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Space Plasma Simulations and Models
  Impulsive penetration mechanism
  Investigation of non-uniform electromagnetic fields with test-particle simulations
  Solutions of the steady-state Vlasov equation by numerical integration of its characteristics
  Magnetohydrostatic model of mirror modes
Solutions of the steady-state Vlasov equation by numerical integration of its characteristics
Starting from our previous numerical simulations with test-particles (Echim, 2002) we have developed a method to integrate numerically the characteristics of the Vlasov equation. In fact we 'upgrade' our numerical simulations from the integration of individual trajectories of particles to trajectories of 'ensembles' of particles, or velocity distribution functions (VDF). A hydrogen plasma is injected into a steady state magnetic field distribution of a 1D tangential discontinuity (TD). The TD model seems to become an 'invariant' of our simulation efforts due to the great impact it has in many aspects of the solar-terrestrial interactions. It is supposed that the TD is an equilibrium solution for the global plasma flow and field. The moving collection/ensemble of particles (or cloud) is a perturbation to this equilibrium configuration and will alter it. This effect is modeled by the distribution of the electric field that we consider. In our steady state model the magnetic field changes orientation and intensity from B1 at the 'left' hand side of the discontinuity to B2 at the 'right' hand side.
The velocities of the protons and electrons are initialized with values consistent with a velocity distribution function described by a displaced Maxwellian. The electric field is initialized with E1=B1 x V0, where V0 is the initial bulk velocity of the two populations. The electromagnetic field depends on only one spatial coordinate, x, normal to the discontinuity. The particles are injected from sources distributed along the x-axis. Their orbits are integrated numerically. In a steady state situation the orbits in the physical space correspond to the characteristics of the Vlasov equation (Delcroix and Bers, 1994). The velocity distribution function is reconstructed using the Liouville theorem. It is assumed that each particle 'carries' a 'part' of the VDF that corresponds to its initial position in the velocity space and do not change along the orbit. That 'part' of the VDF is assigned to the (varying) velocity components of the particle all along its orbit (Curran and Goertz, 1989). Thus it is then possible to compute the VDF of both species inside and outside the discontinuity. With a reasonable number of particles the phase space can be quite well populated such that the spatial resolution of the reconstructed VDF can be increased. We illustrate the results obtained by a study of the variations introduced by the non-uniformities of the electromagnetic field. We were particularly interested to simulate non-gyrotropic VDFs as those observed in the plasmasheet by Wilber et al. (2004) and to understand the mechanism of remote sensing of thin current sheets (Lee et al., 2004) as the TD distribution considered in our simulations. This project is on development and is carried on in collaboration with the Faculty of Physics of the University of Bucharest.
Two-dimensional section of the VDF in the velocity plane normal to the local B-field. This figure illustrate the VDF reconstructed in the cell B-B (see the particles position diagram). One notable feature is the void of particles in the core of the VDF due to a velocity filtering at the edges of the moving cloud. Note also the enhanced anisotropy for increased magnetic field field resembling to ther VDFs reported in the plasma sheet by Wilber et al. (2004).

Echim, M., Test-particle trajectories in ''sheared'' stationary field: Newton-Lorenz and first order drift numerical simulations, Cosmic Research, 40, 534-547, 2002
Curran, D.B. and Goertz, C.K., Particle distributions in a two-dimensional reconnection field geometry, J. Geophys. Res., 94, 272-286, 1989
Delcroix, J.L. and Bers, A., Physique des plasmas, Savoirs Actuels - Inter Editions/CNRS, 1994
Lee, E., M. Wilber, Parks, G.K., et al., Modeling of remote sensing of thin current sheet, Geophysical Research Letters, 31, L21806, 2004
Wilber, M., Lee, E., Parks, G., et al., Cluster observationjs of velocity space-restricted ion distributions near the plasma sheet, Geophysical Research Letters, 31, L24802, 2004
Contact: Dr. Marius Mihai Echim
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