1. Fluid vs Kinetic Models in Fusion Laboratory Plasmas ie Tokamaks Howard Wilson Department of Physics, University of York, Heslington, York

2. Outline Tokamak magnetic geometry – Some basic features Plasma turbulence – in the edge – in the core Reconnection – An “MHD” phenomenon, but you cannot get away from kinetics Plasma eruptions – early days, so an open question

3. Tokamak Magnetic Geometry Rod current ~few MA Solenoid current Toroidal component of magnetic field ~ T and toroidal current ~MA Poloidal component of magnetic field ~ T R B

4. Trapped Particles The magnetic field is weaker on the outboard side than the inboard side – particles with low component of velocity parallel to magnetic field are trapped If trapped particles perform a complete orbit before colliding, trapped particle effects are often important: points towards a kinetic model Grad-B and curvature drifts point straight up (or down) Trapped particle orbit has finite width due to drifts: called a banana orbit

5. Turbulence at the Plasma Edge The plasma near the plasma periphery is often dense and cold(ish) – collisions are frequent, so trapped particle effects are not important – the high collision frequency also means that (2-) fluid models provide a good description – fine-scale filamentary structures are well-produced by turbulence codes (at least qualitatively) Benkadda, et al

6. Turbulence bifurcation: The L-H transition As the plasma heating power exceeds a well-defined threshold, the confinement suddenly increases by a factor of 2 – This is known as the L-H transition This transition remains a mystery – It cannot be reproduced either by kinetic or fluid codes It is due to a sudden drop in the turbulent transport in the plasma edge region, leading to a steepening of the pressure gradient there radius pressure Low performance, Turbulent L-mode state

7. Turbulence bifurcation: The L-H transition As the plasma heating power exceeds a well-defined threshold, the confinement suddenly increases by a factor of 2 – This is known as the L-H transition This transition remains a mystery – It cannot be reproduced either by kinetic or fluid codes It is due to a sudden drop in the turbulent transport in the plasma edge region, leading to a steepening of the pressure gradient there radius pressure High performance, or H-mode

8. Flow shear plays a role? There is strong evidence that flow shear plays a role: We believe that the turbulence itself can drive the flow shear: so-called zonal flows – tears apart turbulent eddies, reducing turbulence correlation length These “transport barriers” can also be triggered in the core of the plasma: is there an overlap with solar phenomena here (the tachocline?) MAST data H Meyer, H-mode Workshop, 2007

9. Illustration of “zonal flows” on Jupiter: Voyager images

10. Turbulence in the hot core plasma For the linear ion-temperature-gradient (ITG) mode, a fluid model is rigorous provided one is well above threshold and the growth rate is strong However, near the threshold, ion Landau damping and finite ion Larmor radius effects are important [A.G. Peeters, et al., NF 42, 1376 (2002) Central versus edge ion temperature AUG Theory predicts ITG unstable when Consequence: central temperature is proportional to edge temperature: Some evidence for this Suggests temperature gradient is tied to marginal stability  kinetic effects are important

11. Adapted from Dimits et al, PoP 7 (2000) 969 Non-linear simulations Early gyro-fluid closure predicts non-linear diffusivity rises sharply with increasing temperature gradient  temperature gradient pinned to marginal More accurate gyro-kinetic model predicts diffusivity does not rise immediately because of “zonal flows”, but then takes off  Dimits shift 12 10 8 6 4 2 0  i L n /  i 2 v ti 0 5 10 15 20 R/L Ti Linear threshold Diffusivity rises sharply IFS-PPPL model (gyro-fluid) LLNL model (gyro-kinetic) Dimits shift Conclusion: kinetic effects are crucial for ITG turbulence But maybe it depends what your turbulence drive is

12. Transport barriers: good for confinement, but trigger damaging instabilities, called ELMs Edge localised modes, or ELMs, are triggered because of the high pressure gradient near the plasma edge – The ELM is a transient “bursty” ejection of heat and particles – Must be controlled to avoid excessive erosion – But we do not fully understand the mechanisms Ideal MHD (ballooning) theory predicts filamentary structures associated with the ELM – subsequently observed in experiment (MAST tokamak, Culham) – is there a link to solar eruptions? Theoretical prediction: filamentsExperimental observation (A Kirk)

13. Eruptions likely involve the both MHD and kinetic processes There appears to be an excellent agreement between onset of ELMs and (linear) ideal MHD – The steep gradients mean that diamagnetic effects are important, but only make a quantitative impact However, the plasma eruption does release large amounts of energy – ideal MHD cannot describe this process – hard to believe it wouldn’t be a kinetic process Possible model for energy loss: – non-linear ideal MHD (with diamagnetic effects, which influence mode structure) could predict filament sizes – Assume filament empties energy by parallel transport along field line – Still left with the duration of the ELM to model

14. Reconnection: neoclassical tearing modes Tokamaks have good confinement because the flux surfaces lie on nested tori If current flows preferentially along certain field lines, magnetic islands form The plasma is then ‘short-circuited’ across the island region As a result, the plasma pressure is flattened across the island region, and the confinement is degraded:

15. MHD or Kinetics? A bit of both We begin by defining the perturbed flux: Away from the rational surface (where a field line maps back onto itself after a finite number of turns around the torus),  is determined by the equations of ideal MHD: a second order differential equation – it predicts that  has a discontinuous derivative at r=r s – this is conventionally parameterised by  : r  rsrs  is almost constant, but has a jump in its derivative

16. Tearing Mode Theory: Ampère’s Law d 2  /dr 2 ~  0 J || (via Ampère’s law) d  /dr We consider a small “layer” around the rational surface: perturbed flux, , is approximately independent of radius, r r  r=r 2 Integrate Ampère’s law across current layer Obtained by matching to solution of ideal MHD Kinetic effects are important for the current in the layer

17. The bootstrap current drive: kinetic, but there is a fluid model Consider two adjacent flux surfaces: The apparent flow of trapped particles “kicks” passing particles through collisions: – accelerates passing particles until their collisional friction balances the collisional “kicks” – This is the bootstrap current – No pressure gradient  no bootstrap current – No trapped particles  no bootstrap current The bootstrap current perturbation can drive the island to large size High density Low density Apparent flow

18. Theory predictions from perturbed bootstrap current Experimental measurement First positive identification of NTMs on TFTR Mode initiated at finite amplitude Both indicate a role for a threshold effect NTMs were first positively identified on TFTR in the mid-90’s, and showed good agreement with theory: Discrepancy as island decays Except

19. The polarisation current: requires a kinetic treatment For islands with width ~ion orbit (banana) width: – electrons experience the local electrostatic potential – ions experience an orbit averaged electrostatic potential  the effective E  B drifts are different for the two species  a perpendicular current flows: the polarisation current The polarisation current is not divergence-free, and drives a current along the magnetic field lines via the electrons Thus, the polarisation current influences the island evolution: – a quantitative model remains elusive – if stabilising, provides a threshold island width ~ ion banana width (~1cm) – this is consistent with experiment A kinetic treatment indicates two collision frequency regimes for poln current E×BE×B J pol

20. Summary The onset of global or fast events associated with thermal particle distributions appear to be well-described by ideal MHD Fluid turbulence models may be able to reproduce features in collisional plasmas (eg the tokamak edge), but probably require 2-fluid effects Kinetic theory is well-developed for core turbulence: computational models based on gyro-kinetic theory are becoming quantitative – understanding the impact (and generation) of flow shear is an important outstanding problem – this means that one must always put in a boundary condition for the temperature at the top of the pedestal (and confinement is very sensitive to this) Some macroscopic features of reconnection may be adequately described by a fluid theory – threshold effects are almost certainly a kinetic effect – indeed, the threshold physics probably requires an understanding of how reconnection and turbulence interact…a challenging issue