Note that for sausage wave modes, where =0, the equations describing the magnetoacoustic and torsional waves (i.e., Equations (2, 3)) are decoupled so that mode conversion and absorption through the Alfvén resonance cannot take place (the slow resonance can still operate; see, e.g., Erdélyi, 1997). For the slab geometry,this corresponds to propagation parallel to the magnetic field, i.e., .

To study resonant absorption the following procedure is often undertaken. Whilst outside of the resonant layer the ideal MHD equations can be applied, inside the resonant layer dissipative effects have, in principle, to be taken into account to ensure that the solution remains regular. But unless one is interested in the details of the solution in the resonant layer, this can be avoided. The thickness of the resonant layer, , is proportional to , where is the length scale over which the Alfvén speed varies around the resonance. In the solar atmospheric context can be assumed to be small. The solution in the resonant layer is replaced by a jump relation, which is based upon a Taylor series expansion of the ideal MHD equations around the resonance (Kappraff and Tataronis, 1977; Ionson, 1978; Hollweg, 1987; Hollweg and Yang, 1988; Sakurai et al., 1991a). It is assumed that the quantities that are conserved across the layer in ideal MHD will remain so in weakly dissipative MHD.

Consider a loop model where the internal and external media are homogeneous, except for a loop edge layer of width where the density varies monotonically from to (see Figure 5). A global wave mode is present, which has a frequency which matches the Alfvén frequency at within the Alfvén continuum of the edge layer (see Figure 6). In the internal and external media the solutions are calculated using, e.g., Equation (6). Those solutions contain arbitrary integration constants, which are related to each other with use of certain jump relations at the edge layer. The width of the edge layer, , is considered to be thin, i.e., , but wider than the resonant layer, i.e., . The jump relations are constructed as follows. Inside the edge layer, the system of Equations (2) is Taylor expanded around the resonance using the small parameter (). Thus, a second order differential equation for is derived, which is of the form It has solutions in the form of modified Bessel functions. The total pressure perturbation is described as which is approximately constant across the layer (Hollweg, 1987; Sakurai et al., 1991a). Since for =0 no resonant coupling can occur, the Equation (10) does not apply for that case. Therefore, the jump relation for the total pressure perturbation is simply , where the square brackets denote the difference between the solutions in the right and left limits of tending to zero, respectively. Note that, when the loop is twisted, the total pressure is no longer a conserved quantity (Sakurai et al., 1991a). Similarly, the solution for the radial displacement perturbation can be found to be approximately where . depends on through a logarithmic term, which diverges for tending to zero. The jump relation for is for which the relation has been used. To match the internal and external solutions, the usual jump conditions of continuity of total pressure and radial displacement are now replaced by the above derived jump relations. Depending on the condition that is imposed at , the trapped and/or leaky eigenmodes of the system may be studied (involving a dispersion relation) or the reflection/absorption problem of an externally driven wave that interacts with the loop.By scanning through the frequency of the collective wave, the wave absorption as a function of is studied (see, e.g., the numerical simulations by Poedts et al., 1989). The fractional absorption spectrum often shows well-defined maxima where the absorption reaches 100%. The spatial structure of the excited wave mode at those maxima shows a combination of localised and global behaviour. It is a global wave (e.g., discrete fast eigenmode) with a frequency that lies in the Alfvén continuum and is, therefore, locally coupled to an Alfvén wave. These types of wave modes are known as quasi-modes and they are natural wave modes of the dissipative and inhomogeneous system (Balet et al., 1982; Steinolfson and Davila, 1993; Ofman et al., 1994; Ofman and Davila, 1995; Tirry and Goossens, 1996). Therefore, it is easily understood why maximum absorption occurs when driving at the frequency of a quasi-mode. Also, the presence of steady flows can significantly change the efficiency of resonant absorption (Erdélyi, 1998).

Furthermore, in the absence of flow, these modes are damped (Poedts et al., 1990; Ofman et al., 1994; Wright and Rickard, 1995). From the point of view of the global nature of the mode, the damping is primarily a conversion of energy from the collective to the local. This damping rate has been calculated for various geometries (see, e.g., Lee and Roberts, 1986; Hollweg, 1987; Goossens et al., 1992). Ruderman and Roberts (2002) calculated the damping time, , of a global kink wave in a long, thin loop () in the limits of weak dissipation () and zero plasma-:

This time scale is generally much shorter than the time scale of the dissipative damping of the small-scale perturbations of the local mode in the resonance layer. In Section 3.4 this aspect of rapid mode conversion will be explored further within the context of the observed rapid damping of transverse loop oscillations (Roberts, 2000; Ruderman and Roberts, 2002; Goossens et al., 2002a; Van Doorsselaere et al., 2004).Ofman et al. (1994) and Ofman and Davila (1995) studied numerically nonlinear resonant absorption and found that the large shear velocities produced at the resonance layer are subject to Kelvin-Helmholtz instabilities. The velocity amplitudes derived from linear theory are much larger than the observed velocities from nonthermal broadening of coronal emission lines. This discrepancy may be explained by a turbulent enhancement of dissipation parameters due to the instabilities. Additional complexity is brought by the effects of boundary conditions in the longitudinal direction, e.g., Beliën et al. (1999) examined numerically the effect of the transition region and chromosphere on the resonant absorption in coronal loops. They found that the nonlinear energy transfer from the Alfvén waves to slow magnetoacoustic waves in the lower atmosphere can much diminish the absorption efficiency compared with models of line-tied loops without a lower atmosphere. The driver they considered was, though, monoperiodic and this study should be extended to include more realistic drivers. Furthermore, the heating of the resonance layer would spread due to thermal conduction and heat the lower atmosphere. The resulting chromospheric evaporation enhances the loop density at the resonance layer and, hence, shifts the Alfvén frequency away from resonance, as well as change the quasi-mode frequencies (see, e.g., the discussion in Ofman and Davila, 1995). Ofman et al. (1998a) considered a broad band Alfvén wave driver (see also DeGroof and Goossens, 2002), and coupling to the chromosphere of the loop density with the use of a quasi-static equilibrium scaling law. They found that the heating is concentrated in multiple resonance layers, rather than in the single layer of previous models, and that these layers drift throughout the loop to heat the entire volume. These properties are in much better agreement with coronal observations that imply multithreaded loop structure.

In the developed stage of phase mixing (when ) the Alfvénic perturbations of different magnetic surfaces become uncorrelated with each other, the perturbations decay according to the law (Heyvaerts and Priest, 1983)

for propagating harmonic Alfvén waves, and for standing harmonic Alfvén waves. The decay time due to phase-mixing is proportional to , similar to the dissipative decay of Alfvén waves due to resonant absorption. Figure 7 shows the dissipative decay of propagating Alfvén waves in a 1D plasma inhomogeneity with Alfvén speed profiles of different steepness, compared with the dissipative decay of Alfvén waves in a homogeneous medium (solid curve).If instead of a monochromatic wave in the longitudinal direction a localised Alfvén pulse is considered, Heyvaerts and Priest’s expression for the exponential decay (17) should be replaced by the power law

see Hood et al. (2002) and Tsiklauri et al. (2003) for two methods of derivation and comparison with results of numerical modelling.In the nonlinear regime, growing transverse gradients induce oblique fast magnetoacoustic waves (Nakariakov et al., 1997). Consequently, phase mixing regions may be characterised by the presence of compressible perturbations.

http://www.livingreviews.org/lrsp-2005-3 |
© Max Planck Society and the author(s)
Problems/comments to |