Solar Photospheric Magnetic Flux Tubes: Theory E NCYCLOPEDIA of a STRONOMY and a STROPHYSICS

Solar Photospheric Magnetic Flux Tubes: Theory E NCYCLOPEDIA OF A STRONOMY AND A STROPHYSICS

Solar Photospheric Magnetic Flux balance, and unless there are substantial temperature differences between the tube and its environment then Tubes: Theory this reduction in plasma pressure is accompanied by a The magnetic field in the photospheric layers of the reduction in plasma density. In other words, the magnetic Sun is found to occur not in a homogeneous form field of a photospheric flux tube partially evaculates but in discrete concentrations of intense field. The the plasma. This is in contrast to the coronal flux most obvious form of magnetic flux (magnetic field tube which is generally a region of plasma enhancement strength times surface area occupied by the field) is and reduced Alfvén speed, achieved without violation seen in SUNSPOTS but it turns out that much smaller of pressure balance because the strong and pervasive arrangements of magnetic field are to be found in the lanes magnetic field of the solar corona can easily accommodate between granules where downdraughts occur (see SOLAR any enhancements or reductions in the plasma pressure. PHOTOSPHERE: GRANULATION). These are the photospheric flux The thermal structure of a photospheric flux tube tubes: small-scale concentrations of intense magnetic field. is complicated. The tube may exchange heat with its The tubes have diameters of a few hundred kilometers surroundings, through radiative transport and convective or smaller and field strengths of 1–2 kG. Generally found energy transport. Detailed numerical studies suggest that in intergranular lanes, they are subject to the dynamical photospheric magnetic flux tubes are hotter than their nature of the photospheric environment with its sound surroundings in their upper layers but cooler than their waves and flows. The tubes are isolated flux tubes in that surroundings in their deeper layers. c c their immediate surroundings are essentially field-free. The speeds t and k of a photospheric flux tube See SOLAR PHOTOSPHERIC MAGNETIC FLUX TUBES: OBSERVATIONS. are sub-Alfvénic, and the slow speed is also sub-sonic; The small-scale tubes of the photosphere may be whether the kink speed is sub-sonic depends on the thought of as the elemental building blocks of solar specific parameters of the tube. A numerical illustration B = magnetism, so an understanding of intense flux tubes has is of interest. In a photospheric flux tube with 0 2kG ρ = . × −4 −3 implications for a wider understanding of solar and stellar and plasma density 0 2 2 10 kg m , the Alfvén v = −1 c magnetic phenomena. Theoretical progress in modelling speed is A 12 km s . Taking the sound speed s of −1 small-scale flux tubes builds on the fact that in such tubes the photosphere to be 8 km s then gives a slow speed c = . −1 variations across the interior of a tube are less important of t 6 7kms . For a flux tube that has depleted its than variations along the tube. plasma density to some 50% of its environment (so that ρ = ρ / The elemental photospheric flux tube is taken to 0 e 2, consistent with an environment sound speed −1 c = . −1 be a cylinder of magnetic flux embedded in a field-free of 9.6 km s ), we obtain k 6 9kms . Thus the slow c environment; the photospheric flux tube is thus a region speed t is some 56% of the Alfvén speed and the kink c of locally high Alfvén speed. Two basic speeds prove speed k is about 57% of the Alfvén speed. The precise central to an understanding of tube dynamics: the slow ordering of the two speeds depends on the details of the tube speed c and the kink speed c . These speeds are modelling. t k c c in turn defined in terms of the sound speed and Alfvén The speeds t and k are associated with the dynamics B of flux tubes, arising in both wave phenomena and flows speed. For a tube of magnetic field strength 0 with plasma ρ p c in tubes. The speed c , made up in equal measure of density 0 and pressure 0, the sound speed s and Alfvén t speed vA of the tube are defined by the sound speed and the Alfvén speed, is evidentally associated with the compressibility of the tube. Any / γp 1/2 B2 1 2 squeezing of a tube results in increases in the magnetic c = 0 v = 0 s ρ A µρ (1) pressure and plasma pressure in the tube, which act to 0 0 restore the undisturbed state. This is a feature shared by where γ (generally taken to be 5/3) is the ratio of specific all elastic tubes and consequently the equivalent of the c heats of the plasma and µ is its magnetic permeability. speed t arises in a variety of other physical situations, c v From the speeds s and A we may construct the slow with the role of the Alfvén speed being played by the c c magnetoacoustic speed t and the kink speed k of a tube: appropriate elastic speed of an elastic tube. For example, v in the case of a blood vessel the equivalent of the speed A / c v ρ 1 2 is simply the elastic speed in the membrane of the blood c = s A c = 0 v t k A (2) c (c2 v2 )1/2 ρ ρ vessel. The speed of sound s in blood is much larger s + A 0 + e than the elastic speed and so the slow speed is close to ρ here e denotes the plasma density of the field-free the elastic speed of the blood vessel. For water in a pipe, environment of the tube. The slow tube speed and the the relative magnitudes of the two basic speeds depend kink speed are important in coronal flux tubes too (see on the material of the pipe. In a metal pipe, the elastic MAGNETOHYDRODYNAMIC WAVES). speed is much larger than the sound speed in water and A photospheric flux tube is typically a region of so the effective slow speed is close to the sound speed in plasma density depletion: the presence of the magnetic water (about 1.4 km s−1). By contrast, in a plastic pipe the field leads to a plasma pressure reduction inside the tube, orderings in the two speeds are reversed and the effective in keeping with the requirement of transverse pressure propagation speed is close to the elastic speed in plastic

(about 10 m s−1), lying far below the speed of sound in Waves in photospheric flux tubes water. To give a detailed description of the magnetohydrodyan- The kink speed involves both the tube and its mic waves that an isolated photospheric flux tube sup- environment. The kink wave which propagates with this ports we consider the equations of ideal MHD. Consider B = B (r)zˆ speed (see below) involves the magnetic tension force an equilibrium magnetic field 0 0 aligned with z (r,θ,z) in the tube and the kink wave displaces both the tube the -axis of a cylindrical polar coordinate system . and its surroundings by about equal measures, hence the To begin with we will ignore the effects of gravity. Then p (r) ρ ρ the equilibrium plasma pressure 0 of a radially struc- combined occurrence of the densities 0 and e. The speed tured magnetic atmosphere is such as to maintain total also arises in the description of magnetoacoustic surface p (r) waves on an interface between two plasmas of different pressure balance: the sum of the plasma pressure 0 and the magnetic pressure B2(r)/2µ is a constant. Where properties. 0 the field is strong, as in the center of a flux tube, the plasma An important question in the physics of photospheric pressure is correspondingly reduced. flux tubes is how are they formed. A flux tube Small-amplitude motions v = (vr ,vθ ,vz) about a with kilogauss field strength requires strong forces structured equilibrium state satisfy the coupled wave to bring about its intense state, with the magnetic equations B2/ µ pressure 0 2 being far in excess of the pressure ∂2 ∂2 ∂2p of converging flows that granulation may provide; 2 T ρ (r) − v (r) vr + = 0 (3) the tube finds equilibrium through evacuation of its 0 ∂t2 A ∂z2 ∂r∂t interior, reducing the plasma pressure and thus allowing ∂2 ∂2 ∂2p magnetostatic pressure balance between the flux tube 2 1 T ρ (r) − v (r) vθ + = 0 (4) and its environment to occur. It is envisaged that 0 ∂t2 A ∂z2 r ∂θ∂t granulation acts to ‘shepherd’ flux tubes into the regions ∂2 ∂2 c2(r) ∂2p 2 s 1 T where downdraughts between the granules occur. The − c (r) vz =− (5) ∂t2 t ∂z2 c2(r) v2 (r) ρ (r) ∂z∂t process is an example of flux expulsion (by advection), s + A 0 p (r,θ,z) whereby the convective flows of the photosphere move with the evolution of T described by magnetic field into intergranular lanes, in keeping with ∂p ∂vz the frozen flux concept of ideal MAGNETOHYDRODYNAMICS. T = ρ (r)v2 (r) − ρ (r) c2(r) v2 (r) v. 0 A 0 [ s + A ]div (6) The process is not of itself sufficient to account for the ∂t ∂z observed field strengths; it brings about the ‘seedlings’ c (r) c (r) v (r) Here t , s and A denote the slow speed, sound of photospheric flux tubes but other effects are needed speed and Alfvén speed within the radially structured to produce the observationally determined intense field ρ (r) plasma, of density 0 . Equations (3)–(5) come from strengths. The additional process that comes into play the components of the momentum equation in which is driven by the super-adiabatic temperature gradient of the magnetic force has been expressed in terms of the the solar convection zone (see SOLAR INTERIOR: CONVECTION p perturbation in total pressure T, defined as the sum of ZONE), which in field-free regions drives the convective the perturbation p in the dynamical plasma pressure and SOLAR PHOTOSPHERE: B (r)B /µ B flows (granules and supergranules; see the magnetic pressure perturbation 0 z , with z SUPERGRANULATION) that distinguish the Sun’s convection. being the component of the perturbed magnetic field in B In flux tubes, convectively driven flows are channelled by the direction of the applied magnetic field 0. Equation the tubes. The fact that photospheric flux tubes reside (6) results from a combination of the isentropic (adiabatic) in a plasma that is stratified by gravity means that any equation and the induction equation of ideal MHD. downdraught generated within a flux tube brings less One solution of the above equations is the torsional dense plasma from the higher reaches of the stratified tube Alfvén wave, which propagates motions vθ , independent to the deeper layers within the tube, allowing the high of θ, according to the wave equation plasma pressure of the environment to squeeze the tube ∂2v ∂2v further. An increase in magnetic field strength results. The θ = v2 (r) θ . A (7) process is referred to as convective collapse. It may be ∂t2 ∂z2 described in mathematical detail for a thin flux tube (see v = v = p = Torsional oscillations, which have r 0, z 0 and T below). 0, have a radial dependence determined by the manner A flow in a magnetic flux tube may also develop in which the oscillations are generated. The oscillations because of pressure differences between the footpoints of exhibit phase mixing by which radial gradients of the the tube. For example, if the footpoints of a tube are rooted motion grow rapidly in regions where the Alfvén speed v (r) r in regions that are magnetically distinct from one another, A varies sharply in . such as one footpoint in a sunspot and the other in a field- In addition to torsional Alfvén waves there are p = p = free region, then plasma pressure differences arise and a motions that are compressive, with 0 and T 0. flow may be driven in the tube. Such motions of the plasma We examine the elemental magnetic flux tube that models within the tube are called siphon flows. photospheric conditions, taking the applied magnetic field

B (r)zˆ 0 to be confined within a cylindrical tube of radius where a prime ( ) denotes the derivative of a modified a p I (m a) ≡ I (x)/ x . The plasma pressure and density within the tube are 0 Bessel function (e.g. n 0 d n d evaluated at ρ x = m a and 0; the corresponding plasma pressure and density in 0 , etc). This is the dispersion relation for tube p ρ the tube’s field-free environment are e and e. The tube is waves in an isolated magnetic flux tube, requiring that m > m2 > in magnetostatic pressure balance with its surroundings: e 0; surface waves arise whenever 0 0, and body p = p B2/ µ c v 2 e 0 + 0 2 . With s and A now denoting the waves occur if m < 0. It is evident that tube waves are c 0 sound speed and Alfvén speed within the tube and e the dispersive, their phase speed c(≡ ω/kz) depending on the sound speed in the tube’s environment, pressure balance wavenumber (in the combination kza). combined with the ideal gas law then shows that the Solution of the dispersion relation (10), for conditions ρ plasma density 0 within the tube is related to the density appropriate for an isolated photospheric flux tube, show ρ e in the tube’s environment through that there are slow body waves (both sausage and kink) c c c ρ c2 with phase speeds that lie between t and s and slow 0 = e . (8) surface waves which have phase speeds that are less than ρ 2 1 2 e c + γv c s 2 A t. Also, there is a surface wave with a phase speed close c c c to the kink speed k and another surface wave with phase Thus, for sound speeds s and e that are roughly c c comparable, a photospheric flux tube is a region of density speed near e. Fluting modes also have speeds close to k. depletion (i.e. ρ <ρ). There is particular interest in waves that are much 0 e k a Now the media inside and outside the elemental longer than a tube diameter, corresponding to z 1. flux tube are uniform, and this affords a considerable This is the thin flux tube limit, of special interest for ω ∼ k v simplification. In a uniform medium, the wave equations photospheric flux tubes. Since generally z A, the k a (3)–(6) may be combined to yield an equation for p . condition z 1 corresponds to requiring that periods T τ(≡ π/ω) τ πa/v τ Writing 2 are such that 2 A, giving 50 s for a tube of radius a = 100 km. P-modes, for example, have p (r,θ,z,t)= p (r) (ωt nθ − k z) T T exp i + z periods around 300 s, roughly meeting this criterion. With kza 1, the slow surface wave has phase speed close to provides a Fourier description of the motions, of frequency c . ω k t , longitudinal wavenumber z and azimuthal number Dispersion relations such as equation (10) are n. The integer n(= 0, 1, 2,...) describes the geometrical n = important in the development of a nonlinear wave theory form of the perturbations. The case 0 gives the for a tube. Indeed, it has been shown that slow (sausage) sausage wave and corresponds to symmetric squeezings surface waves have motions v(z,t) along a thin tube which and rarefactions of the tube; these are compressional (p = T satisfy the nonlinear integrodifferential equation 0) oscillations. The case n = 1 describes the kink wave of the tube; in such waves the motion of the tube resembles ∞ ∂v ∂v ∂v ∂3 v(s,t)ds a wriggling snake, with the whole tube being displaced + c + β v + α = 0. ∂t t ∂z 0 ∂z 0 ∂z3 λ2a2 (z − s)2 1/2 without changing its cross-sectional shape. Finally, there −∞ [ + ] (11) are also waves with n ≥ 2; these are the fluting waves. The constants α , β and λ depend on the various For a uniform tube, p satisfies 0 0 T parameters of the tube. d2p dp Equation (11) is sometimes referred to as the r2 T + r T − (m2r2 + n2)p = 0 (9) dr2 dr 0 T Leibovich–Roberts equation. An explicit form of its solution is not known although it is amenable to numerical where 2 2 2 2 2 2 solution. It is of particular interest because it is associated (kz c − ω )(kz v − ω ) m2 = s A . with soliton behavior. Solitons are nonlinear waves with 0 (c2 v2 )(k2c2 − ω2) s + A z t specific interaction properties that make them worthy of p = This is a form of Bessel’s equation with solution T special study. For the related problem of a magnetic slab— I (m r) ra) describing the weakly nonlinear slow surface wave is the c of the flux tube, where the sound speed is e, a solution Benjamin–Ono equation, i.e. p K (m r) T proportional to the modified Bessel function n e ∂v ∂v ∂v ∂2 ∞ v(s,t) arises, where c β v α s = ω2 + t + 0 + 1 d 0 (12) 2 2 ∂t ∂z ∂z ∂z2 −∞ s − z m = kz − . e c2 e α m2 for constant 1. The Benjamin–Ono equation has been It is usual to examine modes for which e is positive, corresponding to selecting waves that are confined to near studied extensively and its soliton solution is known p v r = a explicitly. Equations (11) and (12) may be derived the tube. Then, matching T and r across results in using thin flux tube theory (see below) or from the I (m a) K (m a) 1 m n 0 1 m n e = full equations of ideal MHD, assuming that motions are 0 + e 0 (10) ρ (k2v2 − ω2) In(m a) ρ ω2 Kn(m a) weakly nonlinear and weakly dispersive. 0 z A 0 e e

Thin flux tube theory z The restriction to thin tubes, in the sense that waves B are much longer than a tube radius (so that kza 1), o leads to a considerable simplification in the description of the dynamics of photospheric flux tubes, so much so that it becomes possible to include a number of other r effects ignored in the discussion of an elemental magnetic flux tube. Of particular importance is the influence of p stratification brought about by gravity. The pressure scale e height in the photosphere is comparable with a tube radius g so the effects of gravity are particularly significant in the photospheric layers of a tube. Other effects, such as Bo radiative transport, may also in principle be included in thin flux tube theory. To describe motions in a thin tube in the presence Figure 1. Equilibrium state of a thin magnetic flux tube in a of gravity the so-called thin flux tube equations are stratified plasma. The tube is confined by the external plasma used. These equations have been used extensively for pressure pe in the field-free environment of the tube. The region both analytical and numerical investigations of flux tube within the tube is generally of lower plasma density than the dynamics. The sausage and kink modes are treated environment. separately. The thin tube equations for the sausage mode (in ideal MHD) are The linear form of the thin tube equations is readily ∂ ∂ found for the equilibrium (18). Ignoring contributions ρA ρvA = from variations in p , longitudinal motions are found to ∂t + ∂z 0 (13) e satisfy the Klein–Gordon equation ∂v ∂v 1 ∂p ∂2Q ∂2Q + v =− − g (14) − c2(z) + '2(z)Q = 0 (19) ∂t ∂z ρ ∂z ∂t2 t ∂z2 S ∂p ∂p γp ∂ρ ∂ρ where Q(z, t) is related to the flow v(z,t) and '2(z) + v = + v (15) S ∂t ∂z ρ ∂t ∂z depends on the details of the equilibrium state. % c '2 BA = In an isothermal atmosphere, 0, t and S are all constant (16) constants, with B2 p = p . c2 9 2 4c2 1 + e (17) '2 = t − + s 1 − . (20) 2µ S %2 γ γv2 γ 4 0 4 A In these equations, B(z,t) is the field strength of a A(z, t) v(z,t) The Klein–Gordon equation (19) then yields the dispersion thin tube with cross-sectional area , and is relation the longitudinal flow speed within the tube, where the plasma pressure and density are p(z, t) and ρ(z,t). The 2 2 2 2 ω = kz c + ' (21) gravitational acceleration is g(=274 m s−2), acting in the t S ' negative z-direction. The external gas pressure p (z, t) which shows that S is a cutoff frequency (for sausage e '2 is calculated on the boundary of the tube and may vary waves in a thin tube). S may be viewed as made up of in response to motions in the tube or its environment; two contributions, the first (corresponding to the first term on the right-hand side of equation (20)) arising from the pressure balance across the tube is maintained at all times geometrical shape of the undisturbed tube and the second (as expressed by equation (17)). (corresponding to the second term on the right of equation In equilibrium (v = 0, ∂/∂t = 0) the thin tube (20)) being determined by the tube’s elasticity. A rigid equations for a magnetic flux tube in temperature balance c = c tube with exponential cross-sectional area (determined with its surroundings (so that s e) yield % according to the equilibrium (18) with 0 constant) has ( / − /γ )1/2c / % % ( ) cutoff frequency 9 4 2 s 2 0, whereas a straight p (z) = p ( ) −N ρ (z) = ρ ( ) 0 0 −N c / % 0 0 0 e 0 0 0 % (z) e and vertical rigid tube has cutoff frequency s 2 0.In 0 general the cutoff frequency for the sausage mode in a N/ −N/ tube is less than the cutoff frequency of a rigid tube. A (z) = A (0) e 2 B (z) = B (0) e 2 (18) 0 0 0 0 The Klein–Gordon equation also describes the kink where mode in a thin tube. Thin tube equations for the kink mode z z N(z) = d may be used to describe the transverse displacements % (z) ξ(z,t) 0 0 of a vertical tube, leading to the wave equation % (z)(≡p (z)gρ (z)) 2 2 is the integrated pressure scale height; 0 0 0 ∂ ξ ∂ ξ ρ − ρ ∂ξ = c2 (z) + g 0 e . (22) is the pressure scale height inside the tube. See figure 1. ∂t2 k ∂z2 ρ ρ ∂z 0 + e

Equation (22) may be cast into an equation of the Klein– With the illustrative boundary conditions that u = 0at Gordon form, just as for the sausage mode. For an levels z = 0 and z =−d, we obtain the approximate isothermal medium, the dispersion relation for the kink solution z πz mode is u(z) = u exp sin (26) ω2 = k2c2 '2 0 % d z k + K (23) 4 0 where the cutoff frequency for the kink mode is with c π 2 γ ' = k . 2 2 1 1 2 K (24) ω = c + + + ω . 4% t d2 %2 v2 c2 g 0 16 0 A 2 s

The forms (19)–(24) allow a direct comparison ω2 < ω2 < Hence, if g 0 then a mode with 0 may arise, between sausage and kink modes. For example, with corresponding to an instability. The instability amounts to again v = 12 km s−1, c = 8kms−1 and % = 140 km, A s 0 a flow of plasma within the stratified tube; a downdraught sausage and kink modes both grow in amplitude by a brings the more tenuous plasma in the higher levels of the factor of e in propagating a distance of 560 km (four scale tube to layers deeper within the tube, allowing the external heights). The sausage wave, propagating with a speed plasma pressure to squeeze the tube. A stronger field c = 6.7kms−1, has a cyclic cutoff frequency, ' /2π,of t S results. This is convective collapse. The process is halted 4.2 mHz (period 240 s). The kink wave has a speed of when the field strength B is sufficiently large to resist c = 6.9kms−1, close to the sausage mode’s, but its cutoff 0 k the compression of the tube, and so quench the instability frequency at 2 mHz (period 500 s) is very different. The (reducing ω2 to zero). A flux tube with only moderate field sausage mode’s behavior is similar to that of a vertically strength finds itself subject to the instability until it has propagating sound wave, which has a propagation speed − strengthened itself, through convective collapse, to a state of8kms 1 and cutoff frequency of 4.5 mHz (period 220 s), where it may resist the process. Consequently, intense and although it e-folds in just 280 km (two scale heights); isolated magnetic flux tubes are the expected norm in any consequently, sound waves are expected to form shocks medium such as the solar photosphere where convective lower in the atmosphere than tube waves. The fact that effects operate in the presence of magnetic field. the cutoff frequency of the kink mode is considerably A determination of the field strength necessary to smaller than the cutoff frequency of the sausage wave or a sound wave suggests that kink waves may survive to quench the instability depends on the precise details of higher levels in the solar atmosphere than sausage waves the modelling. Studies of the process under conditions or sound waves. appropriate for the solar convection zone show that The dispersion relations (21) and (23) imply that an kilogauss field strengths are required before the collapse impulsively generated wave (sausage or kink) results in a is halted. However, what precisely is the ensuing state c c of a collapsed flux tube remains uncertain, mainly as a wavefront which propagates with the speed t or k, the wavefront supporting an oscillating wake which rises and result of the need to take into account a number of detailed ' ' and complicated effects ignored in the above treatment. In falls with the frequency S or K. particular, it is necessary to include the effects of RADIATIVE Convective collapse TRANSFER, nonlinearity and a more realistic choice of flow Convective collapse, the process by which a photospheric boundary conditions. flux tube may be formed with kilogauss field strengths, may be demonstrated from the thin flux tube equations Numerical simulations for the sausage mode. The differential equation satisfied Numerical simulations allow the inclusion of a variety of by the longitudinal flow, Fourier analyzed by writing effects that complicate the description of the basic modes v(z,t) = u(z) exp(iωt),is of behavior of photospheric flux tubes, and this promotes a more direct comparison with solar observations. By B ρ c2 u c2 γc2 0 d 0 d following the motion of a parcel of magnetized fluid (using t + ω2 − ω2 t + t u = 0 (25) ρ z B z g v2 c2 0 d 0 d A 2 s a so-called Lagrangian approach), it proves possible, for example, to explore the linear and nonlinear sausage and where kink modes in a vertical tube or to simulate the convective g γ − 1 ω2 = % collapse process. These problems are usually addressed g % 0 + γ 0 through numerical solution of thin flux tube equations, is the square of the Brunt–Väisäla (or buoyancy) frequency allowing for the important effects of stratification. Broadly, of a plasma element in a stratified atmosphere. In analytic theory for an ideal plasma has provided valuable the atmosphere of a flux tube embedded within the insights for the more complex modelling of flux tubes ω2 solar convection zone, g is negative and this drives an under conditions representative of the solar photosphere. instability within the tube. The magnetic field may quench Thin flux tube theory for the kink mode has also been the instability if it is sufficiently strong. This is readily applied to modelling the behaviour of a nonvertical flux illustrated by an approximate solution of equation (25). tube, addressing in particular the important question of

Copyright © Nature Publishing Group 2001 Brunel Road, Houndmills, Basingstoke, Hampshire, RG21 6XS, UK Registered No. 785998 and Institute of Physics Publishing 2001 Dirac House, Temple Back, Bristol, BS1 6BE, UK 5 Solar Photospheric Magnetic Flux Tubes: Theory E NCYCLOPEDIA OF A STRONOMY AND A STROPHYSICS how magnetic field stored below the convection zone is transported to the solar surface. An important question still under active considera- tion is how to incorporate the environment’s back-reaction on a moving flux tube. The effect has been explored for the sausage mode and leads to the formation of solitons; the influence of stratification on flux tube solitons has not so far been explored. The back-reaction of the environment on the kink mode is somewhat uncertain, with several al- ternative possibilities suggested in the literature. The behaviour of shocks excited within tubes and also the formation of shocks and downflows as a result of the convective collapse of a tube have been modelled in two-dimensional simulations. Numerical simulations have also permitted realistic estimates of the energy flux carried by the various waves, allowing applications to the Sun and stellar atmospheres more generally to be explored.

Bibliography Hollweg J V 1990 MHD waves on solar magnetic flux tubes Physics of Magnetic Flux Ropes (Geophys. Monogr. 58) ed C T Russell, E R Priest and L C Lee (Washington, DC: AGU) p 23 Parker E N 1979 Cosmical Magnetic Fields (Oxford: Oxford University Press) pp 841 Roberts B 1992 Magnetohydrodynamic waves in structured magnetic fields Sunspots: Theory and Obser- vations ed J H Thomas and N O Weiss (Dordrecht: Kluwer) p 303 Roberts B and Ulmschneider P 1997 Dynamics of flux tubes in the solar atmosphere: theory Solar and Heliospheric Plasma Physics ed G M Simnett, C E Alissandrakis and L Vlahos (Berlin: Springer) p 75 Ryutova M P 1990 Waves and oscillations in magnetic flux tubes Solar Photosphere: Structure, Convection, and Magnetic Fields (IAU Symp. 138) ed J O Stenflo (Dordrecht: Reidel) p 229 Schussler ¨ M 1990 Theoretical aspects of small-scale photospheric magnetic fields Solar Photosphere: Structure, Convection, and Magnetic Fields (IAU Symp. 138) ed J O Stenflo (Dordrecht: Reidel) p 161 Spruit H C 1981 Magnetic flux tubes The Sun as a Star (NASA SP-450) ed S Jordan (Washington, DC: NASA) p 385 Spruit H C and Roberts B 1983 Magnetic flux tubes on the Sun Nature 304 401 Spruit H C, Schussler ¨ M and Solanki S K 1991 Filigree and flux tube physics Solar Interior and Atmosphere ed A N Cox, W C Livingston and M S Matthews (Tucson, AZ: University of Arizona Press) p 890

B Roberts