Effect of a uniform magnetic field on nonlinear magnetocenvection in a rotating fluid spherical shell

Abstract

Dynamic interaction between magnetic field and fluid motion is studied through a numerical experiment of nonlinear three-dimensional magnetoconvection in a rapidly rotating spherical fluid shell to which a uniform magnetic field parallel to its spin axis is applied. The fluid shell is heated by internal heat sources to maintain thermal convection. The mean value of the magnetic Reynolds number in the fluid shell is 22.4 and 10 pairs of axially aligned vortex rolls are stably developed. We found that confinement of magnetic flux into anti-cyclonic vortex rolls was crucial on an abrupt change of the mode of magnetoconvection which occurred at Δ = 1 ～ 2, where A is the Elsasser number. After the mode change, the fluid shell can store a large amount of magnetic flux in itself by changing its convection style, and the magnetostrophic balance among the Coriolis, Lorentz and pressure forces is established. Furthermore, the toroidal/poloidal ratio of the induced magnetic energy becomes less than unity, and the magnetized anti-cyclones are enlarged due to the effect of the magnetic force. Using these key ideas, we investigated the causes of the mode change of magnetoconvection. Considering relatively large magnetic Reynolds number and a rapid rotation rate of this model, we believe that these basic ideas used to interpret the present numerical experiment can be applied to the dynamics in the Earth's and other planetary cores.     

Magnetic instabilities in rapidly rotating spherical geometries I. from cylinders to spheres

Abstract

The solution of the full nonlinear hydromagnetic dynamo problem is a major numerical undertaking. While efforts continue, supplementary studies into various aspects of the dynamo process can greatly improve our understanding of the mechanisms involved. In the present study, the linear stability of an electrically conducting fluid in a rigid, electrically insulating spherical container in the presence of a toroidal magnetic field B_o(r,θ)lø and toroidal velocity field U_o(r,θ)lø, [where (r,θ,ø) are polar coordinates] is investigated. The system, a model for the Earth's fluid core, is rapidly rotating, the magnetostrophic approximation is used and thermal effects are excluded. Earlier studies have adopted a cylindrical geometry in order to simplify the numerical analysis. Although the cylindrical geometry retains the fundamental physics, a spherical geometry is a more appropriate model for the Earth. Here, we use the results which have been found for cylindrical systems as guidelines for the more realistic spherical case. This is achieved by restricting attention to basic states depending only on the distance from the rotation axis and by concentrating on the field gradient instability. We then find that our calculations for the sphere are in very good qualitative agreement both with a local analysis and with the predictions from the results of the cylindrical geometry. We have thus established the existence of field gradient modes in a realistic (spherical) model and found a sound basis for the study of various other, more complicated, classes of magnetically driven instabilities which will be comprehensively investigated in future work.     

Alfén waves in a thermally stratified fluid

Abstract

The propagation of Alfvén waves along a uniform horizontal field in a highly conducting incompressible fluid, subject to the convective forces produced by a uniform vertical temperature gradient, is treated in a Boussinesq approximation. It is shown that there are exact solutions with large amplitude but restricted form. Their restricted form means that an arbitrary disturbing force produces other motions as well as Alfvén waves. An arbitrary initial disturbance of small amplitude produces waves whose state of polarization varies along the direction of propagation. For large amplitudes, however, any mixtures of polarization states causes scattering into new modes.     

Magnetic instabilities in rapidly rotating spherical geometries II. more realistic fields and resistive instabilities

Abstract

Magnetic instabilities play an important role in the understanding of the dynamics of the Earth's fluid core. In this paper we continue our study of the linear stability of an electrically conducting fluid in a rapidly rotating, rigid, electrically insulating spherical geometry in the presence of a toroidal basic state, comprising magnetic field B_MB _O(r, θ)1ø and flow U_MU _O(r, θ)1ø The magnetostrophic approximation is employed to numerically analyse the two classes of instability which are likely to be relevant to the Earth. These are the field gradient (or ideal) instability, which requires strong field gradients and strong fields, and the resistive instability, dependent on finite resistivity and the presence of a zero in the basic state B _O(r,θ). Based on a local analysis and numerical results in a cylindrical geometry we have established the existence of the field gradient instability in a spherical geometry for very simple basic states in the first paper of this series. Here, we extend the calculations to more realistic basic states in order to obtain a comprehensive understanding of the field gradient mode. Having achieved this we turn our attention to the resistive instability. Its presence in a spherical model is confirmed by the numerical calculations for a variety of basic states. The purpose of these investigations is not just to find out which basic states can become unstable but also to provide a quantitative measure as to how strong the field must become before instability occurs. The strength of the magnetic field is measured by the Elsasser number; its critical value _c describing the state of marginal stability. For the basic states which we have studied we find _c 200–1000 for the field gradient mode, whereas for the resistive modes _c 50–160. For the field gradient instability, _c increases rapidly with the azimuthal wavenumber m whereas in the resistive case there is no such pronounced difference for modes corresponding to different values of m. The above values of _c indicate that both types of instability, ideal and resistive, are of relevance to the parameter regime found inside the Earth. For the resistive mode, as is increased from _c, we find a shortening lengthscale in the direction along the contour B_O = 0. Such an effect was not observable in simpler (for example, cylindrical) models.     

Nonlinearities in dynamo

By the example of the dynamo model in the rotating plane layer heated from below, the effects are examined that lead to the stabilization of an exponentially growing magnetic field in the magnetostrophic convection in passing from the kinematic dynamo mode to the nonlinear mode. The estimates of the energy redistribution in the spectrum are given, and the mechanisms of suppression of helicity are presented. Equalization of the field of velocity and the magnetic field is analyzed. The modes examined are close to those utilized in the up-to-date models of the planetary dynamo in the cores of planets.     

Geostrophic vs magneto-geostrophic adjustment and nonlinear magneto-inertia-gravity waves in rotating shallow water magnetohydrodynamics

The evolution of localised jets and periodic nonlinear waves in rotating shallow water magnetohydrodynamics (rotating SWMHD) and standard rotating shallow water model (RSW) is compared within the framework of translationally-invariant 1.5-dimensional configurations, which are traditionally used in geophysical fluid dynamics for studying geostrophic adjustment and frontogenesis. Such configurations also allow for exact nonlinear wave solutions in both models. A theory of the magneto-geostrophic adjustment, i.e. adjustment of an arbitrary initial configuration to a state of magneto-geostrophic equilibrium in RSWMHD, is developed and confirmed by numerical simulations with a finite-volume well-balanced code. The code is resolving all kinds of waves in the model and corresponding weak solutions equally well. It is benchmarked by reproducing exact solutions – steady essentially nonlinear Alfvèn and mixed magneto-inertia-gravity waves – and used to demonstrate robustness of these solutions with respect to localised along-wave perturbations. It is also shown how the results on adjustment can be extended to the fully 2-dimensional case.     

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.     

Nonlinear dynamics of boussinesq convection in a deep rotating spherical shell-i

Abstract

Convection in a rotating spherical shell has wide application for understanding the dynamics of the atmospheres and interiors of many celestial bodies. In this paper we review linear results for convection in a shell of finite depth at substantial but not asymptotically large Taylor numbers, present nonlinear multimode calculations for similar conditions, and discuss the model and results in the context of the problem of solar convection and differential rotation. Detailed nonlinear calculations are presented for Taylor number T = 10⁵, Prandtl number P = 1, and Rayleigh number R between 1 |MX 10⁴ and 4 |MX 10⁴ (which is between about 4 and 16 times critical) for a shell of depth 20% of the outer radius. Sixteen longitudinal wave numbers are usually included (all even wave numbers m between 0 and 30) the amplitudes of which are computed on a staggered grid in the meridian plane.

The kinetic energy spectrum shows a peak in the wave number range m = 12–18 at R = 10⁴, which straddles the critical wave number m = 14 predicted by linear theory. These are modes which peak near the equator. The spectrum shows a second strong peak at m = 0, which represents the differential rotation driven by the peak convective modes. As R is increased, the amplitude of low wave numbers increases relative to high wave numbers as convection fills in in high and middle latitudes, and as the longitudinal scale of equatorial convection grows. By R = 3 |MX 10⁴, m = 8 is the peak convective mode. There is a clear minimum in the total kinetic energy at middle latitudes relative to low and high, well into the nonlinear regime, representing the continued dominance of equatorial and polar modes found in the linear case. The kinetic energy spectrum for m > 0 is maintained primarily by buoyancy work in each mode, but with substantial nonlinear transfer of kinetic energy from the peak modes to both lower and higher wave numbers.

For R = 1 to 2 |MX 10⁴, the differential rotation takes the form of an equatorial acceleration, with angular velocity generally decreasing with latitude away from the equator (as on the sun) and decreasing inwards. By R = 4 |MX 10⁴, this equatorial profile has completely reversed, with angular velocity increasing with depth and latitude. Also, a polar vortex which has positive rotation relative to the reference frame (no evidence of which has been seen on the sun) builds up as soon as polar modes become important. Meridional circulation is quite weak relative to differential rotation at R = 10⁴, but grows relative to it as R is increased. This circulation takes the farm of a single cell of large latitudinal extent in equatorial regions, with upward flow near the equator, together with a series of narrower cells in high latitudes. It is maintained primarily by axisymmetric buoyancy forces. The differential rotation is maintained at all R primarily by Reynolds stresses, rather than meridional circulation. Angular momentum transport toward the equator for R = 1–2 |MX 10⁴ maintains the equatorial acceleration while radially inward transport maintains the opposite profile at R = 4 |MX 10⁴.

The total heat flux out the top of the convective shell always shows two peaks for the range of R studied, one at the equator and the other near the poles (no significant variation with latitude is seen on the sun), while heat flux in at the bottom shows only a polar peak at large R. The meridional circulation and convective cells transport heat toward the equator to maintain this difference.

The helicity of the convection plus the differential rotation produced by it suggest the system may be capable of driving a field reversing dynamo, but the toroidal field may migrate with lime in each cycle toward the poles and equator, rather than just toward the equator as apparently occurs on the sun.

We finally outline additions to the physics of the model to make it more realistic for solar application.     

On the normal mode instability of modons and Wu-Verkley waves

Abstract

The normal mode instability of steady Wu-Verkley (1993) wave and modons by Verkley (1984, 1987, 1990) and Neven (1992) is considered. All these flows are solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere. A conservation law for infinitesimal perturbations to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (1953) average spectral number of the amplitude of an unstable mode must be equal to a specific number that depends on the degree of the solution in its inner and outer regions as well as on spectral distribution of the mode energy in these regions. Some properties of the conditions for different types of modons are discussed. The maximum growth (and decay) rate of the modes is estimated, and the orthogonality of the amplitude of each unstable, decaying, or non-stationary mode to the basic solution is shown in the energy inner product.

The new instability conditions confine the unstable disturbances of the WV wave and modon to a hypersurface in the perturbation space and allow interpretation of their energy structure. They are also useful both in estimating the maximum growth rate of unstable modes and in testing the numerical algorithms designed for the linear stability study.     

Hydromagnetic waves in a differentially rotating annulus IV. Insulating boundaries

Abstract

The first three papers in this series (Fearn, 1983b, 1984, 1985) have investigated the stability of a strong toroidal magnetic field B_o =B_o(s?)Φ [where (s?. Φ, z?) are cylindrical polars] in a rapidly rotating system. The application is to the cores of the Earth and the planets but a simpler cylindrical geometry was chosen to permit a detailed study of the instabilities present. A further simplification was the use of electrically perfectly conducting boundary conditions. Here, we replace these with the boundary conditions appropriate to an insulating container. As expected, we find the same instabilities as for a perfectly conducting container, with quantitative changes in the critical parameters but no qualitative differences except for some interesting mixing between the ideal (“field gradient”) and resistive modes for azimuthal wavenumber m=1. In addition to these modes, we have also found the “exceptional” slow mode of Roberts and Loper (1979) and we investigate the conditions required for its instability for a variety of fields B_o(s?) Roberts and Loper's analysis was restricted to the case B_o∝s? and they found instability only for m=1 and ?1 <ω<0 [where ω is the frequency non-dimensionalised on the slow timescale τ_x, see (1.5)]. For other fields we found the necessary conditions to be less “exceptional”. One surprising feature of this instability is the importance of inertia for its existence. We show that viscosity is an alternative destabilising agent. The standard (magnetostrophic) approximation of neglecting inertial (and viscous) terms in the equation of motion has the effect of filtering out this instability. The field strength required for this “exceptional” mode to become unstable is found to be very much larger than that thought to be present in the Earth's core, so we conclude that this mode is unlikely to play an important role in the dynamics of the core.     

Thermal and magnetic instabilities in a rapidly rotating fluid sphere

Abstract

A linear analysis is used to study the stability of a rapidly rotating, electrically-conducting, self-gravitating fluid sphere of radius r ₀, containing a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance from the rotation axis. The Lorentz force is of a magnitude comparable with that of the Coriolis force and so convective motions are fully three-dimensional, filling the entire sphere. We are primarily interested in the limit where the ratio q of the thermal diffusivity κ to the magnetic diffusivity η is much smaller than unity since this is possibly of the greatest geophysical relevance.

Thermal convection sets in when the temperature gradient exceeds some critical value as measured by the modified Rayleigh number R_c. The critical temperature gradient is smallest (R_c reaches a minimum) when the magnetic field strength parameter Λ ? 1. [R_c and Λ are defined in (2.3).] The instability takes the form of a very slow wave with frequency of order κ/r ² ₀ and its direction of propagation changes from eastward to westward as Λ increases through Λ _c ? 4.

When the fluid is sufficiently stably stratified and when Λ > Λ_m ? 22 a new mode of instability sets in. It is magnetically driven but requires some stratification before the energy stored in the magnetic field can be released. The instability takes the form of an eastward propagating wave with azimuthal wavenumber m = 1.     

Hydromagnetic waves in rapidly rotating spherical shells generated by poloidal decay modes

Abstract

This paper presents the first attempt to examine the stability of a poloidal magnetic field in a rapidly rotating spherical shell of electrically conducting fluid. We find that a steady axisymmetric poloidal magnetic field loses its stability to a non-axisymmetric perturbation when the Elsasser number A based on the maximum strength of the field exceeds a value about 20. Comparing this with observed fields, we find that, for any reasonable estimates of the appropriate parameters in planetary interiors, our theory predicts that all planetary poloidal fields are stable, with the possible exception of Jupiter. The present study therefore provides strong support for the physical relevance of magnetic stability analysis to planetary dynamos. We find that the fluid motions driven by magnetic instabilities are characterized by a nearly two-dimensional columnar structure attempting to satisfy the Proudman-Taylor theorm. This suggests that the most rapidly growing perturbation arranges itself in such a way that the geostrophic condition is satisfied to leading order. A particularly interesting feature is that, for the most unstable mode, contours of the non-axisymmetric azimuthal flow are closely aligned with the basic axisymmetric poloidal magnetic field lines. As a result, the amplitude of the azimuthal component of the instability is smaller than or comparable with that of the poloidal component, in contrast with the instabilities generated by toroidal decay modes (Zhang and Fearn, 1994). It is shown, by examining the same system with and without fluid inertia, that fluid inertia plays a secondary role when the magnetic Taylor number T_m ? 10⁵. We find that the direction of propagation of hydromagnetic waves driven by the instability is influenced strongly by the size of the inner core.     

The role of mean circulation in parity selection by planetary magnetic fields

Abstract

The kinematic dynamo problem is considered for certain steady velocity fields with symmetries that are plausible in a rapidly rotating convective system. By generalizing results proved for the mean field dynamo model by Proctor (1977a), it is shown that for a related “comparison problem” with modified boundary conditions, the eigenvalues are degenerate if there is no axisymmetric mean circulation, with modes of dipole and quadrupole parity excited with equal ease. The comparison problem can be shown to be closely similar to the dynamo problem when there is a region unfavourable to dynamo action surrounding the dynamo region. The near-symmetries found by Roberts (1972) for the mean field model are invoked to suggest that a close correspondence is likely even when this region is absent. It is therefore conjectured that such mean motions may be important in explaining the observed preference for solutions of dipole parity by planetary dynamos.     

Instability of flows near the onset of convection in a rotating layer with stress-free horizontal boundaries

We investigate instability of convective flows of simple structure (rolls, standing and travelling waves) in a rotating layer with stress-free horizontal boundaries near the onset of convection. We show that the flows are always unstable to perturbations, which are linear combinations of large-scale modes and short-scale modes, whose wave numbers are close to those of the perturbed flows. Depending on asymptotic relations of small parameters α (the difference between the wave number of perturbed flows and the critical wave number for the onset of convection) and ε (ε² being the overcriticality and the perturbed flow amplitude being O(ε)), either small-angle or Eckhaus instability is prevailing. In the case of small-angle instability for rolls the largest growth rate scales as ε^8/5, in agreement with results of Cox and Matthews (Cox, S.M. and Matthews, P.C., Instability of rotating convection. J. Fluid. Mech., 2000, 403, 153–172) obtained for rolls with k = k _c . For waves, the largest growth rate is of the order ε^4/3. In the case of Eckhaus instability the growth rate is of the order of α².     

Wave propagation in a rotating fluid of spherical configuration

Abstract

An asymptotic approximation to the solution of the time-dependent linearized equations governing the motion of an incompressible, inviscid rotating fluid of spherical configuration having uniform density, variable depth and a free upper surface is obtained using the ray method without a shallow water assumption. This result is then modified to obtain a ray approximation to the solution of the time-reduced problem and the free oscillations of the fluid are studied. Axisymmetric modes covering the whole sphere and asymmetric modes trapped in both equatorial and non-equatorial regions are discovered, and all these modes are shown to have countably many resonance frequencies. A shallow water limit is defined and this limit of the time-reduced approximation is obtained. Most of the modes of free oscillation are lost in this limit and the limiting axisymmetric modes are shown to be trapped in the equatorial region and are singular at the wave region boundaries. The limiting approximation is compared to previous results obtained under a shallow water assumption.     

Magnetic buoyancy instabilities incorporating rotation

Abstract

Recent calculations suggest that the bulk of the solar toroidal field may be stored in a thin, convectively stable region situated between the convection zone proper and the radiative zone. Determining the stability properties of such a field is therefore important with implications for both the generation and escape of magnetic flux. The plane layer, linear stability analysis of Hughes (1985) is extended to incorporate the effects of uniform rotation. Detailed studies are made of interchange, or “axisymmetric” modes and of undular, or wavelike, motions, considering modes of both low and high frequency. The force due to rotation acts to constrain the fluid motions, a feature which is strongly stabilizing for direct modes, but can, in certain circumstances, be destabilizing for oscillatory motions.

For the interchange modes we show that the instability discussed at length by Hughes (1985), driven by fields increasing with height, is still present and indeed may be enhanced by rotational effects. We also study the more conventional instabilities, discussing the transformation between direct and oscillatory modes and considering in detail some peculiar properties of the oscillatory instabilities.

The more relevant instabilities in an astrophysical context are likely to be undular modes. Previous studies of low frequency modes driven by top heavy field gradients are extended to consider modes of various frequencies for a wide range of parameter values. Of particular interest is the occurrence of two distinct modes of instability for bottom heavy field gradients. We also exhibit some of the peculiar stability boundaries which can result when none of the competing influences in the problem is dominant.     

Convective instabilities in a variable viscosity fluid cooled from above

Whether in the mantle or in magma chambers, convective flows are characterized by large variations of viscosity. We study the influence of the viscosity structure on the development of convective instabilities in a viscous fluid which is cooled from above. The upper and lower boundaries of the fluid are stress-free. A viscosity dependence with depth of the form ν₀ + ν₁ exp(?γ.z) is assumed. After the temperature of the top boundary is lowered, velocity and temperature perturbations are followed numerically until convective breakdown occurs. Viscosity contrasts of up to 10⁷ and Rayleigh numbers of up to 10⁸ are studied.For intermediate viscosity contrasts (around 10³), convective breakdown is characterized by the almost simultaneous appearance of two modes of instability. One involves the whole fluid layer, has a large horizontal wavelength (several times the layer depth) and exhibits plate-like behaviour. The other mode has a much smaller wavelength and develops below a rigid lid. The “whole layer” mode dominates for small viscosity contrasts but is suppressed by viscous dissipation at large viscosity contrasts.For the “rigid lid” mode, we emphasize that it is the form of the viscosity variation which determines the instability. For steep viscosity profiles, convective flow does not penetrate deeply in the viscous region and only weak convection develops. We propose a simple method to define the rigid lid thickness. We are thus able to compute the true depth extent and the effective driving temperature difference of convective flow. Because viscosity contrasts in the convecting region do not exceed 100, simple scaling arguments are sufficient to describe the instability. The critical wavelength is proportional to the thickness of the thermal boundary layer below the rigid lid. Convection occurs when a Rayleigh number defined locally exceeds a critical value of 160–200. Finally, we show that a local Rayleigh number can be computed at any depth in the fluid and that convection develops below depth z_r (the rigid lid thickness) such that this number is maximum.The simple similarity laws are applied to the upper mantle beneath oceans and yield estimates of 5 × 10¹⁵?5 × 10¹⁶ m² s^?1 for viscosity in the thermal boundary layer below the plate.     

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.     

Convection in rotating annular channels heated from below. Part 3: Experimental boundary conditions

Results are presented from both linear stability analysis and numerical simulations of three-dimensional nonlinear convection in a Boussinesq fluid in an annular channel, under experimental boundary conditions, rotating about a vertical axis uniformly heated from below. The focus is placed on the Prandtl number Pr = 7.0, representing liquid water at room temperature. The linear analysis shows that, when the aspect ratio is sufficiently small, there exists only one stationary mode that occupies the whole fluid container. When the aspect ratio is moderate or large, however, there exist three different linear solutions: (i) the outer sidewall-localized traveling wave propagating against the sense of rotation; (ii) the inner sidewall-localized traveling wave propagating in the same sense as rotation; and (iii) both the counter-traveling waves occurring simultaneously. Guided by the result of the linear stability analysis, fully three-dimensional simulations are then performed for a channel with a moderate aspect ratio. It is found that neither the prograde nor the retrograde mode is physically realizable near threshold and beyond. The dynamics of nonlinear convection in a rotating channel are chiefly characterized by the interaction between the sidewall-localized waves and the interior convection cells/rolls, producing an interesting and unusual nonlinear phenomenon. In order to compare with the classical Rayleigh–Bénard problem without vertical sidewalls, we also study linear and nonlinear convection at exactly the same parameters but in an infinitely extended layer with periodic horizontal conditions. This reveals that both the linear instability and nonlinear convection in a rotating channel are characteristically different from those in a rotating layer with periodic horizontal conditions.