Low-dimensional chaos and wave turbulence in plasmas

Communications in Nonlinear Science and Numerical Simulation, 2012

Turbulence is one of the key problems of classical physics, and it has been the object of intense research in the last decades in a large spectrum of problems involving fluids, plasmas, and waves. In order to review some advances in theoretical and experimental investigations on turbulence a mini-symposium on this subject was organized in the Dynamics Days South America 2010 Conference. The main goal of this mini-symposium was to present recent developments in both fundamental aspects and dynamical analysis of turbulence in nonlinear waves and fusion plasmas. In this paper we present a summary of the works presented at this mini-symposium. Among the questions to be addressed were the onset and control of turbulence and spatio-temporal chaos.

Journal of Atmospheric and Solar-Terrestrial Physics, 2005

There is increasing observational evidence of nonlinear wave-wave interactions in space and astrophysical plasmas. We first review a number of theoretical models of nonlinear wave-wave interactions which our group has developed in past years. We next describe a nonlinear three-mode truncated model of Alfvén waves, involving resonant interactions of one linearly unstable mode and two linearly damped modes. We construct a bifurcation diagram for this three-wave model and investigate the phenomenon of intermittent chaos. The theoretical results presented in this paper can improve our understanding of intermittent time series frequently observed in space and astrophysical plasmas.

Physics of Plasmas, 1995

Conventional linear stability analyses may fail for fluid systems with an indefinite free-energy functional. When such a system is linearly stable, it is said to possess negative energy modes. Instability may then occur either via dissipation of the negative energy modes, or nonlinearly via resonant wave-wave coupling, leading to explosive growth. In the dissipationless case, it is conjectured that intrinsic chaotic behavior may allow initially nonresonant systems to reach resonance by diffusion in phase space. In this and a companion paper (submitted to Phys. Plasmas), this phenomenon is demonstrated for a simple equilibrium involving cold counterstreaming ions. The system is described in the fluid approximation by a Hamiltonian functional and associated noncanonical Poisson bracket. By Fourier decomposition and appropriate coordinate transformations, the Hamiltonian for the perturbed energy is expressed in action-angle form. The normal modes correspond to Doppler-shifted ion-acoustic waves of positive and negative energy. Nonlinear coupling leads to decay instability via two-wave interactions, and to either decay or explosive instability via three-wave interactions. These instabilities are described for various integrable systems of waves interacting via single nonlinear terms. This discussion provides the foundation for the treatment of nonintegrable systems in the companion paper. 0 199.5 American Institute of Physics.

New Journal of Physics, 2002

Drift wave turbulence, in general a balance between E × B drift turbulence in planes perpendicular to, and dissipative wave dynamics parallel to, a background magnetic field, is a hallmark example of nonlinearity in plasma physics. The turbulence generally has the same basic character in a sheared magnetic field lying in closed surfaces whether linear instabilities are present or not. Only when the linear forcing terms are dominant does this situation not prevail; it is not analogous to neutral fluid turbulence where pure linear forcing is balanced by pure nonlinear mixing and decorrelation. Detailed computations show that two types of nonlinearity are simultaneously present: advection of fluid vorticity by the E × B flows, which tends to have a scattering character, and E × B advection of pressure disturbances, which has the familiar diffusive mixing character. The vorticity nonlinearity excites the turbulence, acting against the mostly linear parallel dynamics which constrains it, while the pressure nonlinearity provides dissipation via transfer to ever smaller scales. The practical result is that the saturated level of the turbulence and the resulting averaged thermal energy transport are controlled principally by these nonlinear mechanisms even when moderate linear instabilities are present. The model is mostly applicable to tokamak edge turbulence, for which the linear forcing effects are sufficiently moderate that the nonlinear physics is allowed to operate. 1. Introduction-drift wave dynamics and the nonlinear instability Drift waves are a class of small amplitude, low frequency eigenmodes [1] which exist in a low beta plasma in a background magnetic field (B) with a pressure gradient (∇p). An MHD equilibrium is assumed, with ∇p ⊥ B. Low beta means the gas pressure is small compared to the magnetic field energy density and low frequency means sufficiently slow dynamics that

Physics of plasmas, 2005

The competition between drift wave and interchange physics in general E-cross-B drift turbulence is studied with computations in three dimensional tokamak flux tube geometry. For a given set of background scales, the parameter space can be covered by the plasma beta and drift wave collisionality. At large enough plasma beta the turbulence breaks out into ideal ballooning modes and saturates only by depleting the free energy in the background pressure gradient. At high collisionality it finds a more gradual transition to resistive ballooning. At moderate beta and collisionality it retains drift wave character, qualitatively identical to simple two dimensional slab models. The underlying cause is the nonlinear vorticity advection through which the self sustained drift wave turbulence supersedes the linear instabilities, scattering them apart before they can grow, imposing its own physical character on the dynamics. This vorticity advection catalyses the gradient drive, while saturation occurs solely through turbulent mixing of pressure disturbances. This situation persists in the whole of tokamak edge parameter space. Both simplified isothermal models and complete warm ion models are treated.

The European Physical Journal Special Topics, 2008

We analyze probe data obtained from a toroidal magnetized plasma configuration suitable for studies of low-frequency gradient-driven instabilities. These instabilities give rise to field-aligned convection rolls analogous to Rayleigh-Benard cells in neutral fluids, and may theoretically develop similar routes to chaos. When using mean-field dimension analysis, we observe low dimensionality, but this could originate from either lowdimensional chaos, periodicity or quasi-periodicity. Therefore, we apply recurrence plot analysis as well as estimation of the largest Lyapunov exponent. These analyses provide evidence of low-dimensional chaos, in agreement with theoretical predictions.

By employing a quantum hydrodynamic (QHD) model with appropriate boundary conditions, the nonlinear self-interaction of an electrostatic surface wave on otherwise homogeneous semibounded plasma with degeneracy effects is investigated. It has been found that a part of the second harmonic generated through self-interaction does not have a true surface wave feature but propagates obliquely away from the plasma-vacuum interface into the bulk of the plasma. Such a situation is obtained in laser-plasma interaction during surface etching, plasma processing, and so on. In this article, we study the harmonics generation mechanism, associated Lagrangian chaos, and harmonic conversation rate when an intense laser is incident on an unmagnetized plasma. The dense plasma displays quantum diffraction effects and other quantum statistical effects. We made use of perturbative analysis for the field quantities. The findings will help workers dealing with laser-plasma interaction so as to enhance the tendency of harmonic generation and energy localization.

Chaos: An Interdisciplinary Journal of Nonlinear Science, 1997

Bench mark test of the code for nonlinear simulation of plasma turbulence is performed. A new code describing four fields (B) is compared to the existing code (A) w.hich treats three fields. Examining the results from two codes under the physically identical conditions, characteristics of the deviations are analyzed. It is found that the infinitesimally small initial noise, due to the cancelling, grows in accordance with the nonlinear development of turbulence mode. Interaction with an intrinsic nonlinearity of the system makes the noise grow, whose contribution becomes similar magnitude to the fluctuation itself of the results. The instantaneous deviation shows the chaotic characteristics. The spectrum analysis is made. These show the intrinsic nonlinearity of the plasma turbulence.

1997

Reviews of Modern Plasma Physics

Intermittent turbulence is key for understanding the stochastic nonlinear dynamics of space, astrophysical, and laboratory plasmas. We review the theory of deterministic and stochastic temporal chaos in plasmas and discuss its link to intermittent turbulence observed in space plasmas. First, we discuss the theory of chaos, intermittency, and complexity for nonlinear Alfvén waves, and parametric decay and modulational wave–wave interactions, in the absence/presence of noise. The transition from order to chaos is studied using the bifurcation diagram. The following two types of deterministic intermittent chaos in plasmas are considered: type-I Pomeau–Manneville intermittency and crisis-induced intermittency. The role of structures known as chaotic saddles in deterministic and stochastic chaos in plasmas is investigated. Alfvén complexity associated with noise-induced intermittency, in the presence of multistability, is studied. Next, we present evidence of magnetic reconnection and in...

Plasma Physics, 1983

Experimental investigations of strong turbulence associated with the radial density gradient of a rotating magnetized plasma column are reported. The experiment is designed to make Taylor's hypothesis effective, in order to allow a simple interpretation of measured frequency spectra in terms of wavenumber spectra. The spectral index of the turbulent potential fluctuations is determined and the variation of the spectral intensity is investigated for varying magnetic fields. The results compare favourably with theoretical predictions. The importance of distinguishing subranges in the turbulent spectrum is demonstrated. Some aspects of the relative diffusion of a test-cloud of charged particles released in the turbulent field 2re discussed.

Physics of Plasmas, 1999

By employing the two-fluid model, a system of nonlinear equations for low-frequency electromagnetic waves in nonuniform collisional magnetoplasmas has been derived. The plasma contains both the equilibrium density gradient and sheared flows. In the linear limit, a local dispersion relation has been obtained and analyzed in several interesting limiting cases. It is found that equilibrium sheared plasma flows cause instabilities of Alfvén-type waves even in the absence of the density gradient. The numerical results also show a large growth rate of electromagnetic parallel velocity shear ͑PVS͒ mode compared to the electrostatic mode for some ionospheric parameters. For this case, the temporal nonlinear behavior of the relevant governing mode coupling equations is governed by six coupled equations, which are a generalization of the Lorenz-Stenflo equations and which admit chaotic trajectories. The results of this investigation should be useful for understanding the linear and nonlinear properties of electromagnetic waves that are generated by sheared plasma flows in magnetized plasmas.

Plasma Physics and Controlled Fusion, 2014

We consider the excitation, nonlinear evolution and saturation of a single coherent mode in a turbulent plasma background. We adopt a generic wave-kinetic description of plasma turbulence, based on a Wigner-Moyal equation, which stays valid well beyond the geometric optics approximation. We illustrate our model by considering the case of a geodesic acoustic mode in a tokamak plasma, in the presence of a broad spectrum of drift wave turbulence.

Nonlinear Processes in Geophysics, 2014

Brazilian Journal of Physics, 2014

Plasma turbulence at the edge of tokamaks is an issue of major importance in the study of the anomalous transport of particles and energy. Although the behavior of a turbulent plasma seems intractable, it turns out that many of its aspects can be described by low-dimensional nonintegrable dynamical models. In this paper, we consider a number of dynamical effects occurring in tokamak plasma edge-in particular the role of internal transport barriers. Furthermore, we present experimental results on turbulentdriven transport for two machines-the Brazilian TCABR tokamak and University of Texas' Helimak-that can be explained by those theoretical models. Dedicated to Professor Wendell Horton for his outstanding contributions to the nonlinear dynamics approach to plasma turbulence.