The Dynamical Effects of Hyperviscosity on Numerical Geodynamo Models

The most fundamental difficulty in the construction of an Earth-like dynamo model is associated with the constraint caused by the rapid rotation of the Earth. To stabilise numerical codes, many workers have introduced hyperviscosity into the governing equations. One of the major effects introduced by hyperviscosity is to offset the rotational constraint, and, consequently, to alter the key dynamics of an Earth-like dynamo. In this paper, an Earth-like convection model with or without the presence of an imposed magnetic field is investigated with or without the effect of hyperviscosity. A nonlinear dynamo model with the mean field approximation is also used to examine the dynamical effect of hyperviscosity. The results suggest that great care should be taken when hyperviscosity is employed in geodynamo models.     

Analytic base of geodynamo-like scaling laws in the planets,geomagnetic periodicities and inversions

Scaling laws for hydromagnetic dynamo in planets initially express the characteristic strength of the magnetic field through the primary values, such as the size of the conductive core of the planet, the angular rotation rate, electrical conductivity and energy flows. Most of the earlier proposed scaling laws based only on observations and assumptions about force balances. Recent and my new approaches to fully take into account the energy and induction balance has additionally expressed here in terms of primary values such important characteristics as forces, magnitudes, energies, scales and orientations of hydromagnetic fields. The direct numerical simulation of the hydromagnetic dynamo and modeling ability in a fairly wide range of parameters for the first time allowed direct test such laws. The obtained numerical geodynamo-like results for the Earth, Jupiter and partially Saturn postulated previously not identified analytically simplest law that predicts the field strength is only depended on the specific energy density of convection and the size of the dynamo area. This simplest and already widely used law was original way analytically grounded here along with other previously known and new laws. This analytic identifies the physics determining geomagnetic periodicities for jerk, secular variations and inversions. Mean period between the inversions is found to be roughly proportional to the intensity of the geomagnetic field that is confirmed by some paleomagnetic researches. Possible dynamos in Mercury, Ganymede, Uranus and Neptune are also discussed.     

The flux ejection dynamo effect

Abstract

The mean-field effects of cyclonic convection become increasingly complex when the cyclonic rotation exceeds ½-π. Net helicity is not required, with negative turbulent diffusion, for instance, appearing in mirror symmetric turbulence. This paper points out a new dynamo effect arising in convective cells with strong asymmetry in the rotation of updrafts as against downdrafts. The creation of new magnetic flux arises from the ejection of reserve flux through the open boundary of the dynamo region. It is unlike the familiar α-effect in that individual components of the field may be amplified independently. Several formal examples are provided to illustrate the effect. Occurrence in nature depends upon the existence of fluid rotations of the order of π in the convective updrafts. The flux ejection dynamo may possibly contribute to the generation of field in the convective core of Earth and in the convective zone of the sun and other stars.     

Modeling of the solar activity double cycle using dynamical systems

The generation and evolution of the Sun’s magnetic field and other stars is usually related to the dynamo mechanism. This mechanism is based on the consideration of the joint influence of the α effect and differential rotation. Dynamo sources can be located at different depths of the convection zone and can have different intensities. Based on such a system, the dynamical system in the case of the stellar dynamo in a two-layer medium has been constructed with regard to meridional fluxes in order to model the double cycle that corresponds to the simultaneous presence of 22-year and quasi-biennial magnetic field oscillations. It has been indicated that the regime of mixed oscillations can originate because a dynamo wave moves oppositely to the meridional flows in the upper layer of the convection zone. This results in the deceleration of the toroidal field propagation and in the generation of slow oscillations. In deeper layers, the directions of a dynamo wave and meridional flows coincide with each other, as a result of which fast magnetic fields originate. Therefore, the total contribution of two oscillations with different frequencies corresponds to the appearance of quasi-biennial cycles against 22-year cycles. It has been indicated that the beating regime, which can be related to the secular oscillations of solar magnetic activity, originates in the system when the meridional flows are weak.     

Dynamo in Asymmetric Square Convection

Linear and nonlinear dynamo action is investigated for square patterns in nonrotating and weakly rotating Boussinesq Rayleigh-Bénard convection in a plane horizontal layer. The square-pattern solutions may or may not be symmetric to up-down reflections. Vertically symmetric solutions correspond to checkerboard patterns. They do not possess a net kinetic helicity and are found to be incapable of kinematic dynamo action at least up to magnetic Reynolds numbers of , 12 000. There also exist vertically asymmetric squares, characterized by rising (descending) motion in the centers and descending (rising) motion near the boundaries, among them such that possess full horizontal square symmetry and others lacking also this symmetry. The flows lacking both the vertical and horizontal symmetries possess kinetic helicity and show kinematic dynamo action even without rotation. The generated magnetic fields are concentrated in vertically oriented filamentary structures. Without rotation these dynamos are, however, always only kinematic, not nonlinear dynamos since the back-reaction of the magnetic field then forces the solution into the basin of attraction of a roll pattern incapable of dynamo action. But with rotation added parameter regions are found where stationary asymmetric squares are also nonlinear dynamos. These nonlinear dynamos are characterized by a subtle balance between the Coriolis and Lorentz forces. In some parameter regions also nonlinear dynamos with flows in the form of oscillating squares or stationary modulated rolls are found.     

On the locality of the kinetic energy transfer in a wave space

For the model of thermal turbulence in the Boussinesq approximation with heating from below a change in the structure of the nonlinear interaction of the harmonics of the velocity field with the appearance of rotation is examined. The parameters of convection with rotation are chosen in such a way to correspond to the typical regimes in the models of the planetary dynamo. For such regimes, the structure of the triadic mechanism of the kinetic energy transfer through the spectrum is investigated.     

Geomagnetic Reversals and Dynamo Bursts in a Simple Geodynamo Model

The question of what exactly happens with the geodynamo process during the reversal of a geomagnetic field is studied in a simple geodynamo model. The geodynamo action is described by the so called dynamo number characterizing the joint action of the main drivers of the geomagnetic field, i.e., the differential rotation and mirror–asymmetric convection. In mirror-asymmetric convection, for instance, in the northern hemisphere, there are more right vortices than left vortices, whereas in the southern hemisphere, there are more left vortices than right vortices. The effect of the magnetic field on the flow is described by the suppression of the mirror asymmetry: due to this suppression, e.g., in the northern hemisphere, the excess of right vortices over left vortices decreases. It is also assumed that due to this suppression, the mirror asymmetry can change its sign; i.e., the number of left vortices in the northern hemisphere can become larger than the number of right vortices. Correspondingly, the dynamo number can also change its sign. It is shown that the short-term changes of the sign of the dynamo number are responsible for the very short time span accommodating the reversal, when compared to the interval between the reversals.     

Towards the interpretation of Uranus's eccentric magnetic field

Abstract

Coriolis forces stimulate dynamo action in a rapidly-rotating fluid by promoting complexities in the pattern of fluid motions, notably departures from symmetry about the axis of rotation. This pattern and its time variations determine the instantaneous form and temporal behaviour of the magnetic field so produced. Instantaneous magnetic fields will usually exhibit in their broad-scale features approximate alignment with the rotation axis. This is borne out by observations of the magnetic fields of the Earth, Jupiter and Saturn, and it is likely on general grounds that Neptune will be found to have an aligned magnetic field. But, as is shown by laboratory and theoretical studies of thermal convection in rapidly-rotating fluids, for some ranges of rotation speed, rate of heating, etc. certain patterns can occur which in electrically-conducting fluids would produce magnetic fields exhibiting departures from alignment with the rotation axis, which instantaneously could be quite pronounced but would average out to very small values over sufficiently long periods of time. These findings indicate obvious strategies for theoretical studies towards the interpretation of Uranus's eccentric magnetic field (which need not invoke departures from axial symmetry in the thermal, mechanical or electrical boundary conditions of the dynamo region within the planet) and for further observational studies.     

On the minimal instability time of hydromagnetic flows in the Earth’s core

The dependence of the intensity of geomagnetic field on the intensity of thermal convection in the liquid core of the Earth, which has been empirically derived by a number of the authors from the results of numerical modeling of convective dynamo, is substantiated theoretically. This dependence is used for estimating the characteristic time scale of jerk evolution.     

Oscillatory large-scale dynamos from Cartesian convection simulations

We present results from compressible Cartesian convection simulations with and without imposed shear. In the former case the dynamo is expected to be of α² Ω type, which is generally expected to be relevant for the Sun, whereas the latter case refers to α² dynamos that are more likely to occur in more rapidly rotating stars whose differential rotation is small. We perform a parameter study where the shear flow and the rotational influence are varied to probe the relative importance of both types of dynamos. Oscillatory solutions are preferred both in the kinematic and saturated regimes when the negative ratio of shear to rotation rates, q?≡??S/Ω, is between 1.5 and 2, i.e. when shear and rotation are of comparable strengths. Other regions of oscillatory solutions are found with small values of q, i.e. when shear is weak in comparison to rotation, and in the regime of large negative qs, when shear is very strong in comparison to rotation. However, exceptions to these rules also appear so that for a given ratio of shear to rotation, solutions are non-oscillatory for small and large shear, but oscillatory in the intermediate range. Changing the boundary conditions from vertical field to perfect conductor ones changes the dynamo mode from oscillatory to quasi-steady. Furthermore, in many cases an oscillatory solution exists only in the kinematic regime whereas in the nonlinear stage the mean fields are stationary. However, the cases with rotation and no shear are always oscillatory in the parameter range studied here and the dynamo mode does not depend on the magnetic boundary conditions. The strengths of total and large-scale components of the magnetic field in the saturated state, however, are sensitive to the chosen boundary conditions.     

From stable dipolar towards reversing numerical dynamos

We are using a three-dimensional convection-driven numerical dynamo model without hyperdiffusivity to study the characteristic structure and time variability of the magnetic field in dependence of the Rayleigh number (Ra) for values up to 40 times supercritical. We also compare a variety of ways to drive the convection and basically find two dynamo regimes. At low Ra, the magnetic field at the surface of the model is dominated by the non-reversing axial dipole component. At high Ra, the dipole part becomes small in comparison to higher multipole components. At transitional values of Ra, the dynamo vacillates between the dipole-dominated and the multipolar regime, which includes excursions and reversals of the dipole axis. We discuss, in particular, one model of chemically driven convection, where for a suitable value of Ra, the mean dipole moment and the temporal evolution of the magnetic field resemble the known properties of the Earth’s field from paleomagnetic data.     

Tuning of the mean-field geodynamo model

Parker’s two-dimensional (2D) dynamo model with an algebraic form of nonlinearity for the α-effect is considered. The model uses geostrophic distributions for the α-effect and differential rotation, which are derived from the three-dimensional (3D) convection models. The resulting configurations of the magnetic field in the liquid core are close to the solutions in Braginsky’s Z-model. The implications of the degree of geostrophy observed in the 3D dynamo models for the behavior of the mean magnetic field are explored. It is shown that the reduction in geostrophy leads to magnetic field reversals accompanied by the relative growth of the nondipole component of the field on the surface of the liquid core. The simulations with a random α-effect which causes turbulent pulsations are carried out. The approach is capable of producing realistic sequences of magnetic reversals.     

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.     

Distribution of magnetic energy in αΩ-dynamos,I: The method

Abstract

In this paper a method for solving the equation for the mean magnetic energy <BB> of a solar type dynamo with an axisymmetric convection zone geometry is developed and the main features of the method are described. This method is referred to as the finite magnetic energy method since it is based on the idea that the real magnetic field B of the dynamo remains finite only if <BB> remains finite. Ensemble averaging is used, which implies that fields of all spatial scales are included, small-scale as well as large-scale fields. The method yields an energy balance for the mean energy density ε ≡ B ²/8π of the dynamo, from which the relative energy production rates by the different dynamo processes can be inferred. An estimate for the r.m.s. field strength at the surface and at the base of the convection zone can be found by comparing the magnetic energy density and the outgoing flux at the surface with the observed values. We neglect resistive effects and present arguments indicating that this is a fair assumption for the solar convection zone. The model considerations and examples presented indicate that (1) the energy loss at the solar surface is almost instantaneous; (2) the convection in the convection zone takes place in the form of giant cells; (3) the r.m.s. field strength at the base of the solar convection zone is no more than a few hundred gauss; (4) the turbulent diffusion coefficient within the bulk of the convection zone is about 10¹⁴cm²s^?1, which is an order of magnitude larger than usually adopted in solar mean field models.     

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.     

Energy geodynamo parameters compatible with analytical,numerical, paleomagnetic models and observations

A hydromagnetic dynamo is only possible at a sufficiently powerful convection. In the Earth’s core, it is probably the nonthermal convection very much in excess of its critical level with the molecular transporr coefficients. However, in the case of medium- or large-scale fields, the critical energy level caused by the turbulent tranport coefficients is likely to be slightly below the actual level. This probably explains both the 22-year success of this type of simplified geodynamo models and the energy scaling laws for hydromagnetic fields, which generalize these models. Also the review of energy-dependent analytical and observational estimates of vortex fields, hydromagnetic scale sizes, and velocities in the core is presented. These typical parameters are partly in a new way linked to the observed and more ancient magnetic variations. New, albeit, simplified and self-evident, substantiation is given to the paleomagnetic hypothesis about the predominance of the axial dipole under a certain time averaging. In (Pozzo et al., 2012) and more recent works, it is shown that the adiabatic heat flow and electrical conductivity in the Earth’s core are severalfold higher than the generally accepted estimates. Here, the dynamo supporting Braginsky’s convection (Braginsky, 1963) (under the crystallization of the heavy fraction of a liquid onto the solid core) started less than 1 Ga ago, whereas the more ancient geodynamo was supported by the compositional convection of another type. The known mechanisms implementing this convection, which differ by the scenarios of magnetic evolution, are reviewed. This may help identify the sought mechanism through the most ancient paleomagnetic estimates of the field’s intensity and through the numerical models. The probable mechanisms of generation and their absence for the primordial and recent magnetic field of the studied terrestrial planets are discussed.     

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.     

Large-scale convective dynamos in a stratified rotating plane layer

We study the effect of stratification on large-scale dynamo action in convecting fluids in the presence of background rotation. The fluid is confined between two horizontal planes and both boundaries are impermeable, stress-free and perfectly conducting. An asymptotic analysis is performed in the limit of rapid rotation (τ???1 where τ is the Taylor number). We analyse asymptotic magnetic dynamo solutions in rapidly rotating systems generalising the results of Soward [A convection-driven dynamo I. The weak field case. Philos. Trans. R. Soc. Lond. A 1974, 275, 611–651] to include the effects of compressibility. We find that in general the presence of stratification delays the efficiency of large-scale dynamo action in this regime, leading to a reduction of the onset of dynamo action and in the nonlinear regime a diminution of the large-scale magnetic energy for flows with the same kinetic energy.     

Helicity–vorticity turbulent pumping of magnetic fields in the solar convection zone

We study the effect of turbulent drift of a large-scale magnetic field that results from the interaction of helical convective motions and differential rotation in the solar convection zone. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. The direction of the drift depends on the sign of helicity and it is defined by the Parker–Yoshimura rule. The analytic calculations are done within the framework of mean-field magnetohydrodynamics using the minimal τ-approximation. We estimate the magnitude of the drift velocity and find that it can be a few m/s near the base of the solar convection zone. The implications of this effect for the solar dynamo are illustrated on the basis of an axisymmetric mean-field dynamo model with a subsurface shear layer. The model shows that near the bottom of the convection zone the helicity–vorticity pumping results mostly from the kinetic helicity contributions. We find that the magnetic helicity contributions to the pumping effect are dominant at the subsurface shear layer. There the magnitude of the drift velocity is found to be a few cm/s. We find that the helicity–vorticity pumping effect can have an influence on the features of the sunspot time–latitude diagram, producing a fast drift of the sunspot activity maximum at the rise phase of the cycle and a slow drift at the decay phase of the cycle.