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A Mathematical Introduction to Magnetohydrodynamics

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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. 119 5.2 Nonlinear shear Alfv´enwaves. 128 5.3 Magnetic field diffusion. 129 A Energy conservation in extended MHD models 137 B Magnetic vector potential in MHD 141 References 145 2 Preamble Magnetohydrodynamics is the theory of electrically conducting, neutral fluids in the low-frequency regime. The basic equations of magnetohydrodynamics (MHD) have been proposed by Hannes Alfv´en[1, 2], who realized the importance of the electric currents carried by a plasma and the magnetic field they generate. Alfv´encombined the equations of fluid dynamics with Faraday's and Amp`ere's laws of electrodynamics, thus obtaining a novel mathematical theory, which helped understanding space plasmas in Earth and planetary magnetospheres, as well as the physics of the Sun, solar wind, and stellar atmospheres. In fusion research, MHD is crucial to the understanding of plasma equilibria and their stability. Liquid metals and electrolytes, like salt water, can also be modeled by MHD equations. Besides the important physical applications, MHD equations exhibit a re- markably beautiful mathematical structure, with connections to geometry and topology that allows us to understand some of the dynamics of magnetic fields in plasmas in terms of topological ideas [3, 4, 5, 6, 7, 8, 9]. As a dynamical system MHD is an example of infinite-dimensional Hamiltonian system [10, 11]. The scope of this lecture is, in this regard, extremely limited. The goal is to introduce MHD equations in a reasonably self-contained way and to discuss some of their most important features. The style of this lectures is quite similar to the mathematical introduction to fluid dynamics by Chorin and Marsden [12] as the title of this note suggests. Particularly, we shall attempt to introduce the physics modeling in a mathematically precise albeit not always rigorous way. The physics literature on the subject is vast. As a reference for further read- ing, the book by Biskamp [13] provides a clear and comprehensive exposition, while the lectures by Schnack [14] offer a more gradual learning curve. For MHD of the solar atmosphere one can refer to Priest [15] as well as to Aschwanden's book on the solar corona [16]. For an introduction to magnetohydrodynamics with emphasis on equilibria and stability of fusion plasmas one can refer to the books by Freidberg [17] and Zohm [18], while Goedbloed, Poedts and Keppens address applications to both astrophysical and fusion plasmas [19, 20]. A nice introduction to MHD with a broader perspective which includes applications to metals can be found in Davidson's book [21]. On the mathematical side, MHD has received considerable attention from ap- plied mathematicians. Its rich mathematical structure has become a paradigm for the application of geometry and topology [9, 22] as well as for structure preserving discretization [23, 24, 25, 26]. As a system of partial differential equations, well-posedness of the Cauchy problem for MHD equations subject to appropriate boundary conditions have been studied first by Duvaut and Lions [27, in French], Sermange and Temam [28], Secchi [29] and more recently, by Chen and coworkers [30] and by Fefferman and coworkers [31, and references therein]. 3 1 Basic elements of fluid dynamics The basic understanding of fluid dynamics is an essential prerequisite to the study of MHD. We shall start by recalling the basic elements thereof, following Chorin and Marsden [12], cf. also Marsden and Hughes [32]. First, we define the physical quantities that describe the dynamical state of a fluid (kinematics) and continue with the equations of motion (dynamics). 1.1 Kinematics of fluids. Fluid dynamics is built on the basis of the con- tinuum hypothesis: A fluid is a distribution of matter occupying a certain region of the continuous three-dimensional space. The considered region of space is a domain (i.e., an open and connected subset) Ω ⊆ R3. We neglect the fact that any fluid is ultimately made of atoms and molecules as we are interested in studying its collective motion on a much larger spatial scale. With the continuum hypothesis, one needs to quantify how matter is dis- tributed in Ω at any time t in a certain time interval I ⊆ R. Thus, the first physical quantity of interest is the mass density, which is a non-negative time- dependent scalar field ρ : I × Ω ! R, such that ρ(t; x) ≥ 0 (1.1) gives the mass per unit of volume at time t 2 I at the spatial location x 2 Ω. By definition, the amount of mass contained in an arbitrary volume W ⊆ Ω (to be referred to as a control volume) is given by Z (mass in W at time t) = ρ(t; x)dx; W which implies that the mass density must be at least locally integrable. Physically we think of an infinitesimal volume of fluid centered around a point x 2 Ω. The volume of this infinitesimal region is mathematically repre- sented by the Lebesgue measure dx in Ω and the mass is represented by the measure ρ(t; x)dx. Such infinitesimal portions of fluid are referred to as fluid elements. In plasma physics the mass density is often replaced by an equivalent non- negative scalar field referred to as the particle number density or simply number density which is defined in terms of the mass density by n(t; x) = ρ(t; x)=m: Here the fluid is regarded as a collection of particles that all have the same mass m; thus, the number of particles contained in a control volume W is given by 1 Z Z (number of particles in W at time t) = ρ(t; x)dx = n(t; x)dx: m W W According to this definition, the number of particles does not need to be an integer, due to the continuum hypothesis. Next we need to describe the motion of the fluid. We introduce a velocity field defined as a time-dependent vector field, u : I × Ω ! R3 such that, u(t; x) 2 R3 (1.2) 4 gives the velocity of the fluid element at the point x 2 Ω and time t 2 I. The vector u(t; x) is referred to as the fluid velocity. The mass times the velocity of the fluid element, namely, ρ(t; x)u(t; x)dx, gives the linear momentum of the fluid element, hence Z (momentum in W at time t) = ρ(t; x)u(t; x)dx: W In addition to the quantities ρ(t; x) and u(t; x), that can be regarded as the counterpart in fluid dynamics of mass and velocity of a particle in mechan- ics, we need to specify another scalar field for the internal energy of the fluid element. Differently from a point-mass particle, a fluid element is a thermody- namical system that can undergo expansions and compressions, thus absorbing and releasing energy. The thermodynamic status of a fluid element is specified by the internal energy density U : I × Ω ! R≥0, where R≥0 denotes the set of non-negative real numbers. If each fluid element is regarded as an ideal gas composed by n(t; x)dx particles, we can equivalently express the internal energy density in terms of a new variable. Specifically the laws of thermodynamics for a perfect gas allow us to write the internal energy of a fluid element as 3 U(t; x)dx = n(t; x)k T (t; x)dx; 2 B where kB is the Boltzmann constant and T : I × Ω ! R is a strictly positive scalar field, such that T (t; x) > 0 (1.3) represents the local temperature of the fluid element at time t 2 I and position x 2 Ω. Therefore, the total energy carried by a fluid element is the sum of the kinetic energy associated to its motion plus the internal energy associated to its thermodynamics, namely, Z 1 2 3 (energy in W at time t) = ρ(t; x)u(t; x) + n(t; x)kBT (t; x) dx: W 2 2 When the law of a perfect gas does not apply we can still define T as above, but now it has the meaning of an effective temperature, which is just a measure of the internal energy and does not imply thermodynamical equilibrium.
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