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Nonlinear effects on torsional Alfven waves
S. Vasheghani Farahani, V.M. Nakariakov,
T. Van Doorsselaere, E. Verwichte
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.
• 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.
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).
Model & equilibrium conditions:Non-twisted and non-rotating magnetic flux
tube embedded in a static plasma with a straight magnetic field.
Extended thin flux tube (Zhugzhda 1996) allows one to study long-wavelength perturbations of twisted and rotating plasma cylinders.
For A=0 it reduces to the thin flux tube model of Roberts and Webb (1979).
We consider a weakly nonlinear torsional wave and restrict our attention to the linear terms of the compressible variables
3 forces induce compressible motions for a torsional wave:
Centrifugal force, magnetic tension force, Ponderomotive force
Driven wave equation for the density perturbation
Where
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.
Propagating torsional wavesWe obtain
for
The nonlinear twist and rotation effects cancel each other out in traveling waves.
With the driven solution
Propagating shear waves (Nakariakov et al. ApJ 2000)
for
We obtain
With the driven solution
• Nakariakov et al. 2000
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
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).
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
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
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.
Thank you for your attention