16. The Alfvén Amplifier

October 21, 2011, from uksp_nug_ed

Author: Youra Taroyan is from the Solar System Physics Group at Aberystwyth University.

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Introduction

Magnetohydrodynamic (MHD) instabilities play a key role in a number of processes occurring in the Sun and in the solar-terrestrial environment: small perturbations become exponentially amplified leading to large scale changes in the system. Well-known examples include the Rayleigh-Taylor and the Kelvin-Helmholtz instabilities.

Since their discovery in the 1940s Alfvén waves have been studied in relation to the heating of laboratory plasmas and the solar atmosphere, the formation of spicules, and the acceleration of the solar wind. Recently, an MHD instability associated with the amplification of incompressible Alfvénic disturbances in compressible plasma flows has been found [1]. We elaborate on the implications of this new instability in the context of Alfvén waves and outflows observed in the solar atmosphere (e.g. [2, 3, 4]). The details of this nugget are presented in [5].

Unstable flux tubes

We analyse the stability of an axisymmetric magnetic flux tube in a steady state (Figure 1).

Figure 1: A cartoon of an expanding flux tube. The tube is gravitationally stratified and permeated by a field-aligned smooth mass flow u0. The s=0 level represents the photospheric footpoint shaken by convective motions.

The flux tube is gravitationally stratified and permeated by a field-aligned isothermal mass flow u0. The flow may be accelerate or decelerate upwards in different sections of the tube. Small-amplitude perturbations are launched from the footpoint by a photospheric driver representing random convective motions. The evolution of torsional perturbations is analysed. We identify solutions that grow exponentially in time, for a range of sub-Alfvénic and subsonic smooth flow profiles. The instability arises when rapid expansion occurs in a fraction of the tube leading to deceleration of the flow upwards. Figure 2 compares stable and unstable tubes with accelerating and decelerating flows.

Figure 2: Comparison of three tubes. The first two remain stable. The third tube is unstable due to rapid expansion in a marked section. The derivative of the flow velocity with respect to the coordinate along the background field is denoted by u0.

The mechanism behind the Alfvén instability

In the case of the Kelvin-Helmholtz instability of incompressible plasmas, compressibility may either stabilise or destabilise the steady state depending on the density jump and the velocity shear. A magnetic field aligned with the flow introduces a threshold flow speed below which the instability is suppressed. In contrast to the Kelvin-Helmholtz instability, the Alfvén instability presented here only arises  in the presence of a compressible flow and a magnetic field. As an example we have considered an expanding flux tube permeated by a moderate subsonic and sub-Alfvénic flow ([5]). No high speed flows or shear are required.

The Alfvénic perturbations travelling from the footpoint are over-reflected back and transmitted forward as the decelerating plasma flow reduces their propagation speed. Over-reflection represents reflection with a coefficient greater than one. Thus the region of rapid expansion acts as an amplifier for the perturbations, where the flow provides the required energy. The following animation shows the over-reflection and amplification process of a single damped pulse in a simplified structure with a uniform background field along which a compressible flow is present.

Figure 3: The process of amplification in a simplified structure with a uniform background field. An exponentially damped pulse launched from s=0 is being over-reflected from an interface at s=L that separates two media with different densities and flow speeds. Only the region from 0 to L is shown. The details are discussed in Taroyan 2008 PRL.

Implications

Observations of bright network regions suggest that two types of magnetic structures coexist: small closed loops in the chromosphere and funnels that are connected to the corona. The second type is usually associated with blue-shifts representing outflows and enhanced line-widths [2]. These enhanced line-widths are best interpreted as nonlinear Alfvén waves passing through the funnel. A number of recent observational studies of chromospheric and transition region lines have found evidence for the presence of Alfvén or Alfvén-like waves and outflows in magnetic structures of the solar atmosphere (e.g. [3], [4]) Estimates of the energy flux carried by these waves indicate that they could accelerate the solar wind and heat the corona.

According to our results (Figure 2), a funnel shaped flux tube permeated by outflowing plasma may be unstable. The funnel acts as an amplifier of Alfvénic disturbances which could then be seen as periodic or non-periodic line-width enhancements. As the amplitudes increase the energy of the disturbances will eventually be converted into heating or acceleration through some dissipative process, e.g., shock heating due to non-linear mode conversion.

Conclusions

Expanding isothermal flux tubes with smooth flow profiles can be unstable with respect to linear torsional perturbations. This is a new ideal MHD instability: a funnel shaped fraction of the tube becomes an amplifier of Alfvénic disturbances. The obtained results could explain the non-thermal broadenings associated with outflows in magnetic regions of the lower solar atmosphere. However, further work is required to establish the dynamic and energetic implications of the Alfvén instability in the solar atmosphere.

References

  • [1] Taroyan, Y. 2008, Physical Review Letters,  101, 5001
  • [2] Peter, H. 2001, Astronomy & Astrophysics, 374, 1108
  • [3] Jess, D. B., Mathioudakis, Mihalis, Erdélyi, R., Crockett, P. J., Keenan, F. P., Christian, D. J. 2009, Science, 323, 1582
  • [4] McIntosh, S. W., de Pontieu, B., Carlsson, M., Hansteen, V., Boerner, P., Goossens, M. 2011, Nature, 475, 477
  • [5] Taroyan, Y. 2011, Astronomy & Astrophysics, 533, A68


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