Space portrait of electrons acted upon by an electrostatic wave and with initial distribution function , where is a function with very small width . (a) The wave amplitude is zero. (b) The wave has grown slowly enough to induce nearly adiabatic motion and has trapped the electrons which, due to action conservation, all lie on very close orbits. (c) In the limit when the periods of the trapped orbits are infinitely small compared to the typical timescale of variation on the EPW amplitude, when this wave decays, half electrons are detrapped with and half with . Then, again due to action conservation, when the EPW amplitude has decreased back to zero the distribution function is .
Space profile of the plasma wave (in arbitrary units), at time (black solid line) and at time (blue dashed line), when the group velocity of the wave packet (whose amplitude is indicated by the arrows) decreases with the EPW amplitude at the rear side, and remains fixed at its minimum nonlinear value at the front side. One clearly sees that the wave packet at time is narrower than at time .
Results obtained with BRAMA, at time ps, showing the profiles of: panel (a), ; panel (b), the transverse component of the group velocity, , induced by collisionless dissipation, and normalized to the modulus of the group velocity, ; panel (c), the total transverse component, , of the EPW group velocity, normalized to ; panel (d), the collisionless damping rate normalized to its linear value. At time ps, when the damping rate is still significant compared to its linear value, , and the variations of tend to increase the transverse size of the wave packet.
Same as Fig. 3 but at time ps. As in Fig. 3, the global effect of the group velocity is to increase the transverse extent of the wave packet. Note that, in spite of this, the transverse profile of the plasma wave packet has shrunk compared to the previous figure because of the inhomogeneity of the SRS growth rate (the same would have occurred for a freely propagating wave because of the inhomogeneity of the collisionless damping rate).
Same as Fig. 3 but at time ps, when Landau damping has nearly vanished. At this time, the focussing effect due to wave front bowing prevails near the axis and at the front side of the wave packet.
Same as Fig. 3 but at time ps, when the component due to dissipation has changed sign close to the axis and at the front side of the wave packet, which keeps on focussing more rapidly. Note the change in shape of the EPW profile between Figs. 5 and 6, resulting from the focussing of the front side.
Acronyms and notations used in this paper.
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