Whistler waves in magnetosheath with observed flat top distributions

Whistler waves having frequencies nearly centered around 100 Hz are frequently observed in the magnetosheath. Cluster observed such low frequency Whistler waves named as Lion roars and electron velocity distribution function within the magnetosheath during its several crossings. The observed electron velocity distribution functions clearly show non-Maxwellian feature such as flat tops at low energies. These observed lion roars were studied and interpreted using bi-Maxwellian distribution function by employing the kinetic theory but could not justify the observations both quantitatively as well as qualitatively. The obvious reason was to employ idealized Maxwellian distribution function instead of using non-Maxwellian distribution function. We derived the dispersion relation of Whistler waves by using generalized (r, q) distribution function which is the generalized form of kappa and Maxwellian distribution functions. It is supposed to be the best distribution when the real distributions contain flat tops. We employed kinetic theory to obtain the modified expressions for real frequency and established the necessary and sufficient condition to achieve growth/damping rates based on the generalized (r, q) distribution function. We then compare our numerical values of real and damping/growth rates with the Cluster observations, a good quantitative and qualitative agreement between the observed and theoretically obtained values have been found.