Fluid vs Kinetic Models
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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

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Fluid vs Kinetic Models in Fusion Laboratory Plasmas

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Fluid vs kinetic models in fusion laboratory plasmas

Fluid vs Kinetic Models

in Fusion Laboratory Plasmas

ie Tokamaks

Howard Wilson

Department of Physics, University of York, Heslington, York


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

Illustration of “zonal flows” on Jupiter:

Voyager images


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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)


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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:


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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


Fluid vs kinetic models in fusion laboratory plasmas

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 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


Fluid vs kinetic models in fusion laboratory plasmas

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


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