1. 1 Tokamak Drift-Wave Instabilities （ ITG ， ETG ， CTEM) ： Nonlinear Theory & Simulations X.Q.Xu Lawrence Livermore National Laboratory USA Institute of Fusion Theory and Simulations Zhejiang University 2009 Chinese Summer School on Plasma Physics Hangzhou, China July 18 – July 28, 2009

2. 2 Acknowledgments We thank Drs. J. Candy, L. Chen, P.H.Diamond, A.Dimits, D.Ernst,T. S. Hahm, G. Hammett, F. Jenko, Z.Lin, W. M. Nevins, H. Qin, and R.Waltz for fruitful physics discussions. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory in part under Contract DE-AC52-07NA27344.

3. Normalized Confinement Time H H =  E /  Empirical Fusion performance depends sensitively on confinement Sensitive dependence on turbulent confinement causes some uncertainties, but also gives opportunities for significant improvements, if methods of reducing turbulence extrapolate to larger reactor scales. Caveats: best if MHD pressure limits also improve with improved confinement. Other limits also: power load on divertor & wall, … 0 5 10 15 20 Q = Fusion Power / Heating Power 3

4. Overview of Mechanisms “Islands” –bootstrap etc.; magnetic flutter Ion Temperature Gradient Driven Mode (ITG), k  i <1 –electromagnetic with finite beta modes –Adiabatic electron model Trapped Electron Mode (TEM), k  i ≥1 –Skin depth fluctuations –Non-adiabatic electron model Electron Temperature Gradient Driven Mode (ETG), k  e 20 –electrostatic + Streamer Critical theoretical issues: –ETG must couple to larger scales a  i c/  ce  e 4

5. Multiple scales in plasma microturbulence ITG/TEM: ion scale turbulence ITG/TEM and ETG scales separated by TEM/ETG: the same type mode with peak growth at different k ┴ –TEM may smoothly connect to ETG at high k ┴ trapped electron modes ETG modes ITG modes Not shown here: - drift waves - ballooning modes 5

6. Nonlinear Saturation, Secondary Flows & Transport Nonlinear saturation due to wave coupling/cascades Turbulent Transport calculated from  =, fluxes Secondary flow generation via envelope modulation of drift- wave packet. –Zonal flow –GAM –Streamer Secondary flows in turn regulate the turbulence and transport linear nonlinear 6

7. Diffusion Semi-empirical transport models –Generally diffusion can be modeled as a random walk of steps of length  and time . If the diffusion is collisional, then  is the mean free path and  is the inverse of the collision frequency. The diffusion coefficient D can be expressed variously as  =  2  7

8. Bohm Diffusion Bohm expressions for the transport coefficients –It was first observed in 1946 by David Bohm, et al, while studying magnetic arcs for use in isotope separation. It has since been observed that many other plasmas follow this law. Bohm diffusion Scaling –turbulent transport is characterized by a radial correlation length L r =(  i a) 1/2 and by a decorrelation rate of the order of the diamagnetic frequency  *i =v ti /a (v ti being the ion thermal velocity and a the minor radius), as expected for the microinstabilities of the drift branch.  B =  i 2  ci =T/eB –the missing 1/16 in front is no cause for concern. Therefore, at least within a factor of order unity, Bohm diffusion is always greater than classical diffusion since ii <<  ci. 8

9. Gyro-Bohm Diffusion gyro-Bohm expressions for the transport coefficients –turbulent transport is characterized by a radial correlation length L r of the order of the ion gyroradius  i and by a decorrelation rate of the order of the diamagnetic frequency  *i =v ti /a (v ti being the ion thermal velocity and a the minor radius), as expected for the microinstabilities of the drift branch.  GB =  i 2  *i For drift-wave,  *i = v Ti /L ni  GB =  i 2 v Ti /L ni 9

10. Mixing length theory Developed by Prandl in the early 20th century only a rough approximation The mixing length is conceptually analogous to the concept of mean free path in thermodynamics: a fluid parcel will conserve its properties for a characteristic length  and time , before mixing with the surrounding fluid. The diffusion coefficient  can be expressed variously as   =  2  Prandtl, L. (1926). "Proc. second int. Congr. appl. Mech.". Zurich.. Temperature is conserved for a certain distance as a parcel moves across a temperature gradient. The fluctuation in temperature that the parcel experienced throughout the process is. 10

11. Simple mixing length estimates for transport  s =  s 2 v Ts /L Ts Local balance between linear drive and nonlinear damping Where  – linear growth rate, k ┴ - perpendicular wavenumber No differentiation between particle and heat transport Mixing length model yields a gyro-Bohm scaling for drift-wave turbulence Where  s  gyroradius for species s v Ts  thermal velocity for species s L Ts  equilibrium temperature gradient scale length for species s D  /k ┴ 2 11

12. Simple mixing length estimates for transport from ITG, TEM, and ETG  s =  s 2 v Ts /L Ts ITG, k  i <1, ETG, k  e 20 –electrostatic The disparity in anomalous electron and ion thermal transport is This disparity is not typically observed in expt. –Suggesting different electron and ion scale physics –Electron transport is driven by ion-scale turbulence In ITB experiments –Electron transport remains high in the absence of ion-scale turbulence  i =  i 2 v Ti /L Ti  e =  e 2 v Te /L Te  i ~ 60  e 12

13. Transport fluxes from ExB drift Particle Flux Heat Flux  s =  n s  v ExB > Q s =  P s  v ExB > 13

14. BASIC INSTABILITY PICTURE ITG-TEM-ETG G.Hammett, APS 2007 review talk 14

15. 1.Intuitive pictures of gyrokinetic turbulence, & how to reduce it analogy w/ inverted pendulum / Rayleigh-Taylor instability reduce turbulence with sheared flows, magnetic shear, … effective gravity Inverted-density fluid  Rayleigh-Taylor Instability 15

16. Stable Pendulum L M F=Mg  =(g/L) 1/2 Unstable Inverted Pendulum  = (-g/|L|) 1/2 = i(g/|L|) 1/2 = i  g L (rigid rod) Density-stratified Fluid stable  =(g/L) 1/2  =exp(-y/L) Max growth rate  =(g/L) 1/2  =exp(y/L) Inverted-density fluid  Rayleigh-Taylor Instability Instability 16

17. “Bad Curvature” instability in plasmas  Inverted Pendulum / Rayleigh-Taylor Instability Top view of toroidal plasma: plasma = heavy fluid B = “light fluid” g eff = centrifugal force R Growth rate: Similar instability mechanism in MHD & drift/microinstabilities 1/L =  p/p in MHD,  combination of  n &  T in microinstabilities. 17

18. The Secret for Stabilizing Bad-Curvature Instabilities Twist in B carries plasma from bad curvature region to good curvature region: UnstableStable Similar to how twirling a honey dipper can prevent honey from dripping. 18

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21. Rosenbluth-Longmire picture 21

22. Rosenbluth-Longmire picture Can repeat this analysis on the good curvature side & find it is stable? 22

23. Gyrokinetic Theory developments Development of gyrokinetic equations one of the triumphs of high- power theoretical plasma physics and asymptotic analysis Key advance: Frieman & Chen show nonlinear version of gyrokinetics possible Other advances: Hamiltonian/Lagrangian derivations, insure conservation properties, easier to go to higher order GK ordering allows capture of drift/micro-instabilities & much of MHD at just order  & not  2 Guided by expts.,  wave scattering, physics insights 23

24. Two length scale separation, locally flatten gradients, quadratic nonlinearities 24

25. Why gyroaverage? Reduce the dimensions to 5D 25

26. Main Comprehensive Gyrokinetic Codes A partial list of  F codes –GTC (Lin & Lee), GTS(Wang & Lee) PIC, global, USA –GS2 (Dorland & Kotschenreuther) continuum, flux-tube, USA –GENE (Jenko, Garching) continuum, flux-tube, Germany –GYRO (Candy & Waltz) continuum, global, USA –GEM (Parker and Chen)  F PIC, global, USA –All of these codes include: toroidal geometry, general axisymmetric plasma shapes, multiple species, trapped and passing non-adiabatic electrons, electromagnetic fluctuations, collision operators, equilibrium scale ExB shear flow, GS2 & GENE use the  *  0 limit local flux-tube (equivalent to thin annulus) A partial list of Full-F global, under developments & to be completed –TEMPEST (Xu &ESL), continuum, USA –XGC (Chang & CPES), PIC, USA –GT5D (Idomura et al), continuum, Japan –GYSELA (Garbet et al), semi-Lagrangian, France These gyrokinetic codes use a number of advanced algorithms 26

27. ITG GYROKINETIC SIMULATION RESULTS Dimits et al., Phys. Plasmas, 7 969 ( 2000). Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang, and R. B. White, Science 281, 1835 (1998). Z. Lin et al., Phys. Rev. Lett. 83, 3645 (1999). Z. Lin et al., Phys. Rev. Lett. 88, 195004 (2002). 27

28. Dimits et al., Phys. Plasmas, 7 969 ( 2000). Ion Temperature Gradient Driven Modes (ITG) In a toroidal system the instability is mainly driven by the magnetic field curvature and the ion temperature gradient ITG is of interest due to the successful interpretations of various experimental trends – related to the observed levels of turbulent transport in tokamak plasmas Theory/Simulations typically with adiabatic e - model 28

29. Dimits et al., Phys. Plasmas, 7 969 ( 2000). Parameterization dependence of the Dimits upshift? ITG drives strong heat transport to force plasma near marginal stability Explicit threshold of the Linear ITG instability, R/L ti,crit Simulations find that the threshold of  i is different from that of linear growth. It is termed “Dimits upshift” 29

30. Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang, and R. B. White, Science 281, 1835 (1998). Zonal flow is found to regulate the turbulence and to play a key role in the Dimits upshift 30

31. Z. Lin et al., Phys. Rev. Lett. 83, 3645 (1999). Ion-ion collision is found to affect the ion heat transport via zonal flow damping 31

32. Zonal flow Zonal flow is a meteorological term the general flow pattern is west to east along the Earth's latitude lines (as opposed to meridional flow). Meridional flow the general flow pattern is cross the latitude lines at a sharp angle 32

33. Zonal flow In toroidally confined fusion plasma experiments the term zonal flow means a plasma flow within a magnetic surface primarily in the poloidal direction. Zonal flows in the toroidal plasma context are further characterized by –being localized in their radial extent transverse to the magnetic surfaces (in contrast to global plasma rotation), –having little or no variation in either the poloidal or toroidal direction -- they are m = n = 0 modes (where and m and n are the poloidal and toroidal mode numbers, respectively), –having zero real frequency when analyzed by linearization around an unperturbed toroidal equilibrium state (in contrast to the geodesic acoustic mode branch, which has finite frequency). They arise via a self-organization phenomenon driven by low-frequency drift-type modes, in which energy is transferred to longer wavelengths by modulational instability or turbulent inverse cascade. From Wikipedia, the free encyclopedia 33

34. 34  =  2 

35. Z. Lin et al., Phys. Rev. Lett. 88, 195004 (2002). Transition from Bohm to gyro-Bohm Scaling 35

36. Global code approaches local flux-tube limit as  *  0 Candy et al. PoP 04 36

37. Moderate amount of turbulence spreading occurs in some cases Waltz, Candy, Petty 2006 PoP 13, 072304 DIII-D L-mode amount of spreading: ~ 0.1 a ~ 2-5 radial correlation lengths ~ 20-50  i see also Hahm et al. 2004, Lin et al. 2002, Garbet et al., Newman, Xu 37

38. Successful Benchmarks of Independent Gyrokinetic Codes Good agreement in  (+/- 10% on long time average t > 1000 a/c s ) between 3 continuum and 2 PIC codes Nevins et al. 2007 Correlation functions agree well 38

39. Gyro movie Evolution of potential fluctuations in a plasma very similar to DIII-D 101381/101391. Simulation is centered at r/a=0.6. Note the strong equilibrium sheared rotation, which leads to a strong reduction in transport. This landmark simulation from 2002 includes kinetic electrons at finite-beta, along with the equilibrium ExB variation. 39

40. TEM GYROKINETIC SIMULATION RESULTS Ernst et al., APS invited talk 2008 Ernst et al., PoP 2004, 2007, 2009 Lang, Chen, Parker, PoP 2007, 2008 Lin et al, TTF talk 2009 40

41. Trapped Electron Modes (TEM) In typical ITG simulations, e - is adiabatic Adding trapped electron effects introduce a new unstable root, the TEM root, –Due to resonances between the mode eigenfrequency and the orbit-time-average magnetic drift frequency (precession frequency) –in addition to increasing the ITG growth rate ITG has a real frequency in the ion diamagnetic drift direction (negative) TEM has a real frequency in the electron diamagnetic drift direction (positive) 41

42. At Very Steep Density Gradients, Mode Structure Changes 42

43. At Very Steep Density Gradients, Mode Structure Changes 43

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50. GTC simulations show CTEM saturation & eddy size regulated by Zonal flow CTEM: collisionless trapped electron modes, ei =0 Linear streamers mostly broken by zonal flows Transport much higher if zonal flows removed Significant tails in correlation function indicate existence of meso-scale eddies, evidences of non-diffusive transport Xiao and Lin, submitted to PRL 2009 50

51. ETG GYROKINETIC SIMULATION RESULTS T.Görler and F. Jenko, PRL 100, 185002 (2008) T. Görler, 1 st September 2008, TTF Waltz, Candy and Fahey, PHYSICS OF PLASMAS 14, 056116 2007 Lin et al, PoP 2005 Lin et al, PRL 2007 51

52. ETG fluctuations (k   i > 1) may account for significant fraction of transport in some plasmas Simple scaling from ITG to ETG:  itg ~ C itg  i 2 v ti /L  etg ~ C itg  e 2 v te /L ~  itg /60 But Dorland & Jenko (2000) showed ETG turbulence larger because: perpendicular adiabatic ions for ETG gives more shielding of zonal electric fields than does parallel adiabatic electrons for ITG. Candy showed ETG will be reduced by kinetic ions, more so if strong ITG turbulence ITG can be weak near marginal stability w/ ExB shear. Görler and Jenko shows ETG / high-k TEM may still be important in some cases. Dorland et al., Phys. Rev. Lett. 85, 5579 (2000). 52

53. Electron Temperature Gradient Mode (ETG) Electron temperature gradient modes are driven by the circulating electrons TEM mode are driven by trapped electrons ETG and TEM mode are the same family of linear electron modes with growth rates peak at different k 53

54. Waltz, Candy, and Fahey, Phys. Plasmas 14, 056116 2007 54

55. Waltz, Candy, and Fahey, Phys. Plasmas 14, 056116 2007 55

56. Largest GYRO simulations used to study interaction of ITG & ETG Turbulence 1280  e x 1280  e x 20 parallel pts/orbit x 8 energies x 16 v || /v electrons + kinetic ions, m i /m e = 20 2 - 30 2 5 days on DOE/ORNL Cray X1E w/ 720 Multi-Streaming Processors Candy, Waltz, et al. JPSC 2007 ETG w/ kinetic ions R/L Ti =0 ETG+ITG R/L Ti =6.9 56

57. Gyro movie Electron density fluctuations in a simulation coupling ITG/TEM instabilities with ETG instabilities. In this case, ITG modes are linearly unstable. 57

58. ETG + kinetic ion GYRO simulation movie large box on right: full simulation domain, 1280  e x 1280  e = 64  i x 64  i small box on lower left: zoom in on a 64  e x 64  e patch http://fusion.gat.com/THEORY/images/1/1f/ETG-ki.mpghttp://fusion.gat.com/THEORY/images/1/1f/ETG-ki.mpg from http://fusion.gat.com/theory/Gyromovies Candy & Waltz 58

59. Gyro movie Electron density fluctuations in a simulation coupling ITG/TEM instabilities with ETG instabilities. In this case, ITG modes are linearly stable. 59

60. ExB shear can affect even ETG GYRO ETG-ki sim. turbulent e flux ~ 1 MW (NSTX expt ~ 2 MW) ExB shearing rate varied from 2X to 1/4 experimental rate. Eddies grow longer (and wider) as shearing rate is reduced. 2X experimental ExB rate 1/2X 1/4X Mikkelsen, NSTX Radial domain ~400  e. 60

61. GENE: a nonlinear gyrokinetic Vlasov code –fully gyrokinetic electrons and ions –fully electromagnetic fluctuations –realistic collision operators –flux tube volume in general toroidal geometry here: electrostatic, collisionless, s-α-model equilibrium reduced mass ratio: from m i /m e = 1836 to m i /m e = 400 –computational effort lowered by one order of magnitude: T CPU ~ (m i /m e ) 3/2 –still more than 100,000 CPUh / simulation Background: Nonlinear gyrokinetic simulations T. Görler, 1 st September 2008 61

62. Physical parameters  Cyclone-like – except for profile gradients Numerical parameters:  perpendicular box size: 64 x 64 ion gyroradii  perpendicular resolution: 1.5 x 3 electron gyroradii Unconventional way of plotting transport spectra conventional plot ‘area preserving’ plot Simulation parameters and plotting conventions 64 ρ i T. Görler, 1 st September 2008 62

63. A.) R/L T i = 6.9, R/L T e = 6.9, R/L n = 2.2 B.) R/L T i = 5.5, R/L T e = 6.9, R/L n = 0.0 C.) R/L T i = 0.0, R/L T e = 6.9, R/L n = 0.0 T. Görler, 1 st September 2008 63

64. Dominant and subdominant modes for case A 64

65. Case A: R/L T i =6.9,R/L T e = 6.9, R/ L n = 2.2 – strong ITG T. Görler, 1 st September 2008 ions electrons ion heat transport – localized at low-k – unrealistically high: Electron heat transport – ETG nonlinearly present – ‘high-k’ contribution to electron heat transport ~10% 65

66. Case A: R/L T i =6.9,R/L T e = 6.9, R/ L n = 2.2 – strong ITG T. Görler, 1 st September 2008 small-scale streamers are subject to large-scale vortex shearing Isotropic spectrum 66

67. A.) R/L T i = 6.9, R/L T e = 6.9, R/L n = 2.2 B.) R/L T i = 5.5, R/L T e = 6.9, R/L n = 0.0 C.) R/L T i = 0.0, R/L T e = 6.9, R/L n = 0.0 T. Görler, 1 st September 2008 67

68. ions ion heat transport still dominat but reduced by one order of magnitude more than 40% of the electron heat transport is in the ‘high-k’ regime! Scale separation between ion and electron heat transport electrons T. Görler, 1 st September 2008 Case B: R/L T i =5.5,R/L T e = 6.9, R/ L n = 0.0 – weak ITG 68

69. T. Görler, 1 st September 2008 Case B: R/L T i =5.5,R/L T e = 6.9, R/ L n = 0.0 – weak ITG Thermal fluxes (2D): Particle flux (2D): ions electrons Particle flux similar to case A (not shown) negative (pinch) restricted to low-k due to increasing ion “adiabaticity” at higher wavenumbers → high-k transport anisotropy which has been absent in case A 69

70. A.) R/L T i = 6.9, R/L T e = 6.9, R/L n = 2.2 B.) R/L T i = 5.5, R/L T e = 6.9, R/L n = 0.0 C.) R/L T i = 0.0, R/L T e = 6.9, R/L n = 0.0 T. Görler, 1 st September 2008 70

71. T. Görler, 1 st September 2008 Case C: R/L T i =0.0,R/L T e = 6.9, R/ L n = 0.0 – no ITG Only TEM and ETG modes unstable (dominant electron heating, high beta values, substantial equilibrium ExB shear, ITBs) More than 50% of the electron heat transport is in the ‘high-k’ regime! ETG transport level ( ) is in line with pure ETG simulations 71

72. ETG streamers and TEM vortices coexist Anisotropic spectrum TEM streamer ETG streamer T. Görler, 1 st September 2008 Case C: R/L T i =0.0,R/L T e = 6.9, R/ L n = 0.0 – no ITG 72

73. T. Görler, 1 st September 2008 Case C: R/L T i =0.0,R/L T e = 6.9, R/ L n = 0.0 – no ITG Thermal fluxes (2D): Particle flux (2D): small & mainly positive still restricted to low-k Ion heat flux negligible Transport fraction: – k y ρ s >1: ~50% – k x ρ s >1: ~30% → high-k transport anisotropic 73

74. Comparison: Density spectra ((x,z,t) average) T. Görler, 1 st September 2008 (Linearly) Pure ITG, TEM and ETG simulation results Multiscale simulation results A second ‘knee’ is observed in multiscale simulations Power laws in pure and mixed turbulence sims: 2 ~k y -α with α~3.5 Disagreement with experimental results (α~6)? 74

75. Comparison: Density spectra (k x =0, (z,t) avg.) T. Görler, 1 st September 2008 Some diagnostics detect e.g. k x ≈0 contribution; asymmetry important! Exp. Results (P.Hennequin et al., PPCF 46, B121) Power laws steeper at k x =0, closer to experiment Similarities between blue (TEM/ETG) curve and exp. results High power law exponent (case C: α~5) does not imply negligible transport contribution 75

76. Comparison: Frequency spectra T. Görler, 1 st September 2008 Case ACase B Case C Nonlinear and linear frequencies match over a wide range except for transition regime from dominant ITG to TEM/ETG Phase velocity on the order of v ph < 5 c s ρ s /R ~ 76

77. Linear and nonlinear cross phases R/L T i = 0.0 R/L T i = 5.5R/L T i = 6.9 Linear Nonlinear nonlinear spread of ITG features T. Görler, 1 st September 2008 77

78. If ETG modes are unstable, – there tends to be a scale separation between ion and electron heat transport (the latter can exhibit substantial or even dominant high-k contributions) [T. Görler and F. Jenko, PRL 100, 185002 (2008)] discharges with dominant electron heating, high beta, large equilibrium ExB shear residual electron heat fluxes in transport barriers – density spectra tend to be anisotropic at higher k and may exhibit a flat region or modified power laws at k y ρ e ~0.15-0.25 (k y ρ s ~9-15 for D plasmas) Linear features (like cross phases or frequencies) tend to survive in the nonlinear simulations; deviations most pronounced in mode-transitional regimes Summary 78

79. Streamers Streamers: Radially extended eddies –k r =0, n=m=0, or k r << k  –Increased correlation length can lead to increased transport Origin – remnants of linear modes (Jenko, Dorland,…) –Radial streamers, which represent ExB velocity fields, are generated by nonlinear toroidal coupling (Lin). –Nonlinear driven convective cells (Champeaux, Diamond, …) Stability –Typical scale of streamers? ITG turbulence acting like Zonal flow for ETG to break up the radial streamers and reduce/suppress the ETG transport (Candy &Waltz) Dorland et al., Phys. Rev. Lett. 85, 5579 (2000). Lin et al, PoP 2005  =  2  79

80. Differences between ITG and ETG ITG –Electrons are Boltzmann due to k || v || >>  ETG – Ions are Boltzmann due to k ┴  i >>  ITG turbulence acting like Zonal flow for ETG to break up the radial streamers and reduce/suppress the ETG transport. 80

81. 81 Problems Using mixing length model, derive ETG thermal transport scaling and compare with simulation results in Dorland et al., Phys. Rev. Lett. 85, 5579 (2000).