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In this paper, we discuss the formation of ring-shaped and gyro-phaserestricted velocity distribution functions (VDFs) at the edges of a cloud ofprotons injected into non-uniform distributions of the electromagnetic field.The velocity distribution function is reconstructed using the forwardtest-kinetic method. We consider two profiles of the electric field: (1) anon-uniform E-field obtained by solving the Laplace equation consistent withthe conservation of the electric drift and (2) a constant and uniform E-field.In both cases, the magnetic field is similar to the solutions obtained fortangential discontinuities. The initial velocity distribution function isLiouville mapped along numerically integrated trajectories. The numericalresults show the formation of an energy-dispersed structure due to theenergy-dependent displacement of protons towards the edges of the cloud by thegradient-B drift. Another direct effect of the gradient-B drift is theformation of ring-shaped velocity distribution functions within thevelocity-dispersed structure. Higher energy particles populate the edges of theproton beam, while smaller energies are located in the core. Non-gyrotropicvelocity distribution functions form on the front-side and trailing edge of thecloud; this effect is due to remote sensing of energetic particles with guidingcenters inside the beam. The kinetic features revealed by the test-kineticsolutions have features similar to in-situ velocity distribution functionsobserved by Cluster satellites in the magnetotail, close to the neutral sheet.

physics of plasmas

Ring-shaped velocity distribution functions in energy-dispersed  structures formed at the boundaries of a proton stream injected into a  transverse magnetic field: Test-kinetic results