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.     

Hydromagnetic waves in a differentially rotating annulus. II. Resistive instabilities

Abstract

In part I of this study (Fearn, 1983b), instabilities of a conducting fluid driven by a toroidal magnetic field B were investigated. As well as confirming the results of a local stability analysis by Acheson (1983), a new resistive mode of instability was found. Here we investigate this mode in more detail and show that instability exists when B(s) has a zero at some radius s=s _c. Then (in the limit of small resistivity) the instability is concentrated in a critical layer centered on s _c . The importance of the region where B is small casts some doubt on the validity of the simplifications made to the momentum equation in I. Calculations were therefore repeated using the full momentum equation. These demonstrate that the neglect of viscous and inertial terms when the mean field is strong does not lead to spurious results even when there are regions where B is small.     

A note on the heat transfer by the axisymmetric thermal convection in a rotating fluid annulus

Abstract

Some new measurements are presented of the axisymmetric heat transport in a differentially heated rotating fluid annulus. Both rigid and free upper surface cases are studied, for Prandtl numbers of 7 and 45, from low to high rotation rates. The rigid lid case is extended to high rotation rates by suppressing the baroclinic waves, that would normally develop at some intermediate rotation rate, with the use of sloping endwalls.

A parameter P is defined as the square of the ratio of the (non-rotating) thermal sidewall layer thickness to the Ekman layer thickness. For small P the heat transport remains unaffected by the rotation, but as P increases to order unity the Ekman layer becomes thin enough to inhibit the radial mass transport, and hence the heat flux. No explicit Prandtl number dependence is observed. Also this scaling allows the identification of the region in which the azimuthal velocity reaches its maximum. Direct comparisons are drawn with previous experimental and numerical results, which show what can be interpreted as an inhibiting effect of increasing curvature on the heat transport.     

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.     

Role of diffusive overturning in nonlinear axisymmetric convection in a differentially heated rotating annulus

Abstract

The “viscous overturning” mechanism, described in its simplest form by the linearized instability theory of the previous paper, is discussed in relation to certain numerical solutions recently obtained by G. P. Williams for steady thermally driven axisymmetric convective flow of water (Prandtl number = 7) in a rotating annulus differentially heated in the horizontal, in the “upper symmetric regime” parameter range. Viscous overturning plays an important and clearly identifiable role in the flows A3B, A4 and A5, which have free‐slip side walls and top surface, and a less clearcut role in A3 and B2, for which only the top surface is free. The discussion leads to various predictions about annulus flows not yet studied in detail.     

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.     

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.     

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.     

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.     

Some aspects of vortices in rotating stratified fluids

Summary Ideas concerning the overturning of unstably stratified, rotating fluids are explored using potential vorticity.A set of equations governing axi-symmetric flow in a quasi-Boussinesq system are found based on the gradient wind approximation, and a transformation analogous to that developed byHoskins [6] is used.The time-development of a linear, thermally unstable vortex under the action of Ekman pumping is studied with these equations. The changing radial scale during amplification of the vortex is well represented.Finally, some exact steady vortex states for stably stratified fluids are found and their possible relevance to atmospheric vortices is discussed.     

Thermal convection in differentially rotating systems

Abstract

The onset of convection in a cylindrical fluid annulus is analyzed in the case when the cylindrical walls are rotating differentially, a temperature gradient in the radial direction is applied, and the centrifugal force dominates over gravity. The small gap approximation is used and no-slip conditions on the cylindrical walls are assumed. It is found that over a considerable range of the parameter space either convection rolls aligned with the axis of rotation or rolls in the perpendicular (azimuthal) direction are preferred. It is shown that by a suitable redefinition of parameters, results for finite amplitude Taylor vortices and for convection rolls in the presence of shear can be applied to the present problem. Weakly nonlinear results for transverse rolls in a Couette flow indicate the possibility of subcritical bifurcation for Prandtl numbers P less than 0.82. Heat and momentum transports are derived as functions of P and the problem of interaction between transverse and longitudinal rolls is considered. The relevance of the analysis for problems of convection in planetary and stellar atmospheres is briefly discussed.     

Nonlinear waves in rotating magnetohydrodynamics

We consider an electrically conducting fluid in rotating cylindrical coordinates in which the Elsasser and magnetic Reynolds numbers are assumed to be large while the Rossby number is assumed to vanish in an appropriate limit. This may be taken as a simple model for the Earth's outer core. Fully nonlinear waves dominated by the nonlinear Lorentz forces are studied using the method of geometric optics (essentially WKB). These waves are assumed to be of the form of an asymptotic series expanded about ambient magnetic and velocity fields which vanish on the equatorial plane. They take the form of short wave, slowly varying wave trains. The first-order approximation is sinusoidal and basically the same as in the linear problem, with a dispersion relation modified by the appearance of mean terms. These mean terms, as well the undetermined amplitude functions, are found by suppressing secular terms in a “fast” variable in the second-order approximation. The interaction of the mean terms with the dispersion relation is the primary cause of behaviors which differ from the linear case. In particular, new singularities appear in the wave amplitude functions and an initial value problem results in a singularity in one of the mean terms which propagates through the fluid. The singularities corresponding to the linear ones are shown to develop when the corresponding waves propagate toward the equatorial plane.     

Convection in a rotating magnetic system and Taylor's constraint

Abstract

A system is considered in which electrically conducting fluid is contained between two rigid horizontal planes and bounded laterally by a circular cylinder. The fluid is permeated by a strong azimuthal magnetic field. The strength of the field increases linearly with distance from the vertical axis of the cylinder, about which the entire system rotates rapidly. An unstable temperature gradient is maintained by heating the fluid from below and cooling from above. When viscosity and inertia are neglected, an arbitrary geostrophic velocity, which is aligned with the applied azimuthal magnetic field and independent of the axial coordinate, can be superimposed on the basic axisymmetric state. In this inviscid limit, the geostrophic velocity which occurs at the onset of convection is such that the net torque on geostrophic cylinders vanishes (Taylor's condition). The mathematical problem which describes the ensuing marginal convection is nonlinear, and was discussed previously for the planar case by Soward (1986). Here new features are isolated which result from the cylindrical geometry. New asymptotic solutions are derived valid when Taylor's condition is relaxed to include viscous effects.     

Nonlinear internal waves in ideal rotating basins

Abstract

Starting from Euler's equations of motion a nonlinear model for internal waves in fluids is developed by an appropriate scaling and a vertical integration over two layers of different but constant density. The model allows the barotropic and the first baroclinic mode to be calculated. In addition to the nonlinear advective terms dispersion and Coriolis force due to the Earth's rotation are taken into account. The model equations are solved numerically by an implicit finite difference scheme. In this paper we discuss the results for ideal basins: the effects of nonlinear terms, dispersion and Coriolis force, the mechanism of wind forcing, the evolution of Kelvin waves and the corresponding transport of particles and, finally, wave propagation over variable topography. First applications to Lake Constance are shown, but a detailed analysis is deferred to a second paper [Bauer et al. (1994)].     

Hydromagnetic waves in a differentially rotating,stratified spherical shell

Abstract

Investigations of an earlier paper (Friedlander 1987a) are extended to include the effect of an azimuthal shear flow on the small amplitude oscillations of a rotating, density stratified, electrically conducting, non-dissipative fluid in the geometry of a spherical shell. The basic state mean fields are taken to be arbitrary toroidal axisymmetric functions of space that are consistent with the constraint of the ‘‘magnetic thermal wind equation''. The problem is formulated to emphasize the similarities between the magnetic and the non-magnetic internal wave problem. Energy integrals are constructed and the stabilizing/destabilizing roles of the shears in the basic state functions are examined. Effects of curvature and sphericity are studied for the eigenvalue problem. This is given by a partial differential equation (P.D.E.) of mixed type with, in general, a complex pattern of turning surfaces delineating the hyperbolic and elliptic regimes. Further mathematical complexities arise from a distribution of the magnetic analogue of critical latitudes. The magnetic extension of Laplace's tidal equations are discussed. It is observed that the magnetic analogue of planetary waves may propagate to the east and to the west.     

位场数据归一化总水平导数垂向导数边缘识别方法(英文)

位场数据边缘识别技术常常用来识别地质体的边缘位置.本文提出了一种新的位场数据边缘识别方法--归一化总水平导数垂向导数,它具有边缘探测和边缘增强两种功能.该方法首先计算位场数据的总水平导数THDR和总水平导数THDR的n阶垂向导数VDRn,并对n阶垂向导数VDRn采用取大于0的阈值技术得到总水平导数峰值PTHDR,该值可以用来进行边缘探测;其次,计算总水平导数峰值PTHDR与总水平导数THDR的比,并用最大值进行归一化得到归一化总水平导数垂向导数,该值可以用来进行边缘增强;最后,通过理论模型和实际资料检验了方法的有效性和可靠性.     

The role of boundary conditions in the simulation of rotating,stratified turbulence

Abstract

In this paper we use the CASL method to explore the role of boundary conditions in determining the long-time behaviour of rotating, stratified, quasi-geostrophic turbulence. We find that initially two-dimensional (sufficiently tall) columns of potential vorticity (PV) break down through three-dimensional instability to give a fully three-dimensional flow consisting of ellipsoidal structures. This is the case both for rigid-lid (isothermal) vertical boundary conditions and for vertically periodic boundaries. However, the rigid boundary case gives rise to semi-ellipsoids at both the top and bottom boundaries, and, for sufficient domain depths, preferred depths for the formation of ellipsoids in the interior. By contrast, in the vertically periodic case, the distribution of ellipsoids is homogeneous in depth.

The role of the horizontal boundaries is indirect, but still significant. In all cases doubly periodic horizontal boundary conditions are imposed. We consider a range of initial conditions where in each case equal numbers of two-dimensional columns of positive and negative vorticity are used, taking up a fixed, but relatively small fraction of the domain (approximately 5%). Thus when there is only a small number of vortices, they have larger radius. When the initial number of vortices is small enough (i.e., when the radius is not small compared with the horizontal domain width), at long time there is a two-dimensionalisation giving rise to a single column of positive PV and a single column of negative PV, as has been observed in some previous simulations. We find the same phenomenon for both vertically periodic and rigid lid boundary conditions, but it occurs over a broader range of initial conditions in the vertically periodic case. However, in all cases fully three-dimensional final states are regained when the number of vortices is increased while keeping the fraction of the domain occupied by vortices fixed, i.e., when the vortex radius/domain width ratio is sufficiently small.     

Resistive instabilities in rapidly rotating fluids: Linear theory of the convective modes

Abstract

This paper analyzes the linear stability of a rapidly-rotating, stratified sheet pinch in a gravitational field, g, perpendicular to the sheet. The sheet pinch is a layer (O ? z ? d) of inviscid, Boussinesq fluid of electrical conductivity σ, magnetic permeability μ, and almost uniform density ρ _o; z is height. The prevailing magnetic field. B _o(z), is horizontal at each z level, but varies in direction with z. The angular velocity, Ω, is vertical and large (Ω ? V_A/d, where V_A = B₀√(μρ₀) is the Alfvén velocity). The Elsasser number, Λ = σB² ₀/2Ωρ₀, measures σ. A (modified) Rayleigh number, R = gβd²/ρ₀V² _A, measures the buoyancy force, where β is the imposed density gradient, antiparallel to g. A Prandtl number, P_K = μσK, measures the diffusivity, k, of density differences.     

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.     

Convection in a rotating magnetic system and taylor's constraint part II,numerical results

Abstract

Results are presented of a numerical study of marginal convection of electrically conducting fluid, permeated by a strong azimuthal magnetic field, contained in a circular cylinder rotating rapidly about its vertical axis of symmetry. To this basic state is added a geostrophic flow U_G (s), constant on geostrophic cylinders radius s. Its magnitude is fixed by requiring that the Lorentz forces induced by the convecting mode satisfy Taylor's condition. The nonlinear mathematical problem describing the system was developed in an earlier paper (Skinner and Soward, 1988) and the predictions made there are confirmed here. In particular, for small values of the Roberts number q which measures the ratio of the thermal to magnetic diffusivities, two distinct regions can be recognised within the fluid with the outer region moving rapidly compared to the inner. Otherwise, conditions for the onset of instability via the Taylor state (U_G 0) do not differ significantly from those appropriate to the static (U_G = 0) basic state. The possible disruption of the Taylor states by shear flow instabilities is discussed briefly.