Fluid vs Kinetic Models in Fusion Laboratory Plasmas

ie Tokamaks

Howard WilsonDepartment of Physics, University of York, Heslington, York

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

Tokamak Magnetic GeometryRod current ~few MA

Solenoid current

Toroidal component of magnetic field ~ Tand toroidal current ~MA

Poloidal component of magnetic field ~ T

R

B

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

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

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

pres

sure

Low performance, Turbulent L-mode state

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

pres

sure

High performance, or H-mode

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 dataH Meyer, H-mode Workshop, 2007

Illustration of “zonal flows” on Jupiter:

Voyager images

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

4~TcL

R

dr

dT

T

R

• Consequence: central temperature is proportional to edge temperature:

• Some evidence for this

• Suggests temperature gradient is tied to marginal stability kinetic effects are important

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

iL n

/i2 v

ti

0 5 10 15 20R/LTi

Linearthreshold

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

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: filaments Experimental observation (A Kirk)

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

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:

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=rs

– this is conventionally parameterised by :

lrrlrr ssdr

d

dr

d

l

1

0

lim

r

rs

is almost constant,but has a jump in itsderivative

rRm

BmB r~

cos~

Tearing Mode Theory: Ampère’s Law

d2/dr2~0J|| (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=r2

Integrate Ampère’s law across current layer

drJdr

d rr

rr

||

2

2

Obtained by matching to solution of ideal MHD

Kinetic effects are important for the current in the layer

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

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

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 EB 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×B

Jpol

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