Modeling of seed magnetic island formation

Transcription

1 EUROFUSION WPMST1-PR(16) IG Miron et al. Modeling of eed magnetic iland formation Preprint of Paper to be ubmitted for publication in 43rd European Phyical Society Conference on Plama Phyic (EPS) Thi work ha been carried out within the framework of the EUROfuion Conortium and ha received funding from the Euratom reearch and training programme under grant agreement No The view and opinion expreed herein do not necearily reflect thoe of the European Commiion.

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3 Modeling of eed magnetic iland formation I.G. Miron National Intitute for Laer, Plama and Radiation Phyic, Magurele-Bucharet, Romania Introduction The dynamic of the neoclaical tearing mode and magnetic iland i hown baed on a quai-analytic model that calculate the 3D perturbation pectrum inide and outide the magnetic iland. The calculation are performed for the cae of an ASDEX-Upgrade plama urrounded by an inhomogeneouly reitive wall. A pectrum of magnetic perturbation (MP) generated by a et of in-veel addle coil () i conidered[1]. The matching condition and the outer olution The 3D model conider a low plama invere apect ratio approximation, a thin urrounding reitive wall and the aumption that the wall and the feedback coil lie on magnetic urface. The ued geometry involve flux coordinate of Hamada type, (r,θ,ϕ), i.e. the radial flux coordinate, the poloidal and the toroidal angle, repectively. A contant local plama toroidal rotation i kept in order to preerve the validity of the perturbed model, i.e. the mall perturbation cale of variation from a tatic equilibrium tate. According to [2], for a perturbed magnetic parametrization of the form b = ϕ ψ, the tearing tability index meauring the jump of the perturbation acro the magnetic iland at the(m,n) magnetic urface i (t)= (2m/r ) [ 1 ψ,ext(t)/ψ (t) ] (1) ψ (t) i the NTM perturbation magnetic flux calculated inide the iland (Fourier decompoition term) from the olving of the magnetic iland perturbed reitive equation. r i the radial flux coordinate of the magnetic urface where the iland develop. ψ,ext (t) i the perturbation calculated outide the magnetic iland. Our calculation of the outer perturbation rely on the parametrization of the perturbed magnetic field in term of φ, where φ/ t i the perturbed calar electric potential, and of the perturbed plama velocity v, for an equilibrium magnetic field B: b/ t = (v B), v=(1/b) ( φ/ t) n+(v n)n, n=b/b. By comparing both parametrization we get within the firt order of the low invere apect ratio approximation that,ext (t)= i(n m/q )φ (t), our calculated outer olution being ψ φ (t)= A + B exp( inω MP t)+ 6L p=1 C p exp(τ p t) (2) q i the afety factor at r. Ω MP i the toroidal rotation angular velocity of the rotating magnetic perturbation pectrum generated by the. τ p are the root of the determinant of the

4 linearized ytem of the Laplace tranformed perturbed equation: (τ p ) =, p = 1,..,6L. L=(m 2 m 1 + 1)(n 2 n 1 + 1), where m 1 m m 2 and n 1 n n 2 (ee [3]). The calculated coefficient in (2) are A l = (τ+ inω MP ), B τ= τ, C τ= inωmp = l p = (τ τ p ) l τ(τ+ inω MP ) (3) τ=τp l i with l = m m (n n 1 )(n 2 n 1 + 1) colu replaced by right hand term vector of the following outer ytem of Laplace tranformed equation (bar indicate the Laplace tranform φ(τ)=l(φ(t))) 4 j,k α= 2 j,k α= τ α( P jkα τ α( W jkα jk φ jk φ + P jkα + W jkα jk φ ) = jk φ ) j,k( 3 α= τ α R jkα 1 = τ j,k( α R jkα α= 1 τ+ ikω MP 1 τ+ ikω MP ) 3 S jkα α= 3 S jkα α= ) (4) P α, P α and W α, W α are the plama parameter and the wall and feedback parameter matrice, repectively. R α, R α and S α, S α are the initial perturbation and the rotating MP pectrum matrice having the toroidal angular velocity Ω MP. Inner olution Following the method from [2], but uing our time-dependent olution derived outide the magnetic iland, we get at early time ψ (t)= im(n m/q { ) πt R t A C p i(n m/q ) τp 2 6L p=1 A t 2 + 2B n 2 Ω 2 MP [1+τ p t exp(τ p t)] } [1 inω MP t exp( inω MP t)] t R and t A are the reitive and the Alfven time, repectively. At later time (FKR and Rutherford regime), a more complicated time dependent olution i obtained { [ ψ (t)= i(n m/q ) A t 5/4 t 5/4 Γ(9/4) ib t 1/4 ( 4+exp( inωmp t)e nω MP Γ(1/4) 3/4 ( inω MP t) ) FKR exp( inω ] [ 6L MPt) C ( inω MP t) 1/4 p t 1/4 ( 4+exp(τp t)e p=1 τ p Γ(1/4) 3/4 (τ p t) ) ]} exp(τ (6) pt) τp 1/4 Γ and E ν are gamma and generalized exponential integral function, repectively. t FKR i the linear tearing mode diffuion time t FKR =(t 3/5 R t2/5 A /m6/5 )[πγ(3/4)/γ(1/4)] 4/5. Modeling of the iland evolution The above calculated olution are ued to analytically derive a time dependent formula of the tearing tability index (1). By olving the modified Rutherford equation the magnetic iland (5)

5 width evolution i obtained. A a general obervation, it hould be noted that the model preented here i a perturbation theoretical model that i obviouly valid a long a the plama equilibrium i not changed. Therefore the model cannot decribe the NTM aturation regime. An on-going confinement degradation invalidate the perturbed dynamic model. Time trace of the iland width are hown in Fig. 1 with and without the boottrap term. An expected detabilizing behavior i obtained. Due to the multimode approach the effect of the adjacent mode to the central NTM perturbation i found. Thi apect i of a pecial interet when a pectrum of external MP (of error field type) i taken w [m] t [] l l + BS into account, uch a the one generated by the ASDEX-Upgrade. Fig. 2 and 3 how the iland evolution in the ingle mode cae along Figure 1: Calculated iland width in the abence v. in the preence of the boottrap term. i the calculated tearing tability index. with the cae when adjacent poloidal mode are conidered. Wherea for the (2, 1) iland the both more negative and more poitive neighboring mode detabilize the central mode (Fig. 2), for the (3,2) iland the more negative neighboring mode ha a more detabilizing effect (Fig. 3). A imilar analyi performed in the toroidal cae prove that the adjacent toroidal mode have a ignificantly lower influence on the central untable mode compared to the poloidal neighboring mode w [m].24 w [m] (1,1) + + (3,1) t [].21 (3,2) (2,2) + (3,2) (3,2) + (4,2) t [] Figure 2: The effect of the neighboring poloidal mode to the iland width dynamic. Figure 3: (3, 2) iland evolution in the preence of neighboring poloidal mode. Time trace of the normalized mode are hown in Fig. 4 for different phae hift between the upper and lower row φ. The coil are witched on between 1.5

6 and 2.5. The ignal pectrum ha a maximum current of I max = 1 ka at f =.5 Hz toroidal frequency. The maximum reonance between the MP and the mode occur for π/2< φ < 3π/ φ= 15 φ=π/ φ=π /2+ 1/2-1/2 - -1/2 15 φ=3π/ t [] φ=π/4 φ=3π/4 φ=5π/4 φ=7π/ t [] Figure 4: NTM normalized for different toroidal phaing φ of the coil current. I max = 1 ka Φ (degree) More preciely, Fig. 5 drawn at different time point how that φ 11 o correpond to the maximum reonance. Plama repone to applied perturbation i explicitly calculated. To conclude, the modeling of the iland evolution i poible within the regime of interet a long a the model validity requirement are fulfilled. The olution derived here could be eaily ued to further calculate the MP induced braking torque that damp the plama rotation and ubequently affect the iland evolution. Acknowledgment normalized mode t = 1 t = 2 t = 3 Figure 5: Calculated normalized NTM veru the toroidal phaing φ of the coil current at t = 1, 2 and 3, repectively. Thi work wa upported by Euratom and carried out within the framework of the European Fuion Development Agreement. Thi work ha alo been carried out within the framework of the EUROfuion Conortium and ha received funding from the Euratom reearch and training programme under grant agreement No The view and opinion expreed herein do not necearily reflect thoe of the European Commiion. Reference [1] W. Suttrop et al., Fuion Eng. and Deign 84, 29 (29) [2] C.C Hegna, J.D. Callen and R.J. LaHaye, Phy. Plama 6, 13 (1999) [3] I.G. Miron, Plama Phy. Control. Fuion 5, 953 (28)