Magnetohydrodynamics (MHD; also called magnetofluid dynamics or hydromagnetics) is a model of electrically conducting fluids that treats all types of charged particles together as a single continuous fluid. It is primarily concerned with the low-frequency, large-scale magnetic behavior of plasmas and liquid metals and has applications in multiple fields, including space physics, geophysics, astrophysics, and engineering.
Etymology The word magnetohydrodynamics is derived from , meaning magnetic field; , meaning water; and , meaning movement. The field of MHD was initiated by Hannes Alfvén, who received the Nobel Prize in Physics in 1970 for his work in the field.
History The MHD description of electrically conducting fluids was first developed by Hannes Alfvén in a 1942 paper published in Nature titled "Existence of Electromagnetic–Hydrodynamic Waves", which outlined his discovery of what are now known as Alfvén waves. Alfvén initially referred to these waves as "electromagnetic–hydrodynamic waves"; however, in a later paper, he noted, "As the term 'electromagnetic–hydrodynamic waves' is somewhat complicated, it may be convenient to call this phenomenon 'magneto–hydrodynamic' waves."
Equations In MHD, motion in the fluid is described using linear combinations of the mean motions of the individual species: the current density and the center-of-mass velocity . In a given fluid, each species has a number density , mass , electric charge , and mean velocity . The fluid's total mass density is then , and the fluid's motion can be described by the current density expressed as
and the center of mass velocity expressed as:
MHD can be described by a set of equations consisting of a continuity equation, an equation of motion (the Cauchy momentum equation), an equation of state, Ampère's law, Faraday's law, and Ohm's law. As with any fluid description of a kinetic system, a closure approximation must be applied to the highest moment of the particle distribution equation. This is often accomplished through approximations to the heat flux under conditions of adiabaticity or isothermality.
In the adiabatic limit, that is, under the assumption of an isotropic pressure and isotropic temperature, a fluid with adiabatic index , electrical resistivity , magnetic field , and electric field can be described by the continuity equation
the equation of state
the equation of motion
the low-frequency Ampère's law
Faraday's law
and Ohm's law
Taking the curl of this equation and applying Ampère's law and Faraday's law yields the induction equation,
where is the magnetic diffusivity.
In the equation of motion, the Lorentz force term can be expanded using Ampère's law and a vector calculus identity to yield
where the first term on the right-hand side is the magnetic tension force and the second term is the magnetic pressure force.
Ideal MHD
The simplest form of MHD, ideal MHD, assumes that the resistive term in Ohm's law is sufficiently small relative to the other terms that it can be taken to be zero. This occurs in the limit of large magnetic Reynolds numbers, in which magnetic induction dominates magnetic diffusion at the velocity and length scales under consideration. Consequently, processes in ideal MHD that convert magnetic energy into kinetic energy, referred to as ideal processes, cannot generate heat or increase entropy.
A fundamental concept underlying ideal MHD is the frozen-in flux theorem, which states that the bulk fluid and the embedded magnetic field are constrained to move together such that one can be said to be "tied" or "frozen" to the other. Therefore, any two points that move with the bulk fluid velocity and lie on the same magnetic field line will continue to lie on that field line even as the points are advected by fluid flows in the system. The connection between the fluid and magnetic field fixes the topology of the magnetic field in the fluid. For example, if a set of magnetic field lines is tied into a knot, then it will remain so as long as the fluid 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 to release the stored energy from the magnetic field.
Ideal MHD equations
In ideal MHD, the resistive term vanishes in Ohm's law, yielding the ideal Ohm's law:
Similarly, the magnetic diffusion term in the induction equation vanishes, yielding the ideal induction equation:
Applicability of ideal MHD to plasmas
Ideal MHD is strictly applicable when: The plasma is strongly collisional, so that the collision time scale is shorter than the other characteristic time scales 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 time over any length scale present in the system must be longer than any time scale of interest. The relevant length scales are much longer than the ion skin depth and the Larmor radius perpendicular to the field, sufficiently long scales along the field to ignore Landau damping, and time scales much longer than the ion gyration time (that is, the system is smooth and slowly evolving).
Importance of resistivity
In an imperfectly conducting fluid, the magnetic field can generally move through the fluid according to a diffusion law, with the resistivity of the plasma serving as a diffusion constant. This means that solutions to the ideal MHD equations are applicable only for a limited time within a region of a given size before diffusion becomes too significant 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 that are large and conductive enough that simple estimates of the Lundquist number suggest that the resistivity can be ignored, resistivity may still be important: many instabilities can increase the effective resistivity of the plasma by factors greater than 109. The enhanced resistivity is usually the result of the formation of small-scale structures such as current sheets or fine-scale magnetic turbulence, introducing small spatial scales into the system over which ideal MHD breaks down and magnetic diffusion can occur rapidly. When this happens, magnetic reconnection may occur in the plasma, releasing 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, such that gentle forces applied to a plasma over long periods of time can cause violent explosions and bursts of radiation.
When the fluid cannot be considered 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 Ohm's law that models the collisional resistivity. Generally, MHD computer simulations are at least somewhat resistive because their computational grid introduces a numerical resistivity.
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 within which the currents are relatively weak. Current sheets in the solar corona are thought to be between a few meters and a few kilometers thick, which is quite thin compared to the magnetic domains, which are thousands to hundreds of thousands of kilometers across. Another example occurs in the Earth's magnetosphere, where current sheets separate topologically distinct domains, isolating most of the Earth's ionosphere from the solar wind.
Waves
The wave modes derived from the MHD equations are called magnetohydrodynamic waves, or MHD waves. There are three MHD wave modes that can be derived from the linearized ideal MHD equations for a fluid with a uniform, and constant magnetic field: Alfvén waves Slow magnetosonic waves Fast magnetosonic waves
These modes have phase velocities that are independent of the magnitude of the wave vector, so they experience no dispersion. The phase velocity depends on the angle between the wave vector and the magnetic field . An MHD wave propagating at an arbitrary angle with respect to the time-independent, or bulk, field satisfies the dispersion relation: where
is the Alfvén speed. This branch corresponds to the shear Alfvén mode. Additionally the dispersion equation yields: where
is the ideal gas speed of sound. The plus branch corresponds to the fast MHD wave mode, and the minus branch corresponds to the slow MHD wave mode. A summary of the properties of these waves is provided below: Mode Type Limiting phase speeds Groupvelocity Direction ofenergy flow Alfvén wave transversal; incompressible Fast magnetosonic wave neither transversal norlongitudinal; compressional equal tophase velocity approx. Slow magnetosonic wave approx.
MHD oscillations are damped if the fluid is not perfectly conducting but has finite conductivity, or if viscous effects are present.
MHD waves and oscillations are popular tools for the remote diagnostics of laboratory and astrophysical plasmas, for example, the Sun's corona (coronal seismology).
Extensions Resistive Resistive MHD describes magnetized fluids with finite electron diffusivity (). This diffusivity leads to a break in the magnetic topology; magnetic field lines can "reconnect" when they collide. Usually, this term is small, and reconnections can be treated as similar to shocks; this process has been shown to be important in Earth-solar magnetic interactions. Extended Extended MHD describes a class of phenomena in plasmas that are higher-order than resistive MHD, but 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 Two-fluid MHD describes plasmas that include a non-negligible Hall electric field. As a result, the electron and ion momenta must be treated separately. This description is more closely tied to Maxwell's equations because an evolution equation for the electric field exists. Hall 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. This concerned the neglect of the "Hall current term" in Ohm's law, a frequent simplification made in magnetic fusion theory. Hall magnetohydrodynamics (HMHD) takes into account this electric-field description of magnetohydrodynamics, and Ohm's law takes the form: where is the electron number density, and is the elementary charge. The most important difference is that, in the absence of field line-breaking, the magnetic field is tied to the electrons rather than to the bulk fluid. Electron MHD Electron magnetohydrodynamics (EMHD) describes small-scale plasmas in which electron motion is much faster than the ion motion. The main effects are changes in conservation laws, additional resistivity, and the importance of electron inertia. Many effects of Electron MHD are similar to those of two-fluid MHD and Hall MHD. EMHD is especially important for z-pinches, magnetic reconnection, ion thrusters, neutron stars, and plasma switches. Collisionless