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Fluid vs Kinetic Models in Fusion Laboratory Plasmas

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Fluid vs Kinetic Models

in Fusion Laboratory Plasmas

ie Tokamaks

Howard Wilson

Department 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 Geometry

Rod current ~few MA

Poloidal component of magnetic field ~ T

Toroidal component of magnetic field ~ T

B

Solenoid current

R

and toroidal current ~MA

Trapped Particles

Grad-B and curvature drifts point straight up (or down)

Trapped particle orbit has finite width due to drifts: called a banana orbit

- 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

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

Low performance,

Turbulent L-mode state

pressure

radius

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

High performance, or H-mode

pressure

radius

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

Illustration of “zonal flows” on Jupiter:

Voyager images

Turbulence in the hot core plasma

Central versus edge ion temperature

AUG

[A.G. Peeters, et al., NF 42, 1376 (2002)

- 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

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

Non-linear simulations

LLNL model (gyro-kinetic)

Dimits shift

- 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

- Conclusion: kinetic effects are crucial for ITG turbulence
- But maybe it depends what your turbulence drive is

Diffusivity rises sharply

12

10

8

6

4

2

0

IFS-PPPL model (gyro-fluid)

ciLn/ri2vti

0 5 10 15 20

R/LTi

Linear

threshold

Adapted from Dimits et al, PoP 7 (2000) 969

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),y is determined by the equations of ideal MHD: a second order differential equation
- it predicts that y has a discontinuous derivative at r=rs
- this is conventionally parameterised by D:

y

y is almost constant,

but has a jump in its

derivative

r

rs

Tearing Mode Theory: Ampère’s Law

- We consider a small “layer” around the rational surface:
- perturbed flux, y, is approximately independent of radius, r

y

r

r=r2

dy/dr

Kinetic effects are important for the current in the layer

d2y/dr2~m0J|| (via Ampère’s law)

Integrate Ampère’s law across current layer

Obtained by matching to solution of ideal MHD

The bootstrap current drive: kinetic, but there is a fluid model

High density

Low density

Apparent flow

- 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

Mode initiated at finite amplitude

Both indicate a role for a threshold effect

Discrepancy as island decays

First positive identification of NTMs on TFTR

- NTMs were first positively identified on TFTR in the mid-90’s, and showed good agreement with theory:

Theory predictions from perturbed bootstrap current

Experimental measurement

Except

The polarisation current: requires a kinetic treatment

Jpol

E×B

- 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

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