LIST OF PARTICIPANTS, TITLES OF THE TALKS AND THE ABSTRACTS (where available)

Armbruster, D. Structurally stable heteroclinic cycles and the dynamo dynamics
Alemany, D.
Ashwin, P.
Benevolenskaya, E. Two-component dynamical model of the solar cycle
Bigazzi, A. Dynamo action in 3D homogeneous turbulence
Brachet, M. Are the Taylor-Green vortex and the von Karman swirling flow good candidates for dynamo action?
Brandenburg, A. Simulations of large-scale dynamo action
Brasch, W.
Brestensky, J. Rotating magnetoconvection in dependence on various diffusive processes and boundary conditions
Brooke, J.M. Symmetries of the solar dynamo: comparing theory with observations
Brummel, N.Linear and nonlinear action in simply driven flows.
Burguete, J. Homogeneous dynamo: numerical analysis of experimental von Karman type flows
Busse, F. Convection driven dynamos in spherical shells
Cattaneo, F. Convectively driven dynamos
Chossat, P.
Covas, E. Nonlinear dynamics underlying axisymmetric mean field dynamo models.
Cupal, I. 3D geodynamo in mean field approximation
Daviaud, F. VKS, an experimental fluid dynamo
Dormy, E. Interaction between numerical simulations and asymptotic studies in the Earth's liquid core
Egbers, C. Thermal flow in a rotating spherical gap with dielectrophoretic central force field
Emonet,T. Recent results on convectively driven dynamos
Fauve, S.
Ferriere, K. The galactic dynamo
Forest, C. Physics goals and initial results from the Madison dynamo experiment
Frick, P. On nonstationary dynamo experiment in a breaked torus
Fuchs, H. On self-killing and self-exciting dynamos
Gabov, A.
Gailitis, A., Riga dynamo experiment
Hejda, P. About the numerical method solving 3D geodynamo in mean field approximation
Hollerbach, R. Magnetically induced shear layers and jets
Hughes, D.W. Large and small scale dynamo action
Ivers, D. Dynamo problems in spherical and nearly spherical geometries
Jault, D. Inviscid geodynamo models
Jones, C. Onset of convection and magnetoconvection in rapidly rotating spherical geometry
Junk, M.
Kaiser, R. Mathematical problems related to antidynamo theorems
Kleeorin, N. Magnetic helicity with zero and nonzero mean magnetic field
Knobloch, E.
Kosovichev, A. Helioseismic tests of dynamo models
Kurts,J.Solar activity cycle in phase synchronized with solar inertial motion
Kuzanyan, K. Asymptotic WKBJ-modelling of solar dynamo waves
Lathrop, D.P. Towards a self-generating magnetic dynamo: the role of turbulence
Laure, P. Generation of magnetic field in the Couette-Taylor system
Lauterbach, R. Heteroclinic cycles in systems with spherical symmetries
Leorat, J.
Matthews, P. Dynamo action in rotating convection
Melbourne, I.Magnetic dynamos in nonlinear RMHD
Mueller, U. An experimental demonstration of a homogeneous two scale dynamo
Nore, C. Dynamo action in the Taylor-Green vortex: thresholds and saturation
Odier, P. Advection of a magnetic field by a turbulent swirling flow
Oprea, I.Transition from kinematic to dynamic regime in an intermediate convective dynamo
Otmianowska-Mazur, K. 3D MHD simulations of magnetic dynamos in barred galaxies
Petrelis, F. Simple models for studying the dynamics of fluid dynamos
Pinton, J-F.
Ponty, Y. Dynamo action in flow driven by shear and convection
Proctor, M. Destabilization of transverse modes by noise
Raedler, K-H.
Rieutord, M.$\alpha$-effect produced by inertial modes in a spherical shell
Roberts, P. On the dynamics of the Earth dynamo
Rogachevskii, I. Nonlinear mean field dynamo: electromotive force for anisotropic turbulence and intermediate nonlinearity
Rosaev, A.E. The variation of Earth's magnetic field by tidal force
Ruediger, G. The simplest consistent dynamo: a non-rigidly rotating sphere
Rucklidge, A. Heteroclinic model of geodynamo reversals
Ruzmaikin, A. The solar dynamo: old problems and new challenges
Seehafer N. Dynamos in rotating and nonrotating convection
Silbert, M.
Simkanin, J. Hydromagnetic instabilities in non-uniformly stratified layer in dependence on boundary conditions and Roberts number
Starchenko
Stepanov
Soward, A. Nonlinear $\alpha^2\omega$ waves in stellar shells
Tavakol, R.K. Intermittency and fragility in axisymmetric dynamo models
Tilgner, A. Precession driven flows and dynamos - progress on the Karlsruhe experiment
Tobias, S.
Weiss, N. O.
Von Rekovski, B. Dynamos in accretion disks with vertical winds.

## Onset of convection and magnetoconvection in rapidly rotating spherical geometry , C.A. Jones, A.I. Mussa and A.M. Soward

• The problem of the onset of thermal convection in a rapidly rotating sphere is a classical problem, but the asymptotic solution in the limit of large Taylor number is not completely straightforward. Since the onset of convection occurs close to a critical cylinder aligned with the rotation axis, and of radius approximately one half the radius of the sphere, a {\bf local} theory seems appropriate, and this was developed by Roberts (1968) and Busse (1970). However, to compute the critical Rayleigh number for onset, a global theory is required, which eliminates the possibility of {\it phase-mixing}. The Roberts-Busse local theory still provides the required dispersion relation, but the global criteria requires that phase-mixing must vanish, so that a point in the complex $s$-plane must be isolated at which $\partial \omega / \partial s=0$, where $\omega$ is the frequency derived from the local dispersion relation and $s$ is distance from the rotation axis. The method by which this double-turning point is found, and hence how the critical Rayleigh number is found, will be discussed. The asymptotic theory is compared with full numerical solutions of the partial differential equations at Taylor numbers up to $10^{12}$;excellent agreement is found. The technique can be extended to the case of magnetoconvection, for simple magnetic fields, provided the Elsasser number is of order $O(E^{1/3})$, where $E$ is the Ekman number. The magnetic case does, however, introduce some new features arising from the boundary conditions. Further problems can arise in the spherical gap problem (convection between concentric rotating spheres) when the critical cylinder at which convection onsets is too close to the inner sphere.
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## The simplest consistent dynamo: a non-rigidly rotating sphere, G\"unther R\"udiger

• The global structure of a self-excited magnetic field arising from the magnetic shear instability has been simulated in spherical geometry by a 3D fully nonlinear approach. In order to model the structure of an non-rigidly rotating star we prescribe a rotation profile of the Brandt-type which is steep in the outer regions but yields rigid rotation at the inner core. A series with different magnetic Reynolds numbers has been performed with an increasing number of modes in spectral space. Starting from arbitrary small perturbations the magnetic and kinetic energies grow by several orders of magnitude as soon as a certain azimuthal resolution of at least $m=15$ has been used at a Reynolds number of order C$_{\Omega}=10^{5}$. Several phases of the transition to turbulence are realized and interpretations are given for the respective effects occurring at each stage. The resulting magnetic field is highly nonaxisymmetric. The flow is almost axisymmetric but shows a Kolmogorov-like behavior for small scales. The outer surface of the shell is penetrated by magnetic field lines in spotlike regions which are located mainly in the equatorial plane. The problem of angular momentum transport is discussed in terms of the Shakura-Sunyaev viscosity-alpha, which adopts values in the range $10^{-(3 ...5)}$. If the parameters are used of a main-sequence star of spectral type A, then the resulting magnetic fields are of order $10^{3...5}$ Gauss, very close to the values known from the observations of Ap-stars. First results are reported for the realistic case that a density stratification stabilizes the magnetorotational instability. ...
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## The Galactic Dynamo, Katia Ferriere

• I will present a model of supernova-driven turbulence in the interstellar medium of our Galaxy, describe the threefold impact of this turbulence on the large-scale Galactic magnetic field (alpha-effect, vertical advection, and magnetic diffusion), and present recent numerical solutions of the dynamo equation, which support the idea that the large-scale magnetic field can be amplified through a combination of large-scale differential rotation and supernova-driven turbulence.
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## Large and small scale dynamo action, D. Hughes

• We have considered the issue of large- and small-scale dynamo action, by conducting numerical simulations of dynamo action in a tall box. The initial magnetic field strength is chosen to be sufficiently weak that the magnetic field first evolves kinematically, before saturating in the nonlinear regime. The magnetic field configuration in the dynamical regime is shown to depend crucially on the choice of initial field. In certain cases the growth and saturation of the large-scale magnetic field can be attributed to an $\alpha$-effect arising from the interactions of a small-scale flow and a small-scale field. To examine the nature of the $\alpha$-effect in more detail and, in particular, its dependence on the mean field and on the magnetic Reynolds number, we performed a further series of simulations of dynamo action in a smaller box with an imposed uniform magnetic field. The most significant finding is that the $\alpha$-effect has a strong dependence on the magnetic Reynolds number; consequently the $\alpha$-effect is influenced by a very weak mean magnetic field.
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## Nonlinear $\alpha^2\Omega$-dynamo waves in stellar shells, A.M.Soward

• Nonlinear $\alpha^2\Omega$-dynamo waves are considered in a thin turbulent, differentially rotating convective stellar shell. Nonlinearity arises from $\alpha$-quenching, while an asymptotic solution is based on the small aspect ratio of the shell. Wave modulation is linked to a latitudinal-dependent local $\alpha$-effect and zonal shear flow magnetic Reynolds numbers $R_\alpha f(\theta)$ and $R_\Omega g(\theta)$ respectively; here $\theta$ is the latitude. The essential picture developed is that of a modulated dynamo wave whose amplitude varies spatially with $\theta$. The linear solution is controlled by the properties of the double turning point $\theta_c$ of the ordinary differential equation for the mode amplitude. Significantly, though $\theta_c$ is real and is located at the local dynamo number maximum in the $\alpha\Omega$-dynamo limit $R_\alpha\to 0$, it migrates into the complex $\theta$-plane once $R_\alpha\not=0$. Linear and weakly nonlinear solutions are found over a limited range of $R_\alpha$ and their qualitative properties are found to be largely similar to those for the $\alpha\Omega$-dynamo limit. One significant astrophysical difference is the fact that the frequency generally decreases with increasing $R_\alpha$. Thus $\alpha^2\Omega$-stellar dynamos may occur with $\alpha\Omega$-dynamo wave characteristics but exhibit significantly longer cycle times increased by a factor roughly two or more. {}Finite amplitude dynamo waves, like those when $R_\alpha \to 0$, are modulated by an envelope which evaporates smoothly at some low latitude but is terminated abruptly by a front at a high latitude $\theta_F$. For given non-zero $R_\alpha$ these frontal solutions are subcritical (a property linked to the complex-value taken by $\theta_c$). This issue and the validity of the frontal solutions is explored for small $R_\alpha$, when the solutions may be regarded as modifications of pure $\alpha\Omega$-dynamo waves. {}For sufficiently large $R_\alpha$ new low frequency modes are introduced that are more closely identified with steady $\alpha^2$-dynamos localised near the pole $\theta=\pi/2$. Up to four distinct finite amplitude states can exist; these can be loosely characterised as $\alpha\Omega$-high frequency, $\alpha^2\Omega$-medium frequency, $\alpha^2$-low frequency and $\alpha^2$-steady modes. In view of the possible mode competition, we comment on the likely realised physical state.
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## Dynamo effect of a helical flow with a superimposed turbulence in a cylinder, R. Stepanov, Institute of Continuous Media Mechanics, Perm, RUSSIA

• The screw dynamo or so-called Ponomorenko dynamo is able to generate a non-axisymmetric magnetic field in an axisymmetric helical flow and is one of the simplest known dynamo models. Basically a laminar helical flow like Couette-Poiseuille can produce the magnetic field self-excitation. A generation of magnetic field is possible in other way which is based on small-scale turbulence. A turbulent helicity (alpha-effect) can amplify a magnetic field in absence of mean field flow. Both generation mechanisms is supposed to act in several astrophysical object and the proposed Perm dynamo experiment. We study interaction of generation mechanisms helical large-scale laminar flow and small-scale turbulent helicity. The crucial parameters of dynamo-process in a different regimes have a strong interest especially in the frame of the Perm dynamo experiment. The simultaneous action of these mechanisms has been studied in the context of this experiment, taking properly into account inhomogeneities and anisotropies of the turbulence. Depending on the relevant parameters they may indeed support or counteract each other.

## On Nonstationary Dynamo Experiment In A Breaked Torus,P.Frick, S. Denisov, S. Khripchenko, V.Noskov, D. Sokoloff, R.Stepanov

• A new experimental project on MHD-dynamo problem is developing in the Institute of Continuous Media Mechanics of Ural Branch of Russian Academy of Sciences, Perm, Russia. The experiment is intended for study the magnetic field evolution in nonstationary turbulent flow of liquid sodium. In contrast to existing experimental projects, which have to realize the stationary flow of liquid sodium and require extremal power, in this project the energy will be accumulated during relatively long acceleration of the toroidal vessel (diameter about 1 m) with liquid sodium and the large power screw flow will be obtained only during the abrupt brake. Internal divertors will enable the required profile of velocity. The expected flow can act as a dynamo provided its magnetic Reynolds $R_m$ number is expected to exceed several tens. In this talk we present the results of experimental study of screw water flows in braked toroidal chanels, the results of numerical simulations of the corresponding dynamo problem which enables to optimize the parameters of the MHD apparatus and the general design of the dynamo apparatus.

## On the application of finite element method to the solution of dynamo equations, P. Hejda, I. Cupal and J. Merinsky

• The geodynamo process of magnetic field generation occurs in the liquid spherical core of the Earth and is described by 3D magnetohydrodynamic equations. Due to the complexity of these equations, the approach to the problem is usually numerical. >From the pioneering calculations of the kinematic dynamo in fifties to the most advanced 3D hydronagnetic simulations in nineties, the numerical process was based on the decomposition of the magnetic and velocity fields into toroidal and poloidal parts. The functions were usually further expanded into spherical harmonics, some of the 2D dynamo computations were carried out by finite difference method. The latter approach was mostly applied to the axisymmetric hydrodynamic models in magnetostrophic approximation. >From the numerical point of view, the magnetostrophic approximation is characterized by the fact that the sharp changes of velocities in the boundary layers are replaced by jumps and the velocity in the bulk of the core is expressed in the form of integrals in {\bf z} direction. The solution thus requires a transition between spherical and cylindrical coordinates. Alternative methods that deal directly with three components of magnetic and velocity fields were developed by Nakajima and Roberts ({\it Phys. Earth Planet. Inter.}, {\bf 91}, 53-61) (1995)) and Hejda and Reshetnyak ({\it Studia geoph. et geod.}, {\bf 43}, 319-325 (1999)). Whereas Hejda and Reshetnyak kept the spherical coordinates, Nakajima and Roberts solved the problem in cylindrical coordinates and avoided the transition between the coordinate systems by a special mapping method. Formulation of the problem in above mentioned curvilinear coordinates has following drawbacks:\\ - the differential operators are singular at the axis of rotation (and in the center od c.s. in spherical coordinates) - the grid is much more denser near the axis of rotation (center of coordinate system); the computer memory and time are thus wasted on the regions which are not of special interrest - extra boundary conditions must be formulated on the artificial boundaries. In the present contribution the application of finite element method will be discussed. It will be shown that all the peculiarities connected with the spherical geometry can be solved on the level of mesh generation. The spherical mesh can be constructed with nearly uniformly distributed nodes. Attention will be paid to the estimation of the mesh quality. As the equations are expressed in cartesian coordinates, they contain no singular coefficients. Moreover, the form of differential operators is much more simple then in curvilinear coordinates. The method will be tested on some simple problems. Eigenvalues of kinematic models will be found by inverse iterations and compared with results obtained by different methods and authors.

## Two-component dynamical model of the solar cycle, E.E. Benevolenskaya, Pulkovo Astronomical Observatory, St.Petersburg, Russia

• The solar magnetic cycle shows a complicated multi-mode behaviour. At least, two components are found in the distribution of the magnetic field on the solar photosphere. The existence of these two components can be explained by a non-linear model based on the Parker's dynamo theory with two sources. The properties of this dynamical system are investigated numerically, and it shown that this model can qualitatively reproduce the observed behaviour of the two dynamo components.

## Helioseismic tests of dynamo models, A.G. Kosovichev, Stanford University

• Helioseismology provides important tests for dynamo theories by measuring the variations of the internal structure and dynamics of the Sun with the activity cycle. Recent results from the GONG network and MDI/SOHO space experiment have revealed correlated variations of the zones of generation of the solar magnetic fields and zonal shear flows in the convection zone. An attempt is made to detect the solar-cycle variations in the tachocline region at the base of the convection zone, which is believed to be the main cite of the solar dynamo. The helioseismic measurements provide both input and test data for dynamo models.

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## On Self--killing and Self--creating Dynamos, H. Fuchs, K.-H. R\"adler, M. Rheinhardt

• Numerical studies of a laminar dynamo model have revealed two remarkable phenomena. A spherical body of an electrically conducting incompressible fluid is considered which is surrounded by free space. The fluid shows an inner motion due to a given force and satisfies the no--slip condition at the boundary. A rotation of the fluid body as a whole is admitted, too. The full interaction of magnetic field and motion is taken into account. Starting from a fluid motion capable of dynamo action and a weak magnetic field it was observed that the growing magnetic field destroys the dynamo property of the motion and then decays, and that the system ends up in a state with another motion incapable of dynamo action and zero magnetic field. The finding of this 'self-killing' of dynamos implies a warning concerning simple parameterizations of the back--reaction of the magnetic field on the motion as used, e.g., in mean--field dynamo theory. In another case with a motion unable to prevent small magnetic fields from decay it proved to be possible that strong magnetic fields deform it so that a dynamo starts to work which enables the system to approach a steady state with a finite magnetic field. This state is a genuine magneto--hydromagnetic one and has no kinematic counterpart. A similar phenomenon has been discussed in the context of Balbus--Hawley instability.

## Destabilization of transverse modes by noise, M.Proctor

• It is shown that transverse perturbations from structurally stable heteroclinic cycles can be destabilized by added noise even when each individual fixed point of the cycle is stable to tranverse modes. A necessary condition for this process is that the tranverse dynamics is non-normal. The phenomenon is illustrated by a simple two-dimensional switching model and by a simulation of a convectively driven dynamo.

## Advection of a magnetic field by a turbulent swirling flow, Philippe Odier

• I will present an experimental study of the magnetic field fluctuations generated in a turbulent flow of liquid metal, in the presence of an externally applied field. We consider the case of a weak seed' field, so that the Lorentz forces do not modify the flow. The velocity gradients induce magnetic field fluctuations at all scales, the description of which pertains to the dynamics of a passive vector' in turbulence, in analogy to the passive scalar case. However this passive vector dynamics involves stretching of magnetic field lines by velocity gradients, analogous to stretching of vorticity lines, and is thus at an intermediate level of complexity between passive scalar advection and fully developed turbulence. In particular, stretching of magnetic field lines by velocity gradients may overcome Joule dissipation and generate a large scale magnetic field by amplification of weak initial disturbances-- this is the dynamo effect. We use the flow created in the gap between two coaxial rotating disks, the von K\'arm\'an swirling flow. At small scales, it is known to produce a very intense turbulence in a compact region of space, allowing a study of statistical properties of the magnetic field, such as scaling laws. At large scales, the mean flow acts upon the magnetic field, modifying the field lines. In the case of counter-rotating disks, this mean flow possesses many features, such as differential rotation or poloidal and toroidal components, which are known to favor dynamo action. In the co-rotating case, the flow contains a coherent vortex which acts upon the magnetic field, such as to expulse it from its core. Experimental results on these various aspects of the magnetic field dynamics will be presented.

## Dynamo action in 3D homogeneous turbulence, Alberto Bigazzi

• The properties of both decaying and forced homogeneous compressible MHD turbulence with helicity are investigated via direct numerical simulations at resolutions up to $240^3$. Particular stress is put on the detection and study of dynamo action and the relation of the alpha-effect with properties of the flow such as the Helicity. The inverse cascade of energy and its connection with the dynamo is also investigated along with some topological properties of the resulting flow such as the alignment properties of the fields.

## Dynamo Problems in Spherical and Nearly Spherical Geometries, D. J. Ivers

• The solution of various forms of the rotating Boussinesq/anelastic non-linear and linearised magnetohydrodynamic (mass, momentum, induction and heat) equations in spherical and nearly-spherical geometries is discussed. Linearisation of the equations may be about an axisymmetric or three-dimensional (steady or possibly non-steady) basic state. The prototype physical model consists of uniformly electrically-conducting spherical solid-inner and spherical-shell fluid-outer cores, and an insulating exterior. Various extensions have been considered, including no inner-core or an insulating inner-core, a non-uniformly electrically-conducting mantle, anistropic turbulent viscous or thermal diffusion, pressure buoyancy, non-spherically-symmetric gravitation and departures from spherical boundaries. The equations are discretised using scalar, vector and tensor spherical harmonic expansions and toroidal-poloidal representations in angle, and second- or fourth-order finite-differences on a variable grid or Chebychev collocation in radius. The equations are solved using eigenvalue, critical-value or time-stepping techniques. These codes have been extensively tested against many known analytical and numerical solutions, including kinematic dynamo problems and thermal-/magneto-convection problems. The main problems of interest are topographic effects on the convection, pressure buoyancy, locking of convection to a non-spherically-symmetric gravitation varying in colatitude and/or azimuth and turbulent viscous and thermal diffusivities due to the effects of the magnetic field and rotation.

## Physics goals and initial results from the Madison Dynamo Experiment, Cary Forest

• A spherical, liquid socium dynamo experiment has been constructed at the Univerisity of Wisconsin. The experiment is constructed with the goals of (1) observing self-excited magnetic eigenmodes and subsequent saturation, of the variety proposed by Gubbins, and Dudley and James; (2) experimental observation of turbulent alpha and beta effects prior to self-excitation; and (3) studies of MHD turbulence and the possibility of a self-excited, small-scale, turbulent dynamo of the type proposed by Kraichnan and Iroshnikov. The experiment consists of a 1 m diameter sphere filled with liquid sodium and two 100 Hp motors driving two impellers in the sodium. Extensive experiments have been performed in a dimensionally identical water experiment, where velocity fields have been measured with Laser Doppler Velocimetry. MHD modelling indicates the measured velocity fields lead to self-excited eigenmodes in the sodium experiment. The infrastructure of the sodium laboratory is completed and experiments are to begin in spring, 2000. Initial results will be presented.

## Inviscid Geodynamo Models, Dominique Jault

• Modelling of the geodynamo mechanism is hindered by the small values of both the Ekman number and the magnetic Prandtl number in the Earth's core. On the other hand, the magnetic field is expected to be predominantly large scale, enabling large scale simulations. I have thus endeavoured to study numerically the balance between the Coriolis acceleration and the Lorentz force density, while all other accelerations and viscous forces are ignored. A basis of divergence-free velocity field densities obeying the no-penetration condition at the boundaries is defined to calculate a magnetic wind'' from a known distribution of magnetic forces. Projecting onto this basis allows to invert the Coriolis operator. Then, the inferred velocity is inserted in the induction equation to update the magnetic field. I will report on the difficulties of this approach and on my most recent results. Finally, the calculated models are used to show how the frozen-flux description becomes valid, at the core surface, with increasing magnetic field strength. This question is of importance because the frozen-flux hypothesis is widely used to infer core dynamics from magnetic field measurements.

## Simple models for studying the dynamics of fluid dynamos, Francois PETRELIS

• We study simple models of fluid dynamos in order to characterize the bifurcation that leads to the dynamo effect (stationary or oscillatory, supercritical or subcritical) and to understand the energy or power balances in the saturated regimes (ratio of kinetic to magnetic energy, ratio of viscous to Joule dissipation). We discuss the experimental realisation of these simple fluid dynamos.

## Dynamo action in the Taylor--Green vortex: thresholds and saturation, C. Nore, M.E. Brachet, H. Politano and A. Pouquet

• Dynamo action is demonstrated numerically in the forced Taylor-Green vortex \cite{BMONMU83} made up of a confined swirling flow composed of a shear layer between two counter--rotating eddies. This flow corresponds to a standard experimental set--up in the study of turbulence called the Von Karman swirling flow. The critical magnetic Reynolds number above which the dynamo sets in depends crucially on the fundamental symmetries of the TG vortex. These symmetries can be broken by introducing a scale separation in the flow, or by letting develop a small non--symmetric perturbation which can be either kinetic and magnetic, or only magnetic \cite{NBPP97}. The nature of the boundary conditions for the magnetic field (either conducting or insulating) is essential in selecting the fastest growing mode. We also present cases where a long term magnetic field produced by dynamo action saturates and discuss the saturation energetics. Implications of our results to a planned laboratory experiment using sodium liquid in the Von Karman swirling flow (Daviaud, Fauve, Pinton, private communication) will be discussed.

## Are the Taylor-Green Vortex and the Von Karman swirling flow good candidates for Dynamo Action?, M. E. Brachet

• The Taylor-Green (TG) vortex is a standard turbulent flow used in numerical computations. It is related to the experimental Von Karman swirling flow. The relation between the experimental flow and the TG vortex is a similarity in overall geometry: a shear layer between two counter-rotating eddies. The TG vortex, however, is periodic with free-slip boundaries while the experimental flow is contained inside a tank between two counter-rotating disks. Experiments in Sodium are planned (Fauve, private communication), for a Von Karman swirling flow in which the magnetic Reynolds number may be close to the critical value $R^m_c$ above which a dynamo sets in. We will show that a forced Taylor-Green vortex is consistent with a long term magnetic field produced by dynamo action and find the critical magnetic Reynolds numbe. A number of symmetries of the TG vortex are found to be {\it spontaneously broken}, in the sense that a small non-symmetric component of the initial data will grow and eventually completely break the symmetry of the solution. The relation between the TG and the Von Karman Dynamos and the effect of boundaries will also be discussed.

## Dynamo action in flows driven by shear and convection, Yannick Ponty

• A numerical investigation is presented of kinematic dynamo action in a dynamically driven fluid flow. The fluid lies in a rotating plane layer and all motions are taken to be two-dimensional, that is, depending on two spatial coordinates, together with time. The base of the layer is given a constant velocity (the top remaining at rest) and this produces a shear flow in the form of a spiral Ekman layer localised close to the lower boundary. This basic state is destabilised by a convective instability through heating the base of the plane layer, or by a purely hydrodynamic instability of the Ekman layer flow. Kinematic dynamo action is studied in the flows that result in various regimes. When the Ekman layer flow is destabilised by the hydrodynamic Ekman layer instability, the fluid flow that results is steady in an appropriately chosen moving frame, and takes the form of a row of cats' eyes. Kinematic magnetic field growth is observed, with some modes growing by the Ponomarenko dynamo mechanism, while other modes appear to be associated with stagnation points and heteroclinic separatrices. When the Ekman layer flow is destabilised thermally, far away from the onset the convective instability creates a flow that is intrinsically time-dependent, that is, is unsteady in any moving frame. The magnetic field is concentrated in magnetic sheets situated around the convective cells and evidence for fast dynamo action is obtained. The presence of an Ekman layer close to the bottom boundary of the layer breaks an up--down symmetry and localises the magnetic field near the lower boundary.

## Magnetic helicity with zero and nonzero mean magnetic field, N. Kleeorin and I. Rogachevskii

• The magnetic helicity is a fundamental quantity in magnetohydrodynamics because it is conserved in the limit of infinite electrical conductivity. The magnetic helicity determines the magnetic part of the $\alpha$-effect. The latter is of great importance in view of nonlinear magnetic dynamo. The evolution of the magnetic helicity tensor for a nonzero mean magnetic field in an anisotropic turbulence is studied. It is shown that the isotropic and anisotropic parts of the magnetic helicity tensor have different characteristic times of evolution. The time of variation of the isotropic part of the magnetic helicity tensor is much larger than the correlation time of the turbulent velocity field. The anisotropic part of the magnetic helicity tensor changes for the correlation time of the turbulent velocity field. The mean turbulent flux of the magnetic helicity is calculated as well. It is shown that even a small anisotropy of turbulence strongly modifies the flux of the magnetic helicity. It is demonstrated that the tensor of the magnetic part of the $\alpha$-effect for weakly inhomogeneous turbulence is determined only by the isotropic part of the magnetic helicity tensor. The magnetic helicity in the case of zero mean magnetic field is also discussed. It is demonstrated that the two-point correlation function of the magnetic helicity in the case of zero mean magnetic field has anomalous scalings for both, compressible and incompressible turbulent helical fluid flow. The magnetic helicity in the limit of very high electrical conductivity is conserved. This implies that the two-point correlation function of the conserved property does not necessarily have normal scaling. The reason for the anomalous scalings of the magnetic helicity correlation function is that the magnetic field in the equation for the two-point correlation function of the magnetic helicity plays a role of a pumping with anomalous scalings. It is shown also that when magnetic fluctuations with zero mean magnetic field are generated the magnetic helicity is very small even if the hydrodynamic helicity is large.

## Nonlinear mean-field dynamo: electromotive force for an anisotropic turbulence and intermediate nonlinearity, I. Rogachevskii, N. Kleeorin, The Ben-Gurion University of the Negev, Israel

• A nonlinear electromotive force for an anisotropic turbulence in the case of intermediate nonlinearity is derived. The intermediate nonlinearity implies that the mean magnetic field is not enough strong in order to affect the correlation time of turbulent velocity field. The nonlinear mean-field dependencies of the hydrodynamic and magnetic parts of the $\alpha$ effect, turbulent diffusion, turbulent diamagnetic and paramagnetic velocities for an anisotropic turbulence are found. It is shown that the nonlinear turbulent diamagnetic and paramagnetic velocities are determined by both, an inhomogeneity of the turbulence and an inhomogeneity of the mean magnetic field ${\bf B} .$ The latter implies that there are additional terms in the turbulent diamagnetic and paramagnetic velocities $\propto {\nabla} B^{2}$ and $\propto ({\bf B} \cdot {\nabla}) {\bf B} .$ These effects are caused by a tangling of a nonuniform mean magnetic field by hydrodynamic fluctuations. This increases inhomogeneity of the mean magnetic field. It is also shown that in an isotropic turbulence the mean magnetic field causes an anisotropy of the nonlinear turbulent diffusion. Two types of nonlinearities in magnetic dynamo determined by algebraic and differential equations are discussed. Nonlinear systems of equations for axisymmetric $\alpha \Omega$ dynamos in both, spherical and cylindrical coordinates are derived. Nonlinear galactic dynamo in the thin-disc approach is discussed.

## Asymptotic WKBJ-modelling of solar dynamo waves: results and perspectives, K. Kuzanian, D. Sokoloff, Moscow, Russia; A. Soward and A. Bassom, Exeter, UK; G. Belvedere, Catania, Italy.

• Systematic observations of tracers of the solar magnetic activity such as sunspots, sunspot groups, active regions, polar faculae etc. indicate that this activity has a form of two travelling waves: low-latitudinal equatorward and high-latitudinal poleward waves in each given hemisphere. The simplest mechanism of this field generation can be given in the framework of mean field dynamo theory. The corresponding dimensionless number (field regeneration rate) is large, which can be used in asymptotic analysis of the governing equations. The use of WKBJ enables us to resolve analytically short waves in thin layers. Even a kinematic approach yields general structure of the solution and fixes the frequency of the cycle. Provided knowledge on the generation sources ($\alpha$ and $\Omega$-effects) we are able to outline the shape of the dynamo wave. Nonlinear development of such solution retains some properties of the linear wave but includes finite wave amplitudes and refines the structure of the wave front. Asymptotic solution of the weakly-nonlinear problem yields the transition (bifurcation) to the finite amplitude state and enables some relation between the period and amplitude of the dynamo wave. Recent advances in helioseismology provide reliable information on the internal rotation of the Sun. These data allow us to estimate gradients of angular velocity , i.e., the strength of the $\Omega$-effect. The 2-D asymptotic dynamo model reveals spatial locations of the maxima dynamo waves in the solar convective zone. Further studies are focused on the interaction between the two waves in one given hemisphere as well as the waves in both the hemispheres with each other. This yields splitting the frequency of the cycle for dipole and quadrupole modes, and thus causes global modulation of the solar cycle. Further nonlinear development of these studies should involve not only algebraic but dynamic nonlinear back-reaction of the magnetic field to rotation and convection (e.g., Malkus-Proctor effect). These produces complicated multiperiodic oscillatory behaviour of mean fields. These theoretical considerations are subject to comparison with available observations.

## Dynamos in Rotating and Nonrotating Convection, Norbert Seehafer, Ayhan Demircan

• We study the dynamo activity of Boussinesq convection in a horizontal plane electrically conducting fluid layer. First the case of nonrotating convection is considered. For the primary convection rolls no dynamo action is found. However, the skewed varicose instability leads from the rolls to a pattern of asymmetric squares which is capable of kinematic dynamo action. For these squares, with rising or with descending motion in the center (and descending or rising motion near the boundary) the up-down reflection symmetry is spontaneously broken. They possess a net kinetic helicity and are generated by two unstable checkerboard (symmetric square) patterns and their nonlinear interaction. The checkerboard patterns possess no net helicity and show no dynamo action. The asymmetric squares are only kinematic, not nonlinear dynamos. Nonlinearly these kinematic dynamos kill themselves by forcing the solution into the basin of attraction of purely hydrodynamic roll patterns (with wave numbers smaller than that of the original rolls). However, the asymmetric squares become nonlinear dynamos if rotation is added to the system.

## Thermal flow in a rotating spherical gap with a dielectrophoretic central force field, Christoph Egbers, Werner Brasch, Markus Junk

• The understanding of thermal convection in spherical gaps under a central force field is important for large scale geophysical motions. Neglecting the magnetic field, the dielectrophoretic force can be used to produce a central force field under microgravity conditions. In a space experiment, currently under construction, thermal convection in a rotating spherical gap with heated inner sphere and cooled outer sphere will be visualized by a Wollaston interferometer. High voltage is used to produce a dielectrophoretic central force field in the gap. The parameters are chosen in analogy to the convection in the earth's inner core. The experiment and its restrictions will be presented as well as numerical predictions for the expected flows. The axial-symmetric flow is calculated on a staggered grid with a finite volume method. The conjugate-gradient method with a pre-conditioner accelerates the approximation. In azimuthal direction a spectral analysis allows a three-dimensional simulation for spherical shells with large radius ratios.

## Interactions between numerical simulations and asymptotics studies in the Earth's liquid core... , Emmanuel Dormy, Dominique Jault and Philippe Cardin

• It is well known that dimensionless numbers relevant to geodynamo modeling are out of reach of present numerical computations. Difficulties are raised by the extremely small values of both the Ekman number ($10^{-15}$) and the magnetic Prandtl number ($10^{-6}$). Asymptotic techniques however can lead to a description of the flow and of magnetic induction, inside boundary and shear layers. These studies give a nice way to validate numerical simulations. We concentrate here on simplified problems and try to achieve a satisfactory agreement with asymptotic studies. The first problem we consider is an apparently simple axisymmetrical problem, where all motions are generated by differentially rotating boundaries in the presence of an imposed magnetic field. In the non-magnetic case, we show for the first time how the solution tends, with decreasing Ekman numbers, to the asymptotic limit of Proudman(1956). We then show the first good quantitative agreement with Stewartson's (1966) description of the shear layer near the tangent cylinder. We then consider the MHD flow in the same geometry with an imposed dipolar magnetic field. The first effects of the Lorentz force is to smooth the change in angular velocity at the tangent cylinder. As the Elsasser number is further increased, the Proudman-Taylor constraint is violated, Ekman layers are changed into Hartmann layers, shear at the inner sphere boundary vanishes, and the flow tends to a bulk rotation together with the inner sphere. Unexpectedly, for increasing strength of the field, we observed a super rotation (the fluid's angular velocity reaches a maximum inside the fluid volume) localized inside an equatorial torus limited by field lines of the imposed magnetic field. At a given field strength, the amplitude of this phenomenon depends on the Ekman number and tends to vanish in the magnetostrophic limit. This demonstrates the decisive role played by the Ekman number on the solutions (even with important magnetic effects). We then concentrate on the asymptotic description of this feature for large Hartmann numbers. An equation has been derived by Soward (University of Exeter), for this MHD shear layer. We present the numerical solution of this equation and a comparison with finite Hartmann number solutions. Finally, we study the properties of the onset of convection in a rotating spherical shell. We study no-slip and free slip boundary conditions and consider different heatings~: uniform heat sources (widely used in previous studies, and in asymptotic studies) and imposed differential temperature gradient. We present a comparison with the asymptotic studies of (Jones et al. 2000 and Roberts 1968/Busse 1970) and show how the pattern of motions at the onset of convection strongly depends on the heating considered. We suggest consequences for numerical simulations of the geodynamo at moderate Ekman numbers.

## Hydromagnetic instabilities in non-uniformly stratified layer in dependence on boundary conditions and Roberts number,J. Simkanin, J. Brestensky, S.Sevcik

• The stability study of the hydromagnetic systems is a significant part of the geodynamo theory. The linear magnetoconvection and its dependence on boundary conditions grants a lot of useful results till now. Marginal instabilities study follows in the model of rapidly rotating non-uniformly stratified fluid layer permeated by an azimuthal magnetic field growing linearly with distance from the vertical rotation axis. More cases of non-uniform stratification are investigated and compared with uniform unstably stratified layer case. The boundaries are perfectly thermally conducting and rigid (no-slip). The dependence on electromagnetic boundary conditions is investigated, i.e. both boundaries are perfect conductors or finitely conducting as well as in the Earth's core. The significant parameter of the presented analysis is Roberts number $q\,\equiv\,\kappa\,/\,\eta\,=\,\tau_{\eta}\,/\,\tau_{\kappa}$ (where $\tau_{\eta}\,,\,\tau_{\kappa}$ are times of ohmic and thermal diffusion), which expresses the coupling of two different processes - thermal and magnetic diffusion. There are two qualitatively different sets of results (from the physical point of view): (1) $q\,\ll\,1,$ (i.e. $\tau_{\eta}\,\ll\,\tau_{\kappa}$). The developed instabilities are extremely slow (of frequencies $\approx \tau_{\kappa}^{-1}$) and are thermally or magnetically driven. (2) $q\,\ge\,O(1),$ (i.e. $\tau_{\eta}\,\ge\,\tau_{\kappa}$). The developed instabilities of the MAC waves type (Soward 1979) are characteristic by period typical for geomagnetic secular variations and they are determined by dynamical balance of Magnetic, Coriolis and Archimedean forces. The hydromagnetic system is able to change into the state of double scale magnetoconvection, i.e. there are more discrete values of Elsasser number, $\Lambda$, for which two instabilities of different critical radial wave numbers with corresponding frequencies coexist at the same Rayleigh number. The developed instabilities and the mean electromotive force, which is produced by these ones, are in both cases dependent on electromagnetic boundary conditions.

## Rotating magnetoconvection in dependence on various diffusive processes and boundary conditions, J. Brestensky, S. Sevcik and J. Simkanin

• Marginal instabilities in horizontal uniformly stratified rapidly rotating fluid planar layer permeated by the azimuthal magnetic field, linearly growing with distance from the vertical rotation axis, are investigated. Arising linear magnetoconvection is studied in viscous and inviscid fluid in dependence on Elsasser number, ${\cal O}(10^{-2})\leq\Lambda \leq {\cal O}(10^3)$, for two sets of Roberts number, $q={\cal O}(1)$ and $q\ll 1$. Both boundaries are perfectly thermally conducting with two types of electromagnetic and/or mechanical boundary conditions. Their electrical conductivities are either infinite or finite Earthlike ones. Each boundary is only either stress free or no-slippery for viscous fluid of various Ekman number, ${\cal O}(10^{-7})\leq E\leq {\cal O}(10^{-3})$. Numerical results were confirmed by asymptotic analysis which indicated physical interpretation, too. Almost all modes of instabilities correspond to Soward (1979) model of the simplest boundary conditions (of stress free perfect electric conductors), and propagate azimutally eastward or westward. Westward modes are always preferred with some exceptions of magnetically driven instabilities for larger $\Lambda={\cal O}(10^3)$. Case $q\ge{\cal O}(1)$, in distinction with case $q\ll 1$, gives the instabilities of MAC-waves kind with frequencies typical for geomagnetic secular variations much greater than frequencies ${\cal O}(\tau_{\kappa}^{-1})$ corresponding to thermal diffusion time for case $q\ll 1$. Due to viscosity various competitive modes arise. Their preference is strongly dependent on $q$ and $E$. Critical frequencies of MAC waves are most sensitive on Roberts number for $q\approx 1$ and lose this sensitivity for increasing $q$ at $q\approx 2.$ Greater frequencies of MAC waves induce their sensitivity to viscosity for greater $\Lambda > {\cal O}(1),$ too. This sensitivity for $\Lambda = {\cal O}(1),$ is apparent only for greater azimuthal wave numbers $m\geq{\cal O}(10)$. However, the preferred modes are not influenced by viscosity and they are sensitive only to electromagnetic boundary conditions. Many qualitative differencies in the dependences on $q$ and $E$ do exist for infinitely and finitely conducting boundaries. E.g. MC-waves were found only for finitely electrically conductive boundaries, and for sufficiently strong field only with $m=1$. They almost do not depend on Roberts number $q$, but the direction of their propagation strongly depends on Ekman number. The eastward MC-waves do occur at very small Ekman number, $0\leq E \leq {\cal O}(10^{-7})$, as well as at small $E \geq {\cal O}(10^{-3})$, while the westward MC-waves do occur only at small $E \geq {\cal O}(10^{-3})$.

## Intermittency and fragility in axisymmetric dynamo models, Reza Tavakol

• Eurico Covas has an extensive abstract that covers various aspects of the joint work we have been doing over the last 3 years. We will talk on different aspects of these works. One of us will focus on the mathematical results and the other on the confirmation of these results in various families of axisymmetric mean field dynamo models.

## Homogeneus dynamo: numerical analysis of experimental von Karman type flows,J .Burguete F. Daviaud J. L\'eorat

• Dynamo action, which converts kinetic energy into magnetic energy, is the manifestation of the coupling between kinetic and magnetic excitations in a conducting fluid. The occurence of the dynamo action is not questionable,but the nonlinear regimes are very poorly known. As for hydrodynamic turbulence, the experimental approach could represent an efficient tool to study the nonlinear effects in MHD flows. Some numerical examples have shown that dynamo action is present in flows at magnetic Reynolds numbers $Rm = \lambda V_{max} L_{max} = 100$, say, where $1/\lambda$ is the magnetic diffusivity and $V_{max},L_{max}$ are respectively the maximal speed of the flow and a characteristical length of the conducting volume. Using the best available fluid conductor ($1/\lambda= 10 m^2 /s$), liquid sodium at 150$^\circ$C, the condition $Rm = 100$ implies that $V_{max} L_{max} = 10 m^2/s$, which represents the main technical challenge to be achieved by any experimental fluid dynamo. We will here concentrate on the feasability study of a peculiar experimental fluid dynamo. Note also that using liquid sodium as a conducting fluid, the kinetic Reynolds number of the flow is about $10^5 Rm$, which shows that it is in a regime of fully developped turbulence. Our approach is to try various forcing mechanisms and geometries in water models to measure the velocity fields which are then introduced in the numerical computation of a kinematic dynamo problem. The experimenatal flows are von Karman type, produced between two counter-rotating disks in a cylindrical container. Using laser doppler and pulsed doppler ultrasonic velocimetries we can retrieve the temporal mean velocity and the turbulence rates at each spatial point. The kinematic approach gives relatively fast answers. Looking at the magnetic energy evolution $E_{B}(t)= e^{\sigma t}$, the dynamo effect appears when this energy grows without external excitation in the form of a magnetic field: $\sigma > 0$ means dynamo effect, and $\sigma < 0$ no dynamo effect. The influence of various parameters, such as the poloidal to toroidal component ratio of the velocity field $V_{pol} / V_{tor}$, an external magnetic field, the influence of time dependent flows or an external conducting layer are being tested.

## Riga Dynamo Experiment, A.Gailitis, O.Lielausis, E.Platacis, G.Gerbeth, F.Stefani

• In the first run of Riga Dynamo experiment an intense flow of liquid sodium produced by an outside driven propeller have generated a slowly growing magnetic field eigenmode. For a slightly decreased flowrate the observed field is slowly decaying. The measured results correspond satisfactory with theoretical predictions for the growth rates and frequencies. In the report will be presented computational base, optimisation, the detailed design of the experiment, current results and next experimental steps.

## Non--Linear Dynamics Underlying Axisymmetric Mean-Field Dynamo Models E. Covas

• Sunspot records dating back to the $17^{\rm th}$ century show an aperiodic cyclic activity. These records show also that this cyclic behaviour was interrupted by epochs of reduced activity, such as the M\"aunder minimum, with intermediate time scales of $\sim 10^2$ years. Furthermore, there is indirect evidence from the study of proxy indicators, such as $^{14}C$ and $^{10}Be$, which demonstrate that such epochs of reduced activity persisted back in time, at least for the last 10\,000 years. In addition, observations of magnetic activity in solar--type stars over the last three decades provide evidence that similar variability can also be present in such stars. This latter class of observations seem to also indicate that similar solar--type stars can exhibit a range of qualitative distinct magnetic behaviours. Such variabilities in the Sun and solar--type stars are thought to be due to dynamos operating in or near the base of stellar convective interiors. Motivated by the above observational evidence and bearing in mind that the underlying regimes in such stars are bound to be non--linear, an attempt is made in this Thesis to study possible mechanisms for such variability by making a study of the generic non--linear dynamics underlying axisymmetric dynamo models. Emphasis on generic features --- namely the presence of invariant subspaces as well as two other technical assumptions generically satisfied by such dynamo models --- is essential, given the unavoidably approximate nature of such models. This study reveals a number of novel results, including a new form of intermittency, referred to as {\em in--out intermittency} --- together with its precise signatures and scalings --- as well as several forms of {\em final state sensitivity}, including the presence of regions of parameter space possessing sensitivity to initial conditions and parameter values. By employing extensive simulations of axisymmetric ODE and PDE mean--field dynamo models, it is shown that these predictions do indeed occur. In particular, it is shown that in--out intermittency does indeed occur in such models, hence substantiating an old proposal --- the {\em intermittency conjecture} --- according to which dynamical intermittency could account for M\"aunder--type variability. In addition, it is shown that more than one type of intermittency can occur in such settings which demonstrates that a more appropriate proposal would be that of {\em multiple intermittency}, each with their own precise signatures and scalings. Furthermore, it is shown that such models can possess multiple coexisting solutions (attractors), as well as what is referred to as {\em fragility}, i.e.\ the property that small changes in the initial conditions as well as in the parameters and the details of the models can produce qualitative changes. If present in real stars, this could, in principle, account for the observational evidence for similar solar--type stars having qualitatively different modes of magnetic behaviour. We call this the {\em fragility conjecture}. Given the genericity of such non--linear behaviours, it is likely that these phenomena will persist in more realistic models. Finally given the related precise signatures and scalings, it is in principle possible to make a comparison with the observational and proxy data.

## Magnetically Induced Shear Layers and Jets, R. Hollerbach, S. Skinner

• We consider numerically the problem of spherical Couette flow in an electrically conducting fluid, and impose a strong magnetic field aligned with the axis of rotation. We show that the resulting flow depends dramatically on whether the boundaries are taken to be electrically insulating or conducting. In both cases the so-called tangent cylinder, the cylinder circumscribing the inner sphere and aligned with the axis of rotation, plays a crucial role, but the details of what happens on it are very different in the two cases. For insulating boundaries, the flow consists of a shear layer right on the tangent cylinder, with the fluid at rest outside, and in essentially solid-body rotation at a rate $\Omega/2$ inside. In sharp contrast, for conducting boundaries, the flow consists of a powerful counter-rotating jet just outside the tangent cylinder. The thickness of both the shear layer and the jet scale as $M^{-1/2}$, where the Hartmann number $M=B_0 L(\sigma /\nu\rho)^{1/2}$ measures the strength of the imposed field. However, whereas the jump across the shear layer remains constant at $\Omega/2$, the magnitude of the jet {\it increases}, roughly as $M^{0.6}$, and thus exceeds $\Omega$ for sufficiently large $M$. Having obtained these --- so far purely axisymmetric --- solutions, we next compute the onset of non-axisymmetric instabilities, and find that the critical Reynolds number scales as $M^{0.66}$ for the shear layer and as $M^{0.16}$ for the jet. The (fully three-dimensional) nonlinear equilibration of these instabilities will also be considered. Finally, time permitting, we will consider what happens if dipole or quadrupole or other non-uniform fields are imposed instead.

## Toward a Self-generating Magnetic Dynamo: the Role of Turbulence, Nicholas L. Peffley, A. B. Cawthorne,Daniel P. Lathrop

• Turbulent flow of liquid sodium is driven toward the transition to self-generating magnetic fields. The approach toward the transition is monitored with decay measurements of pulsed magnetic fields. These measurements show significant fluctuations due to the underlying turbulent fluid flow field. This talk presents experimental characterizations of the fluctuations in the decay rates and induced magnetic fields. These fluctuations have a significant implications on the transition to self-generation which should occur at larger magnetic Reynolds number. Specifically, we predict that the transition will show intermittency.

## Symmetries of the Solar Dynamo: comparing theory with observation, J. M. Brooke, University of Manchester

• It is well known that the symmetry group of a system has important consequences for its dynamical behaviour. The symmetry group of the axisymmetric mean-field dynamo equations and its effects on the bifurcation sequence as the dynamo number increases has been the subject of recent study. There is strong evidence that there was a major departure from equatorial symmetry in the distribution of sunspots at the end of the Maunder minimum. However, the equatorial symmetry of the sunspot cycle over the last 200 years has been the subject of some debate.It is clear from the records that there are large departures from equatorial symmetry over timescales of a few years, and evidence that whole cycles have a dominant hemisphere which can change between cycles. It is not agreed, however, to what extent this is a statistical effect of a noisy component of the sunspot spatiotemporal distribution, or whether there are cyclic changes in the symmetry of the solar field over a time scale of several solar cycles, as predicted by several recent dynamo models. This paper addresses this question by combining insights from the dynamics of symmetric systems, with robust tools for identifying multi-periodicity in complex and noisy time series. We use a time series which supplies the latitude and longitude of daily sunspot observations from 1853-1996. This series extends the Greenwich Photoheliographic Records backwards from 1876 by using the observations of Carrington and Sp\"{o}rer and forwards from 1976 by using the SOON records. An important result of this analysis has been the identification of a long-period oscillation of the solar magnetic equator (defined as the average of the sunspot numbers weighted by latitude) on a period of 90 years, which is the period most closely associated with the Gleissberg cycle. These methods also show that the cycle length also varies on this timescale in accordance with other studies based on the total sunspot counts (without reference to the spatial distribution of the spots). We discuss this finding in the light of two possible models of changes in the symmetry of the solar field, the parity modulation associated with the Type I modulation of Knobloch and Landsberg (1996) and the newly identified form of intermittency described as spiralling'' or in-out'' intermittency. There will also be discussion of the problems of identifying the spatial behaviour of the sun's magnetic field from such a limited time series. Regular observations of sunspot position are not available before 1853, apart from the French observations during the Maunder minimum (1660-1719). Proxy records, such as $^{10}Be$ and $^{14}C$ records, can be extended back for much longer but do not contain information as to the equatorial symmetry of the solar field.

## The variation of the Earth's magnetic field by tidal force, Rosaev A.E., Ufimtceva M.V.

• It is well known, that all greatest planets of Solar system have a magnetic field. The Mars have weak magnetic field too, the Venus, most similar to Earth planet - in contrary have not it. At recent time magnetic fields of Galilean satellites of Jupiter are discovered. The presence of magnetic properties on celestial bodies so different nature is non-direct evidence of external nature of them. The most possible way of explanation of geomagnetic field inversion may be following. The nature of magnetic field related with electric current in Earth's core. On the other hand, the west drift of non-dipole component of this field with velocity 0.2 rad/year is well known. The most simply explanation of this fact is in differential rotation of core relatively daily Earth's surface. Really, such phenomenon - more faster rotation of inner core was discovered recently. On the other side, the increasing of day and, as followed, decreasing of Earth's surface rotation rate is well known. So, in accordance with orbital moment's conservation law, the removing Moon from Earth and Moon's velocity decreasing take place. However, due to Earth's orbit changes by planets perturbations (eccentricity decreasing), the Moons acceleration is observed. The eccentricity show oscillation variations - there are its increasing epoch sometime in past. According of them, the epoch of more strong and more slow decreasing Earth's rotation rate change one another. The core have more inertia relatively outer Earth's layers, it may be described mathematically through 'angle of late'. At present time core keep in memory epoch of more slow, then now epoch of decreasing Earth's rotation rate. It explain inner core leading over daily surface and west drift on secular variations of magnetic fields. The reversed situation is according of inverse epoch, when core rotated more slow then daily surface. It possible, when Earth's orbit eccentricity increased. For complete model the description of way of charges separation is required. Maybe, high temperature or phase changes is able to provide necessary polarization.

## Dynamo effect of a helical flow with a superimposed turbulence in a cylinder" Rodion Stepanov, Peter Frick, Perm, Russia

• The screw dynamo or so-called Ponomorenko dynamo is able to generate a non-axisymmetric magnetic field in an axisymmetric helical flow and is one of the simplest known dynamo models. Basically a laminar helical flow like Couette-Poiseuille can produce the magnetic field self-excitation. A generation of magnetic field is possible in other way which is based on small-scale turbulence. A turbulent helicity (alpha-effect) can amplify a magnetic field in absence of mean field flow. Both generation mechanisms is supposed to act in several astrophysical object and the proposed Perm dynamo experiment. We study interaction of generation mechanisms helical large-scale laminar flow and small-scale turbulent helicity. The crucial parameters of dynamo-process in a different regimes have a strong interest especially in the frame of the Perm dynamo experiment. The simultaneous action of these mechanisms has been studied in the context of this experiment, taking properly into account inhomogeneity and anisotropy of the turbulence. Depending on the relevant parameters they may indeed support or counteract each other.
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