Konovalov, S.V.

Mikhailovskii, A.B.

Shirokov, M.S.

Tsypin, V.S.

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POLARIZATION CURRENT THRESHOLD MODEL OFNEOCLASSICAL TEARING MODES IN THE PRESENCE OFANOMALOUS PERPENDICULAR VISCOSITYM.S.Shirokov,1,2 A.B.Mikhailovskii,1 S.V.Konovalov,1 V.S.Tsypin31 Institute of Nuclear Fusion, RRC “Kurchatov Institute”, Moscow, Russia2 Moscow Institute of Physics and Engineering, Moscow, Russia3 Institute of Physics, University of Sa˜o Paulo, S/N, 05508-900 SP, Brasil1. The primary theory of neoclassical tearing modes(NTMs) [1-3] predicted that, as a result of the boot-strap drive, magnetic islands can be generated intokamak discharges with favorable current profile,i.e. for ∆′ < 0, where ∆′ is the standard parameterof tearing mode theory [4]. Then, the island widthW should be smaller than a maximal one Wmaxproportional to the parameter beta (the ratio ofplasma pressure to the magnetic field pressure), i.e.W ≤Wmax, (1)whereWmax =βCb(−∆′) , (2)Cb is a constant.However, the theory [1-3] did not explain, whichβ should be substituted into this expression forWmax. Then one could suggest that generation ofNTMs is possible for arbitrarily low β. Meanwhile,expreimental data on TFTR [5] have shown thatthese modes are generated only if β exceeds somecritical (threshold) value βcrit,β ≥ βcrit. (3)Then the question arose: how one should modifythe theory [1-3] in order to explain the critical betafor NTM onset.As a result, two threshold models of NTMs, al-lowing one to predict βcrit, have been formulated:the polarization current threshold model [6] andthe transport threshold model [7] (such a termi-nology has been introduced in [8]). The presentwork is addressed to further development of thefirst of these models, while the companion work [9]summarizes recent developments of the transportthreshold model. The goal of the present work isto incorporate anomalous perpendicular viscosityinto the polarization current threshold model.2. The polarization current threshold model ofNTM looks as [6]dWdt∼ ∆′4+ ∆bs +∆p, (4)where ∆bs and ∆p are responsible for the bootstrapdrive and polarization current effect, respectively.Starting expression for ∆p is [10]∆p =1cψ˜(2LsB0)1/2 ∫ −∞−ψ˜+εdψ∮J‖ cos ξdξ(ψ˜ cos ξ − ψ)1/2 .(5)Here J‖ is the oscillatory part of the parallel currentdensity, ψ is the magnetic flux function introducedbyψ = ψ˜ cos ξ − x2B0/2Ls, (6)ψ˜ is a constant related to W byψ˜ = 16W 2B0/Ls, (7)ξ is the island cyclic variable, Ls is the shear length,B0 is the equilibrium magnetic field, c is the speedof light, ε is a positive infinitesimal.To find J‖ one should solve the current continu-ity equation∇‖J‖ +∇⊥ · j⊥ = 0, (8)where j⊥ is the perpendicular current density, ∇‖and ∇⊥ are the parallel and perpendicular gradi-ents. For simplicity we neglect drift effects. Then,for obtaining∇⊥ ·j⊥ one can use the perpendicularprojection of the single-fluid motion equationρ0d0Vdt=1c[j⊥ ×B]−∇p−∇ ·(pi⊥ + pi‖), (9)whered0/dt = ∂/∂t+V ·∇, (10)V is the plasma velocity (its structure is explainedbelow), ρ0 is the equilibrium plasma mass density, pis the plasma pressure, pi⊥ and pi‖ are the perpen-dicular and parallel viscosity tensors, respectively,B is the total magnetic field.In the Braginskii [11] approximation, the per-pendicular viscosity tensor gradient is given by∇ · pi⊥ = −ρ0µ⊥(∇2⊥V⊥ + 4b∇2⊥V‖) , (11)where µ⊥ is the perpendicular viscosity coefficient,b is the unit vector along the total magnetic field.On the other hand, according to equation (19.6) of[12], the parallel viscosity tensor gradient can beexpressed in terms of the parallel viscosity scalarpi‖ by∇ · pi‖ = 32b (b ·∇)pi‖ −12∇pi‖+32pi‖ [b∇ · b+ (b ·∇)b] . (12)This scalar satisfies the relation (see, for details,chapter 19 of [12] and, in particular, Eqs. (19.19),(19.73), (19.78) of [12])32〈pi‖∂ ln√gm∂θ〉θ= rsρ0χˆθ(Vy +qV‖). (13)Here gm is the metric tensor determinant, 〈. . .〉θ isthe averaging over the poloidal angle θ,χˆθ =q21/2(d0dt+νi). (14)The y - direction is defined by the unit vector y =b× x, x is the unit vector along the x - direction,where x = r − rs, rs is the radial coordinate of therational magnetic surface, where the island chain islocalized, Vy is the y - projection of the cross-fieldvelocity V⊥ given byV⊥ = c [b0 ×∇φ] /B0, (15)φ is the electrostatic potential. Remaining defini-tions in (13), (14) are: q is the safety factor,  is theinverse aspect ratio for r = rs, νi is the ion collisionfrequency. The function V‖ is the oscillatory partof the parallel plasma velocity.As a result, according to [12], the value ∇⊥ · j⊥is given by∇⊥ · j⊥ = −cρ0B0∂∂x[χˆθ(Vy +qV‖)]. (16)To find V‖ we use the parallel projection of Eq.(9). Then, allowing for (13), one can find (see, fordetails, [12, 13])(d0dt+2q2χˆθ − 4µ⊥ ∂2∂x2)V‖ +qχˆθVy = 0. (17)3. We consider the problem of interest qualita-tively, i.e. changing ∂2/∂x2 → −1/W 2, d0/dt →−iω. It then follows from (17) thatReV‖ = gqVy/, (18)whereg =ν2i[1 + 3/2(1 +W 2µ/W2)W 2µ/W2]+ 1/2ω2ν2i(1 +W 2µ/W 2)2 + ω2/(19)andWµ '(µ⊥1/2νi)1/2(20)is the characteristic island width governed by theperpendicular viscosity. The functiong = g (νi, ω, µ⊥) (21)characterizes the collisionality dependence of thepolarization current effect. It is introduced by therelation∆p = g∆∞p , (22)where∆∞p = ∆p|νi→∞ . (23)In the limit of vanishing perpendicular viscosity,Wµ/W → 0, Eq. (19) reduces to [14]g =ν2i + 1/2ω2ν2i + ω2/. (24)Then the function g is given by [14]g = 3/2, νi/ (ω) < C0,ν2i /ω2, C0 < νi/ (ω) < −3/2,1, νi/ (ω) > −3/2,(25)whereC0 ' 3/4. (26)In the limit Wµ/W →∞ one has from (19)g = 3/2, (27)which is the same as the first line of the right-handside of (25). Thus, for sufficiently large perpendic-ular viscosity one deals with the minimal value grelevant to the limiting case of weak collisions.4. The expressions for the function g given byEq. (25) were found in the linear approximation[14]. Let us show that they are qualitatively validalso in the nonlinear regime.One can find that Eq. (17) with µ⊥ = 0 and χˆθof form (14) leads to (1/2νi − ω∂h∂x∂∂ξ)V‖=ωkyq1/2(νi − ω∂h∂x∂∂ξ)(∂h∂x−〈∂h∂x〉), (28)where h = h (ψ) is the electrostatic potential profilefunction [10]. In the limit of weak collisions, νi → 0,it hence follows thatV‖ = 1/2qωky(∂h∂x−〈∂h∂x〉). (29)This corresponds to g = 3/2, see the first line ofthe equation (25).In the opposite case of strong collisions, νi →∞, instead of (29), one has from (28)V‖ =qωky(∂h∂x−〈∂h∂x〉). (30)This yields g = 1, see the third line of (25).To analyze (28) for finite νi/ (ω) we representV‖ as the sum of the cosine and sine parts, i.e. theeven and odd parts (with respect to the variable ξ),V‖ = Vc + Vs. (31)Then we arrive at the following two equationsω∂h∂x∂Vc∂ξ= 1/2νiVs + 1/2qω2ky∂h∂x∂2h∂ξ∂x, (32)ω∂h∂x∂Vs∂ξ= 1/2νi[Vc − qωky(∂h∂x−〈∂h∂x〉)].(33)The polarization current is defined by the functionVc.Note that for νi/ (ω) < −3/4 the contribu-tion of Vs into (32) can be neglected. Then Vcproves to be the same as for νi/ (ω) < 1, i.e. Vcis given by the right-hand side of (29). On theother hand, if the ratio νi/ (ω) lies in the interval−3/4 < νi/ (ω) < −3/2, the equation system (32)and (33) reduces toω∂h∂x∂Vc∂ξ= 1/2νiVs, (34)ω∂h∂x∂Vs∂ξ= −νi ωkyq1/2(∂h∂x−〈∂h∂x〉). (35)It hence follows that in order of magnitudeVc ' q ν2iωky(∂h∂x−〈∂h∂x〉). (36)This corresponds to g = ν2i /ω2, cf. the second lineof (25).The above nonlinear substantiation of of the col-lisionality dependence (24) for µ⊥ → 0 allows oneto suggest that by means of more complicated non-linear analysis for µ⊥ 6= 0, one can justify qualita-tively behavior of the function g (νi, ω,W ) given by(19).5. According to [6], dependence of βcrit on thefunction g, βcrit (g), characterizing the collisional-ity dependence of NTMs, is given byβcrit ∼ g1/2. (37)Such a collisionality dependence was the subjectof experimental studies [15]. Following [6], it wasassumed in [15] that the function g has the step-likeform with a jump in a region of sufficiently largeνi/ω. Then the authors of [15] have concludedthat their experimental data corroborate the theory[6]. However, according to [15], the form of thefunction g suggested in [6] is unadequate, so that,instead of g given by [6], one should use g of form(25). Then one can find that experimental data of[15] is in disagreement with the polarization currentthreshold model.According to (19), (27), such a disagreement isredoubled in the presence of anomalous perpendicu-lar viscosity. As a whole, this decreases attractive-ness of the polarization current threshold model.Then, in order to find βcrit one should appeal tothe transport threshold model [9] or to the theoryof β-limiting sub-Larmor modes [16].AcknowledgmentsThis work was supported by the Russian FederalProgram on Support of Leading Scientific Schools,Grant No. 00-15-96526, the Research Support Foun-dation of the State of Sa˜o Paulo (FAPESP), Uni-versity of Sa˜o Paulo, and Excellence Research Pro-grams (PRONEX) RMOG 50/70 Grant from theMinistry of Science and Technology, Brazil.References[1] Qu W X and Callen J D 1985 Nonlinear Growth of aSingle Neoclassical MHD Tearing Mode in a Tokamak.UWPR855 Report of University of Wisconsin[2] Callen J D et al 1987 Plasma Physics and ControlledNuclear Fusion Research vol 2 (Vienna: IAEA) p 157[3] Carrera R et al 1986 Phys. Fluids 29 899[4] Furth H P, Killeen J and Rosenbluth M N, 1963,Phys. Fluids 6 459[5] Chang Z et al 1995 Phys. Rev. Lett. 74 4663[6] Wilson H R et al 1996 Plasma Phys. Control. Fu-sion 38 A149[7] Fitzpatrick R 1995 Phys. Plasmas 2 825[8] ITER Physics Expert Group on Disruptions, PlasmaControl, and MHD, ITER Physics Basis Editors 1999Nucl. Fusion 39 2251[9] Shirokov M S, Mikhailovskii A B, Konovalov S V,Tsypin V S, 2002, Development of Transport ThresholdModel of Neoclassical Tearing Modes, this conference[10] Mikhailovskii A B, Pustovitov V D, Smolyakov AI, and Tsypin V S, 2000 Physics of Plasmas 7 1214[11] Braginskii S I 1965 Reviews of Plasma Physics vol1, ed M A Leontovich (New York: Consultants Bureau)p 205[12] Mikhailovskii A B, 1998 Instabilities in a ConfinedPlasma (Bristol: Institute of Physics)[13] Mikhailovskii A B, Lakhin V P, 1995 , Plasma Phys.Rep. 21 271[14] Mikhailovskii A B, Pustovitov V D, and SmolyakovA I, 2000, Plasma Phys. Contr. Fusion 42 309[15] Marashek M et al., 1999, Plasma Phys. Contr.Fusion 41 L1[16] Shirokov M S, Mikhailovskii A B, Konovalov S V,Tsypin V S, 2002, Spontaneous Generation of Beta-Limiting MHD Modes in Tokamaks, this conference

Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity

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Polarization current threshold model of neoclassical tearing modes in the presence of anomalous perpendicular viscosity

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