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.