Energy conversion between longitudinal and transverse waves by mode-mode coupling in a relativistic plasma (application to Crab nebula)

Abstract

Nonlinear mode-mode coupling between a linearly unstable longitudinal oscillation and two electromagnetic waves in a spatially homogeneous, but anisotropic, relativistic plasma has been investigated in detail. The objective of doing this problem is to see if there is any 'quenching' effect on the linear instability of the longitudinal oscillation. The linear instability occurs when the relativistic component of the plasma is assumed to be a stream of high-energetic protons (cosmic rays) which possess a streaming velocity larger than the phase velocity of the longitudinal oscillation. It is found that 'the imaginary detuning factor' (which is related to the growth rate of the linear instability of the longitudinal mode) plays a dominant role in quenching the linear instability as well as in changing the wave amplitudes. For astrophysical bodies, e.g. in the vicinity of supernovae, having a thermal electron density approximately 102 cm-3 and probably a cosmic proton density approximately 10-3 cm-3 the e-folding time will be much reduced ( approximately 102-3 yr), then the quenching effect and the energy conversion between electrostatic and electromagnetic waves will be significant, with a time scale at the order of 103-4 yr. The application of these nonlinear mechanisms to the typical astrophysical radio source, the Crab nebula, has been demonstrated upon inserting numerical values.