Exact evaluation of the rates of electrostatic decay and scattering off thermal ions for an unmagnetized Maxwellian plasma

Abstract

Electrostatic decay of Langmuir waves into Langmuir and ion sound waves (L→L′+S) and scattering of Langmuir waves off thermal ions (L+i→L′+i′, also called “nonlinear Landau damping”) are important nonlinear weak-turbulence processes. The rates for these processes depend on the quadratic longitudinal response function α(2) (or, equivalently, the quadratic longitudinal susceptibility χ(2)), which describes the second-order response of a plasma to electrostatic wave fields. Previous calculations of these rates for an unmagnetized Maxwellian plasma have relied upon an approximate form for α(2) that is valid where two of the wave fields are fast (i.e., vϕ=ω/k≫Ve where ω is the angular frequency, k is the wavenumber, and Ve is the electron thermal speed) and one is slow (vϕ≪Ve). Recently, an exact expression was derived for α(2) that is valid for any phase speeds of the three waves in an unmagnetized Maxwellian plasma. Here, this exact α(2) is applied to the calculation of the three-dimensional rates for electrostatic decay and scattering off thermal ions, and the resulting exact rates are compared with the approximate rates. The calculations are performed using previously derived three-dimensional rates for electrostatic decay given in terms of a general α(2), and newly derived three-dimensional rates for scattering off thermal ions; the scattering rate is derived assuming a Maxwellian ion distribution, and both rates are derived assuming arc distributions for the wave spectra. For most space plasma conditions, the approximate rate is found to be accurate to better than 20%; however, for sufficiently low Langmuir phase speeds (vϕ/Ve≈3) appropriate to some spatial domains of the foreshock regions of planetary bow shocks and type II solar radio bursts, the use of the exact rate may be necessary for accurate calculations. The relative rates of electrostatic decay and scattering off thermal ions are calculated for a range of parameters using the exact expressions for the rates; electrostatic decay is found to have the larger growth rate over the whole range of parameters, consistent with previous approximate calculations.