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Astron. Astrophys. 354, 334-348 (2000)

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1. Introduction

The mechanism by which the solar corona is heated is still one of the major unsolved problems in solar physics. Reviews of the coronal heating problem have been presented by Narain & Ulmschneider (1990, 1996), Browning (1991) and Zirker (1993). In the open field regions of coronal holes, wave heating mechanisms remain the most attractive possibility but like all proposed heating theories, magnetic wave heating depends on the creation of sufficiently small lengthscales in order for dissipation to play an efficient role. Since it was first realised that Alfvén waves are not easily damped, various effects of the propagation of MHD waves have been investigated. An important property of MHD waves in an inhomogeneous plasma is that individual surfaces can oscillate with their own Alfvén frequency. This implies that a global wave motion can be in resonance with local oscillations on a specific magnetic surface. The resonance condition is that the frequency of the global motion is equal to the local Alfvén frequency of the magnetic surface. In this way, energy is transferred from the large scale motion to the small scale oscillations, i.e. to a lengthscale where dissipation can become effective. This process of resonant absorption was first suggested by Ionson (1978) as a mechanism for heating coronal loops. Since this original work, a lot of studies, both numerically and analytically have been done on resonant absorption (e.g. Goedbloed & Halberstadt 1994; Halberstadt & Goedbloed 1995a, b; Tirry et al. 1997; Berghmans & Tirry 1997; Tirry & Berghmans 1997; Poedts & Boynton 1996).

Heyvaerts and Priest (1983) proposed a simple but promising idea for the behaviour of Alfvén waves when the local Alfvén speed varies across the magnetic field lines. They suggested damping of Alfvén waves due to phase mixing could be a possible source of coronal heating. Basically, phase mixing and resonant absorption are two aspects of the same physical phenomenon, namely that Alfvén waves can exist on individual flux surfaces. Examples of the close interplay between the related phenomena of phase mixing and resonant absorption can be found in e.g. Ruderman et al. (1997a, 1997b). However, in this paper we will not consider resonant absorption and concentrate on damping of Alfvén waves due to phase mixing . The propagation and damping of shear Alfvén waves in an inhomogeneous medium has been studied in more detail (Ireland 1996; Cally 1991; Browning & Priest 1984; Nocera et al. 1984) by relaxing the Heyvaerts and Priest limits of weak damping and strong phase mixing. Recently, Hood et al. (1997a, 1997b) have found analytical, self-similar solutions describing phase mixing of Alfvén waves in both open (coronal holes) and closed (coronal loops) magnetic configurations. Possible observational evidence of coronal heating by phase mixing is discussed by Ireland (1996). Numerical simulations of phase mixing in coronal holes have been performed by Poedts et al. (1997) who found that in coronal holes, the phase mixing of Alfvén waves is speeded up by the flaring out of the magnetic field lines. Ofman & Davila (1995) found that in an inhomogeneous coronal hole with an enhanced dissipation parameter ([FORMULA]), the Alfvén waves dissipate within several solar radii and can provide significant energy for the heating and acceleration of the high-speed solar wind. Ruderman et al. (1998) considered phase mixing of Alfvén waves in planar two-dimensional open magnetic configurations, using a WKB method. However, the validity of the WKB technique requires a particular relationship between the magnetic Reynolds number, the wavelength of the basic Alfvén wave and the coronal pressure scale height. Nakariakov et al. (1997) considered the non linear generation of fast magnetosonic waves by Alfvén wave phase mixing and showed that transversal gradients in the Alfvén wave, produced by phase mixing, lead to the generation of propagating fast waves which are subject to strong damping. This phenomenon may be considered as indirect heating of the coronal plasma by phase mixing. The propagation of magnetohydrodynamic waves in a cold plasma with an inhomogeneous ste ady flow directed along a straight magnetic field was studied by Nakariakov et al. (1998). They found that in regions with transversal gradients in the steady flow, phase mixing of Alfvén waves takes place similarly to classical phase mixing in a static medium with an inhomo geneity in the Alfvén speed. Non linear effects on the dissipation of Alfvén waves have been discussed by Boynton & Torkelsson (1996), and Torkelsson & Boynton (1998) for spherical Alfvén waves, who found that non linear Alfvén waves can steepen to form current sheets which enhance the dissipation rate of the Alfvén waves by several orders of magnitude. This result was confirmed by Nakariakov et al. (1999a) who describe the dynamics of non linear, spherical, linearly polarised small amplitude Alfvén waves in the stratified and dissipative plasma of coronal holes by the spherical scalar Cohen-Kulsrud-Burgers equation. From the analysis of this equation it was found that linearly polarised Alfvén waves of weak amplitude and relatively long periods are subject to non linear steepening and efficient non linear dissipation. Narain & Sharma (1998) found that in the non linear regime the viscous damping of Alfvén waves becomes a viable mechanism of solar coronal plasma heating when strong spreading of the magnetic field is taken into account. From recent TRACE observations, Nakariakov et al. (1999b) estimate the coronal dissipation coefficient to be eight or nine orders of magnitude larger than the theoretically predicted value. This larger dissipation coefficient could solve some of the existing difficulties with wave heating and reconnection theories.

In this paper we aim to study the effect of both vertical and horizontal density stratifications on the phase mixing of Alfvén waves in an open and (radially) diverging magnetic atmosphere. We restrict ourselves to a study of travelling waves, generated by photospheric motions that cause disturbances to propagate outwards from the Sun without total reflection.

In Sect. 2 we describe the basic equilibrium and equations. For simplicity we consider the scale height to be infinity and in Sect. 3 we discuss the effect of the radially diverging background magnetic field on phase mixing of Alfvén waves in the absence of dissipation. In Sect. 4 we add dissipation to our basic model. In Sect. 5 we look at the combined effect of the vertical stratification of the density and the divergence of the background magnetic field, while Sect. 6 contains the discussion and conclusion.

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© European Southern Observatory (ESO) 2000

Online publication: January 31, 2000