Remarks on nonlinear relation among phases and frequencies in modulational instabilities of parallel propagating Alfvén waves

Abstract

Abstract. Nonlinear relations among frequencies and phases in modulational instability of circularly polarized Alfvén waves are discussed, within the context of one dimensional, dissipation-less, unforced fluid system. We show that generation of phase coherence is a natural consequence of the modulational instability of Alfvén waves. Furthermore, we quantitatively evaluate intensity of wave-wave interaction by using bi-coherence, and also by computing energy flow among wave modes, and demonstrate that the energy flow is directly related to the phase coherence generation. We first discuss the modulational instability within the derivative nonlinear Schrödinger (DNLS) equation, which is a subset of the Hall-MHD system including the right-and left-hand polarized, nearly degenerate quasi-parallel Alfvén waves. The dominant nonlinear process within this model is the four wave interaction, in which a quartet of waves in resonance can exchange energy. By numerically time integrating the DNLS equation with periodic boundary conditions, and by evaluating relative phase among the quartet of waves, we show that the phase coherence is generated when the waves exchange energy among the quartet of waves. As a result, coherent structures (solitons) appear in the real space, while in the phase space of the wave frequency and the wave number, the wave power is seen to be distributed around a straight line. The slope of the line corresponds to the propagation speed of the coherent structures. Numerical time integration of the Hall-MHD system with periodic boundary conditions reveals that, wave power of transverse modes and that of longitudinal modes are aligned with a single straight line in the dispersion relation phase space, suggesting that efficient exchange of energy among transverse and longitudinal wave modes is realized in the Hall-MHD. Generation of the longitudinal wave modes violates the assumptions employed in deriving the DNLS such as the quasi-static approximation, and thus Correspondence to: Y. Nariyuki (nariyuki@esst.kyushu-u.ac.jp) long time evolution of the Alfvén modulational instability in the DNLS and in the Hall-MHD models differs significantly, even though the initial plasma and parent wave parameters are chosen in such a way that the modulational instability is the most dominant instability among various parametric instabilities. One of the most important features which only appears in the Hall-MHD model is the generation of sound waves driven by ponderomotive density fluctuations. We discuss relationship between the dispersion relation, energy exchange among wave modes, and coherence of phases in the waveforms in the real space. Some relevant future issues are discussed as well.

Response to “Comment on ‘Kinetically modified parametric instabilities of circularly polarized Alfvén waves: Ion kinetic effects’ ” [Phys. Plasmas 14, 024703 (2007)]

“…2 are already known, and are consistent with the common view on Alfvén wave parametric instabilities, e.g., unstable +f mode exhibits the modulational instability. 9 This is the reason why we did not fully discuss the real frequency, except for confirming that the propagation direction of the daughter waves are in agreement with the past studies.…”

Ion Dynamics During the Parametric Instabilities of a Left-Hand Polarized Alfvén Wave in a Proton-Electron-Alpha Plasma

“…This conclusion can also be confirmed by the bicoherence index in Figures 7(b) and (c). When bc ∼ 1, it means that the wave number modes are in strong resonance (Diamond et al 2000;Nariyuki & Hada 2006;Nariyuki et al 2009). The strong resonance is observed in the region with k 2 k 3 < 0 in Figure 7(b), which is consistent with the parametric decay of the pump wave.…”

“…The phase coherence C φ for the waves with k < 0 is calculated with the same procedure developed by Nariyuki et al (2009). The generation of phase coherence is a typical characteristic of the modulational instability (Nariyuki & Hada 2006), which is indicated by a finite value of C φ (Nariyuki et al 2009). From Figure 7(a), we can find that the modulational instability of the daughter Alfvén waves becomes important during the nonlinear evolution of the parametric instabilities (after about Ω p t = 800), and the wave numbers of the backward Alfvén waves begin to cascade toward larger values as shown in Figure 6(d).…”

Kinetic modulational instability of broadband dispersive Alfvén waves in plasmas

“…The latter include the three-wave decay [13][14][15][16]22] and modulational instabilities [18][19][20][21]. However, the coherent wavewave and wave-particle [15,21] interactions are inappropriate when the DAWs have random phases/broadband spectra.…”

“…Large-amplitude coherent non-dispersive and DAWs are subjected to a great variety of nonlinear effects [2,3,9,[13][14][15][16][17][18][19][20][21][22]. The latter include the three-wave decay [13][14][15][16]22] and modulational instabilities [18][19][20][21].…”