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

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4. Saturation levels of magnetosonic waves

The numerical results obtained using a periodic z direction - the direction of the uniform magnetic field [FORMULA] - are presented in this section. An Alfvén wave is initialized at [FORMULA] and the simulation then evolves from that state without further Alfvén wave driving, i.e. at [FORMULA] we set [FORMULA] and [FORMULA].

As the numerical code evolves in time, the Alfvén wave moves at different local speeds due to the [FORMULA] gradient (Fig. 1). Throughout the numerical simulation the Alfvén wavelength [FORMULA] stays practically constant (the nonlinear modification of the Alfvén wavelength is insignificant in this study because the amplitude is weak and the Alfvén waves remain practically linear, see the discussion in Sect. 2.2), so that phase mixing of the Alfvén waves takes place where the background density gradient (and hence the local Alfvén velocity) changes. Fig. 2 presents two contour plots of the [FORMULA] plane in the beginning of the run and at time t=3. These show the phase mixing region centred to the right of [FORMULA] as expected from Fig. 1.

[FIGURE] Fig. 2. A contour plot of the Alfvén wave. The [FORMULA] amplitude is plotted on the [FORMULA] plane at times t=0.5 and t=3.

Slow magnetosonic waves are generated nonlinearly by the Alfvén wave. These are generated at every position in the x direction. Fast magnetosonic waves are generated where the phase mixing of the Alfvén wave occurs. The amplitude of the fast magnetosonic waves grows linearly with time during the initial stages of the numerical runs. However, they soon saturate at levels that are dependent on the amplitude and wavelength of the Alfvén wave, as well as the scale length of the background density gradient. The saturation levels of the slow and the fast magnetosonic waves are independent of the plasma-[FORMULA].

Fig. 3 shows that as the background density scale length decreases, i.e. [FORMULA] increases, the initial rate of linear growth of [FORMULA] increases. Note that Fig. 3 and Fig. 4 are plots of the maximum value of [FORMULA] along the line [FORMULA] versus time and not plots of [FORMULA] at a point. In all cases the amplitude still saturates at a low level compared to the Alfvén wave component (which has an amplitude of [FORMULA]). Lower values of [FORMULA] show linear growth of the [FORMULA] amplitude for longer and end with a larger saturated amplitude. However, decreasing [FORMULA] by two orders of magnitude only approximately doubles the maximum [FORMULA] amplitude. Fig. 4 shows that the initial linear growth of the [FORMULA] amplitude is influenced by the Alfvén wavelength: shorter Alfvén waves causes stronger growth. Once the [FORMULA] amplitude has saturated, there is little distinction between the different Alfvén wavelengths. Fig. 5 shows that the saturated amplitude varies as the square of the Alfvén amplitude.

[FIGURE] Fig. 3. Time evolution of the fast magnetosonic wave ([FORMULA]) amplitude as a function of the background density gradient parameter [FORMULA]. Lines are for [FORMULA] (dotted curve ), [FORMULA] (dashed curve ) and [FORMULA] (solid curve ).

[FIGURE] Fig. 4. Time evolution of the fast magnetosonic wave ([FORMULA]) amplitude as a function of the Alfvén wave length [FORMULA]. Lines are for [FORMULA] (dashed curve ), [FORMULA] (dotted curve ), [FORMULA] (solid curve ) and [FORMULA] (dot-dashed curve ).

[FIGURE] Fig. 5. The saturation levels of the fast ([FORMULA]) and slow ([FORMULA]) magnetosonic wave amplitudes as a function of the Alfvén wave ([FORMULA]) amplitude.

It is the gradient in the x direction of [FORMULA] which is the source of the fast waves. Initially this is zero but grows linearly in time due to phase mixing. This linear growth of the gradient continues throughout the simulations presented in this paper, i.e. there is no saturation of the basic phase mixing process. The natural questions which then arise are why does the amplitude of the fast wave not also continue to grow linearly and why does it saturate at the level it does? A possible explanation of this can be given in terms a greatly simplified model. From the full simulations it is clear that the fast wave saturates at low amplitude (typically proportional to the square of the Alfvén wave amplitude). To a good approximation we can then assume that the phase mixing acts as a source of fast waves without perturbing the Alfvén waves. We therefore represent the generation of fast waves by the generation of 1D sound waves (or fast waves if P is interpreted as a magnetic pressure) with a specified driving term, [FORMULA]. In this case [FORMULA] is the pressure gradient resulting from phase mixing, i.e. it represents the [FORMULA] term in Eq. (18) at a fixed point in z. The simplified equations are then


where [FORMULA] is determined from the gradient of [FORMULA] with [FORMULA] and [FORMULA]. The advantages of this simplified model are that it contains a driver consistent with the full equations but the only other process included is simple linear wave propagation. Results from this model can therefore only be attributed to the interference of linear waves and it removes any possibility of nonlinear wave interactions being an explanation of the derived solution. This is unlike the full set of MHD equations where all possible ideal MHD process are included. Eqs. (37) and (38) are solved numerically using a second order Maccormack scheme. Fig. 6 shows [FORMULA] and v at three times in the solution. The top figures show that the driving term [FORMULA] has the required properties of a transverse gradient generated by phase mixing: the amplitude is growing linearly in time and the number of oscillations in the x direction also increases secularly. The bottom figures show that the amplitude of the generated waves has grown linearly in time but then saturates. This can be seen more clearly in Fig. 7 which shows the solution for v as a function of time at [FORMULA]. All of these results were produced with [FORMULA] [FORMULA] and [FORMULA]. Varying a confirmed that the saturated amplitude scales as [FORMULA] which is also in agreement with the full MHD results. Calculating the maximum v, [FORMULA] when [FORMULA] is varied does not produce results in clear agreement with the results of Fig. 3. With [FORMULA] and [FORMULA] this simple model gives [FORMULA], while [FORMULA] gives [FORMULA] and [FORMULA] gives [FORMULA]. Thus the general behaviour between [FORMULA] and [FORMULA] is consistent with the full simulations, i.e. as the density scale length increases the final saturated amplitude increases, but the scaling over this range is not the same as found from the full MHD equations.

[FIGURE] Fig. 6. Plots of [FORMULA] and v from the solution to Eqs. 37 and 38 at [FORMULA], [FORMULA] and [FORMULA].

[FIGURE] Fig. 7. v as a function of time at [FORMULA].

From these model equations a natural interpretation of the saturation of fast wave generation by Alfvén waves follows from simple wave interference. At early times there is a single maximum in [FORMULA] whose amplitude grows linearly in time. This generates right and left propagating waves whose amplitudes also grow linearly in time. However, at later times there are many peaks in [FORMULA] and while each of these is growing in amplitude neighbouring peaks are not in phase, i.e. their separation is not an integer multiply of the generated wave's wavelength. Indeed, since the number of maxima in [FORMULA] grows and their separation decreases, it is impossible for these to be coherent emitters of fast waves. Thus while the phase mixing continues to generate shorter and shorter scale lengths the generated fast wave amplitude saturation occurs soon after the phase difference across the density ramp generates a second maximum in the transverse gradient of [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000