Nonlinear treatment for solar radio spikes

Abstract

Linear (temporal and spatial) growth rates of high-frequency normal modes are calculated when an unstable fast electron population with a two-side loss cone and power-law energetic distribution exists in a magnetized plasma. The dominant mode is shown to be: fundamentalX-mode ifY =ω _pe/ω _Be < 0.2∓0.3,lh-mode if 0.2∓0.3 <Y < 0.5∓0.8, fundamentalZ-mode if 0.5∓0.8 <Y < 0.9 for a broad energetic spectrum of electrons (10 keV≲E ≲ 200 keV). These modes are generated by different groups of fast electrons. The results presented can be used for further study of nonlinear effects in ECM problems including quasilinear relaxation of the instability. From the obtained dependence of the growth rates on the bandwidth of the electron spectrum we predict that the spiky emission should have no correlation with electron gamma-ray Bremsstrahlung, contrary to the correlation with hard X-ray emission. We find that the growth rates can grow with the decrease of the electron distribution anisotropy (in a certain range of the anisotropy parameter value). This behaviour explains naturally a super-exponential (gaussian) rise phase of the spikes discovered in observations.