Beidler, C.

Helander, P.

Maaßberg, H.

Marushchenko, N.

English

MPG.PuRe

On the bootstrap current in stellarators and tokamaksP. Helander, J. Geiger, H. MaaßbergMax-Planck-Institut für Plasmaphysik, Greifswald, GermanyThe aim of this paper is to calculate the collisionless limit of the bootstrap current ina way that makes no distinction between tokamaks and stellarators, and also to considerthe correction due to a small but finite collisionality. This correction is proportional to thesquare root of the collision frequency and tends therefore to be important even in fairlylow-collisionality plasmas. We compare the analytical result with a numerical solution ofthe drift kinetic equation and find excellent agreement in the appropriate collisionalityrange.We consider the problem of solving the first-order drift kinetic equationv‖∇‖fa1 − νaDLfa1 = −vda · ∇fa0 −eav‖∇‖φ1Tafa0,where the collision operator has been approximated by pure pitch-angle scattering andthe interaction between different species is thus neglected. The notation is standard andfollows Ref. [1]. Instead of solving this equation directly, we consider the adjoint equationv‖∇‖ga + νaDL(ga) = −v‖B〈B2〉fa0, (1)from whose solution the bootstrap current can be constructed as〈Ja‖B〉=〈eaB∫fa1v‖d3v〉= −ea〈B2〉〈∫gavda · ∇ ln fa0d3v〉. (2)At low collisionality we may expand ga = ga0+ga1+ · · · in the same way as when treatingthe banana regime in a tokamak, givingga0 =σvfa02νaD∫ 1/Bmaxλdλ′〈√1− λ′B〉 . (3)ga1 =∫ llmaxfa0[− B〈B2〉 +1B∂∂λ(H(B−1max − λ) λξ〈ξ〉)]dl′, (4)with ξ =√1− λB . The Heaviside function H appearing in this expression vanishes inthe trapped domain, and lmax denotes the position along the field line where the fieldstrength is the largest, B(lmax) = Bmax . Substituting this expression in Eq. (2) gives thecurrent as a sum of two terms, 〈Ja‖B〉 = Ja1 + Ja2 , corresponding to the two terms inthe integrand of Eq. (4). The first one isJa1 =〈eaB∫ ∂fa0∂ψ2πv2dv∫ 1/B0vda · ∇ψξdλ∫ llmaxBdl′〉,where ψ is the toroidal flux andvda · ∇ψ =mav2eaξ (b×∇ψ) · ∇(ξB). (5)38th EPS Conference on Plasma Physics (2011) P4.045It is straighforward to carry out the integrals over v and λ to obtainJa1 = paA1a〈(B×∇ψ) · ∇(1B2) ∫ llmaxBdl′〉,where we have written A1a = d ln pa/dψ + (ea/Ta)dΦ/dψ , with Φ(ψ) the equilibriumelectrostatic potential. Writingg2(l) = B2∫ llmax(b×∇ψ) · ∇B−2dl′,we obtain Ja1 = −paA1a 〈g2〉 . The second term in Eq. (4) makes the following contribu-tion to the current,Ja2 = −〈B2〉〈ea∫ ∂fa0∂ψ2πv2dv∫ 1/Bmax0Bvda · ∇ψξdλ∫ llmax∂∂λ(λξ〈ξ〉)dl′B〉,where we substitute the drift velocity (5) and integrate by parts in λ to obtainJa1 =3paA1a 〈B2〉4∫ 1/Bmax0〈g4〉〈ξ〉 λdλ,withg4(λ, l) = ξ∫ llmax(b×∇ψ) · ∇ξ−1dl′, (λ < 1/Bmax).The total current of each species is thus〈Ja‖B〉= Ja1 + Ja2 = paA1a3 〈B2〉4∫ 1/Bmax0〈g4〉λdλ〈√1− λB〉 − 〈g2〉 , (6)as originally found by Shaing and Callen [2] using a very different argument.The result (6) is valid in the collisionless limit. Hinton and Rosenbluth [3] consideredthe effect of a small but finite collisionality in the geometry of a large-aspect-ratio toka-mak with circular flux surfaces. We extend their calculation to arbitrary axisymmetricgeometry and find that the result then also applies to an important class of stellarators.The first correction to the bootstrap current is proportional to the square root of thecollisionality and therefore cannot be obtained by simply continuing the expansion ofga = ga0 +ga1 + · · · to higher order. Instead, it is found from a boundary-layer analysis ofthe region around the trapped-passing boundary, λ = 1/Bmax . Because ga0 vanishes inthe trapped region and is equal to Eq. (3) in the passing region, its derivative ∂ga0/∂λ isdiscontinuous at the boundary, making the collision operator infinte, since the pitch-anglescattering operator contains two λ -derivatives. Collisions therefore need to be retainedin a boundary layer whose width is proportional to the square root of the collisionality.In this layer, Eq. (1) can be replaced byv‖∇‖ga + νaDL(ga) = 0,with boundary conditions obtained by asymptotic matching to the collisionless solution(3) away from λ = 1/Bmax . Remarkably, this boundary-value problem is identical to onearising in the theory of transport across magnetic islands [4]. The solution, which can beobtained with the Wiener-Hopf method, has the asymptotic behaviorga(x, y) → ±(c0 + c1x), x→ −∞38th EPS Conference on Plasma Physics (2011) P4.045on the circulating side of the boundary layer. Here c0/c1 =√2(√2−1)ζ(1/2) ≃ −0.855 ,ζ denotes the Riemann zeta function, andx(λ) =λBmax − 1√νa0,νa0 =2νaDπv∮ √1− b(l) dlb(l). (7)with b = B/Bmax . Hence it is possible to compute the bootstrap current〈Ja‖B〉= −Ipaι{(1− f effc (v)) [A1a +(mav22Ta− 52)A2a]},where I = RBϕ and A2a = d lnTa/dψ . Here the brackets denote a velocity-space average[1]{· · ·} =∫(· · ·) mav23Tafa0nad3v,and f effc = fc + δfc(v) is an effective fraction of circulating particles. Its collisionlessvalue isfc =3〈B2〉4∫ 1/Bmax0λdλ〈√1− λB〉 , (8)and the correction from the boundary layer isδfc = 0.51〈b2〉√√√√ νaDv〈√1− b〉∮ dlb(9)In a large-aspect-ratio tokamak with circular cross section, B = B0(1−ǫ cos θ) , we obtainf effc ≃ 1− 1.46ǫ1/2 +1.35ν∗1/2ǫ1/4, (10)where ν∗ = νaDR/ιv .This calculation applies to an axisymmetric tokamak of arbitrary cross section butappears difficult to generalize to general stellarator geometry. Of course, it immediatelyapplies to quasiaxisymmetric and quasihelically symmetric configurations, since their neo-classical properties are identical (in leading order) to those in tokamaks. However, thereis another important but less trivial case amenable to a similar analysis, and this is thecase of a perfectly quasi-isodynamic stellarator [5], i.e., a stellarator where the contoursof constant |B| are poloidally closed and the bounce-averaged radial drift vanishes for allparticle orbits. The latter property is also referred to as omnigeneity in the literature. Insuch a magnetic field, it can be shown that if B = ∇ψ ×∇α , then the radial excursion(in terms of ψ ) for passing particle orbits can be written as [5]∆a = −µ0J(ψ)2πv‖Ωa+∂∂α∫ BmaxBk∂∂B′(v′‖Ω′a)dB′, λ < 1/Bmax, (11)where J(ψ) is the toroidal current enclosed by the flux surface ψ , and the functionk(ψ, α,B) encapsulates all other necessary geometric information of the magnetic field.As shown in Ref. [5], it then follows that the first-order distribution function is equal tofa1 = −∆a∂fa0∂ψ+ ha,38th EPS Conference on Plasma Physics (2011) P4.045where as a result of the α -derivative in Eq. (11), which disappears on an orbit average, theequation for ha is the same as in a tokamak, with the replacement I(ψ) → −ιµ0J(ψ)/2π .The solution therefore coincides with that in a tokamak, not only in lowest order [5] butalso in the trapped-passing boundary layer. In particular, if the flux surface in questiondoes not enclose any net toroidal current, J(ψ) = 0 , which is the usual situation in astellarator, then ha = 0 and the bootstrap current vanishes. Thus we conclude thatthe bootstrap current in a quasi-isodynamic stellarator vanishes to a very high degreeof accuracy: not only is there no current in the collisionless limit [5], but the leadingcorrection due to finite collisionality is also absent.Finally, we verify our results numerically, by using the DKES code [6] for a high-beta-poloidal NSTX equilibrium [7]. Figure 1 shows the bootstrap current coefficientcomputed numerically for three different radii as well as the result obtained from theanalytic representation (8)-(9) of f effc and the large-aspect-ratio limit given by the lastterm in Eq. (10) but using the exact fc value (8) instead of the first two terms. TheDKES results clearly confirm the boundary-layer analysis. With increasing value of thelocal inverse aspect ratio ǫ , the plateau regime shrinks and is shifted to higher ν∗ . Thedifference bewteen the full analytical result (solid lines) and the large-aspect-ratio limit(dashed lines) increases with ǫ .Figure 1: The mono-energetic bootstrap current coefficient vs. collisionality, ν∗ , forǫ = 0.0335, 0.119 and 1.047 (from top to bottom at low ν∗ ) with DKES results (fullcircles), the analytical form of f effc from Eqs. (8)-(9) (solid lines) and the limit of a large-aspect-ratio tokamak with circular cross section for δfc (dashed lines). The dotted linesrepresent interpolations to the DKES data.[1] P. Helander and D.J. Sigmar, Collisional transport in magnetized plasmas (CambridgeUniversity Press, 2002).[2] K.C. Shaing and J.D. Callen, Phys. Fluids 26, 3315 (1983).[3] F.L. Hinton and M.N. Rosenbluth, Phys. Fluids 16, 836 (1973).[4] R.D. Hazeltine, P. Helander, and P.J. Catto, Phys. Plasmas 4, 2920 (1997).[5] P. Helander and J. Nührenberg, Plasma Phys. Control. Fusion 51, 055004 (2009).[6] W.I. van Rij and S.P. Hirshman, Phys. Fluids B1, 563 (1989).[7] S.P. Gerhardt et al., Nucl. Fusion 51, 033004 (2011).38th EPS Conference on Plasma Physics (2011) P4.045

On the bootstrap current in stellarators

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On the bootstrap current in stellarators

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