Magnetohydrodynamics

The main magnetic field on the Earth is generated by, and has been maintained throughout Earth’s history by, a fluid dynamo operating in the Earth’s electrically conducting core. The author gives his personal view of how understanding of this ‘geodynamo’ grew during his lifetime, and he includes recollections of some of the scientists involved. The remarkable evolution of the subject from simple applications of electromagnetic theory to today’s sophisticated magnetohydrodynamic theory is outlined. The importance of Coriolis forces in core MHD is not fully appreciated even today, but it transforms MHD into an essentially different subject that is briefly reviewed here. Proposals are made to give it its own special name.

Electric discharges in gases were subject to a rather limited research effort during the first three decades of the twentieth century. There was, however, a turning point when Hannes Alfvén [1,2] started his studies on electromagnetic phenomena in astrophysics at the end of the 1930s, and especially after his discovery of magnetohydrodynamic waves in 1942.

The long history of laboratory experiments on homogeneous dynamo action is delineated. It is worked out what sort of insight can be expected from experiments, and what not. Special focus is laid on the principle and the main results of the Riga dynamo experiment which is shown to represent a genuine hydromagnetic dynamo with a non-trivial saturation mechanism that relies mainly on the fluidity of the electrically conducting medium.

Mean-field dynamo theory has proved to be a useful tool for understanding the generation of magnetic fields in the Earth and the Sun, in stellar bodies, and even in galaxies. It provides a basis for the elaboration of detailed dynamo models of these objects. Fundamentals of this theory were developed in the 1960s of the last century in the Institute for Magnetohydrodynamics in Jena in Germany under the directorship of Max Steenbeck. The 21st of March 2004, a date close to that of the Coventry meeting, would have been his 100th birthday. Let me say first some words about his life and his contributions to various fields in physics. This will lead naturally to the early ideas of meanfield dynamo theory, to its remarkable findings and to some of the problems of its further development.

Halley, working before the discoveries of Oersted, Ampère, Faraday, and Henry, pictured the Earth’s interior with two massive blocks of permanently magnetized material (lodestone): an outer shell and a concentric inner nucleus. From the observed magnetic variations, he inferred the existence of a fluid domain.

Magnetic fields permeate the Universe. They are found in planets, stars, accretion discs, galaxies, clusters of galaxies, and the intergalactic medium. While there is often a component of the field that is spatially coherent at the scale of the astrophysical object, the field lines are tangled chaotically and there are magnetic fluctuations at scales that range over orders of magnitude. The cause of this disorder is the turbulent state of the plasma in these systems. This plasma is, as a rule, highly conducting, so the magnetic field lines are entrained by (frozen into) the fluid motion. As the fields are stretched and bent by the turbulence, they can resist deformation by exerting the Lorentz force on the plasma. The turbulent advection of the magnetic field and the field’s back reaction together give rise to the statistically steady state of fully developed MHD turbulence. In this state, energy and momentum injected at large (object-size) scales are transfered to smaller scales and eventually dissipated.

Some researches are described concerning early work on the transient self-magnetic “pinch effect” in plasmas at both high and low pressures. Theoretical work motivated by interest in shock waves led to the discovery in 1958 of solitary waves in plasma physics, they were later to become known as solitons.

How the study of magnetohydrodynamics came to the Engineering Department of Cambridge University was never fully recorded. What is known is that an interest in the heat-transfer properties of liquid metals began with the research undertaken by L.M. Trefethen, who entered the Department as a research student in 1946. Trefethen’s earlier education had been in the United States, culminating in a master’s degree from the Massachusettes Institute of Technology. His subject of research at Cambridge was approved initially as “Gas turbines – turbine blade cooling”.

The realization of controlled thermonuclear fusion could lead to a significant contribution to future energy demands. The reaction between the fuel components tritium and deuterium requires temperatures above 108 K so that any confinement using solid walls is excluded. At these temperatures the fuels are ionized and form an electrically highly conducting plasma that can be confined by strong magnetic fields to a defined volume. During the past decades different concepts of magnetic confinement have been investigated and a number of conceptual designs for commercial or experimental fusion reactors have been studied.

Low magnetic Reynolds number magnetohydrodynamic (MHD) flows and low Rossby number rotating flows share a number of common features. Both are subjected to a strong linear force: Lorentz or Coriolis. From an energetic point of view, they are very different, as Lorentz forces are dissipative in nature (Joule dissipation adding to viscous dissipation) while Coriolis forces are purely conservative. Both forces however tend to favour a two-dimensional (2D) flow, independent of the direction of the applied magnetic field or rotation axis.

This essay provides a personal account of the development of the subject of magnetohydrodynamic (MHD) turbulence from its birth in 1950 to its “coming-of-age” in 1971, following the development of mean-field electrodynamics, a major breakthrough of the 1960s. The discussion covers the early ideas based on the analogy with vorticity, the passive vector problem, the suppression of turbulence by an applied magnetic field, and aspects of the turbulent dynamo problem.

A brief account is presented on analogies between the processes of evolution of vorticity and magnetic field and related problems starting from the very beginning and including the most recent results. The emphasis is made on essential differences as contrasted to similarities. This is seen already on a purely kinematic level which is the main theme of this communication.

This paper is an attempt to summarize the most important results and established ideas on magnetohydrodynamic (MHD) turbulence in flows of liquid metals when the magnetic Reynolds number is significantly smaller than unity. It is written on the basis of the round-table discussion organised during the Coventry meeting, with additions introduced by the authors, coming from their own vision of the subject, or raised during their exchanges with other specialists. It covers the turbulent regimes observable in rather well controlled laboratory experiments as well as in metal processes where electromagnetic devices are used for different purposes (stirring, pumping, refining, etc). A number of still not-understood points are mentioned and some needs of new efforts are underlined.

Numerical simulations of turbulent phenomena in fluids have made considerable progress with the emergence of large parallel computers. For simple geometries, very efficient numerical methods have been developed to provide accurate numerical solutions to the equations of fluid dynamics. These approaches are referred to as direct numerical simulation (DNS) and their predictions are often regarded as almost as reliable as the experimental data.

The beginning of the rapid development of theoretical and applied magnetohydrodynamics (MHD) during the end of the 1950s can be understood only by following the activities of a single talented, creative, and dedicated individual – a man who was appointed executive director of the newly established Institute of Physics at the Latvian Academy of Sciences. This was an unusual appointment because the person we are referring to, Professor Igor Mikhailovich Kirko, had just celebrated his 30th birthday. To the best of my knowledge, Kirko was the first to direct a scientific institution that made a broad experimental investigation on different phenomena of magnetohydrodynamics.

Analysis and control of fluid flows, often subsidiary to industrial design issues, require measurements of the flow field. For classical transparent fluids such as water or gas a variety of well-developed techniques (laser Doppler and particle image velocimetry, Schlieren optics, interferometric techniques, etc.) have been established. In contrast, the situation regarding opaque liquids still lacks almost any commercial availability. Metallic and semiconductor melts often pose additional problems of high temperature and chemical aggressiveness, rendering any reliable determination of the flow field a challenging task. This review intends to summarise different approaches suitable for velocity measurements in liquid metal flows and to discuss perspectives, particularly in view of some recent developments (ultrasound, magnetic tomography). Focusing mainly on local velocity measurements, it is subsequently distinguished between invasive and non-invasive methods, leaving entirely aside the acquisition of temperature, pressure, and concentration, for which [1] may serve as a comprehensive reference.

The possibility to act on a fluid flow in a contactless way, offered by magnetohydrodynamics (MHD), stimulated the imagination of aerodynamists and naval engineers relatively early.

History of electromagnetic processing of materials (EPM) is described and several functions utilized in EPM are reviewed. Main activities of EPM are summarized with the view on mass production and applications of high magnetic fields related to nanotechnology. Future trends and prospects of EPM are discussed.

Magnetic fields are often used in materials processing, especially in solidification processing for metallic alloys and semiconductors. For example, both static and alternating magnetic fields have been used extensively to control melt flow and solidified structures in the continuous casting of steels [1, 2]. Static magnetic fields are used as electromagnetic brakes and alternating magnetic fields are used as electromagnetic stirrers. Alternating magnetic fields with rather high frequencies are used to hold melt pools and to achieve soft contacts with molds. It is well known that electromagnetic processing can significantly improve the quality of products.

Electromagnetic processing of materials (EPM) is one of the most widely practiced and fast growing applications of magnetic and electric forces to fluid flow. EPM is encountered in both industrial processes and laboratory investigations. Applications range in scale from nano-particle manipulation to tonnes of liquid metal treated in the presence of various configurations of magnetic fields. Some of these processes are specifically designed and made possible by the use of the electromagnetic force, like the magnetic levitation of liquid droplets, whilst others involve electric currents essential for electrothermal or electrochemical reasons, for instance, in electrolytic metal production and in induction melting. An insight for the range of established and novel EPM applications can be found in the review presented by Asai [1] in the EPM-2003 conference proceedings.

We may define three main categories of crystal growth techniques: growth from solid, vapour, and melt. These three main categories of crystal growth methods need careful control of the phase change. We may introduce a subcategory, growth from the solution, which is strictly already included in the above definitions, and which represents crystal growth processes of solute from an impure melt.

Magnetoelectrochemistry (MEC) is electrochemistry in the presence of an imposed magnetic field. This relatively new branch of electrochemistry has seen rapid development during the last years [1], the potential applications being very promising even if not industrially realized up to now. Several studies have been performed with the objective to elucidate the effect of a magnetic field on the electrolyte properties, on the mass transfer processes and, at a smaller scale, on the electrochemical kinetics and on the structure and quality of the deposit.