Fig. : QDB density vs. rottion with the best fit. n ( 9 m - ) u ( 5 m/s) 5 JT-U n JT-U Best n.... Fig. 5: JT-U density profile nd the best fit. u ( 5 m/s).... Fig.: An emple of fit with slightly positive second vrition vlue..5.5 JT-U u_fit.... Fig. : JT-U rottion profile. Similr tests hve been conducted ginst JT-U dischrges with ITB []. The JT-U stedy stte density profiles with non-bo-type ITB stisfy the condition given by Eq. () very well nd the second order vritions clculted from Eq. () re confirmed to be negtive. An emple of (#E75 t. s) is shown in Fig. 5. The blue circle points with error brs re the mesured density nd the red curve represents the best fit stisfying Eq. () ( χ =.7 nd
G =.). In this cse the centrl density is chosen so s to mtch the mesured line density since the direct Thomson mesurement ner the center is missing. The mesured line density must be higher by bout 5% (bout % higher in the centrl density) in order to give slightly positive G vlue. Therefore, it is very likely tht the JT-U dischrges with ITB re in the minimum energy stte lso. In these nlysis ITB plsms with etremely steep density grdients (bo type ITB modes) re ecluded since the mesured impurity rottion velocity my not represent the min ion (deuteron) rottion velocity. Recently, control of the QDB density profiles by pplying centrl electron cyclotron heting (ECH) hs been reported []. A very intriguing observtion is tht the toroidl rottion profile significntly flttens s the density profile flttens fter ECH while the ion temperture profile does not chnge much. It should lso be noted tht the ECH does not provide ny ngulr momentum input either. n ( 9 m - ) ECH n ECH Best n u ( 5 m/s) QDB u_fit ECH u_fit.... Fig. 7: QDB density profiles nd the best fits just before (blue) nd ms fter (red) ECH..... Fig. : QDB rottion profiles just before (blue) nd ms fter (red) ECH. ECH 5 n ECH 5 Best n ECH 5 u_fit n ( 9 m - ) u ( 5 m/s).... Fig. 9: QDB density profile t 5 ms fter ECH..... Fig. : QDB rottion profile t 5 ms fter ECH. An emple is illustrted in Fig. 7 nd Fig.. The blue circle dt points in Fig. 7 represent the QDB density profile just before the ECH ppliction (ectly the sme s Fig. ). The red squres re tken t ms fter the ECH ppliction where the density profile settles down to new stedy stte condition. Figure 7 shows the corresponding toroidl rottion profiles. The nlysis of the dt t ms fter the ECH ppliction indictes tht the condition Eq. () lso holds between the density profile nd the rottion profile s shown by the red best fit curve in Fig. 7. The second order vrition is confirmed to be negtive ( G =.7). Figure 9 nd Fig. respectively shows the density nd toroidl rottion profiles tken during the trnsition period t 5 ms fter the ECH ppliction. The density profile hs nerly settled down to the new stedy stte profile, but the rottion profile hs not. These dt t 5 ms fter ECH seemingly stisfy the condition Eq. () s indicted by the red curve in Fig. 9. However, the χ vlue is
5 somewht higher compred to the other two cses (. compred to. nd.) nd the G vlue is only mrginlly negtive (.). This suggests tht the QDB plsm once settled down in the minimum energy stte before ECH shifts into nother minimum energy stte fter ECH is pplied, but the condition for the minimum energy stte is not well stisfied during the trnsition.. Discussions The result indictes tht the density profile for the ITB plsm corresponds to very prticulr rottion profile. In these tokmk plsms toroidl rottion is supposed to be generted by neutrl bem injection. Since the locl momentum source term is determined from the injected neutrl bem distribution nd the density profile, there is no gurntee tht the rottion profile mtches to the desired prticulr profile. Therefore it is very likely tht the ITB plsms re self-orgnized or self-djusted. This eplins stiff profiles for ITB plsms climed in mny tokmk devices. A possible physicl mechnism for this process is s follows: An instbility occurs which chnges the density profile. Accordingly, the momentum nd prticle deposit profiles chnge. This loop continues until the plsm finds the prticulr minimum energy stte condition between the density nd toroidl rottion profiles. Therefore this seems consistent with some theoreticl clims tht the plsm profile corresponds to mrginlly stble condition for certin instbility, such s trpped prticle mode [5]. Acknowledgement The uthors would like to thnk P. Gohil nd M. Chu (Generl Atomics) nd M Murkmi (ORNL) for their vluble suggestions nd discussions. References. TAMANO, T. nd KATANUMA, I., "Plsm confinement with trnsport brrier", Nucl. Fusion () -.. TAMANO, Teruo nd KATANUMA, Iso, "Minimum energy stte of tokmk plsms with n internl trnsport brrier", Plsm Phys. nd Contr. Fusion () A7-A.. GOHIL, P., et l., "Dynmics of Formtion, Sustinment, nd Destruction of Trnsport Brriers in M gneticlly Contined Fusion Plsms", Plsm Phys. nd Control. Fusion () A7. GOHIL, P., et l., "Development of Methods to Control Internl Trnsport Brriers in DIII-D plsms", Bull. Am. Phys. Soc. () 9.. KAMADA, K., JT- Tem, "Etended JT-U plsm regimes for high integrted performnce", Nucl. Fusion (). 5. R. Wltz, et l., Phys. Plsms (997).