Note on the scalar dynamo model

Abstract

Bayly (1993) introduced and investigated the equation (? _t + v·▽-η ▽²)S = RS as a scalar analogue of the magnetic induction equation. Here, S(r, t) is a scalar function and the flow field v(r, t) and “stretching” function R(r, t) are given independently. This equation is much easier to handle than the corresponding vector equation and, although not of much relevance to the (vector) kinematic dynamo problem, it helps to study some features of the fast dynamo problem. In this note the scalar equation is considered for linear flow and a harmonic potential as stretching function. The steady equation separates into one-dimensional equations, which can be completely solved and therefore allow one to monitor the behaviour of the spectrum in the limit of vanishing diffusivity. For more general homogeneous flows a scaling argument is given which ensures fast dynamo action for certain powers of the harmonic potential. Our results stress the singular behaviour of eigenfunctions in the limit of vanishing diffusivity and the importance of stagnation points in the flow for fast dynamo action.     

Fast dynamo action in the Ponomarenko dynamo

Abstract

An analysis of small-scale magnetic fields shows that the Ponomarenko dynamo is a fast dynamo; the maximum growth rate remains of order unity in the limit of large magnetic Reynolds number. Magnetic fields are regenerated by a “stretch-diffuse” mechanism. General smooth axisymmetric velocity fields are also analysed; these give slow dynamo action by the same mechanism.     

Inverse dynamo problem in a cylinder

An inverse dynamo problem is presented in which we search for either kinematic dynamos which produce the same external magnetic fields or an invisible dynamo. The existence of flows which produce the same external magnetic fields is proved. However, we have not found general conditions necessary for such kind of dynamos. An “invisible dynamo” operates in an electrically conducting region surrounded by vacuum and generates a magnetic field trapped in the electrically conducting region so that no magnetic field exists in the vacuum. Invisible magnetic decay modes exist in cylinders, but no invisible growing field supported by the dynamo mechanism has been found.     

A nonlinear dynamo driven by rapidly rotating convection

In Kim et al. (Kim, E., Hughes, D.W. and Soward, A.M., “An investigation into high conductivity dynamo action driven by rotating convection”, Geophys. Astrophys. Fluid Dynam. 91, 303–332 ().) we investigated kinematic dynamo action driven by rapidly rotating convection in a cylindrical annulus. Here we extend this work to consider self-consistent nonlinear dynamo action in which the back-reaction of the Lorentz force on the flow is taken into account. In particular, we investigate, as a function of magnetic Prandtl number, the evolution of an initially weak magnetic field in two different types of convective flow – one chaotic and the other integrable. On saturation, the latter shows a systematic dependence on the magnetic Prandtl number whereas the former appears not to. In addition, we show how, in keeping with the findings of Cattaneo et al. (Cattaneo, F., Hughes, D.W. and Kim, E., “Suppression of chaos in a simplified nonlinear dynamo model”, Phys. Rev. Lett. 76, 2057–2060 ().), saturation of the growth of the magnetic field is brought about, for the originally chaotic flow, by a strong suppression of chaos.     

A shell model for a large-scale turbulent dynamo

This article addresses the interesting and important problem of large-scale magnetic field generation in turbulent flows, using a self-consistent dynamo model recently developed. The main idea of this model is to consider the induction equation for the large-scale magnetic field, integrated consistently with the turbulent dynamics at smaller scales described by a magnetohydrodynamic shell model. The questions of dynamo action threshold, magnetic field saturation, magnetic field reversals, nature of the dynamo transition and the changes of small-scale turbulence as a consequence of the dynamo onset are discussed. In particular, the stability curve obtained by the model integration is shown in a very wide range of values of the magnetic Prandtl number not yet accessible by direct numerical simulation but more realistic for natural dynamos. Moreover, from our analysis it is shown that the large-scale dynamo transition displays a hysteretic behaviour and therefore a subcritical nature. The model successfully reproduces magnetic polarity reversals, showing the capability to generate persistence times which are increasing for decreasing magnetic diffusivity. Moreover, when the system reaches a statistically stationary dynamo state, where the large-scale magnetic field can abruptly reverse its polarity (magnetic reversal state) or not, keeping the same polarity (steady state), it shows an unmistakable tendency towards the energy equipartition for the turbulence at small scale.     

On a dynamo driven topographically by longitudinal libration

Variation in the angular velocity Ω of a planetary body is called libration or longitudinal libration when the Ω-axis is fixed in direction. This motion of the body's solid mantle drives motions in its fluid core, either by viscous coupling across the core-mantle interface S, or topographically when S is asymmetric with respect to the Ω-axis, the only case considered in this article. A significant topographically-driven flow is identified having uniform vorticity within S and no component parallel to the Ω-axis. Its dynamic stability depends on the amplitude, Ω ₁, of the sinusoidally varying part of Ω and on the ratio, b/a, of the lengths of the principal axes of S, assumed spheroidal. In (Ω ₁/Ω ₀, b/a) parameter space where Ω ₀ is the average Ω, islands are shown to exist where the constant vorticity states are dynamically unstable. These are surrounded by a sea in which they are stable. When the fluid is slightly viscous, a state in the stable sea retains its uniform vorticity structure except in a viscous boundary layer on S in which the flow acquires a component parallel to the Ω-axis. For (Ω ₁/Ω ₀, b/a) on an island where the uniform vorticity state is unstable, an “alternative flow” exists, which is three-dimensional and is examined here. Assuming that the core is electrically conducting, kinematic dynamos are sought. Uniform vorticity flow appears to be non-regenerative but, when it is stable and viscosity acts to create a sufficiently strong boundary layer flow, dynamo action may occur. It is shown that the alternative flow that exists on an instability island in (Ω ₁/Ω ₀,?b/a) space can be vigorously regenerative.     

Kinematic dynamo induced by helical waves

We investigate numerically the kinematic dynamo induced by the superposition of two helical waves in a periodic box as a simplified model to understand the dynamo action in astronomical bodies. The effects of magnetic Reynolds number, wavenumber and wave frequency on the dynamo action are studied. It is found that this helical-wave dynamo is a slow dynamo. There exists an optimal wavenumber for the dynamo growth rate. A lower wave frequency facilitates the dynamo action and the oscillations of magnetic energy emerge at some particular wave frequencies.     

Kinematic dynamo action in spherical Couette flow

We investigate numerically kinematic dynamos driven by flow of electrically conducting fluid in the shell between two concentric differentially rotating spheres, a configuration normally referred to as spherical Couette flow. We compare between axisymmetric (2D) and fully 3D flows, between low and high global rotation rates, between prograde and retrograde differential rotations, between weak and strong nonlinear inertial forces, between insulating and conducting boundaries and between two aspect ratios. The main results are as follows. Azimuthally drifting Rossby waves arising from the destabilisation of the Stewartson shear layer are crucial to dynamo action. Differential rotation and helical Rossby waves combine to contribute to the spherical Couette dynamo. At a slow global rotation rate, the direction of differential rotation plays an important role in the dynamo because of different patterns of Rossby waves in prograde and retrograde flows. At a rapid global rotation rate, stronger flow supercriticality (namely the difference between the differential rotation rate of the flow and its critical value for the onset of nonaxisymmetric instability) facilitates the onset of dynamo action. A conducting magnetic boundary condition and a larger aspect ratio both favour dynamo action.     

Nonlinear dynamics of a stellar dynamo in a spherical shell

Abstract

A simple mean-field model of a nonlinear stellar dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the only nonlinearity retained is the influence of the Lorentz forces on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.'s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.     

Convection in a rotating,laterally heated annulus the wave number transitions

Abstract

The radial temperature differences at which the transitions from one wave number to the next occur have been measured with either increasing or decreasing positive radial temperature gradients, at five different rotation rates, with the fluid being always in thermal equilibrium and being in contact with an upper rigid lid. Hysteresis has been observed in all wave number transitions, and also in the transition to upper symmetry. There are, nevertheless, regions in the stability diagram where the wave number is unique. There is an excluded region where the wave number four cannot be obtained through quasi-steady procedure. There is a reversal of the sense of the hysteresis of the transitions. At low ΔT, a wave number transition with increasing radial temperature difference occurs at a higher ΔT, than the same transition with decreasing temperature difference. On the other hand, at large values of ΔT, a wave number transition with increasing radial temperature difference occurs at a lower ΔT, than the same transition with decreasing temperature difference. Wave number transitions with increasing ΔT, occur spontaneously out of amplitude oscillations. Wave number transitions with decreasing ΔT, occur via slow wave splitting in association with phase modulations of the waves. The uniqueness of the wave number in the unique areas of the stability diagram has been confirmed by sudden start experiments.     

Numerical simulation of Martian historical dynamo: Impact of the Rayleigh number on the dynamo state

The observed Mars remnant magnetism suggests that there was an active dynamo in the Martian core. We use the MoSST core dynamics model to simulate the Martian historical dynamo, focusing on the variation of the dynamo states with the Rayleigh number Ra (a non-dimensional parameter describing the buoyancy force in the core). Our numerical results show that the mean field length scale does not vary monotonically with the Rayleigh number, and the field morphology at the core mantle boundary changes with Rayleigh number. In particular, it drifts westward with a speed decreasing with Rayleigh number. Supported by National Natural Science Foundation of China (Grant No. 40328006)     

The galactic dynamo: Axisymmetric and non-axisymmetric modes

Abstract

We discuss recent developments in the theory of large-scale magnetic structures in spiral galaxies. In addition to a review of galactic dynamo models developed for axisymmetric disks of variable thickness, we consider the possibility of dominance of non-axisymmetric magnetic modes in disks with weak deviations from axial symmetry. Difficulties of straightforward numerical simulation of galactic dynamos are discussed and asymptotic solutions of the dynamo equations relevant for galactic conditions are considered. Theoretical results are compared with observational data.     

Baroclinically-driven flows and dynamo action in rotating spherical fluid shells

The dynamics of stably stratified stellar radiative zones is of considerable interest due to the availability of increasingly detailed observations of Solar and stellar interiors. This article reports the first non-axisymmetric and time-dependent simulations of flows of anelastic fluids driven by baroclinic torques in stably stratified rotating spherical shells – a system serving as an elemental model of a stellar radiative zone. With increasing baroclinicity a sequence of bifurcations from simpler to more complex flows is found in which some of the available symmetries of the problem are broken subsequently. The poloidal component of the flow grows relative to the dominant toroidal component with increasing baroclinicity. The possibility of magnetic field generation thus arises and this paper proceeds to provide some indications for self-sustained dynamo action in baroclinically-driven flows. We speculate that magnetic fields in stably stratified stellar interiors are thus not necessarily of fossil origin as it is often assumed.     

Axially asymmetric velocities in the boundary layer of the nearly symmetric hydromagnetic dynamo

Abstract

An attempt has been made to include the axially asymmetric velocities into the calculation of Braginsky's Z-model of the nearly symmetric hydromagnetic dynamo. In this axisymmetric non-linear model dominated by Lorentz and Coriolis forces and maintained by a specified convection, the α-effect is prescribed. An example is shown of the axially asymmetric Archimedean buoyancy, which can imply an arbitrary alpha effect in the model with viscous core-mantle coupling. The formalisms of Tough and Roberts (1968) is also discussed and a modified α-effect in the Z-model is suggested.     

On the inverse dynamo problem in a simplified mean-field model

Inverse dynamo theory seeks to gain information about the motion of a liquid conductor from measurements of the magnetic field in the surrounding vacuum. We consider here a highly simplified model problem, namely a steady α²-dynamo in plane geometry with an α-field varying only in the z-direction normal to the conductor–vacuum interface. Based on perturbation theory about constant-α solutions, we find as many integral conditions on α(z) as modes are present in the vacuum field. This result is corroborated by the complete solution of a special case.