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Astron. Astrophys. 342, 300-310 (1999)

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5. Nonlinear MHD slab waves

The single magnetic slab model already briefly introduced in Sect. 2 is used to investigate the behavior of MHD body and surface waves. In the model shown in Fig. 6, the magnetic slab with the physical parameters [FORMULA], [FORMULA], [FORMULA] and [FORMULA] is located at the center of the computational domain, with its axis aligned along the y-axis and with its thickness denoted by b. The external medium is field-free, [FORMULA], and has the same temperature as the slab, [FORMULA]. The magnetic field inside the slab is uniform and the pressure balance in the x-direction is satisfied with the gas density [FORMULA] lower than [FORMULA]. The surface and body waves are generated by imposing longitudinal and transverse perturbations on the slab. In the performed numerical simulations, the slab thickness is assumed to be [FORMULA], which corresponds to two thirds of the acoustic wavelength, [FORMULA], and [FORMULA] grid points are chosen per acoustic wavelength to achieve a good numerical resolution of the generated waves. In addition, the CFL number (see the RHS of Eq. 11) is taken to be 0.4 and the numerical viscosity is chosen to be 0.01; the latter effectively minimizes numerical oscillations.

[FIGURE] Fig. 6. Schematic description of single magnetic slab model used in numerical simulations described in the main text.

5.1. Longitudinal perturbations

The following longitudinal velocity perturbations are imposed on the slab: [FORMULA], at the center of the computational domain ([FORMULA]), with [FORMULA]. Recall that t is the dimensionless time. The computational domain is given by [FORMULA] points with grid distance [FORMULA]. The perturbation extends over the entire width [FORMULA] (of 20 points) of the magnetic slab region. As a result of these perturbations, [FORMULA] is initially zero. The velocity perturbation amplitude is assumed to be perc=0.2, that is [FORMULA] of the sound speed, which means that finite (nonlinear) amplitude waves are generated by the imposed motions.

The snapshots of the resulting velocity field and magnetic field perturbations at [FORMULA] are shown in Figs. 7 to 9. Fig. 7 shows the logarithm of the velocity square, that is, the quantity [FORMULA], where [FORMULA]. It is clearly seen in Fig. 7 that three types of waves are excited, namely, the internal body wave which is confined to the slab ([FORMULA]), the surface wave which propagates along the slab boundaries, and the external acoustic wave ([FORMULA] and [FORMULA]) which is propagating isotropically in the external medium.

[FIGURE] Fig. 7. Square of the velocity plotted on a logarithmic scale for a nonlinear wave generated by the longitudinal perturbation inside the magnetic slab with [FORMULA] at the center of the computational domain.

From a more detailed inspection of the velocity as seen in Fig. 8, one can see that the velocity inside the slab is essentially in the y-direction, which is the propagation direction of the internal body wave. This indicates that the internal body wave is longitudinal as expected from the imposed perturbation. The figure also shows some asymmetry in the resulting velocity field. This is caused by the fact that the longitudinal perturbations, which resemble a piston moving up and down, are imposed first in the upward direction leading thereby to the observed asymmetry in the velocity field. Note also that despite the fact that the initial perturbation is imposed only inside the slab there is significant wave energy leakage to the external medium; the latter is discussed separately in Sect. 6. The magnetic field perturbations displayed in Fig. 9 in the vicinity of the slab axis show the typical behavior of the sausage mode. Clearly, the longitudinal perturbation, which is in the direction of the magnetic field, does not disturb very much the magnetic field lines inside the slab but only at its boundaries, where the surface wave is generated.

[FIGURE] Fig. 8. The wave-induced velocity field generated by longitudinal perturbations with [FORMULA] imposed on a single magnetic slab. Same as in Fig. 7, the snapshot is taken at the dimensionless time [FORMULA].

[FIGURE] Fig. 9. The same as Figs. 7 and 8 but for the magnetic field perturbation.

Fig. 10 shows a comparison of the velocity [FORMULA] on the slab axis between the linear and nonlinear cases. The initial perturbation is imposed at [FORMULA]. Longitudinal body waves are seen propagating along the slab axis in the upward ([FORMULA]) and downward ([FORMULA]) direction with respect to the location of the wave source. The steepening of the initially sinusoidal waves is evident in the case of the nonlinear perturbation ([FORMULA]) but it is not present for the linear case when [FORMULA]. The observed steepening is a consequence of the finite amplitude of the perturbations and will eventually result in the development of sawtooth shock waves. Note also that the decrease of the wave amplitude with distance from the wave source is caused by wave energy leakage to the external medium and the resulting excitation of external acoustic waves (see Sect. 6 for details).

[FIGURE] Fig. 10. Longitudinal velocity [FORMULA] is plotted versus y for linear and nonlinear waves generated by the longitudinal perturbation with [FORMULA] and [FORMULA], respectively. The results are shown for the dimensionless time [FORMULA] and at the location [FORMULA].

5.2. Transverse perturbations

In this numerical simulation the physical parameters are the same as described above. The only difference is that the velocity perturbation is now transverse instead of longitudinal, which means that [FORMULA] and [FORMULA]), with [FORMULA] and [FORMULA]. The obtained results for the velocity field and magnetic field perturbations are shown in Figs. 11 to 13, respectively. As seen in Fig. 11, three types of waves are excited: the internal body wave, the surface wave on the slab boundaries and the external acoustic wave. The internal body wave is essentially transverse (see Fig. 12), which is easily understood considering the form of the imposed perturbation. Note that the surface wave is mainly longitudinal although the imposed perturbation is transverse. The latter and the "vortex structure" seen in this plot are caused by the transition of the fluid motion from the transverse body wave to the external acoustic waves. There is also a prominent wave energy leakage to the external medium; see Sect. 6 for more details.

[FIGURE] Fig. 11. Square of the velocity plotted on a logarithmic scale for a nonlinear wave generated by the transverse perturbation inside the magnetic slab with [FORMULA] at the center of the computational domain.

[FIGURE] Fig. 12. The wave-induced velocity field generated by transverse perturbations with [FORMULA] imposed on a single magnetic slab. Same as in Fig. 11, the snapshot is taken at the dimensionless time [FORMULA].

The observed asymmetry in the x-direction of the magnetic field can be explained by the fact that the very first perturbation is imposed to the right with respect to the slab axis. The swaying of the magnetic field lines which is a characteristic behavior of the kink mode is shown in Fig. 13. It is seen that the field lines act in unison, however, lines located closer to the slab boundaries lag behind. This is a typical behavior of the slow surface mode (see Roberts 1981b). Note that fast surface waves can only exist when the background medium is not isothermal, which is not the case considered here.

[FIGURE] Fig. 13. The same as Figs. 11 and 12 but for the magnetic field perturbation.

Fig. 14 shows the velocities [FORMULA] and [FORMULA] at the slab axis for the downward ([FORMULA]) and upward ([FORMULA]) propagating waves excited at [FORMULA]. The amplitude of the transverse wave [FORMULA] decreases with distance from the wave source as a result of wave energy leakage to the external medium and due to the generation of longitudinal waves [FORMULA] that propagate inside the slab. The process responsible for the excitation of these longitudinal waves is called nonlinear mode coupling and its efficiency has been investigated by Ulmschneider et al. (1991) in their one-dimensional numerical studies of the propagation of magnetic flux tube waves in the solar atmosphere. They suggested that the main reason for the appearance of longitudinal waves is the curvature force that results from the swaying of the magnetic field lines and is perpendicular to these lines. The consequence of this process is that the generated longitudinal waves take away part of the energy carried by the internal transverse body waves and, as a result, the latter are damped when they propagate. It must be noted that the longitudinal waves may form shocks and dissipate the wave energy leading thereby to the heating of the background plasma.

[FIGURE] Fig. 14. Transverse [FORMULA] and longitudinal [FORMULA] velocities are plotted versus y for nonlinear waves generated by the transverse perturbation with [FORMULA]. The results are shown for the dimensionless time [FORMULA] and at the location [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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