Runaway Electrons and Current Dynamics During Tokamak Disruptions

Transcription

1 Runaway lctrons and Currnt Dynamics During Tokamak Disruptions P. Hlandr, D. Andrson, F. Andrsson, L.-G. riksson 3, M. Lisak uratom/ukaa Fusion Association, Culham Scinc Cntr, UK Dpt of lctromagntics, Chalmrs Univ. of Tchnology, Götborg, Swdn 3 CA Cadarach, St Paul Lz Duranc, Franc This work was fundd by URATOM undr Association Contracts with Swdn, Franc and UK, and by th UK nginring and Physical Scincs Rsarch Council

2 Ovrviw Chalmrs Univrsity of Tchnology Cntral qustion: How many runaways ar gnratd in a disruption, and what is thir radial profil? Gnration of runaway lctrons Primary (Dricr gnration Scondary (avalanch gnration Tokamak disruptions Mont Carlo simulations: th ARA cod Smi-analytical modl umrical solution Prdicting th final runaway currnt profil Conclusions

3 Friction on an lctron Chalmrs Univrsity of Tchnology on-monotonic function of vlocity Runaway acclration of som lctrons if > C Massiv runaway if > D D 3 n ln Λ, 4πε T c D T mc << 3

4 Chalmrs Univrsity of Tchnology 4 Primary gnration of runaways Fokkr-Planck quation for suprathrmal lctrons Quasi-stady stat solution has runaway tail xtnding to infinity. Runaway gnration rat for << D (Gurvich; Kruskal and Brnstin, 96 ( ξ ξ ξ τ π u f u f u u f u f C f m t f 4 3 ( 3 v τ // // 3/8 // 4 xp ~ n dt dn D D D r

5 Chalmrs Univrsity of Tchnology Scondary runaway gnration - avalanching Th Fokkr-Planck collision oprator nglcts clos collisions, whr v/v O(. Boltzmann oprator ndd. In such a collision an xisting runaway can throw a thrmal lctron abov th runaway thrshold Runaway population growth rat: n r dn dt r π, 3τ ln Λ / c (Sokolov, 98; Jayakumar t al., 993; Rosnbluth and Putviniski, 997 Mor important than primary runaway production if I p > a fw MA. 5

6 Chalmrs Univrsity of Tchnology Tokamak disruptions A stability limit is rachd. Plasma intracts with wall and cools quickly. Rsistivity incrass, η~t-3/ Larg lctric fild is inducd, trying to maintain th plasma currnt. Runaway lctrons ar gnratd, which ar acclratd to ~ MV. carry much of th original currnt. usually hit th wall > hard X-rays. can caus srious damag. JT: Gill t al, ucl. Fusion ( 6

7 Mont Carlo simulation Chalmrs Univrsity of Tchnology Th ARA cod (Analysis of Runaway lctrons by umrical Algorithms solvs th orbit-avragd drift kintic quation 3D (V radius toroidal gomtry fully rlativistic radiation raction forc wighting schm to nhanc accuracy in high-nrgy tail Coupld to FM solvr of induction quation t ( µ ( σ j, r ( n τ σ.96 ε m B 3 v f d v B T (t spcifid, fast-lctron distribution function and (r,t volvd. j r (riksson and Hlandr, Comp. Phys. Comm. 3 7

8 Mont Carlo simulation: ITR Chalmrs Univrsity of Tchnology Thrmal qunch prscribd T ( r, t T t / t [ T ( r T ( r ] ( r ITR paramtrs: I 5 MA, n.. m -3, T kv (-.9 r/b, T 5 V Initial and final currnt profil 8

9 Smi-analytical modl Chalmrs Univrsity of Tchnology Modl: Runaway production primary scondary Induction law: µ j / t Plasma currnt Ohmic currnt runaway currnt, (all runaways mov at vlocity c ormalisd qs: Cylindrical plasma with normalisd radius xr/b n F( n t x α x x x t ( σ n whr n( x, t n c / j r ( x, t / C normalisd runaway lctron currnt dnsity normalisd lctric fild 9

10 Smi-analytical modl, cont d Chalmrs Univrsity of Tchnology n F( n t x α x x x t ( σ n Primary runaway rat: F 3/8 T ( x, t ( xp 4u u u( x, t << mc F Scondary runaway rat: n α 3/ (π 3ln Λ ja I A >>, I A 4πmc µ 7 ka σ( x, t normalisd conductivity (input

11 Typical numrical solution: JT Chalmrs Univrsity of Tchnology umrical simulation using finit diffrncs. Disruption is triggrd by prscribing a rapid tmpratur drop, with Dnsity T ( x, t T T ( x.4 kv (.9x T ( x 5 V (.9x t. ms t / t [ T ( x T ( x ] ( x 9-3 n ( x 5 m (.9x / Rsults: Convrsion ratio ~4% Currnt dnsity doubls on axis!

12 Chalmrs Univrsity of Tchnology Typical numrical solution, cont d Primary gnration producs a sd for th avalanch. Most runaways gnratd by th avalanch. Runaway production asir on axis than lswhr hollow (r diffusion of lctric fild into th cntr pakd runaway currnt profil.

13 Chalmrs Univrsity of Tchnology 3 Final currnt profil Primary gnration dominats initially if and producs a sd of runaways which is amplifid by th scondary mchanism. Primary gnration can b nglctd aftr som short tim t. For t>t Lt ln n and intgrat onc in tim: Final runaway dnsity profil dtrmind by whr (x(x,t is th sd profil from primary gnration, and j (xj(x,t. ( x s t t t t σ α σ α t n n t n ln ( j α, n

14 Chalmrs Univrsity of Tchnology Final currnt profil: gnral proprtis Runaway currnt < initial plasma currnt, I( t < I ( t, d( ( j x dx α dx Howvr, th runaway currnt dnsity can locally xcd that of th initial plasma currnt. On can show that if f ( x j ( x ( x α is a monotonically dcrasing function, thn (x is pakd at x. x Uppr limit to th paking ratio can b calculatd by using < α ( j n( j ( α ( 4

15 Small-scal variations Chalmrs Univrsity of Tchnology Suppos α ( j with εsin kx, ε << << k Writ and linaris. Thn so that for k >> α εk α sin kx k ( sin kx ε A similar rsult holds for suddn jumps in : thy ar inhritd by th final currnt profil. n( x n ( x ( x ( x ( x ( x n( x n ( x 5

16 Small-scal variations Chalmrs Univrsity of Tchnology Fin-scal variations ar inhritd by th final currnt profil Could xplain why X-ray mission is bursty. 6

17 Th runaway sd profil Chalmrs Univrsity of Tchnology n F( n t x α x x x t ( σ n Assumptions: Diffusion is nglctd on th short tim scal of primary gnration. n σ j( x indpndnt of t for t < t Th sd is small, i.., most runaways ar vntually producd by scondary gnration. Th thrmal qunch is takn to b infinitly fast. [ σ ( x σ ( x ] θ( σ σ( x t j( x ( x, 3 ( x >>, ~ σ ( x 7

18 Chalmrs Univrsity of Tchnology Th runaway sd profil, cont d Dfin t as th tim at which primary and scondary gnration ar qual. Thn (x (x, t and n (x n(x, t ar dtrmind by so that F( n and n j σ n( x F( ln n D 4 D t larg and ngativ Although α >>, currnt paking occurs sinc in j α ( α ~ 8

19 Chalmrs Univrsity of Tchnology Th runaway sd profil, cont d OD to dtrmin th runaway currnt profil ( j 4u u α j j Analytically calculatd sd profil and numrically calculatd runaway currnt profil from abov OD 9

20 xprimntal confirmation? Chalmrs Univrsity of Tchnology R.D. Gill t al (ucl. Fusion masurd th post disruption currnt profil in JT from X-ray mission causd by K-shll vacancy production by runaways in i. Cntral currnt dnsity vry high! q.5 -.6

21 Whn ar runaways important? Chalmrs Univrsity of Tchnology -dimnsional modl obtaind by rplacing Maxwll s quation by induction law giving V loop πr L di dt p dn dt F( n d dt ( σ n α α π 3 L µ R I A I ln Λ ~ 4I [MA] Critrion for significant runaway production H α 4 7 T ln 4 m c >, πt m D D D / 3µ nqr B (Hlandr, riksson, and Andrsson,

22 Conclusions Chalmrs Univrsity of Tchnology Slf-consistnt modlling of th volution of currnt (thrmal runaway and lctric fild following th thrmal qunch of a disruption, through 3D Mont Carlo simulation smi-analytical modl Most runaways ar gnratd by th scondary (avalanch mchanism in JT and ITR. Typically ~ 5% convrsion of thrmal currnt to runaways in JT; mor in ITR. Th runaway currnt profil is asily corrugatd. mor pakd than pr-disruption currnt profil. Possibl implications for runaway bam stability.

23 Chalmrs Univrsity of Tchnology Typical numrical solution, cont d Runaway production in th cntr of discharg. Rduction of fild whr runaway production is larg. 3

24 Chalmrs Univrsity of Tchnology Typical numrical solution, cont d Larg runaway production in th cntr diffusion of lctric fild into cntr hollow (r mor pakd j(r 4

25 xprimntal commnts Chalmrs Univrsity of Tchnology Runaway production favourd by low dnsity nhancs sd low post-disruption tmpratur nhancs sd high plasma currnt nhancs avalanch R.D. Gill t al (ucl. Fusion masurd th post disruption currnt profil in JT from X-ray mission causd by K-shll vacancy production by runaways in i. Cntral currnt dnsity vry high! q