Nonlinear effects on torsional Alfven waves

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Nonlinear effects on torsional Alfven waves

• S. Vasheghani Farahani, V.M. Nakariakov,
• T. Van Doorsselaere, E. Verwichte

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• Motivation
• Study the compressible flows induced by nonlinear
  Alfven waves, both standing and propagating.
• Study the three main forces responsible
  (centrifugal, magnetic tension, ponderomotive) .
• The study of the initial stage of the nonlinear
  cascade in the corona requires consideration of
  non-planer waves.
• Is the excitation of compressible perturbations
  by plane shear Alfven waves and torsional waves
  the same? And how do they depend on the plasma
  beta.
• Do 1D models of solar wind acceleration by Alfven
  waves (e.g. Suzuki et al., Torkellson et al.
  Nakariakov et al.) reqiure modification.

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• Moriyasu et al. 2004 Nonlinear torsional waves
  as source for nanoflares.

Observed spiky intensity profiles due to
impulsive energy releases could be obtained by
nonlinear torsional waves. Thus, growing interest
in nonlinear effects in torsional waves.
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• Why isn't the plane wave model sufficient to
  study the long wave-length Alfven waves for the
  dynamics of the lower atmosphere?
• For a period of 10 minutes and an Alfven speed of
  1 Mm, the longitudinal wave-length is 600 Mm
  (c.f. solar radius).
• For a plane wave the transverse wave-length
  should be much larger than the longitudinal
  wave-length not observed.
• Hence, for the generation of a plane wave of a 10
  minute period, the wave driver should be of the
  size exceeding the solar diameter.
• There should be no transverse structuring of the
  plasma in the Alfven speed, otherwise the
  wave-front of a plane Alfven wave is distorted
  (phase mixing).

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• Model equilibrium conditions
• Non-twisted and non-rotating magnetic flux tube
  embedded in a static plasma with a straight
  magnetic field.

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• Extended thin flux tube (Zhugzhda 1996) allows
  one to study long-wavelength perturbations of
  twisted and rotating plasma cylinders.
• For A0 it reduces to the thin flux tube model of
  Roberts and Webb (1979).

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• We consider a weakly nonlinear torsional wave and
  restrict our attention to the linear terms of the
  compressible variables

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• 3 forces induce compressible motions for a
  torsional wave
• Centrifugal force, magnetic tension force,
  Ponderomotive force

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• Driven wave equation for the density perturbation
  
• Where

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• In our consideration
• we neglect the higher order terms of r
• The first term on the RHS has 2 terms associated
  with the nonlinear torsional wave, hence there
  combined effect on the compressible flow depends
  on the phase relation between the twist and
  rotation of the plasma in the torsional waves.

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• Propagating torsional waves
• We obtain
• for
• The nonlinear twist and rotation effects cancel
  each other out in traveling waves.
• With the driven solution

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• Propagating shear waves (Nakariakov et al. ApJ
  2000)
• for
• We obtain
• With the driven solution

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• Nakariakov et al. 2000

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• Standing torsional waves
• We obtain
• First term on RHS is the ponderomotive force
  effects and the second team is the magnetic and
  centrifugal forces effects

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• This means that standing torsional Alfven waves
  similar to standing shear waves induce growing
  perturbations (Tikhonchuk 1994, Verwichte et al
  1999, Litwin Rosner 1998) and like standing
  kink waves (Terradas Ofman 2004).

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• The standing wave solution
• General form of Tikhonchuk Phys. Plasmas 1994
  obtained for shear Alfven waves.
• In the zero beta the secular growth comes in to
  play

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• The highest value for density perturbations is
• If we consider a loop with length L the
  longitudinal wave number would be
• The highest value for the density perturbation is
  reached at the time
• Where the growth is with the time scale
  

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• Conclusions
• Long wave-length torsional waves induce
  nonlinerly compressible perturbations by the
  ponderomotive, centrifugal and magnetic twist
  forces. The perturbations have double the
  frequency of the inducing torsional wave.
• The efficiency of the generation of compressible
  perturbations by propagating torsional waves is
  independent of the plasma beta. This is different
  from the excitation of compressible perturbations
  by plane shear Alfven waves. This is because the
  tube speed is always lower than the Alfven speed.
• There are 2 kinds of compressible perturbations
  induced by standing torsional waves growth with
  and perturbations oscillating with
  double the frequency of the driving torsional
  mode.
• The growing density perturbation saturates at a
  level inversely proportional to the sound speed.

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• Thank you for your attention