Fast Growth and Sheared Flow Generation in Nonlinear Development of Double Tearing Modes

Transcription

1 1 TH/P9-5 Fast Gowth and Sheaed Flow Geneation in Nonlinea Development of Double Teaing Modes J. Q. Dong 1,, Y. X. Long 1, Z. Z. Mou 1, Z.X. Wang 3, J.Q. Li,3, Y. Kishimoto 3 1, Southwesten Institute of Phsics, P.O. Box 43, Chengdu, China, Institute fo Fusion Theo and Simulation, Zhejiang Univesit, Hangzhou, China 3, Depatment of Fundamental Eneg, Koto Univesit, Gokasho, Uji, Japan jiaqi@swip.ac.cn Abstact The nonlinea development of double teaing modes (DTMs) mediated b paallel electon viscosit in a lage aspect atio tous is simulated. The emphasis is placed on the mechanisms fo the fast gowth and sheaed flow geneation in the development of the DTMs. Fou nonlinea developing stages: the eal gowth, tansition, fast gowth and deca, ae found. In compaison with the helical magnetic flux contou, the fast gowth is evealed to be associated with the econnection and annihilation of the magnetic islands fomed b the econnection of equilibium magnetic field in eal stage of the mode development. The quantitative scaling of the gowth ate and the flow shea with espect to electon viscosit is pesented. The stabilization effect of a votex flow geneated in teaing mode development on ion tempeatue gadient modes is also demonstated stonge than that of a mean o steame flow of same magnitude. 1. Intoduction A non-monotonic safet facto q pofile is desiable fo advanced tokamak (AT) opeation with intenal tanspot baies (ITBs). The latte ae fomed pefeentiall in poximities of low safet facto q ational flux sufaces and often coincide with occuence of magnetohdodnamics (MHD) activities in expeiments [1-6]. The double teaing modes (DTMs) ae known to develop pefeentiall aound the suface of minimum q close to a ational value. On the othe hand, ITB fomation is commonl accepted as a esult of tubulence suppession b sheaed flows in AT dischages [7-9]. Theefoe, the elation between low q ational flux sufaces, MHD activities, flow lae fomation, tubulence suppession, and ITB fomation is essential fo undestanding the expeiments. The inteaction between MHD activities and tanspot inducing tubulence, in geneal, and possible causal elationship between DTM and the ITB tiggeing, in paticula, is one of eseach focuses [9-1]. In addition, chaacteistics of nonlinea DTMs ae impotant fo undestanding tokamak plasma behavios in dischages with negative magnetic shea (NMS) [13, 14]. The nonlinea fast gowth and flow lae fomation in development of DTMs and single teaing mode (STM) have been studied in slab geomet [15-19]. The fast gowth and flow lae fomation in nonlinea development of DTMs is studied fo lage aspect atio tooidal plasmas in this wok. The mechanisms fo the fast gowth and sheaed flow geneation in the mode development ae investigated in detail. The effects of the sheaed helical o votex flows, fomed in the teaing mode development, on the ion tempeatue gadient modes (ITG) ae also investigated in a sheaed slab. 1

2 TH/P9-5. Phsics model and basic equations We conside a lage aspect atio tous with a majo adius R 0 and mino adius a. A clindical coodinate (, θ, ϕ ) is emploed. The magnetic field and plasma velocit ae expessed in tems of two scala potentials: the poloidal flux function ψ (, θϕ, ), B= Bϕ 0 ψ ϕ, (1) and the steam function φ(, θϕ, ), V = vϕ 0 + φ ϕ. () With electon viscosit included, the Ohm s law becomes 1 meμ e 4 E = η j v B j. (3) c nee Using Faada s law and Eq.(1), it is staightfowad to get ψ ηc meμec 4 B0 = v ψ + ψ ψ + E w φ (4) t 4π 4πnee R0 ϕ fom the ϕ -component of Eq.(3). Hee, η is the plasma esistivit, μ e is the paallel electon viscosit diffusion coefficient, m e is electon mass, n e is electon densit, and e and c ae, espectivel, the electon chage and speed of light, B 0 is the tooidal magnetic field at the magnetic axis, and v 0 is equilibium tooidal velocit. The ϕ -component of the cul of the equation of plasma motion ma be witten as v0 1 μ 4 ( φ) = ( v ) φ φ+ [ ( ψ) ψ] ϕ φ (5) t ρ R0 ϕ 4πρ ρ whee ρ is the plasma mass densit and μ is the fluid viscosit. Then, the basic govening educed MHD (RMHD) equations ae deived fom Eqs. (1), (4) and (5) as ψ B0 φ 1 1 = [ ψφ, ] + S ψ Re χ+ Ew, (6) t R0 ϕ v0 U B0 ψ 1 ( φ) = [ U, φ] + [ ψ, ψ] + U. (7) t R0 ϕ R0 ϕ Ri Hee, χ = ψ, U = φ, and the Laplacian and Poisson backet ae defined as, a b b a = + +, () and [ ab, ] = ( ). (9) θ θ θ The usual nomalizations ae emploed such that lengths ae nomalized to a, magnetic fields ae to Bθ ( a), time is to τ pa = a 4 πρ / Bθ ( a) etc. In addition, S = τ / τ pa is the magnetic Renolds numbe with τ = 4 πa / ηc being the magnetic diffusion time, 4 Re = τ v / τ pa is the viscosit diffusion Renolds numbe, while τ v = 4 πne a e / meμec is the electon viscosit diffusion time, and R = τ / τ withτ = ρa / μ. i i pa 3. Fast gowth and velocit Shea lae fomation i i

3 3 TH/P9-5 The equilibium configuation with non-monotonic q pofile is emploed [14], λ 1/ λ δ B0 d dq( ) q ( ) = qc{1 + ( ) } [1+ Aexp{ ( ) }], ψ 0,0( ) =, s ( ). δ R = q( ) q( ) d The paametes ae chosen such q c =1.05, λ = 1, 0 = 0.41, δ = 0.73, δ = 0, A= 3 that two esonant q=3 sufaces ae located at 1 =0.31 and =0.53. v 0 = 0 is assumed. The numeical code fo solving the nonlinea patial diffeential equations (6, 7) is benchmaked with an eigenvalue code in the linea stage and with the esults in [14] fo nonlinea simulations. The eigenfunctions fom the eigenvalue code, with the lowest m and n, multiplied with a small facto, ae taken as the initial conditions. The initial values fo the othe components ae all taken to be zeo. The bounda conditions ae that the petubations and thei adial deivatives ae zeo at the cente (=0) and the plasma bounda (=1). The maximum 000 gid Fig. 1. The linea gowth ate and the scaling index α of γ vesus 1/R. points ae emploed in the adial diection and the esults ae checked to be independent of the numbe of the gid points. The DTMs mediated b electon viscosit ae studied. Shown in Fig. 1 ae the linea gowth ate and the scaling index α of γ R α vesus 1/R. The tansition fom the scaling of 1/3 1/5 γ R to γ R is cleal demonstated fo inceasing 1/R [15]. 6 7 The time evolution of the petubed total kinetic and magnetic enegies fo R=5 10, 10, and 10, and of enegies of the lowest seven hamonics fo R= 10 is given in Fig.. Afte a shot linea gowth, the total petubed kinetic eneg goes though fou developing stages: the eal 7 7 gowth, tansition, fast gowth and deca when R 5 10 in contast to the cases of R 10 whee thee ae no fast gowth stages. In addition, thee ae no fast gowth stages in the petubed R α Fig., The time evolutions of the petubed total kinetic and magnetic enegies fo R=5 10, 10, 5 10 and 10, and the enegies of the fist seven hamonics fo R= 10. magnetic eneg eithe. The hamonic analsis indicates that the fast gowth is dominated b the m=3/n=1 hamonic but ma be elevant with the appeaance of the highe hamonics at the tansition stage. The appeaance aound t=50 and significant gowth of 3

4 4 TH/P9-5 Fig. 3. The helical magnetic flux contous at t=00, 400, 40, 440, 460, 40, 500, and 50. the m=0/n=0 magnetic petubation, which induces vaiation of the equilibium magnetic configuation, seem to be the cause of the tansition o tansient satuation. Shown in Fig. 3 ae the contous of the helical magnetic flux at t=00, 400, 40, 440, 460, , and 50 fo R= 10. Thee ae two chains of magnetic islands located at the two q= 3 ational sufaces befoe t=400. The magnetic islands gow athe fast with the inne and oute chains moving outwads and inwads, espectivel. The islands of the two chains stat to econnect with each othe afte the flux between them is all econnected out due to the fact that the fluxes inside the two chains ae opposite to each othe. It is this seconda econnection and the annihilation of the magnetic islands fomed in the fist econnection of equilibium magnetic field in the eal development of the mode that is esponsible fo the fast gowth of the kinetic eneg fom t=450 to 500. The m=0/n=0 magnetic petubation is satuated and the highe hamonics stat deca afte t=500. Fig.4. The steam function contous at t=00, 400, 40, 440, 460, 40, 500, and 50. The contous of the steam function at t=00, 400, 40, 440, 460, 40, 500, and 50 ae given in Fig.4 fo R= 10. Thee is not an appaent two chain stuctue, illustating the appoximate constant displacement amplitude of the DTM ove the egion between the two ational sufaces. In addition, the flow and is shea ae mainl confined in the 4

5 5 TH/P9-5 vicinit of the =0.53 ational suface fo t 40 when the magnetic islands keep gowing. The contous begin distotion when the two island chains stat to econnect with each othe and to annihilate. To see this moe cleal, the pofiles of the amplitudes of the m=3/n=1 hamonic of the steam function φ (a), the velocit v θ (b) and its adial gadient (c) ae given in Fig. 5. It is cleal shown that significant m=3/n=1 hamonic velocit and its shea appea aound the two esonant flux sufaces at t=300 and then gow ve fast. In compaison, the amplitude of the m=6/n= hamonic given in Fig. 5(d) is moe than one ode of magnitude smalle than that of the m=3/n=1 hamonic. This is an impotant indicato that the flow with the lagest poloidal scale length dominates in the entie pocess, especiall in the eal stage, of the mode development. φ 3/ R=10 t=300 t=400 t=450 t= dφ 3/1 /d Fig. 5. The pofiles of the amplitude of the m=3/n=1 hamonic of the steam function (a), the velocit (b) and its shea (c), and the pofile of the amplitude of the m=6/=1hamonic of the steam function (d). Shown in Fig. 6 ae the amplitude pofiles of the m=3/n=1 hamonic of the poloidal flux function, the extensions of which ae equivalent to the pofiles of the petubed adial magnetic field and elevant to the widths of the magnetic islands. Keeping this in mind and in compaison with Fig. 5, it is athe evident that the flow shea laes ae mostl confined in the egions just outside the magnetic islands. This is an inevitable esult due to the facts that magnetic eneg eleased in nonlinea development of the modes convets to kinetic eneg and the coheent plasma motion, pependicula to magnetic field, is pohibited in ideal MHD egion, and, theefoe, is confined in the teaing lae. 4. Effects of helical flow on ion tempeatue gadient modes R=10 t=300 t=400 t=500 t= Fig. 6. The amplitude pofiles of the m=3/n=1 hamonic of the poloidal flux function. Self-consistent stud of the effects of the helical sheaed flow and magnetic islands spontaneousl geneated in the development of teaing modes on mico-instabilit and tubulence is a big challenge. The effects of a helical flow on ITG modes, neglecting magnetic island stuctues, ae discussed in a sheaed slab as the fist step in this Section. The helical flows educe to votex flows (VFs) in the slab. The VF is consideed static and expessed with the steam function, φt = φtx( x)sin( kt) = φmf( x)sin( kt), (10) whee f ( x ) is the nomalized adial pofile of the VF and k T is its poloidal wave vecto. φ m is the amplitude. The nonlinea electostatic gofluid model in a -dimensional (D) slab ( z = 0 ) is d φ 3/1 /d t=300 t=400 t=450 t=500 R=10 5

6 6 TH/P9-5 emploed with the above extenal VF included [0], k k k V φ k k k 4 k φ φ = (1 + K ) + v [ φ, φ ] + μ φ + t x (11) k k i f ( x) k [ k i V V φm ik V ik V ] ( ikt ) φm f ( x ) +, x x x k v k k k k k = ( φ + p ) [ φ, v ] + η v t k k (1) i φm f( x) k [ k i v v φm ik v ik v ] ( ikt ) φmf ( x ) +, x x x k k p φ k k k k k k = K Γ v ( Γ 1) k ( p φ ) [ φ, p ] + χ p t π k k (13) i f( x) k [ k i p p φm ik p ik p ] ( ikt ) φmf ( x ) +, x x x k k whee, k + k = k + kt = k kt = k, K = 1+ ηi and V = ( + δ 1) φ, δ = 1 (0) fo zonal flow (fluctuating components). The notation and nomalization as well as the magnetic configuation ae the same as that in [0]. This set of mode coupling equations deived in the same wa as that in [0] cleal shows the inteaction between a VF and ITG modes though both adial and poloidal couplings b the last two tems on the ight hand side (RHS) of each equation. The Equations (11)-(13) ae numeicall solved b using of a D initial value code. The simulation box is Lx = 100ρi and L = 0πρ i. Fixed and peiodic bounda 1 conditions ae imposed at x =± Lx and =0, L, espectivel. The othe paametes ae η i =., μ = η = χ = 0.1, and k T = 0.1. The epesentative stuctue of the VF with two votices on each side of the ational suface, geneated in development of a single teaing mode, is shown in Fig. 7. Besides the VF, a steame-like flow (SF) of φt = φmsin( kt) and a mean flow (MF) of φ = φ ( ) ae f x T m intoduced in helping to undestand the complicated stabilization mechanisms of the VF with compaison. Fig. shows the effects of the thee extenal flows on the ITG mode linea gowth ate fo diffeent magnetic sheas, s = L / L. The gowth ate inceases n s Fig. 7. The adial pofile f(x) of the steam function φ = φ f ( x)sin( k ) (a) and T m T fist and then deceases when the amplitude of the MF inceases fo the thee cases as is well documented [0]. The stabilization effect 6

7 7 TH/P9-5 of the SF deceases damaticall with deceasing magnetic shea. The effect of the VF is the stongest and the mechanism is elativel complicated. As indicated in Eqs. (13-15), a MF, dependent on x, couples a mode to othe modes of diffeent adial scale lengths though the fist tem of the last two tems on the RHS of each equation. Wheeas a SF, dependent on, couples a mode to othe modes of diffeent poloidal wave vecto k s though the second tem of the last two tems on the RHS of each equation. Geneall speaking, coupling to moe unstable modes inceases the gowth ate of the unstable mode unde consideation and thus the whole coupled sstem, and vice vese. Spectum analsis (not shown hee) fo φ m = 3 and s=0.4 shows that most unstable modes ae limited appoximatel to the egions of 0.1 < < 5 and k x 0.3 < k < 1.1 in the wave vecto specta with the maximum gowth ate γ = 0.99 at k ~0.7 in the absence of an flows. The unstable k -spectum does not change significantl wheeas the k x -spectum expands to the egion of k x >.5, whee Fig., The linea gowth ate of the ITG modes vesus the magnitude φ m of the MF, SF and VF (the lines with ed squaes, blue dots and geen tiangles), fo magnetic shea s=0.4, 0., and 0.1. the gowth ates of the modes ae lowe, when thee is a MF. The esult is the opposite with a SF; the unstable k x -spectum does not change significantl while the k -spectum expands to both 0. > k and k >1.1 egions, whee the gowth ates of the modes ae lowe. In the case of VF, the unstable k x -spectum expands to highe k x egion as it does in the case of a MF, wheeas the unstable k -spectum expands to both 0. > k and k >1.1 egions as it does in the case with a SF. In addition, the both expansions ae boade pobabl due to enhancement of each othe. Theefoe, the stongest stabilization effects of a VF on ITG modes than those of a MF o a SF of same magnitude in most of the paamete egimes studied ae demonstated and phsicall undestandable. 5. Summa and discussion The numeical stud of the electon viscosit DTMs and the geneation of the helical MHD flows in the development the modes is extended fom quasi-linea simulation in a slab [15, 16] to nonlinea simulation in a lage aspect atio tous. The chaacteistics, such as stong poloidal sheaed flows located at the boundaies of the magnetic islands in the nonlinea stage, found in the slab geomet ae qualitativel confimed. Quantitativel speaking, in-out asmmet of the induced helical velocit pofiles is found. The shea at the oute bounda of the islands is highe than that at the inne, in contast to the slab case [15]. The shea of the velocit was estimated as~10 5 /s fo Bθ ( a) =1.7T, a=0.4m, n e =10 19 /m 3 fom the numeical esults fo R=10. This is 7

8 TH/P9-5 compaable with the gowth ate of the dift instabilities in tokamak plasmas. As an impotant pat of multi-scale dnamics, the inteaction between teaing modes and tubulence is unde intensive investigation in plasma phsics [10-1]. The subject includes two aspects, the effect of tubulence on teaing modes, and the influence of teaing modes on tubulence and tubulent tanspot. This is a challenge since it concens multi- spatial and tempoal scales. Phsicall, a teaing mode consists of two elements: magnetic petubation, foming magnetic islands, and plasma motion including the helical flows studied in this wok. The ultimate aim of the subject is to stud the dift wave tubulence dnamics in a ealistic tooidal configuation with magnetic islands and plasma flows. Befoe being able to do so, as a fist step, we t to undestand the chaacteistics of the flows and then to put the flows as an input fo a tubulence stud in a configuation without magnetic islands in this wok. The pelimina esults show that the stabilization effect on a votex flow on ITG modes is highe than that of a mean o steame flow of same magnitude. This ma povide insight into the elation between low safet facto q ational sufaces, MHD activities, flow lae and ITB fomation in evesed magnetic shea configuations. Woks on linea and non-linea development of teaing modes and tubulence self-consistentl, taking into account the multi- spatial (fom 0.1mm to 0.1m) and tempoal (fom 0.01 ms to 1ms) scales ae in pogess. Acknowledgement This wok is suppoted b the NSFC Gant No and No Refeences [1] Y. Koide et al., Phs. Rev. Lett. 7, 366 (1994). [] C.M. Geenfield et al., Plasma Phsics and Contolled Nuclea Fusion Reseach (IAEA, Vienna, 199), Vol. 1, P. 13. [3] M.G. Bell et al., Plasma Phs. Contol. Fusion 41, A719 (1999). [4] E. Jofin et al., Nucl. Fusion 43, 1167 (003) and the efeences theein. [5] E. Joffin et al., Nucl. Fusion 4, 35 (00). [6] G.M.D. Hogeweij et al., Plasma Phs. Contol. Fusion 44, 1155 (00). [7] P.W. Te, Rev. Mod. Phs. 7, 109 (000). [] P. H. Diamond et al., Plasma Phs. Contolled Fusion 47, R35 (005). [9] J.Conno et al., Nucl. Fusion 44, R1 (004). [10] C.J. McDevitt and P.H.Diamond, Phs. Plasmas 13, 0330 (006). [11] Y.Kishimoto et al., Nucl.Fusion 40, 667(000). [1] F. Militello et al., [J] Phs. Plasmas 15, (00). [13] B. Caeas et al., Phs. Fluids 3(9), (190). [14] Y. Ishii, M. Azumi, G. Kuita, and T. Tuda, Phs. Plasmas 7, 4477 (000). [15] J.Q. Dong, S.M. Mahajan and W. Hoton, Phs. Plasmas 10, 3151 (003). [16] J.Q. Dong, Y. X. Long, and Z.Z. Mou et al., Phs. Plasmas 14, (007). [17] Z. X. Wang, X. G. Wang, and J. Q. Dong et al., Phs. Rev. Lett. 99, (007). [1] N. F. Loueio, S. C. Cowl et al., Phs. Rev. Lett. 95, (005). [19]D. Gaso et al., Plasma Phs. & Contolled Fusion 4, L97 (006). [0]] J. Li and Y. Kishimoto, Phs. Plasmas 10, 63 (003).