Mode analysis of numerical geodynamo models

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, antisymmetric with respect to the equator and non-periodic fundamental dynamo mode. 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

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

Context. The recently developed test-field method permits 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. So far, this has been done 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 may be appropriately described by corresponding mean-field results. Also, it is pointed out under which conditions mean-field models do not match 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 fast 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 time-dependence leads to an improved scale separation and thus to a better agreement between direct numerical simulations and mean-field results.Comment: 6 pages, 6 figure

Topology and field strength in spherical, anelastic dynamo simulations

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, 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. 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. 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. 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.Comment: 14 pages, 11 figure

Saturation and time dependence of geodynamo models

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 & Tobias (2009) and is closely related to the test field method by Schrinner et al. (2005, 2007). 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 & 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.Comment: 6 pages, 8 figure

Oscillatory dynamos and their induction mechanisms

Context: Large-scale magnetic fields resulting from hydromagnetic dynamo action may differ substantially in their time dependence. Cyclic field variations, characteristic for the solar magnetic field, are often explained by an important omega-effect, i.e. by the stretching of field lines due to strong differential 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 coefficients have been computed with the help of the so-called test-field method. Subsequently, these coefficients have been used in a mean-field calculation in order to explore the underlying dynamo mechanism. Results: Although the rather strong differential rotation present in this model influences the magnetic field, the omega-effect alone is not responsible for its cyclic time variation. If the omega-effect is suppressed, the resulting alpha^2-dynamo remains oscillatory. Surprisingly, the corresponding alpha-omega dynamo leads to a non-oscillatory magnetic field. Conclusions: The assumption of an alpha-omega mechanism does not explain the occurrence of magnetic cycles satisfactorily

Stability problem in dynamo

It is shown, that the saturated $\alpha$-effect taken from the nonlinear dynamo equations for the thin disk can still produce exponentially growing magnetic field in the case, when this field does not feed back on the $\alpha$. For negative dynamo number (stationary regime) stability is defined by the structure of the spectra of the linear problem for the positive dynamo numbers. Stability condition for the oscillatory solution (positive dynamo number) is also obtained and related to the phase shift of the original magnetic field, which produced saturated $\alpha$ and magnetic field in the kinematic regime. Results can be used for explanation of the similar effect observed in the shell models simulations as well in the 3D dynamo models in the plane layer and sphere

Alpha effect due to buoyancy instability of a magnetic layer

A strong toroidal field can exist in form of a magnetic layer in the overshoot region below the solar convection zone. This motivates a more detailed study of the magnetic buoyancy instability with rotation. We calculate the alpha effect due to helical motions caused by a disintegrating magnetic layer in a rotating density-stratified system with angular velocity Omega making an angle theta with the vertical. We also study the dependence of the alpha effect on theta and the strength of the initial magnetic field. We carry out three-dimensional hydromagnetic simulations in Cartesian geometry. A turbulent EMF due to the correlations of the small scale velocity and magnetic field is generated. We use the test-field method to calculate the transport coefficients of the inhomogeneous turbulence produced by the layer. We show that the growth rate of the instability and the twist of the magnetic field vary monotonically with the ratio of thermal conductivity to magnetic diffusivity. The resulting alpha effect is inhomogeneous and increases with the strength of the initial magnetic field. It is thus an example of an "anti-quenched" alpha effect. The alpha effect is nonlocal, requiring around 8--16 Fourier modes to reconstruct the actual EMF based on the actual mean field.Comment: 14 pages, 19 figures 3 tables (submitted to A & A

Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo

A comparison is made between mean-field models and direct numerical simulations of rotating magnetoconvection and the geodynamo. The mean-field coefficients are calculated with the fluid velocity taken from the direct numerical simulations. The magnetic fields resulting from mean-field models are then compared with the mean magnetic field from the direct numerical simulations

The fratricide of alpha-Omega dynamos by their alpha-squared siblings

Context. Helically forced magneto-hydrodynamic shearing-sheet turbulence can support different large-scale dynamo modes, although the {\alpha} - {\Omega} mode is generally expected to dominate because it is the fastest growing. In an {\alpha} - {\Omega} dynamo, most of the field amplification is produced by the shear. As differential rotation is an ubiquitous source of shear in astrophysics, such dynamos are believed to be the source of most astrophysical large-scale magnetic fields. Aims. We study the stability of oscillatory migratory {\alpha} - {\Omega} type dynamos in turbulence simulations. Methods. We use shearing-sheet simulations of hydromagnetic turbulence that is helically forced at a wavenumber that is about three times larger than the lowest wavenumber in the domain so that both {\alpha} - {\Omega} and {\alpha}2 dynamo action is possible. Results. After initial dominance and saturation, the {\alpha} - {\Omega} mode is found to be destroyed by an orthogonal {\alpha}2 mode sustained by the helical turbulence alone. We show that there are at least two processes through which this transition can occur. Conclusions. The fratricide of {\alpha} - {\Omega} dynamos by its {\alpha}2 sibling is discussed in the context of grand minima of solar and stellar activity. However, the genesis of {\alpha} - {\Omega} dynamos from an {\alpha}2 dynamo has not yet been found.Comment: 10 pages, 12 figures, 3 table

Kinematic alpha effect in isotropic turbulence simulations

Using numerical simulations at moderate magnetic Reynolds numbers up to 220 it is shown that in the kinematic regime, isotropic helical turbulence leads to an alpha effect and a turbulent diffusivity whose values are independent of the magnetic Reynolds number, \Rm, provided \Rm exceeds unity. These turbulent coefficients are also consistent with expectations from the first order smoothing approximation. For small values of \Rm, alpha and turbulent diffusivity are proportional to \Rm. Over finite time intervals meaningful values of alpha and turbulent diffusivity can be obtained even when there is small-scale dynamo action that produces strong magnetic fluctuations. This suggests that small-scale dynamo-generated fields do not make a correlated contribution to the mean electromotive force.Comment: Accepted for publication in MNRAS Letter