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Astron. Astrophys. 332, 786-794 (1998)

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1. Introduction

The solar atmosphere, from the photosphere to the corona, is a highly structured inhomogeneous medium, e.g. recent observations of X-ray emission from the solar corona by Yohkoh (Shibata et al. 1992) have emphasized the complex, highly structured nature of the corona. Commonly observed features such as the photospheric flux tubes, coronal holes, coronal loops, magnetic arcades indicate the existence of pronounced local nonuniformities at their boundaries. Another key observation of the highly inhomogeneous solar atmosphere is the presence of steady flows. Bulk motions are observed along or nearly along the magnetic field lines which outline the magnetic structures (Doyle et al. 1997). The nonuniformities are often in the form of transition layers that separate regions of larger extent with different, comparatively uniform physical characteristics.

In particular, these sharply structured interfaces can support magneto-acoustic surface waves, which may be classified as either fast or slow modes (Roberts 1981).

An important property of MHD waves in an inhomogeneous plasma is that a global wave motion can be in resonance with local oscillations of a specific magnetic surface. The resonance condition is that the frequency of the global motion should be equal to either the local Alfvén or the local cusp frequency of the magnetic surface. In this way energy is transferred from the large scale motion to oscillations which are highly localized to the neighbourhood of the Alfvén or cusp singular surface. In dissipative MHD this behaviour is mathematically recovered as eigenmodes which are exponentially damped in time. Due to their global character (oscillating with the same frequency throughout the plasma) these modes are called 'global modes'. For ideal MHD such damped oscillations cannot be eigenmodes of the system, and for this reason they are often called 'quasi-modes' (see Tirry & Goossens 1996 and references therein). Since quasi-modes turn out to be global natural oscillations of the magnetic structures they will probably be most easily observed. It is therefore important to know how the frequencies of the quasi-modes are related to the distribution of the physical quantities and the geometry of the structure. For the same reason the quasi-mode plays a central role in the resonant absorption process as possible heating mechanism.

Since the work of Ionson (1978), resonant absorption of MHD surface modes has been recognized as a suitably efficient damping mechanism, and much recent work has been devoted to this process (Rae & Roberts 1982; Lee & Roberts 1986; Davila 1987; Hollweg & Yang 1988; Hollweg et al. 1990; Sakurai, Goossens & Hollweg 1991; Goossens, Ruderman & Hollweg 1995; Erdélyi & Goossens 1996). Except for the paper by Hollweg et al. (1990) and Erdélyi & Goossens (1996), these prior studies and many others of resonant absorption have assumed that there is no background velocity shear in the plasma. However in the paper by Hollweg et al., the authors introduced velocity shear and investigated its effects on the rate of resonant absorption of MHD waves supported by thin "surfaces" in an incompressible plasma. They found that the velocity shear can either increase or decrease the resonant absorption rate and that for certain values of the velocity shear the absorption rate goes to zero. In addition, they also found that there can be resonances which do not absorb energy from the surface wave but rather give energy back to it, leading to instabilities, even at velocity shear, which are below the threshold for the Kelvin-Helmholz instability (Chandrasekhar 1961).

Ryutova (1988) has considered a closely related problem. She studied the propagation of kink waves along thin magnetic flux tubes in the presence of a homogeneous parallel flow outside of the tube. She was the first to introduce the concept of negative energy waves to solar physics and suggested that the resonant instability can be interpreted in terms of negative energy waves. However, according to Hollweg et al. (1990), her work contains an inconsistency.

Erdélyi & Goossens (1996) studied the p-mode resonant interaction with steady inhomogeneous flux tubes (sunspots) and found that for steady flows at around 10 percent of the local Alfvén speed, sunspots emitted energy. This phenomenon was again associated with negative energy waves.

Joarder et al. (1997) applied the concept of negative energy waves for simple homogeneous solar configurations. They showed that backward propagating waves found in these magnetic structures of the solar atmosphere are negative energy waves.

Ruderman & Goossens (1995) studied the stability of an MHD tangential discontinuity in an incompressible plasma where viscosity is taken into account at one side of the discontinuity. The instability, which occurs for velocity shears smaller than the threshold value for the onset of the Kelvin-Helmholz instability, can be explained in terms of a negative energy wave which becomes unstable because of the presence of a dissipation mechanism (viscosity).

Since these instabilities may play a role in the development of turbulence in regions of strong velocity shear such as in the solar wind and the Earth's magnetosheath, but also, for example, in the interaction of p-modes with sunspots, the assumption of incompressibility has to be dropped in the spectral study.

In addition, in order to deal in a mathematically but also physically consistent manner with the resonant wave excitation (the singularity in the ideal MHD equations) in solving the eigenvalue problem for the MHD surface modes, we analytically derive the dissipative solution close to the resonant magnetic surface in resistive MHD.

Hence in this paper we consider the spectral problem of the MHD surface mode on a transitional nonuniform layer that separates two uniform regions, and in the presence of a velocity shear (discontinuity) in the background flow. The mass flow is assumed to be uniform in one of the uniform regions and in the transitional layer whereas the other uniform region is static.

By deriving the solution in resistive MHD analytically around the resonance, the effect of the velocity shear on the damping rate can easily be investigated and it clearly shows how and when the resonant instability occurs. In the presence of a background flow it can be anticipated that the damping of the MHD surface mode due to resonant wave transformation is altered, since the flow does not only Doppler shift the continuum frequencies but it also affects the energy of the eigenmodes. At the resonance the flow could drain energy away from the surface mode, which additionally increases the wave damping, but the flow could also be an energy source so that the surface mode gains energy and becomes unstable.

The paper is organized as follows. In Sect. 2 the stationary equilibrium configuration is described while in Sect. 3 the linearized MHD equations for small perturbations are discussed. The analytical dissipative solutions around the resonant surfaces are derived in Sect. 4. In Sect. 5 we shortly describe the numerical procedure for solving the spectrum of the MHD surface modes. The results are presented in Sect. 6 whereas the correspondence between the overstabilities and the negative energy waves is illustrated in Sect. 7. In Sect. 8 we give a summary.

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© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998