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

2. Basic model assumptions

Test particle trajectories in the presence of large amplitude magnetic field fluctuations are obtained by solving the equation of motion for single ions in given electric and magnetic fields of SLAMS in front of quasi-parallel shock waves. The calculations are performed in the shock rest frame under the following assumptions (cf. Fig. 2): 1) The ions are reflected off a plane shock front with vectors (unperturbed upstream magnetic field), (shock normal) and (upstream plasma bulk velocity) lying in the (x,z)-plane. 2) Then, the ions move upstream under the combined influence of the ambient magnetic field and the motional electric field (along the y-axis) superimposed with the magnetic and electric field of the SLAMS and (i.e., and ). 3) The SLAMS are considered as planar simple waves moving in the negative x-direction, i.e., the magnetic field components can be described as (). The inclination between the propagation direction and the unperturbed magnetic field should be smaller than (see e.g. Mann et al. 1994). The electric field is computed using Faraday's law . 4) A self consistent generation and modification of the SLAMS by the reflected ions is not taken into consideration.

Fig. 2. Geometry of a supercritical, quasi-parallel shock wave with approaching SLAMS in the upstream region: The shock transition zone (grey shaded area) is at rest, the upstream and downstream region are lying at the right and left hand side of this area, respectively. The unperturbed upstream magnetic field and the shock normal spread the (x,z)-plane; x, y, and z are chosen mutually orthogonal. The SLAMS move along the x-axes and are described by their magnetic field compression (, see text) and by their distance from the transition zone. , , and are the angles between and , between the propagation direction of the SLAMS and , and between the starting velocity of the test particle and , respectively.

The equation of motion can be written in dimensionless quantities using natural scales T and L for time- and space-coordinates, where is the inverse proton cyclotron frequency (, proton mass and e, elementary charge) and the proton inertial length (, proton plasma frequency, , upstream particle number density, c, velocity of light and , dielectric field constant). Thus, we obtain

with and . The velocity is normalized to the Alfvén velocity . Using the geometrical conditions depicted in Fig. 2 the dimensionless upstream magnetic field in the shock rest frame is given by

with the unperturbed magnetic field as . The magnetic field of the SLAMS is given by and the second term in the -component, with . and p () denote the maximum field compression and the polarisation of the SLAMS, respectively. For plane wave structures varying only with the x-coordinate the -component must be a constant because of . The -term represents a noncoplanar field component of SLAMS. An example for the magnetic field behaviour at SLAMS is shown in Fig. 1.

The dimensionless electric field can be written as a sum of the motional electric field , superimposed with the electric field induced by the SLAMS motion. Using Faraday's law we obtain

The integration constants have been chosen in such a way that the pure motional electric field is obtained outside the SLAMS. Furthermore, it should be noted that the induction equation gives no information about the -component.

In general, the behaviour of test particles in the electromagnetic fields given by Eqs. 2 and 3 shows two different effects. The first one leads to a small population of clearly superthermal particles accelerated due to the wave electric field parallel to the local magnetic field. Mechanisms of this kind are described for example in Claßen & Mann (1997). Furthermore, the hybrid simulations of Kucharek and Scholer (1991) showed that this process leads to a population of diffusive, superthermal ions, with a ratio of accelerated particles in the order of 1 % . The main effect however is a reflection of the incoming ions with only a slight change of the particle velocity due to a grad-B-drift in the magnetic field of approaching SLAMS. Since we are interested in the behaviour of just this weakly superthermal ions we can simplify our description assuming a vanishing noncoplanar component, i.e., and . Thus, we choose in Eq. 2 and in Eq. 3. This assumption is in general not a substantial restriction. Even if we take a polarisation of SLAMS into account the reflection conditions and the energy gained after reflection are not changed. The only effect caused by the noncoplanar component is a energy gain inside the SLAMS (see e.g. Claßen and Mann 1997), but this energy is lost while the particles move out of the SLAMS.

As a special realisation of such modelled SLAMS we took a function given by

Here denotes the wave number of the ULF waves (wavelength ) from which the SLAMS develop and L () is a typical length scale of SLAMS. In correspondence with the statistical analysis of 18 SLAMS observed at the earth's bow shock (Mann et al. 1994) we took and . Furthermore, the statistical analysis showed that the lateral dimension of SLAMS (y-direction in Fig. 2) is about 100 proton inertial lengths justifying the assumption of SLAMS as plane structures. These scales seem to be typical for steepened wave structures and were confirmed by numerical simulations (e.g. Omidi & Winske 1990, Scholer et al. 1992). Finally, it should be mentioned that the basic results presented in this paper depend on the used length scales and not on the explicit shape of given by Eq. 4 (see Sect. 4).

© European Southern Observatory (ESO) 1998

Online publication: January 8, 1998
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