DOCSLIB.ORG

Introduction to Magnetohydrodynamics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Magnetohydrodynamics JOHANNESKEPLER UNIVERSITÄTLINZ Netzwerk für Forschung, Lehre und Praxis Introduction to Magnetohydrodynamics Bakkalaureatsarbeit zur Erlangung des akademischen Grades Bakkalaureus der Technischen Wissenschaften in der Studienrichtung Technische Mathematik Angefertigt am Institut für Numerische Mathematik Betreuung: DI Peter Gangl Eingereicht von: Andreas Schafelner Linz, März 2016 Johannes Kepler Universität A-4040 Linz · Altenbergerstraße 69 · Internet: http://www.jku.at · DVR 0093696 Abstract Magnetohydrodynamics denotes the study of the dynamics of electrically conducting fluids. It establishes a coupling between the Navier-Stokes equations for fluid dynamics and Maxwell’s equations for electromagnetism. The main concept behind Magnetohy- drodynamics is that magnetic fields can induce currents in a moving conductive fluid, which in turn create forces on the fluid and influence the magnetic field itself. This Bachelor thesis is concerned with the mathematical modelling of Magnetohydro- dynamics. After introducing and deriving Maxwell’s Equations and the Navier-Stokes equations, their coupling is described and the governing equations for Magnetohydro- dynamics are obtained. Eventually some applications of Magnetohydrodynamics are described, i.e., in industrial processes, geophysics and astrophysics. i Contents 1 Introduction 1 1.1 What is Magnetohydrodynamics? . .1 1.2 Notation . .1 2 Electromagnetism 4 2.1 Introduction . .4 2.1.1 Electromagnetic quantities . .4 2.2 Maxwell’s Equations . .5 2.2.1 The Ampère-Maxwell law . .5 2.2.2 The magnetic Gauss law . .5 2.2.3 Faraday’s law . .6 2.2.4 Gauss’s law for electric fields . .6 2.2.5 Constitutive equations and material laws . .7 2.2.6 The Lorentz force . .8 2.2.7 The conservation of charge . .8 2.2.8 Summary of Maxwell’s Equations . .9 2.3 Simplifications for MHD . .9 2.3.1 Maxwell’s Equations, constitutive laws and the Laplace force . .9 2.3.2 The induction equation . 12 3 Fluid dynamics 14 3.1 Introduction . 14 3.2 Representation of a flow . 14 3.2.1 Lagrangian representation . 14 3.2.2 Eulerian representation . 15 3.3 The transport theorem . 16 3.4 The continutiy equation . 17 3.5 Equations of motion . 18 3.6 The Navier-Stokes equations . 19 3.6.1 Non-dimensionalisation and the Reynolds number Re ...... 21 4 Magnetohydrodynamics (MHD) 22 4.1 The governing equations . 22 4.2 Non-dimensionalisation . 23 ii CONTENTS iii 4.2.1 Summary . 24 4.3 Diffusion and Convection . 24 4.3.1 Diffusion of the magnetic field . 25 4.3.2 Convection of the magnetic field . 28 5 Applications of MHD 30 5.1 Simplifications for low Rem ........................ 30 5.2 MHD generators and pumps . 31 5.2.1 Theory of Hartmann layers . 31 5.2.2 Applications of Hartmann layers . 34 6 Conclusion 36 Chapter 1 Introduction 1.1 What is Magnetohydrodynamics? In nature as in industrial processes, we can observe magnetic fields influencing the behaviour of fluids and flows. In the metallurgical industry, magnetic fields are used to stir, pump, levitate and heat liquid metals. The earth’s magnetic field, protecting the surface from deadly radiation, is generated by the motion of the earth’s liquid core. Sunspots and solar flares are generated by the solar magnetic field and the galactic magnetic field which influences the formation of stars from interstellar gas clouds. We use the word Magnetohydrodynamics (MHD) for all of these phenomena, where the magnetic field B and the velocity field u are coupled, given there is an electrically conducting and non-magnetic fluid, e.g. liquid metals, hot ionised gases (plasmas) or strong electrolytes. The magnetic field can induce currents into such a moving fluid and this creates forces acting on the fluid and altering the magnetic field itself. 1.2 Notation In this short section of the thesis all facts about notation are presented. First of all, if not specified otherwise, a normal letter x represents a scalar value, whereas a bold T letter x = (x1, x2, x3) indicates a three dimensional vector. This holds for variables, constants and functions. Functions are either of static or dynamic nature. Static functions depend only on space coordinates, e.g., T (x) gives the temperature in the point x, whereas dynamic functions additionally have a time-dependent component t, e.g., T (x, t) gives the temperature in the point x at time t. The symbol · represents the euclidean inner product and × stands for the cross product. With | . | we denote either the absolute value of a scalar or the euclidean norm for vectors. Throughout the thesis the following differential operators will occur: 1 CHAPTER 1. INTRODUCTION 2 Definition 1.1. Let u : R3 × [0,T ] → R3 and f : R3 × [0,T ] → R be sufficiently smooth. Then the gradient ∇, the divergence ∇·, the curl ∇× and the Laplacian ∆ are defined as ∂f ∂x1 ∂u2 − ∂u3 ∂f ∂x3 ∂x2 ∇f = ∂u ∂u ∂x2 ∇ × u = − 3 + 1 ∂f ∂x1 ∂x3 ∂u ∂u ∂x3 2 − 1 2 2 2 ∂x1 ∂x2 ∂ f ∂ f ∂ f ∆f = 2 + 2 + 2 ∆u1 ∂x1 ∂x2 ∂x3 ∆u = ∆u2 ∂u1 ∂u2 ∂u3 ∇ · u = + + ∆u3 ∂x1 ∂x2 ∂x3 Remark 1.2. As we can see in Definition 1.1, the differential operators are only using the partial derivatives of the space coordinates, but not the time coordinate. Usually the operators are defined for every variable of a function. Apart from differential operators, we will encounter different types of integrals, which are also summarized in the following definition. Definition 1.3. Let u : R3 × [0,T ] → R3 be a sufficiently smooth vector field. We define the line integral as u · dl := u(φ(τ)) · φ0(τ) dτ, (1.1) ˆC ˆI with φ : I → R3 a parametrisation of the curve C ⊂ R3 and I = [a, b] ⊂ R. Furthermore we define the surface integral ∂ψ ∂ψ ! u · ~n dS := u(ψ(σ, τ)) · × d(σ, τ), (1.2) ˆS ˆK ∂σ ∂τ where ψ : K → R3 is a parametrisation of the surface S ⊂ R3 and K ⊂ R2 is a reference domain. The volume integral is defined as u dx = u(x1, x2, x3, t) d(x1, x2, x3), (1.3) ˆV ˆV where V ⊂ R3 is a volume. With these definitions, we can formulate two important theorems, which relate the different types of integrals to each other. Theorem 1.4 (Stokes’ Theorem). Let S be a surface in R3, parametrised by ϕ : M → R3, where M is a superset of a normal range K in R2, with ϕ(K) = S. Let ϕ ∈ C2(M, R3) and ∂S be the boundary curve of S. Let u ∈ C1(ϕ(M), R3). Then it holds that ∇ × u · ~n dS = u · dl. (1.4) ˆS ˆ∂S CHAPTER 1. INTRODUCTION 3 Proof. See Thm. 8.50 in [2]. Theorem 1.5 (Divergence Theorem). Let V be a sufficiently smooth subset of R3 and let ∂V its boundary. Let M ⊇ V and u ∈ C1(M, R3). Then it holds that u · ~n dS = ∇ · u dx. (1.5) ˆ∂V ˆV Proof. See Thm. 8.58 and Remark 8.59 in [2] Chapter 2 Electromagnetism - Maxwell’s Equations 2.1 Introduction 2.1.1 Electromagnetic quantities Throughout this thesis, we will deal with a set of physical quantities to describe elec- tromagnetic processes. These are summarized in Table 2.1. However, some quantities need further explanation. The electric field E = Es + Ei consists of two parts, the electrostatic field Es and the induced electric field Ei which have a similar effect but a different structure. The electric current density J = J c + J i consists of the conduct current density J c and the induced current density J i. The magnetisation M quanti- fies the strength of a permanent magnet and the electric polarisation P is its electric counterpart. Variable Physical quantity Unit E electric field intensity [V/m] D electric flux density [As/m2] H magnetic field density [A/m] B magnetic flux density [V s/m2] J electric current density [A/m2] 2 ρc electric charge density [As/m ] M magnetisation [V s/m2] P electric polarisation [As/m2] Table 2.1: Overview of electromagnetic quantities 4 CHAPTER 2. ELECTROMAGNETISM 5 2.2 Maxwell’s Equations We continue now by deriving Maxwell’s Equations. We start with the integral form of each equation and will then extract the differential form. Both forms are commonly used, but for the following chapters, the latter will be used. The content of this section is based on [3] and [4]. 2.2.1 The Ampère-Maxwell law Ampère’s law, or more correctly Ampère-Maxwell’s law describes the relation of a magnetic field and an electric current. In terms of mathematics, we can write the relation as ∂D H · dl = (J + ) · ~n dS, (2.1) ˛C ˆS ∂t where C is a closed curve bounding a surface S. Now applying Stokes’ theorem to the left hand side of (2.1) gives us ∂D ∇ × H · ~n dS = (J + ) · ~n dS, ˆS ˆS ∂t and as this holds for any surface S, we obtain ∂D ∇ × H = J + . (2.2) ∂t We call (2.2) the Ampère-Maxwell law in differential form. In the static case, (2.2) reduces to ∇ × H = J, which is commonly known as Ampère’s law. 2.2.2 The magnetic Gauss law Gauss’s law for magnetic fields gives information about the magnetic flux. It states that the total magnetic flux through any closed surface is zero [3]. We can write this in terms of mathematics as B · ~n dS = 0, (2.3) ˛S with S a closed surface surrounding a volume V. Using the divergence theorem we obtain 0 = B · ~n dS = ∇ · B dx, ˛S ˆV which holds for arbitrary volumes V with closed surface S.
Load more

Recommended publications
  • Small-Scale Dynamo Action in Rotating Compressible Convection
    Small-scale dynamo action in rotating compressible convection B. Favier∗and P.J. Bushby School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK Abstract We study dynamo action in a convective layer of electrically-conducting, compressible fluid, rotating about the vertical axis. At the upper and lower bounding surfaces, perfectly- conducting boundary conditions are adopted for the magnetic field. Two different levels of thermal stratification are considered. If the magnetic diffusivity is sufficiently small, the con- vection acts as a small-scale dynamo. Using a definition for the magnetic Reynolds number RM that is based upon the horizontal integral scale and the horizontally-averaged velocity at the mid-layer of the domain, we find that rotation tends to reduce the critical value of RM above which dynamo action is observed. Increasing the level of thermal stratification within the layer does not significantly alter the critical value of RM in the rotating calculations, but it does lead to a reduction in this critical value in the non-rotating cases. At the highest computationally-accessible values of the magnetic Reynolds number, the saturation levels of the dynamo are similar in all cases, with the mean magnetic energy density somewhere be- tween 4 and 9% of the mean kinetic energy density. To gain further insights into the differences between rotating and non-rotating convection, we quantify the stretching properties of each flow by measuring Lyapunov exponents. Away from the boundaries, the rate of stretching due to the flow is much less dependent upon depth in the rotating cases than it is in the corre- sponding non-rotating calculations.
    [Show full text]
  • Magnetohydrodynamics - Overview
    Center for Turbulence Research 361 Proceedings of the Summer Program 2006 Magnetohydrodynamics - overview Magnetohydrodynamics (MHD) is the study of the interaction of electrically conduct- ing fluids and electromagnetic forces. MHD problems arise in a wide variety of situations ranging from the explanation of the origin of Earth's magnetic field and the prediction of space weather to the damping of turbulent fluctuations in semiconductor melts dur- ing crystal growth and even the measurement of the flow rates of beverages in the food industry. The description of MHD flows involves both the equations of fluid dynamics - the Navier-Stokes equations - and the equations of electrodynamics - Maxwell's equa- tions - which are mutually coupled through the Lorentz force and Ohm's law for moving electrical conductors. Given the vast range of Reynolds numbers and magnetic Reynolds numbers characterising astrophysical, geophysical, and industrial MHD problems it is not feasible to devise an all-encompassing numerical code for the simulation of all MHD flows. The challenge in MHD is rather to combine a judicious application of analytical tools for the derivation of simplified equations with accurate numerical algorithms for the solution of the resulting mathematical models. The 2006 summer program provided a unique opportunity for scientists and engineers working in different sub-disciplines of MHD to interact with each other and benefit from the numerical capabilities available at the Center for Turbulence Research. In materials processing such as the growth of semiconductor crystals or the continuous casting of steel turbulent flows are often undesirable. The application of steady magnetic fields provides a convenient means for the contactless damping of turbulent fluctuations and for the laminarisation of flows.
    [Show full text]
  • Effects of a Uniform Magnetic Field on a Growing Or Collapsing Bubble in a Weakly Viscous Conducting Fluid K
    Effects of a uniform magnetic field on a growing or collapsing bubble in a weakly viscous conducting fluid K. H. Kang, I. S. Kang, and C. M. Lee Citation: Phys. Fluids 14, 29 (2002); doi: 10.1063/1.1425410 View online: http://dx.doi.org/10.1063/1.1425410 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v14/i1 Published by the American Institute of Physics. Related Articles Dynamics of magnetic chains in a shear flow under the influence of a uniform magnetic field Phys. Fluids 24, 042001 (2012) Travelling waves in a cylindrical magnetohydrodynamically forced flow Phys. Fluids 24, 044101 (2012) Two-dimensional numerical analysis of electroconvection in a dielectric liquid subjected to strong unipolar injection Phys. Fluids 24, 037102 (2012) Properties of bubbled gases transportation in a bromothymol blue aqueous solution under gradient magnetic fields J. Appl. Phys. 111, 07B326 (2012) Magnetohydrodynamic flow of a binary electrolyte in a concentric annulus Phys. Fluids 24, 037101 (2012) Additional information on Phys. Fluids Journal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors Downloaded 07 May 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions PHYSICS OF FLUIDS VOLUME 14, NUMBER 1 JANUARY 2002 Effects of a uniform magnetic field on a growing or collapsing bubble in a weakly viscous conducting fluid K. H. Kang Department of Mechanical Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Pohang 790-784, Korea I.
    [Show full text]
  • A Mathematical Introduction to Magnetohydrodynamics
    Typeset in LATEX 2" July 26, 2017 A Mathematical Introduction to Magnetohydrodynamics Omar Maj Max Planck Institute for Plasma Physics, D-85748 Garching, Germany. e-mail: [email protected] 1 Contents Preamble 3 1 Basic elements of fluid dynamics 4 1.1 Kinematics of fluids. .4 1.2 Lagrangian trajectories and flow of a vector field. .5 1.3 Deformation tensor and vorticity. 14 1.4 Advective derivative and Reynolds transport theorem. 17 1.5 Dynamics of fluids. 20 1.6 Relation to kinetic theory and closure. 24 1.7 Incompressible flows . 32 1.8 Equations of state, isentropic flows and vorticity. 34 1.9 Effects of Euler-type nonlinearities. 35 2 Basic elements of classical electrodynamics 39 2.1 Maxwell's equations. 39 2.2 Lorentz force and motion of an electrically charged particle. 52 2.3 Basic mathematical results for electrodynamics. 57 3 From multi-fluid models to magnetohydrodynamics 69 3.1 A model for multiple electrically charged fluids. 69 3.2 Quasi-neutral limit. 74 3.3 From multi-fluid to a single-fluid model. 82 3.4 The Ohm's law for an electron-ion plasma. 86 3.5 The equations of magnetohydrodynamics. 90 4 Conservation laws in magnetohydrodynamics 95 4.1 Global conservation laws in resistive MHD. 95 4.2 Global conservation laws in ideal MHD. 98 4.3 Frozen-in law. 100 4.4 Flux conservation. 105 4.5 Topology of the magnetic field. 109 4.6 Analogy with the vorticity of isentropic flows. 117 5 Basic processes in magnetohydrodynamics 119 5.1 Linear MHD waves.
    [Show full text]
  • Stability Study of the Cylindrical Tokamak--Thomas Scaffidi(2011)
    Ecole´ normale sup erieure´ Princeton Plasma Physics Laboratory Stage long de recherche, FIP M1 Second semestre 2010-2011 Stability study of the cylindrical tokamak Etude´ de stabilit´edu tokamak cylindrique Author: Supervisor: Thomas Scaffidi Prof. Stephen C. Jardin Abstract Une des instabilit´es les plus probl´ematiques dans les plasmas de tokamak est appel´ee tearing mode . Elle est g´en´er´ee par les gradients de courant et de pression et implique une reconfiguration du champ magn´etique et du champ de vitesse localis´ee dans une fine r´egion autour d’une surface magn´etique r´esonante. Alors que les lignes de champ magn´etique sont `al’´equilibre situ´ees sur des surfaces toriques concentriques, l’instabilit´econduit `ala formation d’ˆıles magn´etiques dans lesquelles les lignes de champ passent d’un tube de flux `al’autre, rendant possible un trans- port thermique radial important et donc cr´eant une perte de confinement. Pour qu’il puisse y avoir une reconfiguration du champ magn´etique, il faut inclure la r´esistivit´edu plasma dans le mod`ele, et nous r´esolvons donc les ´equations de la magn´etohydrodynamique (MHD) r´esistive. On s’int´eresse `ala stabilit´ede configurations d’´equilibre vis-`a-vis de ces instabilit´es dans un syst`eme `ala g´eom´etrie simplifi´ee appel´ele tokamak cylindrique. L’´etude est `ala fois analytique et num´erique. La solution analytique est r´ealis´ee par une m´ethode de type “couche limite” qui tire profit de l’´etroitesse de la zone o`ula reconfiguration a lieu.
    [Show full text]
  • Magnetohydrodynamics 1 19.1Overview
    Contents 19 Magnetohydrodynamics 1 19.1Overview...................................... 1 19.2 BasicEquationsofMHD . 2 19.2.1 Maxwell’s Equations in the MHD Approximation . ..... 4 19.2.2 Momentum and Energy Conservation . .. 8 19.2.3 BoundaryConditions. 10 19.2.4 Magneticfieldandvorticity . .. 12 19.3 MagnetostaticEquilibria . ..... 13 19.3.1 Controlled thermonuclear fusion . ..... 13 19.3.2 Z-Pinch .................................. 15 19.3.3 Θ-Pinch .................................. 17 19.3.4 Tokamak.................................. 17 19.4 HydromagneticFlows. .. 18 19.5 Stability of Hydromagnetic Equilibria . ......... 22 19.5.1 LinearPerturbationTheory . .. 22 19.5.2 Z-Pinch: Sausage and Kink Instabilities . ...... 25 19.5.3 EnergyPrinciple ............................. 28 19.6 Dynamos and Reconnection of Magnetic Field Lines . ......... 29 19.6.1 Cowling’stheorem ............................ 30 19.6.2 Kinematicdynamos............................ 30 19.6.3 MagneticReconnection. 31 19.7 Magnetosonic Waves and the Scattering of Cosmic Rays . ......... 33 19.7.1 CosmicRays ............................... 33 19.7.2 Magnetosonic Dispersion Relation . ..... 34 19.7.3 ScatteringofCosmicRays . 36 0 Chapter 19 Magnetohydrodynamics Version 1219.1.K.pdf, 7 September 2012 Please send comments, suggestions, and errata via email to [email protected] or on paper to Kip Thorne, 350-17 Caltech, Pasadena CA 91125 Box 19.1 Reader’s Guide This chapter relies heavily on Chap. 13 and somewhat on the treatment of vorticity • transport in Sec. 14.2 Part VI, Plasma Physics (Chaps. 20-23) relies heavily on this chapter. • 19.1 Overview In preceding chapters, we have described the consequences of incorporating viscosity and thermal conductivity into the description of a fluid. We now turn to our final embellishment of fluid mechanics, in which the fluid is electrically conducting and moves in a magnetic field.
    [Show full text]
  • Application of High Performance Computing for Fusion Research Shimpei Futatani
    Application of high performance computing for fusion research Shimpei Futatani Department of Physics, Universitat Politècnica de Catalunya (UPC), Barcelona, Spain The study of the alternative energy source is one of the absolutely imperative research topics in the world as our present energy sources such as fossil fuels are limited. This work is dedicated to the nuclear fusion physics research in close collaboration with existing experimental fusion devices and the ITER organization (www.iter.org) which is an huge international nuclear fusion R&D project. The goal of the ITER project is to demonstrate a clean and safe energy production by nuclear fusion which is the reaction that powers the sun. One of the ideas of the nuclear fusion on the earth is that the very high temperature ionized particles, forming a plasma can be controlled by a magnetic field, called magnetically confined plasma. This is essential, because no material can be sustained against such high temperature reached in a fusion reactor. Tokamak is a device which uses a powerful magnetic field to confine a hot plasma in the shape of a torus. It is a demanding task to achieve a sufficiently good confinement in a tokamak for a ‘burning plasma’ due to various kinds of plasma instabilities. One of the critical unsolved problems is MHD (MagnetoHydroDynamics) instabilities in the fusion plasmas. The MHD instabilities at the plasma boundary damages the plasma facing component of the fusion reactor. One of techniques to control the MHD instabilities is injection of pellets (small deuterium ice bodies). The physics of the interaction between pellet ablation and MHD dynamics is very complex, and uncertainties still remain regarding the theoretical physics as well as the numerical modelling point of view.
    [Show full text]
  • Identification of the Sequence of Steps Intrinsic to Spheromak Formation
    Identification of the Sequence of Steps Intrinsic to Spheromak Formation P.M. Bellan, S. You, and G. S. Yun California Institute of Technology Pasadena, CA 91125, USA Abstract. A planar coaxial electrostatic helicity source is used for studying the relaxation process intrinsic to spheromak formation. Experimental observations reveal that spheromak formation involves: (1) breakdown and creation of a number of distinct, arched, filamentary, plasma-filled flux loops that span from cathode to anode gas nozzles, (2) merging of these loops to form a central column, (3) jet-like expansion of the central column, (4) kink instability of the central column, (5) conversion of toroidal flux to poloidal flux by the kink instability. Steps 1 and 3 indicate that spheromak formation involves an MHD pumping of plasma from the gas nozzles into the magnetic flux tube linking the nozzles. In order to measure this pumping, the gas puffing system has been modified to permit simultaneous injection of different gas species into the two ends of the flux tube linking the wall. Gated CCD cameras with narrow-band optical filters are used to track the pumped flows. Keywords: spheromak, magnetohydrodynamics, kink, jet PACS: 52.55.Ip, 52.35.Py, 52.55.Wq, 52.59.Dk INTRODUCTION which is rich in toroidal field energy (typical poloidal currents are 100 kA) but poor in poloidal field energy. Spheromaks are toroidal MHD equilibria incorporating Spheromak formation involves conversion of toroidal nested poloidal magnetic flux surfaces suitable for magnetic flux into poloidal magnetic flux while confining a fusion plasma. Unlike tokamaks, conserving magnetic helicity (but not conserving spheromaks are a self-organizing, minimum-energy magnetic energy).
    [Show full text]
  • Toward a Self-Generating Magnetic Dynamo: the Role of Turbulence
    PHYSICAL REVIEW E VOLUME 61, NUMBER 5 MAY 2000 Toward a self-generating magnetic dynamo: The role of turbulence Nicholas L. Peffley, A. B. Cawthorne, and Daniel P. Lathrop* Department of Physics, University of Maryland, College Park, Maryland 20742 ͑Received 6 July 1999͒ Turbulent flow of liquid sodium is driven toward the transition to self-generating magnetic fields. The approach toward the transition is monitored with decay measurements of pulsed magnetic fields. These mea- surements show significant fluctuations due to the underlying turbulent fluid flow field. This paper presents experimental characterizations of the fluctuations in the decay rates and induced magnetic fields. These fluc- tuations imply that the transition to self-generation, which should occur at larger magnetic Reynolds number, will exhibit intermittent bursts of magnetic fields. PACS number͑s͒: 47.27.Ϫi, 47.65.ϩa, 05.45.Ϫa, 91.25.Cw I. INTRODUCTION Reynolds number will be quite large for all flows attempting to self-generate ͑where Re ӷ1 yields Reӷ105)—implying The generation of magnetic fields from flowing liquid m turbulent flow. These turbulent flows will cause the transition metals is being pursued by a number of scientific research to self-generation to be intermittent, showing both growth groups in Europe and North America. Nuclear engineering and decay of magnetic fields irregularly in space and time. has facilitated the safe use of liquid sodium, which has con- This intermittency is not something addressed by kinematic tributed to this new generation of experiments. With the dynamo studies. The analysis in this paper focuses on three highest electrical conductivity of any liquid, sodium retains main points: we quantify the approach to self-generation and distorts magnetic fields maximally before they diffuse with increasing Rem , characterize the turbulence of induced away.
    [Show full text]
  • Stability and Instability of Hydromagnetic Taylor-Couette Flows
    Physics reports Physics reports (2018) 1–110 DRAFT Stability and instability of hydromagnetic Taylor-Couette flows Gunther¨ Rudiger¨ a;∗, Marcus Gellerta, Rainer Hollerbachb, Manfred Schultza, Frank Stefanic aLeibniz-Institut f¨urAstrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany bDepartment of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom cHelmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr. 400, D-01328 Dresden, Germany Abstract Decades ago S. Lundquist, S. Chandrasekhar, P. H. Roberts and R. J. Tayler first posed questions about the stability of Taylor- Couette flows of conducting material under the influence of large-scale magnetic fields. These and many new questions can now be answered numerically where the nonlinear simulations even provide the instability-induced values of several transport coefficients. The cylindrical containers are axially unbounded and penetrated by magnetic background fields with axial and/or azimuthal components. The influence of the magnetic Prandtl number Pm on the onset of the instabilities is shown to be substantial. The potential flow subject to axial fieldspbecomes unstable against axisymmetric perturbations for a certain supercritical value of the averaged Reynolds number Rm = Re · Rm (with Re the Reynolds number of rotation, Rm its magnetic Reynolds number). Rotation profiles as flat as the quasi-Keplerian rotation law scale similarly but only for Pm 1 while for Pm 1 the instability instead sets in for supercritical Rm at an optimal value of the magnetic field. Among the considered instabilities of azimuthal fields, those of the Chandrasekhar-type, where the background field and the background flow have identical radial profiles, are particularly interesting.
    [Show full text]
  • Peaceful Uses of Atomic Energy, Held in Point of Discussions Back in 1958
    INTERNATIONAL ATOMIC ENERGY AGENCY Wagramer Strasse 5, P.O. Box 100 A-1400 Vienna, Austria www.iaea.org CONTENT Guenter Mank, Head, Physics Section Alan Aqrawi, Physics Section Division of Physical and Chemical Sciences Division of Physical and Chemical Sciences Department of Nuclear Sciences and Applications Department of Nuclear Sciences and Applications CREDITS Photography: UNOG Archives Geneva (page 4,5,6,16,17,20,24,25,26), Nuclear Fusion Institute RRC "Kurchatov Institute" Russia (page 9,18,19), KFA Kernforschungsanlage Juelich Germany (page 10,11,15), ORNL Oak Ridge National Laboratory USA (page 6), IAEA Archives Vienna (page 8). DESIGN Alan Aqrawi © IAEA, 2008 INDEX FOREWORD Foreword “Celebrating fifty years of fusion… Foreword 2 …entering into the burning plasma era.” Energy in all its forms has always driven been achieved every 1.8 years. New human development. New technologies in developments in science and engineering energy production, starting from the use of fire have led to an optimized magnetic prototype The 2nd Geneva Conference - Introduction 4 itself, have driven economic and social reactor, with corresponding cost savings. development. In the mid-1950s, nuclear Inertial confinement experiments have energy created new hope for an abundant achieved similar progress. The culmination of source of that energy for the world. international collaborative efforts in fusion is the start of construction of the International Declassification 6 To promote this groundbreaking technology, Thermonuclear Experimental Reactor (ITER), and to host a neutral ground for substantive the biggest scientific endeavour the world has scientific debate, the United Nations in 1955 ever seen involving States with more than half organized the first of a series of conferences of the world’s population.
    [Show full text]
  • Plasma Physics and Controlled Fusion Research During Half a Century Bo Lehnert
    SE0100262 TRITA-A Report ISSN 1102-2051 VETENSKAP OCH ISRN KTH/ALF/--01/4--SE IONST KTH Plasma Physics and Controlled Fusion Research During Half a Century Bo Lehnert Research and Training programme on CONTROLLED THERMONUCLEAR FUSION AND PLASMA PHYSICS (Association EURATOM/NFR) FUSION PLASMA PHYSICS ALFV N LABORATORY ROYAL INSTITUTE OF TECHNOLOGY SE-100 44 STOCKHOLM SWEDEN PLEASE BE AWARE THAT ALL OF THE MISSING PAGES IN THIS DOCUMENT WERE ORIGINALLY BLANK TRITA-ALF-2001-04 ISRN KTH/ALF/--01/4--SE Plasma Physics and Controlled Fusion Research During Half a Century Bo Lehnert VETENSKAP OCH KONST Stockholm, June 2001 The Alfven Laboratory Division of Fusion Plasma Physics Royal Institute of Technology SE-100 44 Stockholm, Sweden (Association EURATOM/NFR) Printed by Alfven Laboratory Fusion Plasma Physics Division Royal Institute of Technology SE-100 44 Stockholm PLASMA PHYSICS AND CONTROLLED FUSION RESEARCH DURING HALF A CENTURY Bo Lehnert Alfven Laboratory, Royal Institute of Technology S-100 44 Stockholm, Sweden ABSTRACT A review is given on the historical development of research on plasma physics and controlled fusion. The potentialities are outlined for fusion of light atomic nuclei, with respect to the available energy resources and the environmental properties. Various approaches in the research on controlled fusion are further described, as well as the present state of investigation and future perspectives, being based on the use of a hot plasma in a fusion reactor. Special reference is given to the part of this work which has been conducted in Sweden, merely to identify its place within the general historical development. Considerable progress has been made in fusion research during the last decades.
    [Show full text]