E NCYCLOPEDIA OF A STRONOMY AND A STROPHYSICS

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1 Magnetohydrodynamics Magnetohydrodynamics (or MHD for short) is the study of the interaction between a magnetic field and a plasma treated as a continuous medium (e.g. Cowling 1957, Roberts 1967, Priest 1982, 1994). Most of the universe is not a normal gas but is instead a plasma. We are all familiar on Earth with the three states of matter (solid, liquid and gas). You change from one state to another (such as ice to water) by heating, and if you raise the temperature of gas sufficiently it changes to the fourth state of matter, namely plasma. In the plasma state the atoms have split into positive ions and negative electrons, which can flow around freely, so the gas becomes electrically conducting and a current can flow. (MHD can also be used to describe the behavior of an electrically conducting liquid.) Many dynamical processes in the universe are caused by the subtle nonlinear relationship between a magnetic field and a plasma. In a normal gas such as the air we breathe, there is virtually no interaction with a magnetic field. But in a plasma the extremely close coupling with the magnetic field means that whatever the plasma is doing intimately affects the magnetic field and vice versa. Indeed, on Earth we are in an extremely unusual part of the cosmos, a tiny island of solid, liquid and gas. But, as soon as we go up to the ionosphere, the plasma universe begins, including the region between Earth and Sun, the whole of the Sun itself, as well as the interstellar and intergalactic media and the stars and galaxies contained in them. In MHD we are not concerned with individual particles but treat the plasma as a continuous medium. It builds partly on electromagnetism and partly on fluid mechanics. The assumption of a continuous medium is valid for length-scales much larger than the mean-free path for particle collisions ( ) 2 ( ) 1 T n λ mfp 300 m, 10 6 K m 3 which is typically 3 cm in the solar chromosphere and 30 km in the solar corona. The magnetic field has several physical effects: (a) It exerts a force, which may accelerate plasma or create structure. (b) It stores energy, which may later be released as, for example, a solar eruption or a solar flare. (c) It acts as a thermal blanket, which, when wrapped around a cool solar prominence, say, may protect it from the surrounding corona. (d) It channels fast particles and plasma. (e) It drives instabilities and supports waves. MHD is important in a wide variety of cosmic phenomena. The different aspects of MHD are described in the articles that follow and the applications are discussed in many of the articles throughout the encyclopedia, notably in those about the Sun, where Figure 1. Segment of a magnetic flux tube. MHD phenomena are widespread. In this introductory article, we first describe a magnetic flux tube and then introduce the fundamental equations. Particular attention is given to the induction equation and the Lorentz force with which a magnetic field acts on a plasma. Then we describe the force balance of a plasma at rest, known as magnetohydrostatics, and mention briefly the possible wave modes and instabilities. Magnetic flux tubes A magnetic field line is a curve such that the tangent at any point is in the direction of the magnetic field. Its equation, for a two-dimensional magnetic field having components (B x,b y ), is dy dx = B y B x or dx/b x = dy/b y = dz/b z in three dimensions. A magnetic flux tube is the surface generated by the set of magnetic field lines which intersect a simple closed curve. Flux tubes are the building blocks of a magnetic configuration, but they must not be thought of as independent isolated structures. The strength (F ) of a flux tube is the amount of magnetic flux crossing a section (S), i.e. F = B ds. s Consider a finite segment of a flux tube bounded by plane sections S 1 and S 2 (figure 1). There is no flux across the walls of the tube, and so physically, if no flux is created inside the tube, the flux (F 1 ) entering through section S 1 equals the flux (F 2 ) leaving through section S 2. In other words, the strength (F ) is constant along the tube. Mathematically, integrating over the whole surface (S V ) of the segment of the tube between S 1 and S 2 gives, since B ds vanishes on the curved part B ds = B ds + B ds = F 1 + F 2 S V S 1 S 2 where F 1 is the flux from left to right and, on S V,dS is in the direction of the outwards normal. However, the Gauss divergence theorem gives B ds = B dv = 0 S V V since B = 0. Thus F 1 = F 2 as required, and therefore B = 0 implies that no flux is indeed created and the strength is constant along the flux tube. Dirac House, Temple Back, Bristol, BS1 6BE, UK 1

2 y x where B = µh, D = ɛe and j and ρ c are the electric current and charge densities present in the plasma. (Sometimes a subscript 0 is placed on µ and ɛ.) Here H is the magnetic field, B is the magnetic induction (although we shall loosely refer to it as the magnetic field), µ the magnetic permeability of free space, E the electric field, D the electric displacement, ɛ the permittivity of free space, ρ c the charge density, j the electric current density. These are supplemented by Ohm s law E = j/σ (3) where σ is the electrical conductivity. The equations of fluid mechanics for a perfect gas, on the other hand, are Figure 2. Field lines for the magnetic field (B x,b y ) = (y, x). ρ dv = p (4a) If the cross-sectional area (A) of a flux tube is small, then F BA. Thus, as the magnetic field lines become closer together, so A becomes smaller and, since F is constant, B increases in value, and vice versa. When sketching field lines from expressions for the field components there are three stages: (a) (b) (c) Evaluate the expressions for the field lines and sketch a typical one. Decide the directions of the arrows. Decide the spacings of other field lines. Thus, for example, if B x = y and B y = x, the field lines are given by dy/dx = x/y or y 2 x 2 = constant. When the constant is zero we obtain the two field lines (y =±x) through the origin (called separatrices); when the constant is positive (negative) we have branches of a rectangular hyperbola intersecting the y-axis (x-axis). From the orientations of the axes in figure 2 we have decided that the positive directions are to the right and upward; thus, for instance, on the positive x-axis the field is simply B y = x, which is positive and so the arrow is directed upwards. Also, as one moves out along the x-axis the magnitude of the field increases and so the field lines become more closely spaced. The origin is a special point, an X-type neutral point, where the field vanishes and the topology of nearby magnetic field lines is hyperbolic; it represents a weak spot in a configuration where magnetic energy tends to be released after the formation of a current sheet. Fundamental equations The MHD equations are a unification of the equations of slow electromagnetism and fluid mechanics. Maxwell s equations are H = j + D B = 0 (1) E = D = ρ c (2) dρ + ρ v = 0 (4b) p = RρT (4c) and an energy equation (see below), where ρ is the plasma density (the mass per unit volume), v the plasma velocity, p the plasma pressure, T the temperature, R the gas constant. The operator d/ = / + v is the total (or material) derivative and represents the time rate of change moving with an element of plasma, in contrast to / which represents the time rate of change at a fixed point of space. Equation (4a) is the equation of motion, which says that the mass times acceleration of a moving element of plasma equals the sum of the forces acting on the element. Here we have included just the pressure gradient ( p). Equation (4b)is the equation of mass continuity and is simply a mathematical expression of the physical fact that no plasma is being created or destroyed: thus, if for example mass is flowing outwards away from an element (so that the flow is diverging and the divergence v is positive), then the density of the element must be decreasing so that from (4b) dρ/ must be negative. Equation (4c) is the perfect gas law, stating that the pressure of a plasma is proportional to its density and temperature. In principle (4a) determines the velocity v, (4b) the density ρ, while (4c) and an energy equation determine the pressure p and temperature T. The (internal) energy equation may be written ρ de + p v = (κ T ) Q ν + Q r where p e = (γ 1)ρ is the internal energy density, κ is the thermal conductivity tensor, Q ν is the heating by viscous dissipation, Q r is the radiative energy loss and γ is the ratio of specific heats. In Dirac House, Temple Back, Bristol, BS1 6BE, UK 2

3 many astrophysical and solar applications the (optically thin) radiative loss term can be expressed as Q r = ρ 2 Q(T ) where Q(T ) is a function describing the temperature variation of the radiative loss. In a neutral gas the electromagnetic (1) (3) and fluid dynamic (4) equations are decoupled and so the electromagnetic and fluid properties are independent. However, in MHD we modify the above equations in four ways: (a) (b) (c) (d) A plasma feels an extra force, the so-called Lorentz force (j B), which is added on to the right-hand side of (4a). It is this term which couples the fluid equations to the electromagnetic equations. It is well known that an element dl of wire carrying a current J in a magnetic field feels a force J dl B perpendicular to the wire and to the field, and so it is natural that a plasma element of volume dv carrying a current of density j per unit volume should feel a force j dv B. The presence of the electric current adds an ohmic heating term (j 2 /σ ) to the energy equation. Ohm s law states that the electric field in a frame moving with the plasma is proportional to the current, but the total electric field on moving plasma is E + v B, where E is the field seen in a frame where the observer is at rest, so (3) is modified by adding v B to the left-hand side. This too couples the electromagnetic equations to the fluid equations. We consider processes with plasma and wave speeds much slower than the speed of light (v c), so that the displacement current ( D/) in (1) is negligible. This in turn means that we don t need to consider D = ρ c since it just determines ρ c if needed. There are at least a dozen different approximations that are commonly used for the MHD system of equations. Some of the more well-known ones are: incompressibility (when ρ is constant following the motion or v = 0), which requires that the flow speed be much smaller than the sound speed and Alfvén speed; a steady state ( / = 0 for all variables); the Boussinesq approximation (filtering out sound waves by including density variations only in the gravitational term in the equation of motion); an isothermal state (when T = constant); an ideal MHD state (when η = ν = 0, R m = R e =, see below); an inviscid state (when the viscous effects are negligible); an irrotational (potential) flow (when the vorticity, v, vanishes); an isentropic state (when the entropy is constant); a force-free field (j B = 0); a potential magnetic field (j = 0); a strong magnetic field (β 1); a weak magnetic field (β 1, where β is defined by equation (15) below); a supersonic flow regime (plasma velocity larger than sound speed); a subsonic flow regime (velocity smaller than sound speed). Validity of MHD in collisionless plasmas Many cosmic plasmas are collisionless, where the mean free-path for binary collisions between particles is much greater than the characteristic length-scale of the system. MHD ignores particle interactions, and so the reader may at first think that it is of little use in collisionless plasmas. In practice, however, MHD and its two-fluid variants often describe the behavior of collisionless plasmas surprisingly well. For instance, the solar wind beyond 10 solar radii is completely collisionless, but MHD models describe its global velocity, temperature and density quite well, including such time-dependent aspects as shock waves, stream interactions and turbulence. In principle, the appropriate theory for collisionless plasmas is kinetic plasma theory derived from Vlasov s equation. However, due to its mathematical complexity, it is rarely used to construct a global model of the solar wind and is instead confined mainly to calculating effective transport coefficients and modeling localized effects such as shock structure. MHD has been successful in describing collisionless plasmas for several reasons. Firstly, conservation of mass, momentum and energy are principles common to both ideal MHD and collisionless systems. It is only when dissipation processes are considered that classical MHD becomes problematical. Secondly, ionized particles undergo a gyro-motion about the magnetic field which prevents them from traveling unimpeded in the direction perpendicular to the field. Thus, long-range interactions occur only along the field, while short-range interactions across the field may be described by MHD-like equations. Finally, many plasmas are perturbed by waves which interact with particle motion, even along the field, scattering them in all directions and so re-introducing some form of effective collisionality. In ideal MHD the magnetic field is frozen to the plasma, but this is also a natural consequence of gyromotion in a collisionless plasma whenever the E B drift of the gyro-centers dominates all other drifts. Thus, if a collisionless plasma contains particles undergoing gyromotion and the E B drift dominates, it will obey the ideal MHD Ohm s law. The MHD Lorentz force (j B) is recovered in a collisionless plasma whenever the particle gyro-velocity is much smaller than the speed of light. When the Lorentz force on individual particles is added together it gives a net force per unit volume of F = ρ c E + j B where ρ c is the electric charge density. However, the ratio of the two terms on the right is ( ) 2 ρ c E j B V c which is much less than unity when the particle speed (V ) is non-relativistic. In other words, ρ c E is usually Dirac House, Temple Back, Bristol, BS1 6BE, UK 3

4 negligible compared with the (j B) force for the lowenergy particles. In a collisionless plasma, the most important new effect arises in the pressure and viscous terms in the momentum equation, since the gas pressure is generally anisotropic relative to the magnetic field. The plasma exerts a pressure (p ) along the field which is usually different from the pressure (p ) perpendicular to the field. Similarly, the viscous stress tensor is also anisotropic and dependent on the magnetic field. If p and p can be expressed in terms of the local bulk properties of the plasma, then the MHD equations can still be applied by just adding the appropriate forms of the pressure and viscous stress tensors. In a collisionless plasma equation (6a) in the next section is replaced by the generalized Ohm s law. For a fully ionized plasma it takes the form ( ) E = v B + j σ + m e j + (vj + jv) j B p e ne 2 ne ne where vj and jv are dyadic tensors and p e is the electron stress tensor. The first term on the left-hand side of this equation is the convective electric field, while the second term is the field associated with Ohmic dissipation caused by electron ion collisions. The next three terms describe the effects of electron inertia, while the penultimate term is the Hall effect and the last term includes the electron gyroviscosity. For a partially ionized plasma, collisions between charged particles and neutrals lead also to ambipolar diffusion. Induction equation With the above assumptions, equations (1) (3) become j = B/µ (5a) are primary variables, with the current driven by electric fields, and then the magnetic field is a secondary variable produced by currents. However, in MHD the basic physics is quite different, since the plasma velocity (v) and magnetic field (B) are the primary variables, determined by the induction equation and the equation of motion, while the resulting current (j) and electric field (E) are secondary and may be deduced from (5a) and (6a) if required. If V 0, L 0 are typical velocity and length-scale, the ratio of the first to the second term on the right-hand side of (7) is, in order of magnitude, the magnetic Reynolds number R m = L 0V 0 η. Thus, for example, in a solar active region where η 1m 2 s 1, L m, V ms 1 we find R m 10 9 and so the second term on the right of (7) is completely negligible. In turn, equation (6a) reduces to E = v B to a very high degree of approximation. This is the case in almost all of the solar atmosphere, indeed in almost all of the universe the only exception is in regions (such as current sheets) where the length-scale is extremely small, so small that R m 1 and the second term on the right of (7) becomes important. If R m 1, the induction equation reduces to = η 2 B (8) and so B is governed by a diffusion equation, which implies that field variations (irregularities) on a scale L 0 diffuse away on a time-scale of τ d = L2 0 η = E (5b) which is obtained simply by equating the orders of magnitude of both sides of (8). The corresponding speed at which they diffuse is where E = v B + j/σ B = 0 (6a) (6b) We may therefore eliminate j and E by substituting for j from (5a)in(6a) and for E from (6a)in(5b), with the result that / = (v B) η ( B), where η = 1/(µσ ) is the magnetic diffusivity, here assumed uniform. By expanding out the triple vector product in the last term and using (6b), we obtain finally = (v B) + η 2 B (7) which is known as the induction equation. This is the basic equation for the magnetic behavior in MHD: it determines B once v is known. In electromagnetism the electric current and electric field ν d = L 0 τ d = η L 0. With η 1 m 2 s 1, the decay time for a sunspot is (with L 0 = 10 6 m) sec = yr, so that the process whereby sunspots disappear in a few days cannot just be diffusion (and is probably instead decaying by the convection away from the spot of many small flux tubes). Similarly, the diffusion time for a magnetic field pervading the Sun as a whole (with L 0 = m) is s = yr. This is of the order of the age of the universe, so a magnetic field in a star at its formation has not had time to diffuse much. Consider, for example, a one-dimensional magnetic field (B(x, t)ŷ) satisfying = η 2 B x 2 (9) Dirac House, Temple Back, Bristol, BS1 6BE, UK 4

5 B B 0 t = 0 t = t 1 t = t 2 plasma C 1 motion C 2 2 ηt 1 2 ηt 2 x -B 0 t 1 t 2 (a) Figure 3. The magnetic field (B) in a diffusing one-dimensional current sheet as a function of distance (x) for times t = 0, t 1, t 2. and suppose that initially the field is B(x,0) = B 0 for x > 0 and B( x,0) = B(x,0), as shown in figure 3. Physically, what do we expect to happen? Since (9) has the form of a heat conduction equation and we know that heat tends to flow from a hot region to a cool one and smooth out a temperature gradient, we expect the same diffusive process to occur for our magnetic field and for the initially steep magnetic gradient at x = 0 to smooth out, as shown in figure 3. Mathematically, the required solution of (9) turns out to be, in terms of the error function (erf(ξ)) ( ) x B(x,t) = B 0 erf = 2B x/ (4ηt) 0 e u2 du. (4ηt) π This solution does indeed have the form shown in figure 3 and it may be verified a posteriori by substituting back into (9). The resulting field lines diffuse through the plasma and cancel at x = 0. The main reason for variations in R m from one phenomenon to another is variations in the appropriate length-scale (L 0 ). If R m 1, the induction equation becomes = (v B) and Ohm s law reduces to E + v B = 0 so that the total electric field in a frame of reference moving with the plasma vanishes. Then, if we consider a curve C (bounding a surface S) which is moving with the plasma, in a time an element ds of C sweeps out an element of area v ds. The rate of change (d/ B ds) of magnetic flux through C then consists of two parts, namely S ds + B v ds. C As C moves, so the flux changes partly because the magnetic field is changing with time (the first term) and partly because of the motion of the boundary (the second 0 P 1 P 2 plasma motion P 1 P 2 t 1 t 2 (b) Figure 4. (a) Magnetic flux conservation if a curve C 1 is distorted into C 2 by a plasma motion, the flux through C 1 at t 1 equals the flux through C 2 at t 2.(b) Magnetic field line conservation if plasma elements P 1 and P 2 lie on a field line at t 1, then they will lie on the same line at t 2. term). Then, by putting B v ds = v B ds and applying Stokes theorem to the second term, we obtain d ( ) B ds = (v B) ds S which vanishes in the present approximation. Thus, the total magnetic flux through C remains constant as it moves with the plasma. In other words we have proved magnetic flux conservation (figure 4(a)). It follows that plasma elements that form a flux tube initially do so at all later times (figure 4(a)). There is also magnetic field line conservation (or conservation of magnetic connectivity), namely that, if two plasma elements lie on a field line initially they will always do so (figure 4(b)). At t = t 1, say, suppose elements P 1 and P 2 lie on a field line, which may be defined as the intersection of two flux tubes. Then, at some later time (t = t 2 ) by magnetic flux conservation P 1 and P 2 will still lie on both tubes, and so they will lie on the field line defined by their intersection. We interpret the above results by saying that the magnetic field lines move with the plasma we say that they are frozen into the plasma and plasma can move freely along field lines, but in motion perpendicular to them they are dragged with the plasma or vice versa. Dirac House, Temple Back, Bristol, BS1 6BE, UK 5

6 The Lorentz force One equation relating our fundamental variables v and B is the induction equation; the other is the equation of motion, whose most common form is y ρ dv = p + j B (10) although other forces such as viscous or gravitational may sometimes be important too. The term p is the plasma pressure gradient: it acts from regions of high plasma pressure to low pressure and is perpendicular to the isobars (curves of constant pressure). The magnetic force (j B) is perpendicular to the magnetic field, and so any plasma acceleration parallel to the magnetic field must be caused by other forces. By substituting for j from Ampère s law (5a) and using an identity for the triple vector product, we can write it as ( ) j B = ( B) B/µ = (B ) B µ B 2. 2µ Magnetic Pressure Force Magnetic Tension Force (a) x The two terms on the right-hand side have important physical interpretations. Since the second term has the same form as p, we can say that it represents the effect of a magnetic pressure of magnitude B 2 /(2µ). It gives a force when B 2 varies with position, and the direction of the force is from regions of high magnetic pressure to low magnetic pressure. The first term represents the effect of a magnetic tension parallel to the magnetic field of magnitude B 2 /µ per unit area. It gives a force when the field lines are curved, just like an elastic band or rope. By writing B = Bŝ in terms of the unit vector (ŝ) along the magnetic field, the tension term ((B )B/µ) may be written B µ d B2 dŝ (Bŝ) = ds µ ds + B db µ ds ŝ = B2 µ ˆn R + d ds ( B 2 2µ ) ŝ where ˆn is the principal normal and R is the radius of curvature. The second term on the right of this equation is irrelevant, since it cancels with the component of (B 2 /2µ) parallel to B, as it must since j B is perpendicular to B. However, the first term on the right is the magnetic tension term, which shows that when the radius of curvature is small the tension force is large. Thus, we have shown that the magnetic force has two distinct effects: the magnetic pressure tries to compress the plasma and produces a net force if B varies with position; the magnetic tension tends to make the field lines shorten themselves like elastic bands and gives a net force if the field lines are curved. As an example, consider the field with components B x = y,b y = x, with X-type field lines shown in figure 5(a). At a point on the positive x-axis, the curvature of the field lines suggest a magnetic tension force to the right, while the fact that the magnetic pressure is increasing (like x 2 ) away from the origin suggests a magnetic pressure force (b) Figure 5. Magnetic forces due to the fields (a) B = y ˆx + xŷ and (b) B = r ˆθ. to the left. However, from the field lines alone, it is not evident which dominates. Now ( )( ) (B ) B µ = y x + x y y µ ˆx + x µ ŷ = x µ ˆx + y µ ŷ Dirac House, Temple Back, Bristol, BS1 6BE, UK 6

7 which becomes x ˆx/µ on the x-axis and so does indeed act away form the origin. Also, the magnetic pressure force is ( ) ( ) B 2 x 2 + y 2 = = x 2µ 2µ µ ˆx y µ ŷ which becomes x ˆx/µ on the x-axis and so does act towards the origin. We find therefore that the tension and pressure forces are in precise balance, as they must be since the electric current is ( ) j = 1 y µ x x ẑ = 0 y and so the Lorentz force j B vanishes everywhere. Also, consider next the field B = r ˆθ, for which the field lines are the circles r = constant, becoming more closely spaced as one moves outwards (figure 5(b)). Thus, physically the magnetic tension and pressure forces are expected both to act inwards. Mathematically, we find the magnetic pressure force is ( ) ( ) B 2 r 2 = = r 2µ 2µ µ ˆr which does indeed act inwards. Also, the magnetic tension is ( )( ) ( ) (B ) B µ = B θ B θ ˆθ = r ˆθ r θ µ θ µ and now r/µ does not vary with θ but ˆθ does! In fact ˆθ/ θ = ˆr and so (B )B/µ = rˆr/µ, which again acts inwards, as expected. Magnetohydrostatics Consider the equation of motion when pressure, magnetic and gravitational forces are acting ρ dv = p + j B + ρg (11) where j = B/µ. (12) If L 0,v 0,L 0 /v 0 are typical values for the length-scale, plasma velocity and time-scale respectively, the order of magnitude of the current from (12) is j 0 = B 0 /µl 0. Then, in terms of the typical density (ρ 0 ), pressure (p 0 ) and magnetic field (B 0 ), the sizes of each term in equation (11) are ρ 0 v0 2/L 0, p 0 /L 0, B0 2/µL 0, ρ 0 g, respectively. Now, provided the magnetic term is of the same order as the largest force term, we have force balance if the first term is much smaller than the third, namely v 2 0 B2 0 µρ 0 v 2 A where v A is known as the Alfvén speed. Then (11) reduces to the equation for magnetohydrostatic force balance 0 = p + j B + ρg (13) where j = B/µ, B = 0 and ρ = p/(rt). In this equation, when gravity is negligible we have magnetostatic balance 0 = p + j B (14) if the fourth term in (11) is much smaller than the third, namely L 0 B 2 0 /(µρ 0g) = 2H/β, where H = p 0 ρ 0 g = RT 0 g is the pressure scale-height and p 0 β = B0 2/(2µ) (15) in the plasma beta, namely the ratio of plasma to magnetic pressure. In turn, in (14) the magnetic force dominates if 2β 1, and then (14) reduces further to the equation j B = 0 (16) for a FORCE-FREE MAGNETIC FIELD, in which the magnetic field is in equilibrium with itself under a balance between magnetic pressure and magnetic tension forces. Suppose now that gravity is directed vertically downwards in the negative z-direction (g = gẑ). Then the component of (13) parallel to a particular magnetic field line is 0 = dp/ds ρg cos θ, where s is measured along the field, or, since ds cos θ = dz, we have 0 = dp/dz ρg. After putting ρ = p(rt) 1 this becomes dp dz = p g RT which may be integrated to give ( ) z g p = p 0 exp 0 RT dz where p 0 is the pressure at the base (z = 0) of the field line. If the variation (T (z)) of the temperature with height is known, this determines the pressure and therefore the density. If, in particular, the temperature is uniform (T = T 0 ) then p = p 0 e z/h so that the pressure (and density) decrease exponentially with height, with the scale-height H being the vertical distance over which the pressure falls off by a factor e. Down in the solar photosphere, where T 0 = 6000 K, the scale-height is about 150 km or less and so, over for instance a vertical distance of 1.5 Mm, the pressure and density would fall off by a factor of e By contrast, in the solar corona where T 0 = K, say, the scale-height is about 100 Mm, and so the density falls off much more slowly. Indeed, for many purposes, in the corona we can neglect the effect of gravity, i.e. when the vertical scales of interest are 100 Mm or less. Dirac House, Temple Back, Bristol, BS1 6BE, UK 7

8 Potential fields If the pressure gradient and gravitational force are negligible, (16) holds and a particular case of interest is when the current vanishes, so that B = 0 (17) where B = 0. (18) Equation (17) may be satisfied identically by putting B = ψ and then (18) gives Laplace s equation 2 ψ = 0. Thus, many of the general results of potential theory may be applied and the magnetic field is said to be current-free or potential. Magnetic reconnection MAGNETIC RECONNECTION is a fundamental process in a plasma, whereby the magnetic connectivity of plasma elements changes and magnetic energy is converted into other forms such as heat, kinetic energy and fast-particle energy. It involves a localized breakdown of ideal MHD in a small region due to, for example, resistive effects. It is responsible for many dynamic processes in the universe, such as SOLAR FLARES and GEOMAGNETIC STORMs and probably solar coronal heating (see CORONAL HEATING MECHANISMS). Magnetic helicity For a perfectly conducting plasma in a closed volume (V ), the MAGNETIC HELICITY is K = A B dv V where B = A, and the state of minimum energy for fixed K is a linear force-free field. K is a measure of the sum of the twist and the linkage of flux tubes. Thus, for two linked tubes K = 1 F F LF 1F 2 where 1 and 2 are the twists of the two tubes F 1,F 2 their magnetic fluxes and L the linking number. Reconnection can convert linkage helicity to twist helicity, but it produces only a small change in the total helicity in a plasma of large global magnetic Reynolds number. Magnetohydrodynamic waves and shock waves In a gas there are sound waves which propagate equally in all directions at the sound speed c s = ( γp 0 ρ 0 ) 1/2. In a plasma there are also waves, but they are of several types. Waves are very important in the solar atmosphere and throughout the cosmos. For example, they may be seen propagating out of sunspots or away from large solar flares. They are also a prime candidate for heating the solar atmosphere. In MHD there is an incompressible mode known as the Alfvén wave, which propagates at the Alfvén speed along the magnetic field. Also, there are two compressible modes known as slow and fast magnetoacoustic waves, which propagate slower and faster respectively than the Alfvén speed. These MHD waves have propagation characteristics that depend on the direction of propagation relative to the magnetic field. Small-amplitude sound waves propagate without change of shape, but when the amplitude is finite the crest can move faster than its trough, causing a progressive steepening. Ultimately, the gradients become so large that dissipation becomes important, and a steady shock wave shape may be attained with a balance between the steepening effect of the nonlinear convective term and the broadening effect of dissipation. The dissipation inside the shock front converts the energy being carried by the wave gradually into heat. The effect of the passage of the shock is to compress and heat the gas. In MHD the slow magnetoacoustic wave can steepen to form a slow-mode shock and the fast wave to form a fast-mode shock. Magnetohydrodynamic instabilities Equilibrium magnetic fields in the universe can go sometimes unstable to a wide variety of instabilities. Sometimes the effect is to create fine-scale structure and at other times it is to produce dynamic events such as solar flares and dramatic eruptions from the Sun called SOLAR CORONAL MASS EJECTIONs. The two main ways of analysing stability are by investigating the natural (or normal) modes of variation or by an energy (or variational) method. The instabilities that commonly occur include: the Rayleigh Taylor instability, when heavy fluid is supported above light fluid; kink instability or sausage instability of a magnetic flux tube when its twist is too large; resistive instability such as the tearing-mode instability of a current sheet or a sheared structure; convective instability of a plasma that is heated sufficiently from below; and flow instabilities that occur when the shear in a flow is too great (such as the Kelvin Helmholtz instability and in accretion discs the Balbus Hawley instability). Conclusion It is important to recognize that magnetohydrodynamics builds on the tools of both fluid dynamics and electromagnetism, but it possesses many new features that are present in neither. Furthermore, it is advisable to build both a physical and mathematical understanding of magnetic field behavior, since they both complement one another and together give a deeper understanding than either alone. In the following articles, various aspects and applications of MHD theory are developed in detail. In a companion article, Parker describes the key physical Dirac House, Temple Back, Bristol, BS1 6BE, UK 8

9 processes involved in LARGE-SCALE PLASMA DYNAMICS. Then basic DYNAMO THEORY for the generation and maintenance of cosmical magnetic fields is developed by Moffatt and in particular for the GEODYNAMO by Proctor and for solar and stellar dynamos by Rosner. Then several important aspects of MHD theory are described, including FORCE- FREE MAGNETIC FIELDS (Low), MAGNETIC HELICITY (Berger) and MHD WAVES (Roberts). The theory for MHD INSTABILITIES is set up by Hood and applied in particular to MAGNETIC BUOYANCY by Hughes and to MHD: MAGNETOCONVECTION by Cattaneo. The key process of MAGNETIC RECONNECTION is reviewed by Schindler and Hörnig and its role in MHD: MAGNETIC RECONNECTION AND TURBULENCE is described by Matthaeus. Finally, two astrophysical environments where MHD plays a crucial role are discussed, namely the MHD OF ACCRETION DISKS by Brandenburg and the MHD OF ASTROPHYSICAL WINDS by Heyvaerts. Acknowledgments We are most grateful to colleagues in St Andrews and Durham and to Jean Heyvaerts and Karl Schindler for helpful comments. Bibliography Cowling T G 1957 Magnetohydrodynamics (New York: Interscience) Priest E R 1982 Solar Magnetohydrodynamics (Dordrecht: Reidel) Priest E R 1994 Magnetohydrodynamics Plasma Astrophysics ed J G Kirk, D B Melrose and E R Priest (Berlin: Springer) pp Priest E R and Forbes T G 2000 Magnetic Reconnection: MHD Theory and Applications (Cambridge: Cambridge University Press) Roberts P H 1967 An Introduction to Magnetohydrodynamics (London: Longman) Eric R Priest and Terry G Forbes Dirac House, Temple Back, Bristol, BS1 6BE, UK 9