Convection driven dynamos

Various models of thermal convection in rapidly rotating fluids permeated by strong magnetic fields are discussed. Particular attention is paid to the possibility that the magnetic field can be maintained by dynamo action rather than by externally applied electric currents. Two dynamo models are given particular attention. They are the plane layer model of Childress and Soward (1972) and the annulus model of Busse (1975). Though these models do not totally resolve the geodynamo problem, they do highlight important features of hydromagnetic dynamos. As a result some speculations are made about the true character of the geodynamo.     

Are planetary dynamos driven by gravitational settling?

The basic features of gravitationally driven convection are summarized, and an expression for the power available to drive a hydromagnetic dynamo is given. Some thermal consequences of gravitationally driven convection are presented.     

Self-consistent dynamo models driven by hydromagnetic instabilities

The dynamics of the Earth's core are dominated by a balance between Lorentz and Coriolis forces. Previous studies of possible (magnetostrophic) hydromagnetic instabilities in this regime have been confined to geophysically unrealistic flows and fields. In recent papers we have treated rather general fields and flows in a spherical geometry and in a computationally simple plane-layer model. These studies have highlighted the importance of differential rotation in determining the spatial structure of the instability. Here we have proceeded to use these results to construct a self-consistent dynamo model of the geomagnetic field. An iterative procedure is employed in which an α-effect is calculated from the form of the instability and is then used in a mean field dynamo model. The mean zonal field calculated there is then input back into the hydromagnetic stability problem and a new α-effect calculated. The whole procedure is repeated until the input and output zonal fields are the same to some tolerance.     

Thermally driven hydromagnetic instabilities in a non-constantly stratified layer

Summary The convection in a rapidly rotating electrically conducting, fluid horizontal layer of non-constant stratification, permeated by an inhomogeneous magnetic field, is studied. In this connection, a temperature model of the layer is constructed, which creates a structure such that part of the layer is unstably and a part stably stratified. The results obtained are applied to the conditions in the fluid Earth's core.

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Control volume method for hydromagnetic dynamos in rotating spherical shells: Testing the code against the numerical dynamo benchmark

Hydromagnetic dynamos in rotating spherical shells are investigated using the control volume method. We present a validation of our code against the numerical dynamo benchmark. It is successfully benchmarked and we are able to conclude that the control volume method is another numerical method available for numerical modelling of self-consistent dynamos. In addition, the efficiency of our numerical code is tested. Computations provide conclusions that dynamo codes based on the spectral methods are much more efficient than our code based on the control volume method at the study of global fields on small and medium size parallel computers. However, our code could be much more efficient than codes based on the spectral methods on very large parallel computers, especially at the study of turbulence.     

A search for axisymmetric dynamos

Some general theorems on the necessary level of symmetry in the velocity field of a kinematic dynamo have been proposed: the analytic difficulties of direct proofs of these appear insurmountable at present. Explicit axisymmetric counter-examples are here sought in an attempt to disprove one such general hypothesis numerically. The usual Bullard-Gellman technique, and a new time-dependent extension of it, are used. Despite extensive computation on a number of different models, no axisymmetric dynamo is found: possible reasons for this are discussed.     

Nonlinear galactic dynamos

Abstract

We present trial calculations for a simple nonlinear ‘‘thick disk'’ galaxy dynamo. The nonlinearity is a simple α-quenching. Our strictly axisymmetric solution demonstrates the possibility of a nonlinear interaction between modes of opposite parity. We suggest that a three dimensional model might exhibit a similar persistent interaction between axisymmetric and non-axisymmetric modes.     

Dipole moment scaling for convection-driven planetary dynamos

Scaling laws are derived for the time-average magnetic dipole moment in rotating convection-driven numerical dynamo models. Results from 145 dynamo models with a variety of boundary conditions and heating modes, covering a wide section of parameter space, show that the time-average dipole moment depends on the convective buoyancy flux F. Two distinct regimes are found above the critical magnetic Reynolds number for onset of dynamo action. In the first regime the external magnetic field is dipole-dominant, whereas for larger buoyancy flux or slower rotation the external field is dominated by higher multipoles and the dipole moment is reduced by a factor of 10 or more relative to the dipolar regime. For dynamos driven by basal heating, the dipole moment M increases like M  F^1/3 in the dipolar regime. Reversing dipolar dynamos tend to cluster near the multipolar transition, which is shown to depend on a local Rossby number parameter. The geodynamo lies close to this transition, suggesting an explanation for polarity reversals and the possibility of a weaker dipole earlier in Earth history. Internally heated dynamos generate smaller dipole moments overall and show a gradual transition from dipolar to multipolar states. Our scaling yields order of magnitude agreement with the dipole moments of Earth, Jupiter, Saturn, Uranus, Neptune, and Ganymede, and predicts a multipolar-type dynamo for Mercury.     

An explicit solution for static unbounded helical dynamos

Abstract

The Lortz dynamo with helical symmetry is re-examined. It is shown that by imposing appropriate boundary conditions the set of possible solutions can be broken down into various classes characterized by the behavior of the mean magnetic field. It is found that, as the cylindrical radius, s, tends to zero, <B_Φ> ～ 0(s^j), <B_z> ～ const + 0(s^j?i), where j>5. It is proved that the azimuthal wavenumber associated with the j=5 class is necessarily equal to 2. The existence of at least one cylindrical surface inside which the dynamo is self-sustained is demonstrated. A new simple explicit solution is obtained. The topology the magnetic field is studied and three-dimensional pictures of the magnetic field lines are exhibited. Finally, a criterion for reversal of the magnetic field as a function of radius is ohtained and is applied to our solution.     

Magnetic Helicity in fast dynamos

The behaviour of magnetic helicity in kinematic dynamos at large magnetic Reynolds number is considered. Hughes, et al . [ Phys. Lett. A 223 , 167-172 (1996)] observe that the relative helicity tends to zero in the limit of large magnetic Reynolds number. This paper gives upper bounds on the helicity, by relating the helicity spectrum to the energy spectrum. These bounds are confirmed by numerical simulation and the distribution of helicity over scales is considered. Although it is found that the total helicity becomes small in the limit of high conductivity, there can remain significant, but cancelling, helicity at large and small scales of the field. This is illustrated by considering the evolution of helicity in the stretch-twist-fold dynamo picture.     

Numerical simulations of MHD dynamos

The generation of magnetic fields in space plasmas and in astrophysics is usually described within the framework of magnetohydrodynamics. Turbulent helical flows produce magnetic fields very efficiently, with correlation length scales larger than those characterizing the flow. Within the context of the solar magnetic cycle, a turbulent dynamo is responsible for the so-called alpha effect, while the Omega effect is associated to the differential rotation of the Sun.We present direct numerical simulations of turbulent magnetohydrodynamic dynamos including two-fluid effects such as the Hall current. More specifically, we study the evolution of an initially weak and small-scale magnetic field in a system maintained in a stationary regime of hydrodynamic turbulence, and explore the conditions for exponential growth of the magnetic energy. In all the cases considered, we find that the dynamo saturates at the equipartition level between kinetic and magnetic energy, and the total energy reaches a Kolmogorov power spectrum.     

Magnetic field reversals and galactic dynamos

We argue that global magnetic field reversals similar to those observed in the Milky Way occur quite frequently in mean-field galactic dynamo models that have relatively strong, random, seed magnetic fields that are localized in discrete regions. The number of reversals decreases to zero with reduction of the seed strength, efficiency of the galactic dynamo and size of the spots of the seed field. A systematic observational search for magnetic field reversals in a representative sample of spiral galaxies promises to give valuable information concerning seed magnetic fields and, in this way, to clarify the initial stages of galactic magnetic field evolution.     

Biomorphological scaling laws from convectively accelerated streams

Worldwide convectively accelerated streams flowing in downstream-narrowing river sections show that riverbed vegetation growing on alluvial sediment bars gradually disappears, forming a front beyond which vegetation is absent. We revise a recently proposed analytical model able to predict the expected longitudinal position of the vegetation front. The model was developed considering the steady state approximation of 1-D ecomorphodynamics equations. While the model was tested against flume experiments, its extension and application to the field is not trivial as it requires the definition of proper scaling laws governing the observed phenomenon. In this work, we present a procedure to calculate vegetation parameters and flow magnitude governing the equilibrium at the reach scale between hydromorphological and biological components in rivers with converging boundaries. We collected from worldwide rivers data of section topography, hydrogeomorphological and riparian vegetation characteristics to perform a statistical analysis aimed to validate the proposed procedure. Results are presented in the form of scaling laws correlating biological parameters of growth and decay from different vegetation species to flood return period and duration, respectively. Such relationships demonstrate the existence of underlying selective processes determining the riparian vegetation both in terms of species and cover. We interpret the selection of vegetation species from ecomorphodynamic processes occurring in convectively accelerated streams as the orchestrated dynamic action of flow, sediment and vegetation characteristics. © 2019 John Wiley & Sons, Ltd.     

Residual fields from extinct dynamos

Abstract

The paper explores some of the many facets of the problem of the generation of magnetic fields in convective zones of declining vigor and/or thickness. The ultimate goal of such work is the explanation of the magnetic fields observed in A-stars. The present inquiry is restricted to kinematical dynamos, to show some of the many possibilities, depending on the assumed conditions of decline of the convection. The examples serve to illustrate in what quantitative detail it will be necessary to describe the convection in order to extract any firm conclusions concerning specific stars.

The first illustrative example treats the basic problem of diffusion from a layer of declining thickness. The second adds a buoyant rise to the field in the layer. The third treats plane dynamo waves in a region with declining eddy diffusivity, dynamo coefficient, and large-scale shear. The dynamo number may increase or decrease with declining convection, with an increase expected if the large-scale shear does not decline as rapidly as the eddy diffusivity. It is shown that one of the components of the field may increase without bound even in the case that the dynamo number declines to zero.     

Bounds on the infimum decay rate for axisymmetric incompressible dynamos

Abstract

The variational lower bound v > 0.39π² determined by Ivers (1984) for the infimum decay rate v of axisymmetric poloidal magnetic fields is corroborated numerically and an upper bound v < 0.66°² established. This is achieved by correcting and extending results for certain flows considered by Chandrasekhar (1956).     

Some new results on spherical dynamos

Small-scale motions in the Earth's liquid core are likely to be highly anisotropic because of the effects of rotation. Guided by physical considerations, the “-effect” described by an anisotropic tensor _ik is formulated and the corresponding boundary value problem for a sphere is solved for a variety of boundary conditions. A converged solution has been obtained only in the case of the Fermi condition of an infinitely conducting exterior of the sphere. In a second part of the paper, some remarks are made on the hypothetical upper bound on magnetic field strengths in planetary cores originally proposed by Busse (1976).     

Evolution of aligned states within nonlinear dynamos

ABSTRACT

The Archontis dynamo is a rare example of an MHD dynamo within which forcing drives a dynamo where the flow and magnetic fields are almost perfectly aligned and the energies are approximately equal. In this paper, I expand upon our knowledge of the dynamo by showing that the intermediate steady states of the kinetic and magnetic energies observed by Cameron and Galloway are not a necessary feature of aligned dynamos. Furthermore, I show that the steady state into which the flow and magnetic fields eventually evolve is remarkably robust to the addition of time dependence and asymmetry to the forcing.