Dynamo and Dynamics, a Mathematical Challenge

An experiment for study magnetic field evolution in nonstationary turbulent flow of liquid sodium is discussed. Kinetic energy is accumulating 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 diverters 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. The experiment is developing in the Institute of Continuous Media Mechanics of Ural Branch of Russian Academy of Sciences, Perm, Russia.

It is widely believed that almost all magnetic fields in a natural environment are a result of the dynamo process—field generation in a moving nearly homogeneous electroconducting fluid of celestial bodies. Such are fields of the Earth, most of the planets, Sun, another stars and even galaxies. The believe is based on the theory and numerical simulation. Until very recently no direct laboratory experiment was supporting this important point.

The governing equation for the magnetic field B in an electrically conducting fluid with conductivity σ and velocity v is the so-called induction equation (1)$$ \frac{{\partial B}} {{\partial t}} = curl(v{\text{ x B) + }}\frac{1} {{\mu _0 \sigma }}\Delta B $$) which follows from Maxwell equations and Ohm’s law. The solution B = 0 may become unstable for some critical value Re_mc of the magnetic Reynolds number, (2)$$ \operatorname{Re} _m = \mu _0 \sigma LV, $$) L and V being respectively typical length and velocity scales.

Conversion of kinetic energy into magnetic energy within a conducting fluid is but an example of various physical phenomena occuring in MHD turbulence at magnetic Reynolds numbers R_m greater than a few tens. While the first numerical evidence of this phenomenon has been obtained since almost 45 years, it was only at the end of the year 1999 that it could be experimentally verified in liquid sodium flows. There is still a great gap between such experimental approaches and the understanding of natural dynamos, which are responsible for example of solar magnetic activity or of the geomagnetic field. To get closer to the real MHD problems, new designs of experimental devices and new numerical codes are needed in order to achieve greater Rm, study the turbulent characteristics, the nonlinear saturation regime and the influence of large-scale flow configuration and boundaries.

The magnetism of many astrophysical objects, such as various stars or planets, galaxies, the intergalactic medium, etc, is attributed to the motion of conducting fluid in their interiors. It has been first proposed by Larmor [17] that a flow of conducting fluid generates the magnetic field of the sun by maintaining the corresponding electric current against ohmic dissipation. Such a generation of electromagnetic energy from mechanical work using a self-excited dynamo has been known since Siemens [34] and is the most basic mechanism of electrical engineering. However, in industrial dynamos the path of the electric currents are constrained by a complex wiring, which even in the most elementary device, the homopolar dynamo [3], breaks mir ror symmetry. In addition, magnetic field lines are usually canalized using a high magnetic permeability material. No such well controlled external constraints on the field or on the current lines exist in“natural” dynamos, and for a long time, it has been far from obvious that the dynamo effect was the correct explanation for solar or earth magnetism. It has been even shown that a lot of flow and / or field configurations with enough symmetries cannot behave as fluid dynamos [for a review on anti-dynamo theorems, see Kaiser et al., these proceedings].

Dynamo action is demonstrated numerically in the forced Taylor-Green (TG) vortex [1] made up of a confined swirling flow composed of a shear layer between two counter-rotating eddies and corresponding to a standard experimental set-up in the study of turbulence called the von Kármán (VK) swirling flow. The critical magnetic Reynolds number above which the dynamo sets in depends crucially on the fundamental symmetries of the TG vortex which 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 [2]. We present cases where a long term magnetic field produced by dynamo action saturates. Implications of our results to VK sodium experiments are discussed.

The cause of spontaneous generation of magnetic fields in conducting bodies (such as plasmas) is a longstanding, major problem in plasma astrophysics, geophysics, and laboratory plasmas. It is observed that magnetic fields exist in the Earth, Sun and other stars (and perhaps in galaxies), that cannot be explained as surviving primordial fields, and generally believed that such magnetic fields are generated by plasma flow (or flow of liquid metal for the Earth). The question of how magnetic fields are generated by unconstrained flows of conducting fluids and plasma is referred to as the“dynamo” problem; theoretical research into dynamo mechanisms has been actively pursued for several decades. However, until quite recently our probing of the dynamo problem has been limited to analytic calculations, numerical modelling and observational studies; experimental validation (the critical test for any theory) of aspects of the theory and experimental studies of laboratory dynamos have been scarce.

The kinematic dynamo problem is rather well understood in the case of laminar flows [1]. Several simple but clever examples have been found in the past [2, 3, 4, 5] and more realistic geometries can be easily studied numerically [6]. However, most flows of liquid metal are fully turbulent before reaching the dynamo threshold: indeed, the magnetic Prandtl number, Pm = μ₀σν, where μ₀ is the magnetic permeability of vacuum, σ is the electric conductivity and ν is the kinematic viscosity, is smaller than 10-5 for all liquid metals. Since the dynamo action requires a large enough magnetic Reynolds number, Pm = μ₀σLU, where U is the fluid characteristic velocity and L is the characteristic scale, one expects to observe the dynamo effect when the kinetic Reynolds number, Re = UL/ν, is larger that 106. The kinematic dynamo problem with a turbulent flow is much more difficult to solve. A theoretical approach exists only when the magnetic neutral modes grow at large scale. It has been shown that the role of turbulent fluctuations may be twofold: on one hand, they decrease the effective electrical conductivity and thus inhibits dynamo action by increasing Joule dissipation. On the other hand, they may generate a large scale magnetic field through the“alpha effect” or higher order similar effects [7]. Consequently, it is not known whether turbulent fluctuations inhibits or help dynamo action. More precisely, for a given configuration of the moving solid boundaries generating the flow, the behavior of the critical magnetic Reynolds number Rm _c for the dynamo threshold, as a function of the flow Reynolds number Re (respectively Pm) in the limit of large Re (respectively small Pm), is not known.

The Ekman layer becomes hydrodynamically unstable at sufficiently large Reynolds number Re. For the case of purely vertical rotation, the Ekman layer instability has been studied experimentally by Faller [4] and Caldwell & Van Atta [1], and numerically by Faller & Kaylor [5] , Lilly [10], Melander [11] and Ponty et al. [13]. The linear and nonlinear behaviour of Ekman- Couette instabilities in a plane layer has been discussed by Hoffmann et al. [8]. The transition between the Taylor-Couette instability and the Ekman layer instability is explored in Hoffmann & Busse [9]. Two different Ekman layer instabilities are distinguished in these studies, which for historical reasons are now referred to as types I and II. Type II occurs when the Reynolds number Re* defined using the Ekman layer thickness, exceeds the experimentally measured value of Re*c ≃ (or 124.5 for type I). We will focus on the type II travelling wave, which has the smaller critical Reynolds number and so is easier to study numerically.

In attempting to create a laboratory scale dynamo, an experimentalist is faced with a daunting question: What sort of flow can I produce that will yield a dynamo? We present eight variations of a flow motivated by the s2t2 flow numerically studied by Dudley and James [1]. Pulse decay measurements of an externally applied magnetic field are used to quantify the approach to transition to dynamo action.

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 are 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 preconditioner accelerates the approximation. In azimuthal direction a spectral analysis allows a three-dimensional simulation for spherical shells with a wide gap.

Turbulent transport plays an important role in the evolution of astrophysical magnetic fields. The underlying idea is that the small-scale, turbulent motions increase the magnetic diffusivity above the value due to the molecular diffusion . However, it has been shown that in two dimensions the turbulent diffusion of large-scale magnetic fields is reduced significantly once the large-scale field strength exceeds a critical value, which can be much less than equipartition. The question that naturally arises is whether this holds true in three dimensions or whether magnetic fields of equipartition energy are required in order to suppress the effective turbulent diffusion. In order to elucidate this question we performed numerical calculations based on the full, incompressible MHD equations, that allow us to measure the effective magnetic diffusivity for different values of the large-scale magnetic field strength.

The dynamo experiment in Karlsruhe (Busse et al., 1998) has been successful in generating a self sustained magnetic field (Stieglitz and Müller, 2000). The focus of theoretical investigations now shifts from the study of the conditions for the onset of dynamo action to the mechanisms responsible for saturation. One might suspect that due to the guiding mechanical structures present in this experiment, saturation simply happens in that the magnetic field grows until the Lorentz force reduces the volumetric flow rate to its critical value, without any significant change to the shape of the velocity field. Any pressure applied by the pumps in addition to the critical pressure would be balanced by the Lorentz force according to this scenario. Experimental observations show however that volumetric flow rates continue to increase as a function of the applied pressure even when a steady dynamo field is present, which indicates that the velocity profile inside individual spin generators changes in response to the magnetic field.

The effect of shear on the growth of large scale magnetic fields in helical turbulence is investigated. The resulting large-scale magnetic field is also helical and continues to evolve, after saturation of the small scale field, on a slow resistive time scale. This is a consequence of magnetic helicity conservation. Because of shear, the time scale needed to reach an equipartition-strength large scale field is shortened proportionally to the ratio of the resulting toroidal to poloidal large scale fields.

The emergence of a large scale magnetic field from randomly forced isotropic strongly helical flows is discussed in terms of the inverse cascade of magnetic helicity and the α-effect. In simulations of such flows the maximum field strength exceeds the equipartition field strength for large scale separation. However, helicity conservation controls the speed at which this final state is reached. In the presence of open boundaries magnetic helicity fluxes out of the domain are possible. This reduces the timescales of the field growth, but it also tends to reduce the maximum attainable field strength.

Instabilities of MAC-waves type influenced by three diffusive processes in planar rapidly rotating stratified fluid layer permeated by the azimuthal magnetic field (1b) were investigated in dependence on Elsasser, Ekman, Roberts numbers and stratification parameter (7) for various mechanical and electrically conductive boundaries (8, 9). Among corresponding MC-waves the westward ones significantly determined by viscosity were revealed.

We study the fully nonlinear magnetohydrodynamic (MHD) equations in 3-D driven by a specific time-dependent forcing function, characterised by a frequency, Ω. Stable hydrodynamic solutions exist that are or are not dynamos depending on Ω. We examine the effect of such flows on an imposed large-scale (mean) field and measure the associated α-effect. We find that a significant non-zero α-effect can exist, indicating possible dynamo action on scales larger than the velocity scale, when sometimes no dynamo action exists at the velocity scale itself. Flows exist which are not small-scale dynamos, yet may be large-scale dynamos.

Many dynamical processes occuring on large scales in planets and in stars can be modeled surprisingly well on a laboratory scale, and experimental measurements and observations have made major contributions to our understanding of flows in the atmosphere, in the oceans or on the surface of the sun. The similarity of dynamical processes on the laboratory and on planetary or even stellar scales has been rationalized by the concept of eddy diffusivities which is based on the idea that small scale turbulence acts on large scale fluid motions in approximately the same way as molecular diffusivities affect the dynamics of fluids in a laboratory experiment. It is generally expected that the concept of eddy diffusivity also holds for the dynamo process of the generation of planetary and stellar magnetic fields. In fact, in the case of the Earth’s core there is no need to use the eddy version of the magnetic diffusivity since the magnetic Reynolds number may not be larger than a few hundreds which can be easily accommodated in numerical simulations. Since dynamical processes often depend sensitively on the ratio of diffusivities, the fact that numerical simulations of the geodynamo have not been able to reach magnetic Prandtl numbers P_m much less than unity, is more worrying. In this review we shall consider some typical results for convection driven dynamos in rotating spherical shells down to values P _m = 0.4.

We study the dynamo properties of asymmetric square patterns in Boussinesq Rayleigh-Bénard convection in a plane horizontal layer. Cases without rotation and with weak rotation about a vertical axis are considered. There exist different types of solutions distinguished by their symmetry, among them such with flows possessing a net helicity and being capable of kinematic dynamo action in the presence as well as in the absence of rotation. In the nonrotating case these flows are, however, always only kinematic, not nonlinear dynamos. Nonlinearly the back-reaction of the magnetic field then forces the solution into the basin of attraction of a roll pattern incapable of dynamo action. But with rotation added parameter regions are found where the Coriolis force counteracts the Lorentz force in such a way that the asymmetric squares are also nonlinear dynamos.

We present results from a systematic numerical survey of the interaction between turbulent convection and magnetic fields, as the net vertical magnetic flux through a large aspect ratio computational box is increased from low to high values. Different regimes are identified and discussed.

The geodynamo process of magnetic field generation occurs in the outer core of the Earth and is described by 3D MHD-equations. The focus of this paper is on a numerical solution and thus the equations will be considered, without the loss of generality, in a simple Boussinesq approximation. Denoting B the magnetic field, v the velocity, p the pressure and T the temperature, the dimensionless equations read (1)$$ \frac{{\partial B}} {{\partial t}}{\text{ = }}\nabla {\text{ }} \times {\text{ (v }} \times {\text{ B) + }}\nabla ^2 B, $$(2)$$ R_0 (\frac{{\partial v}} {{\partial t}}{\text{ + v}} \cdot \nabla v){\text{ = }}\nabla p + F + E\nabla ^2 {\text{v,}} $$(3)$$ \frac{{\partial T}} {{\partial t}}{\text{ + v}} \cdot \nabla {\text{ (}}T{\text{ + }}T_0 {\text{) = }}q\nabla ^2 T{\text{,}} $$(4)$$ \nabla \cdot {\text{B = 0,}} $$(5)$$ \nabla \cdot {\text{v = 0,}} $$ where F is the sum of Archimedean, Corioliss and Lorenz forces $$ {\text{F = }}qR_a Tr1_r {\text{ - (v}} \times {\text{1}}_z {\text{) + }}\Lambda {\text{(}}\nabla \times {\text{B)}} \times {\text{B,}} $$ and where the following dimensionless numbers are introduced $$ \begin{gathered} E = \frac{v} {{2\Omega L^2 }}{\text{ [Ekman number], R}}_0 = \frac{\eta } {{2\Omega L^2 }}{\text{ [Rossby number],}} \hfill \\ \Lambda {\text{ = }}\frac{{B_0^2 }} {{2\Omega \eta \mu \rho }}{\text{ [The Elsasser number], q = }}\frac{\kappa } {\eta }{\text{ [Roberts number],}} \hfill \\ R_a = \frac{{\alpha g_0 \Theta L}} {{2\Omega \kappa }}[Raleigh number]. \hfill \\ \end{gathered} $$

Spherical Couette flow is the flow induced in a spherical shell by fixing the outer sphere and rotating the inner one. Magnetic Couette flow is then the natural magnetohydrodynamic extension in which the fluid is taken to be electrically conducting, and a magnetic field is imposed. For the very strong fields we will consider here, the topology of this imposed field — which field lines thread both boundaries, and which only one or the other — turns out to be crucial. In this work we will therefore present a systematic survey of the possible field configurations, and in each case study the effect on the resulting flow.

For an idealized model of helically forced flow in an extended domain, we have investigated the interaction between dynamo action on different spatial scales. The evolution of the magnetic field is studied numerically; from an initial state of weak magnetization, through the kinematic and into the dynamic regime. We show how the choice of initial conditions is a crucial factor in determining the structure of the magnetic field at subsequent times. Furthermore, with initial conditions chosen to favour the growth of the small-scale field, the evolution of the large-scale magnetic field can be described in terms of the α-effect of mean field magnetohydrodynamics.

Hybrid vector spherical harmonic / poloidal-toroidal spherical spectral forms of the linearised magnetohydroynamic equations are described. The equations are highly structured with relatively few terms and form the basis of computer codes, which implement a wide range of dynamo problems in spherical and nearly spherical geometries.

A self-consistent anelastic planetary/satellite MHD system is optimally scaled. This scaling identifies key properties of MHD generators. Those are primarily located in thin (∼ r/R _n ) buoyancy layers at the liquid core boundary. Here n = 1/3 at the onset of convection, n = 1/2 for the developed magneto-convection and R, which is defined via the preliminary Reference State of the planet/satellite, is about the ‘turbulent’ Reynolds or/and magnetic Reynolds number. Simple diffusion and heat equations together with non-inertia state for magnetic and velocity equations are proposed in order to solve the real 3D MHD problems in planet or satellite. Boussinesq and anelastic approaches are compared.

The purpose of this paper is to present a model of Galactic dynamo driven by supernova explosions. I first describe, in physical and mathematical terms, the threefold impact of supernova-driven turbulence on the large-scale Galactic magnetic field, namely, the alpha-effect, the vertical advection, and magnetic diffusion. I then present recent numerical solutions of the Galactic 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.

Turbulent diamagnetism results in an expulsion of large-scale magnetic field from regions with a high intensity of turbulent motions (see, e.g. [1]): a turbulent conductive fluid behaves like a diamagnetic with an effective magnetic permeability μ ∝ β−1/2, with β being the turbulent magnetic diffusivity. This phenomenon can be described as a transport of the largescale magnetic field at a velocity which is proportional to — ▽β.

In the present paper, we investigate the influence of the Coriolis force and magnetic reconnection on the evolution of the Parker instability in galactic disks. We apply a model of a local gas cube, permeated by an azimuthal large-scale magnetic field and solve numerically resistive 3D MHD equations with the contribution of Coriolis force. We introduce a current dependent resistivity which switches on the magnetic reconnection above a certain critical current density. Our main goal is to study the magnetic field topology and the formation of large scale poloidal magnetic fields from the initial azimuthal field. Our simulations demonstrate that the Parker instability leads to the formation of helically twisted magnetic flux tubes which are next agglomerated by reconnection forming significant poloidal magnetic field component on the scale of the whole cube. Such an evolution represents a kind of the fast dynamo process as proposed by (1992).

We demonstrate that by including a flux of helicity in the magnetic helicity balance in the nonlinear galactic dynamo equations, the magnetic field dynamics are changed radically. The large-scale magnetic field now saturates at approximately the equipartition level. This is in contrast to the situation without the flux of helicity, when the magnetic helicity is conserved locally, leading to substantially subequipartition values for the equilibrium large-scale magnetic field.

Helioseismology provides important input and test data for dynamo theories of solar activity by measuring 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 obtained in 1996-2000 cover the period of transition from the ‘old’ solar cycle 22 to the ‘new’ cycle 23. These data have revealed correlated variations of zones of generation of the solar magnetic fields and zonal shear flows in the convection zone. An attempt is made to detect 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. However, the current results are controversial. By comparing the internal rotation profile with the rotation rates of the ‘old’ and ‘new’ magnetic fluxes it has been suggested that both fluxes were generated in a low-latitude zone of the tachocline. Zonal flows and structures migrating towards the equator and probably associated with dynamo waves have been detected in the solar interior. They show a curious sudden displacement towards higher latitudes in the second-half of 1999. This is not explained by the current dynamo theories and indicates a complex dynamical behavior of the solar dynamo. Recently developed methods of local helioseismology have allowed us to investigate processes of formation of active regions and sunspots. In particular, converging vortex flows have been found beneath a sunspot in agreement with Parker’s theory.

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 a travelling dynamo wave. It consists of two wings over a given hemisphere at every 11 yr sunspot cycle. They are low-latitudinal equatorward and high-latitudinal poleward waves whose propagation time approximately equal to the period of the cycle and which are shifted in time with respect to each other by approximately a half of the period. The waves represent toroidal and poliodal magnetic fields generated in the solar convective zone.

We present preliminary results of integrations of the induction equation using the velocity field produced by inertial modes in a spherical shell. The results indicate a possible dynamo action at rather large Reynolds number (R _e > 500).

Two types of nonlinearities (algebraic and dynamic) are discussed. The algebraic nonlinearity implies a nonlinear dependence of the mean electromotive force on the mean magnetic field. The dynamic nonlinearity is determined by a differential equation for the magnetic part of the α-effect. It is shown that the algebraic nonlinearity alone (which includes the nonlinear α-effect, the nonlinear turbulent diffusion, etc) cannot saturate the dynamo generated mean magnetic field while both, the algebraic and dynamic nonlinearities limit the mean magnetic field growth.The nonlinear mean electromotive force is calculated for an anisotropic background turbulence (i.e., the turbulence with zero mean magnetic field) with one preferential direction. It is shown that the toroidal and poloidal magnetic fields have different nonlinear turbulent diffusion coefficients. It is demonstrated that even for homogeneous turbulence there is an effective nonlinear velocity which exhibits diamagnetic or paramagnetic properties depending on anisotropy of turbulence and level of magnetic fluctuations in the background turbulence. The diamagnetic velocity results in the field is pushed out from the regions with stronger mean magnetic field, while the paramagnetic velocity causes the magnetic field tends to be concentrated in the regions with stronger field. Analysis shows that anisotropy of turbulence strongly affects the nonlinear turbulent diffusion coefficients and the nonlinear effective velocity.

The Sun is a natural site for a dynamo. In fact, the dynamo concept was introduced by Larmor in his 1919 report to the British Association for Advanced Science entitled”How could a rotating body as the Sun become magnetic?” Cowling’s famous anti-dynamo theorem appeared in his paper ”Magnetic fields of sunspots” (MNRAS. 94, 39 ,1934). Yet the origin of the Sun’s magnetic field is not well understood. Some scientists still challenge the dynamo as the source of solar magnetic field [7].

The generally accepted explanation of the sunspot cycle is in terms of a kinematic αΩ-dynamo wave propagating with fixed period from the pole to the equator (see Parker [1]); for a recent review see Rüdiger and Arlt [2]. Since such oscillatory behaviour is a robust feature generic to all αΩ- dynamo models, the simplicity of the idea is compelling. Both solar and stellar dynamos generally operate in convective spherical shells. There are two limiting cases, namely thick or thin shells as characterised by the ratio e of the shell thickness to shell radius. In the thick shell limit, it is necessary to consider the full partial differential equations involving the radial and latitudinal dependence. Conversely in the thin shell limit ε ≪ 1, it is possible to average the dynamo equations radially leaving a one-dimensional system dependent on the latitude θ alone.

We study the effects of vertical outflows on mean-field dynamos in disks. These outflows could be due to thermal winds or magnetic buoyancy. We analyse numerical solutions of the nonlinear mean-field dynamo equations using a two-dimensional finite-difference model. Contrary to expectations, a modest vertical velocity can enhance dynamo action. This can lead to super-exponential growth of the magnetic field and to higher magnetic energies at saturation in the nonlinear regime.

Heteroclinic cycles, i.e. trajectories that connect a finite number of saddle points of a dynamical system until they eventually come back to the same saddle point, are structurally unstable. They occur as bifurcation phenomena. However it has been shown that additional structure in the dynamical systems may lead to structurally stable behavior of these cycles. This is typically the case for Hamiltonian systems where it has been well known for a long time. In addition, symmetry in the equations will also force heteroclinic cycles to be structurally stable. This fundamentally is accomplished by the fact that symmetric systems will have invariant subspaces. Hence a connection between two saddles will become structurally stable if the restriction of one of the saddles to an invariant subspace leads to a sink in that subspace and hence the restriction of the flow to the invariant subspace may generate a saddle — sink connection. For a bibliography on the subject see [8] in the present volume and for a comprehensive introduction, see [6]. The prototypical example has been studied by Busse et al [4] and analysed as a robust heteroclinic cycle by Guckenheimer and Holmes [7]. Figure 1 illustrates the case: We consider a 3-d system where all coordinate planes and all coordinate axes are invariant subspaces. This can for instance be obtained for a system that has the symmetry group generated by reflections through the planes of coordinates and by cyclic permutation of these axes of coordinates. Now, assume that there exists a saddle on a coordinate axis with a 2-d stable manifold and a 1-d unstable manifold. By the permutation symmetry we will have saddles on each of the axes and if there is a heteroclinic orbit connecting 2 saddles in one invariant subspace there will be a whole cycle of these orbits connecting a saddle back to itself (see figure 1). In some parameter regimes the cycle is attracting. A time series for any of the three variables will show the variable to level off at a particular value until it transits to another value in a very short time where it will stay again for a long time etc. Upon addition of noise a“stochastic limit cycle” is created, whereby the transition times between saddles is exponentially distributed and unlike an attracting heteroclinic cycle without noise, there exists a finite mean period [11]. This dynamical behavior of relatively long quiescent behavior randomly followed by a quick transition to another long quiescent behavior makes this an attractive model for the behavior of magnetic reversals. The following sections will discuss our program to flesh out this model with more and more concrete physical details.

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.

Since the discovery by Schwabe in 1843 of the cyclic nature of sunspot activity, there has been a great deal of attention paid to the nature of these cycles. In 1849 Wolf established a measure of sunspot activity, the Wolf number, defined as W = k(10g + f where g is the number of sunspot groups, f is the total number of spots and k is a factor allowing comparison between different observation series. Based on this measure we have the Zurich series of Wolf numbers which extends back to 1749. Since the counts for W are taken over the whole visible surface of the sun, all information about the spatial distribution of the sunspot activity is lost. Realising the importance of such spatial information, Carrington in 1853 began the recording of the latitudes and longitudes of spots and this eventually resulted in the Greenwich series of photographic recordings of the solar surface which lasted from 1874 to 1976. Maunder introduced the famous Maunder or butterfly diagram (so called because the pattern of the cycles resembles the wings of a butterfly) which shows spot activity as a travelling wave beginning in the higher latitudes and proceeding to the equator.

Numerical studies of a laminar dynamo model have revealed two remarkable phenomena. We consider a spherical body of an electrically conducting incompressible fluid 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. For some investigations in addition to the forcing a rotation of the body is also considered. 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 can destroy the dynamo property of the motion so that it decays, and that the system ends up in a state with another motion incapable of dynamo action. However, for sufficient high magnetic Reynolds numbers a dynamo with a different symmetry may switch on after the ‘suicide’ of the original one. 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 ‘parthenogenetic’ dynamo is a genuine magneto-hydromagnetic state and hats no kinematic counterpart.

In their pioneering work BUSSE & HEIKES [2] looked at the classical Bénard problem in a rotating frame. They observed rolls with a certain axial direction which seem to be stable, ie they remain unchanged for long periods of time, but suddenly the behavior changes: new rolls appear which are rotated with respect to the original rolls by approximately 60 degrees. GUCKENHEIMER & HOLMES [15] looked at the Busse-Heikes problem from a theoretical point of view. They derived a three dimensional ODE exhibiting heteroclinic cycles. Thereafter many papers have dealt with various aspects of cycles: existence, stability, bifurcations and structural stability under certain settings: [1, 3, 4, 5, 7, 8, 16, 17, 18, 20, 21, 22, 25, 23, 24, 26, 27, 28, 29]. Due to the fact that solutions which pass near steady states remain there for a long time heteroclinic cycles serve as a model for metastable behavior. We see such a metastable behavior if we look at the polarity reversals of the magnetic field of the Earth, see for example GHIL & CHILDRESS [13]. Since the origin of the magnetic field and the mechanism of its reversals are unknown we try to look at it from a dynamical systems point of view and ask ourselves whether there are heteroclinic cycles in this problem. Here, we only look at the fluid mechanical part and not at full MHD-equations. In this paper we emphasize the role of rotation and the question what happens to heteroclinic cycles in the presence of rotations. Of course this approach dictates to look at the non rotating case first and treat the rotating case as a perturbation of the non rotating one. It is not clear whether such a approach is reasonable for studying the Earth’ field but on the other hand numerical computations indicate that the region of validity of the results which are presented here exceeds the marginal speeds of rotation allowed by the usual perturbation methods.

This paper describes the weakly nonlinear behaviour of a dynamo driven by rotating convection in the form of two-dimensional rolls. The linear problem is separable and the onset of dynamo action occurs at small wavenumbers m with a growth rate proportional to m2. In the weakly nonlinear regime a band of wavenumbers is unstable, and an amplitude equation is obtained describing the nonlinear interactions between modes of different wavenumber. The behaviour of the amplitude equation shows an inverse cascade, with the mode of fastest growth rate giving way to solutions with longer and longer wavelength, over long timescales.

The Earth’s magnetic field is by and large a steady dipole, but its history has been punctuated by intermittent excursions and reversals. This is at least superficially similar to the behaviour of differential equations containing structurally stable heteroclinic cycles. We present a model of the geodynamo that is based on the symmetries of velocity fields in a rotating spherical shell, and that contains such a cycle. Patterns of excursions and reversals that resemble the geomagnetic record can be obtained by introducing small symmetry-breaking terms.

Observations of the Sun and solar-type stars continue to reveal phenomena whose understanding is very likely to require a nonlinear framework. Here we shall concentrate on two such phenomena, namely the grand minima type behaviour observed in the Sun and solar-type stars and the recent dynamical variations of the differential rotation in the solar convection zone, deduced from the helioseismic observations, and discuss how their explanations have recently motivated the development/employment of novel ideas from nonlinear dynamics.

Modulation of cyclic magnetic activity associated with grand minima, as well as modulation associated with breaking of dipole or quadrupole symmetry, can be represented in low-order normal form equations. The behaviour found is robust and can be related to similar spatio-temporal patterns in mean field dynamo models.