Threshold for electron trapping nonlinearity in Langmuir waves

Abstract

We assess when electron trapping nonlinearity is expected to be important in Langmuir waves. The basic criterion is that the inverse of the detrapping rate νd of electrons in the trapping region of velocity space must exceed the bounce period of deeply trapped electrons, τB≡(ne/δn)1/22π/ωpe. A unitless figure of merit, the “bounce number” NB≡1/νdτB, encapsulates this condition and defines a trapping threshold amplitude for which NB=1. The detrapping rate is found for convective loss (transverse and longitudinal) out of a spatially finite Langmuir wave. Simulations of driven waves with a finite transverse profile, using the 2D-2V Vlasov code LOKI, show trapping nonlinearity increases continuously with NB for transverse loss, and is significant for NB≈1. The detrapping rate due to Coulomb collisions (both electron-electron and electron-ion) is also found, with pitch-angle scattering and parallel drag and diffusion treated in a unified manner. A simple way to combine convective and collisional detrapping is given. Application to underdense plasma conditions in inertial confinement fusion targets is presented. The results show that convective transverse loss is usually the most potent detrapping process in a single f/8 laser speckle. For typical plasma and laser conditions on the inner laser cones of the National Ignition Facility, local reflectivities ∼3% are estimated to produce significant trapping effects.