Towards a Simple Model of Compressible Alfvénic Turbulence

The theory of compressible MHD (e.g., Alfvénic) turbulence has been a topic of interest for some time [1]. Alfvén wave turbulence presents several novel challenges, due to the fact the k–ω selection rules preclude three Alfvén-wave resonance. Thus, in incompressible MHD, two Alfvén waves can interact only with the vortex (i.e., eddy) mode. Compressibility relaxes this constraint by allowing interaction with accoustic and ion-ballistic modes (i.e., Landau damping), along with waveform steepening. This naturally leads to the formation of Alfvénic shocklets. Thus, one approach, which is analogous to the noisy-Burgers model in hydrodynamics, is based on the study of nonlinear wave evolution equations with external noise drive [e.g., the noisy derivative nonlinear Schrödinger equation (DNLS) equation, in space physics]. Such theories describe turbulence as an ensemble of nonlinear structures, e.g., shocks, discontinuities, and high-amplitude waves, which are typically observed in compressible (e.g., interplanetary [2]) plasmas. This course of investigation was pursued computationally to study the noisy-DNLS equation [3]. Stationarity was achieved by inserting ad-hoc viscous damping (later linked to finite plasma conductivity [4]) into the otherwise conservative DNLS equation. The DNLS model fails, however, for the important case of β ∼ 1 (β = 4πp/B 0 is the ratio of plasma pressure to magnetic pressure, B 0 is an external magnetic field) and the electron-to-ion temperature ratio Te/Ti ∼ 1 (for instance, in the solar wind plasma), when Alfvén waves couple to strongly damped ion acoustic modes. As a consequence, the kinetically modified DNLS [5,6], referred to as the kinetic nonlinear Schrödinger equation (KNLS), which exhibits intrinsically dissipative nonlinear coupling, emerges as the superior basic model. Numerical solution of the KNLS reveals a new class of dissipative structures, which appear through the balance of nonlinear steepening with collisionless nonlinear damping. These structures include arc-polarized and S polarized rotational discontinuities [7], observed in the solar wind plasma and not predicted by other models. The resulting quasi-stationary structures typically have narrow spectra. Here, we present the first analytical study of the noisy-KNLS equation as a generic model of collisionless, largeamplitude Alfvénic shocklet turbulence. Indeed, this is, to our knowledge, the first structure-based theory of compressible MHD turbulence in a collisionless system. Stationarity is maintained via the balance of noise and dissipative nonlinearity. Dissipation here results from ion Landau damping, which balances the parallel ponderomotive force produced by modulations of the compressible Alfvén wave train. A one-loop renormalization group (RG) calculation (equivalent [8] to a direct interaction approximation [9] closure) is utilized. Although the KNLS describes both quasi-parallel and oblique waves [5], we consider here the simpler case of quasi-parallel propagation. The general case will be addressed on future publication. The noisy-KNLS is, thus, a generic model of strong, compressible Alfvénic turbulence and may be relevant to the solar wind, interstellar medium, shock acceleration as well as to compressible MHD theory, as a whole. Note that this perspective is analogous to that of the noisy-Burgers equation model of compressible fluid turbulence [10]. Several features which are not common in standard MHD turbulence theories appear in this model. It is shown that the dissipative integral coupling renormalizes the wave train velocity, in addition to inducing nonlinear damping and dispersion. Moreover, consideration of the resulting solvability condition for a stationary state in the hydrodynamic limit (ω, k → 0) suggests that KNLS turbulence can exist in one of two different states or phases. In the hydrodynamic regime, turbulence consists of large-scale, smooth (ω, k → 0) waveforms and dissipative structures. In the regime when the hydrodynamic limit does not exist, one may expect a small-scale, spikey, intermittent (ω, k 6→ 0) shocklet turbulence. This hypothesis, however, needs further (e.g., numerical) study. The “noisy-KNLS” equation is