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List of chapters and authors. Updated 11 January 2003

TITLE OF CHAPTER	AUTHOR(s)
1.  The field, the mean and the meaning	Peter Hoyng [Utrecht]   p.hoyng [AT] sron.nl
2.  Fast dynamos	David Galloway [Sydney]    dave [AT] maths.usyd.edu.au
3.  On the theory of convection in the Earth's core	Stanislav Braginsky [Los Angeles]  Paul Roberts [Los Angeles]    roberts [AT] math.ucla.edu
4.  Dynamo action of magnetostrophic waves	Dieter Schmitt [Lindau]   schmitt [AT] linmpi.mpg.de
5.  Magnetic flux tubes and the dynamo problem	Manfred Schüssler [Lindau]   msch [AT] linmpi.mpg.de   Antonio Ferriz-Mas[Oulu]   antonio.ferriz [AT] oulu.fi
6.  Physics of the solar cycle	Günther Rüdiger [Potsdam]   gruediger [AT] aip.de   Rainer Arlt [Potsdam]   rarlt [AT] aip.de
7.  Highly supercritical convection in strong magnetic fields	Keith Julien [Boulder]  julien [AT] colorado.edu    Edgar Knobloch [Berkeley]    knobloch [AT] physics.Berkeley.EDU   Steve Tobias [Leeds]   smt11 [AT] damtp.cam.ac.uk
8.  Thin aspect ratio alpha-omega-dynamos in galactic discs and stellar shells	Andrew Soward [Exeter]   soward [AT] maths.exeter.ac.uk
9.  Computational aspects of astrophysical MHD and turbulence	Axel Brandenburg [Copenhagen]   brandenb [AT] nordita.dk
10.  Topological quantities in magnetohydrodynamics	Mitch Berger [London]    m.berger [AT] ucl.ac.uk

This article is about the fundamentals of mean field theory, with an emphasis on the statistical aspects of the theory. The equations for passive vectorial transport (dynamo equation) and passive scalar transport are derived on a par, as applications of the theory of stochastic equations with multiplicative noise. Only approximate transport equations for mean quantities exist, and the approximations are scrutinized in relation to the ensemble average and the azimuthal average. A summary of the elementary physics of the dynamo equation is followed by an analysis of the influence of mean shear flows and resistive effects on the transport coefficients $\alpha$ and $\beta$. It is argued that the dynamo equation contains in practical situtions no higher than second order spatial derivatives. The physics of turbulent diffusion and the information content of the dynamo equation is analysed, and it is concluded that transport equations for the mean can only make probabilistic statements about the physical systems to which they apply. They cannot predict the evolution for all times. This is elucidated for ensemble and azimuthal averages at the hand of examples, referring to the phase memory and magnetic energy losses of the solar dynamo, and to the variability and reversals of the geodynamo. The accent is on clarity of presentation, and on linear theory. A few remarks on nonlinear theory are made.

A dynamo is termed fast if it grows on the turnover timescale of the flow rather than on the ohmic diffusion timescale. In astrophysics the latter is often much too long to allow significant build-up of magnetic fields during the lifetime of the Universe, and for this reason fast dynamos are sought to explain the generation of fields in stars and larger-scale objects.

This chapter reviews the topic of fast dynamos, mainly from a numerical point of view (this emphasis reflects the author's own background, and also the existence of an excellent book by Childress and Gilbert dealing mainly with analytic aspects). In the first section, the history and motivation of the search for fast dynamos are established, and the necessity for the flow to be chaotic (in the sense of dynamical systems theory) is explained. The second section reviews the various numerical experiments which have been mounted to provide evidence that fast dynamos actually exist, and describes the physical processes whereby the field is generated. The third, short section mentions some of the fast dynamos for which analytic solutions have been found: these flows ``cheat'' the chaos by localising it in artificial features outside which piecewise integration is possible, often using asymptotic methods.

The final section addresses what happens when fast dynamos enter the nonlinear regime and the Lorentz force is allowed to limit the growth of the field strength by reacting back on the motion. A number of calculations are described and for some of these it is possible to advance scaling laws to determine the ultimate strength of the fields. So far these examples all suggest that the effectiveness of the dynamo goes down as the (kinetic) Reynolds number of the flow goes up. The difficulties this poses for astrophysical dynamos are confronted but remain largely unsolved, even at high magnetic Reynolds number.

The stability of magnetostrophic waves is considered in a horizontal thin plane layer of a perfectly conducting fluid, which is stratified according to gravitation and permeated by a variable toroidal magnetic field. The layer rotates rigidly around an axis inclined to the horizontal plane. It is shown that unstable magnetostrophic waves, driven by magnetic buoyancy, are capable of inducing an electromotive force parallel to the toroidal magnetic field. This dynamic effect is an alternative to the kinematic $\al$-effect and of importance for the dynamo theory of strong magnetic fields. For parameters of the lower convection zone of the sun the process leads to an effective $\al$ of a few cm/s. Especially the dynamo action for various angles of inclination is discussed.

The observed properties of the magnetic field in the solar photosphere and theoretical studies of magneto-convection in electrically well-conducting fluids suggest that the magnetic field in stellar convection zones is quite inhomogeneous: magnetic flux is concentrated into magnetic flux tubes embedded in significantly less magnetized plasma. Such a state of the magnetic field potentially has strong implications for stellar dynamo theory since the dynamics of an ensemble of flux tubes is rather different from that of a more uniform field and new phenomena like magnetic buoyancy appear.

If the diameter of a magnetic flux tube is much smaller than any other relevant length scale, the MHD equations governing its evolution can be considerably simplified in terms of the thin-flux-tube approximation. Studies of thin flux tubes in comparison with observed properties of sunspot groups have led to far-reaching conclusions about the nature of the dynamo-generated magnetic field in the solar interior. The storage of magnetic flux for periods comparable to the amplification time of the dynamo requires the compensation of magnetic buoyancy by a stably stratified medium, a situation realized in a layer of overshooting convection at the bottom of the convection zone. Flux tubes stored in mechanical force equilibrium in this layer become unstable with respect to an undular instability once a critical field strength is exceeded, flux loops rise through the convection zone and erupt as bipolar magnetic regions at the surface. For parameter values relevant for the solar case, the critical field strength is of the order of $10^5\,$G. A field of similar strength is also required to prevent the rising unstable flux loops from being strongly deflected poleward by the action of the Coriolis force and also from `exploding' in the middle of the convection zone. The latter process is caused by the superadiabatic stratification.

The magnetic energy density of a field of $10^5\,$G is two orders of magnitude larger than the kinetic energy density of the convective motions in the lower solar convection zone. This raises serious doubts whether the conventional turbulent dynamo process based upon cyclonic convection can work on the basis of such a strong field. Moreover, it is unclear whether solar differential rotation is capable of generating a toroidal magnetic field of $10^5\,$G; it is conceivable that thermal processes like an entropy-driven outflow from exploded flux tubes leads to the large field strength required.

The instability of magnetic flux tubes stored in the overshoot region suggests an alternative dynamo mechanism based upon growing helical waves propagating along the tubes. Since this process operates only for field strengths exceeding a critical value, such a dynamo can fall into a `grand minimum' once the field strength is globally driven below this value, for instance by magnetic flux pumped at random from the convection zone into the dynamo region in the overshoot layer. The same process may act as a (re-)starter of the dynamo operation. Other non-conventional dynamo mechanisms based upon the dynamics of magnetic flux tubes are also conceivable.

The theory of the solar/stellar activity cycles is presented, based on the mean-field concept in magnetohydrodynamics. A new approach to the formulation of the electromotive force as well as the theory of differential rotation and meridional circulation is described for use in dynamo theory. Activity cycles of dynamos in the overshoot layer (BL-dynamo) and distributed dynamos are compared, with the latter including the influence of meridional flow. The overshoot layer dynamo is able to reproduce the solar cycle periods and the butterfly diagram only if $\alpha=0$ in the convection zone. The problems of too many magnetic belts and too short cycle times emerge if the overshoot layer is too thin. The distributed dynamo including meridional flows with a magnetic Reynolds number ${\rm Rm}\gsim 20$ (low magnetic Prandtl number) reproduces the observed butterfly diagram even with a positive dynamo-$\alpha$ in the bulk of the convection zone.

The nonlinear feedback of strong magnetic fields on differential rotation in the mean-field conservation law of angular momentum leads to grand minima in the cyclic activity similar to those observed. The 2D model described here contains the large-scale interactions as well as the small-scale feedback of magnetic fields on differential rotation and induction in terms of a mean-field formulation ($\Lambda$-quenching, $\alpha$-quenching). Grand minima may also occur if a dynamo occasionally falls below its critical eigenvalue. We expressed this idea by an on-off $\alpha$ function which only exists non-zero in a certain range of magnetic fields near the equipartition value. We never found any indication that the dynamo collapses by this effect after it had once been excited.

The full quenching of turbulence by strong magnetic fields in terms of reduced induction ($\alpha$) and reduced turbulent diffusivity ($\eta_{\rm T}$) is studied with a 1D model. Its cycle period depends much more on its dynamo number than it results with the simpler $\alpha$-quenching model -- which hardly gives a significant relation.

Also the temporal fluctuations of $\alpha$ and $\eta_{\rm T}$ from a random-vortex simulation were applied to a dynamo model. Then the low `quality' of the solar cycle can be explained with a relatively small number of giant cells acting as the dynamo-active turbulence. The simulation contains the transition from almost regular magnetic oscillations (many vortices) to a more or less chaotic time series (very few vortices).

When an astrophysical dynamo is confined to a region with small aspect ratio $\epsilon$, asymptotic approximations may lead to analytic solutions.

In the case of galactic discs, for which $\epsilon$ is the ratio of the disc width $2H_0$ to its diameter $2L$, analytic progress is possible when both the $\alpha$ and $\Omega$-effects vary on the long radial length scale $L$. As a prototype example, we will consider Stix's (1975,1978) oblate spheroid, which is characterised by a local dynamo number ${\cal D}(s)$ that increases from zero at the symmetry axis $s=0$ to a maximum (extremum) $D$ at some radius $s=s_E$ and decreases to zero at the disc's edge $s=L$. Steady dynamo modes have a radial length scale large compared to $H_0$ but short compared to $L$. Consequently, the effect of radial-$s$ diffusion inside the disc is negligible when compared to the potential field coupling outside. Both the steady quadrupole and `forgotten' dipole modes lead to integral equations, which may be solved by Fourier transform methods. The oscillatory quadrupole mode, which has a radial scale comparable to the disc height, is resolved by WKBJ methods, which highlight the importance of global stability criteria, namely the vanishing of both the group velocity and phase mixing.

In the case of stellar shells, $\epsilon$ is the ratio of the shell thickness to radius. Here the radial structure across the disc is averaged leading to dependence on the latitude $\theta$ alone. The resulting one-dimensional $\alpha\Omega$-dynamo is characterised by a local dynamo number ${\cal D}(\theta)$ that increases from zero at the Equator $\theta=0$ to a maximum $D$ at some latitude $\theta=\theta_E$ and decreases to zero at the North Pole $\theta=\pi/2$. The ensuing latitudinally modulated kinematic dynamo wave is again resolved by WKBJ methods. In the case of quenching, its weakly nonlinear development signals a complicated and almost spontaneous (in terms of the maximum dynamo number $D$) transition to full nonlinearity. That state is characterised by a localised finite amplitude Parker wave, which emerges smoothly at a low latitude and terminates abruptly across a front at a high latitude. At lowest order the frequency and front location are fixed by the Dee \& Langer (1983) criterion that the group velocity vanishes ahead of the front. We remark on other recent developments.

Astrophysical MHD and turbulence simulations: The advantages of high-order finite difference scheme for astrophysical MHD and turbulence simulations are highlighted. A number of one-dimensional test cases are presented ranging from various shock tests to Parker-type wind solutions. Applications to magnetised accretion discs and their associated outflows are discussed. Particular emphasis is placed on the possibility of dynamo action in three-dimensional turbulent convection and shear flows, relevant to stars and discs, are considered. The generation of large scale fields is discussed in terms of an inverse magnetic cascade.

Increasingly, observations, experiments, and numerical simulations reveal a rich non-linear world full of complex structures. Geometric and topological techniques can provide valuable assistance in making sense of these structures. This chapter describes crossing numbers and helicity integrals, which measure different aspects of topological complexity in vector fields. These measures can be applied to vortex structures in fluid mechanics and to magnetic field line structures in magnetized fluids. We also discuss in detail astrophysical applications concerning magnetic fields in the sun and solar atmosphere.

Helicity integrals measure the linking of field lines, and are pseudo-scalar under spatial reflection. They are conserved for vector fields dragged by fluid motions. Crossing numbers, on the other hand, do not measure the sign of linkings. They are not conserved, but have a positive lower bound for a given field topology which provides a good measure of complexity. Simple relations exist between crossing numbers and magnetic energy. If crossing numbers can be predicted for a field, then the energy stored in that field can also be predicted. Crossing numbers will be described for both magnetic fields inside a sphere, and braided fields connecting two parallel planes.

To illustrate these ideas, we apply crossing numbers and braid theory to the problem of solar coronal heating and the source of coronal microflares. Observations of structures such as x-ray loops in the solar atmosphere have gained in resolution over the past few decades, revealing more and more small scale structure. This small scale structure originates in the turbulent motions below the surface. Simple models predict how much structure (as measured by crossing numbers) is generated by the turbulence, and hence how much energy is available for heating and flaring. These models provide sufficient energy to heat the corona.

Helicity integrals quantify vector field features such as shear, twist, and braiding. In ideal magnetized plasmas, helicity is absolutely conserved. Even with resistivity the decay of helicity obeys rather severe bounds. No other knot-like invariants of magnetic field lines possess similar bounds. Approximate helicity conservation implies that reconnecting flux tubes become twisted. This result has relevance to the observation of twisted field structures in solar and space physics. Recently space scientists have become highly interested in helicity: observations show that active regions on the sun, and magnetic clouds in the solar wind, have preferential signs of magnetic helicity. The solar dynamo is probably responsible for this asymmetry. We discuss the contributions to helicity generation from both the $\alpha$ effect and from differential rotation.

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Last modified: 11 January 2003

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