Y. Nariyuki

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

Abstract. Nonlinear relations among frequencies and phases in modulational instability of circularly polarized Alfven 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 Alfven 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 Schrodinger (DNLS) equation, which is a subset of the Hall-MHD system including the right- and left-hand polarized, nearly degenerate quasi-parallel Alfven 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 long time evolution of the Alfven 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.

Observations of linear and nonlinear processes in the foreshock wave evolution

Waves in the foreshock region are studied on the basis of a hypothesis that the linear process first excites the waves and further wave-wave nonlinearities distribute scatter the energy of the primary waves into a number of daughter waves. To examine this wave evolution scenario, the dispersion relations, the wave number spectra of the magnetic field energy, and the dimensionless cross helicity are determined from the observations made by the four Cluster spacecraft. The results confirm that the linear process is the ion/ion right-hand resonant instability, but the wave-wave interactions are not clearly identified. We discuss various reasons why the test for the wave-wave nonlinearities fails, and conclude that the higher order statistics would provide a direct evidence for the wave coupling phenomena.

Kinetically modified parametric instabilities of circularly polarized Alfvén waves: Ion kinetic effects

Parametric instabilities of parallel propagating, circularly polarized, finite amplitude Alfven waves in a uniform background plasma is studied, within a framework of one-dimensional Vlasov description for ions and massless electron fluid, so that kinetic perturbations in the longitudinal direction (ion Landau damping) are included. The present formulation also includes the Hall effect. The obtained results agree well with relevant analysis in the past, suggesting that kinetic effects in the longitudinal direction play essential roles in the parametric instabilities of Alfven waves when the kinetic effects react “passively.” Furthermore, existence of the kinetic parametric instabilities is confirmed for the regime with small-wave-number daughter waves. Growth rates of these instabilities are sensitive to ion temperature. The formulation and results demonstrated here can be applied to Alfven waves observed in the solar wind and in the earth’s foreshock region.

Parametric instabilities of circularly polarized Alfvén waves in plasmas with beam protons

[1]xa0The proton velocity distribution functions (VDFs) observed in the solar wind often show beam components. In this study, by means of one-dimensional hybrid simulation, parametric instabilities of circularly polarized Alfven waves in core proton-electron-beam proton plasmas are discussed. Numerical results show that nonlinear evolution of parametric instabilities in core proton-electron-beam proton plasmas are different from those in core proton-electron plasmas. Furthermore, numerical solutions of the linear dispersion relations suggest the importance of the proton kinetic effects, which agrees with the past studies. In the case that the beams are stable, according to the excitation of the broadband Alfven waves by the parametric instabilities, protons are diffused in the velocity space, resulting in the scattering and broadening of the beam components. Such time evolution of the proton VDFs changes the wave dispersion relation and affects the properties of the parametric instabilities, in agreement with the past theoretical studies. As a result, even if a decay instability is dominant at the linear stage, the parent wave can mainly be dissipated by the modulational instability at the nonlinear stage. Phase coherent turbulence is generated corresponding to the occurrence of the modulational instability. Parametric instabilities are also observed in plasmas with unstable beams. When left-hand polarized (LH-) Alfven wave is initially given, both backward propagating right-hand polarized (RH-) and LH- Alfven waves are excited by the decay instabilities in our runs. As time elapses, phase coherent turbulence is generated by the modulational instabilities, as in the case of stable beams. Some of the beam protons are perpendicularly accelerated by cyclotron resonance with the primary wave.

Parametric instabilities of parallel propagating incoherent Alfvén waves in a finite ion beta plasma

Large amplitude, low-frequency Alfven waves constitute one of the most essential elements of magnetohydrodynamic (MHD) turbulence in the fast solar wind. Due to small collisionless dissipation rates, the waves can propagate long distances and efficiently convey such macroscopic quantities as momentum, energy, and helicity. Since loading of such quantities is completed when the waves damp away, it is important to examine how the waves can dissipate in the solar wind. Among various possible dissipation processes of the Alfven waves, parametric instabilities have been believed to be important. In this paper, we numerically discuss the parametric instabilities of coherent/incoherent Alfven waves in a finite ion beta plasma using a one-dimensional hybrid (superparticle ions plus an electron massless fluid) simulation, in order to explain local production of sunward propagating Alfven waves, as suggested by Helios/Ulysses observation results. Parameter studies clarify the dependence of parametric instabilities ...

Consequences of finite ion temperature effects on parametric instabilities of circularly polarized Alfvén waves

[1]xa0Parametric instabilities of finite amplitude, circularly polarized, parallel propagating Alfven waves in a homogeneous plasma is discussed analytically, taking into account the ion Landau damping and the ion finite Larmor radius (FLR) effects. A hybrid kinetic fluid model is systematically derived from one-dimensional Vlasov equation for longitudinal ion motion and the FLR-Hall magnetohydrodynamic (MHD) equations for transverse directions. The longitudinal kinetic effects are retained in the model, whereas transverse kinetic effects such as ion cyclotron damping is neglected. Validity of the model is justified as far as the collisionless damping is concerned, since the ion cyclotron damping for typical quasi-parallel Alfven waves in the solar wind is considered to be negligibly small. As already shown in a number of past studies, inclusion of the kinetic effects let some new instabilities emerge, while that reduces the growth rates of fluid instabilities in general. Damping rates computed by a model using collision-like (local) damping terms deviate from results of the present model, suggesting the importance of using the exact Landau-type interactions. Furthermore, as a consequence of the FLR effects, the growth rates of the fluid decay and beat instabilities of the left-hand (right-hand) polarized mode are reduced more strongly (weakly) in the FLR-Hall-MHD model (FHM model) than in the Hall-MHD model (HM model). A hybrid simulation is carried out to confirm that the FHM model is in better agreement than the HM model with the simulation results. When the initial parent wave amplitude is relatively small, the simulation results quantitatively agree with the linear analysis. Furthermore, some arguments are given to the observed ion relaxation and the energy oscillation of the parent waves observed in the hybrid simulations. Here again, quantitatively better explanation is obtained by using the FHM model rather than the HM model, suggesting that it is important to include the FLR effects for correctly describing the Alfven wave parametric instabilities in finite ion beta plasmas, such as the solar wind near the Earth and the foreshock plasma.

On nonlinear evolution of Alfvénic turbulence in low beta plasmas

Nonlinear evolution of broadband spectrum Alfvenic turbulence is studied by using one-dimensional hybrid simulations. Numerical solutions indicate that the important characteristics in the solar wind magnetohydrodynamic (MHD) turbulence are naturally reproduced by nonlinear evolution of Alfvenic turbulence with more realistic power spectrum than one used in the past studies. Namely, (i) the modulational instability can dissipate the magnetic energy of Alfvenic turbulence and preserve the Alfvenicity even in low beta plasmas. (ii) Nonlinear evolution of the Alfvenic turbulence can induce the generation of the localized structures in the solar wind MHD turbulence.

Parametric instabilities of large-amplitude parallel propagating Alfvén waves: 2D PIC simulation

We discuss the parametric instabilities of large-amplitude parallel propagating Alfven waves using the 2D PIC simulation code. First, we confirmed the results from a previous study (Sakai et al 2005 New J. Phys.xa07 233) that the electrons are heated due to the modified two-stream instability and that the ions are heated by the parallel propagating ion acoustic waves. However, although the past study argued that such parallel propagating longitudinal waves are excited by transverse modulation of the parent Alfven wave, we consider these waves are more likely to be generated by the usual, parallel decay instability. Further, we performed other simulation runs with different polarization of the parent Alfven waves or different ion thermal velocity. Numerical results suggest that electron heating by the modified two-stream instability due to the large amplitude Alfven waves is unimportant with most parameter sets.

The truncation model of the derivative nonlinear Schrödinger equation

The derivative nonlinear Schrodinger (DNLS) equation is explored using a truncation model with three resonant traveling waves. In the conservative case, the system derives from a time-independent Hamiltonian function with only one degree of freedom and the solutions can be written using elliptic functions. In spite of its low dimensional order, the truncation model preserves some features from the DNLS equation. In particular, the modulational instability criterion fits with the existence of two hyperbolic fixed points joined by a heteroclinic orbit that forces the exchange of energy between the three waves. On the other hand, numerical integrations of the DNLS equation show that the truncation model fails when wave energy is increased or left-hand polarized modulational unstable modes are present. When dissipative and growth terms are added the system exhibits a very complex dynamics including appearance of several attractors, period doubling bifurcations leading to chaos as well as other nonlinear pheno...

Magnetohydrodynamic parametric instabilities of parallel propagating incoherent Alfvén waves

Numerical experiments for parametric instabilities of incoherent Alfvén waves in the context of one dimensional Hall-MHD equation sets are studied. The reason why the decay instability of incoherent waves can be explained in terms of that of the coherent wave is understood through analysis on nonlinearly driven finite amplitude density fluctuations. Numerical results suggest the importance of modulational instabilities of left-hand polarized Alfvén waves.