Contribution to the nonlinear theory of plasma wave excitation inside the electron foreshock region

The broadband electrostatic waves inside the electron foreshock region well away from its boundaries were explained as a beam mode for which the beam velocity is lower than the thermal velocity of the background plasma. We discuss here the nonlinear saturation of this instability. The waves propagating along the beam are stabilized by a plateau formation on the electron distribution function. The waves that propagate at an angle to the beam are stabilized in a different way. In the course of the development of the instability the beam velocity spreads and an initially negative energy mode, which is initially unstable, experiences increased damping by the beam particles and finally is stabilized due to this effect. We give an estimation of the wave energy in both cases, and we present a numerical modelling to substantiate our analytical estimates. The saturated wave energy level is found to be comparable in both cases and agrees with the levels observed in the foreshock.