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