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Astron. Astrophys. 330, 381-388 (1998)

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1. Introduction

In magnetohydrodynamics (MHD) shock waves are idealized as discontinuities without any internal structure, while real measurements show a more or less rich inner life of these structures ordered by the shock velocity and geometry (see Kennel et al. 1985, as a review). From a microscopic point of view the difference between quasi-parallel and quasi-perpendicular shock waves can be understood analysing the ion movement. Thus, the simple MHD description of streaming fluids becomes modified because of ion reflections at the shock transition zone. The reasons for reflection are magnetic mirror effects due to magnetic field compressions (e.g. Sonnerup 1969) and/or an electric potential rise inside the transition zone of the shock (e.g. Goodrich & Scudder 1984). The reflection process itself can be described either as adiabatic processes conserving the "magnetic moment" [FORMULA] or not [FORMULA] -conserving reflections (e.g. specularly reflected ions, Paschmann et al. 1982). For steady, uniform magnetic fields on both sides of the shock the guiding center velocity of ions specularly reflected at the shock transition is directed upstream if the angle between the unperturbed upstream magnetic field and the shock normal [FORMULA] is smaller than [FORMULA] and downstream if [FORMULA] (Gosling et al. 1982). For shock waves with [FORMULA] the gyromotion of the reflected ions causes them to reencounter the shock even though the guiding center velocity is directed upstream (Schwartz et al. 1983).

Most of the information from shock waves in space plasmas comes from the various satellite missions investigating the bow shock of the earth. They found that the upstream region of the bow shock is populated with different types of superthermal ion distributions. At least four distinct classes have been identified: "reflected ions" (Asbridge et al. 1968), "intermediate ions" (Paschmann et al. 1979), "diffuse ions" (Gosling et al. 1978) and "gyrating ions" (Gosling et al. 1982, Eastmann et al. 1981). Furthermore, a variety of low-frequency magnetic field fluctuations are closely related with these different ion populations backstreaming from the bow shock (for a review see Russell & Hoppe 1983).

Gyrating ion distributions are typically associated with large-amplitude monochromatic MHD-like waves (Thomsen 1985) and the diffuse and intermediate protons with ULF (ultra-low-frequencies in the order of [FORMULA]) waves (Paschmann et al. 1979), so-called shocklets (Hoppe et al. 1981) and SLAMS (Schwartz et al. 1992). Shocklets appear as steepened waves with amplitudes in the order of the backgound magnetic field (Hoppe et al. 1981). SLAMS were characterized by Schwartz et al. (1992) as well defined single magnetic structures with large amplitudes of about two or more times the background magnetic field and short durations of typically 10 s. Fig. 1 shows the magnetic field and density behaviour at a typical SLAMS measured by the AMPTE/IRM satellite.

[FIGURE] Fig. 1. Magnetic field and density behaviour associated with typical SLAMS occurring on October 30, 1984. The first three panels show the magnetic field components in a minimum variance frame; the fourth panel shows the magnitude of the magnetic field. The fifth panel displays a rise in the electron particle number density associated with the SLAMS (measurement of the AMPTE/IRM satellite).

The relationship between superthermal ions and the different kinds of waves has been reviewed by Thomsen (1985) from the observational point of view and was analysed by numerical particle simulations of supercritical, quasi-parallel shock waves. Diffuse ions in the far-upstream region generate ULF waves propagating along the ambient magnetic field lines. Because of the super-Alfvénic speed of the solar wind they are convected back to earth. During their approach to the bow shock they steepen into shocklets and subsequently into SLAMS. The numerical simulations of Omidi & Winske (1990) showed that the original polarisation of the small-amplitude waves changes from an elliptical (right-handed in the plasma rest frame) to a linear behaviour. Further simulations (Scholer et al. 1992) also confirmed the observed quasi-planar structure of ULF waves and SLAMS (Mann et al. 1994). An analytical approach describing low frequency plasma waves with finite amplitudes and their steepening into SLAMS was carried out by Malara & Elaoufir (1991) and Mann (1995) using non-linear MHD wave theory. These investigations showed that SLAMS can be regarded as simple magnetohydrodynamic waves, i.e., the magnetic field components can be described as functions of the form [FORMULA] ([FORMULA]), where [FORMULA] is the propagation velocity of the SLAMS.

Looking at this scenario it should be clear that the aforementioned assumption of a steady uniform magnetic field outside the shock transition zone as assumed by Schwartz et al. (1983) is no longer valid. Some of the ions reflected at the shock transition will be reflected back by the incoming wave structures. Especially, coherent bunches of nearly specularly reflected ions are observed in the supercritical, quasi-parallel region of the earth's bow shock (Gosling et al. 1989). Such reflected ions are not always present and seem to be related with the occurence of density fluctuations with shock like features (Onsager et al. 1990).

The aim of this paper is to study the ion motion in such arrangements of moving magnetic mirrors (SLAMS) to get insight in the basic microphysical processes. Thus, our investigations are closely related to an analysis of Fuselier et al. (1986). In both approaches the scattering magnetic field fluctuations that modify the ion movement at quasi-parallel shock waves are based on plasma wave observations at the earth's bow shock. Fuselier et al. (1986) studied the ion movement of specularly reflected ions under the influence of monochromatic MHD-waves with a finite amplitude ([FORMULA]), while our paper is addressed to steepened wave packets with large amplitudes compared to the background magnetic field ([FORMULA]). In the first case the ions move under a permanent influence of a weak magnetic field perturbation and in the second case the ions are only affected occasionally by the contact with single wave packets. The model for the description of the SLAMS, i.e., the electric and magnetic fields used for our test particle calculations, is presented in the next section. The results of the test particle calculations are presented in Sect. 3, where we also discuss our results in the framework of adiabatic theory. Furthermore, we compare our results with those obtained by Fuselier et al. (1986), discuss the validity of adiabatic theory and try to explain the observed coherent bunches of reflected ions (see Sect. 4).

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© European Southern Observatory (ESO) 1998

Online publication: January 8, 1998
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