Electricity and magnetism

Transcription

1 Physics: Physics: Electricity and magnetism Fluid mechanics Magnetohydrodynamics he 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. Magnetic fields can also be spontaneously generated in inertial confinement experiments, making magnetohydrodynamics relevant in this area. See also: Cosmic rays; Geomagnetism; Ion propulsion; Magnetohydrodynamic power generator; Magnetosphere; Nuclear fusion; Plasma (physics); Solar wind; Sun 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 also: Electric current; Magnetic field 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 also: Electricity; Generator; Magnetism; Motor 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 also: 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 B 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. Ideal magnetohydrodynamics Ideal magnetohydrodynamics refers to the study of magnetohydrodynamics for a perfectly conducting fluid, that is, one obeying the form of Ohm's law, given by Eq. (1), (1)

2 where E is the electric field, B the magnetic field, and v the center-of-mass velocity of the fluid. This equation means that the electric field in the rest frame of the fluid is zero. See also: Electric field; Relativistic electrodynamics Frozen flow Equation (1) can be combined with Faraday's law, Eq. (2), to give Eq. (3), (2) (3) which shows that the magnetic flux, B ds, where ds is an element of area in the fluid, is conserved in the frame of reference of the fluid. See also: Calculus of vectors; Faraday's law of induction It can thus be said that the magnetic field and the fluid are frozen together. This result also implies that magnetic field lines immersed in a perfectly conducting fluid cannot change their topology. Alfvén waves If a uniform plasma of density 0 is immersed in a uniform magnetic field B 0, then if there is a periodic transverse displacement of the frozen-in plasma and magnetic field (Fig. 1), it can be shown that the displacement propagates as a wave with a velocity c A = (B 2 0/ 0 0) 1/2, where 0 is the permeability of empty space. This is called an Alfvén wave. Physically this is analogous to a wave on a loaded string, the magnetic field lines being the string with tension B 2 0/ 0, and the mass loading of the string being 0 per unit volume [Eq. (16)]. This result can easily be derived by using perturbation techniques. The equation of motion for the plasma is Eq. (4), where p is the pressure and J is the current density. For the situation shown in Fig. 1 this becomes Eq. (5), where J y1 is the perturbed current density and x1 is the x-component of the perturbed velocity. The plasma pressure p does not contribute to Eq. (5) for purely transverse motion. Ampère's law (neglecting the displacement current which is negligible at such low frequencies) is Eq. (6), (6) and relates the perturbation current of the wave with the perturbed magnetic field through Eq. (7). Combining Eqs. (1) and (2) yields Eq. (8), and in perturbation form this becomes Eq. (9). Differentiating Eq. (9) with respect to time t and employing Eqs. (5) and (7) gives Eq. (10), which is the wave equation for a wave propagating in the z direction with a wave velocity c A, the Alfvén speed. See also: Alfvén waves; Ampère's law; Wave motion Fig. 1 Periodic transverse displacement of a uniform magnetic field B 0 and a uniform, frozen-in plasma. (4) (5) (7) (8) (9) (10)

3 Magnetosonic waves For the more general case when the wave vector k is at some arbitrary angle to the equilibrium magnetic field B 0, the plasma pressure also contributes to the restoring force of the waves because, in general, compression of the plasma now occurs. Therefore, in addition to the Alfvén wave there are two magnetosonic waves with phase velocity given by Eq. (11), (11) where c s = ( p 0 / 0 ) 1/2 is the sound speed, being the ratio of specific heats. The plus sign refers to the fast magnetosonic wave in which the plasma pressure perturbations and the magnetic force perturbations act in phase, while the minus sign refers to the slow magnetosonic wave in which the two forces are out of phase. The first (+) magnetosonic wave is the only one that can propagate perpendicular to B 0, and then both the magnetic pressure B [Eq. (16)] and the plasma pressure p 0 contribute to the restoring force of this compressional longitudinal wave, and it propagates with a velocity (c 2 A + c 2 S) 1/2. See also: Sound Shock waves When the amplitude of a wave is large, it tends to steepen in the direction of propagation (Fig. 2a). The reason is that the local phase velocity increases where the temperature and magnetic field are higher. The steepening process increases with wave amplitude until a sharp shock (Fig. 2b) occurs. The shock thickness and structure are determined by dissipative processes such as viscosity, thermal conduction, and electrical resistivity, discussed below. Conservation laws for mass, momentum, energy, and magnetic flux can be applied in the frame of reference of the shock front in which a steady state can be assumed. For a shock in which the magnetic field is transverse to the direction of propagation, the conservation relations between the upstream region (indicated by subscript 1) and downstream region (indicated by subscript 2) are given by Eqs. (12). (12) The first three of these equations describe conservation of mass, momentum, and energy in that order, the fourth relation being conservation of magnetic flux, derived from Eq. (8). See also: Conservation of energy; Conservation of mass; Conservation of momentum Fig. 2 Evolution of a large-amplitude wave. (a) Steepening of wave. (b) Formation of shock wave.

4 The shock velocity 1 in the laboratory frame can be found from Eqs. (12) in terms of the density ratio 1 / 2 from Eq. (13). (13) An important result from this is that the maximum density ratio that can occur in a single shock compression is a factor of 4 for = 5/3. For this reason the high compression that is required in inertial confinement fusion cannot be attained by a single shock wave, but requires a tailored compression wave that is equivalent to many weaker shocks following each other. More generally, it is possible to drive a current in the shock front which causes other components of the magnetic field to be generated or removed. Such shocks are called switch-on or switch-off shocks, respectively. See also: Shock wave Collisionless shock At low density and high temperature the mean free path of electrons and ions is too large for collisional processes to determine the shock structure. Instead, within the shock structure microscopic kinetic instabilities occur because the relative drift velocity of electrons and ions (associated with the net current density in the shock) exceeds some critical value such as the ion sound speed, c S [Eq. (11)]. These instabilities grow to give a turbulent state yielding corresponding anomalous transport coefficients. See also: Transport processes; Turbulent flow Magnetostatic equilibria If the center-of-mass velocity v is very small, a stationary equilibrium can exist in which the pressure gradient in Eq. (4) is balanced by the magnetomotive force, J B; that is, Eq. (14) (14) is satisfied. This nonlinear equation together with Eq. (6) represents the set of equations describing magnetostatic equilibria. Two important results follow from Eq. (14). Taking the scalar product first with B and then with J yields Eqs. (15), (15) which shows that the pressure is constant along both a magnetic field line and the current density direction. Applying this to the case of a toroidal axisymmetric equilibrium (Fig. 3), it is found that magnetic field lines with toroidal B and poloidal B p components form nested surfaces on each of which the pressure is constant. The current density J lies in the plane of each magnetic surface and is related to B through Eq. (6). Thus Eq. (14) can be written as Eq. (16), (16) where signifies the component perpendicular to the magnetic field, and b = B/ B is the unit vector in the direction of the magnetic field. Then it is possible to identify the magnetic pressure, B 2 /(2 0), and the tension in the magnetic field lines, B 2 / 0, which causes a force through the curvature of the lines of force, (b )b. The quantity 2 0p/B 2 defines, the ratio of plasma pressure to magnetic pressure. Fig. 3 Perfectly conducting fluid in toroidal axisymmetric equilibrium. The solution of Eq. (16) for the axisymmetric large aspect ratio toroidal equilibrium shown in Fig. 3 gives nested circular magnetic surfaces that are not concentric; the inner ones are shifted out, away from the major axis of the torus, so that the surfaces are closer together on the outside of the torus.

5 The physical reason for this is that the toroidal current I which produces the poloidal magnetic field B p flows as in a hoop, leading to an outward force on the plasma. (Each current segment experiences a repulsive force from the oppositely flowing current in the segment opposite it.) This toroidal configuration is the basis of the tokamak approach to magnetic fusion energy. Stability of ideal plasmas An equilibrium state as described above is not necessarily stable. The most elementary configuration of a magnetically confined plasma is a Z-pinch (Fig. 4a) in which a cylindrical column carrying an axial current I is confined by the self-magnetic field produced by the current I itself. This was the earliest concept for confinement of plasma for controlled fusion. Consider an azimuthally symmetric perturbation (Fig. 4b) that makes the configuration look rather like a string of sausages. The magnetic field B at the surface of radius a is 0I/2 a, so that the magnetic field is stronger where the radius a is narrower, and vice versa. Hence the plasma column (unless it is very diffuse) can be unstable to this perturbation because of the enhanced J B force where a is small. See also: Pinch effect Fig. 4 Instabilities of a Z-pinch configuration. (a) Initial configuration of magnetically confined plasma. (b) Sausage instability. (c) Kink instability. Another type of instability is the kink mode (Fig. 4c) where the column is bent one way and the other in an alternating way. Again the driving force for the instability can be seen from an examination of the magnetic lines of force. In the kink mode they are concentrated together on the concave side and are further apart on the convex side of each kink. This leads to a net force (as shown in Fig. 4c) which increases the amplitude of the kink. Such instabilities have been seen experimentally, and led to the introduction of another magnetic field component, an axial magnetic field B z, in order to stabilize the configuration. With both B z and B components present, the magnetic field lines are now helices. If the pitch of the helices varies across the radius of the plasma column, then this shear of the field can stabilize the discharge against localized modes. The necessary stability criterion is given by Eq. (17), where r is the distance from the axis of the column and, the pitch, is defined by Eq. (18). For a confined plasma the second term involving the pressure gradient, dp/dr, is always a negative contribution, and so this condition determines the shear d /dr required for stability. When the configuration is bent into a torus, the second term is multiplied by a factor (1 - q 2 ), where q, the safety factor, is rb /RB ; this leads to stability of these internal modes for q (17) (18) 1. Here R is the major radius of the torus, and B is the toroidal component of magnetic field. A purely hydrodynamic instability, found also in dynamic Z-pinches and exploding supernovae, but also of vital concern in the concept of inertial confinement, is the Rayleigh-Taylor instability. The original problem concerned the stability of a heavy fluid above a light fluid under gravitational acceleration g. A small perturbation of the interface (Fig. 5) is clearly unstable, and one with a wave number k (where the wavelength = 2 /k) can be shown to have a growth rate = (kg) 1/2. Short wavelengths thus have the highest growth rates, and in inertial confinement the shell thickness is likely to determine the most dangerous mode. The reason why the Rayleigh-Taylor instability applies here is that the effect of g is the same as a fluid acceleration in the opposite direction. Fig. 5 Rayleigh-Taylor instability, involving a perturbation in the interface between a light fluid and a heavy fluid above it.

6 Energy principle Two theoretical methods have been developed for examining the linear stability of an equilibrium configuration; one is a normal mode analysis which directly determines the growth rate, and the other uses an energy principle. When more complex physical processes such as the nonvanishing ion Larmor radius or the Hall effect are included, the former method is applicable; but where the complexity is entirely in the geometrical configuration of the magnetically confined plasma, the energy principle is the more appropriate technique. In ideal magnetohydrodynamics, Eqs. (1), (2), (4), and (6) can be combined to linearly relate the acceleration of a fluid element to the force F( ), where is the local fluid displacement and the two dots above means the second derivation with respect to time. Because of the adjointness of F, the stability is ensured if the change in potential energy associated with the displacement, given by Eq. (19), is positive, where the integral is taken over the entire volume of the configuration. Equation (17) was derived from the energy principle. [By adjointness is meant that F satisfies Eq. (20), where obeys the same boundary conditions as.] Thus the rate of increase of kinetic energy (that is, the rate of doing work) is F( )d 3 r. By virtue of conservation of total energy and the adjointness of F( ), it follows that the corresponding change in potential energy W is given by Eq. (19). If the kinetic energy is increasing through growth of the instability, there is a corresponding decrease in potential energy. Conversely, if W is positive, the system is stable. More strictly, if W is found to be positive for all allowable displacements, the plasma is stable; but if W is negative for any allowable displacement, the plasma is unstable. The most simple example of the use of the energy principle is the interchange mode. Figure 6 illustrates two flux tubes each having a magnetic flux = B A, where A is the local area of cross section. The volume of a flux tube, V, is the integral Adl, where dl is an element of length along the tube, and therefore the plasma thermal energy in a flux tube is given by Eq. (21). When the plasma on the two flux tubes is interchanged, W is found to be positive if dl/b decreases with the spatially decreasing plasma pressure as one moves toward the walls of the confining vessel. This leads to the concept of a minimum-b confinement configuration, that is, one in which the magnitude of the magnetic field is a minimum at the center of the plasma and increases in all directions away from the center. Fig. 6 Two flux tubes in a magnetic field configuration. (19) (20) (21) Resistive magnetohydrodynamics An electrically conducting fluid acquires new properties when its resistivity is taken into account. Stability of resistive plasmas

7 While the resistivity of thermonuclear plasmas at a temperature of 10 kev is very low (approximately 10-9 ohm m), it is nevertheless sufficient to permit the magnetic field configuration to change its topology in a thin layer inside which the wave number k of the perturbation mode and the magnetic field vector B satisfy the condition k B = 0. The surface that satisfies this condition is called a resonant surface since the magnetic field lies in a plane of constant phase of the plasma displacement. In a torus, such resonant surfaces occur at rational values of q, the safety factor. Figure 7a illustrates the plane containing k and the coordinate r perpendicular to the magnetic surface, surrounding the resonant surface, k B = 0. With nonvanishing resistivity, Ohm's law is given by Eq. (22), (22) replacing Eq. (1). With the presence of the J term, the magnetic field lines can break and reconnect to form a different configuration consisting of magnetic islands centered on the k B = 0 surface. There are corresponding X-type and O-type neutral points (Fig. 7b), the former lying at the intersection of the separatrix and the k B = 0 surface. The growth of such a magnetic island structure to a finite size is of major importance in understanding the magnetohydrodynamic behavior of toroidal magnetically confined plasma, particularly with regard to disruptions in tokamaks. The phenomenon of magnetic reconnection is also important in the evolution of solar flares. The nonlinear evolution of the new closed magnetic lines topology, called magnetic islands, is the subject of much research. For example, the Sweet-Parker model suggests that the open field lines drift slowly in toward the k B = 0 plane, while the plasma flows in the k direction within the reconnection layer have an amplitude of the order of the Alfvén velocity. It is also possible that the high current density induced in the reconnection layer will, as in collisionless shocks, lead to the triggering of microinstabilities with an associated anomalous resistivity, resulting in a fast reconnection. This is particularly applicable to astrophysical reconnection processes. Fig. 7 Effect of nonvanishing resistivity on plasma stability. (a) Location of resonant surface k B = 0. (b) Breaking and reconnection of magnetic field lines about the resonant surface in a resistive plasma. Helicity The possibility of magnetic reconnection even when the resistivity is small but nonzero allows the overriding thermodynamic tendencies to dominate. Hence, for example, in the toroidal reversedfield pinch, the plasma and magnetic field can relax to a state of minimum energy. In doing so, a three-dimensional dynamo occurs as a result of the resistive MHD instabilities and the associated plasma flows, causing the toroidal magnetic field to peak on the minor axis and to reverse in direction at the plasma edge. During this process, it can be reasonably postulated that the magnetic helicity. A BdV is approximately conserved. Here A is the magnetic vector potential defined by B = A. The reason for this conservation is that the local helicity density A B obeys Eq. (23), where is the electrostatic potential. If the flux of helicity, ( B + E A), is zero, then the source or sink of helicity is the term -2E B, which is small everywhere for a plasma of high conductivity, and is only significant in the reconnection layer associated with resistive MHD instabilities. (In evaluating helicity, care must be taken with the choice of gauge for A.) Diffusion Finite resistivity also allows the plasma to diffuse across magnetic field lines. Classical diffusion of a low plasma across an axial magnetic field B z can be described in terms of the pressure balance of Eq. (14), which here becomes Eq. (24). In this the azimuthal current density is driven by the radial diffusion velocity r through Ohm's law, Eq. (22), which here becomes Eq. (25). (25) Thus, the flux of particles of density n in an isothermal plasma is given by Eq. (26). (23) (24)

8 (26) This is Fick's law of diffusion, with a diffusion coefficient D = p/b 2. The B -2 dependence is characteristic of classical diffusion. Plasmas that are microscopically turbulent tend to obey Bohm diffusion where the diffusion coefficient k B T/(16eB) [where k B is Boltzmann's constant, T the absolute temperature, and e the electronic charge] is larger by a factor of e e /16; e is the electron cyclotron frequency, e is the electron-ion collision time, and the product, e e, is called the Hall parameter, being equal to B/(ne ). See also: Diffusion; Electron motion in vacuum; Hall effect; Particle accelerator; Relaxation time of electrons In a toroidal configuration with high safety factor q, greater than 1 (usually called a tokamak), classical diffusion is enhanced by two effects. The first is connected with the magnetohydrodynamic equilibrium itself, as defined by Eq. (14), in which, to balance the outward hoop force of the toroidal current itself, a vertical magnetic field component is required which causes the magnetic surfaces to be shifted outward. Consistent with this, the toroidal current must increase with increasing distance R from the major axis. However, the applied electric field E (equal to - A / t, where A is the toroidal component of the electromagnetic vector potential) varies as R -1. Therefore an outward diffusion velocity R occurs, so that the current density J in Ohm's law, here given by Eq. (27), has the required spatial dependence. This diffusion is 2q 2 times the classical value. See also: Potentials The second effect is important at high temperature when the plasma is relatively collisionless, and when the trajectories of particles have large excursions from magnetic surfaces. Because the toroidal magnetic field varies as 1/R, the magnitude of the total field is greater near the inside of the torus. A guiding center for the motion of a particle can therefore be reflected near the inside of the torus and, in continuing its drift motion, executes a trajectory which is banana-shaped when projected onto a poloidal plane (Fig. 8). The width of the "banana" is 2m /(ZeB p ), where B p is the poloidal component of the magnetic field, m and Z are the particle's mass and charge number, and is the component of its velocity parallel to the magnetic field. This width is much greater than the Larmor radius m /(ZeB ), and when such particles have even only weak collisions, an enhanced diffusion occurs; here is the component of velocity perpendicular to the magnetic field. See also: Charged particle optics Fig. 8 Projection of a guiding center orbit onto a poloidal plane of a tokamak. (27) Magnetic Reynolds number The relative strength of the J and v B terms in Ohm's law can be written with the aid of Eq. (6) as the ratio R M given by Eq. (28), the magnetic Reynolds number, where L is a characteristic length. When R M is much greater than 1, the magnetic field is frozen to the conducting fluid, while for R M much less than 1 the fluid very easily penetrates an applied magnetic field (as in magnetohydrodynamic power generators). When resistive damping is included in the theory of Alfvén waves and magnetohydrodynamic stability theory, a corresponding dimensionless parameter, the Lundquist parameter, given by Eq. (29), occurs, where is the wavelength. When S is much greater than 1 the resistive damping or thickness of the tearing layer is small, while for S less than 1 no Alfvén wave propagation occurs. See also: Dimensionless groups Electrically conducting fluids Electrically conducting fluids are either liquid metals or plasmas. (28) (29)

9 Liquid metals Mercury and liquid sodium are two familiar examples of electrically conducting, incompressible fluids. The core of the Earth is also interpreted to be in the liquid-metal state and able to generate currents and magnetic fields. In particular, in a three-dimensional interaction involving the Coriolis force and the v B force associated with the Earth's rotation, a dynamo action is postulated to account for the Earth's magnetic field. A similar mechanism could generate the solar magnetic field. See also: Coriolis acceleration The necessity of involving a three-dimensional model for dynamo action can be seen from consideration of an axisymmetric, two-dimensional system (Fig. 9), in which only an azimuthal current and poloidal magnetic field components exist. It has been shown that such a system cannot be sustained by fluid motion alone in a steady state. The argument is as follows: the azimuthal electric field must vanish in a steady state so that J = v B; therefore the neutral ring NR where B is zero will also have J = 0 and hence B = 0. If this condition is satisfied for example, if, in the vicinity of the neutral ring, B r for 1, where r is the distance from the neutral ring then Ampère's law gives J r -1, but Ohm's law then yields 1/r which is an impossible singularity. Fig. 9 Axisymmetric, two-dimensional, liquid-metal system in which only an azimuthal current and poloidal magnetic field components exist. Another theorem for the same axisymmetric system is as follows: Because the current is purely azimuthal, the curl of J has no azimuthal component. For uniform resistivity, this means that the azimuthal component of curl (v B) vanishes, which, with the Maxwell equation (30), (30) gives B = 0, where r =. This means that the fluid at all points on a magnetic field line rotates about the axis with a uniform angular velocity, r being the distance from the axis. Plasmas Most work in magnetohydrodynamics has concerned ionized gases or plasmas. Plasma physics concerns the collective interaction of ions and electrons in a usually quasi-neutral system. In such a plasma there are many particles in a Debye sphere. This follows from the condition that for the gas to remain fully ionized, the mean kinetic energy k BT e (where k B is Boltzmann's constant, and T e is the electron temperature) must be larger than the mean potential energy n 1/3 e e 2 / 0, where 0 is the permittivity of empty space, and n e is the number of electrons per cubic meter, so that the average distance between particles is n -1/3 3 e. Raising this condition to the power 3/2 yields the condition n e D 1, where D (the Debye length) is [ 0k BT e/(n ee 2 )] 1/2. In contrast to this condition, there is increasing 3 interest in "strongly coupled" plasmas where n e D 1, as these are relevant to the early stages of inertial confinement when the plasma is very dense and relatively cold. Figure 10 illustrates the regimes in n e - T e parameter space, and shows three quantum-mechanical effects: (1) The classical minimum impact parameter b 0 (the Landau parameter), given by Eq. (31), is less than the de Broglie wavelength, given by Eq. (32), where is Planck's constant h divided by 2, and m e is the electron mass, above a temperature of 27 ev (1 ev = K). [The effective minimum impact parameter is then the de Broglie wavelength, though strictly the reason for the departure from classical Rutherford scattering is that the presence of Debye shielding modifies the r -1 dependence of the Coulomb potential to r -1 exp(-r/ D), which quantum-mechanically leads to the maximum impact parameter being reduced from D to D b 0/. Furthermore, electrons colliding with ions with an impact parameter of have a maximum (31) (32)

10 cross section for radiating their energy due to their large acceleration during the collision. This radiation loss is hence called bremsstrahlung, literally, braking radiation. The radiation is characterized by photon energy hv of order k BT.] (2) Below 1 ev a typical gas is only weakly ionized. (3) For dense plasmas when n -1/3 e is less than, the electron gas becomes degenerate. Above 0.5 MeV the electron energy is relativistic (k B T e m e c 2, where m e is the electron mass and c is the speed of light). See also: Bremsstrahlung; Nonrelativistic quantum theory; Quantum mechanics; Relativistic quantum theory Fig. 10 Plot of electron temperature T e and density n e in a plasma, showing values of these parameters where various types of behavior occur. The classical collision cross section for a single 90 deflection of an electron by an ion is b 2 0. However, since there are many particles in a Debye sphere, each particle simultaneously interacts with n 3 D particles. This leads to an increase in collision cross section by a factor 8 ln, where is D/b 0 (or D/ for T e 10 5 K). Apart from the weak logarithmic dependence, the mean free path is then found to be proportional to T 2 /n and the mean collision frequency to nt -3/2 ; for an electron of velocity, the mean free path is proportional to 4 /n, and its collision frequency (with stationary ions) is proportional to n -3. This velocity dependence of the collision frequency is responsible for the penetration and preheating by more energetic electrons in laser-induced compression experiments, for so-called runaway electrons in ohmically heated toroidal plasmas, and in linear transport theory for the existence of large thermoelectric and Nernst terms in Ohm's law, discussed below. When the plasma is only partially ionized (for example, in a magnetohydrodynamic power generator), the effective collision frequency is the sum of the electron-ion and electron-neutral collision frequencies. See also: Kinetic theory of matter For a fully ionized plasma the electrical conductivity (equal to -1 ) is ne 2 /(m e ) and therefore is independent of electron density and proportional to T 3/2. Similarly the thermal conductivity and the viscosity are proportional to T 5/2 in the absence of a magnetic field. See also: Conduction (electricity); Conduction (heat); Viscosity Fluid equations for a plasma The full set of fluid equations for a plasma in a magnetic field displays a large number of terms which give many interesting effects in magnetohydrodynamics. This set can be derived from the Boltzmann kinetic equation. See also: Boltzmann transport equation Conservation equations The first of these equations, Eq. (33), describes the conservation of mass, where is the mass density and v is the fluid center-of-mass velocity. That this does indeed give conservation, which means no creation or destruction of mass, can be shown by integrating Eq. (33) over a volume V and employing Gauss' theorem, Eq. (34), so that the mathematical result of Eq. (35) (33) (34) (35)

11 means that the rate of change of mass in the volume V is equal to the net inward flux, - v, integrated over the closed surface ds enclosing the volume V. See also: Equation of continuity; Gauss' theorem The conservation of total momentum is of similar form, given by Eq. (36). Here v is the momentum of the fluid per unit volume, and to this must be added the momentum of the electromagnetic fields themselves per unit volume, D B. Since flux is a vector and momentum is also a vector, the flux of momentum is in general a tensor. For example, there can be a momentum component x transported with a velocity component y. Thus the terms in the divergence operator are tensors: vv is the flux of momentum associated with the center-of-mass velocity; P, called the stress tensor, is the flux of momentum associated with the random or thermal motion of the fluid particles; and T is the flux of electromagnetic momentum usually called the Maxwell stress tensor. In most cases, Eq. (35) can be reduced to a more simple form that does not involve tensors, as given in Eq. (41). See also: Tensor analysis The conservation of total energy also obeys a similar equation, Eq. (37), (36) (37) where 2 and 2 v are the kinetic energy per unit volume and flux of kinetic energy, respectively; U = 3/2nk B T is the internal energy density, n being the particle number density, k B Boltzmann's constant, and T the temperature; and q is the heat flux. The terms E D and H B are the energy per unit volume in electric and magnetic fields, respectively, and E H is the electromagnetic energy flux, commonly called Poynting's vector, and equal to c 2 D B for vacuum fields. See also: Poynting's vector Application of Maxwell's equations Maxwell's equations are Eqs. (38), (2), (39), and (30), (30) where q is the charge density. From these equations it can be shown that the electromagnetic force per unit volume is given by Eq. (40), where T is given by Eq. (41) (41) and 1 is the unit tensor. The left-hand side of Eq. (40) can also be considered as the rate at which electromagnetic momentum per unit volume is given to the fluid or plasma, or minus the rate at which fluid momentum density is given to the electromagnetic fields. Through the effects of magnetic field tension and pressure, the electromagnetic stress T provides the means, for example, of transferring the equal and opposite momentum onto a field coil when a plasma is being accelerated. Using Eq. (40) and subtracting v times Eq. (33), Eq. (36) can now be written simply as Eq. (42), where the stress tensor has been approximated by the isotropic scalar pressure p = nk BT. It has also been assumed here that the plasma is quasi-neutral, that is, that there are many particles in a Debye sphere so that the residual charge density q is small, causing q E to be a negligible force per unit volume. For this reason Eq. (38) is employed only for describing high-frequency phenomena. The operator d/dt refers to the absolute rate of change with time as seen by a moving fluid element. Maxwell's equations can also be used to derive Eq. (43) (38) (2) (39) (40) (42)

12 for J E, the rate of transfer of energy per unit volume from electromagnetic to plasma energy. For a dynamo or a generator (for example, a magnetohydrodynamic generator) J E must be negative, while for dissipation in or Joule heating of a stationary plasma, J E will be positive. Viscous effects When the stress tensor is retained in the equation of motion, kinetic theory shows that in addition to the isotropic pressure there are contributions proportional to gradients or shear in velocity v. For an isotropic plasma (that is, one without large magnetic field effects) the equation of motion (42) is modified to give Eq. (44), (43) (44) where is the ion viscosity, equal to the ion pressure divided by the collision frequency for ion-ion collisions. The viscous terms in Eq. (44) are the characteristic Navier-Stokes terms, the second one vanishing for an incompressible fluid. See also: Fluid-flow principles; Navier-Stokes equation; Viscosity Other dimensionless numbers The ratio of (v )v to 2 v in Eq. (44) gives Eq. (45) for the (viscous) Reynolds number, where L is the characteristic length over which the velocity shear exists. R describes the ratio of convection of vorticity, v, to diffusion. A large value of R of the order of 10 3 leads to hydrodynamic turbulence. See also: Reynolds number The Peclet or thermal Reynolds number R T is the ratio of the enthalpy flow (U + p)v to the heat flow q. If q is given by Fourier's law, Eq. (46), where is the thermal conductivity, the thermal Reynolds number R T describes in Eq. (47) the ratio of convection of temperature to diffusion. (46) The combined effect of convection and diffusion can lead to boundary layers being established. The relative thickness of the thermal viscous and magnetic (or skin current) boundary layers is described by Prandtl numbers which are the ratio of the various Reynolds numbers or diffusivities. The original Prandtl number is R T/R, while the magnetic Prandtl number is R M/R, where R M is the magnetic Reynolds number defined in Eq. (28). See also: Boundary-layer flow; Skin effect (electricity) The relative strength of J B and 2 v in Eq. (44) leads to the definition of the Hartmann number H by Eq. (48), where J is taken to be of order v B/. Setting H approximately equal to 1 defines the scale length over which magnetic and viscous forces are comparable in magnitude. Diamagnetism The mean center-of-mass velocity v for a species should not be confused with the mean guidingcenter velocity v gc associated with the motion of individual particles of the species. Ignoring inertial effects and collisions, v in Eq. (34) can be shown to be the sum of v gc and the diamagnetic velocity 1/Zen curl (p B/B 2 ), where curl is the component of the curl perpendicular to the magnetic field, and p is the component of the anisotropic pressure perpendicular to the magnetic field. The physical meaning of the diamagnetic velocity can be understood from Fig. 11, which shows particle orbits in the plane perpendicular to the magnetic field B z in a frame of reference in which the mean guiding-center motion is zero. It can be seen that if the density or temperature increases in the x direction, or if B z decreases in the x direction, there will be a net contribution to momentum in the y direction in the shaded element due to the time-averaged statistical contribution from particles whose orbits intersect the element. Fig. 11 Particles orbits in the plane perpendicular to a magnetic field B z in a frame of reference in which the mean guiding center motion is zero. (45) (47) (48)

13 Transport equations for a two-fluid plasma in a magnetic field By solving the Boltzmann equation for the case when the electric field E and gradients of pressure, temperature, and velocity are small, transport equations can be derived relating the current density J and the heat flux q to these quantities. Thermodynamically the fluxes J and q are driven by the thermodynamic forces, E and T, so that the irreversible generation of entropy is always positive locally. See also: Entropy In general, cross-phenomena occur; for example, current can flow as a result of a temperature gradient. The physical reason for this is that the collision frequency is velocity-dependent. Thus, when a temperature gradient exists, higher-velocity particles in the hotter region will flow with fewer collisions, so that there will be not only a heat flow but also a current. If a current is prevented from flowing, as in the symmetric radially inward heat flow in laser fusion targets, an electric field will develop instead to retard the hotter electrons and also to drive an equal and opposite cold electron current. The presence of the electric field will also affect the heat flow. See also: Thermoelectricity If there is also a magnetic field which is orthogonal to an electric field or to a temperature gradient, the electrons will cause a current or heat flow not only parallel to E or T but also orthogonal to both E or T and the magnetic field. This is because of the cycloidal motion of charged particles in a magnetic field. See also: Galvanomagnetic effects; Thermomagnetic effects When the center-of-mass velocities and temperature of the electrons and ions are considered to be different, separate equations for the two fluids have to be employed. Using the notation of Braginskii, the transport equations for the electron fluid (subscript e) can be written as Eqs. (49) and (50), (49) (50) where E* is a generalized electric field given by Eq. (51); b is the unit vector in the direction of the magnetic field; the coefficients,, and so forth are functions of ( e e ), the Hall parameter; is the value of when e e = 0, and so forth; and J, J, T e, and T e are given by Eqs. (52)-(55). (52) (53) (54) (55) In Eq. (49) the Hall electric field is principally included in the J B term in the definition of E* [Eq. (51)], while the correction has been shown to vary as ( e e) -2/3 for values of ( e e) much greater than 1. The main property of the Hall effect comes from a comparison of the term and the J B term in E*: This is that the components of current density J and the general electric field E + v B + p e/(n ee) in the plane perpendicular to the magnetic field are at an angle tan -1 ( e e) to each (51)

14 other. Thus in a magnetically confined plasma there is an electric field component orthogonal to the magnetic surface. In magnetohydrodynamic power generators the electrodes have to be segmented in order that the current flow is orthogonal to the gas flow, so that the J B force is directed to oppose this flow, and the generator impedance is minimized. The segmentation of the electrodes permits an electric field to be set up between the segments. When an electric field E is applied to a plasma in a direction orthogonal to a magnetic field so that the current can flow freely at the angle tan -1 ( e e ) to E, the component of current density orthogonal to E is called the Hall current. This can be used to give a J B acceleration of the ions in a suitable magnetic field configuration such as in magnetically insulated diodes (as used for light ion-beam production). The Hall term modifies the ideal magnetohydrodynamic frozen-flow motion so that magnetic fields are now frozen to the electron center-of-mass velocity. The last term in Eq. (49) describes the thermoelectric term, and through this it is possible to generate a current parallel to T e (Seebeck effect) or perpendicular to T e (Nernst effect). The Nernst effect can approximately be represented as an electic field equal to q e B/(3/2n ek BT e) showing by analogy with Eqs. (1) and (3) that the magnetic field is convected in the direction of heat flow with a velocity given by Eq. (56). In the case of laser fusion, this can lead to a magnetic field being convected from the critical surface to the ablation surface, and, furthermore, amplified because of the increase in number density n which reduces the velocity v T causing the transverse magnetic field lines to bunch up. This further modification of frozen flow implies that the magnetic field is frozen to the hotter (and less collisional) electrons that carry the heat. Indeed, it is a special feature of inertial confinement fusion that megagauss magnetic fields can spontaneously be generated in nonuniformly irradiated targets through the p e/(n ee) term in E* [Eq. (50)] from Eq. (2), so that Eq. (57) holds. (In laser fusion the ideal situation is one in which laser radiation is uniformly irradiated onto and absorbed by a perfectly symmetrical spherical target. The above magnetic fields are generated only through a lack of symmetry arising from nonuniform irradiation or through instabilities such as the Rayleigh-Taylor instability discussed above.) In Eq. (50), it is found that while is proportional to T 5/2 e, varies as ( e e ) -2 for values of e e much greater than 1. Therefore, in magnetically confined plasmas the electron thermal conduction across magnetic surfaces is considerably reduced, and indeed ion thermal conduction across field lines is [m i /(Zm e )] 1/2 times larger. The heat flow parallel to a current flow essentially represents the flux of enthalpy of the current carriers and is called the Peltier effect. The heat flux defined by Eq. (50), however, is defined in the center-of-mass frame of the electrons, and so the and terms refer to the corrections to the enthalpy flow caused by the velocity dependence of the collision frequency. The Ettingshausen effect is the heat flow in the direction of the Hall electric field, while the Righi-Leduc effect is the heat flow orthogonal to T e and B. and are proportional to e e for e e much less than 1, and to ( e e ) -1 for e e much greater than 1, while varies as ( e e ) -5/3 for e e much greater than 1. It follows that, (56) (57), and all vanish as the magnetic field tends to 0. The peculiar fractional powers in and occur because for a value of e e much larger than 1 it is found that a fraction of electrons with a velocity less than ( e e) -1/3 times the mean thermal speed (2k B T e /m e ) have in fact unmagnetized orbits. This is because they have a low velocity so that their collision frequency which is proportional to -3 exceeds their cyclotron frequency. The ion stress tensor is also modified by a strong magnetic field, and for i i 1 leads to the persistence of terms in velocity gradients with a coefficient p i/ i, where p i is the ion pressure and i the ion cyclotron frequency. These residual terms describe finite ion Larmor radius effects, which can be a beneficial correction to ideal magnetohydrodynamic stability theory. Malcolm G. Haines How to cite this article Suggested citation for this article: Malcolm G. Haines, "Magnetohydrodynamics", in AccessScience@McGraw-Hill, DOI / , last modified: August 8, For Further Study Topic Page: Physics: Electricity and magnetism Topic Page: Physics: Fluid mechanics

15 Bibliography G. Bateman, MHD Instabilities, 1980 D. Biskamp, Nonlinear Magnetohydrodynamics, 1993 F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, vol 1: Plasma Physics, 2d ed., 1984 E. M. Epperlein and M. G. Haines, Physics of Fluids, 29:1029, 1986 N. A. Krall and W. Trivelpiece, Principles of Plasma Physics, 1973, reprint 1986 K. Miyamoto, Plasma Physics for Nuclear Fusion, 1976, rev. ed.1989 J. A. Wesson, Tokamaks 1987, 2d ed., 1997 Customer Privacy Notice Copyright The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use and Notice. Additional credits and copyright information. For further information about this site contact Last modified: Sep 30, 2003.