Turbulence in Tokamak Plasmas

Transcription

1 ASDEX Upgrade Turbulence in Tokamak Plasmas basic properties and typical results B. Scott Max Planck Institut für Plasmaphysik Euratom Association D Garching, Germany Uni Innsbruck, Nov 2011

2 Basics of Turbulence in a Tokamak Basics of Low-frequency Drift Dynamics low frequency basics, energy transfer in turbulence, equilibrium Methods numerical techniques, mathematical treatment of magnetic geometry Electromagnetic Nonlinear Character energy transfer, nonlinear saturation, turbulence is not a collection of instabilities Some Results parameter scaling scale separation, role of long-wavelength dissipation range experimental comparisons, benchmarking Some Important Lessons

3 Magnetic Confinement J x B = c p B p MHD equilibrium strong magnetic field, small gyroradius closed magnetic flux surfaces --> confined plasma however... turbulence --> losses eddies, few gyroradii

4 Some Scales Plasma interaction radius e 2 /4πǫ 0 r 0 = mv 2 /2 T e hence r 0 = e 2 /4πǫ 0 T e Debye screening length λ 2 D = ne2 /ǫ 0 T e large number of free particles in Debye sphere nλ 3 D 1 and λ D r 0 quantumlimitoninteraction: debroglieradiusr h = h/ m e T e ( > r 0 fort e > 10eV) collisional theory: λ = log(λ D /r h ) 1 Some Scales Magnetised Plasma electron and ion gyroradius: ρ e,i = (c/eb) m e,i T e,i neutral ( quasineutral ) plasma dynamics: ρ i λ D magnetised plasma: ρ i logp 1 or ρ i L

5 Background Debye Screening bare charge in a vacuum, static (do the integral from 0 to r with sphere symmetry, with φ 0 as r ) ǫ 0 2 φ = eδ 3 (r) φ = e 4πǫ 0 r allow a (small amplitude) reaction of the electrons charge density is eδn e force balance: T e δn e = n e e φ hence δn e = (n e e 2 /T e )φ ǫ 0 2 φ = eδ 3 (r) n ee 2 T e φ φ = e 4πǫ 0 r exp[ r/λ D] Debye length λ D (exercise: show δn e n e for r r 0 = e 2 /4πǫ 0 T e ) λ 2 D = ǫ 0T e n e e 2 or λ D = V e ω pe with V 2 e = T e m e main point: test charge is not felt by plasma for r > λ D (time varying: assumes ω ω pe )

6 Background Gyromotion and Drifts motion of a charge under Lorentz force in constant magnetic field m dv dt = ee+ e c v B dx dt = v fast motion: assume E is small and integrate time once to find mv = e c x B constant force perpendicular to v, hence circular motion frequency: ω = Ω = eb mc radius: x = 1 BΩ B v hence ρ = v Ω drifts: include E and d/dt as corrections, solve for v on right side in 1st line: exercise: what are the necessary assumptions about time scales? ExB drift and polarisation drift: v = c mc2 de B2E B+ eb 2 dt

7 Tokamak Magnetic Field B B ζ axisymmetric MHD equilibrium B θ toroidal, poloidal components mainly toroidal ratio of components --> pitch parameter q B ζ / B θ

8 Tokamak Equilibrium axisymmetric divergence free vector for the magnetic field B = I ϕ+ ψ ϕ Ampere s law and magnetic force balance B = µ 0 J J B = p result: p and I and ψ are functions of each other, and the Grad-Shafranov Equation I I ψ +R2 ψ R 2 +µ 0R 2 p ψ = 0 interior to a fixed, closed boundary, find nested flux surfaces ψ =constant, hence also p =constant on these

9 z r = a r η R 0 R

10 Ampere s law and magnetic force balance MHD vs low frequencies B = µ 0 J J B = p manipulate factors of B using B 2 = B B and b = B/B and = (1 bb) J B p = B2 µ 0 b b B 2 2µ 0 p hence curvature of the field lines is a restoring force b b = 1 r c r c with curvature radius r c result: transverse Alfvén waves ( shear Alfvén waves ) with Alfvén velocity v 2 A = B 2 /µ 0 ρ M

11 show the MHD forces again MHD vs low frequencies (2) J B p = B2 µ 0 b b B 2 2µ 0 p magnetic energy acts as a pressure but only across field lines result is magnetosonic waves acting like pressure waves, with same velocity v A across field lines, scale is L, the gradient scale (few cm) usually have v A / t hence perpendicular force balance along field lines, scale is L (few m) turbulence driven by p often has v A / t hence, parallel and perp dynamics is very different origin of 2D nature of magnetised plasma turbulence

12 Magnetic Confinement J x B = c p B p MHD equilibrium strong magnetic field, small gyroradius closed magnetic flux surfaces --> confined plasma however... turbulence --> losses eddies, few gyroradii

13 observations why it is unfair to say fusion is always 30 years in the future MHD equilibrium/confinement solved only in the 1970s (Soviet tokamak) we thought we were at the finish line in mid/late-1970s tokamak transport was found to be anomalous small, 1%-level fluctuations ñ e at cm-scale always observed frequencies below 10 6 Hz while Ω i = eb/m i c 10 8 rad/sec well outside existing paradigm ( macroscopic = MHD equilibrium and stability) theory had been investigating the universal mode (1st phase: ) driven by p in either electrons or ions (various models) unlike most kinetic instabilities, active for Maxwellian background seemed to work in electrostatic limit, not involving MHD in any way this was the beginning of the study of microturbulence

14 a couple of results high-t neoclassical equilbrium reported in the 1970s anomalous (non-collisional) transport found almost immediately major reviews: 1978, 1983, 1985, and Phys Plamas special issue in 1990 in the tokamak edge, information on fluctuations in the ExB velocity are accessible comparison between fluctuations and power balance of transport: 1987 anything in the core other than ñ e, even today, requires model assumptions note: 3D electromagnetic models were only mature at the end of the 1990s 3D kinetic electromagnetic models are maturing only now global EM gyrokinetic simulation is just beginning, with many premature claims the field is still very active many currently-held views will fail

15 Low Frequency Drift Motion magnetic field general sense of gyration for ions magnetic field drift of gyrocenters (v << v ) low frequencies ω << Ω v-space details: gyrokinetic few moments: gyrofluid

16 Low Pressure (Beta) Dynamics low beta low frequencies 2 p << B /8 π ω << k v A flute mode vortices/filaments k << k magnetic field B pressure disturbance magnetic disturbance (parallel to B) p B --> strict perpendicular force balance (p + 4 π BB) ~ 0 ω k v A --> electromagnetic parallel dynamics

17 Sense of Coordinate Geometry p (x,y) phase shift > transport y φ (x,y) p v E B B computations: align coordinates to magnetic field (sheared, curved) (only one contravariant component of B is nonvanishing) (nonorthogonal, takes advantage of slowly varying B) (S Cowley et al Phys Fluids B 1991, B Scott Phys Plasmas 1998, 2001)

18 ExB Drift at Finite Gyroradius c c v E = B x u E = B x B 2 φ B k ρ << 1 ρ k ~ 1 J 2 0 φ φ(x,y) > 0 φ(x,y) > 0 u > v E E u E

19 Phase Shifts and Transport p p and phi in phase --> no net transport phase shift --> net transport phase shift --> net transport down gradient --> free energy drive

20 e e Role of Parallel Forces on Electrons equation of motion for electrons parallel to B. ( _ 1 A φ + η J ) = p e + inertia n + c Alfven (MHD) coupling adiabatic (fluid compression) coupling static balance of gradients --> adiabatic electrons general: response of currents to static imbalance a two fluid effect controls possible phase shifts ~ p <--> e ~ φ

21 why is low frequency turbulence not desribed by MHD? forces on the electrons E+v B = 1 n e e p e in MHD the left side is assumed to vanish ( p e is somehow small ) in general, along the field lines E = 1 n e e p e 0 this is the adiabatic response of the electrons in general forces are weak enough to keep these components comparable we will see: coupling among the dependent variables in otherwise 2D turbulence departures from MHD have a central role in the dynamics

22 Drift (Alfven) Wave Dynamics y p electron current ion current p ~ drift ~ p sound waves ~ φ x B --> φ ~ --> ~ φ coupled to p ~ through Alfven dynamics continually excites p ~ in the gradient (M Wakatani A Hasegawa Phys Fluids 1984) (B Scott Plasma Phys Contr Fusion 1997) --> structure drifts

23 Scales of Motion broad range of both time and space scales to ion gyroradius 10 2 k ρ y s 10 0 slowest time scale reflect flow/equilibrium component for equal temperatures, space scale range includes ion gyroradius high resolution, long runs (> 1000 "gyro Bohm" times) are necessary 10 3 (B Scott Plasma Phys Contr Fusion 2003) ω L /c T s 0 10

24 Turbulence Properties it s really turbulence: no coherent relation between any degrees of freedom energy transfer goes both ways and it is a cascade it is also a turbulent cascade in some cases (tokamak edge), result of dual energy cascade: eddy vorticity > drive rate consequence: linear eigenmodes don t form, linear scaling does not persist we find out these things by measuring nonlinear energy transfer direct statistical diagnosis in the simulation, not in a cartoon of it the linear versus nonlinear part comes from direct quantitative comparison energy transfer and drive rates in linear terms, in linear and turbulent phases

25 energy transfer spectra (Camargo, Scott, Biskamp, Phys Plasmas 1995) ExB and δf free energy, and mean squared vorticity transfer is from k to k, shown where positive direct cascade for U n and W,... inverse cascade for U E cascade dynamics not changed by linear forcing this is the indication that the transfer is a local cascade: it s mostly 1/2 < k /k < 2

26 energy transfer statistics (should have been in B Scott, New J Phys 2002, but wasn t) transfer (ExB, δf, vorticity) from k y ρ s 0.5 to 0.3 in drift Alfvén turbulence M/σ values: Gaussian PDF shape, std dev (σ) always much larger than mean (M) this is the indication that the cascade is turbulent

27 linear to nonlinear (turbulence) transition (B Scott, Phys Plasmas June 2005) linear: where growth rate (Γ T ) is near maximum (which defines the linear value γ L ) saturation: where Γ T hits zero turbulence: at late times, Γ T 0 and the drive rate Γ + is smaller than γ L lots of energy is in stable modes, maintained by the turbulence RMS vorticity is much larger than γ L or Γ +

28 Energy Transfer: electromagnetic turbulence low k ~ φ nonlinear high k ~ φ entire spectrum a unit sink J ~ J ~ p ~ nonlinear p ~ sink thermal gradient DW: direction for J determined by NL (B Scott Phys Fluids B 1992, Plasma Phys Contr Fusion 1997) (S Camargo et al Phys Plasmas 1995 and 1996)

29 Scale Separation Look and Feel electromagnetic core cases with a/ρs of 50, 100, and 200, non-axisymmetric part if you can see the eddies on a global plot they re too large! in the edge you have L /ρs < 100 but 2πa/q > 103 ρs

30 Scale Separation and the Spectrum density and vorticity spectra for the three cases ion heat source and sink spectra for the three cases

31 Relevance Range for Linear Instabilities dispersion space bounded by ideal interchange and diamagnetic rates ω L /c 0 10 T s ω γ I * k ρ c /L y s s T c 2/L R s T k ρ y s if the linear growth rate is above the red line then the instability is relevant usually, this is not the case anywhere in the spectrum (unless: MHD threshold) 0 10 this situation is a direct consequence of very large R/L >> 1 in the edge γ L T (B Scott New J Phys 2002, Phys Plasmas 2005) (drawn for a severely resistive case)

32 Comparison -- Fluctuation Statistics probability distribution of cross phase for each Fourier mode unified spectrum, phase shifts between 0 and π/4, in code and TJK experiment basic signature of drift wave mode structure (parallel current dynamics) (B Scott Plasma Phys Contr Fusion 2003) (U Stroth F Greiner C Lechte et al Phys Plasmas 2004)

33 Comparison -- Fluctuation Statistics wavelet analysis of fluctuation induced transport in code and TJK experiment both results show same phenomenology: regime break in spectrum evidence of nonlinear cascade overcoming drive? (N Mahdizadeh et al Phys Plasmas 2004)

34 Transport due to ExB Turbulence the turbulence causes a finite average advective transport, in general... Q = Q e +Q i Q e = 3 2 p eve x Q i = 3 2 p ive x in a confined plasma, the equilibrium is maintained by a source the time scales are very different; typical values: δ 10 2 τ turb 200 L c s τsource few δ 2 L c s profiles evolve slowly, turbulence in quasistatic statistical equilibrium it is a good approximation to consider turbulence in the presence of a prescribed gradient

35 Important Lessons note we have not discussed instabilities, with good reason: turbulence is about nonlinearity, with forcing in a secondary role cascade dynamics goes both ways, through different nonlinearities linear coupling mechanisms tie the spectrum together as a unit issues of scale separation enter in a big way easy to understand: time scales (eddy turnover, saturation, transport) also: global versus local dynamics (dissipation occurs at both large/small scale) also: anisotropy, meaning L x L y in the 2D drift-plane interaction with confinement phenomena ( barriers ) mostly not understood well resolved, mature global simulation needed, just beginning also: ongoing improvements of analysis of the equations this is far from a solved field