Rotational and magnetic instability in the diffusive tachocline

In this article we study the linear instability of the two-dimensional strongly stratified model for global MHD in the diffusive solar tachocline. Gilman and Fox [Gilman, P.A. and Fox, P., Joint instability of the latitudinal differential rotation and toroidal magnetic fields below the solar convection zone. Astrophys. J., 1997, 484, 439–454] showed that for ideal MHD, the observed surface differential rotation becomes more unstable than is predicted by Watson's [Watson, M., Shear instability of differential rotation in stars. Geophys. Astrophys. Fluid Dyn., 1981, 16, 285–298] nonmagnetic analysis. They showed that the solar differential rotation is unstable for essentially all reasonable values of the differential rotation in the presence of an antisymmetric toroidal field. They found that for the broad field case B _φ～sinθcosθ, θ being the co-latitude, instability occurs only for the azimuthal m?=?1 mode, and concluded that modes which are symmetric (meridional flow in the same direction) about the equator onset at lower field strengths than the antisymmetric modes. We study the effect of viscosity and magnetic diffusivity in the strongly stably stratified case where diffusion is primarily along the level surfaces. We show that antisymmetric modes are now strongly preferred over symmetric modes, and that diffusion can sometimes be destabilising. Even solid body rotation can be destabilised through the action of magnetic field. In addition, we show that when diffusion is present, instability can occur when the longitudinal wavenumber m?=?2.     

Convective dynamos with intermediate and strong fields

Abstract

The weak-field Benard-type dynamo treated by Soward is considered here at higher levels of the induced magnetic field. Two sources of instability are found to occur in the intermediate field regime M ～ T ^1/12, where M and T are the Hartmann and Taylor numbers. On the time scale of magnetic diffusion, solutions may blow up in finite time owing to destabilization of the convection by the magnetic field. On a faster time scale a dynamic instability related to MAC-wave instability can also occur. It is therefore concluded that the asymptotic structure of this dynamo is unstable to virtual increases in the magnetic field energy.

In an attempt to model stabilization of the dynamo in a strong-field regime we consider two approximations. In the first, a truncated expansion in three-dimensional plane waves is studied numerically. A second approach utilizes an ad hoc set of ordinary differential equations which contains many of the features of convection dynamos at all field energies. Both of these models exhibit temporal intermittency of the dynamo effect.     

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.     

Analytical configurations of a force-free magnetic cylinder in the solar wind

The problem of a smooth field configuration, which should be an initial configuration in modeling (using the method of coarse particles) the problem of a stationary solar wind flow around a magnetic cloud in the case of a spatially two-dimensional statement (when a magnetic cloud is considered as a force-free magnetic cylinder with a finite radius) is considered. It has been indicated that such a statement is possible only when the magnetic field in the solar wind is parallel to the cylinder axis. The method for finding the magnetic field of a force-free cylinder with a finite radius, when some field component is specified and another component is determined based on this one (which makes it possible to construct fields with preassigned properties), has been proposed. The variant for constructing the initial field configuration in the transition region around a cylinder has been proposed. This variant makes it possible to gradually pass from homogeneous crossed fields in the solar wind to a force-free magnetic and zero electric fields within a cylinder, an electric field being potential and orthogonal to a magnetic field (in the reference system related to a magnetic cloud).     

Specific features of development of two-dimensional convection in a rotating compressible conducting fluid

An axisymmetric model of convection in a rotating cylinder in an external uniform magnetic field has been considered. In the considered model, the meridional circulation is created by a nonuniform rotation of the lower boundary relative to the other boundaries. In the considered model, the time of formation of the stationary regime in the magnetic field considerably increases if the vertical density (compressibility) inhomogeneity is taken into account for Ekman numbers of E = E _M = 3 × 10⁻³. This example shows that the compressibility of a medium should be taken into account in the convection and dynamics of the magnetic field when the magnetohydrodynamics of the Earth is analyzed.     

Dynamos with a flat α-effect distribution

Abstract

In order to obtain a better insight into the excitation conditions of magnetic fields in flat objects, such as galaxies, we have calculated critical dynamo numbers of different magnetic field modes for spherical dynamos with a flat α-effect distribution. A simple but realistic approximation formula for the rotation curve is employed. In most cases investigated a stationary quadrupole-type solution is preferred. This is a consequence of the flat distribution of the α-effect. Non-axisymmetric fields are in all cases harder to excite than axisymmetric ones. This seems to be the case particularly for flat objects in combination with a realistic rotation curve for galaxies. The question of whether non-axisymmetric (bisymmetric) fields, which are observed in some galaxies, can be explained as dynamos generated by an axisymmetric αω-effect is therefore still open.     

Magnetic field structure in differentially rotating discs

Abstract

In order to gain a better understanding of the processes that may give rise to non-axisymmetric magnetic fields in galaxies, we have calculated field decay rates for models with a realistic galactic rotation curve and including the effects of a locally enhanced turbulent magnetic diffusivity within the disc. In all cases we have studied, the differential rotation increases the decay rate of non-axisymmetric modes, whereas axisymmetric ones are unaffected. A stronger magnetic diffusivity inside the disc does not lead to a significant preference for non-axisymmetric modes. Although Elsasser's antidynamo theorem has not yet been proved for the present case of a non-spherical distribution of the magnetic diffusivity, we do not find any evidence for the theorem not to be valid in general.     

Forward computation of the magnetic field of a 3D body with arbitrary boundary and continually varying magnetization

The forward computation of the gravitational and magnetic fields due to a 3D body with an arbitrary boundary and continually varying density or magnetization is an important problem in gravitational and magnetic prospecting. In order to solve the inverse problem for the arbitrary components of the gravitational and magnetic anomalies due to an arbitrary 3D body under complex conditions, including an uneven observation surface, the existence of background anomalies and very little or no a priori information, we used a spherical coordinate system to systematically investigate forward methods for such anomalies and developed a series of universal spherical harmonic expansions of gravitational and magnetic fields. For the case of a 3D body with an arbitrary boundary and continually varying magnetization, we have also given the surface integral expressions for the common spherical harmonic coefficients in the expansion of the magnetic field due to the body, and a very precise numerical integral algorithm to calculate them. Thus a simple and effective method of solving the forward problem for magnetic fields due to 3D bodies of this kind has been found, and in this way a foundation is laid for solving the inverse problem of these magnetic fields. In addition, by replacing the parameters and unit vectors in the spherical harmonic expansion of a magnetic field by gravitational parameters and a downward unit vector, we have also derived a forward method for the gravitational field (similar to that for the magnetic case) of a 3D body with an arbitrary boundary and continually varying density.     

Evolution of Magnetic Fields in Galaxies with Spiral Structure

We present simulations of the 3D nonlinear induction equation in order to investigate the temporal evolution of large-scale magnetic fields in spiral galaxies. Our model includes differential rotation, ambipolar diffusion and, based on small-scale turbulence, eddy diffusivity and the tensorial -effect with magnetic feedback. The nonaxisymmetric spiral pattern and – if considered – the vertical stratification of the galaxy are represented in its density and turbulence profile. Neglecting vertical stratification the lifetime and geometry of an initial magnetic field depend on the correlation time of interstellar turbulence _corr . Short correlation times increase the lifetime of the initial magnetic field, but the field is rapidly wound up. Its pitch-angles develop to zero. The magnetic field has disappeared after at most 1 to 1.5 Gyr. A resonance like phenomenon is found by tuning the pattern velocity of the galactic spiral. The simulations then show an exceptional amplification of the magnetic field in the case that the pattern speed and a magnetic drift velocity have similar values. Considering a vertical stratification we achieve sufficiently long living grand-designed magnetic fields excited by dynamo action. The behaviour and geometry of the resulting field is again significantly influenced by the correlation time _corr . Small values of _corr lead to axisymmetric fields with small pitch-angles and field-concentration between the spiral arms. Increasing the correlation time the solutions show larger pitch-angles; and depending on very large correlation times the galactic dynamo rather generates fields clearly within the spiral arms and having a bisymmetric structure.     

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.     

On the modified Taylor constraint

A dynamo driven by motions unaffected by viscous forces is termed magnetostrophic. Although such a model might describe magnetic field generation in Earth’s core well, a magnetostrophic dynamo has not yet been found even though Taylor [Proc. R. Soc. Lond. A 1963, 274, 274–283] devised an apparently viable method of finding one. His method for determining the fluid velocity from the magnetic field and the energy source involved only the evaluation of integrals along lines parallel to the Earth’s axis of rotation and the solution of a second-order ordinary differential equation. It is demonstrated below that an approximate solution of this equation for a broad family of magnetic fields is immediate. Furthermore inertia, which was neglected in Taylor’s theory, is restored here, so that the modified theory includes torsional waves, whose existence in the Earth’s core has been inferred from observations of the length of day. Their theory is reconsidered.     

Kinematic dynamo action in a sphere: Effects of periodic time-dependent flows on solutions with axial dipole symmetry

Choosing a simple class of flows, with characteristics that may be present in the Earth's core, we study the ability to generate a magnetic field when the flow is permitted to oscillate periodically in time. The flow characteristics are parameterised by D, representing a differential rotation, M, a meridional circulation, and C, a component characterising convective rolls. The dynamo action of all solutions with fixed parameters (steady flows) is known from earlier studies. Dynamo action is sensitive to these flow parameters and fails spectacularly for much of the parameter space where magnetic flux is concentrated into small regions, leading to high diffusion. In addition, steady flows generate only steady or regularly reversing oscillatory fields and cannot therefore reproduce irregular geomagnetic-type reversal behaviour. Oscillations of the flow are introduced by varying the flow parameters in time, defining a closed orbit in the space ( D,?M ). When the frequency of the oscillation is small, the net growth rate of the magnetic field over one period approaches the average of the growth rates for steady flows along the orbit. At increased frequency time-dependence appears to smooth out flux concentrations, often enhancing dynamo action. Dynamo action can be impaired, however, when flux concentrations of opposite signs occur close together as smoothing destroys the flux by cancellation. It is possible to produce geomagnetic-type reversals by making the orbit stray into a region where the steady flows generate oscillatory fields. In this case, however, dynamo action was not found to be enhanced by the time-dependence. A novel approach is being taken to solve the time-dependent eigenvalue problem where, by combining Floquet theory with a matrix-free Krylov-subspace method, we can avoid large memory requirements for storing the matrix required by the standard approach.     

Magnetic instabilities in rapidly rotating spherical geometries. III. The effect of differential rotation

Abstract

We investigate the influence of differential rotation on magnetic instabilities for an electrically conducting fluid in the presence of a toroidal basic state of magnetic field B ₀ = B_MB₀(r, θ)1 _φ and flow U₀ = U_MU₀ (r, θ)1_φ, [(r, θ, φ) are spherical polar coordinates]. The fluid is confined in a rapidly rotating, electrically insulating, rigid spherical container. In the first instance the influence of differential rotation on established magnetic instabilities is studied. These can belong to either the ideal or the resistive class, both of which have been the subject of extensive research in parts I and II of this series. It was found there, that in the absence of differential rotation, ideal modes (driven by gradients of B ₀) become unstable for A_c ? 200 whereas resistive instabilities (generated by magnetic reconnection processes near critical levels, i.e. zeros of B₀) require A_c ? 50. Here, Λ is the Elsasser number, a measure of the magnetic field strength and Λ_c is its critical value at marginal stability. Both types of instability can be stabilised by adding differential rotation into the system. For the resistive modes the exact form of the differential rotation is not important whereas for the ideal modes only a rotation rate which increases outward from the rotation axis has a stabilising effect. We found that in all cases which we investigated Λ_c increased rapidly and the modes disappeared when R_m ≈ O(Λ_C), where the magnetic Reynolds number R_m is a measure of the strength of differential rotation. The main emphasis, however, is on instabilities which are driven by unstable gradients of the differential rotation itself, i.e. an otherwise stable fluid system is destabilised by a suitable differential rotation once the magnetic Reynolds number exceeds a certain critical value (R_m )_c. Earlier work in the cylindrical geometry has shown that the differential rotation can generate an instability if R_m ) ?O(Λ). Those results, obtained for a fixed value of Λ = 100 are extended in two ways: to a spherical geometry and to an analysis of the effect of the magnetic field strength Λ on these modes of instability. Calculations confirm that modes driven by unstable gradients of the differential rotation can exist in a sphere and they are in good agreement with the local analysis and the predictions inferred from the cylindrical geometry. For Λ = O(100), the critical value of the magnetic Reynolds number (R_m )_c Λ 100, depending on the choice of flow U₀ . Modes corresponding to azimuthal wavenumber m = 1 are the most unstable ones. Although the magnetic field B ₀ is itself a stable one, the field strength plays an important role for this instability. For all modes investigated, both for cylindrical and spherical geometries, (R_m )_c reaches a minimum value for 50 ≈ Λ ≈ 100. If Λ is increased, (R_m )_c ∝ Λ, whereas a decrease of Λ leads to a rapid increase of (R_m )_c, i.e. a stabilisation of the system. No instability was found for Λ ≈ 10 — 30. Optimum conditions for instability driven by unstable gradients of the differential rotation are therefore achieved for ≈ Λ 50 — 100, R_m ? 100. These values lead to the conclusion that the instabilities can play an important role in the dynamics of the Earth's core.     

On the theory of the geomagnetic dynamo based on mean field electrodynamics

A survey is given on the present state of the geomagnetic dynamo theory which is based on the electrodynamics of mean fields in electrically conducting turbulent media. Models of both the ²-type and the ω-type are taken into consideration, i.e. models where the -effect alone, or together with a differential rotation, gives rise to the regeneration of magnetic fields. Results of special investigations are given and discussed. Models of the ²-type apply to the Earth provided that it shows no noticeable differential rotation. It is pointed out that such models can explain not only the generation of a magnetic field with an axisymmetric dipole-like structure, but also the occurrence of non-axisymmetric components and their westward drift. If, however, a noticeable differential rotation exists, models of the ω-type have to be envisaged. Such models also allow an understanding of the existence of a magnetic field with an axisymmetric dipole-like structure, but they do not provide for a straightforward interpretation of the occurrence and behaviour of the non-axisymmetric components.     

Scaling Law for Galactic Magnetic Fields

A nonlinear mean field dynamo in turbulent disks and spherical shells is discussed. We use a nonlinearity in the dynamo which includes the effect of delayed back-reaction of the mean magnetic field on the magnetic part of the — effect. This effect is determined by an evolutionary equation. The axisymmetric case is considered. An analytical expression (in a single-mode approximation) is derived which gives the magnitude of the mean magnetic field as a function of rotation and the parameters for turbulent disks. The value obtained for the mean magnetic field is in agreement with observations for galaxies.     

Shear flow instability of rotating hydromagnetic fluids

Abstract

The effect of an axial magnetic field on the linear stability of shear flows in rotating systems is examined by extending Busse's analysis of the nonmagnetic case to fluids of high magnetic diffusivity in the presence of a magnetic field. The shear is caused by differential rotation which creates slight deviations from a state of rigid rotation, corresponding to a small Rossby number. It is found that the Rossby number for the onset of instability is larger when a magnetic field is present than when it is absent.     

Influence of magnetic interactions on the anisotropy of magnetic susceptibility: the case of single domain particles packed in globule aggregates

The influence of magnetic interactions on the anisotropy of magnetic susceptibility (AMS) have been largely studied by several theoretical models or experiments. Numerical models have shown that when magnetostatic interactions occur, the distributions of particles over the volume rather than their individual orientations control the AMS. We have shown recently from a comprehensive rock magnetic study and from a theoretical 2-dimensional (2-D) model that single domain particles closely packed in globule aggregates could produce strong local random interaction magnetic fields which could influence the magnetic susceptibility and decrease the degree of anisotropy. In this paper, we first present in detail this 2-D theoretical model and then we extend it to the 3-D case. The possible distribution function of the magnetostatic interaction fields comprises two extreme states: it is either isotropic or ordered. The former case corresponds to the thermal-demagnetized state while the second case corresponds to the alternating field (AF) demagnetized state. We show that when easy axes of magnetization are not uniformly distributed, the degree of anisotropy decreases as the interaction field increases in both AF- and thermal-demagnetized states in 2-D and 3-D geometry. Thus we conclude that random magnetic fields generated by a random arrangement of magnetic particles over the sample volume decrease the degree of anisotropy of AMS and may alter the magnetic fabric.     

The effects of different parameter regimes in geodynamo simulations

Over the past 10 years, geodynamo simulations have grown rapidly in sophistication. However, it is still necessary to make certain approximations in order to maintain numerical stability. In addition, models are forced to make assumptions about poorly known parameters for the Earth's core. Different magnetic Prandtl numbers have been used and different assumptions about the presence of radiogenic heating have been made. This study examines some of the consequences of different approximations and assumptions using the Glatzmaier–Roberts geodynamo model. Here, we show that the choice of magnetic Prandtl number has a greater influence on the character of the magnetic field produced than the addition of a plausible amount of radiogenic heating. In particular, we find that prescribing a magnetic Prandtl number of unity with Ekman number limited by current computing resources, results in magnetic fields with significantly smaller intensities and variabilities compared with the much more Earth-like results obtained from simulations with large magnetic Prandtl numbers. A magnetic Prandtl number of unity, with both the viscous and magnetic diffusivities set to the Earth's magnetic diffusivity, requires a rotation rate much smaller than that of the Earth for currently reachable Ekman numbers. This results in a reduced dominance of the Coriolis forces relative to the buoyancy forces, and therefore, a reduction in the magnetic field intensity and the variability compared to the large Prandtl number cases.     

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.