Nonlinear Ponderomotive Force by Low Frequency Waves and

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12th International Workshop on Spherical Torus
2006 Oct.11-13 Chengdu China

• Nonlinear Ponderomotive Force by Low Frequency
  Waves and
• the Possibility of Alfvén Wave Current Drive in
  Spherical Tori
• Zhe Gao
• Department of Engineering Physics, Tsinghua
  University, Beijing 100084, China
• Electronic mail gaozhe_at_mail.tsinghua.edu.cn
• in collaboration with
• Prof. N. J. Fisch, Dr. Hong Qin
• Princeton Plasma Physics Laboratory, Princeton,
  NJ08543, USA
• and Prof. Yexi He
• Department of Engineering Physics, Tsinghua
  University, Beijing 100084, China

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Low frequency wave current drive

• RF resonant current drive

V
Vph ?/k
Resonance condition

• Low-frequency waves, Alfvén waves or fast waves,
  were considered as an attractive mechanism of
  driving plasma current because of its potential
  high efficiency, no density limit and the
  convenience of high power RF generating and
  launching.
• Electron trapping may dramatically reduce the
  current drive efficiency in the subthermal
  resonant regime

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Helicity inject current drive

• Alternatively, the possibility of increasing the
  current drive efficiency by helicity injection
  has been proposed (Ohkawa 1989)
• This idea was later described as helicity balance
  between the input by wave and the dissipation by
  resistivity and viscosity,(Taylor 1988, Mett1989)
  and developed for arbitrary polarization waves.
  (Chan et al 1990)
• Helicity conversation
• current form and sustain during plasma relaxation
  in toroidal pinches, spheromaks and low aspect
  ratio tokomaks (Taylor 1986, Jarboe et al 1994,
  2002)
• Is it possible to use RF wave to drive
  nonresonant current in steady status?

helicity flux by DC, AC RF
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Ponderomotive force by RF

• helicity injection current driven by low-f waves
  can be referred to as a ponderomotive force of
  the waves
• Nonresonant force both in fluid (Klima 1980,
  Elfimov et al 1994, Tsypin et al 1995) and
  kinetic (Chan and Chiu 1993, Fukuyama et al 1993)
  model
• If true, this nonresonant current drive in the
  toroidal geometry does not depend on trapped
  particle effects and the current drive efficiency
  is expected to be strongly increased.
• Some computations for AWCD design in ST were
  based on the nonresonant current drive scheme.
  (Cuperman 1998, Elfimov 2001 )

Chan and Chiu 1993
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An apparent incongruity exists

• The well-known ponderomotive forces only in the
  direction of the gradient of second-order field
  quantities. (Gaponov and Miller 1958) Under the
  condition of ,
• However, for pure propagating waves without
  dissipation, the RF field is fully symmetric in
  the propagating direction.
• Where is the nonresonant force from? Is it
  ture?

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Single particle picture

• cancellation between the electric force and the
  magnetic force in the symmetry directions
• consider simplified fields
• and

cancellation
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Lagrangian fluid element analysis

• Lagrangian fluid element (at a fixed spatial
  location)
• the force due to the displacement of single
  particle, disappears
• a new electric field force acting on the charged
  fluid element appears.
• the total EM force on the fluid element the
  nonresonant force

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Lagrangian fluid element analysis (II)

• But, the stress must be included in the fluid
  momentum equation.
• Then, the total parallel force reduces to zero.
  The fluid picture becomes consistent with the
  single particle picture.
• The fluid helicity of the fluctuations fully
  compensates the dissipation of the EM helicity

For cold plasma, the stress reduces to the
Reynolds stress
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Lagrangian fluid element analysis (III)

• For hot plasmas, the thermal pressure and
  viscosity should be included in the fluid
  picture.
• The reason for the confusion among previous
  results is whether the stress force is considered
  completely (including thermal pressure, viscosity
  and Reynolds stress)
• correct treatment a model for the equation of
  state needs to be given, e. g. the CGL model was
  used and it is found the steady-state
  collisionless dynamo effect is absent in the
  double-adiabatic MHD theory. (Litwin 1994)
• But, many nonlinear modifications make
  things complicated and different closure schemes
  may give different conclusions

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Lagrangian fluid element analysis (IV)

• incomplete treatment no stress contribution
  (Chan and Chiu 1993, Fukuyama et al 1993) or only
  the Reynolds stress (Elfimov et al 1994) or the
  Reynolds stress and viscosity (Tsypin et al 1995)
  consideration.
• the Reynolds stress gives a similar trend as
  the kinetic stress force, but it cannot cancel
  all the terms related to the EM force
• A big difficulty in the fluid description is the
  evaluation of the thermal pressure
• Instead of adopting a particular closure scheme,
  we will carry out a nonlinear kinetic analysis in
  the following for hot plasmas.

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Nonlinear RF force in kinetic formalism

• Expand the distribution function in powers of
  electric field
• the second-order, time-averaged Vlasov Eq.

the equilibrium distribution, usually assumed to
be Maxwellian
the linear response to the RF field
the slowly varying in time, second-order response
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Nonlinear RF force in kinetic formalism (II)

• the time-averaged parallel momentum equation

quasi-linear EM force
Nonlinear stress force
Except for the J0 terms, all the components
in the quasi-linear EM force are cancelled by the
kinetic stress gradient force.
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Nonlinear RF force in kinetic formalism (III)

• under the assumption of low frequency
• Retaining the first order of , we get

for real , , and
Same as in Chan and Chiu 1993
Nonresonant force divergence of helicity flux
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Nonlinear RF force in kinetic formalism (IV)

• the total nonlinear RF force

or

• Unless the wavenumber or is complex,
  all the driving terms are multiplied by
• that is, only the Landau resonant forces survive.
  
• For steady state using RF driven current in
  toroidal systems, the frequency and the toroidal
  and poloidal wavenumbers are real. Therefore, the
  collisionless nonresonant force vanishes

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Alfvén Wave Current Drive in Spherical Tori

• In ST, most of electrons are trapped and more
  resonant electron for low frequency wave are
  trapped.
• For example, R/a1.3, r/a0.3 ft68
  ( fu32)
• For vp/vte0.1,the fraction of untrapped
  resonant electrons is
• The fractional energy absorbed by the
  untrapped electrons exceeds their abundance
  almost by a factor of 13 (Elfimov NF 1990)

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redistribution of the momentum

• Considering the redistribution of the momentum
  from trapped electrons to the bulk plasma over a
  period . The final
  fraction of the original wave momentum to the
  untrapped electrons is (Puri and Wilhelm, Conf.
  Proc. No. 190 APS 1989)
• 
  (for
  Zeff1.5)
• (for conventional Tokamak, this
  fraction can up to above 50)

• Energy input produces a current
  of duration of
• low phase velocity increases the
  current-drive efficiency, while low untrapped
  (moment) fraction decreases it

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Ware pinch and bootstrap current

• the moment absorbed by trapped electrons
• Ware pinch will be clear when Alfven wave is
  injected.
• Bootstrap current can completely recover the
  moment carried by trapped electrons?
• Strong peak of density due to the wave
  inject will change plasma equilibrium.
  (self-consistent equilibrium)

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Summary

• In collisionless plasmas, the RF force by low f
  wave, in the picture of single particle, is shown
  to be consistent with that in the fluid and
  kinetic theory.
• In the parallel direction, for a single particle,
  
• and for a Lagrangian fluid element,
• Therefore, in collisionless plasmas, only the
  Landau resonant forces survive in the parallel
  direction, and none of the ponderomotive forces
  by low frequency waves can drive nonresonant
  current. Trapped electron effects must be faced.
  
• In ST, low untrapped (moment) fraction may
  dramatically decrease the drive efficiency,
  however, the key of Alfven wave current drive in
  ST is whether/how the collisional and
  neoclassical effects be of real effect.

the Lorentz force
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ACKNOWLEDGEMENT

• This work is supported by
• PRC-US Fusion Cooperation Program,
• National Science Foundation of China (Grant
  No. 10535020), and
• Foundation for the Author of National
  Excellent Doctoral Dissertation of PR China