Arguments against modulational instabilities of Langmuir waves in Earth's foreshock

Abstract

The nonlinear dispersion equation for monochromatic pump Langmuir waves is solved analytically and numerically and the results are applied to Langmuir‐like waves growing in Earth's foreshock. It is shown that modulational instabilities and parametric decay instabilities occur in distinct regions of W − κλD space, confirming and extending earlier work. Here W is the ratio of electric field to thermal energy density and κ is the pump wavenumber. In particular, for W ≲ 10−3 and Te ≳ 3Ti modulational instability occurs only for waves with κλD ≲ (me/9Mi)1/2, where me and Mi are electron and ion masses; decay is relevant at higher wavenumbers. Beam‐driven waves in the foreshock are shown to lie well outside the region of parameter space for which modulational instability can proceed. In contrast, the beam‐driven waves have wavenumbers appropriate for the decay. The beam‐driven waves are also shown theoretically and observationally to have bandwidths much larger than the nonlinear growth rate for modulational instability and the parametric decay. The absence of a random‐phase version of modulational instability and the decrease in growth rate caused by finite bandwidth effects provide two more arguments against modulational instability occurring frequently in the foreshock. Modulational instability may, however, be possible for very rare, intense (W ≳ 10−2) wave packets with small wavenumbers (κλD ≲ (me/9Mi)1/2) and small bandwidths (Δω/ωp ≲ 10−3). Furthermore, the large Langmuir bandwidths predicted and observed require that the parametric decay cannot proceed, except perhaps for very intense wave packets with W > 10−2. Instead the random‐phase, weak turbulence decay is predicted to occur, consistent with some previous observations and theoretical suggestions. Nonlinear plasma theory and our current understanding of foreshock electron beams and waves thus argue strongly against modulational instability and/or parametric decay proceeding or being important for the great majority of beam‐driven Langmuir wave packets in Earth's foreshock.