6. Coupled Alfvén and Kink Oscillations Propagating in the Solar Corona

December 3, 2010, from uksp_nug_ed

Author: David Pascoe is a Research Fellow at the School of Mathematics and Statistics at the University of St Andrews.

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Introduction


Figure 1: Cartoon of the mode coupling mechanism for propagating kink wavepackets. The blue line represents the coronal loop density profile.

Observations reveal that oscillations are ubiquitous in the solar corona. Transverse velocity perturbations with periods of about 5 minutes propagate upwards through the solar atmosphere and are damped over length scales of a few hundred megametres [1]. Different interpretations of these waves, as Alfvén waves or kink waves [2,3], have consequences for the inferred coronal magnetic field strength in MHD coronal seismology, and the energy budget calculations for the coronal heating problem. In this nugget we set out to investigate, by means of numerical simulations, the way that wave energy may be transferred from the kink mode into the Alfvén mode so that in fact both modes may be present.

The Simulations

The corona has highly structured magnetic field and density profiles, thus a highly structured profile of Alfvén speeds. Regions of lower Alfvén speed can act as a waveguide for magnetohydrodynamic (MHD) waves [4].

We model a coronal loop with a density profile having a linear transition between an internal (ρ0) and external (ρe) density (see Figure. 1). The transition happens in an inhomogeneous shell of finite thickness l, across which the Alfvén speed varies continuously in the inhomogeneous layer. Where the phase speed of a kink mode matches the local Alfvén speed a process called mode coupling happens, which is able to transfer energy from one mode to another. We use the Lare3D code [5] to simulate the 3D evolution [6,7].

Single Period Wavepacket

We launch a small-amplitude propagating kink wavepacket into our loop at one foopoint. The driver is applied from t = 0 to t = P0. Figure 2 shows vx at the axis of the cylinder at time t = 1.5P0. The density profile, outlined by the vertical lines, is defined by ρ0/ρe = 2 and l/a = 0.5. The kink wavepacket propagates along the flux tube at a group velocity and also has a phase velocity associated with it. Efficient mode coupling between kink and Alfvén waves occurs on field lines in the inhomogeneous regions where the kink wavepacket phase speed equals the local Alfvén phase speed. Note that there is no resonant singularity or harmonic driving frequency in this process. Our work thus supports an interpretation of the observed waves as a coupling of the kink and Alfvén modes. Additionally, the coupling can account for the observed predominance of outward wave power in longer coronal loops [6].


Figure 2: vx at y = 0 (left) and x = 0 (right). Energy is transferred from the kink mode in the core region to the Alfvén mode in the inhomogeneous shell region by mode coupling.

This coupling of the wavetrain to a local Alfvén mode causes a decrease in kink wave energy and damps the transverse tube oscillation. Also, since the Alfvén mode is in an inhomogeneous layer, it phase mixes [8]. The corresponding characteristic spatial scale decreases with time.

Non-axisymmetric Coronal Loops

Next we consider what happens in the case that the coronal loop is not an axisymmetric cylinder. Figure 3 shows the wave energy (integrated over the entire z-range) at t = 4P0 for the case of an undistorted cylinder (left) and a cylinder with multiple distortions in the inhomogeneous layer (right). The energy has been coupled in to Alfvén modes in the inhomogeneous region. Wave energy is located along the dashed line – i.e. wherever the mode coupling condition is satisfied (apart from at the Alfvén mode nodes), so that the basic result is not affected by this change of geometry [7].


Figure 3: The wave energy integrated over the propagation direction z at t = 4P0 for a wavepacket propagating through an undistorted (left) and distorted (right) cylinder. The solid lines are contours of density. The dashed line represents the location where the mode coupling condition is satisfied.

Oscillations Driven by Random Footpoint Motions

It is somewhat unlikely that waves will be driven by a regular driver, so we simulate the case in which the loop is continuously driven in the x-direction with random amplitudes and periods. Figure 4 is a movie showing the velocity field at a point some distance from the footpoint. The contours show the density profile of the coronal loop. The first signal to arrive are the phase-mixed m = 1 Alfvén modes in the inhomogeneous shell. Later, transverse kink oscillations of the core arrive.


Figure 4: A movie of the velocity field in (x,y) at a point some distance from the footpoint. The contours show the density profile of the coronal loop. A direct link to the movie file (m4v format) can be found {link:http://www.uksolphys.org/wp-content/uploads/nuggets/nug6/200.m4v}here{/link}.

Conclusions

According to our models, the transverse waves are an intrinsically coupled mode. This will be true even for a very weak density contrast. The properties of the observed Doppler shifts will predominantly resemble a kink mode. The coupled Alfvén mode component is generally unresolved by modern solar instruments, and so it will contribute to the Doppler shifts incoherently. However, though hard to observe directly it is needed to explain the rapid damping. So it appears that a more complete explanation of the observations require both kink and Alfvén modes.

References

  • [1] Tomczyk, S., & McIntosh, S.W. 2009, ApJ, 697, 1384
  • [2] Van Doorsselaere, T., Nakariakov, V. M., & Verwichte, E. 2008, ApJ, 676, L73
  • [3] Erdélyi, R., & Fedun, V. 2007, Science, 318, 1572
  • [4] Edwin, P. M., & Roberts, B. 1983, Solar Physics, 88, 179
  • [5] Arber, T. D., Longbottom, A. W., Gerrard, C. L., & Milne, A. M. 2001, JCP, 171, 151
  • [6] Pascoe, D.J., Wright, A.N., & De Moortel, I. 2010, ApJ, 711, 990
  • [7] Pascoe, D.J., Wright, A.N., & De Moortel, I. 2011, ApJ, submitted
  • [8] Heyvaerts, J., & Priest, E. R. 1983, ApJ, 117, 220


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