Magnetohydrodynamics

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Image:HD-Rayleigh-Taylor.gif 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 Alfvén, 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 all but the most trivial cases. For real-world problems, numeric solutions are found using supercomputers. 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.

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Ideal MHD

Image:T3e troy.jpeg The most common simplification of MHD is to assume that the fluid is a perfect conductor with little or no resistivity; this simplification is called ideal MHD. In ideal MHD, Lenz's law dictates that magnetic field lines cannot move through the fluid, instead remaining attached to the same small piece of fluid at all times. Under such conditions, most electric currents tend to be compressed into thin, nearly-two-dimensional ribbons termed current sheets. This has the effect of dividing the fluid into magnetic domains, each of which may carry a small electric current in the direction of the magnetic field lines themselves, with most current contained in current sheets between the domains.

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.

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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.

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Limits of ideal MHD

There are few perfect conductors; ideal MHD is an imperfect description of almost all physical systems. In reality, any physical system has some non-ideal behavior. In particular, the magnetic field can generally move through the plasma, 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 before diffusion becomes too important to ignore. Solar active regions, for example, have diffusion times of hundreds to thousands of years, much longer than the actual lifetime of a sunspot -- so they can be treated as ideal MHD systems. By contrast, a meter-sized volume of seawater has a magnetic diffusion time measured in milliseconds.

Under ideal MHD, all "sheets" of current are infinitely thin and have infinitely high current density. In reality, the thickness of a current sheet is limited both by the resistivity of the plasma and by the density of the individual charge carrier particles (such as electrons) in the plasma, and by the Larmor radius of those particles' motion in the magnetic field. 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).

Alfvén described MHD as a "magnetic field description". But based on his experimental work, Alfvén's also applied an "electric current description" to plasmas, whose properties are less well-known, such as Birkeland currents (field-align currents), double layers (charge separation regions), certain classes of plasma instabilities, and chemical separation in space plasmas <ref>Hannes Alfvén, Cosmic Plasma ((Reidel, 1981) ISBN 9027711518</ref>. An extended version of MHD encompassing an electric field description and some of these more complex phenomena is called Hall-magnetohydrodynamics (Hall-MHD or HMHD).

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Breakdown of ideal MHD

Image:Magnetic-rope.gif Even in physical systems that can be treated as ideal, the assumptions of MHD can break down. Many instabilities exist that can increase the effective resistivity of the plasma by factors of more than a billion. When this happens, the electric current sheets that separate different magnetic domains can collapse rapidly, causing magnetic reconnection in the plasma and releasing stored magnetic energy as waves, bulk mechanical acceleration of material, particle acceleration, and heat. Magnetic reconnection in near-ideal MHD 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.

Additionally, MHD "completely excludes the possibility of field aligned potentials for the simple reason that electrons travelling along the field lines would be accelerated by such potentials and quickly redistribute the charge" [3].

Hannes Alfvén, who won the Nobel Prize for his development of magnetohydrodynamics, and co-author Carl-Gunne Fälthammar, wrote in their book Cosmical Electrodynamics (1952, 2nd Ed.): "It should be noted that the fundamental equations of magnetohydrodynamics rest on the assumption that the conducting medium can be considered as a fluid. This is an important limitation, for if the medium is a plasma it is sometimes necessary to use a microscopic description in which the motion of the constituent particles is taken into account. Examples of plasma phenomena invalidating a hydromagnetic description are ambipolar diffusion, electron runaway, and generation of microwaves" <ref>Hannes Alfvén and Carl-Gunne Fälthammar, Cosmical Electrodynamics (1952, 2nd Ed.)</ref>. In other words, MHD may not lead to correct results when applied to low-density cosmic plasmas.

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Extensions to magnetohydrodynamics

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Resistive MHD

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

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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.

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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.

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Hall MHD

In 1960, M. J. Lighthill criticized the applicability of ideal or resistive MHD theory for plasmas <ref>M. J. Lighthill, "Studies on MHD waves and other anisotropic wave motion," Phil. Trans. Roy. Soc., London, vol. 252A, pp. 397-430, 1960.</ref>. 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 <ref>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.</ref>

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Applications

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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.

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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.

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.

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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.

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Trivia

The ebbing salty water flowing past Tower Bridge interacts with the Earth's magnetic field to produce about one volt potential difference between the two river-banks.

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Footnotes

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See also

MHD in fiction:

Template:Fusion power

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References

fr:Magnétohydrodynamique it:Magnetoidrodinamica pl:Magnetohydrodynamika fi:Magnetohydrodynamiikka vi:Từ thủy động lực học

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