KdV equation for kinetic Alfvén waves and ionospheric solitons

The Korteweg–de Vries (KdV) equation is derived for nonlinear kinetic Alfvén waves (KAWs) under the framework of the reductive perturbation method in single ion and bi-ion plasmas. It is pointed out that the KdV equation can be derived following the same normalization of spatial coordinates, which was used to obtain an exact solution of the equations for arbitrary amplitude KAWs [Hasegawa and Mima, Phys. Fluids 21, 87 (1978)]. The KdV equation for KAWs is derived assuming Maxwell velocity distribution for electrons to highlight the appropriate normalization procedure of the nonlinear equations for KAWs in the small amplitude limit. Then, the Kappa distribution of electrons is also considered to investigate the effects of non-thermal particles on linear and nonlinear wave dynamics. The results are applied to single ion oxygen and bi-ion oxygen–hydrogen plasmas of the upper ionosphere. It is found that the presence of 0.4% of protons in oxygen plasma of the ionosphere does not affect the shape of the soliton but the high-energy electrons reduce its amplitude. Present theoretical calculations predict the frequencies of KAWs to lie in the range of 10–30 m and widths of solitons to be larger than 100 m. These estimates are in agreement with the Freja satellite observations [Wahlund et al., Geophys. Res. Lett. 21, 1831 (1994)].