Per Helander. Contributions from: R. Kleiber, A. Mishchenko, J. Nührenberg, P. Xanthopoulos. Wendelsteinstraße 1, Greifswald

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1 Rotation and zonal flows in stellarators Per Helander Wendelsteinstraße 1, Greifswald Contributions from: R. Kleiber, A. Mishchenko, J. Nührenberg, P. Xanthopoulos

2 What is a stellarator? In a tokamak (Tamm and Sakharov, 1951), the magnetic field lines twist around the torus because of the toroidal plasma current. In the stellarator (Spitzer, 1951), this twist is imposed by external coils. Magnetic field is necessarily 3D. No toroidal current is necessary. Less free energy in the plasma. Greater degree of control Tokamak Stellarator

3 Theoretician s stellarator

4 Questions in this talk Can a stellarator plasma rotate? Effect of turbulence? If so,how quickly? Fast rotation V v T i = ion thermal speed Slow rotation V δv Ti, δ = ρ i L Empirically in tokamas, toroidal rotation tends to be fast and poloidal rotation slow expected theoretically in stellarators, the rotation tends to be slow

5 Fast rotation Theorem I: Fast equilibrium rotation is only possible in certain (so-called quasisymmetric) magnetic fields if δ<<1. This conclusion holds independently of any turbulence that may be present, as long as the fluctuations are small.

6 Fast rotation in stellarators Proof: The Vlasov equation f t +(V+v) f + e m µ E +(V + v) B V t (V + v) V f v = C(f) and the ordering Ω 1 ρ i 1 t lead to the drift kinetic equation (Hazeltine and Ware, 1978) f 0 t +(v kb + V) f 0 + ẇ f 0 w = C(f 0) V(r,t)=V k b + B Φ 0 B 2 ẇ = ee kv k mv kv V b mv k 2 b V b + μbv ln B

7 Fast rotation in stellarators An equilibrium with t δv T L must have the the following properties: isothermal flux surfaces either et e V << ion thermal speed or B = f(ψ,l)+o(δ) corrections, l = arc length along B (1) Since this follows in 0th order in δ, it is independent of any turbulence! In a scalar-pressure equilibrium, (1) means that B is quasisymmetric.

8 Terminology Special magnetic fields: Quasisymmetric B symmetic in Boozer coordinates: B = B(ψ,mθ nϕ) omnigenous Neoclassical properties identical to those in a tokamak Isomorphism between stellarator and tokamak drift kinetic equations (Boozer 1984) quasisymmetric Omnigenous (Hall and McNamara 1972) No radial magnetic drift on a bounce average Z τb v d ψ dt =0 0

9 Particle orbits Particle motion in WEGA (an unoptimised stellarator) High Field Strength Low Field Strength

10 Transport due to helically trapped particles Complicated parallel variation of the magnetic field strength Local trapping Unconfined orbits ² h Random walk: Step length r v d t Time between steps t ² h /ν Diffusion coefficient D ² 1/2 r 2 ² 3/2 h v2 d Temperatures beyond which v d D 1/ν ² h t ν χ e > 1 m 2 /s Always dominates over turbulence at high enough temperature, since D 1/ν m1/2 T 7/2 nb 2 R 2

11 Transport vs collisionality D ν 1/2 Gyro-Bohm ν 1 ν 0 ν Tokamak ν

12 Slow rotation Theorem II: Gyrokinetic turbulence cannot affect the macroscopic equilibrium rotation in a stellarator, except if B is quasisymmetric. This rotation is instead determined by neoclassical theory.

13 Slow rotation Projections of the momentum equation hρv Ji = h (ρvv + π + M) Ji t hρv Bi = h (ρvv + π + M) Bi t where ρvv = Reynolds stress π = viscous stress Ã! B2 M = 1 B 2 μ 0 2 I B B = Maxwell stress

14 Gyrokinetics The gyrokinetic ordering gives V f a B v βb e φ a δ 1, k ρ i = O(1) T a f a T a π neocl (ρvv) turb M turb π turb π neocl =(p k p )(bb I/3) O(δp) The turbulent gyroviscous force dominates locally.

15 Global rotation However, on a volume average over a volume ΔV between two flux surface several gyroradii apart ρ r r neoclassical viscosity dominates Z µ Z µ J J ρv dv ' π t B neocl : B V V dv Hence, gyrokinetic turbulence cannot affect macroscopic rotation! one exception

16 Non-ambipolar neoclassical transport Toroidal plasma rotation corresponds to a radial electric field V ϕ E r /B p If the neoclassical transport is not intrinsically ambipolar, there is a radial current µ d ln n J r ned + e dφ, D ν i ρ 2 i dr T dr producing a toroidal torque J r B p ν i ρ 2 i ne 2 E r B p T that slows down the rotation on the time scale m i n i V ϕ ν 1 i J r B p

17 Intrinsic ambipolarity The radial neoclassical current hj ψi = hπ k : Ji/p 0 (ψ) vanishes in case of intrinsic ambipolarity (for any E r ). In what configurations does intrinsic ambipolarity hold? Theorem III: B is intrinsically i i ambipolar if, and only if, it is quasisymmetric. i Boozer, PoP P 1983 our present concern

18 Intrinsic ambipolarity = quasisymmetry Proof: From the drift kinetic equation v k k fa1 + v d f a0 = C a ( f a1 ) (1) In an isothermal plasma follows an entropy production law 0 hj ψi = X Z φ 0 (ψ) T a0 But then the H theorem implies a a0 À d 3 vf f C f a1 a (f a1 )/f a0 0. hj ψi =0 f a1 =(α a + β a v k + γ a v 2 )f a0 Re-insert into (1): µ (B ψ) ln B k = 0 k B B is quasisymmetric

19 Proximity to quasisymmetry How close to perfect quasisymmetry must a stellarator come, in order to have tokamak-like rotation? depends on collisionality and on the radial length scale considered In the 1/ν regime, the diffusion coefficient is D ² 3/2 The current becomes h δ2 T i m i ν i hj ψi ² 3/2 h δ2 i p i Ω i ν i and its torque exceeds the Reynolds stress, averaged over the volume between two flux surfaces N ion gyroradii apart if ² h > ³ ν 2/3 ³ ν N

20 Zonal flows

21 Zonal flow oscillations Rosenbluth-Hinton problem: perturb the plasma density and radial electric field at t=0 watch the linear evolution E Analytical preditiction: geodesic acoustic modes (GAMs) damping of initial perturbation because of banana-orbit polarisation finite residual perturbation level as t Qualitatively different in stellarators different types of orbits end state oscillatory

22 Mathematical problem The linearised drift kinetic equation f a1 e a φ 0 + (v k b + v d ) f a1 = (v d r)f a0 t Ta is to be solved on time scales exceeding the bounce time, coupled to gyrokinetic quasineutrality X µ ma n a φ À n a e a + = 0 a n a t = 1 V 0 r B 2 Laplace transformation gives Z À V 0 f a (v d r)d 3 v L(p)φ) ˆφ 0 (p) = φ 0 0 L(p) =p + X e 2 a T a T a Z v r 2 + p 2 δr 2 f a0 p + ik v ikv r d3 v À À r 2 X / ma n a B 2 a v d r = v r + v k k δ r

23 Inversion of the Laplace transform Im p Weakly damped oscillations Re p Finite residual φ(t) = φ0 0 2πi 0 Z σ+i t σ i a j =Res[1/L(p j )] e pt dp L(p) = φ0 0 X a j e p jt j

24 Zonal flow oscillations Analytical predictions: Oscillation frequency < GAM frequency and Ω 0 in the limit of perfect confinement. Landau damping of zonal flow oscillations. Damping rate γ exp( Ω/k v r ) is sensitive to magnetic geometry: higher in LHD than in W7-X Residual level dependent on radial wavelength φ(t) µ1 αq2 β²1/2 1 lim = 1 + t φ + 0 ² 1/2 k 2 ρ 2 i Confirmation by EUTERPE and GENE: differences between LHD and W7-X qualitatively in line with expectations

25 Zonal flows and GAMs in W7-X and LHD LHD W7-X: GAMs weakly Landau damped GAMs strongly Landau damped Zonal-flow oscillations strongly damped Clear zonal-flow oscillations Landau damping of these GENE Simulations

26 LHD (global simulations) Dependence on k Fourier transform lim t φ(t) ) = φ 0 µ1 αq2 β²1/ q ² + β 1/2 k 2 ρ 2, i

27 Physical picture Consider the plasma between two flux surfaces C v d Ripple wells = C L L R t i(t) =L R t 0 u(t0 )dt 0

28 Physical picture Consider the plasma between two flux surfaces C v d Ripple wells C L = R L phase mixing or collisions R t i(t) =L R t 0 u(t0 )dt 0

29 Summary Only in quasi-symmetric configurations can the plasma rotate rapidly or freely very robust result: insensitive to turbulence In all other stellarators E r and rotation are clamped at the value required for neoclassical ambipolarity gyrokinetic turbulence only matters on small scales momentum transport unimportant on large scales It is much easier to calculate l E r in a stellarator t than in a tokamak! k! Linear zonal-flow physics qualitatively different in stellarators oscillatory response to an applied electric field References: Theorem I: Helander PoP 2007 Theorems II and III: Helander and Simakov PRL 2008 Zonal flows: Sugama, Watanabe et al, PoP P and PRL , 2010 Mischenko, Helander and Könies, PoP 2008 Helander, Mischenko, Kleiber and Xanthopouls, unpublished (2010)

30 Extra material

31 NCSX Monoenergetic particle diffusivity teau D11/Dplat D 11 = D ν = νr 0 ιv

32 Tokamak rotation Tokamak plasmas rotate freely, even spontaneously, in the toroidal direction. Mach numbers ~1/3, up to ~1 in spherical tokamaks with unbalanced NBI Poloidal rotation much slower Damped by collisions Friction between trapped and passing ions Toroidal rotation ti determined d by E r V = u(ψ)b µ dφ dpi dψ + 1 i R ˆϕ ne dψ small Rice et al, Nucl. Fusion 2007

33 Rotation in LHD Much smaller rotation in LHD Yoshinuma et al, Nucl. Fusion 2009