Dynamo Properties Research Articles

Shear-driven dynamo waves at high magnetic Reynolds number

Astrophysical magnetic fields often display remarkable organization, despite being generated by dynamo action driven by turbulent flows at high conductivity. An example is the eleven-year solar cycle, which shows spatial coherence over the entire solar surface. The difficulty in understanding the emergence of this large-scale organization is that whereas at low conductivity (measured by the magnetic Reynolds number, Rm) dynamo fields are well organized, at high Rm their structure is dominated by rapidly varying small-scale fluctuations. This arises because the smallest scales have the highest rate of strain, and can amplify magnetic field most efficiently. Therefore most of the effort to find flows whose large-scale dynamo properties persist at high Rm has been frustrated. Here we report high-resolution simulations of a dynamo that can generate organized fields at high Rm; indeed, the generation mechanism, which involves the interaction between helical flows and shear, only becomes effective at large Rm. The shear does not enhance generation at large scales, as is commonly thought; instead it reduces generation at small scales. The solution consists of propagating dynamo waves, whose existence was postulated more than 60 years ago and which have since been used to model the solar cycle.

Revisiting the ABC flow dynamo

The Arnol'd–Beltrami–Childress flow is a prototype for fast dynamo action, essential to the origin of magnetic field in large astrophysical objects. Probably the most studied configuration is the classical 1 : 1 : 1 flow. We investigate its dynamo properties varying the magnetic Reynolds number Rm. We identify two kinks in the growth rate, which correspond, respectively, to an eigenvalue crossing and to an eigenvalue coalescence. The dominant eigenvalue becomes purely real for a finite value of the control parameter. Finally, we show that even for Rm = 25 000, the dominant eigenvalue has not yet reached an asymptotic behaviour. It still varies very significantly with the controlling parameter. Even at these very large values of Rm the fast dynamo property of this flow cannot yet be established.

Instability of the magnetohydrodynamics system at vanishing Reynolds number

The aim of this note is to study the dynamo properties of the magnetohydrodynamics system at vanishing R m . Improving the analysis in Gérard-Varet (SIAM J Math Anal 37(3):815–840, 2006), we shall establish a generic Lyapunov instability result.

Self-consistent simulations of a von Kármán type dynamo in a spherical domain with metallic walls

We have performed numerical simulations of boundary-driven dynamos using a three-dimensional nonlinear magnetohydrodynamical model in a spherical shell geometry. A conducting fluid of magnetic Prandtl number Pm=0.01 is driven into motion by the counter-rotation of the two hemispheric walls. The resulting flow is of von Kármán type, consisting of a layer of zonal velocity close to the outer wall and a secondary meridional circulation. Above a certain forcing threshold, the mean flow is unstable to non-axisymmetric motions within an equatorial belt. For fixed forcing above this threshold, we have studied the dynamo properties of this flow. The presence of a conducting outer wall is essential to the existence of a dynamo at these parameters. We have therefore studied the effect of changing the material parameters of the wall (magnetic permeability, electrical conductivity, and thickness) on the dynamo. In common with previous studies, we find that dynamos are obtained only when either the conductivity or the permeability is sufficiently large. However, we find that the effect of these two parameters on the dynamo process are different and can even compete to the detriment of the dynamo. Our self-consistent approach allow us to analyze in detail the dynamo feedback loop. The dynamos we obtain are typically dominated by an axisymmetric toroidal magnetic field and an axial dipole component. We show that the ability of the outer shear layer to produce a strong toroidal field depends critically on the presence of a conducting outer wall, which shields the fluid from the vacuum outside. The generation of the axisymmetric poloidal field, on the other hand, occurs in the equatorial belt and does not depend on the wall properties.

MAGNETIC DYNAMO ACTION IN RANDOM FLOWS WITH ZERO AND FINITE CORRELATION TIMES

Hydromagnetic dynamo theory provides the prevailing theoretical description for the origin of magnetic fields in the universe. Here we consider the problem of kinematic, small-scale dynamo action driven by a random, incompressible, non-helical, homogeneous and isotropic flow. In the Kazantsev dynamo model the statistics of the driving flow are assumed to be instantaneously correlated in time. Here we compare the results of the model with the dynamo properties of a simulated flow that has equivalent spatial characteristics as the Kazantsev flow but different temporal statistics. In particular, the simulated flow is a solution of the forced Navier-Stokes equations and hence has a finite correlation time. We find that the Kazantsev model typically predicts a larger magnetic growth rate and a magnetic spectrum that peaks at smaller scales. However, we show that by filtering the diffusivity spectrum at small scales it is possible to bring the growth rates into agreement and simultaneously align the magnetic spectra.

Dynamo efficiency in compressible convective dynamos with and without penetration

We investigate dynamo action in compressible convection via numerical simulations in a Cartesian domain. We directly compare the dynamo properties of a fully convective domain with the same domain extended to include an underlying stable region. These simulations extend models of fully convective domains with open lower boundary conditions to a more self-consistent model. We examine whether the extremely slow recirculation of the lower region affects the dynamo properties in the convection zone. We find that the dynamo properties of the upper convective region are essentially unchanged by the addition of the lower stable region. After a transient period, dynamo action in the convective region not only proceeds as normal, but also extends into the region of overshooting flow in the stable region. Downward magnetic pumping, long recirculation times and the low percentage of rising elements that transit the vertical extent of the domain all fail to eliminate the dynamo. Sufficient magnetic field is recirculated or remains in the convective region to fuel the local dynamo there. The independence of the convective layer from the conditions of the lower layer makes the dynamo truly local.

The influence of stratification upon small-scale convectively-driven dynamos

AbstractIn the quiet Sun, convective motions form a characteristic granular pattern, with broad upflows enclosed by a network of narrow downflows. Magnetic fields tend to accumulate in the intergranular lanes, forming localised flux concentrations. One of the most plausible explanations for the appearance of these quiet Sun magnetic features is that they are generated and maintained by dynamo action resulting from the local convective motions at the surface of the Sun. Motivated by this idea, we describe high resolution numerical simulations of nonlinear dynamo action in a (fully) compressible, non-rotating layer of electrically-conducting fluid. The dynamo properties depend crucially upon various aspects of the fluid. For example, the magnetic Reynolds number (Rm) determines the initial growth rate of the magnetic energy, as well as the final saturation level of the dynamo in the nonlinear regime. We focus particularly upon the ways in which the Rm-dependence of the dynamo is influenced by the level of stratification within the domain. Our results can be related, in a qualitative sense, to solar observations.

A SPHERICAL PLASMA DYNAMO EXPERIMENT

We propose a plasma experiment to be used to investigate fundamental properties of astrophysical dynamos. The highly conducting, fast-flowing plasma will allow experimenters to explore systems with magnetic Reynolds numbers an order of magnitude larger than those accessible with liquid-metal experiments. The plasma is confined using a ring-cusp strategy and subject to a toroidal differentially rotating outer boundary condition. As proof of principle, we present magnetohydrodynamic simulations of the proposed experiment. When a von K\'arm\'an-type boundary condition is specified, and the magnetic Reynolds number is large enough, dynamo action is observed. At different values of the magnetic Prandtl and Reynolds numbers the simulations demonstrate either laminar or turbulent dynamo action.

Dynamo properties of the turbulent velocity field of a saturated dynamo

In order better to understand how dynamo systems saturate, we study the kinematic dynamo properties of velocity fields that arise from nonlinearly saturated dynamos. The technique is implemented by solving concurrently, in addition to the momentum equation, two induction equations, one for the actual magnetic field and one for an independent passive vector field. We apply this technique to two illustrative examples: convectively driven turbulence and turbulence represented by a shell model. In all cases we find that the velocity remains an efficient kinematic dynamo even after nonlinear saturation occurs. We discuss the implications to the process of dynamo saturation.

Convective Dynamos with Penetration, Rotation, and Shear

We investigate the dynamo properties of Boussinesq, penetrative convection in the presence of rotation and large-scale velocity shear flows. Several numerical experiments are conducted in a local Cartesian computational domain in which the relative stability of the lower stable layer, the rotation rate, the supercriticality of the convection, and the strength of the imposed shear flow are varied. Once a statistically steady hydrodynamic state is achieved for any set of parameters, a weak seed magnetic field is added and the subsequent dynamo evolution is followed. In all cases studied, the weak seed field is initially amplified exponentially and then eventually saturates in a stationary magnetohydrodynamic (MHD) state. Even in the presence of penetration, rotation, and shear, the field is predominantly small-scale. We analyze the reasons for this and make suggestions as to possible further mechanisms that may lead to large-scale field generation.

Limited Role of Spectra in Dynamo Theory: Coherent versus Random Dynamos

We discuss the importance of phase information and coherence times in determining the dynamo properties of turbulent flows. We compare the kinematic dynamo properties of three flows with the same energy spectrum. The first flow is dominated by coherent structures with nontrivial phase information and long eddy coherence times, the second has random phases and long-coherence time, the third has nontrivial phase information, but short coherence time. We demonstrate that the first flow is the most efficient kinematic dynamo, owing to the presence of sustained stretching and constructive folding. We argue that these results place limitations on the possible inferences of the dynamo properties of flows from the use of spectra alone, and that the role of coherent structures must always be accounted for.

The α-effect in 2D fast dynamos

We look at the large-scale dynamo properties of spatially periodic, time dependent, helical 2D flows of the form u(x, t) = (∂ y ψ (x, y, t), −∂ x ψ (x, y, t), −ψ (x, y, t). These flows act as kinematic fast dynamos and are able to generate a mean magnetic field uniform and constant in the xy-plane but whose direction varies periodically along z with wavenumber k. Using Mean Field Electrodynamics, the generation mechanism can be understood in terms of a k-dependent α-effect, which depends on the magnetic Reynolds number, R m . We calculate this effect for different motions and investigate how its limit as k → 0 depends on R m and on the properties of the flows such as their spatial structure or correlation time. This work generalises earlier studies based on 2D steady flows to motions with time dependence.

Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields

SUMMARY We study numerically an extensive set of dynamo models in rotating spherical shells, varying all relevant control parameters by at least two orders of magnitude. Convection is driven by a fixed temperature contrast between rigid boundaries. There are two distinct classes of solutions with strong and weak dipole contributions to the magnetic field, respectively. Non-dipolar dynamos are found when inertia plays a significant role in the force balance. In the dipolar regime the critical magnetic Reynolds number for self-sustained dynamos is of order 50, independent of the magnetic Prandtl number Pm. However, dynamos at low Pm exist only at sufficiently low Ekman number E. For dynamos in the dipolar regime we attempt to establish scaling laws that fit our numerical results. Assuming that diffusive effects do not play a primary role, we introduce non-dimensional parameters that are independent of any diffusivity. These are a modified Rayleigh number based on heat (or buoyancy) flux Ra ∗ , the Rossby number Ro measuring the flow velocity, the Lorentz number Lo measuring magnetic field strength, and a modified Nusselt number Nu ∗ for the advected heat flow. To first approximation, all our dynamo results can be collapsed into simple power-law dependencies on the modified Rayleigh number, with approximate exponents of 2/5, 1/2 and 1/3 for the Rossby number, modified Nusselt number and Lorentz number, respectively. Residual dependencies on the parameters related to diffusion (E, Pm, Prandtl number Pr) are weak. Our scaling laws are in agreement with the assumption that the magnetic field strength is controlled by the available power and not necessarily by a force balance. The Elsasser number � , which is the conventional measure for the ratio of Lorentz force to Coriolis force, is found to vary widely. We try to assess the relative importance of the various forces by studying sources and sinks of enstrophy (squared vorticity). In general Coriolis and buoyancy forces are of the same order, inertia and viscous forces make smaller and variable contributions, and the Lorentz force is highly variable. Ignoring a possible weak dependence on the Prandtl numbers or the Ekman number, a surprising prediction is that the magnetic field strength is independent both of conductivity and of rotation rate and is basically controlled by the buoyancy flux. Estimating the buoyancy flux in the Earth’s core using our Rossby number scaling and a typical velocity inferred from geomagnetic secular variations, we predict a small growth rate and old age of the inner core and obtain a reasonable magnetic field strength of order 1 mT inside the core. From the observed heat flow in Jupiter, we predict an internal field of 8 mT, in agreement with Jupiter’s external field being 10 times stronger than that of the Earth.

Vortex dynamos

We investigate the kinematic dynamo properties of interacting vortex tubes. These flows are of great importance in geophysical and astrophysical fluid dynamics: for a large range of systems, turbulence is dominated by such coherent structures. We obtain a dynamically consistent is large enough, because of the formation of coherent structures.

Linear and nonlinear dynamo properties of time-dependent ABC flows

The linear and nonlinear dynamo properties of a class of periodically forced flows is considered. The forcing functions are chosen to drive, in the absence of magnetic effects (kinematic regime), a time-dependent version of the ABC flow with A = B = C = 1. The time-dependence consists of a harmonic displacement of the origin along the line x = y = z = 1 with amplitude ϵ and frequency Ω. The finite-time Lyapunov exponents are computed for several values of ϵ and Ω. It is found that for values of these parameters near unity chaotic streamlines occupy most of the volume. In this parameter range, and for moderate kinetic and magnetic Reynolds numbers, the basic flow is both hydrodynamically and hydromagnetically unstable. However, the dynamo instability has a higher growth rate than the hydrodynamic one, so that the nonlinear regime can be reached with negligible departures from the basic ABC flow.In the nonlinear regime, two distinct classes of behaviour are observed. In one, the exponential growth of the magnetic field saturates and the dynamo settles to a stationary state whereby the magnetic energy is maintained indefinitely. In the other the velocity field evolves to a nondynamo state and the magnetic field, following an initial amplification, decays to zero. The transition from the dynamo to the nondynamo state can be mediated by the hydrodynamic instability or by magnetic perturbations. The properties of the ensuing nonlinear dynamo states are investigated for different parameter values. The implications for a general theory of nonlinear dynamos are discussed.

Linear and nonlinear dynamo action

We discuss a nonlinear forced magnetohydrodynamic dynamo simulation that exhibits a surprising range of behavior, including multiple hydrodynamic states with different dynamo properties. Magnetic fields may decay after initial kinematic amplification. This behavior emphasizes that initial conditions must be taken into account when classifying nonlinear dynamos.