Nonlinear instabilities driven by coherent phase-space structures
Résumé
In the presence of wave dissipation, phase-space structures emerge in nonlinear Vlasov dynamics. Our theory gives a simple relation between the growth of these coherent structures and that of the wave energy. The structures can drive the wave by direct momentum exchange, which explains the existence of nonlinear instabilities in both barely unstable and linearly stable (subcritical) regimes. When dissipation is modeled by a linear term in the field equation, simple expressions of a single-hole growth rate and of the initial perturbation threshold are in agreement with numerical simulations. Instability dynamics [1,2] is of great interest in the context of pattern formation [3], the onset of turbulence [4], and many other subjects. While instabilities are central to virtually every field of physics, in collisionless or weakly collisional plasmas the disparate roles of resonant and nonresonant particles offers an interesting variation on time-honored methods and approaches. In this respect, it has long been realized that wave and instability dynamics and evolution in a collisionless plasma can be described in terms of coupled, interpenetrating ensembles of resonant and nonresonant particles or, equivalently, resonant particles and a gas of plasmon quasiparticles. While the linear theory of the Vlasov plasma is well established, its nonlinear theory is a rich and still-evolving subject. Rather little, however, is understood about nonlinear, or subcritical, Vlasov stability, in which the growth process circumvents linear theory [5]. One idea concerning subcritical processes derives from the properties of phase-space granulations or structures, which can exchange momentum via channels which differ from that of familiar wave-particle resonance, and so can tap free energy when wave excitation cannot [6]. Such granulations are self-bound aggregations of resonant particles, which constitute a novel collective exciton. In this Rapid Communication, we present a theory of subcritical Vlasov plasma instability formulated in terms of the evolution of waves and phase-space density correlations. Not surprisingly, the theory for one-dimensional (1D) Vlasov plasmas has considerable overlap with those describing the evolution of flows in a quasigeostrophic fluid. Both are two-dimensional (2D) systems which support waves, and are constrained by two invariants: energy and enstrophy in the fluid case, wave energy and phasestrophy in the Vlasov case. The mechanisms involved are relevant to many laboratory and space plasmas, in particular, in the context of energetic particle interaction with Alfvén waves, collisionless trapped electron modes, and trapped ion ITG instabilities. To illustrate our theory, we choose two simple models that treat one-dimensional plasmas. The first model is the bump-on-tail instability, which is a fundamental paradigm for the basic process of Langmuir waves driven by a suprathermal population. The Berk-Breizman (BB) extension of the bump-on-tail model includes an external wave damping γ d to account for linear dissipative mechanisms of the wave energy to the background plasma [7]. The second model is the current-driven ion-acoustic (CDIA) instability, which is a fundamental paradigm for sound waves driven by a velocity drift between thermal ions and thermal electrons. In both models, finite wave damping (externally applied in the BB model; due to ion Landau damping in the CDIA model) allows for the spontaneous creation of self-trapped structures (called holes and clumps) in the 2D phase space, whose median velocity evolves in time, resulting in spectral components with a frequency shift δω(t) (chirping). The growth of phase-space structures results from momentum exchange between the structure and the wave, or between species, which is due to the dissipation acting on structures. The evolution of holes and clumps is a self-organization process, which provides the energy required to balance dissipation. Subcritical instabilities have been observed in BB simulations [7,8] and CDIA simulations [9]. Based on the theory, we explain the mechanism of subcritical instabilities as follows. Landau damping generates a seed phase-space structure, whose growth rate can be positive if the growth due to momentum exchange overcomes decay due to collisions. In addition, our theory predicts the persistence of nonlinear instability in the marginally linear unstable regime. The theoretical arguments are in good agreement with results from high resolution numerical simulations. For the first model, we adopt a perturbative approach, and cast the BB model in a reduced form, which describes the time evolution of the beam particles only [7,10]. In this sense, we note that the BB model with extrinsic dissipation is also applicable to the traveling wave tube "quasilinear experiment" with a lossy helix [11]. In this model, a single electrostatic wave with a wave number k is assumed and the real frequency of the wave is set to ω = ω p , the Langmuir plasma frequency. The evolution of the beam distribution, f (x,v,t), is given by a kinetic equation
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