The generation of 150 km echoes through nonlinear wave mode coupling

A fundamental problem in turbulence is understanding how energy cascades across multiple scales. In this paper, a new weak turbulence theory is developed to explain how energy can be transferred from Langmuir and Upper-Hybrid waves (~10 MHz frequencies, 20-cm wavelengths) to ion-acoustic waves (~kHz frequencies, 3-meter wavelengths). A kinetic approach is used where the electrostatic Boltzmann equations are Fourier-Laplace transformed, and the nonlinear term is retained. A unique feature of this approach is the ability to calculate power spectra at low frequencies, for any wavelength or angle to the magnetic field. The results of this theory explain how 150-km echoes are generated in the ionosphere. First, peaks in the suprathermal electron velocity distribution drive a bump-on-tail like instability. This instability excites the Upper-Hybrid mode, and the nonlinear mode coupling theory shows that the instability generates a ~10 dB enhancement of the ion-acoustic mode: matching the observed enhancement in 150-km echoes.