In this thesis we study a variety of problems in superfluid turbulence, princi-pally in two dimensions. A summary of the main results of our studies is given below; we indicate the Chapters in which we present these. In Chapter 1, we provide an overview of several problems in superfluid turbulence with special emphasis on background material for the problems we study in this thesis. In particular, we give: (a) a brief introduction of fluid turbulence; (b) an overview of superfluidity and the phenomenological two-fluid model; (c) a brief overview of experiments on superfluid turbulence; (d) an introductory accounts of the phenomenological models used in the study of superfluid turbulence. We end with a summary of the problems we study in subsequent Chapters of this thesis. In Chapter 2, we present a systematic, direct numerical simulation of the two-dimensional, Fourier-truncated, Gross-Pitaevskii equation to study the turbulent evolutions of its solutions for a variety of initial conditions and a wide range of parameters. We find that the time evolution of this system can be classified into four regimes with qualitatively different statistical properties. First, there are transients that depend on the initial conditions. In the second regime, power- law scaling regions, in the energy and the occupation-number spectra, appear and start to develop; the exponents of these power laws and the extents of the scaling regions change with time and depend on the initial condition. In the third regime, the spectra drop rapidly for modes with wave numbers k > kc and partial thermalization takes place for modes with k < kc ; the self-truncation wave number kc(t) depends on the initial conditions and it grows either as a power of t or as log t. Finally, in the fourth regime, complete thermalization is achieved and, if we account for finite-size effects carefully, correlation functions and spectra are consistent with their nontrivial Berezinskii-Kosterlitz-Thouless forms. Our work is a natural generalization of recent studies of thermalization in the Euler and other hydrodynamical equations; it combines ideas from fluid dynamics and turbulence, on the one hand, and equilibrium and nonequilibrium statistical mechanics on the other. In Chapter 3, we present the first calculation of the mutual-friction coefficients α and α (which are parameters in the Hall-Vinen-Bekharevich-Khalatnikov two-fluid model that we study in chapter 5) as a function of temperature in a homogeneous Bose gas in two-dimensions by using the Galerkin-truncated Gross-Pitaevskii equation, with very special initial conditions, which we obtain by using the advective, real, Ginzburg-Landau equation (ARGLE) and an equilibration procedure that uses a stochastic Ginzburg-Landau equation (SGLE). We also calculate the normal-fluid density as a function of temperature. In Chapter 4, we elucidate the interplay of particles and fields in superfluids, in both simple and turbulent flows. We carry out extensive direct numerical simulations (DNSs) of this interplay for the two-dimensional (2D) Gross-Pitaevskii (GP) equation. We obtain the following results: (1) the motion of a particle can be chaotic even if the superfluid shows no sign of turbulence; (2) vortex motion depends sensitively on particle charateristics; (3) there is an effective, superfluid-mediated, attractive interaction between particles; (4) we introduce a short-range repulsion between particles, with range rSR, and study two- and many-particle collisions; in the case of two-particle, head-on collisions, we find that, at low values of rSR, the particle collisions are inelastic with coefficient of restitution e = 0; and, as we in-crease rSR, e becomes nonzero at a critical point, and finally attains values close to 1; (5) assemblies of particles and vortices show rich, turbulent, spatio-temporal evolution. In Chapter 5, we present results from our direct numerical simulations (DNSs) of the Hall-Vinen-Bekharevich-Khalatnikov (HVBK) two-fluid model in two dimensions. We have designed these DNSs to study the statistical properties of inverse and forward cascades in the HVBK model. We obtain several interesting results that have not been anticipated hitherto: (1) Both normal-fluid and superfluid energy spectra, En(k) and Es(k), respectively, show inverse- and forward-cascade regimes; the former is characterized by a power law Es(k) En(k) kα whose exponent is consistent with α 5/3. (2) The forward-cascade power law depends on (a) the friction coefficient, as in 2D fluid turbulence, and, in addition, on (b) the coefficient B of mutual friction, which couples normal and superfluid compo-nents. (3) As B increases, the normal and superfluid velocities, un and us, re-spectively, get locked to each other, and, therefore, Es(k) En(k), especially in the inverse-cascade regime. (4) We quantify this locking tendency by calculating the probability distribution functions (PDFs) P(cos(θ)) and P(γ), where the angle θ ≡ (un • us)/( |un||us|) and the amplitude ratio γ = |un|/|us |; the former has a peak at cos(θ) = 1; and the latter exhibits a peak at γ = 1 and power-law tails on both sides of this peak. (4) This locking increases as we increase B, but the power-law exponents for the tails of P(γ) are universal, in so far as they do not depend on B, ρn/ρ, and the details of the energy-injection method. (5) We characterize the energy and enstrophy cascades by computing the energy and enstrophy fluxes and the mutual-friction transfer functions for all wave-number scales k. In Chapter 6, we examine the multiscaling of structure functions in three-dimensional superfluid turbulence by using a shell-model for the three-dimensional HVBK equations. Our HVBK shell model is based on the GOY shell model. In particular, we examine the dependence of multiscaling on the normal-fluid fraction and the mutual-friction coefficients. We hope our in silico studies of 2D and 3D superfluid turbulence will stimulate new experimental, numerical, and theoretical studies.

Turbulence is ubiquitous in the flows of fluids and plasmas. This thesis is devoted to studies of the statistical properties of turbulence in the three-dimensional (3D) Hall magnetohydrodynamic (Hall-MHD) equations, the two-dimensional (2D) MHD equations, the one-dimensional (1D) hyperviscous Burgers equation, and the 3D Navier-Stokes equations. Chapter 1 contains a brief introduction to statistically homogeneous and isotropic turbulence. This is followed by an over-view of the equations we study in the subsequent chapters, the motivation for the studies and a summary of problems we investigate in chapters 2-6. In Chapter 2 we present our study of Hall-MHD turbulence [1]. We show that a shell-model version of the 3D Hall-MHD 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 low-k and high-k power-law ranges of 3D Hall-MHD, and find that the extended-self-similarity procedure is helpful in extracting the multiscaling nature of structure functions in the high-k regime, which otherwise appears to display simple scaling. Our results shed light on intriguing solar-wind measurements. In Chapter 3 we present our study of the inverse-cascade regime in two-dimensional magnetohydrodynamic turbulence [2]. We present a detailed direct numerical simulation (DNS) of statistically steady, homogeneous, isotropic, two-dimensional 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, stream-function, and magnetic-vector-potential 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 Okubo-Weiss 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 stream-function 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 self-consistent picture of how dissipative systems, under certain conditions, start behaving like conservative systems, and hence elucidate the subtle connection between equilibrium statistical mechanics and out-of-equilibrium turbulent flows. In Chapter 6 we show how to use asymptotic-extrapolation and Richardson extrapolation methods to extract the exponents ξ p that characterize the dependence of the order-p moments of the velocity gradients on the Reynolds number Re. To use these extrapolation methods we must have high-precision data for such moments. We obtain these high-precision data by carrying out the most extensive, quadruple precision, pseudospectral DNSs of the Navier-Stokes equation.