# magnetohydrodynamics

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## magnetohydrodynamics

(măgnē'tōhī'drōdīnăm`ĭks), study of the motions of electrically conducting fluids and their interactions with magnetic fields. The principles of magnetohydrodynamics are of particular importance in plasma**plasma,**

in physics, fully ionized gas of low density, containing approximately equal numbers of positive and negative ions (see electron and ion). It is electrically conductive and is affected by magnetic fields.

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the energy stored in the nucleus of an atom and released through fission, fusion, or radioactivity. In these processes a small amount of mass is converted to energy according to the relationship

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## Magnetohydrodynamics

The interaction of electrically conducting fluids with magnetic fields. The fluids can be ionized gases (commonly called plasmas) or liquid metals. Magnetohydrodynamic (MHD) phenomena occur naturally in the Earth's interior, constituting the dynamo that produces the Earth's magnetic field; in the magnetosphere that surrounds the Earth; and in the Sun and throughout astrophysics. In the laboratory, magnetohydrodynamics is important in the magnetic confinement of plasmas in experiments on controlled thermonuclear fusion. Magnetohydrodynamic principles are also used in plasma accelerators for ion thrusters for spacecraft propulsion, for light-ion-beam powered inertial confinement, and for magnetohydrodynamic power generation. *See* Nuclear fusion, Plasma (physics)

The conducting fluid and magnetic field interact through electric currents that flow in the fluid. The currents are induced as the conducting fluid moves across the magnetic field lines. In turn, the currents influence both the magnetic field and the motion of the fluid. Qualitatively, the magnetohydrodynamic interactions tend to link the fluid and the field lines so as to make them move together. *See* Electric current

The generation of the currents and their subsequent effects are governed by the familiar laws of electricity and magnetism. The motion of a conductor across magnetic lines of force causes a voltage drop or electric field at right angles to the direction of the motion and the field lines; the induced voltage drop causes a current to flow as in the armature of a generator.

The currents themselves create magnetic fields which tend to loop around each current element. The currents heat the conductor and also give rise to mechanical ponderomotive forces when flowing across a magnetic field. (These are the forces which cause the armature of an electric motor to turn.) In a fluid, the ponderomotive forces combine with the pressure forces to determine the fluid motion. *See* Electricity, Magnetism

Magnetohydrodynamic phenomena involve two well-known branches of physics, electrodynamics and hydrodynamics, with some modifications to account for their interplay. The basic laws of electrodynamics as formulated by J. C. Maxwell apply without any change. However, Ohm's law, which relates the current flow to the induced voltage, has to be modified for a moving conductor. *See* Electrodynamics, Hydrodynamics, Maxwell's equations, Ohm's law

It is useful to consider first the extreme case of a fluid with a very large electrical conductivity. Maxwell's equations predict, according to H. Alfvén, that for a fluid of this kind the lines of the magnetic field move with the material. The picture of moving lines of force is convenient but must be used with care because such a motion is not observable. It may be defined, however, in terms of observable consequences by either of the following statements: (1) a line moving with the fluid, which is initially a line of force, will remain one; or (2) the magnetic flux through a closed loop moving with the fluid remains unchanged.

If the conductivity is low, this is not true and the fluid and the field lines slip across each other. This is similar to a diffusion of two gases across one another and is governed by similar mathematical laws.

As in ordinary hydrodynamics, the dynamics of the fluid obeys theorems expressing the conservation of mass, momentum, and energy. These theorems treat the fluid as a continuum. This is justified if the mean free path of the individual particles is much shorter than the distances that characterize the structure of the flow. Although this assumption does not generally hold for plasmas, one can gain much insight into magnetohydrodynamics from the continuum approximation. The ordinary laws of hydrodynamics can then easily be extended to cover the effect of magnetic and electric fields on the fluid by adding a magnetic force to the momentum-conservation equation and electric heating and work to the energy-conservation equation.

## magnetohydrodynamics

(mag-nee-toh-hÿ-droh-dÿ-**nam**-iks) (MHD) The study of the behavior of electrically conducting fluids, i.e. a plasma or some other collection of charged particles, in a magnetic field. The collective motion of the particles gives rise to an electric field that interacts with the magnetic field and causes the plasma motion to alter. This coupling between hydrodynamic forces and magnetic forces means that the magnetic field is effectively ‘frozen into’ the plasma; the field lines flow with the plasma, and can be stretched, squeezed, or looped. One consequence is that the frozen-in field lines of two plasmas prevent them from mixing. MHD has contributed to the understanding of the solar wind and its interaction with planetary magnetospheres, of solar flares and prominences, the sunspot cycle, the formation and heating of the solar corona, star formation, and many other processes. See also Alfvén waves.

## Magnetohydrodynamics

the science of the motion of electrically conducting liquids and gases in the presence of a magnetic field; a branch of physics that developed at the boundary of hydrodynamics and classical electrodynamics. Characteristic objects of study in magnetohydrodynamics are plasma (since magnetohydrodynamics is sometimes considered as a branch of plasma physics), liquid metals, and electrolytes.

The first research in magnetohydrodynamics dates from the time of M. Faraday, but magnetohydrodynamics developed as an independent field of knowledge in the 20th century in connection with the requirements of astrophysics and geophysics. It was established that many heavenly bodies have magnetic fields. For example, fields having an intensity of the order of 10,000 oersteds have been observed in stellar atmospheres (the field intensity of the sun is up to 5,000 oersteds), and, according to current concepts, the field intensities in the pulsars discovered in 1969 may reach 10^{12} oersteds.

The dynamic behavior of plasma in such fields is radically altered, since the energy density of the magnetic field becomes comparable to or exceeds the density of the kinetic energy of plasma particles. This criterion is also valid for weak cosmic magnetic fields having an intensity of 10^{-3}-10^{-5} oersted (in interstellar space and in the earth’s field in the upper atmosphere and beyond) if the concentration of charged particles in the region occupied by them is low. Thus, it became necessary to create a special theory of the motion of cosmic plasma in magnetic fields (which has come to be called cosmic electrodynamics) and, in the case where plasma may be considered as a continuous medium, cosmic magnetohydrodynamics (cosmic MHD).

The basic premises of magnetohydrodynamics were formulated in the 1940’s by H. Alfven, who was awarded the 1970 Nobel Prize in physics for the development of magnetohydrodynamics. He theoretically predicted the existence of the specific wave motions of a conducting medium in a magnetic field, which are called Alfven waves. The methods of magnetohydrodynamics, which took shape as the science of the behavior of cosmic plasma, soon extended to conducting media under terrestrial conditions (mainly media created in research and in production operations). In the early 1950’s national research programs (in the USSR, the USA, and Great Britain) on the problem of controlled thermonuclear fusion gave powerful impetus to the development of magnetohydrodynamics and to plasma physics as a whole. Numerous technical applications of magnetohydrodynamics, such as MHD pumps, generators, separators, accelerators, and plasma motors that are promising for space flight, appeared and are being rapidly improved.

Two groups of physical laws underlie magnetohydrodynamics: the equations of hydrodynamics and the equations of the electromagnetic field (the Maxwell equations). The former describe flows of a conductive liquid or gaseous medium; however, in contrast to conventional hydrodynamics, these flows are associated with electric currents that are distributed throughout the medium. The presence of a magnetic field leads to the appearance in the equation of an additional term that corresponds to the volumetrically distributed electrodynamic force that acts on the currents. The currents themselves in the medium and the distortion of the magnetic field caused by them are defined by the second group of equations. Thus, in magnetohydrodynamics the equations of hydrodynamics and electrodynamics are essentially interrelated. It should be noted that displacement currents in the Maxwell equations may almost always be disregarded in magnetohydrodynamics (nonrelativistic magnetohydrodynamics).

In the general case the equations of magnetohydrodynamics are nonlinear and are extremely complex to solve, but in practical problems it is often possible to treat only various extreme modes, in the evaluation of which a dimensionless quantity called the magnetic Reynolds number is an important parameter:

(1) *R _{m}* =

*LV/v*=

_{m}*4 π σ LV/c*

^{2}where *L* is the dimension characteristic of the flow of the medium; *V* is the characteristic rate of flow; *v _{m} =* c

^{2}/4Δσ is the magnetic viscosity, which describes dissipation of the energy of the magnetic field; σ is the conductance of the medium; and

*c*is the speed of light in a vacuum (here and hereafter the absolute Gaussian system of units is used).

When *R _{m}* « 1, which is usually the case for laboratory conditions and technical applications, the flow of a conductive medium slightly distorts a magnetic field, which therefore may be considered to be defined by external sources. Such a flow may be used, for example, to generate electric current—the energy of the hydrodynamic motion of the medium is converted into the energy of the current in an external circuit. Conversely, if the current in the medium is maintained by an external electromotive force, then the presence of an external magnetic field causes the appearance of the volume electrodynamic force mentioned above, which creates a pressure gradient in the medium and sets it in motion. This effect is used in MHD pumps (for example, to pump molten metal) and in plasma accelerators. The volume electrodynamic force also makes possible the generation of a controlled expulsive (buoyancy) force that acts on bodies placed in a conductive fluid. The operation of MHD separators is based on this important effect.

Such are the main technical applications of magnetohydrodynamics. Certain problems of ordinary hydrodynamics and gas dynamics, including flow around bodies and the boundary layer, also find a natural extension in magnetohydrodynamics. In many cases (for example, during the flight of spacecraft in the ionosphere and in ducts through which conductive media flow), the properties of the corresponding flows may be significantly influenced by using a magnetic field.

However, the most interesting and varied effects are characteristic of another extreme class of media that are examined in magnetohydrodynamics (media with *R _{m}* » 1—that is, with high conductivity and/or large dimensions). These conditions are usually satisfied in media that are studied in the geophysical and astrophysical applications of magnetohydrodynamics, as well as hot plasma (such as thermonuclear plasma). Flows in such media have an extraordinarily strong effect on the magnetic field within them. The freezing-in of a magnetic field is one of the most important effects under these conditions. In a good conductor (or, strictly speaking, an ideal conductor), electromagnetic induction causes the appearance of currents that impede any change whatever in the magnetic flux through any physical circuit. In a moving MHD medium with

*R*» 1 this is valid for any circuit formed by its particles. As a result, the magnetic flux through any moving component of a medium that changes its dimensions remains unchanged (the greater the quantity

_{m}*R*, the higher the degree of accuracy), and it is in this sense that one speaks of the “freezing-in” of a magnetic field. In many cases this makes it possible to obtain a qualitative picture of the flows in a medium and the deformations of a magnetic field, using simple concepts and without resorting to cumbersome calculations; it is necessary only to consider magnetic lines of force as elastic threads on which the particles of the medium are strung. More rigorous examination of this “elastic” action of a magnetic field on a conductive medium shows that it reduces to the isotropic (that is, identical in all directions) “magnetic” pressure

_{m}*PM = B*, which is added to the normal gas-dynamic pressure

^{2}/8π*p*of the medium, and to the magnetic tension

*T = B*, which is directed along the field’s lines offeree (the magnetic permeability of all media that are of consequence for magnetohydrodynamics is equal to 1 with a high degree of accuracy, and both the magnetic induction

^{2}/4π*B*and intensity

*H*may be used with equal validity).

The presence of additional “elastic” stresses in MHD media leads to a specific oscillatory (wave) process, Alfven waves, which result from the magnetic tension *T* and propagate along the lines of force (like waves traveling along an elastic thread) at a rate

where p is the density of the medium. Alfvén waves are described by the exact solution of the nonlinear equations of magnetohydrodynamics for an incompressible medium. Because of the complexity of these equations, very few such exact solutions have been obtained for large *R _{m}*. Still another of these solutions describes the flow of an incompressible fluid (p = const.) having the same Alfvén velocity (2) along an arbitrary magnetic field.

An exact solution is also known for MHD discontinuities, which include contact, tangential, and rotational discontinuities, as well as fast and slow shock waves. In a contact discontinuity the magnetic field intersects the interface of two different media, impeding their relative motion (in the near-boundary layer the media are immobile with respect to each other). In a tangential discontinuity the field does not intersect the interface of the two media (the field component normal to the boundary is equal to zero), and the media may be in relative motion. A neutral current layer that separates magnetic fields of equal magnitude and opposite direction is a particular case of a tangential discontinuity. Magnetohydrodynamics shows that under certain conditions a magnetic field stabilizes a tangential discontinuity of velocity, which is absolutely unstable in ordinary hydrodynamics.

Rotational discontinuity, in which the magnetic induction vector is rotated about the normal to the surface of the discontinuity without changing in magnitude, is specific to magnetohydrodynamics (it has no analogue in the hydrodynamics of nonconductive media). In this case the magnetic tensions set the medium in motion in such a way that the rotational discontinuity propagates in the direction of the normal to the surface at the Alfven velocity (2), if *B* in (2) is understood to mean the normal component of induction. Fast and slow shock waves in magnetohydrodynamics differ from ordinary shock waves in that, after passage of the wave front, momentum tangential to the front is imparted to the particles of the medium because of magnetic tensions (since the magnetic lines of force may be considered as elastic threads; *see above*). In a fast shock wave the magnetic field beyond the wave front is amplified and the jump in magnetic pressure on the front acts in the same direction as the jump in gas-dynamic pressure; therefore, the speed of such a wave is greater than the speed of sound in the medium. In a slow shock wave, on the contrary, the field grows weaker after the wave passes, and the drops in the gas-dynamic and magnetic pressure on the wave front are opposite in direction. The speed of a slow wave is less than the speed of sound. In magnetohydrodynamics the number of theoretically conceivable irreversible shock waves is found to be much greater than the number of real waves. The selection of solutions that conform to reality is made by using the evolutionary condition, which follows from examination of the stability of shock waves upon interaction with oscillations of low amplitude.

However, the known exact solutions do not exhaust the content of the theoretical magnetohydrodynamics of media with *R _{m}* » 1. A broad class of problems may be studied in terms of approximations. Two main approaches are possible in such an investigation: the weak-field approximation, in which the magnetic pressure and tension are small in comparison with other dynamic factors (gas-dynamic pressure and inertial forces), and the strong-field approximation, in which

where *v* is the velocity of the medium and *p* is its gas-dynamic pressure.

In the weak-field approximation the flow of the medium is determined by conventional gas-dynamic factors (the influence of magnetic tensions is disregarded). Here the computation of changes in the field in a medium that moves according to a given law is required. The very important problem of the hydromagnetic dynamo and the problem of MHD turbulence fall within this class. The former consists in finding laminar flows of conductive media that can generate, amplify, and maintain a magnetic field. The problem of the hydromagnetic dynamo is the basis of the theory of geomagnetism and of the magnetism of the sun and stars. Simple kinematic models exist that show that a hydromagnetic dynamo may be realized in principle by special selection of the velocity distributions of the medium. However, as yet there has been no rigorous proof that such distributions are achieved in reality.

Elucidation of the behavior of a weak initial magnetic field (“seed” field) in a turbulent conducting medium is the main part of the problem of MHD turbulence. Proof exists for the increase in the mean square of the intensity of a randomly occurring weak initial field—that is, the increase in magnetic energy in the initial stage of the process. However, the problem of the steady turbulent state associated with the origin of magnetic fields in outer space, in particular in our galaxy and other galaxies, remains open.

The strong-field approximation, in which magnetic tensions are decisive, is used in the study of rarefied atmospheres of magnetic heavenly bodies, such as the sun and earth. There are grounds to assume that this approximation will prove useful in the study of processes in distant astrophysical objects, such as supernovas, pulsars, and quasars. Under conditions corresponding to (3), changes in a magnetic field near its sources (such as the appearance of active regions and spots on the sun and the displacement of the magnetopause in the earth’s magnetic field under the influence of the solar wind) are conveyed at the Alfven velocity (2) along the field, inducing corresponding displacements of the plasma. Such characteristic motions as outbursts and prominences, fanlike structures and streamers on the sun, and the magnetic tail of the earth appear as a result of the action of magnetic forces.

Particularly interesting phenomena take place in the vicinity of the points of a strong field at which the field becomes zero. In such regions, thin current layers that separate magnetic fields of opposite direction (so-called neutral layers) are formed. The process of the “annihilation” of magnetic energy—that is, the release and transformation of the energy into other forms—takes place in these layers. In particular, strong electric fields arise that accelerate charged particles. The annihilation of a magnetic field in neutral current layers is responsible for the appearance of chromospheric flares on the sun and substorms in the earth’s magnetosphere. Many other markedly nonstationary processes in the universe that are accompanied by the generation of accelerated charged particles and hard radiation are probably related to annihilation. From the standpoint of magnetohydrodynamics, neutral currents are discontinuities of the magnetic field (like shock waves and tangential discontinuities). However, the processes in current layers—and particularly the instabilities, which lead to the appearance of strong accelerating electric fields—go beyond the framework of magnetohydrodynamics and fall within the range of precision problems of plasma physics that as yet have not been fully resolved.

### REFERENCES

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*Uspekhi fizicheskikh nauk*, 1957, vol. 62, fasc. 3.

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*Magnitnaia gidrodinamika*. Moscow, 1962.

Shercliff, J.

*Kurs magnitnoi gidrodinamiki*. Moscow, 1967. (Translated from English.)

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*Uspekhi fizicheskikh nauk*, 1960, vol. 72, fasc. 1.

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*Osnovy kosmicheskoi elektrodinamiki*. Moscow, 1966. Dungey, J.

*Kosmicheskaia elektrodinamika*. Moscow, 1961. (Translated from English.)

Anderson, E.

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*Elektrodinamika sploshnykh sred*. Moscow, 1959. (

*Teoreticheskaia fizika*.)

S. I. SYROVATSKII