Summer College on Plasma Physics August Drift Wave Turbulence and Zonal Flows. Part I

Transcription

1 Summer College on Plasma Physics August 2009 Drift Wave Turbulence and Zonal Flows Part I George R. Tynan UCSD University of California San Diego USA

2 Drift Wave Turbulence and Zonal Flows Part I G.R. Tynan UCSD This talk is made possible by the many discussions and contribution of materials and results from collaborators and colleagues: J. Boedo, M. Burin, P.H. Diamond, R. Fonck, A. Fujisawa, O. Gurcan, C. Holland, K. Itoh, S. Itoh, G. McKee, R. Moyer, J. Yu, S. Zweben

3 Background and Motivation Review of Drift Waves Comments on Transition to Turbulence Why Is Turbulence of Interest in Fusion? What are Zonal Flows? Why Care About the Nonlinear Interactions Between Turbulence and Zonal Flows?

4 Fundamental Origins of Drift Waves & Instability pol V i Perpendicular Ion Polarization Drift = 1 B Ci de dt = 1 B Ci d dt Parallel Electron Motion Gives Boltzmann Relation V e = kt e m e e eff n e n e e kt e d dt t + V i ExB Charge Conservation ij = ij ien i V i pol = ien e V e Quasi-neutrality n i n e

5 Simple Linear Collisional Drift Wave Instability Slab Result r = * 1 + k 2 S 2, * = k y V de, S = C S Ci i = eff e k 2 2 v the *2 ( 1 + k 2 2 ) > 0 3 S

6 Physical Picture of Drift Waves See e.g. F.F. Chen Isodensity Curve for a Single Mode z >> because Electron M.F.P. << z << V THRM(elec)

7 The Basic Picture of Turbulent Transport Ref: T. Ribiero IPP-Garching 2005 Summer School

8 The Basic Picture of Turbulent Transport Ref: T. Ribiero IPP-Garching 2005 Summer School

9 Characteristics of Drift Waves: 1. Electrostatic Fluctuation: V THI << /k z << V TH E 2. << Ci, Ce 3. z >> 6. ~n 1 peaks near maximum n 0 7. Boltzmann Relation: ~ ~ n 10.Propagates in Electron Diamagnetic Drift Direction 11. Growth Rate Increases with: 12. (theory) (expt.) REAL REAL B, e

10 From Waves to Turbulence Multiple Unstable Eigenmodes Can Exist Linear Theory NEGLECTS Fluctuating Convective Derivative Term (Higher Order) What Happens When Can No Longer Neglect This Term Get Exchange of Energy Across Spatiotemporal Scales

11 Effect of Convective Derivative Nonlinearity Configuration Space n t + nv = 0 v + vv =... t n k 2 t v k 2 t Fourier Space + n * k v k1 n k2 = 0 k 1,k 2 k =k 1 +k 2 + v * k ( v k1 )v k2 =... k 1,k 2 k =k 1 +k 2 Different Spatio-temporal scales interact, or exchange energy

12 Flow Generation from Turbulence: Fourier Space Usual Reynolds Stress Term in Simplified Momentum Eqn (ala Diamond et al. PRL 1994 and others) u t + u r u r = damp u Radial Transport of Angular Momentum Consider a Zonal Flow to Have: u = u Z k r Z = k rz ˆr Z k r << ~ 1/ Z k 1, k 2 F.T., Write as KE, and Average Energy Eqn over Z-flow scales: 1 2 u Z 2 where t ( ) k Z P turb kz = μ u 2 Z ( k Z ) P turb kz = Re u * Z k Z k 1 k 2 k Z =k 1 ±k 2 ( ) u k 1 Z Z k r u >> ( ( )i )u k 2 ( ) t corr NL Energy Transfer

13 Flow Generation from Turbulence: Fourier Space k Power Spectrum, k 2 ( k ), k T k 1, 2 k Z a>1 T k Large Scale Damped Flow Inertial Range Power Transferred, P k ( k k ) 0 1, 2 > k 0 L n ~0.1 Unstable Region k i ~1 k FLR Dissipation > 0 Free Energy Source Releases Energy On One Scale Nonlinear Energy Transfer Moves Energy to Dissipation Region Shear Flows Develop Via Transfer of Energy to LARGE SCALES (small k) 0 = μ k < ; k T k ( k k ) 0 1, 2 <

14 Images of Turbulence in Tokamaks GYRO DIII-D J. Yu C-III Emission (UCSD)

15 Imaging of Turbulent Density Fluctuations in the Core Region of DIII-D Tokamak DIII-D Ref: G. McKee, private communication

16 G.R. Tynan, 2008 Int'l Summer School, KAIST August 2008

17 Result: Turbulent Transport in Confined Plasmas Ref: Lackner, DEISY Talk 2005

18 Turbulence Leads to Cross-field Transport n t + = 0 = n B B 2 v q mn + ì R + = t m ( E + v B) ( nt ) ì R ~ ~ B B 2 B B nt t n T t + Q cond + Q conv = q class + P rad ~ Q cond 3 n 2 ~ ~ T 2 Neglecting DC Convection, Magnetic Fluctuations, Parallel flow fluctuations, Viscosity, B B ~ Q conv 3 T 2 ~ n~ 2 B Assuming electrostatic ExB dynamics for velocity B

19 Turbulent Fluxes are (Roughly) Consistent with Global Confinement Hallock, PRL 1987 p V A nd 3 x r da p exp ~ few msec

20 Turbulent Electron Heat Flux Consistent with Global Confinement (at least at edge) Ritz, PRL 1989 E 3 2 ntd 3 x V q r da A E exp

21 Background and Motivation Review of Drift Waves Comments on Transition to Turbulence Why Is Turbulence of Interest in Fusion? What are Zonal Flows? Why Care About the Nonlinear Interactions Between Turbulence and Zonal Flows?

22 Drift Turbulence Fluctuations Elongated Along B Grulke PoP 2006 ALCATOR C-Mod

23 Origin of Zonal Flow Lies in 2D Dynamics Navier-Stokes Eqn for Rotating Fluid: t + v v + 2 v = 1 p + 2 v For v << v = 0 = 0 v = 0 I.e. no velocity gradient along direction of axis of rotation

24 Flow Generation from Turbulence: the Vortex Merging Picture 3-D 2-D z v z = 0 Vortex Merging z z v z 0 z z + ==> Vortex Stretching Generate Flows on Smaller Scales z

25 2D Dynamics in Magnetized Plasmas & Rotating Fluids Navier-Stokes Eqn for Rotating Fluid: t + v v + 2 v = 1 p + 2 v Momentum Eqn for Magnetized Plasma: t + v v + B v = 1 + p I.e. momentum conservation same form for rotating fluid And magnetized plasma > DYNAMICS ARE SAME! 2 v

26 Zonal Flows are Common & Effect Transport in Many Systems

27 Zonal Flows are Common & Effect Transport in Many Systems CASSINI Imaging Team, NASA

28 Result: Strong Similarity Between Planetary Flows & Magnetized Plasma Flows ITER ExB flows m=n=0, k r = finite Zonal flow on Jupiter Ref: K. Itoh, APS 2005 Invited Talk, PoP May 2006 From GYRO ZFs are "mode", but: 1. Turbulence driven 2. No linear instability 3. No direct radial transport

29 Large Scale Sheared Flows Can Develop 7 =0 n ~ 0 V V ZF GAM = c s /R New feature: geodesic acoustic coupling (GAC) B D Scott 2003

30 Zonal flows really do exist! CHS Dual HIBP System 90 degree apart P E (a.u.) Z. F. GAM? A. Fujisawa, PRL / i 1 r 1 =12cm 0.5 E r (r,t) C (r 1,r 2 ) E r (r,t) High correlation on magnetic surface, Slowly evolving in time, Rapidly changing in radius r 2 (cm) Radial distance August 2008

31 GAM-ZF Observed in Edge Region BES measures localized, long-wavelength (k i < 1) density fluctuations Can be radially scanned shot to shot to measure turbulence profiles Recent upgrades allow for BES to measure core fluctuations Time-delay estimation (TDE) technique uses cross-correlations between two poloidally separated measurements to infer velocity DIII-D n ( f ) 2 V ( f ) 2 GAM McKee PoP 2001 August 2008

32 Measured V Spectra Exhibit Signatures of Both ZMF Zonal Flows and GAMs in DIII-D Spectra indicate broad, low-frequency structure with zero measurable poloidal phase shift Consistent with low-m (m=0?) Peaks at/near zero frequency GAM also clearly observed near 15 khz ZMF zonal flow has radial correlation length comparable to underlying density fluctuations Necessary for effective shearing of turbulence Gupta et al., PRL (2006) August 2008

33 Observe Transition from ZMF-Dominated Core to GAM-Dominated Edge Velocity spectra show broad ZF spectrum for r/a < 0.8 ZMF flow Superposition of broad spectrum and GAM peak near r/a = 0.85 GAM dominates for r/a > 0.9 Consistent with theory/simulation expectations that GAM strength increases with q Increase in GAM strength with q 95 also observed (McKee et al., PPCF 2006) GAM is highly coherent, with correlation time GAM > 1 ms, two orders of magnitude larger than turbulence decorrelation turb ~ 10 s Indicates GAM is slow relative to edge turbulence timescales, and so can effectively interact with turbulence (Hahm et al., PoP 99) McKee IAEA 2006

34 Regulation of particle transport by ZF 150 Fujisawa, PPCF 2006 HIBP on CHS = 1 E B, n f (khz) f (khz) ZF (V) time (ms) 70 Time (ms) August 2008

35 Regulation of transport by ZF HIBP on JFT-2M Ido, NF 2006 GAMs DW t Time (ms) August 2008

36 Schematic of NONLINEAR drift turbulence-zonal Flow interactions Ref: Itoh APS 2005 T, n... DRIVE Drift wave turbulence ~ ~ <v x p> Transport Suppression of DW by shearing Shearing Zonal flows <v ~ ~ x v y > energy return Generation By vortex tilting Collisional flow damping SUPPRESS Nonlinear flow damping Damping by collisions Tertiary instability

37 SUMMARY of Part I Reviewed Drift Wave Picture Discussed How Waves Turbulence Introduced Zonal Flows & Summarized Interactions with Drift Turbulence In Part II Look at Basic DW Experiments Transition to Turbulence from DWs Onset of Nonlinear Energy Transfer Development of Zonal Flows & Back Reaction on Turbulence Impact on Flux vs Gradient Relations