Validity of quasilinear theory: refutations and new numerical confirmation

Abstract

The validity of quasilinear (QL) theory describing the weak warm beam-plasma instability has been a controversial topic for several decades. This issue is tackled anew, both analytically and by numerical simulations which benefit from the power of modern computers and from the development in the last decade of Vlasov codes endowed with both accuracy and weak numerical diffusion. Self-consistent numerical simulations within the Vlasov-wave description show that QL theory remains valid in the strong chaotic diffusion regime. However, there is a non-QL regime before saturation, which confirms previous analytical work and numerical simulation, but contradicts another analytical work. We show analytically the absence of mode coupling in the saturation regime of the instability where a plateau is present in the tail of the particle distribution function. This invalidates several analytical works trying to prove or to contradict the validity of QL theory in the strongly nonlinear regime of the weak warm beam-plasma instability.

How to Face the Complexity of Plasmas?

“…This brings a small spatial modulation to the particle density which provides a source term for the Langmuir waves. However, if the plateau is broad, the evolution of the wave spectrum is slow, which brings only a small change to the previous simplistic picture of a uniform density (see section 2.2 of [15]). Therefore, there is almost no density fluctuation to drive the wave evolution as defined by the self-consistent dynamics: the wave spectrum is frozen.…”

“…However, the simulations brought an unexpected clue to elucidate it: the variation of the phase of a given wave with time was found to be almost non fluctuating with the random realizations of the initial wave phases [16]. Therefore the simulations showed that the randomness of the final wave phases was a mere consequence of that of initial phases.…”

“…As yet, MHD simulations remain the main way to address theoretical issues, since the analytical description of these states is just in its infancy 15 [18,25]. In many other fields of plasma physics too, numerical simulations are the irreplaceable tool for investigating complexity 16 . This has made the issue of verification and validation of these codes a crucial one [118].…”

“…Therefore its position has a weak dependence over any two random phases, which justifies the quasilinear estimate. If s ≫ 1, it can be shown ( [58], section 6.8.2 of [15], and Appendix 2) that the position of a particle has a weak dependence over any two random phases over a time on the order of τ QL = τ spread ln(s). Therefore, if s ≫ 1, the quasilinear estimate holds over a time τ QL ≫ τ spread .…”

“…In order to find out the nature of the wave spectrum at saturation, numerical simulations were performed using a semi-Lagrangian code for the Vlasov-wave model [15]. This model is the mean-field limit of the granular dynamics defined by the self-consistent Hamiltonian: waves are still present as M harmonic oscillators, but particles are described by a continuous distribution function.…”