Modeling of Noble Gas Injection into Tokamak Plasmas

Transcription

1 IT/P3-7 Modeling of Noble Gas Injection into Tokamak Plasmas D.Kh. Moozov ), E.I. Yuchenko ), V.E. Lukash ), E.O. aonova ), V.A Rozhansky ), I.Yu. Senichenkov ), I.Yu. Veselova ) and R. Schneide 3) ) Nuclea Fusion Institute, RRC Kuchatov Institute, Moscow, Russia ) St-Petesbug State Polytechnical Univesity, St.Petesbug, Russia 3) Max-Plank Institute fü Plasmaphysik, Teilinsttitut Geifswald, Euatom Association, Geifswald, Gemany. contact of main autho: moozov@nfi.kiae.u Abstact. Noble gas injection fo mitigation of the disuption in DIII-D is simulated. The simulation of the fist two stages is pefomed: of the neutal gas jet penetation though the backgound plasmas, and of the themal quench. In ode to simulate the fist stage the.5-dimensional numeical code LLP with impoved adiation model fo noble gas is used. It is demonstated that the jet emains mainly neutal and thus is able to penetate to the cental egion of the tokamak in accodance with expeimental obsevations. Plasma cooling at this stage is povided by the enegy exchange with the jet. The adiation is elatively small, and the plasma themal enegy is spent mainly on the jet expansion. The magnetic sufaces in contact with the jet ae cooled significantly. The cooling font popagates towads the plasma cente. The simulations of the plasma column dynamics in the pesence of moving jet is pefomed by means of the fee bounday tanspot modeling DINA code. It has been shown that the cooling font is accompanied by stongly localized shak fin-like petubation in tooidal cuent density pofile. Afte few milliseconds the jet (togethe with the cuent petubation) achieves the egion whee safety facto is slightly highe than unity and a new type of the non-local kink mode develops. The unstable kink petubation is non-esonant fo any magnetic suface, both inside the plasma column, and in the vacuum space. The mode distubs mainly the coe egion. The gowth time of the shak fin-like mode is highe than the Alfven time by a facto of fo DIII-D paametes. Hence, the simulation descibes the DIII-D expeimental esults, at least, qualitatively.. Intoduction One of the most impotant poblems fo ITER is the poblem of disuptions. Recent expeiments in DIII-D tokamak have shown the mitigation of the deleteious effects fom tokamak disuption by a high-pessue noble gas injection []. At the fist stage of the expeiment the penetation of the jet into plasma coe without any significant MHD activity has been obseved. Magnetic petubations ise apidly afte jet aival to some citical intenal magnetic suface. The MHD instability causes the themal quench at the second fast stage. This instability mixes the cental plasma egion and bings the noble gas ions into plasma cente in a shot time. The cental tempeatue deceases by a facto of o 3 afte mixing. Stong adiation and following fast plasma cooling is typical fo the thid stage. The thid stage has been investigated in Ref. [] with zeo-dimensional code KPRAD as well as the elativistic electon geneation. The jet deep penetation into plasma coe togethe with plasma-impuity mixing has not been discussed ealie. Jet expansion along and acoss magnetic fields by means of.5-d LLP code was simulated in []. Results of this simulation demonstate that the jet emains mainly neutal and is able to penetate deep to the backgound plasma. The jet penetation to the plasma column cools plasma on the magnetic sufaces contacting with the jet. The coesponding petubed tempeatue pofile initiates cuent edistibution. elow the cuent pofile edistibution is analyzed by two means. Fist it has been shown analytically that the tempeatue font is accompanied by the stongly localized cuent petubation. Second, the numeically simulated shak fin-like fom in the cuent pofile has been obtained by means of DINA code [3]. With MHD-stability theoy

2 IT/P3-7 it has been shown that the shak fin-like pofile becomes unstable when achieves the egion with the safety paamete close to. The new type of kink mode aises. The mode is localized mainly inside the plasma column. The gowth time of the shak fin-like mode is highe than the Alfven time by a facto of fo DIII-D paametes. The mode mixes the plasma column inteio and caies noble gas ions into plasma cente. The suggested patten might explain expeimental esults []. The estimations show that the plasma at the thid stage is optically thin, at least if the tempeatue is highe than few tens of ev. Hence, the KPRAD model [] descibes adiation cooling of plasmas adequately. In this pape we pay attention mainly on the jet penetation and plasma mixing unde DIII- D conditions. One can expect the simila esults fo ITER tokamak. Howeve, the coesponding conclusions can be caied out afte additional studying.. Model of jet penetation The initially neutal jet in a tokamak plasma expands in -paallel and -pependicula diections and becomes patially ionized. The ionized paticles expand feely along the magnetic field lines while the coss-filed motion is affected by an inteaction with the magnetic field. At some time instant, ionization sets in at the jet peiphey and the expansion in the poloidal diection comes to a full stop. The espective stopping adius should be consideed as the tansvese jet size fo futhe field-aligned simulations. Jet motion in the adial diection is povided by the polaization of the jet and the adial E dift (the initial jet velocity is E V = ). The polaization electic field of dipole type = V should appea (see Fig.). Fig.. Foce balance and electic field in the jet penetating into the tokamak. E Calculations ae pefomed fo the paametes typical fo DIII-D and ITER tokamaks. Jet paametes fo DIII-D ae taken fom the expeiment []. Jet expansion stats at t = with given cloud density n = 4 m (DIII-D), n = 4 m (ITER), tempeatue T = 3K, cloud size z = = 7. 5cm (cylindical symmety is assumed) and ionization degee α = (this value coesponds to a jet tempeatue of about 3K). Jet velocity is V = 5m / s. The modeling is pefomed until the jet eaches the tokamak cente, povided thee is no adial deceleation, t cente = a / V, whee a is the tokamak mino adius. (Fo DIII-D it means.5 ms, fo ITER ms). ackgound density and tempeatue pofiles ae supposed to be known. Numeical simulations ae pefomed in the diections along and acoss magnetic field lines. Results obtained show that the jet do not feely expands in poloidal diection because the

3 3 IT/P3-7 ionization degee of the ode of 5% eached at the end of the calculation at the jet peiphey is enough to confine the jet due to inteaction with magnetic field. This conclusion is veified by the time evolution pofile of the tansvese jet size pesented in Fig.. Hee and below all esults ae shown fo DIII-D backgound plasma paametes. Fom Fig. one can see that the tansvese size does not change significantly (by a facto of two) duing jet penetation. (cm) n e = 3x 9 m -3, T e =.5 kev t (µs) Fig. Tempoal evolution of the tansvese jet size. Jet size along magnetic field, z (m) t (ms) Fig. 3 Tempoal evolution of the longitudinal jet size. n (m -3 ) x 5 4 x 3 n a n e n a n e - t =.5 ms - t = ms 3 - t = ms z (m) Fig. 4 Density pofiles along fo diffeent moments. n a n e 3 T e (kev) ms -.5 ms 3 -. ms ms 5 -. ms x (cm) Fig. 5 Tempeatue pofiles fo diffeent moments. 3 Simulation esults along ae pesented in Figs. 3, whee shown is the evolution of the longitudinal jet size. The electon n e and neutal density n a pofiles ae shown in Fig. 4. It is seen that fo the paametes of expeiment [] the most pat of the jet emains neutal. The neutal jet, initially injected in the adial diection, is able to move futhe in the adial diection with the constant velocity. The mechanism is connected with polaization poloidal electic field and is discussed in []. While penetating into the cental tokamak egions the jet cools the hot backgound plasma. Resulting backgound tempeatue pofiles ae pesented in Fig. 5, whee x = coesponds to the plasma bounday. Simila pofiles ae used then as the input ones fo the simulation of the plasma cuent pofile evolution with DINA code.

4 4 IT/P Shak fin-like cuent pofile simulation With DINA code a simulation of plasma cuent pofile evolution in #94 DIII-D shot and the Stat of un opeation point in ITER 5 MA plasma inductive scenaio have been pefomed. Step-like popagation of cooled electon tempeatue pofile coesponding to Fig. 5. is used. The 9 value Z eff = was taken. Plasma density 6 m -3 9 fo DIII-D and 8 m -3 fo ITER ae supposed to be unifom in computation aea. Plasma conductivity includes neoclassical effect of tapping paticles. In Fig. 6 the time evolution of plasma cuent pofile fo this case is pesented. Duing evolution a total plasma cuent is assumed to be fixed t= t=.97 ms t=.37 ms t=.97 ms t=.97 ms j appox Fig. 6 Cuent pofile evolution Fig. 7 The shak fin-like petubation of the initial cuent pofile One can see the cuent spike popagating togethe with the tempeatue font. The spike shape is tansfomed fom tiangula fo ealie moments to the shak fin fo latest moments (Fig. 7). The shak fin may be appoximated by the expession 4 4 (.5( ρ / b ); if ρ b δj (A/m ) (3.).5 exp ( C( ρ b) ); if b ρ (lue line in Fig. 7) Hee ρ = / a, b detemines the position of the spike, C 35. In ITER pofile fo selected scenaio the spike shape keeps the tiangula fom. The elative magnitude is small. The appeaance of the spike may be undestood by a simple linea theoy fo the ideally conducting plasma. Using Maxwell equations µ cul = j, cul E =, V plasma momentum equation Mn = p + [ j ], and assuming the infinite conductivity, E + V = one can find: [ ]

5 5 IT/P3-7 V p Mn = + jθ j θ, ( V ) = µ j. ( V ) θ = µ jθ. Hee the subscipts, θ, and mak adial, poloidal and tooidal components, espectively. The cylindical symmety is assumed. One can assume the initial magnetic field and cuent to be independent on at the spike size. Also, the cylindical suface cuvatue may be ignoed to the zeo appoximation. Hence, all spike functions can be assumed to be the functions of the new vaiable = V t. It yields fo petubed value of the cuent density: δj x θ ca θ = V ( ) + ( ) x δp. Hee c A = >> V jet. µ M n One can see that the spike magnitude is detemined by the pessue petubation gadient and almost independent on V. Hence, the magnitude of the spike coincides with the tooidal component of the diamagnetic cuent. Estimation of this cuent spike using the simulation paametes esults in the value of about % of the Ohmic cuent.. The accuate shape of the shak fin is obtained fom the non-linea numeical simulation (ed line in Fig 7.) This cuent petubation is the tooidal component of the tansvesal cuent. The tansvesal cuent cannot be compensated by the inductive paallel one. On the othe hand, the local bootstap cuent inducing by the step-pessue gadient is compensated by the paallel inductive cuent. Time scale of the step pessue gadient location is much smalle than the skin time. Hence, the bootstap cuent effect may be ignoed. 4. Ideal MHD stability. The cuent column stability in a stong longitudinal magnetic field >> θ has been investigated by many authos. The geneation of the localized flute-like modes, intenal kink modes, and teaing modes is possible if the esonant helical distubance inside the plasma column ( < < a) appeas, k ( ) = [ mµ ( ) n] =. R (4.) Hee R is the majo adius, and Rθ µ =. If the esonance exists in the vacuum space ( a < < w, whee w is the conducting wall adius) the non-local kink mode with the petubed bounday may be diven. The lage scale MHD instability in DIII-D noble gas injection expeiments is a eal challenge fo the stability theoy. The initial cuent pofile is stable. Shak fin-like

6 6 IT/P3-7 petubation is not able to distub the cuent pofile sufficiently to poduce the esonant suface in the vacuum space. Hence, the extenal kink mode cannot be excited. The esonant suface may occu nea the magnetic axis if the spike appoaches vey closely to it (see Fig. 8). Hence, teaing and othe modes may be excited. Howeve, teaing modes ae too slow. Othe instabilities ae the small scale ones. As is shown below, the new type of the ideal kink mode with petubed bounday may be excited. The unstable petubation is nonesonant fo any magnetic suface, both inside the plasma column and in the vacuum space. Fo simplicity, modes m ae consideed in ode to neglect tooidal effects. In this case the helical petubation of magnetic field may be descibed by the following dimensionless equation [4]: µ µ µ x Fig. 8. Radial dependence of the initial (solid line) and petubed (dotted line) µ values γˆ ψ + ρ θ ψ ρ m dj ˆ = + d d ln + dρ Hee ψ is the nomalized electomagnetic potential, ˆ = k R, µ = ρ k ρ µµ θ V A ˆ~ j ρdρ~ d d m, ρ = / a, and = ρ. ρ dρ dρ ρ m dp ψ. (4.) dρ γr µ R γ ˆ =,γ is the gowth ate, jˆ = j, c Fo the maginal stability γ = the equation (4.) is educed to the Schödinge equation. It is moe convenient to tansfom it into the fist ode nonlinea equation: Φ m dy m djˆ d ln θ = + µµ + dy dy θ d + Φ m togethe with the bounday condition Φ ( ) = m. Hee Φ = d lnψ / dy, and y = ln ρ. The stability citeion based on the geneal quantum mechanics pinciple may be fomulated as following: plasma cuent column is unstable if and only if the solution of equation (4.3) has a negative singulaity in any point between and zeo. Fo the fist time the citeion has been intoduced in Ref. [5]. dp dy (4.3)

7 7 IT/P3-7 The maginal stability has been found numeically with the equation (4.3) unde the assumption d / dy = fo the mode m =, n =. The initial cuent pofile has been appoximated by the expession: whee δ j is detemined by (3.)..35 ( ρ ) δj j ( ρ ) = j +, Fig. 9. Eigenfunction coesponding to the maginal stability. Fig.. Dimensionless gowth ate v/s the spike position The eigenfunction coesponding to the maginal stability has been obtained fom (4.). It is shown in Fig.9. The gowth ate of the mode may be estimated solving (4.3) with the ight hand multiplied by facto γ ˆ / k ˆ min + k. Hee k min coesponds to the minimal magnitude of ˆk inside the spike. The esult is shown in Fig. (solid line). The dispesion of points is connected with numeical eos. Dotted line shows the appoximation γ ˆ =.3.65b. One can see that the instability appeas when the tempeatue font penetates inside plasma column to the distance fom the magnetic axis ρ.. The esult coesponds qualitatively to the expeimental one ρ. 3 []. Dimensional gowth ate may be estimated as γ mkmin τa. Hee τ A is the Alfven time. Fo ou example, k min, A, and A 7 5 τ s. Hence γ s. This esult is in ageement with expeimental data, at least, qualitatively. 5. Summay.The noble gas jet emains mainly neutal and thus is able to penetate up to the cental egion of the tokamak in accodance with expeimental obsevations on DIII-D.. The shak fin-like spike on plasma cuent pofile should be fomed due to the step-like plasma tempeatue pofile fomed by the jet. 3. Shak fin-like spike excites the new type of instability afte achieving the intenal plasma 5 egion. Fo DIII-D conditions the gowth time is estimated as s.

8 8 IT/P The mode mixes the plasma column inteio and caies noble gas ions into the plasma cente. The suggested patten might explain expeimental esults on DIII-D. 6. Acknowledgments The authos ae gateful to Pof. V.S. Mukhovatov initiated this wok. The wok is suppoted by the Russian Fedeal pogam on suppot of leading scientific school eseaches, Gant No Refeences [] D.G. Whyte et al. Jounal of Nuclea Mateials (3) 39 [] V. Rozhansky, I. Senichenkov, I. Veselova, D. Moozov, R. Schneide. Poceedings of 3th EPS Confeence on Plasma Phys., London, 4, ECA Vol. 8, (Euopean Physical Society), Pape No. P-4.6 (4). [3] Khayutdinov, R.R. and Lukash V.E., "Studies of Plasma Equilibium and Tanspot in a Tokamak Fusion Device with the Invese-Vaiable Technique", Jounal of Comp. Physics, 9 (993) 93. [4] Kadomtsev,.., and Pogutse O.P., Tubulence in tooidal systems, in Reviews of Plasma Physics, vol. 5, edited by Acad. M.A. Leontovich, Moscow, Atomizdat, (97) 49. [5] lekhe, P.M., Zueva, N.M., Yuchenko, E.I., Cuent pofile influence on kink instabilities of the cuent column with petubed boundaies, Pepint of M.V. Keldysh institute, Moscow, (98). (In Russian).