Magnetohydrodynamics

Magnetohydrodynamics (MHD) (magnetofluiddynamics or hydromagnetics) is the academic discipline which studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water. The word magnetohydrodynamics (MHD) is derived from magneto- meaning magnetic field, and hydro- meaning fluid, and -dynamics meaning movement. The field of MHD was initiated by Hannes AlfvénAlfven, H., "Existence of electromagnetic-hydrodynamic waves" (1942) Nature, Vol. 150, pp. 405, for which he received the Nobel Prize in 1970.

The set of equations which describe MHD are a combination of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism. These differential equations have to be solved simultaneously. This is too complex or impossible to do symbolically in most cases, but there are important classes of analytical solutions (for example, the Solov'ev equilibria). For real-world problems in complex geometries, numeric solutions are found using computers. Because MHD is a fluid theory, it cannot treat kinetic phenomena, i.e., those in which the existence of discrete particles, or of a non-thermal distribution of their velocities, is important.

Ideal and Resistive MHD

The simplest form of MHD, Ideal MHD, assumes that the fluid has so little resistivity that it can be treated as a perfect conductor. In ideal MHD, Lenz's law dictates that the fluid is in a sense tied to the magnetic field lines. To be more precise, in ideal MHD, a small rope-like volume of fluid surrounding a field line will continue to lie along a magnetic field line, even as it is twisted and distorted by fluid flows in the system. The connection between magnetic field lines and fluid in ideal MHD fixes the topology of the magnetic field in the fluid -- for example, if a set of magnetic field lines are tied into a knot, then they will remain so as long as the fluid/plasma has negligible resistivity. This difficulty in reconnecting magnetic field lines makes it possible to store energy by moving the fluid or the source of the magnetic field. The energy can then become available if the conditions for ideal MHD break down, allowing magnetic reconnection that releases the stored energy from the magnetic field.

Ideal MHD Equations

The ideal MHD equations consist of the continuity equation, the momentum equation, Ampere's Law in the limit of no electric field and no electron diffusivity, and a temperature evolution equation. As with any fluid description to a kinetic system, a closure approximation must be applied to highest moment of the particle distribution equation. This is often accomplished with approximations to the heat flux through a condition of adiabaticity or isothermality.

Applicability of Ideal MHD to plasmas

Ideal MHD is only strictly applicable when:

The plasma is strongly collisional, so that the time scale of collisions is shorter than the other characteristic times in the system, and the particle distributions are therefore close to Maxwellian.
The resistivity due to these collisions is small. In particular, the typical magnetic diffusion times over any scale length present in the system must be longer than any time scale of interest.
We are interested in length scales much longer than the ion skin depth and Larmor radius perpendicular to the field, long enough along the field to ignore Landau damping, and time scales much longer than the ion gyration time (system is smooth and slowly evolving).

The importance of resistivity

In an imperfectly conducting fluid, the magnetic field can generally move through the the fluid, following a diffusion law with the resistivity of the plasma serving as a diffusion constant. This means that solutions to the ideal MHD equations are only applicable for a limited time for a region of a given size before diffusion becomes too important to ignore. One can estimate the diffusion time across a Solar active region (from collisional resistivity) to be hundreds to thousands of years, much longer than the actual lifetime of a sunspot -- so it would seem reasonable to ignore the resistivity. By contrast, a meter-sized volume of seawater has a magnetic diffusion time measured in milliseconds.

Even in physical systems which are large and conductive enough that simple estimates suggest that we can ignore the resistivity, resistivity may still be important: many instabilities exist that can increase the effective resistivity of the plasma by factors of more than a billion. The enhanced resistivity is usually the result of the formation of small scale structure like current sheets or fine scale magnetic turbulence, introducing small spatial scales into the system over which ideal MHD is broken and magnetic diffusion can occur quickly. When this happens, Magnetic Reconnection may occur in the plasma to release stored magnetic energy as waves, bulk mechanical acceleration of material, particle acceleration, and heat. Magnetic reconnection in highly conductive systems is important because it concentrates energy in time and space, so that gentle forces applied to a plasma for long periods of time can cause violent explosions and bursts of radiation.

When the fluid cannot be considered as completely conductive, but the other conditions for ideal MHD are satisfied, it is possible to use an extended model called resistive MHD. This includes an extra term in Ampere's Law which models the collisional resistivity. Generally MHD computer simulations are at least somewhat resistive because their computational grid introduces a numerical resistivity.

The importance of kinetic effects.

Another limitation of MHD (and fluid theories in general) is that they depend on the assumption that the plasma is strongly collisional (this is the first criterion listed above), so that the time scale of collisions is shorter than the other characteristic times in the system, and the particle distributions are Maxwellian. This is usually not the case in fusion, space and astrophysical plasmas. When this is not the case, or we are interested in smaller spatial scales, it may be necessary to use a kinetic model which properly accounts for the non-Maxwellian shape of the distribution function. However, because MHD is very simple, and captures many of the important properties of plasma dynamics, it is often qualitatively accurate, and is almost invariably the first model tried.

Effects which are essentially kinetic and not captured by fluid models include double layers, a wide range of instabilities, chemical separation in space plasmas and electron runaway.

Structures in MHD systems

In many MHD systems, most of the electric current is compressed into thin, nearly-two-dimensional ribbons termed current sheets. These can divide the fluid into magnetic domains, inside of which the currents are relatively weak. Current sheets in the solar corona are thought to be between a few meters and a few kilometers in thickness, which is quite thin compared to the magnetic domains (which are thousands to hundreds of thousands of kilometers across). Another example is in the earth's magnotosphere, where current sheets separate topologically distinct domains, isolating most of the earth's ionosphere from the solar wind.

Extensions to magnetohydrodynamics

Resistive MHD

Resistive MHD describes magnetized fluids with non-zero electron diffusivity. This leads to a breaking in the magnetic topology.

Extended MHD

Extended MHD describes a class of phenomena in plasmas that are higher order than resistive MHD, but which can adequately be treated with a single fluid description. These include the effects of Hall physics, electron pressure gradients, finite Larmor Radii in the particle gyromotion, and electron inertia.

Two-Fluid MHD

Two-Fluid MHD describes plasmas that include a non-negligible electric field. As a result, the electron and ion momenta must be treated separately. This description is more closely tied to Maxwell's equations as an evolution equation for the electric field exists.

Hall MHD

In 1960, M. J. Lighthill criticized the applicability of ideal or resistive MHD theory for plasmas M. J. Lighthill, "Studies on MHD waves and other anisotropic wave motion," Phil. Trans. Roy. Soc., London, vol. 252A, pp. 397-430, 1960.. It concerned the neglect of the "Hall current term", a frequent simplification made in magnetic fusion theory. Hall-magnetohydrodynamics (HMHD) takes into account this electric field description of magnetohydrodynamics E.A. Witalis, "Hall Magnetohydrodynamics and Its Applications to Laboratory and Cosmic Plasma", IEEE Transactions on Plasma Science (ISSN 0093-3813), vol. PS-14, Dec. 1986, p. 842-848.

Applications

Geophysics

The fluid core of the Earth and other planets is theorized to be a huge MHD dynamo that generates the Earth's magnetic field due to the motion of the molten rock. Such dynamos work by stretching magnetic field lines that thread through turbulent or sheared flows in a conductive fluid: the total length of magnetic field line in a particular volume determines the strength of the magnetic field, so stretching the field lines increases the magnetic field.

Astrophysics

MHD applies quite well to astrophysics since over 99% of the matter content of the Universe is made up of plasma, including stars, the interplanetary medium (space between the planets), the interstellar medium (space between the stars), nebulae and jets. Many astrophysical systems are not in local thermal equilibrium, and therefore require an additional kinematic treatment to describe all the phenomena within the system (see Astrophysical plasma).

Sunspots are caused by the Sun's magnetic fields, as Joseph Larmor theorized in 1919. The solar wind is also governed by MHD. The differential solar rotation may be the long term effect of magnetic drag at the poles of the Sun, an MHD phenomenon due to the Parker spiral shape assumed by the extended magnetic field of the Sun.

Previously, theories describing the creation of the Sun and planets could not explain how the Sun has 99% of the mass, yet only 1% of the angular momentum in the solar system. In a closed system such as the cloud of gas and dust from which the Sun was formed, mass and angular momentum are both conserved. That conservation would imply that as the mass concentrated in the center of the cloud to form the Sun, it would spin up, much like a skater pulling their arms in. The high speed of rotation predicted by early theories would have flung the proto-Sun apart before it could have formed. However, magnetohydrodynamic effects transfer the Sun's angular momentum into the outer solar system, slowing its rotation.

Breakdown of ideal MHD (in the form of magnetic reconnection) is known to be the cause of solar flares, the largest explosions in the solar system. The magnetic field in a solar active region over a sunspot can become quite stressed over time, storing energy that is released suddenly as a burst of motion, X-rays, and radiation when the main current sheet collapses, reconnecting the field.

Engineering

MHD is related to engineering problems such as plasma confinement, liquid-metal cooling of nuclear reactors, and electromagnetic casting (among others).

In early 1990s, Mitsubishi built a boat, the 'Yamoto', which uses a magnetohydrodynamic drive, is driven by a liquid helium-cooled superconductor, and can travel at 15 km/h.

Trivia

The ebbing salty water flowing past London's Waterloo Bridge interacts with the Earth's magnetic field to produce a potential difference between the two river-banks. Michael Faraday tried this experiment in 1832 but the current was too small to measure with the equipment at the time, Dynamos in Nature by David P. Stern, and the river bed contributed to short-circuiting the signal. However, by the same process, Dr. William Hyde Wollaston was able to measure the voltage induced by the tide in the English Channel in 1851.McKetta J McKetta, "Encyclopedia of Chemical Processing and Design: Volume 66" (1999)

History

The first recorded use of the word magnetohydrodynamics is by Hannes Alfvén in 1942:

"At last some remarks are made about the transfer of momentum from the Sun to the planets, which is fundamental to the theory (§11). The importance of the magnetohydrodynamic waves in this respect is pointed out." Alfvén, H., "On the cosmogony of the solar system III", Stockholms Observatoriums Annaler, vol. 14, pp.9.1-9.29

The US was working on a MHD generator but they gave it up as too expensive. Problems with the power taps in the plasma field. The Russians took it over and have continued the work until the Soviet Union crumbled back in the 80s.

Notes

References

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Tabar, M. R. Rahimi, and S. Rouhani, "Turbulent Two Dimensional Magnetohydrodynamics and Conformal Field Theory". Department of Physics, Sharif University of Technology. Institute for Studies in Theoretical Physics and Mathematics. Tehran, Iran. arXiv:hep-th/9503005 v1 1 Mar 1995.
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Sutton, G. W., and A. Sherman, "Engineering Magnetohydrodynamics", McGraw-Hill Book Company, New York, 1965.
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Dr. James B. Calvert, "Magnetohydrodynamics", 2002-10-20
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Bansal, J.L. "Magneto Fluid Dynamics of Viscous Fluids". (1994) Jaipur Publishing House, Jaipur, India.

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