Lecture Series in Energetic Particle Physics of Fusion Plasmas

1
Lecture Series in Energetic Particle Physics of
Fusion Plasmas

• Guoyong Fu
• Princeton Plasma Physics Laboratory
• Princeton University
• Princeton, NJ 08543, USA

IFTS, Zhejiang University, Hangzhou, China, Jan.
3-8, 2007
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A series of 5 lectures

• (1) Overview of Energetic Particle Physics in
  Tokamaks (today)
• (2) Tokamak equilibrium, shear Alfven wave
  equation, Alfven eigenmodes (Jan. 4)
• (3) Linear stability of energetic
  particle-driven modes (Jan. 5)
• (4) Nonlinear dynamics of energetic
  particle-driven modes (Jan. 6)
• (5) Summary and future direction for research in
  energetic particle physics (Jan. 8)

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Overview of Energetic Particle Physics in Tokamaks

• Tokamak basics
• Roles of energetic particles in fusion plasmas
• Single particle confinement
• Alfven continuum and shear Alfven eigenmodes
• Energetic particle-driven collective
  instabilities discrete AE and EPM
• Nonlinear dynamics single mode saturation and
  multi-mode effects

4
Tokamak basics

• Both fields are necessary to confine charged
  particles or plasmas
• Safety factor q
• Particle orbits trapped particles and
  circulating particles, banana orbit, bounce
  frequency, drift frequency
• Neutral beam heating, RF wave heating and fusion
  alpha particles.

5
Roles of energetic particles in fusion plasmas

• Heat plasmas via Coulomb collision
• Stabilize MHD modes
• Destabilize shear Alfven waves via wave-particle
  resonance
• Energetic particle redistribution/loss can affect
  thermal plasma confinement, degrade plasma
  heating, and damage reactor wall

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Single Particle Confinement

• For an axi-symmetric torus, particles are
  confined as long as orbit width is not too large.
  (conservation of toroidal angular momentum.)
• Energetic particles slow down due to collisions
  with electrons and ions and heat thermal
  particles. For typical parameters, energetic
  particles mainly heat electrons.
• Toroidal field ripple (due to discrete coils) can
  induce stochastic diffusion.
• Symmetry-breaking MHD modes can also cause
  energetic particle anomalous transport.

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Shear Alfven spectrum, continuum damping, and
discrete modes

• Shear Alfven wave dispersion relation
• Continuum spectrum
• Initial perturbation decays
  due to phase mixing at time scale of
• Driven perturbation at w is resonantly absorbed
  at ? continuum damping
• Phase mixing and resonant absorption has exact
  analogy with Landau damping for Vlasov plasma.

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Discrete Alfven Eigenmodes can exist near
continuum accumulation point due to small effects
such as toroidicity, shaping, magnetic shear, and
energetic particle effects.
Coupling of m and mk modes breaks degeneracy of
Alfven continuum K1 coupling is
induced by toroidicity K2 coupling is induced by
elongation K3 coupling is induced by
triangularity.
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Discrete Alfven Eigenmodes versus Energetic
Particle Modes

• Discrete Alfven Eigenmodes (AE)
• Mode frequencies located outside Alfven
  continuum (e.g., inside gaps)
• Modes exist in the MHD limit
• energetic particle effects are often
  perturbative.
• Energetic Particle Modes (EPM)
• Mode frequencies located inside Alfven continuum
  and determined by energetic particle dynamics
• Energetic effects are non-perturbative
• Requires sufficient energetic particle drive to
  overcome continuum damping.

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Shear Alfven Equation

• Assume low-beta, large aspect ratio, shear Alfven
  wave equation can be written as

G.Y. Fu and H.L. Berk, Phys. Plasmas 13,052502
(2006)
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Shear Alfven Eigenmodes

• Cylindrical limit ? Global Alfven Eigenmodes
• Toroidal coupling ? TAE and Reversed shear Alfven
  eigenmodes
• Elongation ? EAE and Reversed shear Alfven
  eigenmodes
• Triangularity ? NAE
• FLR effects?KTAE

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GAE can exist below shear Alfven continuum due to
magnetic shear
wA(r)
U
wGAE
r
rmin
r
rmin
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Toroidal Alfven Eigenmode (TAE) can exist
inside continuum gap
TAE mode frequencies are located inside the
toroidcity-induced Alfven gaps TAE modes peak at
the gaps with two dominating poloidal harmonics.
C.Z. Cheng, L. Chen and M.S. Chance 1985, Ann.
Phys. (N.Y.) 161, 21
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Reversed shear Alfven eigenmode (RSAE) can exist
above maximum of Alfven continuum at qqmin
U
q
wA
wRSAE
r
rmin
r
rmin
r
rmin
w (n-m/qmin)/R
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Linear Stability

• Energetic particle destabilization mechanism
• Kinetic/MHD hybrid model
• TAE stability energetic particle drive and
  dampings
• EPM stability fishbone mode
• Summary

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Destabilize shear Alfven waves via wave-particle
resonance

• Destabilization mechanism (universal drive)
• Wave particle resonance at
• For the right phase, particle will lose energy
  going outward and gaining energy going inward. As
  a result, particles will lose energy to waves.
• Energetic particle drive

Spatial gradient drive
Landau damping Due to velocity space gradient
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Kinetic/MHD Hybrid Model
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Quadratic form
G.Y. Fu et al. Phys. Fluids B5, 4040 (1993)
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Drift-kinetic Equation for Energetic Particle
Response
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Perturbative Calculation of Energetic Particle
Drive
G.Y.Fu and J.W. Van Dam, Phys. Fluids B1, 1949
(1989) R. Betti et al, Phys. Fluids B4, 1465
(1992).
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Dampings of TAE

• Ion Landau damping
• Electron Landau damping
• Continuum damping
• Collisional damping
• radiative damping due to thermal ion gyroradius

G.Y.Fu and J.W. Van Dam, Phys. Fluids B1, 1949
(1989) R. Betti et al, Phys. Fluids B4, 1465
(1992). F. Zonca and L. Chen 1992, Phys. Rev.
Lett. 68, 592 M.N. Rosenbluth, H.L. Berk, J.W.
Van Dam and D.M. Lindberg 1992, Phys. Rev. Lett.
68, 596 R.R. Mett and S.M. Mahajan 1992, Phys.
Fluids B 4, 2885
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Fishbone dispersion relation
L. Chen, R.B. White and M.N. Rosenbluth 1984,
Phys. Rev. Lett. 52, 1122
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Nonlinear dynamics single mode saturation

• Saturation mechanism
• Wave particle trapping leading to flattening
  of distribution function and mode saturation
• Collisions tend to restore the original unstable
  distribution. Balance of nonlinear flattening and
  collisional restoration leads to mode saturation.

H.L. Berk and B.N. Breizman 1990, Phys. Fluids B
2, 2235
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H.L. Berk et al, Phys. Plasmas 2, 3007 (1995).
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Transition from steady state saturation to
explosive nonlinear regime
B.N. Breizman et al Phys. Plasmas 4, 1559 (1997).
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Hole-clump creation and frequency chirping

• For near stability threshold and small collision
  frequency, hole-clump will be created due to
  steepening of distribution function near the
  boundary of flattening region.
• As hole and clump moves up and down in the phase
  space of distribution function, the mode
  frequency also moves up and down.

H.L. Berk et al., Phys. Plasma 6, 3102 (1999).
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Saturation due to mode-mode coupling

• Fluid nonlinearity induces n0 perturbations
  which lead to equilibrium modification, narrowing
  of continuum gaps and enhancement of mode
  damping.
• D.A. Spong, B.A. Carreras and C.L. Hedrick 1994,
  Phys. Plasmas 1, 1503
• F. Zonca, F. Romanelli, G. Vlad and C. Kar 1995,
  Phys. Rev. Lett. 74, 698
• L. Chen, F. Zonca, R.A. Santoro and G. Hu 1998,
  Plasma Phys. Control.
• Fusion 40, 1823
• At high-n, mode-mode coupling leads to mode
  cascade to lower frequencies via ion Compton
  scattering. As a result, modes saturate due to
  larger effective damping.

T.S. Hahm and L. Chen 1995, Phys. Rev. Lett. 74,
266
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.
Multiple unstable modes can lead to resonance
overlap and stochastic diffusion of energetic
particles
H.L. Berk et al, Phys. Plasmas 2, 3007 (1995).
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Nonlinear Hybrid Simulation of Fishbone
instability

• Particle/MHD hybrid model
• Use M3D code
• Observed dynamic distribution flattening as mode
  frequency decreases.

G.Y. Fu et al, Phys. Plasmas 13, 052517 (2006)
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M3D XMHD Model
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Simulation of fishbone shows distribution
fattening and strong frequency chirping
distribution
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Summary I Discrete Alfven Eigenmodes

• Mode coupling induces gaps in shear Alfven
  continuum spectrum.
• Discrete Alfven eigenmodes can usually exist near
  Alfven continuum accumulation point (inside gaps,
  near continuum minimum or maximum).
• Existence of Alfven eigenmodes are due to small
  effects such as magnetic shear, toroidicity,
  elongation, and non-resonant energetic particle
  effects.

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Summary II linear stability

• For discrete modes such as TAE, the stability can
  usually be calculated perturbatively. For EPM, a
  non-perturbative treatment is needed.
• For TAE, there are a variety of damping
  mechanisms. For instability, the energetic
  particle drive must overcome the sum of all
  dampings.
• For EPM to be unstable, the energetic particle
  drive must overcome continuum damping.

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Summary III nonlinear dynamics

• Single mode saturates due to wave-particle
  trapping or distribution flattening.
• Collisions tend to restore original unstable
  distribution.
• Near stability threshold, nonlinear evolution can
  be explosive when collision is sufficiently weak
  and result in hole-clump formation.
• Mode-mode coupling can enhance damping and induce
  mode saturation.
• Multiple modes can cause resonance overlap and
  enhance particle loss.

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Important Energetic Particle Issues

• Linear Stability basic mechanisms well
  understood, but lack of a comprehensive code
  which treats dampings and energetic particle
  drive non-perturbatively
• Nonlinear Physics single mode saturation well
  understood, but lack of study for multi-mode
  dynamics
• Effects of energetic particles on thermal
  plasmas needs a lot of work (integrated
  simulations).