A&A 429, 767-777 (2005)
DOI: 10.1051/0004-6361:20041494
B. M. Shergelashvili^{1,}^{} - T. V. Zaqarashvili^{2} - S. Poedts^{1} - B. Roberts^{3}
1 - Centre for Plasma Astrophysics, Katholieke
Universiteit Leuven, Celestijnenlaan 200B, 3001
Leuven, Belgium
2 - Abastumani
Astrophysical Observatory, Kazbegi Ave. 2a, Tbilisi 380060,
Georgia, USA
3 - School of
Mathematics and Statistics, University of St. Andrews, St.
Andrews, Fife, KY16 9SS, Scotland, UK
Received 18 June 2004 / Accepted 23 August 2004
Abstract
The recently suggested swing interaction between fast
magnetosonic and Alfvén waves (Zaqarashvili & Roberts
2002a) is generalized to inhomogeneous media. We show that
the fast magnetosonic waves propagating across an applied
non-uniform magnetic field can parametrically amplify the Alfvén
waves propagating along the field through the periodical variation
of the Alfvén speed. The resonant Alfvén waves have half the
frequency and the perpendicular velocity polarization of the fast
waves. The wavelengths of the resonant waves have different values
across the magnetic field, due to the inhomogeneity in the
Alfvén speed. Therefore, if the medium is bounded along the
magnetic field, then the harmonics of the Alfvén waves, which
satisfy the condition for onset of a standing pattern, have
stronger growth rates. In these regions the fast magnetosonic
waves can be strongly "absorbed'', their energy going in
transversal Alfvén waves. We refer to this phenomenon as "Swing Absorption''. This mechanism can be of importance in
various astrophysical situations.
Key words: magnetohydrodynamics (MHD) - magnetic fields - waves - Sun: sunspots - Sun: oscillations
The interaction between different MHD wave modes may occur either due to nonlinearity or inhomogeneity. The nonlinear interaction between waves as resonant triplets (multiplets) is well developed (e.g., Galeev & Oraevski 1962; Sagdeev & Galeev 1969; Oraevski 1983). Moreover, the nonlinear interaction of two wave harmonics in a medium with a steady background flow can also be treated as a resonant triplet, with the steady flow being employed as a particular wave mode with zero frequency (Craik 1985). Other mechanisms of wave damping (or energy transformation) are related to the spatial inhomogeneity of the medium, including phase mixing (Heyvaerts & Priest 1983; Nakariakov et al. 1997) and resonant absorption (Ryutova 1977; Ionson 1978; Rae & Roberts 1982; Hollweg 1987; Poedts et al. 1989; Ofman & Davila 1995). Due to inhomogeneity, different regions of the medium have different local frequencies. Resonant absorption arises in those regions where the frequency of an incoming wave matches the local frequency of the medium, i.e. where , with the frequency of the incoming wave and the local frequency. The mechanical analogy of this process is the ordinary mathematical pendulum forced by an external periodic action on the pendulum mass. When the frequencies of the periodic external force and the pendulum are the same the force can resonantly amplify the pendulum oscillation.
An additional issue, linked to MHD wave coupling, is the possibility of mutual wave transformations due to an inhomogeneous background flow. In numerous studies it has been shown that the non-modal (Kelvin 1887; Goldreich & Linden-Bell 1965) temporal evolution of linear disturbances (due to the wave number "drift'' phenomenon) brings different types of perturbations into a state in which they satisfy the resonant conditions so that wave transformations occur (e.g. see Chagelishvili et al. 1996; Rogava et al. 2000).
Recently, another kind of interaction between different MHD wave modes, based on a parametric action, has been suggested (Zaqarashvili 2001; Zaqarashvili & Roberts 2002a,b; Zaqarashvili et al. 2002,2004). In this case the mechanism of wave interaction originates from a basic physical phenomenon known in classical mechanics as "parametric resonance'', occurring when an external force (or oscillation) amplifies the oscillation through a periodical variation of the system's parameters. The mechanical analogy of this phenomenon is a mathematical pendulum with periodically varying length. When the period of the length variation is half the period of the pendulum oscillation, the amplitude of the oscillation grows exponentially in time. Such a mechanical system can consist of a pendulum (transversal oscillations) with a spring (longitudinal oscillations). A detailed description of such a mechanical system is given in Zaqarashvili & Roberts (2002a) (hereinafter referred to as Paper I).
When an external oscillation causes the periodical variation of either the wavelength (Zaqarashvili et al. 2002,2004) or the phase velocity (Zaqarashvili 2001; Zaqarashvili & Roberts 2002b, and Paper I) of waves in the system, then the waves with half the frequency of the external oscillations grow exponentially in time. Energy transfer occurs between different MHD wave modes in different situations. One specific case of wave coupling takes place between fast magnetosonic waves propagating across an applied magnetic field and Alfvén waves propagating along this field. It has been shown that the standing fast magnetosonic wave, which manifests itself through harmonic variations of the density and the magnetic field, amplifies the Alfvén waves with a perpendicular velocity polarization (Paper I). The frequency of the resonant Alfvén waves was found to be half of the fast wave frequency. In Paper I the ambient medium was considered to be homogeneous. Therefore, the resonant Alfvén waves had the same wavelength everywhere. However, if the medium is inhomogeneous across the magnetic field, the wave length of the resonant Alfvén waves will have different values in different regions. If the medium is bounded along the magnetic field lines, as, for example, in photospheric line-tying of coronal magnetic field lines, then Alfvén waves will be amplified in specific regions only, viz. where the conditions for the onset of a standing pattern are satisfied. In these regions the fast waves can be strongly absorbed by Alfvén waves with half their frequency. To study this phenomenon, we first consider fast magnetosonic waves propagating across a non-uniform magnetic field. Then we study the absorption of these waves by perpendicularly polarized Alfvén waves propagating along the field.
Figure 1: Schematic view of the inhomogeneous background configuration and the directions of wave propagation (polarization). | |
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We consider an equilibrium magnetic field
directed along the z axis of a Cartesian coordinate system,
.
The equilibrium magnetic
field
and density
are inhomogeneous in x.
The
force balance condition, from Eq. (2), gives the
total (thermal + magnetic) pressure in the equilibrium to be
a constant:
(6) |
Equations (1)-(4) are linearized around the static ( ) equilibrium state (5). This enables the study of the linear dynamics of magnetosonic and Alfvén waves. A schematic view of the equilibrium configuration is shown in Fig. 1.
The solution of this equation for different particular equilibrium conditions can be obtained either analytically or numerically. Equation (11) describes the propagation of a fast wave with speed , where is the sound speed and is the Alfvén speed.
We study Eq. (11) in accordance with the boundary
conditions:
(14) |
(15) |
(16) |
(17) |
(18) |
Combining Eqs. (20) and (21) we obtain the
following second order partial differential equation:
(22) |
(23) |
(28) |
Figure 2: The profiles of the equilibrium quantities plotted against dimensionless x/L_{x} coordinate: density (panel A)), pressure (panel B)), the magnetic field (panel C)) and plasma beta (panel D)). | |
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Equation (29) has a resonant solution when the
frequency of the Alfvén mode
is half of :
(32) |
(35) |
Figure 3: Sample solutions of the standing fast magnetosonic modes. Panel A) zeroth-order harmonic n=0, period T=5.6069 min, ; Panel B) second-order harmonic n=2, period T=1.8514 min, . | |
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In this section we consider in detail the process of swing
absorption of fast waves into Alfvén waves. We consider a
numerical study of Eq. (11) subject to the boundary
conditions (12). We obtained a numerical solutions of
Eqs. (20) and (21) by using the standard
Matlab numerical code for solving sets of ordinary differential
equations. We study, as an example, the case of a polytropic
plasma when both the thermal and magnetic pressures are linear
functions of the x coordinate:
(40) |
Table 1: Values of the constant parameters used in the calculation of our illustrative solutions. The dimension of C isg^{1/2} cm /dyn .
The values of all constant parameters are given in Table 1. We took arbitrary values of parameters, but they are somewhat appropriate to the magnetically dominated solar atmosphere (say the chromospheric network). In Fig. 2 corresponding equilibrium profiles of the density (panel A), pressure (panel B), the magnetic field (panel C) and plasma beta (panel D) are shown. In Fig. 3 we show the profiles of for the standing wave solutions, for two cases with different modal "wavelength''. Panels A and B, respectively, correspond to the characteristic frequencies: s^{-1} (period 5.61 min) and s^{-1} (period 1.85 min).
For the configuration described by the equilibrium profiles (37)-(39) the resonant condition
(36) yields the areas of swing absorption (located
along the x axis) as the solutions of the following equation:
(41) |
In order to examine the validity of the approximations we made during the analysis of the governing Eq. (24), we performed a direct numerical solution of the set of Eqs. (20) and (21) and obtained the following results. The Alfvén mode m=3 is amplified effectively close to the resonant point . This is shown on panel C_{1} of Fig. 4. Far from this resonant point, the swing interaction is weaker, as at the point (panel B_{1}). For the Alfvén mode m=4 we have the opposite picture: the area of "swing absorption'' is situated around the point x_{0,4} (see panel B_{2}, Fig. 4) and the rate of interaction between modes decreases far from this area, as at point x_{0,3} (panel C_{2}, Fig. 4). In these calculations we took .
Figure 4: Results of numerical simulations for the standing fast magnetosonic mode, shown on panel A in Fig. 3. Panel A) the curves of F_{1} (Eq. (42)) (solid line) and F_{2}(Eq. (43)) (dotted line) vs. fractional distance x/L_{x} for the zeroth-order standing fast mode n=0 and the standing Alfvén mode with m=3; panel B) the numerical solution of the Alfvén mode m=3 amplitude at location ; panel C_{1}) the numerical solution for mode m=3 at ; panel A) as in panel A) for mode m=4; panel B) as in panel B) for mode m=4; panel C) as in panel C_{1}) for mode m=4. | |
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Figure 5: As in Fig. 4 for the second-order harmonic of the fast magnetosonic mode shown on Panel B in Fig. 3 (n=2) and the standing Alfvén with m=8 and m=10. The numerical solutions respectively are given at distances (panels B) and B)), (panels C) and C)). Dashed line in panel A) is curve of F_{1} (42) corresponding to the mode m=9. | |
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Figure 6: As in Fig. 5 for the standing Alfvén modes with m=11 and m=12 at respective resonant points (panels B) and B)), (panels C) and C)). | |
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The results of direct numerical calculations for these Alfvén wave harmonics are shown separately in Figs. 5-6. In Fig. 5 (panel A_{1}), we show the curves corresponding to the mode numbers n=2, m=8. The calculations were performed at two points (panel B_{1}) and (panel C_{1}). In addition, the results of calculations for the mode m=10, under the same conditions, are shown in panels A_{2}, B_{2} and C_{2}. It can be seen from these plots that the m=8 Alfvén mode is amplified effectively within the resonant area close to the resonant point x_{2,8}(see panel C_{1}), which corresponds to the harmonic of the standing Alfvén mode with half the frequency. On the other hand, in panels B_{1} we see that the action of the fast mode on the Alfvén mode is almost negligible outside the swing amplification area. Correspondingly, we get similar results for the Alfvén modes m=10 (Fig. 5, panels A_{2}, B_{2} and C_{2}), and for m=11 and m=12 (Fig. 6) (in the latter case the calculations were done at points and ). Again, all these standing Alfvén modes gain their energy from the fast magnetosonic mode n=2 only at the resonant points x_{2,10} (for m=10, panel B_{2} in Fig. 5), x_{2,11} (for m=11, panel C_{1} in Fig. 6) and x_{2,12} (for m=12, panel B_{2} in Fig. 6), respectively.
As a conclusion from the above analysis one can claim that the zeroth-order harmonic of the standing fast magnetosonic mode propagating across the magnetic field lines, in the system with characteristic length scales L_{x} and L_{z}=6.5L_{z}, can be effectively absorbed only by the standing Alfvén modes with modal numbers m=3 and m=4 within the resonant areas, respectively, around the point of swing absorption x_{0,3} and x_{0,4}. On the other hand, the second-order harmonic of the standing fast mode n=2 is absorbed by four harmonics of the standing Alfvén modes, m=8, 10, 11, and 12, with respective locations of the resonant areas of swing absorption at x_{2,8}, x_{2,10}, x_{2,11} and x_{2,12}. A similar analysis can be performed for the case of any other equilibrium configuration and corresponding harmonics of the standing fast magnetosonic modes.
The most important characteristic of swing absorption is that the velocity polarization of the amplified Alfvén wave is strictly perpendicular to the velocity polarization (and propagation direction) of fast magnetosonic waves. This is due to the parametric nature of the interaction. For comparison, the well-known resonant absorption of a fast magnetosonic wave can take place only when it does not propagate strictly perpendicular to the magnetic flux surfaces and the plane of the Alfvén wave polarization. In other words, the energy in fast magnetosonic waves propagating strictly perpendicular (i.e. ) to the magnetic flux surfaces cannot be resonantly "absorbed'' by Alfvén waves with the same frequency polarized in the perpendicular plane. This is because the mechanism of resonant absorption is analogous to the mechanical pendulum undergoing the direct action of an external periodic force. This force may resonantly amplify only those oscillations that at least partly lie in the plane of force. On the contrary, the external periodic force acting parametrically on the pendulum length may amplify the pendulum oscillation in any plane. A similar process occurs when the fast magnetosonic wave propagates across the unperturbed magnetic field. It causes a periodical variation of the local Alfvén speed and thus affects the propagation properties of the Alfvén waves. As a result, those particular harmonics of the Alfvén waves that satisfy the resonant conditions grow exponentially in time. These resonant harmonics are polarized perpendicular to the fast magnetosonic waves and have half the frequency of these waves. Hence, for standing fast magnetosonic waves with frequency , the resonant Alfvén waves have frequency .
In a homogeneous medium all resonant harmonics have the same wavelengths (see Paper I). Therefore, once a given harmonic of the fast and Alfvén modes satisfies the appropriate resonant conditions (Eqs. (23) and (25) in Paper I), then these conditions are met within the entire medium. Thus, in a homogeneous medium the region where fast modes effectively interact with the corresponding Alfvén waves is not localized, but instead covers the entire system. However, when the equilibrium is inhomogeneous across the applied magnetic field, the wavelengths of the resonant harmonics depend on the local Alfvén speed. When the medium is bounded along the unperturbed magnetic field (i.e. along the zaxis), the resonant harmonics of the standing Alfvén waves (whose wavelengths satisfy condition (36) for the onset of a standing pattern) will have stronger growth rates. This means that the "absorption'' of fast waves will be stronger at particular locations across the magnetic field. In the previous section we showed numerical solutions of standing fast magnetosonic modes for a polytropic equilibrium ( ) in which the thermal pressure and magnetic pressure are linear functions of x. Further, we performed a numerical simulation of the energy transfer from fast magnetosonic waves into Alfvén waves at the resonant locations, i.e. the regions of swing absorption.
The mechanism of swing absorption can be of importance in a veriety of astrophysical situations. Some possible applications of the mechanism are discussed briefly in the following subsections.
We have shown that swing interaction (Zaqarashvili 2001; Zaqarashvili & Roberts 2002a) may lead to fast magnetosonic waves, propagating across a non-uniform equilibrium magnetic field, transforming their energy into Alfvén waves propagating along the magnetic field. This process differs from resonant absorption. Firstly, the resonant Alfvén waves have only half the frequency of the incoming fast magnetosonic waves. Secondly, the velocity of the Alfvén waves is polarized strictly perpendicular to the velocity and propagation direction of the fast magnetosonic waves. The mechanism of swing absorption can be of importance in the dynamics of the solar atmosphere, the Earth's magnetosphere, and in astrophysical plasmas generally.
Here we presented the results of a numerical study of the problem, illustrating the process for two particular harmonics of the standing fast modes in a specific equilibrium configuration. Further numerical and theoretical analysis is needed in order to explore the proposed mechanism of swing absorption in the variety of inhomogeneous magnetic structures recorded in observations. Such tasks are to be the subject of future studies.
Acknowledgements
This work has been developed in the framework of the pre-doctoral program of B. M. Shergelashvili at the Centre for Plasma Astrophysics, K. U. Leuven (scholarship OE/02/20). The work of T. V. Zaqarashvili was supported by the NATO Reintegration Grant FEL.RIG 980755 and the grant of the Georgian Academy of Sciences. These results were obtained in the framework of the projects OT/02/57 (K. U. Leuven) and 14815/00/NL/SFe(IC) (ESA Prodex 6). We thank the referee, Dr. R. Oliver, for constructive comments on our paper.