Martin Schrinner

Weak‐ and strong‐field dynamos: from the Earth to the stars

Observations of magnetism in very low mass stars recently made important progress, revealing characteristics that are now to be understood in the framework of dynamo theory. In parallel, there is growing evidence that dynamo processes in these stars share many similarities with planetary dynamos. We investigate the extent to which the weak \emph{vs} strong field bistability predicted for the geodynamo can apply to recent observations of two groups of very low mass fully-convective stars sharing similar stellar parameters but generating radically different types of magnetic fields. Our analysis is based on previously published spectropolarimetric and spectroscopic data. We argue that these can be interpreted in the framework of weak and strong field dynamos.

Oscillatory dynamos and their induction mechanisms

Context. Large-scale magnetic fields resulting from hydromagnetic d ynamo action may differ substantially in their time dependence. Cyclic field variations, characteristic for the solar magne tic field, are often explained by an important -effect, i.e. by the stretching of field lines due to strong di fferential rotation. Aims. The dynamo mechanism of a convective, oscillatory dynamo model is investigated. Methods. We solve the MHD-equations for a conducting Boussinesq fluid in a rotating spherical shell. For a resulting oscillatory model, dynamo coeffi cients have been computed with the help of the so-called test-field method. Subsequently, these coe ffi cients have been used in a mean-field calculation in order to explore the u nderlying dynamo mechanism. Results. The oscillatory dynamo model under consideration is of � 2 -type. Although the rather strong differential rotation present in this model influences the magnetic field, the -effect alone is not responsible for its cyclic time variation. I f the -effect is suppressed the resulting � 2 -dynamo remains oscillatory. Surprisingly, the corresponding �-dynamo leads to a non-oscillatory magnetic field. Conclusions. The assumption of an �-mechanism does not explain the occurrence of magnetic cycles satisfactorily.

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.

Global dynamo models from direct numerical simulations and their mean-field counterparts

Context. The test-field method permits us to compute dynamo coefficients from global, direct numerical simulations. The subsequent use of these parameters in mean-field models enables us to compare self-consistent dynamo models with their mean-field counterparts. This has been done to date for a simulation of rotating magnetoconvection and a simple benchmark dynamo, which are both (quasi-)stationary. Aims. It is shown that chaotically time-dependent dynamos in a low Rossby number regime can be appropriately described by corresponding mean-field results. We also identify conditions under which mean-field models disagree with direct numerical simulations. Methods. We solve the equations of magnetohydrodynamics (MHD) in a rotating, spherical shell in the Boussinesq approximation. Based on this, we compute mean-field coefficients for several models with the help of the previously developed test-field method. The parameterization of the mean electromotive force by these coefficients is tested against direct numerical simulations. In addition, we use the determined dynamo coefficients in mean-field models and compare the outcome with azimuthally averaged fields from direct numerical simulations. Results. The azimuthally and time-averaged electromotive force in rapidly rotating dynamos is sufficiently well parameterized by the set of determined mean-field coefficients. In comparison to the previously considered (quasi-)stationary dynamo, the chaotic timedependence leads to an improved scale separation and thus to a closer agreement between direct numerical simulations and mean-field results.

Saturation and time dependence of geodynamo models

SUMMARY In this study 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. This experiment has been performed by Cattaneo and Tobias and is closely related to the test field method by Schrinner et al. 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. 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.

Rotational threshold in global numerical dynamo simulations

Magnetic field observations of low-mass stars reveal an increase of magnetic activity with increasing rotation rate. The so-called activity–rotation relation is usually attributed to changes in the underlying dynamo processes generating the magnetic field. We examine the dependence of the field strength on rotation in global numerical dynamo models and interpret our results on the basis of energy considerations. In agreement with the scaling law proposed by Christensen and Aubert, the field strength in our simulations is set by the fraction of the available power used for the magnetic field generation. This is controlled by the dynamo efficiency calculated as the ratio of ohmic to total dissipation in our models. The dynamo efficiency grows strongly with increasing rotation rate at a Rossby number of 0.1 until it reaches its upper bound of 1 and becomes independent of rotation. This gain in efficiency is related to the strong rotational dependence of the mean electromotive force in this parameter regime. For multipolar models at Rossby numbers clearly larger than 0.1, on the other hand, we do not find a systematic dependence of the field strength on rotation. Whether the enhancement of the dynamo efficiency found in our dipolar models explains the observed activity–rotation relation needs to be further assessed.

Mode analysis of numerical geodynamo models

Abstract It has been suggested in Hoyng (2009) that dynamo action can be analysed by expansion of the magnetic field into dynamo modes and statistical evaluation of the mode coefficients. We here validate this method by analysing a numerical geodynamo model and comparing the numerically derived mean mode coefficients with the theoretical predictions. The model belongs to the class of kinematically stable dynamos with a dominating axisymmetric, dipolar and non-periodic fundamental dynamo mode. Our present study supports that contributions from higher order modes to the magnetic field result from the deformation of the fundamental mode by the turbulent flow. The analysis requires a number of steps: the computation of the so-called dynamo coefficients, the derivation of the temporally and azimuthally averaged dynamo eigenmodes and the decomposition of the magnetic field of the numerical geodynamo model into the eigenmodes. For the determination of the theoretical mode excitation levels the turbulent velocity field needs to be projected on the dynamo eigenmodes. We compare the theoretically and numerically derived mean mode coefficients and find reasonably good agreement for most of the modes. Some deviation might be attributable to the approximation involved in the theory. Since the dynamo eigenmodes are not self-adjoint, a spectral interpretation of the eigenmodes is not possible.

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.

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...