Ludovic Petitdemange

Topology and field strength in spherical, anelastic dynamo simulations

Context: Numerical modelling of convection driven dynamos in the Boussinesq approximation revealed fundamental characteristics of the dynamo-generated magnetic fields and the fluid flow. Because these results were obtained for an incompressible fluid of constant density, their validity for gas planets and stars remains to be assessed. A common approach is to take some density stratification into account with the so-called anelastic approximation. Aims: The validity of previous results obtained in the Boussinesq approximation is tested for anelastic models. We point out and explain specific differences between both types of models, in particular, with respect to the field geometry and the field strength, but we also compare scaling laws for the velocity amplitude, the magnetic dissipation time, and the convective heat flux. Methods: Our investigation is based on a systematic parameter study of spherical dynamo models in the anelastic approximation. We make use of a recently developed numerical solver and provide results for the test cases of the anelastic dynamo benchmark. Results: The dichotomy of dipolar and multipolar dynamos identified in Boussinesq simulations is also present in our sample of anelastic models. Dipolar models require that the typical length scale of convection is an order of magnitude larger than the Rossby radius. However, the distinction between both classes of models is somewhat less explicit than in previous studies. This is mainly due to two reasons: we found a number of models with a considerable equatorial dipole contribution and an intermediate overall dipole field strength. Furthermore, a large density stratification may hamper the generation of dipole dominated magnetic fields. Previously proposed scaling laws, such as those for the field strength, are similarly applicable to anelastic models. It is not clear, however, if this consistency necessarily implies similar dynamo processes in both settings.

Influence of the mass distribution on the magnetic field topology

Context. Three-dimensional spherical dynamo simulations carried out within the framework of the anelastic approximation have revealed that the established distinction between dipolar and multipolar dynamos tends to be less clear than it was in Boussinesq studies. This result was first interpreted as a direct consequence of the existence of a larger number of models with a high equatorial dipole contribution, together with an intermediate dipole field strength. However, this finding has not been clearly related to specific changes that would have been introduced by the use of the anelastic approximation. Aims. In this paper, we primarily focus on the effects of choosing a different mass distribution. Indeed, it is likely to have as large consequences as taking a stratified reference state into account would, especially when comparing our results to previous Boussinesq studies. Methods. Our investigation is based on a systematic parameter study of weakly stratified anelastic dynamo models. Results. We show that the tendencies highlighted in previous anelastic dynamo simulations are already present in the Boussinesq limit. Thus they cannot be systematically related to anelastic effects. Actually, a central mass distribution can result in changes in the magnetic field topology that are mainly due to the concentration of convective cells close to the inner sphere.

Dipolar dynamos in stratified systems

Observations of low-mass stars reveal a variety of magnetic field topologies ranging from large-scale, axial dipoles to more complex magnetic fields. At the same time, three-dimensional spherical simulations of convectively driven dynamos reproduce a similar diversity, which is commonly obtained either with Boussinesq models or with more realistic models based on the anelastic approximation, which take into account the variation of the density with depth throughout the convection zone. Nevertheless , a conclusion from different anelastic studies is that dipolar solutions seem more difficult to obtain as soon as substantial stratifications are considered. In this paper, we aim at clarifying this point by investigating in more detail the influence of the density stratification on dipolar dynamos. To that end, we rely on a systematic parameter study that allows us to clearly follow the evolution of the stability domain of the dipolar branch as the density stratification is increased. The impact of the density stratification both on the dynamo onset and the dipole collapse is discussed and compared to previous Boussinesq results. Furthermore, our study indicates that the loss of the dipolar branch does not ensue from a specific modification of the dynamo mechanisms related to the background stratification, but could instead result from a bias as our observations naturally favour a certain domain in the parameter space characterized by moderate values of the Ekman number, owing to current computational limitations. Moreover, we also show that the critical magnetic Reynolds number of the dipolar branch is scarcely modified by the increase of the density stratification, which provides an important insight into the global understanding of the impact of the density stratification on the stability domain of the dipolar dynamo branch.

Two-dimensional non-linear simulations of the magnetostrophic magnetorotational instability

We have shown that a simple, modified version of the Magnetorotational Instability (MRI) can, in principle, develop in the Earths outer liquid core in the presence of a background shear (see Petitdemange, Dormy and Balbus, MagnetoStrophic MRI in the Earths outer core. Geophys. Res. Lett. 2008, 35 15305). We refer to this instability as the Magnetostrophic MRI (MS-MRI). In this article, we extend our investigations to the nonlinear regime and present results from global axisymmetric simulations in spherical geometry. We show that as the angular momentum is transported outward, the MS-MRI saturates by rapidly changing the initial shear profile. Therefore, the saturation process differs substantially from traditional MRI applications (e.g. accretion disks) in which the background shear is essentially fixed. We show that the MS-MRI appears as a new constraint which limits the maximum differential rotation. To illustrate this mechanism, we apply this work to a Jupiter-like planet, and argue that the magnetic field eventually destabilises the conducting zone of this planet. According to these results, purely hydrodynamic models for the deep origin of the banded structure of Jupiter may need to be modified.

Mechanisms of planetary and stellar dynamos

We review some of the recent progress on modeling planetary and stellar dynamos. Particular attention is given to the dynamo mechanisms and the resulting properties of the field. We present direct numerical simulations using a simple Boussinesq model. These simulations are interpreted using the classical mean-field formalism. We investigate the transition from steady dipolar to multipolar dynamo waves solutions varying different control parameters, and discuss the relevance to stellar magnetic fields. We show that owing to the role of the strong zonal flow, this transition is hysteretic. In the presence of stress-free boundary conditions, the bistability extends over a wide range of parameters.

Systematic parameter study of dynamo bifurcations in geodynamo simulations

Abstract We investigate the nature of the dynamo bifurcation in a configuration applicable to the Earth’s liquid outer core, i.e. in a rotating spherical shell with thermally driven motions with no-slip boundaries. Unlike in previous studies on dynamo bifurcations, the control parameters have been varied significantly in order to deduce general tendencies. Numerical studies on the stability domain of dipolar magnetic fields found a dichotomy between non-reversing dipole-dominated dynamos and the reversing non-dipole-dominated multipolar solutions. We show that, by considering weak initial fields, the above transition disappears and is replaced by a region of bistability for which dipolar and multipolar dynamos coexist. Such a result was also observed in models with free-slip boundaries in which the geostrophic zonal flow can develop and participate to the dynamo mechanism for non-dipolar fields. We show that a similar process develops in no-slip models when viscous effects are reduced sufficiently. The following three regimes are distinguished: (i) Close to the onset of convection ( Ra c ) with only the most critical convective mode (wave number) being present, dynamos set in supercritically in the Ekman number regime explored here and are dipole-dominated. Larger critical magnetic Reynolds numbers indicate that they are particularly inefficient. (ii) in the range 3 Ra / Ra c Ra c , the bifurcations are subcritical and only dipole-dominated dynamos exist. (iii) in the turbulent regime ( Ra / Ra c > 10 ) , the relative importance of zonal flows increases with Ra in non-magnetic models. The field topology depends on the magnitude of the initial magnetic field. The dipolar branch has a subcritical behavior whereas the multipolar branch has a supercritical behavior. By approaching more realistic parameters, the extension of this bistable regime increases. A hysteretic behavior questions the common interpretation for geomagnetic reversals. Far above the dynamo threshold (by increasing the magnetic Prandtl number), Lorentz forces contribute to the first order force balance, as predicted for planetary dynamos. When Ra is sufficiently high, dipolar fields affect significantly the flow speed, the flow structure and heat transfer which is reduced by the Lorentz force regardless of the field strength. This physical regime seems to be relevant for studying geomagnetic processes.

Three branches of dynamo action

In addition to the weak-dipolar state and to the fluctuating-multipolar state, widely discussed in the literature, a third regime has been identified in Dormy (2016). It corresponds to a strong-dipolar branch which appears to approach, in a numerically affordable regime, the magnetostrophic limit relevant to the dynamics of the Earths core. We discuss the transitions between these states and point to the relevance of this strong-dipolar state to Geodynamo modelling.

Gravity darkening in late-type stars. I. The Coriolis effect

Context. Recent interferometric data have been used to constrain the brightness distribution at the surface of nearby stars, in particular the so-called gravity darkening that makes fast rotating stars brighter at their poles than at their equator. However, good models of gravity darkening are missing for stars that posses a convective envelope. Aims. In order to better understand how rotation affects the heat transfer in stellar convective envelopes, we focus on the heat flux distribution in latitude at the outer surface of numerical models. Methods. We carry out a systematic parameter study of three-dimensional, direct numerical simulations of anelastic convection in rotating spherical shells. As a first step, we neglect the centrifugal acceleration and retain only the Coriolis force. The fluid instability is driven by a fixed entropy drop between the inner and outer boundaries where stress-free boundary conditions are applied for the velocity field. Restricting our investigations to hydrodynamical models with a thermal Prandtl number fixed to unity, we consider both thick and thin (solar-like) shells, and vary the stratification over three orders of magnitude. We measure the heat transfer efficiency in terms of the Nusselt number, defined as the output luminosity normalised by the conductive state luminosity. Results. We report diverse Nusselt number profiles in latitude, ranging from brighter (usually at the onset of convection) to darker equator and uniform profiles. We find that the variations of the surface brightness are mainly controlled by the surface value of the local Rossby number: when the Coriolis force dominates the dynamics, the heat flux is weakened in the equatorial region by the zonal wind and enhanced at the poles by convective motions inside the tangent cylinder. In the presence of a strong background density stratification however, as expected in real stars, the increase of the local Rossby number in the outer layers leads to uniformisation of the surface heat flux distribution.

Action of differential rotation on the large-scale magnetic field of stars and planets

Magnetic fields are present in many different astrophysical objects, such as accretion discs, stars, and planets. They influence the evolution and dominate the interior dynamics of these objects, in particular their evolutionary stages. The presence of a weak (subthermal) magnetic field plays a crucial role to drive turbulence in accretion discs thus leading to the stresses needed for accretion and angular momentum transport. This instability is known as the Magneto Rotational Instability (MRI) and it has been studied intensively for the last two decades. Recent numerical results show the importance of understanding the dynamo process in accretion discs. Small-scale dynamo action could prevent the saturation of MRI modes whereas the generation of large-scale magnetic fields provides a suitable coherent field for the angular-momentum transport by MRI modes. Observations show a huge variety of stellar and planetary magnetic fields. Cosmic magnetic fields differ in their magnitude, topology and time dependen...