Nonlinear and selfconsistent treatment of ECRH

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Nonlinear and self-consistent treatment of ECRH

• Christos Tsironis
• Loukas Vlahos

EURATOM Association Hellenic Republic
Department of Physics Aristotle University of
Thessaloniki
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Outline

• Introduction - Motivation
• Self-consistent model for wave-particle
  interaction
• Application EC absorption in a finite tokamak
  slab
• Numerical results
• Conclusions

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Electromagnetic waves in fusion

• Auxiliary plasma heating
• Extra heating (apart from Ohmic) is required to
  burn the plasma
• Electric current drive
• Toroidal current induced by transformer ? Pulse
  operation
• Steady-state operation requires non-inductive
  current
• Plasma diagnostics
• Control instabilities, confinement, magnetic
  topology etc.

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Wave-particle interaction and ECRH

• Wave-particle interaction plays a key role in
  heating and current drive experiments (Erckmann
  and Gasparino, 1994)

Flux power density is small in modern
experiments
Linear theory (Ray tracing codes)
Quasilinear theory (Fokker-Planck codes)
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Need to consider nonlinear effects

• In cases where nonlinear effects are
  important, the validity of these theories becomes
  questionable

• Nonlinear effects already appear at power below
  the ones of the strongly nonlinear regime
  (Westerhof, 2004)
• Nonlinear kinetic modelling (Kamendje et. al.,
  2003) ? Deviations can be strong for present
  day experiments
• High RF power ? Quasilinear theory breaks down
  due to resonant islands in phase-space (Tsironis
  and Vlahos, 2005)

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Results from experiments

• Transmission measurements during EC
  propagation

• TFR ? Close to linear theory (Erckmann and
  Gasparino, 1994)
• MTX ? Deviations for high-power FEL (Nevins et.
  al., 1987)
• W7-AS ? Disagreement for intense ECRH (Laqua
  et. al., 2004)

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Nonlinear self-consistent treatment

• Self-consistent analysis of the nonlinear
  interaction of magnetized relativistic electrons
  with electromagnetic waves
• Equations of electron motions are coupled with
  the wave equation for the vector potential
• The particle motions drive the wave amplitude and
  frequency evolution through the current density

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Self-consistent model Overview

• The electromagnetic wave (?,k) propagates in
  the x-z plane at an angle ? to the uniform

• magnetic field

Vector potential

• Wave phase
• px, py, pz Integers that determine the wave
  polarization
• A0, ? depend on time ? Wave-particle coupling

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Equations of the model (1)

• Equations of electron motion
• Wave equation
• Normalizations
• A0 ? Normalized with mec2/e
• Spatial coordinates ? c/?c ? Time ? ?c-1
• Frequencies ? ?c ? Wave-vectors ? ?c/c
• p (Relativistic mechanical momentum) ? mec
• j (Current density) ? enec

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Equations of the model (2)

• One may obtain equations for A0, ?
• Substitute the vector potential in the wave
  equation
• Assume the form
  for j
• Eliminate dependence on r through f, j ?
  Average over r

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Geometry Finite tokamak slab

• Wave-particle interaction occurs in a finite
  region
• propagation ? Width corresponds to 2
  variation of the toroidal field, assuming a
  typical profile B0(1x/Rmaj)-1
• ? propagation ? Defined by the projection of
  the beam width onto the magnetic axis
• Dimensions of the slab are small ? Plasma
  parameters (B0, ne, Te) constant in the slab
  region

• The particles are initialized in the slab region
  and interact with the wave while inside the slab

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Comparison with linear theory

• Refraction index, wave-number ? Solutions of
  the cold plasma linear dispersion relation
• Linear absorption coefficient aL ? Hot plasma
  effects are taken into account (Bornatici, 1982)

• Proper comparison with linear absorption
• Follow the particles for the time tint needed by
  the beam, propagating with group velocity c, to
  cross the slab region
• Compare our results with the linear rate

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Application EC absorption in AUG

• Major-minor radius Rmaj165cm, Rmin60cm
• Magnetic field B02.5T
• Plasma density-temperature ne1013cm-3, Te1KeV
• Initial velocity distribution ? Relativistic
  Maxwellian
• Number of particles N1000000
• ECRH power ? 2nd harmonic X-mode
• Injections at angles ?700, 900 (pxpypz1)
• Beam width w2cm
• Wave power P01MW (gyrotrons), 1GW (high-power
  FEL)

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Numerical results ?700, P01MW (1)

• Wave is absorbed by the particles with a rate
  moderately smaller from what predicted by the
  linear theory
• Energy gain of the electrons due to EC wave
  absorption

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Numerical results ?700, P01MW (2)

• Wave frequency remains confined near the initial
  value
• The initial Maxwellian form of the particle
  distribution (black curve) is not strongly
  affected by the ECRH

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Numerical results ?700, P01GW

• High RF power ? Deviation from linear theory is
  larger
• The distribution differs from its initial form
  (high-energy tail)
• Consistency with energy conservation

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Numerical results ?900, P01MW

• Significant reduction in the absorption, in
  accordance with other results on the importance
  of nonlinear effects on ECRH
• Increase of perpendicular energy due to
  wave-particle interaction

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Conclusions

• Within the limits of our model, the absorption of
  the EC wave is in disagreement with the linear
  theory. This disagreement may in some cases (eg.
  using high-power EC beams) be significant.
• There is a need to reconsider the importance of
  nonlinear effects on ECRH, especially when the
  wave power increases dramatically, as it will be
  the case for the ITER experiments.
• Our current work focuses on more realistic plasma
  and wave beam geometries.