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Astron. Astrophys. 363, 1186-1194 (2000)

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5. Results with Alfvén wave driven at z = 0

In this section a sinusoidal Alfvén wave is driven at the [FORMULA] boundary. In order to prevent the x boundaries influencing the solution where the phase mixing occurs, the x axis is lengthened to [FORMULA] and the [FORMULA] profile of Fig. 1 is shifted so that [FORMULA] is at position [FORMULA].

Fig. 8 shows the oblique propagating fast wave generation by phase mixing. The Alfvén wave generated at [FORMULA] propagates in the direction of increasing z values and experiences phase mixing. Fig. 8 presents the generation of transversal gradients in the initially plane Alfvén wave. The gradients are confined to the region where the Alfvén speed experience the variation. These gradients are a source term on the right hand side of Eq. (27). The fast waves excited by the source, propagate across the field with the fast magnetosonic speed [FORMULA], also shown by Fig. 8. Initially the fast wave growth rate is high, according to previous theoretical expectations. However, it saturates later during the simulation and the fast wave amplitude exhibits periodic modulations. Only every fourth point in each direction was used to produce Fig. 8 as this gave the clearest images. All structures are resolved in these simulations even though some of this resolution is clearly lost in using only one out of 16 available data points.

[FIGURE] Fig. 8. The growth of (a ) transverse gradients [FORMULA] in the Alfvén wave due to phase mixing and (b ) induced fast mode perturbations [FORMULA]. Here the Alfvén wave is driven at [FORMULA].

Fig. 9 contains the oscillations of the driven Alfvén wave ([FORMULA] and [FORMULA]) along the z axis at [FORMULA], as well as the magnetosonic waves generated by the driven wave ([FORMULA], [FORMULA], [FORMULA] and [FORMULA]). The time during the simulation can be read directly from the z axis using the relation [FORMULA] where [FORMULA] is a constant in the z direction, so that as z increases, the time increases. Note that because the amplitudes of the fast and slow magnetosonic components saturate at such a low amplitude there is only a small leakage of energy from the Alfvén wave component into these modes. Thus [FORMULA] and [FORMULA] in Fig. 9 appear as constant amplitude oscillations because the decrease in amplitude of these components is too small to be apparent from these plots.

[FIGURE] Fig. 9. z profiles at time [FORMULA] of the Alfvén wave ([FORMULA] and [FORMULA]), fast magnetosonic wave ([FORMULA], [FORMULA] and [FORMULA]), and the slow magnetosonic wave ([FORMULA]).

By performing a linear analysis similar to Nakariakov et al. (1995 , 1997) on the MHD Eqs. (1) to (4) using the ground state (5) and (6), it is easy to show that the oscillations in [FORMULA], [FORMULA], [FORMULA] and [FORMULA] are double the driven frequency [FORMULA]. Eqs. (31) and (33) show that the oscillations of [FORMULA] and [FORMULA] are determined by the behavior of [FORMULA] in the linear limit.

Fig. 9 shows the Alfvén wave ([FORMULA] and [FORMULA]) propagating in the direction of the magnetic field. At the start of the numerical run the driven amplitude is ramped up to 0.001 over 4 wavelengths. The oscillations in [FORMULA] show the progress of a sound wave. Up till [FORMULA] the oscillations are centered around zero, and in front of the sound wave the oscillations are positive. The interpretation of this phenomenon is trivial: there are two kinds of the longitudinal motions in the system. The first is produced by a slow wave. They are generated by the driver at [FORMULA] and propagate to [FORMULA] with the sound speed [FORMULA], which gives [FORMULA] at [FORMULA]. These perturbations are regular acoustic waves with positive and negative motions of the plasma along the field. The second type of the longitudinal perturbations is driven by the Alfvén wave ponderomotive force (see (34)) and are always positive for an Alfvén wave propagating in the positive direction of z. These nonlinearly driven motions propagate with the Alfvén wave speed [FORMULA] and exist even when [FORMULA] (see Verwichte et al. 1999; Nakariakov et al. 2000 for more detailed discussion). At time [FORMULA] the Alfvén wave has traveled to position [FORMULA] along the z axis.

The [FORMULA] boundary conditions on the velocity components are [FORMULA], [FORMULA] and [FORMULA]. Zero gradient conditions are used on the magnetic field components. These conditions do not drive a pure Alfvén wave but also generate an acoustic wave component. This driven acoustic wave has an amplitude which is proportional to the square of the Alfvén wave. It should be pointed out that the generation of this acoustic wave is by no means an error in the boundary conditions or in the simulation's validity. Photospheric motions will not be generators of pure Alfvén waves and all that is required of driving conditions is that they are physically realistic. The presence of boundary driven acoustic modes does of course introduce another wave into the full set of coupled MHD equations. However, this mode is a second order effect and will not significantly affect the primary Alfvén wave phase mixing. Furthermore, by looking at the results of Fig. 9 beyond [FORMULA], or the periodic boundary results of Sect. 4, we can see the result of phase mixing without this acoustic mode.

The amplitude of [FORMULA] saturates (as shown is Sect. 4) and after saturation it experiences a smooth modulation, the wavelength of which is independent of the driven amplitude. The other parameters that can be changed in the numerical simulation are the frequency [FORMULA] of the driven Alfvén wave and the unperturbed density [FORMULA] gradient. Fig. 10 shows the dependency of the modulation wavelength [FORMULA] on both these parameters. As the driven wavelength is doubled (i.e. [FORMULA] is halved) [FORMULA] doubles in length. When the characteristic length scale is doubled, i.e. [FORMULA] is halved, the wavelength of the modulation [FORMULA] increases by 3/2.

[FIGURE] Fig. 10. Dependence of the modulation in the [FORMULA] wavelength [FORMULA] on the driven frequency [FORMULA] and on [FORMULA].

This modulation, which is also present in the periodic results shown in Fig. 3, is both unexpected and as yet unexplained. A natural interpretation would be that the nonlinearly generated fast mode is beating with some other boundary driven or generated wave. However, in this 2D geometry with a non-sheared magnetic field, this equilibrium cannot support ideal MHD trapped fast waves if [FORMULA] (Roberts 1981) and we see no evidence of the localized structure (in the x direction) that might be expected from a driven quasi-mode (Poedts & Kerner 1991). This leaves no clear candidate to explain this modulation and we must defer its explanation to a later study.

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Online publication: December 5, 2000