## 4. Saturation levels of magnetosonic wavesThe numerical results obtained using a periodic As the numerical code evolves in time, the Alfvén wave moves at different local speeds due to the gradient (Fig. 1). Throughout the numerical simulation the Alfvén wavelength 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 plane in the beginning of the run and at time t=3. These show the phase mixing region centred to the right of as expected from Fig. 1.
Slow magnetosonic waves are generated nonlinearly by the
Alfvén wave. These are generated at every position in the
Fig. 3 shows that as the background density scale length decreases, i.e. increases, the initial rate of linear growth of increases. Note that Fig. 3 and Fig. 4 are plots of the maximum value of along the line versus time and not plots of 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 ). Lower values of show linear growth of the amplitude for longer and end with a larger saturated amplitude. However, decreasing by two orders of magnitude only approximately doubles the maximum amplitude. Fig. 4 shows that the initial linear growth of the amplitude is influenced by the Alfvén wavelength: shorter Alfvén waves causes stronger growth. Once the 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.
It is the gradient in the where is determined from the
gradient of with
and
. 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 and
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 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 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 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 . © European Southern Observatory (ESO) 2000 Online publication: December 5, 2000 |