Time evolution of the magnetic energy for the dominant Fourier mode m = 3 in a logarithmic scale. WI proceeds through the linear stage for , in which the magnetic field strength grows exponentially in time, followed by the nonlinear saturation stage where magnetic trapping takes place.
On top panel: Corresponding time evolution of the mode m = 3 of the inductive part of the electric field Ey showing a strong decrease of the electric energy when the instability saturates. On bottom panel: corresponding electrostatic part Ex showing that the field is non negligible at the saturation. Thus, the mode m = 3 results from the mixture of both magneticand electrostatic contributions. The perpendicular temperature is here and we have used .
Phase space representation of a selected “stream” of the ring in the phase space. The distribution of the trapped particle population exhibits a three-vortex structure as the result of the magnetic trapping. The perpendicular temperature is here and we have used .
Representation of the central stream in phase space for the same system shown in Fig. 3 . We can see clearly the formation and the growth of vortices induced by the presence of both modes m = 3 (i.e., the mode k 0) and m = 6 (the harmonic ). Notice that the plasma “heating” is weaker for the plasma bulk. Numerical results correspond to and to an anisotropy factor of .
Time history of the mode m = 2 for the magnetic field Bz component on top panel, and of the electric field Ex on bottom panel. The evolution of the magnetic field is plotted in a logarithmic scale. The simulation was performed for a higher value of the perpendicular temperature of using an anisotropy factor of .
Plots of the electron distribution function in the phase space for a selected stream of ring. We see clearly the formation of a three-vortex structure on the middle panel, emerging for the growth of the most unstable mode m = 3. The phase space structure is however unstable and the coalescence of two vortices takes place at time t = 75 on bottom panel. The perpendicular temperature is here and we have used .
Phase space representation of the central stream. Particles of the central stream experiences a trapping mechanism induced by the electric potential. The simulation has been performed for and .
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