Weak Turbulence Closures in Incompressible Magnetohydrodynamics by Augustus Antanas Azelis Bachelor of Astronomy and Astrophysics Aerospace, Physics, and Space Sciences Florida Institute of Technology 2016 Bachelor of Aerospace Engineering Aerospace, Physics, and Space Sciences Florida Institute of Technology 2016 A thesis submitted to the College of Engineering and Science at Florida Institute of Technology in partial fulfillment of the requirements for the degree of Master of Science in Physics Melbourne, Florida May, 2019 c Copyright 2019 Augustus Antanas Azelis ⃝ All Rights Reserved The author grants permission to make single copies. We the undersigned committee hereby approve the attached thesis Weak Turbulence Closures in Incompressible Magnetohydrodynamics by Augustus Antanas Azelis Jean Carlos Perez, Ph.D. Assistant Professor Aerospace, Physics, and Space Sciences Committee Chair Gnana Bhaskar Tenali, Ph.D Professor Mathematical Sciences Outside Committee Member J´er´emy A. Riousset, Ph.D. Assistant Professor Aerospace, Physics, and Space Sciences Committee Member Daniel Batcheldor, Ph.D. Professor and Department Head Aerospace, Physics, and Space Sciences ABSTRACT Title: Weak Turbulence Closures in Incompressible Magnetohydrodynamics Author: Augustus Antanas Azelis Major Advisor: Jean Carlos Perez, Ph.D. In this work we apply the weak turbulence closure to the governing equation of incompressible homogeneous magnetohydrodynamics (MHD). It was originally thought that the method lacked applicability to linearly dispersive waves how- ever MHD presents a unique exception to this notion. A perturbative scheme is used in combination with multiple scale analysis to find a dynamic equation for the wavenumber energy spectrum corresponding to weakly non-linear shear Alfv´enic turbulence. We begin by deriving an equation describing the time evolu- tion of the Fourier transformed Els¨asser fields associated with linear propagation and non-linear interactions between counter-propagating Alfv´en waves. A cumu- lant hierarchy is then constructed from products of field amplitudes and the well known problem of closure is encountered. Closure of the energy spectrum is then 2 sought for timescales of order ϵ− via a series of perturbative expansions where ϵ is a small parameter representing the strength of nonlinear effects. Terms are encountered corresponding to resonant interactions between Alfv´en waves that are unbounded in time and result in an expansion that is not well ordered. Multiple scale analysis is then used to modify the slow time behavior of the cumulants to reorder the expansion, yielding an equation for the evolution of the energy spec- trum. The formal mathematical framework of weak turbulence is well discussed in iii the literature however the present applications to shear Alfv´en waves are lacking in rigor. This work forgoes the aforementioned brevity favored by the literature to demonstrate with clarity the closure for MHD. Explicit detail of the formalism results in heightened understanding of the underlying physics as well as the ability to generalize the theory for two time closures. iv Table of Contents Abstract iii List of Figures vii Acknowledgments viii Dedication ix 1 Introduction 1 1.1 ThePlasmaState............................ 1 1.2 TheSolarWind............................. 3 1.3 TheSolarCorona............................ 4 1.4 Alfv´enWavesandTurbulence . 5 1.5 OverviewandObjectives. 6 2 Magnetohydrodynamics 8 2.1 Magnetohydrodynamics (MHD) Model Equations . 8 2.2 Fourier Analysis of Shear Alfv´enic Turbulence . ... 12 3 Turbulence Kinematics and Dynamics 16 3.1 IntroductionandOverview. 16 3.2 PhysicalOriginsofTurbulence. 17 v 3.3 One-Pointformulation . 18 3.4 MomentHierarchyandProblemofClosure . 19 3.5 ElementaryTurbulencePhenomenology. 20 3.6 Two-PointClosureTheory . 23 3.7 Two-Time Correlations & the Taylor Hypothesis . .. 24 4 A Weakly Nonlinear Oscillator Problem 28 4.1 IntroductionandOverview. 28 4.2 NaiveSolutionoftheModelEquation. 29 4.3 DevelopmentofMulti-ScaleAnalysis . 31 4.4 AnAlternativeFormulationofMSA. 33 5 Weak Turbulence Closure for Incompressible Magnetohydrody- namics 35 5.1 IntroductionandOverview. 35 5.2 TheClosureProblemforSystemsofWaves. 36 5.3 WeakTurbulenceDiscussion . 37 5.4 Weak Turbulence Closure for Incompressible MHD . 39 5.4.1 Cumulants............................ 39 5.4.2 ConstructionoftheCumulantHierarchy . 41 5.4.3 Asymptotic Expansion of Cumulant Hierarchy . 47 5.4.4 ImplementationofMSAandSolution . 51 6 Conclusions 55 References 57 vi List of Figures 1.1 PhotooftheSunwithvisiblecorona[1]. 5 3.1 Arbitraryturbulentenergyspectrum[2] . 22 3.2 Contours of constant correlation for Taylor’s Frozen Flow Hypoth- esis[3] .................................. 26 vii Acknowledgements I would like to express my deepest gratitude to Dr. Jean-Carlos Perez for his support and mentorship as well as making me realize that I am capable of far more than I ever thought I could be. viii Dedication For my pupper, Annelisa. ix Chapter 1 Introduction 1.1 The Plasma State Contrary to casual quotidian observation, the vast majority of visible matter in the universe exists in a plasma state. A plasma is a gas of which the constituents are positively and negatively charged particles that collectively obey a set of dynam- ical equations, and whose coupling to Maxwell’s equations results in remarkably complex behavior. Overall plasma properties can be formulated explicitly in terms of number density n and temperature T , yielding a set of spatial, temporal, and density related parameters that presuppose the presence of plasma phenomena [4]. One of these quantities is a characteristic lengthscale known as the Debye length. The Debye length λD describes the ability of a plasma to shield charges and bring about electrical neutrality, defined as 1 ϵ kT 2 λ = 0 , (1.1) D ( ne2 ) 1 where ϵ0 is the vacuum permitivity, k is the Boltzmann constant, and e is the electron charge. The electric potential due to the presence of a charged particle will in general be inhibited by the other ones around it, causing it to decay much more rapidly than the Coulomb’s potential. The potential at a distance r from a charge q in a plasma is known as a Debye potential ϕD and expressed as q r ϕD = exp . (1.2) 4πϵ0r (−λD ) The Debye length directly determines the distance over which a given charge will be able to influence others in the plasma. Over lengths much larger than λD the electrical effects of any individual charge are not felt and it is as though the plasma is neutral. Collective shielding is one particular characteristic of a plasma that separates it from an ionized gas. Consequentially the first requirement of a plasma is that the scale size of the system L is much larger than the Debye length L λ . (1.3) ≫ D A plasma is only able to shield a potential when there are sufficient charges present to do so. This is quantified by the plasma parameter Λ. Λ characterizes the number of particles within a sphere of radius λD and as such we necessitate Λ= nλ3 1 (1.4) D ≫ in order for a system to exhibit plasma behavior. The final criterion is temporal in nature. When a plasma is disturbed from a neutral state, the electrons oscillate about an equilibrium in an attempt to restore charge neutrality. It can be shown 2 that the frequency of this oscillation is given by 1 ne2 2 ωp = (1.5) (meϵ0 ) 1 and is known as the plasma frequency. The reciprocal of the plasma frequency ωp− is a timescale associated with the restoration of charge neutrality. In temporal analogy to the Debye length, a system can be considered a plasma if it is observed 1 on a timescale τ that is much longer than ωp− [4]. Therefore we obtain one more requirement for plasma behavior 1 τ ω− . (1.6) ≫ p 1.2 The Solar Wind Plasma is ubiquitous in our universe. We do not observe it in everyday life because there is a lack of ionization of the gases in Earth’s lower atmosphere, however all matter above 100 km must be analyzed using plasma physics [4]. Experiments exist directly in laboratory facilities in which plasma can be created and stud- ied, however these are fairly limited in capability due to technological constraints. Fortunately, there are an abundance of naturally occurring plasmas in the solar system susceptible to experimentation and study. Among these is a continuous outflow of charged particles from the Sun known as the solar wind. The existence of the solar wind (SW) was postulated by Parker [5] in 1958 despite severe skepticism by the scientific community. Its presence was confirmed by measurements from the Soviet lunar mission Luna 1 in 1959 and the subsequent Explorer 10 satellite in 1961 [6]. The wind manifests as a radially accelerating 3 stream of solar particles originating in the solar atmosphere which pervades the heliosphere. The Sun expels these particles into the surrounding solar system because the pressure gradient in its atmosphere is too large to maintain a state of hydrostatic equilibrium. The imbalance initiating the outflow occurs in a region known as the corona. 1.3 The Solar Corona From the Latin for “crown,” the corona appears as a dim aura of high temperature and low density plasma around the sun, as seen in Figure 1.1. The corona is the quintessential example of a naturally occurring plasma specimen that presents a vast array of physical phenomena to study. Problems involving the explanation of coronal loops, solar wind acceleration, and the structure and heating of this region are large, open fields of plasma physics research [7]. Of particular interest to this work, is the resolution of a quandry in solar physics known as the coronal heating problem. The coronal heating problem refers to the massive dissimilarity in the tem- peratures of the 6000 K photospheric surface and over 1,000,000 K corona. The general consensus is that the heating is associated with magnetic field based phe- nomena: magnetic energy is either built up in the corona or transported to it from activity on the solar surface. The two primary models currently in explanatory contention are those of magnetohydrodynamic (MHD) waves and magnetic recon- nection [8].