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Astron. Astrophys. 326, 1241-1251 (1997)

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1. Introduction

The solar atmosphere contains a wide variety of magnetic structures which can support MHD waves. Waves that are generated either by turbulent motions in the photosphere and chromosphere, or by global solar oscillations or by local releases of energy in reconnection events interact with these magnetic structures.

Propagating waves transport energy away from the region where they are generated into the ambient plasma, while dissipation causes a deposition of part of the wave energy in the plasma. The classic viscous or resistive damping of for example Alfvén waves in a uniform plasma is known to be a very inefficient way to transform wave energy irreversibly into heat in weakly dissipative plasmas with large values of the viscous and magnetic Reynolds numbers as in the solar plasma.

However, a more efficient mechanism for dissipating wave energy can occur in nonuniform magnetic plasmas, where resonant slow and resonant Alfvén waves can exist. In ideal MHD these resonant waves are confined to an individual magnetic surface without any interaction with neighbouring magnetic surfaces. Since each magnetic surface has its own local slow frequency and its own local Alfvén frequency, a nonuniform magnetic plasma can have two continuous ranges of frequencies corresponding to resonant slow waves and resonant Alfvén waves.

Dissipative effects cause coupling of the resonant magnetic surface to neighbouring magnetic surfaces. For large values of the viscous and the magnetic Reynolds numbers as in the solar atmosphere this coupling is weak and the local resonant slow oscillations and the local resonant Alfvén oscillations are characterized by steep gradients accross the magnetic surfaces. Excitation of these local slow oscillations or local Alfvén oscillations provides a means for dissipating wave energy which is far more efficient in weakly dissipative plasmas than classical resistive or viscous MHD wave damping in a uniform plasma. This mechanism of resonant wave damping was first put forward as a possible mechanism for heating the solar corona by Ionson (1978).

The excitation of the local resonant oscillations can be direct by driving motions in the magnetic surfaces or indirect by driving motions normal to the magnetic surfaces. The indirect excitation relies on global motions that transfer energy across the magnetic surfaces up to the resonant magnetic surface where the frequency of the global motion equals the local Alfvén frequency or local slow frequency. The excitation of the localised resonant waves is indirect since we need magnetosonic waves that propagate across the magnetic surfaces to excite them.

Most studies on heating by indirect excitation have considered incoming fast magnetosonic waves that couple to local resonant Alfvén waves in a nonuniform plasma (Poedts, Goossens and Kerner (1989), Poedts, Goossens and Kerner (1990), Okreti and ade (1991)). Fast magnetosonic waves propagate almost isotropically as long as their high frequencies are above the cutoff frequency for fast waves. Slow magnetosonic waves are not considered an important means for transferring energy into the corona and for heating the plasma there, primarily because of the limited range of frequencies of the slow continuum waves. However, they can play a role when slow and fast magnetosonic waves that are generated in the photosphere interact with chromospheric magnetic structures and when slow and fast magnetosonic waves that are generated locally in the corona interact with coronal magnetic fields.

In this paper the focus is on the absorption of slow and fast incoming magnetosonic waves by coupling to local resonant slow magnetosonic waves in a nonuniform plasma. This process is studied for an equilibrium configuration which makes the mathematical analysis as simple as possible but still contains the basic physical ingredients.

The equilibrium model is specified in Sect. 2. In Sect. 3 we present the set of ideal MHD equations that govern the linear motions in a planar 1D equilibrium model. In this Sect. we also discuss the singularities in the differential equations of linear ideal MHD and their role for resonant wave damping. The solutions of the differential equation of linear ideal MHD in the two uniform plasma layers are also given. In Sect. 4 we explain how the analysis of resonant waves in dissipative layers (Sakurai, Goossens and Hollweg (1991), Goossens and Hollweg (1993), Goossens, Ruderman and Hollweg (1995) in what follows SGH, GH and GRH respectively) can be applied to obtain the solutions in the vicinity of the mathematical singularity of the ideal MHD equations. The definition of the absorption coefficient and the method for its computation are given in Sect. 5. In Sect. 6 we present our results and conclusions.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998