## 1. IntroductionA possible role of magnetohydrodynamic (MHD) waves in accelerating
the high speed solar wind and the heating of plasma in the open
magnetic structures of the solar corona has been discussed since the
pioneering works of Parker (1965). The presence of MHD waves in the
open structures of the solar corona is now beyond doubt. Recently,
upwardly propagating compressive waves were detected in polar plumes
(DeForest & Gurman 1998) with SOHO EIT and interpreted as slow
magnetoacoustic waves (Ofman et al. 1999); non-thermal broadening of
coronal emission lines found with SOHO UVCS is most probably
associated with MHD waves (Ofman & Davila 1997b, Ofman et al.
1998). Also, Alfvén waves are observed In this study, we concentrate our attention on the evolution of Alfvén waves during their propagation through the Sun's atmosphere. A theory of Alfvén waves in coronal holes has to incorporate the effects of the spherical geometry and transversal structuring, along with nonlinear and dissipative effects. Some of these effects are already well-understood. In homogeneous plasmas, elliptically polarized plane Alfvén waves of a finite amplitude are described by the vector Cohen-Kulsrud equation (Cohen & Kulsrud 1974). Solutions of this equation show nonlinear self-interaction of the waves through forced density perturbations and longitudinal motions. Note that while circularly polarized Alfvén waves in the homogeneous plasmas are an exact, although unstable, nonlinear solution of magnetohydrodynamics, Alfvén waves of linear and elliptic polarization are not. They drive second-order fluctuations in velocity and density (see, e.g., Hollweg 1971), which leads to the self-interaction. Including dissipative processes into the model, Kennel et al. (1990) have generalized the Cohen-Kulsrud equation to the so-called Cohen-Kulsrud-Burgers equation. In particular, the Cohen-Kulsrud-Burgers equation describes evolution of the Alfvèn waves into a rotational discontinuity. In the limiting case of a linearly polarized wave, the pair of equations of Cohen-Kulsrud-Burgers decouples and waves of different polarization are described by two independent scalar equations. These are called the scalar Cohen-Kulsrud-Burgers equations. In inhomogeneous plasmas, Alfvén waves are not the exact solution of the MHD set of equations and always evolve. Structuring of the plasma in the direction transverse to the magnetic field leads to Alfvén wave phase mixing (Heyvaerts & Priest 1983), accompanied by nonlinear generation of magnetoacoustic waves (Nakariakov et al. 1997, 1998). Obliquely propagating fast magnetoacoustic waves are generated with secular efficiency, while the slow wave generation is similar to the homogeneous case. However, the efficiency of fast wave generation is determined by the Alfvén wave amplitude, longitudinal wave number and steepness of the transverse inhomogeneity. Consequently, if the transverse gradients are not very high, this effect can be neglected in an initial stage, with respect to the generation of slow waves. Linear aspects of the MHD wave dynamics in the spherically stratified open corona are well understood. The density stratification and magnetic field divergence affect the radial dependence of the Alfvén speed. In particular, this leads to reflection of low frequency Alfvén waves from the inhomogeneity (e.g. An et al. 1990) or, more precisely, from regions where the Alfvén speed increases steeply with distance from the Sun. Alfvén waves of higher frequencies are propagating through the inhomogeneity without significant reflection and can be described by the single wave approximation, or WKB (see, e.g. Hollweg 1990, Barnes 1992, and discussion therein). The stratification affects the propagating waves too, changing the wave amplitude. Moreover, recent consideration of Alfvén wave phase mixing in weakly stratified open magnetic structures has shown that the efficiency of Alfvén wave phase mixing can be dramatically affected by stratification (Ruderman et al. 1998). The main progress in the understanding of nonlinear spherical Alfvén waves is connected with numerical experiments. Ofman & Davila (1997a) have simulated numerically the evolution of initially monochromatic weakly nonlinear Alfvèn waves in a radially divergent, gravitationally stratified magnetic structure in an isothermal plasma with an enhancement in the Alfvèn speed in the transverse direction. Such a structure can be considered as a vertically stratified magnetic flux tube. It was found that the Alfvèn waves quickly generate longitudinal motions, with sharp asymmetric gradients, which possessed solitary wave properties. The longitudinal waves propagate at supersonic phase speed. Peaks of perturbations of the longitudinal velocity and density were found to be in phase. The distance between the peaks increases with the Alfvèn speed. These soliton-like structures were interpreted by authors as a sequence of slow magnetoacoustic solitons. Parametric studies showed that the solitary waves (which, actually, should be called "cnoidal" waves) occur in a broad range of simulation parameters (Ofman & Davila 1998). Torkelsson & Boynton (1998) have numerically simulated spherical nonlinear linearly polarized Alfvén waves in a spherically stratified static atmosphere. The attention was concentrated on 1D effects. Wave frequencies exceeded the Alfvén wave cut-off frequency, so the waves were propagating. As in the simulations of Ofman & Davila (1997a, 1998), the Alfvén waves were found to self-interact through nonlinearly generated longitudinal motions and perturbations of the density, steepening into shock waves. Torkelsson & Boynton (1998) argued against the interpretation of these waves as solitons, based on the fact that the same kind of waves showed up in one-dimensional simulations, in which solitary waves could not appear. (However, the solitary waves can be actually auto-waves or dissipative solitons (see, e.g. Nakariakov & Roberts 1999), formed by a balance between geometrical amplification and dissipation.) A significant fraction of the Alfvén wave energy was found to be dissipated. The authors also reported the appearance of oscillations at frequency lower than the driven frequency, which were interpreted as beating between the frequencies of the incident and nonlinearly generated reflected Alfvén waves. In the nonlinear case, in the presence of the induced longitudinal motions, these frequencies do not coincide with each other and the beating appears. In this paper, we investigate the combined effect of spherical stratification, finite amplitude and dissipation on the propagation of Alfvén waves. These three effects are supposed to be weak, modifying the wave amplitude and shape slowly with height and time. This allows us to apply the method of slowly varying amplitudes and derive an evolutionary equation for the waves. The analytical approach allows us to extract the main features of the dynamics and investigate the physical mechanisms responsible for the wave evolution. In Sect. 2, the model and governing equations considered are discussed. The derivation of an evolutionary equation, which is a spherical analog of the Cohen-Kulsrud-Burgers equation is presented in Sect. 3. Solutions of the equation are discussed in Sect. 4. In Sect. 5, a discussion of results obtained is presented. © European Southern Observatory (ESO) 2000 Online publication: December 17, 1999 |