One of the most important processes, responsible for many dynamical
phenomena observed in the Sun, is the emergence of magnetic flux from the
solar interior in active regions and the modification of the coronal
magnetic field in response to the emergence. In fact, magnetic flux
emergence might be responsible for the appearance of small-scale events
(e.g., compact flares, plasmoids, active-region-associated X-ray
brightenings) and large-scale events (e.g., X-class flares and CMEs).

However, it is clear that the question of how exactly the magnetic fields,
which are generated by the solar dynamo in the convection zone, emerge
through the photosphere and chromosphere into the corona has still not been
solved. Studying the process of flux emergence is an important step towards
the understanding of the dynamic coupling between the solar interior and
the outer solar atmosphere.

In this talk, we review the recent progress and discuss what further
developments are required, to better understand this magnetic coupling in
the Sun.

We address the question of stability of the Euler flow with elliptical
streamlines in a rotating frame, interacting with uniform, external
magnetic field perpendicular to the plane of the flow. Our motivation for
this study is of astrophysical nature, since many astrophysical objects,
such as stars, planets and accretion disks are tidally deformed through
gravitational interaction with other bodies. Therefore, the ellipticity of
the flow models the tidal deformations in the simplest way. The joint
effect of the magnetic field and the Coriolis force is studied here
numerically and analytically in the limit of small elliptical (tidal)
deformations $\left(\zeta\ll1\right)$, using the analytical technique
developed by Lebovitz \& Zweibel (2004). We find that the effect of
background rotation and external magnetic field is quite complex. Both
factors are responsible for new destabilizing resonances as the vortex
departs from axial symmetry $\left(\zeta\ll1\right)$, however just like in
the non-rotating case, there are three principal resonances leading to
instability in the leading order. The presence of the magnetic field is
very likely to destabilize the system with respect to perturbations
propagating in the direction parallel to the magnetic field, if the basic
vorticity and the background rotation have opposite signs (i.e.
for\emph{anticyclonic}background rotation).

We present the dependence of the growth rates of the modes on various
parameters describing the system, such as the strength of the magnetic
field $\left(h\right)$, the inverse of the Rossby number
$\left(\mathcal{R}_{v}\right)$, the ellipticity of the basic flow
$\left(\epsilon\right)$ and the direction of propagation of modes
$\left(\vartheta\right)$. Our analytical predictions agree well with the
numerical calculations.

We present results of pseudo-spectral simulations of two-dimensional MHD
turbulence in bounded domains. An efficient way to compute these flows,
using a penalization method, was recently presented in [1]. The simulations
are performed in circular, square and elliptic domains. It is observed that
the tendency to spin-up, i.e., to spontaneously generate angular momentum,
is significantly increased in non-axisymmetric geometries. An increased
strength of the magnetic fluctuations is shown to enhance the generation of
angular momentum.

[1] S. Neffaa, W.J.T. Bos, and K. Schneider. The decay of magnetohydrodynamic turbulence in a
confined domain. Phys. Plasmas, 15:092304, 2008.

[2] W.J.T. Bos, S. Neffaa, and K. Schneider. Rapid generation of angular momentumin bounded
magnetized plasma. Phys. Rev. Lett., 101:235003, 2008.

We have modelled, based on 2-D flux transport dynamo models, the magnetism and cyclic activity of solar type stars rotating from 0.25 up to 10 times the solar rate. Our models make use of recent 3-D simulations (Brown et al. 2008) to infer the meridional circulation amplitude and profiles as a function of rotation rate. We find that in order to reconcile the observations of faster cycle vs rotation rate, two cells in latitude for the meridional circulation must be invoked otherwise flux transport models cannot reproduce the observations and lead to the odd result of slower cycle!

The elliptical (or tidal) instability takes place in planetary cores elliptically deformed by gravitational effects. It has been studied by our group for several years. Here, we present the first numerical study of a tidal dynamo in a triaxial ellipsoidal geometry. We first validate our MHD numerical approach by comparing its results to the theoretical and experimental ones already obtained regarding the hydrodynamics of the elliptical instability (Lacaze et al., 2005), its interaction with thermal effects (Le Bars et al., 2006) and the induction phenomena in presence of an imposed external magnetic field (Herreman et al., 2009). We then present our first results on the existence of a tidal dynamo, showing that the elliptical instability is able to generate and maintain a magnetic field in a deformed spherical shell, hence validating the initial suggestion of Malkus in the seventies.

[1] Lacaze, L., Le Gal, P. and Le Dizès, S., Elliptical instability of a flow in a rotating shell Physics of The Earth and Planetary Interiors, Volume 151, Issues 3-4, 15 August 2005, Pages 194-205.

[2] Le Bars, M. and Le Dizès, S. Thermo-elliptical instability in a rotating cylindrical shell Journal of Fluid Mechanics (2006), 563:189-198 Cambridge University Press.

[3] W Herreman, M Le Bars, P Le Gal On the effects of an imposed magnetic field on the elliptical instability in rotating spheroids, Phys. Fluids 21 (2009)

We consider the linear stability of nonlinear magnetohydrodynamic basic states to long-wavelength 3D perturbations. The basic states are obtained from a specific forcing function in the presence of an initially uniform mean field of strength $\mathcal{B}$. By extending to the nonlinear regime the kinematic analysis of Roberts (1970), we show that it is possible to predict the growth rate of these perturbations by applying mean field theory to \textit{both} the momentum and the induction equations. If $\mathcal{B}=0$, these equations decouple and large-scale magnetic and velocity perturbations may grow via the kinematic $\alpha$-effect and the AKA instability respectively. However, if $\mathcal{B} \neq 0$, the momentum and induction equations are coupled by the Lorentz force; in this case, we show that four transport tensors are now necessary to determine the growth rate of the perturbations. We illustrate these situations by numerical examples based on 2D basic states as in H ughes \& Proctor (2009). Time permitting, we will address the case of 3D basic states in more details. The implications of our results for the study of $\alpha$-quenching are discussed.

Turbulent flows at high magnetic Reynolds numbers have been shown to act as efficient small-scale dynamos, with the magnetic energy predominantly on the same scale as the turbulent velocity, and with negligible magnetic energy at large scales. This poses a problem in cases such as the solar dynamo, which acts as a large-scale dynamo. Here we investigate an alternative dynamo mechanism, by considering whether flows susceptible to magnetic buoyancy instability can generate an electromotive force on the scales required. We consider a plane layer of fluid in a configuration unstable to the magnetic buoyancy instability and calculate the emf generated by the resulting flows.

I present a short review about reversals of the magnetic field generated by the dynamo process and emphasize that similar features are observed in palaeomagnetism, the VKS experiment and numerical simulations, although these three systems involve strongly different flows. I then discuss low dimensional models for the large scale dynamics of the magnetic field.

In many astrophysical and geophysical systems, the small scale motions are
strongly influenced by rotation and the presence of a large-scale magnetic
field. The separate effects of the Coriolis force and of the Lorentz force
on homogeneous turbulence have been extensively studied. We consider here
both effects.

We present the results of direct numerical simulations of incompressible
homogeneous magnetohydrodynamic (MHD) turbulence submitted to a uniform
magnetic field and to solid-body rotation. The magnetic field $\bm{B}_0$ is
vertical and is given in Alfv\'en-speed units. The rotation rate
$\bm{\Omega}$ is, in a first approach, aligned with $\bm{B}_0$ so that the
configuration remains axisymmetric. The magnetic Reynolds number $R_M$
varies from $1$ to $100$ whereas the Reynolds number is about $100$. The
Elsasser number $\rm{\Lambda}=B_0^2/2\Omega\eta$, characterizing the
relative importance of the Lorentz force to the Coriolis force, varies from
$10^{-1}$ to $10$. We are particularly interested in low Rossby number and
high interaction parameter regimes.

We will discuss the main energetic properties of such flows, and different
statistical indicators (Shebalin angles, angular energy spectra,
correlation lengths...) will be presented to quantify the anisotropy
developing in such flows. Some preliminary results about Lagrangian
statistics and non-axisymmetric configuration (\textit{i.e.}
$\bm{\Omega}\perp\bm{B}_0$) will also be proposed.

In a first approximation, Jupiter is made of two fluid layers: a deep
metallic hydrogen layer where the jovian dynamo is generated and a
superficial "atmospheric" non metallic envelope of approximately 10,000
km depth (10-20% of the total radius of the planet). Recent numerical
simulations of three-dimensional rotating convection in a relatively thin
spherical shell modelling the atmospheric layer of Jupiter reproduce zonal
winds similar to the bands visible on Jupiter's surface [1]. The simulated
flow displays a quasi two-dimensional structure aligned with axis of
rotation. Thus [1] suggests that the zonal winds may be "deep rooted"
within Jupiter's interior. These zonal winds are believed to be damped
within the deep metallic hydrogen layer [2]. The main question that leads
to our work is simple: can the external forcing created by the zonal winds
at the top of the metallic hydrogen region drive a dynamo? The external
zonal winds generate geostrophic shear layers inside whi ch may lead to
non-axisymmetric hydrodynamic instabilities. Such instabilities are known
to excite dynamo action [3], [4] and the jovian dynamo will be discussed
following these ideas.

[1] Heimpel, M.H., Aurnou, J.M., Wicht, J., 2005. Simulation of equatorial
and high-latitude jets on Jupiter in a deep convection model. Nature 438,
193-196.

[2] Kirk, R.L., Stevenson, D.J., 1987. Hydromagnetic constraints on deep zonal flow in the
giant planets. Astrophys. J. 316, 816-846

[3] Guervilly C. and Cardin P., 2009. Numerical simulations of dynamos generated in spherical
Couette flows, submitted to Geophys. Astrophys. Fluid Dyn.

[4] Schaeffer, N. and Cardin, P., 2006. Quasi-geostrophic kinematic dynamos at low magnetic
Prandtl number. Earth Planet. Sci. Lett., 245, 595-604.

Dynamo action is a mechanism by which a magnetic field is self generated by the turbulent flow of an electrically conducting fluid. I present results of 3D direct numerical simulations for different flow geometries used to try to generate experimental dynamos. In the case of the VKS experiment, I show how boundaries with a high magnetic permeability lead to a significant decrease of the critical magnetic Reynolds number, thus allowing the observation of dynamo action. I also understand the mechanism leading to experimentally observed geometries of the magnetic field. Finally, I will present turbulent dynamo simulations showing chaotic reversals of the magnetic field. These simulations show that the nature of the reversals is strongly modified by the value of the magnetic Prandtl number. In the low Pm regime, the behavior of the reversing magnetic field is understood using a simple model of amplitude equations derived by symmetry arguments.

Short wavelength, fast growing m=0 interchange instabilities can lead at saturation to ion viscous heating on the Alfven transit time-scale if the magnetic Prandtl number is large. The example of a stainless-steel wire array Z-pinch at 18MA at Sandia National Laboratory gives 200 to 300keV ion temperature at stagnation which was confirmed by Doppler broadenied spectra. In high Z plasmas electron viscosity can also be important if the equipartition time is much less than the Alfven transit time. Experiments at lower currents are however unlikely to have a high magnetic Prandtl number, and the MHD spectrum is limited by resistivity. Then this powerful heating mechanism is absent. The mechanism can be considered as a negative dynamo, and magnetic energy is rapidly converted to ion thermal energy via the kinetic energy of the compressional and sheared flow resulting from the MHD instabilities.

The elliptical or tidal instability has been proposed as an alternative mecanism to drive flows in planetary interiors, and consequently their dynamos. Yet, in between idea and plausible mecanism, much work still needs to be done. Up to now, most studies concentrated on the linear instability mecanism, which is now well understood as a parametric instability. On the other hand, we do not understand much about the late nonlinear stages of this instability, even though it is essential to estimate how important tidally driven flows can become. Neither do we know whether such flows are capable of dynamo action. We adress the first problem here, and present theoretical and experimental results obtained for and with the IMAGINE set-up. This experiment realises a cylindrical set-up of the elliptical instability of a liquid metal, under an imposed magnetic field. The magnetic field has a double interest. We can follow inertial waves excited by elliptical instability through the fields they induce. Secondly, we use Joule damping to control the transition towards the complex nonlinear dynamics associated with the elliptical instability. This allows us to highlight the importance of small frequency detuning effects on the criticality of the bifurcation. We discuss the transition scenario, and further quantify how inertial wave amplitudes behave far from the linear instability threshold under varying excentricity and imposed rotation speed.

Traditionally, large- and small-scale dynamos have been tackled theoretically independently. Indeed the physical ideas behind them are very different: the former, treated within the framework of mean field electrodynamics, requires flows that lack reflectional symmetry (e.g. helical flows); the latter, treated as fast dynamos, need chaotic fluid trajectories. In astrophysics the fluid flows are both chaotic and helical, and the magnetic Reynolds number is very large, so the ingredients for both large- and small-scale dynamo action are present. I will discuss what might happen in this case, and highlight some of the theoretical difficulties in generating large-scale magnetic fields and also a possible way out of these difficulties.

From inversions of the geomagnetic secular variation, we have found a torsional oscillation, inside the Earth's core, recurring every 6 years. It accounts well, writing the conservation of the total angular momentum of the Earth, for both the phase and the amplitude of the 6 yrs length-of-day signal detected over the second half of the twentieth century. These torsional waves propagate through the entire Earth's outer core in a few years only. This directly translates into a slowness of Alfv\'en waves that gives a mean field strength in the cylindrically radial direction of about 1mT. Scaling laws, obtained from geodynamo numerical simulations, between the dipole field strength at the dynamo surface and its interior had previously yielded typical rms total intensity of the order of 2mT within the Earth's fluid core. Our present work thus reconciles geodynamo simulations and studies of geostrophic motions in the Earth's core from geomagnetic data. Previous estimates of torsional oscillations period were much longer, about 60 yrs, and were associated with rms strength of the field in the cylindrically radial direction and calculated for all length scales of 0.2mT only, even smaller than the rms strength of the large scale (harmonic degree $n \le 13$) visible field at the core-mantle boundary. There are indeed pulsations of the total energy of the geomagnetic SV on 60-80 yrs periods. We argue, however, that on these longer timescales, the magnetic and velocity fields remain in a Taylor state. This assumption may be used to further constrain core flow inversions from geomagnetic data.

Geodynamo models based on convection-driven flow in a rapidly rotating spherical shell frequently give rise to strong stable dipolar magnetic fields. This is in sharp contrast to convection-driven flows with no rotation or slow rotation, where the large scale field is often rather weak compared to the small scale field. In particular, parameter regimes in spherical dynamos where the inertial terms play a limited role are often strongly dipolar. Kinematic dynamo theory makes a distinction between the onset of dipolar and quadrupolar modes, but for the types of flow arising in rotating convection-driven dynamos, the onset of dynamo action for quadrupolar modes often occurs close to the onset of dipolar modes, and indeed in some reasonable models occurs before the onset of dipolar modes. Here we explore nonlinear mechanisms due to the action of Lorentz force which may give rise to a strong preference for dipolar modes. The coherent structures that arise in nonlinear rapidly rota ting convection are affected by magnetic field in ways which significantly enhance their ability to generate more magnetic field by strongly increasing their helicity. This mechanism is expected to lead to subcritical behaviour in the dynamo. There is already numerical evidence for subcriticality in dynamo models, and this may be connected with the sudden demise of the Martian dynamo.

We present the first 3D MHD study in spherical geometry of the non-linear dynamical evolution of magnetic flux tubes in a turbulent rotating convection zone. These numerical simulations use the anelastic spherical harmonic (ASH) code. We seek to understand the mechanism of emergence of strong toroidal fields through a turbulent layer from the base of the solar convection zone to the surface as active regions, with a particular focus on the effects of self-consistently generated mean flows. Weak field cases indicate that downflows and upflows control the rising velocity of particular regions of the rope and could in principle favour the emergence of flux through \Omega-loop structures. For these cases, we focus on the orientation of bipolar regions and find that sufficiently arched structures are able to create bipolar regions with a predominantly East-West orientation. Meridional circulation seems to determine the trajectory of the magnetic rope when the field strength has been significantly reduced near the top of the domain. Local field emergence also feeds back on the horizontal flows thus perturbing the meridional circulation via Maxwell stresses. Finally differential rotation makes it more difficult for tubes introduced at low latitudes to emerge at the surface. \\ We reintroduce these 3D results in 2D mean-field Babcock-Leighton flux-transport dynamo models, and in particular the time-delay caused by the emergence of toroidal structures from the base of the CZ to the surface. We find that these time delays introduce a strong modulation of the solar cycle amplitude, even when strong and thus rapidly rising flux tubes are considered. This modulated activity and the resulting butterfly diagram are then more compatible with observations than the standard Babcock-Leighton model.

We investigate the $\alpha$ effect and the turbulent transport of magnetic flux ($\gamma$ effect) by considering a turbulence driven by an isotropic external forcing in the presence of a background shear flow. We show that for weak shear, both $\gamma$ and $\alpha$ effects are enhanced. On the other hand, a sufficiently strong shear leads to severe quenching of $\gamma$ and $\alpha$ effects due to shear stabilisation. One of our findings is the presence of $\gamma$ effect even in the case of an isotropic forcing due to the anisotropy induced by shear flows.

The magnetorotational instability (MRI) is believed to be an efficient way
of transporting angular momentum in accretion discs. It has also been
suggested to amplify magnetic fields in discs, the instability acting as a
nonlinear dynamo. These nonlinear (or subcritical) dynamos appear only for
\emph{finite amplitude} magnetic field perturbations, making the ordinary
kinematic (and linear) approach inappropriate.

In this talk, I
present new results aiming at describing the nonlinear dynamo operating in
accretion discs. I will show that a large-scale magnetic field can be
driven by the MRI of the same field, via a regeneration cycle. This
regeneration cycle can be understood as an ordinary $\omega$ effect
combined with a nonlinear feedback, regenerating the poloidal field under
certain conditions. I will describe in details this feedback using
numerical results and a quasi-linear analysis of localized MRI modes. In
particular, I will show that the Lorentz force is a \emph{required}
ingredient to get the dynamo loop in these systems and I will compare these
findings to more classical kinematic dynamo theories.

The solar magnetic field is maintained by a hydromagnetic dynamo, and it is generally accepted that strong toroidal magnetic field is produced at the tachocline below the solar convection zone. The location of production of the poloidal field via the $\alpha$-effect is still in question. One such location is the solar surface, where decay of bipolar magnetic regions produces poloidal magnetic field, this is known as the Babcock-Leighton mechanism. Another possible location is the base of the solar convection zone. Recent work has looked at the competition in the solar dynamo between surface and deep-seated $\alpha$-effects. The $\alpha$-effects studied were local effects. However, flux-transport dynamo calculations use non-local poloidal source terms, that is, they act at the surface but are proportional to the toroidal field at the base of the convection zone. We will discuss the effects of non-local competing $\alpha$-effects on the dynamo equations to explore the validity of this type of model. We will also include the effects of meridional circulation.

It is well known that the effect of shear on a weak random vorticity perturbation is to selectively amplify modes that vary slowly in the streamwise direction -- the mechanism by which 'streamwise vortices' are produced. This is a transient instability, ultimately controlled by weak viscosity. The manner in which this instability is influenced by a spanwise magnetic field is investigated.

Numerical calculations of dynamo action in a rapidly rotating spherical shell are performed. We report cases of multistability, i.e. situations for which several solutions with different kinetic energies and different abilities to produce self-sustained dynamos are observed for a given set of driving parameters. We then introduce a modulation (in time) of the thermal forcing. These calculations are carried out for parameters in the vicinity of the ones used in the bistability cases. It is indeed likely that the modulation will have a stronger effect in these regions of the parameter space. In all our simulations, the convective flow roughly follows the modulation. However interesting effects develop in a number of cases. If the imposed modulation is large enough to imply a crossing of the dynamo threshold, magnetohydrodynamic interactions are able to change the dynamics of the system with different hydro and magnetic modes being selected during each oscillation, leading to an overall very complex behavior of the system. Applying a noise-like modulation of relatively low amplitude and of high frequency (compared to the inverse magnetic diffusion time), we observe sudden bursts or drops in magnetic activity. These events are associated with sudden and dramatic changes in the flow geometry.

We study numerically the behaviour of coherent enstrophy for decaying flows in periodic and circular domains, when the initial Reynolds number tends to infinity. The computation is done using a Fourier pseudo-spectral scheme with volume penalization. Wavelet filtering is applied to split enstrophy into coherent and incoherent contributions. In both cases, we find that coherent enstrophy dissipation does not vanish when Reynolds number tends to infinity. In the circular case, coherent enstrophy diverges due to the boundary layer, but after a certain time its derivative seems to remain bounded independently of Reynolds number, indicating that a balance has established between production at the wall and dissipation in the bulk.

I discuss the behaviour of the magnetic field generated by dynamo effect in the vicinity of the dynamo threshold. Taking into account two modes of magnetic field, random reversals are predicted when the modes are close to achieve a saddle-node bifurcation. If the two modes are of dipolar and quadrupolar symmetry, the reversals occur when the flow breaks equatorial symmetry. Hemispherical dynamos also find a simple explanation within this scenario. This is illustrated with measurements of the Von Karman Sodium experiment and with the solution of a kinematic $\alpha^2$ dynamo.

We use a simple model of Bullard-type disc dynamo, in which the disc rotation rate is subject to harmonic oscillations, to analyze the generation of magnetic field by the parametric resonance mechanism. The problem is governed by a damped Mathieu equation. The Floquet exponents, which define the magnetic field growth rates, are calculated depending on the amplitude and frequency of the oscillations. Firstly, we show that the dynamo can be excited at significantly subcritical disc rotation rate when the latter is subject to harmonic oscillations with a certain frequency. Secondly, at supercritical mean rotation rates, the dynamo can also be suppressed but only in narrow frequency bands and at sufficiently large oscillation amplitudes.

Inside a periodic box, we choose a specific volume forcing which produce a large scale vortex velocity, able to induce a Ponomarenko dynamo. The linear and the non linear aspects of the different dynamo modes will be presented, and we will also focus on the role of the fluctuated velocity artificially added.

The transition to an intermittent mean--field dynamo is studied using numerical simulations of isotropic magnetohydrodynamic turbulence driven by a helical forcing. The low-Prandtl number regime is investigated by keeping the kinematic viscosity fixed while the magnetic diffusivity is varied. Just below the critical parameter value for the onset of dynamo action, a transient mean--field with low magnetic energy is observed. After the transition to a sustained dynamo, the system is shown to evolve through different types of intermittency until a large--scale coherent field with small--scale turbulent fluctuations is formed. Prior to this coherent field stage, a new type of intermittency is detected, where the magnetic field randomly alternates between phases of coherent and incoherent large--scale spatial structures. The relevance of these findings to the understanding of the physics of mean--field dynamo and the physical mechanisms behind intermittent behaviour observed in stellar magnetic field variability are discussed.

We address the question under which conditions a saturated velocity field stemming from geodynamo simulations leads to an exponential growth of the magnetic field in a corresponding kinematic calculation. We perform global self-consistent geodynamo simulations and calculate the evolution of a kinematically advanced tracer field. The self-consistent velocity field enters the induction equation in each time step, but the tracer field does not contribute to the Lorentz force. We find two dynamo regimes in which the tracer field either grows exponentially or approaches a state aligned with the actual self-consistent magnetic field after an initial transition period. Both regimes can be distinguished by the Rossby number and coincide with the dipolar and multipolar dynamo regimes identified by Christensen and Aubert (2006). Dipolar dynamos with low Rossby number are kinematically stable whereas the tracer field grows exponentially in the multipolar dynamo regime. This difference in the saturation process for dynamos in both regimes comes along with differences in their time variability. Within our sample of 20 models, solely kinematically unstable dynamos show dipole reversals and large excursions. The complicated time behaviour of these dynamos presumably relates to the alternating growth of several competing dynamo modes. On the other hand, dynamos in the low Rossby number regime exhibit a rather simple time dependence and their saturation merely results in a fluctuation of the fundamental dynamo mode about its critical state.

We discuss a fluctuation dynamo model based on directly modelled magnetic reconnections rather than magnetic diffusion. The plasma heating rate is significantly larger than that in comparable diffusion-based dynamos. The probability distribution function of the energy released in individual reconnection events has the slope of -3. We demonstrate that this scaling is not sensitive to the specific velocity field used. If applied to the solar corona, this implies coronal heating by nanoflares, as suggested by Parker (1963). Nonlinear states of the dynamo are consistent with the idea that it saturates via supprression of the (effective) magnetic diffusivity.

Previous theoretical work has speculated about the existence of double-diffusive magnetic buoyancy instabilities of a dynamically evolving horizontal magnetic layer generated by the interaction of forced vertically-sheared velocity and a background vertical magnetic field. Here, we confirm numerically that if the ratio of the magnetic to thermal diffusivities is sufficiently low then such instabilities can indeed exist, even for high Richardson number shear flows. Magnetic buoyancy may therefore occur via this mechanism for parameters that are likely to be relevant to the solar tachocline, where regular magnetic buoyancy instabilities are unlikely.

We discuss some of the computational challenges in constructing a 3D magnetohydrodynamic code in a finite cylindrical geometry, such as that of the VKS experiment. We generalize techniques used in the spherical geometry, notably the the toroidal-poloidal decomposition to construct velocity and magnetic fields which are divergence-free, i.e. incompressible and free of magnetic monopoles, respectively. We also use the formalism of the Dirichlet-to-Neumann mapping to formulate boundary conditions coupling the magnetic field inside the cylinder to that in the external vacuum, without calculating the external field.

We investigate the interaction of a fluctuating $\alpha$-effect with large-scale shear in a simple 1-d dynamo wave model. We extend the calculations of Proctor (2007) to include spatial variation of the fluctuations, and find that there can be a mechanism for magnetic field generation, even when the mean $\alpha$ is zero, provided the spatiotemporal spectrum of the fluctuations has an appropriate form. We investigate dynamo action when the new term arising from the fluctuations is non-zero, and present results of the stability and parity of the system.

The first spots of the new sunspot cycle appeared in January 2008 but since then it has made a very feeble start. The grand maximum that we have all experienced is apparently coming to an end, with an estimated life expectancy of a further decade or two at most. There is then a 40\% probability that we shall experience a new grand minimum. These changes correspond to long term modulation of cyclic dynamo action with a characteristic period of about 200 years. The cause could either be stochastic (with massive statistical fluctuations in convection, differential rotation or meridional circulation) or else deterministic (with transitions from periodic to quasiperiodic and then to chaotic behaviour). The presence of a persistent long-term periodicity makes the latter explanation seem more likely.

Convective turbulence is known to maintain a state of differential rotation within the Sun's outer convective envelope, and also drives the Sun's meridional circulation. Yet the radiation zone beneath is found to rotate uniformly, separated from the convection zone by a thin shear-layer called the tachocline. This uniform rotation can be explained by the presence of a global scale magnetic field within the radiation zone, provided that (i) this magnetic field remains confined to the radiation zone against diffusion and (ii) the meridional circulations within the convection zone and tachocline are prevented from burrowing into the radiation zone. In high latitudes, the meridional circulation is expected to be downward within the tachocline, and can hold the internal magnetic field in advective--diffusive balance across a thin "magnetic confinement layer" at the base of the tachocline (Gough & McIntyre 1998). We will extend the Gough & McIntyre model to cover the entire pol ar region, and describe modifications to the model arising from compositional stratification.