1. An overview of turbulent transport in tokamaks
Michael Barnes
Rudolf Peierls Centre for Theoretical Physics
University of Oxford

2. Turbulence and confinement
What determines particle, momentum, and energy transport in magnetically confined plasmas? (spoiler: it’s microturbulence)
How big is turbulent transport and can we reduce it? (practical MCF question)
What are the properties of microturbulence in (nearly) collisionless, magnetized plasma?(fundamental physics question, interesting in laboratory/space/astro plasmas)

3. Turbulence and transport in hot, magnetized plasmas

4. Turbulence and transport in hot, magnetized plasmas

5. Toroidal drift instability (ITG/ETG)

6. Toroidal drift instability (ITG/ETG)

7. Toroidal drift instability (ITG/ETG)+ + + vE E − − − vE E + + + vE E − − −

8. Good curvature vs. bad curvature

9. Good curvature vs. bad curvature

10. Turbulence stable on inside+ + + vE E − − − vE E + + + vE E − − −

11. Instability has a critical gradient
Growth rate of instability in bad curvature region increases with magnitude of temperature gradient:
Plasma in bad curvature region is swept along field lines into good curvature region, making a competition between stabilization and destabilization. Expect net instability when:
This implies a critical temperature gradient necessary to sustain instability:

12. What wavelengths are unstable?
Instability depends on charge separation via magnetic drifts
Growth rate of instability related to rate at which charge separation occurs:
If gyroradius large compared to perturbation scale, reduces instability drive

13. What wavelengths are unstable?+ + + E − − − E + + + E − − −

14. Turbulent fluctuation amplitude small
Density (n)
Eddy size ~ ρ L
Position

15. Estimate for turbulent transport
Random walk step size ~ eddy size ~ gyration radius
Average time between steps ~ eddy turnover time
( )
Combine with diffusion equation to get estimate for energy confinement time:
105
10-5 s

16. Estimate for turbulent transport
Random walk step size ~ eddy size ~ gyration radius
Average time between steps ~ eddy turnover time
( )
Combine with diffusion equation to get estimate for energy confinement time:

17. How good must the confinement be?

18. Kinetic corrections
So far, picture has been essentially fluid, even though collisional mean free path is long; what about kinetic effects?
Not all particles travel at sound speed along field
Significant number of particles travel slower
Some particles are trapped in bad curvature region due to magnetic mirroring

19. Turbulence determines plasma profiles
Turbulence diffusion coefficient depends strongly on temperature:
Core
Diffusion coefficient large in core where temperature is high, making it difficult to go above critical temperature gradient
Energy input
Edge
Edge temperature is very important!
Critical T’
T’/T

20. Range of scales Time scale (seconds) 10-12 10-10 10-8 10-6 10-4 10-2 1
Gyromotion
Turbulence
Mean profiles
Collisions
Space scale (meters)
10-5
10-3
10-1 10
103
Gyromotion
Mean Profiles
Turbulence
Collisions

21. Formation of small scales in v-space

22. Formation of small scales in v-space

23. Formation of small scales in v-space
Drift velocity =
Particles with Larmor orbits separated by turbulence wavelength ‘see’ different averaged potential
Drift velocities decorrelated, thus phase mixing
Schekochihin et al., PPCF 2008

24. Kinetic description of dynamics

25. Numerical expense (brute force)
Time scale (seconds)
10-12
10-10
10-8
10-6
10-4
10-2 1
Temporal grid: ~1013 time steps
Collisions
Space scale (meters)
10-5
10-3
10-1 10
103
Spatial grid: ~106 grid points x 3-D = 1018 grid points
Velocity grid: ~10 grid points x 3-D = 103 grid points
Total: ~1034 total grid points

26. Gyrokinetic description of dynamics
Average over fast gyro-motion and follow ‘guiding center’ position
Eliminates fast time scale and gyro-angle variable (6-D  5-D)

27. Numerical expense (gyrokinetics)
Time scale (seconds)
10-12
10-10
10-8
10-6
10-4
10-2 1
Temporal grid: ~109 time steps
gyrofrequency
Space scale (meters)
10-5
10-3
10-1 10
103
Perpendicular spatial grid: ~106 grid points x 2-D = 1012 grid points
Parallel spatial grid: ~10 grid points x 1-D = 10 grid points
Velocity grid: ~10 grid points x 2-D v-space = 102 grid points
Total: ~1024 total grid points (1010 savings)

28. Multi-scale gyrokinetics
Decompose f into mean and fluctuating components:
Mean varies perpendicular to mean field on system size while fluctuations vary on scale of gyro-radius:
Fluctuations are anisotropic with respect to the mean field:
=> Turbulent fluctuations are low amplitude:
=> Mean profile evolution slow compared to turbulence:

29. Gyrokinetic-Poisson system
Gyrokinetic equation:
Quasineutrality:
Equilibrium:

30. Multi-scale gyrokinetics
Turbulent fluctuations calculated in small regions of fine space-time grid embedded in coarse grid for mean quantities
GS2
Trinity

31. Numerical expense (multi-scale GK)
Time scale (seconds)
10-12
10-10
10-8
10-6
10-4
10-2 1
Temporal grid: ~105 time steps
mean-fluctuation
separation
Space scale (meters)
10-5
10-3
10-1 10
103
Perpendicular spatial grid: ~105 grid points x 2-D = 1010 grid points
Parallel spatial grid: ~10 grid points x 1-D = 10 grid points
Velocity grid: ~10 grid points x 2-D v-space = 102 grid points
Total: ~1018 total grid points (106 savings)

32. Simulating gyrokinetic turbulence

33. Simulating gyrokinetic turbulencev
v||

34. Can we use the machinery just developed (gyrokinetics) to determine basic turbulence properties?

35. Turbulent heating
Assume turbulent plasma dominated by electrons and hydrogenic ions
How much are passive minority ions heated by turbulence?
Gyrokinetics restricts us to isotropic heating because magnetic moment is conserved
Turbulent heating mechanism is Joule heating:

36. Cartoon of mass-dependent fluctuations
Acceleration =
heavy ions F
acceleration
light ions
v||

37. Cartoon of mass-dependent fluctuations
heavy ions F
acceleration
light ions
v||

38. Heating from GK simulation
Simple picture consistent with GK simulations and solar wind observations
Minority ion temperatures in solar wind
Heating from GK simulation
Schmidt et al., Geophys. Res. Lett. (1980).
Barnes et al. PRL (2012)

39. Fundamental properties of GK turbulence
Take GK and make a few simple conjectures:
Isotropy in plane perpendicular to B-field
Position and velocity space scales linked
Parallel streaming time and nonlinear turnover time comparable at all scales (critical balance)
Parallel length at outer scale set by system size (connection length)

40. Smooth, isotropic v-space (kρ < 1)
Assume
Quasineutrality:

41. Definition of nonlinear time scale:
Critical balance
Physical idea: two points along field correlated only if information propagates between them before turbulence decorrelated in perpendicular plane
Definition of nonlinear time scale:
Critical balance:

42. Outer scale

43. Outer scale Injection rate comparable to nonlinear decorrelation time:
Combine with critical balance:

44. Outer scale Take parallel length scale to be connection length
Connection length = distance along field line from outside to inside ~ qR
q = number of toroidal transits per poloidal transit. measures pitch of field line

45. Outer scale Take parallel length scale to be connection length
With and , can solve for and
and

46. Heat flux
and
Assume most energy contained in outer scale and use diffusive estimate for heat flux:
with

47. Turbulence scaling tests
Barnes et al., PRL 2011

48. Inertial range

49. Inertial range (kρ < 1)
Flux of free energy (nonlinear invariant) scale-independent in inertial range:
Free energy:

50. Matching
Match solution from outer and inertial scales

51. Inertial range (kρ << 1)
Flux of free energy (nonlinear invariant) scale-independent in inertial range:
Free energy:

52. Inertial range (kρ << 1)
Convert expression for into 1D spectrum
Use critical balance with to relate and

53. Inertial range spectra
Barnes et al., PRL 2011

54. Correlation function:
Critical balance test
Correlation function:
Barnes et al., PRL 2011

55. Inertial range critical balance
Barnes et al., PRL 2011

56. Turbulence properties
Barnes et al., PRL 2011
Schekochihin et al., ApJ 2009

57. Dissipation scale
Dissipation range

58. Dissipation scale
At dissipation scale, dissipation rate comparable to nonlinear decorrelation rate
Decorrelation in v-space due to finite Larmor radius