Helical motion of magnetic flux tubes in the solar atmosphere
Photospheric granulation may excite transverse kink pulses in anchored vertical magnetic flux tubes. The pulses propagate upwards along the tubes with the kink speed, while oscillating wakes are formed behind the wave front in a stratified atmosphere. The wakes oscillate at the kink cut-off frequency of stratified medium and gradually decay in time. When two or more consecutive kink pulses with different polarizations propagate in the same thin tube, then the wakes corresponding to different pulses may superimpose. The superposition sets up helical motions of magnetic flux tubes in the photosphere/chromosphere as seen by recent Hinode movies. The energy carried by the pulses is enough to heat the solar chrmosphere/corona and accelerate the solar wind.
Recent high resolution movies obtained by Hinode spacecraft clearly show continuous helical motions of spicule axes (De Pontieu et al., 2007). This phenomenon is known for a long time (Beckers 1972), but no satisfactory explanation has been done yet. De Pontieu et al. (2007) suggested that these motions are caused by Alfvén waves excited in the photosphere by granular motions or acoustic oscillations. Photospheric magnetic field is concentrated in thin flux tubes and therefore may support the propagation of kink and torsional Alfvén waves (the tubes also support the propagation of sausage waves, but here we consider only transverse waves). The kink wave is a tube wave i.e. the whole tube oscillates with one frequency even if the Alfvén speed changes across the tube, while the frequency of torsional waves is different at different surfaces (Van Doorsselaere et al., 2008). Therefore, the excitation of torsional waves in photospheric magnetic tubes is complicated. The oscillations of spicule axis have also been explained in terms of kink waves (Kukhianidze et al., 2006; Zaqarashvili et al., 2007), but the chromospheric magnetic field has rather expanded structure. Therefore, De Pontieu et al. (2007) suggested through Monte-carlo simulation that spicules are not wave guides for kink waves. The problem is currently under debate and more observations are needed to understand the real process.
On the other hand, granular buffeting on an anchored magnetic tube may easily excite a transverse kink pulse. The pulse propagates through the stratified photosphere with the kink speed, but the oscillating wake is formed behind the wave front. The wake oscillates at cut-off frequency of kink waves and decays as time progresses (Roberts, 1981, 2004; Rae & Roberts, 1982; Spruit & Roberts, 1983; Hasan & Kalkofen, 1999; Musielak & Ulmschneider, 2001). However, the magnetic tube undergoes continuous buffeting of granular cells from different sides. Therefore, another pulse polarized in a different plane may quickly follow. The second pulse again propagates with the kink speed and form another wake oscillating with the same cut-off frequency. Therefore, the superposition of the two oscillations may set up the helical motion of the tube axis in the photosphere. The helical motion will have the photospheric kink cut-off period 7-8 min.
However, when the pulse penetrates into the chromosphere, then two possible scenarios can be developed. If chromospheric magnetic field has a tube structure, then the pulse continues to propagate as the kink one. But if the magnetic field is not concentrated in tubes, then it will be transformed into the Alfvénic pulse. Yet the pulse will have the same main properties in both cases: it will propagate at either the kink or Alfvén speed and the wake oscillating at the chromospheric cut-off frequency will be formed behind the pulse. The superposition of the wakes corresponding to different pulses may set up helical motions of magnetic field lines just as in the photospheric case. These helical motions may be responsible for the oscillations of spicule axes as seen by Hinode movies.
Here we study the phenomenon using the Klein-Gordon equation for wave propagation in the stratified atmosphere.
2 Helical kink waves in the photosphere
Kink wave propagation along vertical thin magnetic flux tube embedded in the stratified field-free atmosphere is governed by the Klein-Gordon equation (Rae & Roberts, 1982; Spruit & Roberts, 1983; Roberts, 2004)
where , is the kink speed, is the density scale height and is the gravitational cut-off frequency for isothermal atmosphere (temperature inside and outside the tube is assumed to be the same and homogeneous). Here is the transversal displacement of the tube, is the tube magnetic field, and are the plasma densities inside and outside the tube respectively (the magnetic field and densities are functions of , while the kink speed is constant in the isothermal atmosphere).
Eq. (1) yields simple harmonic solutions with the dispersion relation
where is the wave frequency and is the wave number. The dispersion relation shows that the waves with higher frequency than may propagate in the tube, while the lower frequency waves are evanescent.
Kink waves cause the transverse displacement of whole tube. The displacement of tube in a simple harmonic kink wave is polarized arbitrarily and the polarization plane depends on the excitation source. Then the superposition of two or more kink waves polarized in different planes may give rise to the complex motion of the tube. The process is similar to the superposition of two plane electromagnetic waves, where the waves with the same amplitudes lead to the circular polarization, while the waves with different amplitudes lead to the elliptical polarization. Consider, for example, two harmonic kink waves with the same frequency but polarized in and planes: and . The superposition of these waves sets up the helical wave with circular polarization if . As a result, the tube axis rotates around the vertical, while the displacement remains constant (Fig. 1). If then the resulting wave is elliptically polarized. The superposition of few harmonics with different frequencies and polarizations may lead to more complex motion of tube axis.
However, simple harmonic kink waves hardly be excited in the photosphere. The more realistic process is the impulsive buffeting of granules on an anchored magnetic flux tube. For the sake of simplicity, we consider the simplest impulsive forcing in both time and coordinate. Then Eq. (1) looks as
where , , is constant and the pulse is set at , .
where and are Bessel and Heaviside functions respectively. Eq. (4) shows that the wave front propagates with the kink speed , while the wake oscillating at the cut-off frequency is formed behind the wave front and it decays as time progresses (Rae & Roberts, 1982; Spruit & Roberts, 1983; Hasan & Kalkofen, 1999; Roberts, 2004). Fig. 2 shows the plot of transverse displacement , where is expressed by Eq. (4). The rapid propagation of the pulse is seen, which is followed by the oscillating wake (the time is normalized by the cut-off period ). Just after the propagation of the pulse, the tube begins to oscillate with the cut-off period at each height. The amplitudes of pulse and wake increase upwards due to the density reduction, but the oscillations at each height decay in time.
Hence, the transverse impulsive action on the magnetic tube at moment near the base of photosphere (set at ) excites the upward propagating kink pulse, while the tube in the photosphere oscillates at the photospheric kink cut-off frequency, , which depends on the plasma parameter () inside the tube. In the case of temperature balance inside and outside the tube, the kink speed can be expressed as , where is the sound speed and is the ratio of specific heats ( for adiabatic process). Then the photospheric sound speed of 7.5 km/s and gives 6.5 km/s for the kink speed. Consequently, we may estimate the kink cut-off period as 8 min using the photospheric scale height of 125 km. Hence the magnetic tube will oscillate with 8 min period in the photosphere. If the external pulse is directed along, say, the axis, then the tube will oscillate in the plane.
However, the anchored magnetic tube undergoes the granular buffeting from different sides. Therefore, suppose that after time another granular cell acts on the same tube along the axis. The solution governing the pulse propagation is
Thus the rapidly propagating pulse is again excited with the oscillating wake behind the front. The wake oscillates with the same cut-off frequency, but the oscillation is polarized in plane. Hence there are two transverse oscillations with the same frequency, but polarized in perpendicular planes. The time interval between consecutive buffeting (say, granular life time) is comparable to the photospheric kink cut-off period. Therefore, these oscillations will be superimposed, because the oscillation excited by the previous pulse still exists in the same tube. The superposition will set up the helical motion of the tube axis with photospheric cut-off period 8 min. Figure 3 shows the superposition of the solutions (4) and (5) at the height of 250 km above the photosphere. The first pulse is imposed along the direction, which is followed by another pulse in the direction. The upper panel corresponds to the same amplitudes of both pulses, but the lower panel corresponds to the case when the first pulse is twice stronger than the second. We see that the tube rotates along nearly circular spiral in the first case and along elliptical spiral in the second case. The displacement gradually decreases with time. Therefore, the granular buffeting with same amplitudes excites the nearly circular motion of the tube, while the buffeting with different amplitudes excites the elliptical motion.
The wave length of oscillations km is quiet long comparing to the width of the photosphere. Therefore, the photospheric magnetic tube will just rotate around the vertical without additional wave nodes. The tube displacement increases with height due to the decreasing density (Fig. 2). Thus the observations should show that the upper part of the tube rotates with larger amplitude than the lower part. We believe that the high resolution observations will reveal the similar behavior of photospheric magnetic tubes.
3 Propagation of transverse pulse through the chromosphere
The situation is changed when the pulse crosses the photosphere and penetrates into the chromosphere. Photospheric magnetic tubes may expand in the chromosphere giving rather different geometry than thin tubes. On the other hand, chromospheric spicules seem to behave like magnetic tubes. But recent Monte-carlo simulations (De Pontieu et al. 2007) suggest that the spicules are not wave guides for tube waves. Therefore, this question is currently under debate and more observations are needed to clarify the intrinsic process (Erdélyi & Fedun 2007). The transverse pulse retains its properties in any case. It continues to be the kink pulse in structured magnetic field, but probably is transformed into the Alfvénic one in the case of smooth transverse profile of the magnetic field.
The photosphere and the chromosphere can be approximated as two different regions with different isothermal temperatures, densities and other plasma parameters. Then the propagation of pulse in the chromosphere is governed by Eq. (3), but with different phase speed and scale height ( now corresponds to the base of the chromosphere). It must be mentioned, however, that the phase speed remains constant only if the magnetic field is expanded with height. This necessarily requires the horizontal component of the magnetic field, which is neglected in the equation. Therefore, the equation (3) is valid only near the tube axis, where the magnetic field is predominantly vertical.
Then the photospheric solution (4) can be directly applied here, but with chromospheric phase speed and scale height. The chromospheric scale height can be estimated as 500 km for 25 000 K temperature. The value of phase speed determines the wave cut-off frequency. For example, Alfvén wave cut-off frequency is (Roberts, 2004), which gives the cut-off period of 250 s for the Alfvén speed of 50 km/s. On the other hand, a cavity with higher density concentrations (for example spicules) may guide kink waves with smaller phase speed. This increases the cut-off period. For example, the kink speed of 25 km/s yields the cut-off period of 500 s.
Therefore, the transverse pulse may set up the oscillating wake in the chromosphere with the period of 250-500 s. The two perpendicularly polarized transverse pulses may form the helical motion in the chromosphere as observed by Hinode (De Pontieu et al. 2007). De Pontieu et al. (2007) argued that the helical motion is caused by Alfvén waves directly excited in the photosphere. The estimated energy flux of the waves was enough to power the solar wind and to heat the quiet corona. However, if observed oscillations of spicule axes are caused by wakes formed after the transverse pulse propagation, then the energy transported into the chromosphere/corona can be much higher: as almost whole energy of initial perturbation is carried by the pulse, while the energy of the wake is much smaller.
The energy flux stored in initial transverse pulse at the photospheric level is , where is the granular velocity being 1-2 km/s. Then for photospheric values of electron density and kink speed, the estimated energy flux is erg cm s. Almost whole energy is carried by the pulse, therefore even if the filling factor of magnetic tubes is , the energy flux is more than enough to heat the solar chromosphere/corona.
We suggest that the propagation of consecutive transverse pulses in the stratified atmosphere, which are excited by photospheric granular buffeting, may set up the helical motions of magnetic flux tubes through the superposition of oscillating wakes formed behind the wave fronts. This scenario may explain the continuous motions of spicule axes seen in recent Hinode movies. The pulses carry almost whole energy of initial perturbations, while the energy in wake oscillations is much smaller. Therefore, the energy carried into corona by transverse pulses can be much higher than it is estimated by observed oscillations. More observations and numerical/analytical works need to look further into this problem.
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