The mechanics of electrically conducting fluids, such as liquid metals and plasmas, when subject to electric and magnetic fields; also called magneto-fluid-mechanics. The study is relevant to plasma nuclear fusion, liquid metal cooling systems, and electrical power generation from hot plasmas. Magnetohydrodynamic propulsion systems for ships and submarines pass an electric current through sea water in the presence of a powerful magnetic field to produce a jet of water which provides thrust.
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 idea of MHD is that magnetic fields can induce currents in a moving conductive fluid, which create forces on the fluid, and also change the magnetic field itself. The set of equations which describe MHD are a combination of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism.
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.
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. 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 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.
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.
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.
Structures in MHD systems
In many MHD systems, most of the electric current is compressed into thin, nearly-two-dimensional ribbons termed current sheets.
Extensions to magnetohydrodynamics
Resistive MHD
Resistive MHD describes magnetized fluids with non-zero electron diffusivity.
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.
Two-Fluid MHD
Two-Fluid MHD describes plasmas that include a non-negligible electric field.
Hall MHD
In 1960, M. Lighthill criticized the applicability of ideal or resistive MHD theory for plasmas . Hall-magnetohydrodynamics (HMHD) takes into account this electric field description of magnetohydrodynamics
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.
Sunspots are caused by the Sun's magnetic fields, as Joseph Larmor theorized in 1919. 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.
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).
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.
The US was working on a MHD generator but they gave it up as too expensive.
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