Some Studies of Statistical Properties of Turbulence in Plasmas and Fluids
Abstract
Turbulence is ubiquitous in the flows of fluids and plasmas. This thesis is devoted to studies of the statistical properties of turbulence in the threedimensional (3D) Hall magnetohydrodynamic (HallMHD) equations, the twodimensional (2D) MHD equations, the onedimensional (1D) hyperviscous Burgers equation, and the 3D NavierStokes equations. Chapter 1 contains a brief introduction to statistically homogeneous and isotropic turbulence. This is followed by an overview of the equations we study in the subsequent chapters, the motivation for the studies and a summary of problems we investigate in chapters 26.
In Chapter 2 we present our study of HallMHD turbulence [1]. We show that a shellmodel version of the 3D HallMHD equations provides a natural theoretical model for investigating the multiscaling behaviors of velocity and magnetic structure functions. We carry out extensive numerical studies of this shell model, obtain the scaling exponents for its structure functions, in both the lowk and highk powerlaw ranges of 3D HallMHD, and find that the extendedselfsimilarity procedure is helpful in extracting the multiscaling nature of structure functions in the highk regime, which otherwise appears to display simple scaling. Our results shed light on intriguing solarwind measurements.
In Chapter 3 we present our study of the inversecascade regime in twodimensional magnetohydrodynamic turbulence [2]. We present a detailed direct numerical simulation (DNS) of statistically steady, homogeneous, isotropic, twodimensional magnetohydrodynamic (2D MHD) turbulence. Our study concentrates on the inverse cascade of the magnetic vector potential. We examine the dependence of the statistical properties of such turbulence on dissipation and friction coefficients. We extend earlier work significantly by calculating fluid and magnetic spectra, probability distribution functions (PDFs) of the velocity, magnetic, vorticity, current, streamfunction, and magneticvectorpotential fields and their increments. We quantify the deviations of these PDFs from Gaussian ones by computing their flatnesses and hyperflatnesses. We also present PDFs of the OkuboWeiss parameter, which distinguishes between vortical and extensional flow regions, and its magnetic analog. We show that the hyperflatnesses of PDFs of the increments of the streamfunction and the magnetic vector potential exhibit significant scale dependence and we examine the implication of this for the multiscaling of structure functions. We compare our results with those of earlier studies.
In Chapter 4 we compare the statistical properties of 2D MHD turbulence for two different energy injection scales. We present systematic DNSs of statistically steady 2D MHD turbulence. Our two DNSs are distinguished by kinj, the wave number at which we inject energy into the system. In our first DNS (run R1), kinj = 2 and, in the second (run R2) kinj = 250. We show that various statistical properties of the turbulent states in the runs R1 and R2 are strikingly different The nature of energy spectrum, probability distribution functions, and topological structures are compared for the two runs R1 and R2 are found to be strikingly different.
In Chapter 5 we study the hyperviscous Burgers equation for very high α, order of hyperviscosity [3]. We show, by using direct numerical simulations and theory, how, by increasing α in equations of hydrodynamics, there is a transition from a dissipative to a conservative system. This remarkable result, already conjectured for the asymptotic case α →∞ [U. Frisch et al., Phys. Rev. Lett. 101, 144501 (2008)], is now shown to be true for any large, but finite, value of α greater than a crossover value α crossover. We thus provide a selfconsistent picture of how dissipative systems, under certain conditions, start behaving like conservative systems, and hence elucidate the subtle connection between equilibrium statistical mechanics and outofequilibrium turbulent flows.
In Chapter 6 we show how to use asymptoticextrapolation and Richardson extrapolation methods to extract the exponents ξ p that characterize the dependence of the orderp moments of the velocity gradients on the Reynolds number Re. To use these extrapolation methods we must have highprecision data for such moments. We obtain these highprecision data by carrying out the most extensive, quadruple precision, pseudospectral DNSs of the NavierStokes equation.
Collections
 Physics (PHY) [301]
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