François Pétrélis

Generation of a magnetic field by dynamo action in a turbulent flow of liquid sodium

We report the observation of dynamo action in the von Kármán sodium experiment, i.e., the generation of a magnetic field by a strongly turbulent swirling flow of liquid sodium. Both mean and fluctuating parts of the field are studied. The dynamo threshold corresponds to a magnetic Reynolds number R(m) approximately 30. A mean magnetic field of the order of 40 G is observed 30% above threshold at the flow lateral boundary. The rms fluctuations are larger than the corresponding mean value for two of the components. The scaling of the mean square magnetic field is compared to a prediction previously made for high Reynolds number flows.

Magnetic field reversals in an experimental turbulent dynamo

We report the first experimental observation of reversals of a dynamo field generated in a laboratory experiment based on a turbulent flow of liquid sodium. The magnetic field randomly switches between two symmetric solutions B and -B. We observe a hierarchy of time scales similar to the Earths magnetic field: the duration of the steady phases is widely distributed, but is always much longer than the time needed to switch polarity. In addition to reversals we report excursions. Both coincide with minima of the mechanical power driving the flow. Small changes in the flow driving parameters also reveal a large variety of dynamo regimes.

Simple Mechanism for Reversals of Earth's Magnetic Field

We show that a model, recently used to describe all the dynamical regimes of the magnetic field generated by the dynamo effect in the von Kármán sodium experiment, also provides a simple explanation of the reversals of Earths magnetic field, despite strong differences between both systems. The validity of the model relies on the smallness of the magnetic Prandtl number.

The von Karman Sodium experiment: Turbulent dynamical dynamos

The von Karman Sodium (VKS) experiment studies dynamo action in the flow generated inside a cylinder filled with liquid sodium by the rotation of coaxial impellers (the von Karman geometry). We first report observations related to the self-generation of a stationary dynamo when the flow forcing is R-pi-symmetric, i.e., when the impellers rotate in opposite directions at equal angular velocities. The bifurcation is found to be supercritical with a neutral mode whose geometry is predominantly axisymmetric. We then report the different dynamical dynamo regimes observed when the flow forcing is not symmetric, including magnetic field reversals. We finally show that these dynamics display characteristic features of low dimensional dynamical systems despite the high degree of turbulence in the flow.

Magnetohydrodynamics measurements in the von Kármán sodium experiment

We study the magnetic induction in a confined swirling flow of liquid sodium, at integral magnetic Reynolds numbers up to 50. More precisely, we measure in situ the magnetic field induced by the flow motion in the presence of a weak external field. Because of the very small value of the magnetic Prandtl number of all liquid metals, flows with even modest Rm are strongly turbulent. Large mean induction effects are observed over a fluctuating background. As expected from the von Karman flow geometry, the induction is strongly anisotropic. The main contributions are the generation of an azimuthal induced field when the applied field is in the axial direction (an Ω effect) and the generation of axial induced field when the applied field is the transverse direction (as in a large scale α effect). Strong fluctuations of the induced field, due to the flow nonstationarity, occur over time scales slower than the flow forcing frequency. In the spectral domain, they display a f−1 spectral slope. At smaller scales (and larger frequencies) the turbulent fluctuations are in agreement with a Kolmogorov modeling of passive vector dynamics.

On the magnetic fields generated by experimental dynamos

We review the results obtained by three successful fluid dynamo experiments and discuss what has been learnt from them about the effect of turbulence on the dynamo threshold and saturation. We then discuss several questions that are still open and propose experiments that could be performed to answer some of them.

Low-frequency noise controls on-off intermittency of bifurcating systems.

A bifurcating system subject to multiplicative noise can display on-off intermittency. Using a canonical example, we investigate the extreme sensitivity of the intermittent behavior to the nature of the noise. Through a perturbative expansion and numerical studies of the probability density function of the unstable mode, we show that intermittency is controlled by the ratio between the departure from onset and the value of the noise spectrum at zero frequency. Reducing the noise spectrum at zero frequency shrinks the intermittency regime drastically. This effect also modifies the distribution of the duration that the system spends in the off phase. Mechanisms and applications to more complex bifurcating systems are discussed.

THE DYNAMO EFFECT

We first present basic results about advection, diffusion and amplification of a magnetic field by the flow of an electrically conducting fluid. This topic has been initially motivated by the study of possible mechanisms to explain the magnetic fields of astrophysical objects. However, self-generation of a magnetic field by an electrically conducting fluid, the so-called dynamo effect, is also a typical bifurcation problem that involves many interesting aspects from the viewpoint of dynamical system theory: the effect of the flow geometry on the nature of the bifurcation, the effect of turbulent fluctuations on the threshold value, the saturation mechanisms above threshold, the dynamics of the generated magnetic field and the statistical properties of its fluctuations with respect to the ones of the turbulent flow. We have tried to emphasize some of these problems within the general presentation of the subject and more particularly in sections 6 and 7. These notes should not be considered as a review article. There exist many well known books and reviews on dynamo theory . For a general presentation of the subject, we refer to “Magnetic field generation in electrically conducting fluids” by Moffatt . The generation of large scale magnetic fields by small scale turbulent motions is reviewed in “Mean-field magnetohydrodynamics and dynamo theory” by Krause and Rädler . The problem of “fast dynamos” in the limit of large magnetic Reynolds number is studied by Childress and Gilbert in “Stretch, twist, fold: the fast dynamo” . Finally, we refer to “Magnetic fields in astrophysics” by Zeldovich, Ruzmaikin and Sokoloff 10 for a detailed review on magnetic fields of astrophysical objects and their possible role in the early evolution of the universe.

Reversals of a large-scale field generated over a turbulent background

We present a study of several systems in which a large-scale field is generated over a turbulent background. These large-scale fields break a symmetry of the forcing by selecting a direction. Under certain conditions, the large-scale field displays reversals so that the symmetry of the forcing is recovered statistically. We present examples of such dynamics in the context of the dynamo instability, of two-dimensional turbulent Kolmogorov flows and of turbulent Rayleigh–Bénard convection. In these systems reversals occur respectively for the dynamo magnetic field, for the large-scale circulation generated by a periodic forcing in space and for the large-scale roll generated by turbulent thermal convection. We compare the mechanisms involved and show that their properties depend on some symmetries of the system and on the way they are broken.

Dynamo regimes and transitions in the VKS experiment

Abstract. The Von Kármán Sodium experiment yields a variety of dynamo regimes, when asymmetry is imparted to the flow by rotating impellers at different speed F1 and F2. We show that as the intensity of forcing, measured as F1+F2, is increased, the transition to a self-sustained magnetic field is always observed via a supercritical bifurcation to a stationary state. For some values of the asymmetry parameter θ = (F1–F2)/(F1+F2), time dependent dynamo regimes develop. They are observed either when the forcing is increased for a given value of asymmetry, or when the amount of asymmetry is varied at sufficiently high forcing. Two qualitatively different transitions between oscillatory and stationary regimes are reported, involving or not a strong divergence of the period of oscillations. These transitions can be interpreted using a low dimensional model based on the interactions of two dynamo modes.