As already mentioned, the investigations presented in the previous section are closely related to an analysis of Fuselier et al. (1986). Before drawing a comparison between the results of these two papers and their application to the observations two brief remarks about the approach used in our paper should be made. The first one concerns the special shape used for our mathematically modelled SLAMS in Eq. 4. As can be seen from Eq. 6 the reflection behaviour in adiabatic theory is purely determined by the magnetic field compression and not by the particular shape of the reflecting mirror. The same statement holds for the velocity gain calculated in Eq. 7. These simple relations are modified as soon as the gyroradius of the ions under consideration becomes comparable to the scales or . There are two possibilities for this modification: either a large wave steepening or a large gyroradius, i.e., a large kinetic energy of the incoming particle. The second remark is related to the standard theory of the first-order Fermi process using a diffusion-convection equation (e.g. Parker 1967). While this method is based on quasi-linear theory, i.e., small magnetic field fluctuations with and a stochastical distribution, the approach in our paper starts with the opposite extreme, i.e., single, large amplitude fluctuations with . The finite amplitude waves or waves packets are additional, non-linear steps in the description of a simple first-order Fermi process. On the other hand Kang & Jones (1997) recently showed a good agreement between the continuum approach and Monte Carlo methods using test particles calculations in turbulent magnetic fields where it is assumed that the scattering process is elastic and keeps the velocity distribution isotropic.
The investigations carried out by Fuselier et al. (1986) refered to dependencies of the ion motion on wavelength, phase and amplitude of the waves convected into the shock. In our case the wavelength corresponds to the width of the SLAMS and the dependence on this width corresponds to the statement that the proton dynamics is influenced by the ratio of proton gyroradius and the scalelength of the reflecting mirror. The phase dependence mentioned by Fuselier et al. (1986) corresponds to the dependence of the starting position . Furthermore, it should be emphasized that this "phase" ( in our case) determines the probability of the SLAMS-reflected protons to penetrate into the downstream region.
Looking at the reflection condition from Eq. 6 it is not surprising that the proton behaviour depends on the magnetic field compression of the reflecting mirror. On the other hand it is surprising that the adiabatic behaviour underlying Eq. 6 is still visible in Figs. 4 and 5. There may be two reasons for this rather simple reflection behaviour in comparison with the results found by Fuselier et al. (1986) (see e.g. Fig. 3 in their paper). Firstly, the protons are only occasionally affected by single mirroring structures and for the rest of the time they move in a nearly undisturbed medium and secondly, the magnitude of the compression ratio is larger than the amplitude of the monochromatic MHD waves used by Fuselier et al. (1986). The influence of the magnitude of magnetic field compression on non adiabatic reflected particles can be obtained by a modification of the -conserving reflection in Eq. 6. Assuming that the magnetic moment is changed about an amount this conservation law can be modified according to
with as kinetic energy perpendicular and parallel to the local magnetic field seen by the particle at the time t. From this equation and we obtain for the kinetic energy parallel to the local magnetic field
Furthermore, the reflection condition can be obtained demanding at a certain time . Setting the modified reflection condition can be written as
In comparison to Eq. 6 this equation contains an additional term . Thus, Eq. 10 illuminates, that the critical loss cone angle is fewer affected by for large values . For the calculations presented in this paper the value of lies between 1 and 3, while the relative change of the magnetic moment is about 0.4 according to our test particle calculations with a ratio . This result is an good agreement with the theoretical considerations of Hellwig (1955) who showed that the magnetic moment is a constant at least to terms of order . This result in connection with Eq. 10 explains the rather simple reflection behaviour in Fig. 4 and 5.
Furthermore, Figs. 4 and 5 show the following results: First, the presence of SLAMS in the upstream regions of quasi-parallel shocks prevents the protons reflected at the shock transition from escaping into the upstream region. With increasing magnetic field compression more and more ions are thrown back into the downstream direction and return with a larger energy than their starting energy (see Eq. 7). Thus, the dividing line at (see vertical line in Fig. 4) for shocks without upstream wave phenomena is shifted to smaller angles . The physical meaning of this result is an increase of available free energy in the downstream region, i.e., the upstream wave phenomena lead to a downstream heating. On the other hand it should be mentioned that the velocity change for test particles with a high starting velocity differs from the value computed by Eq. 7 which used the conservation of the magnetic moment. From this point of view the assumption of elastic scattering in the de Hoffmann-Teller frame is not trivial for large amplitude magnetic field fluctuations with a steep rise in the magnetic field.
Secondly, the simple reflection behaviour at single SLAMS depicted in Fig. 4 helps to understand the observations of coherent bunches of cold ions in the vicinity of the quasi-parallel bow shock. This means that as long as the ion velocity is not to high, e.g. holds for and means , these particles are reflected according to a rather simple reflection law (cf. Eq. 6). Furthermore, it becomes clear from Eq. 10 that a high magnetic field compression is rather insensitive to small changes in the magnetic moment during the reflection process. Thus, it seems natural to address the occasional presence of cold ion bunches to a reflection at the steepened waves structures investigated in the previous sections.
© European Southern Observatory (ESO) 1998
Online publication: January 8, 1998