The onset of magnetoconvection at large Prandtl number in a rotating layer I. Finite magnetic diffusion

Abstract

This paper develops further a convection model that has been studied several times previously as a very crude idealization of planetary core dynamics. A plane layer of electrically-conducting fluid rotates about the vertical in the presence of a magnetic field. Such a field can be created spontaneously, as in the Childress—Soward dynamo, but here it is uniform, horizontal and externally-applied. The Prandtl number of the fluid is large, but the Ekman, Elsasser and Rayleigh numbers are of order unity, as is the ratio of thermal to magnetic diffusivity. Attention is focused on the onset of convection as the temperature difference applied across the layer is increased, and on the preferred mode, i.e., the planform and time-dependence of small amplitude convection. The case of main interest is the layer confined between electrically-insulating no-slip walls, but the analysis is guided by a parallel study based on illustrative boundary conditions that are mathematically simpler.     

The structure of untwisted magnetic flux tubes

Abstract

The equilibrium structure of an axisymmetric magnetic flux tube confined by an external pressure p_e(Z) that varies along the length of the tube is studie. In the past, most work has concentrated on slender flux tubes, where the effects of transverse structure and radial fields are neglected. Here the aim is to explore the properties of thick tubes, in order to see to what extent the slender tube theory is valid. The main results are:     

Non-linear marginal convection in a rotating magnetic system

Abstract

An inviscid, electrically conducting fluid is contained between two rigid horizontal planes and bounded laterally by two vertical walls. The fluid is permeated by a strong uniform horizontal magnetic field aligned with the side wall boundaries and the entire system rotates rapidly about a vertical axis. The ratio of the magnitudes of the Lorentz and Coriolis forces is characterized by the Elsasser number, A, and the ratio of the thermal and magnetic diffusivities, q. By heating the fluid from below and cooling from above the system becomes unstable to small perturbations when the adverse density gradient as measured by the Rayleigh number, R, is sufficiently large.

With the viscosity ignored the geostrophic velocity, U, which is aligned with the applied magnetic field, is independent of the coordinate parallel to the rotation axis but is an arbitrary function of the horizontal cross-stream coordinate. At the onset of instability the value of U taken ensures that Taylor's condition is met. Specifically the Lorentz force, which results from marginal convection must not cause any acceleration of the geostrophic flow. It is found that the critical Rayleigh number characterising the onset of instability is generally close to the corresponding value for the usual linear problem, in which Taylor's condition is ignored and U is chosen to vanish. Significant differences can occur when q is small owing to a complicated flow structure. There is a central interior region in which the local magnetic Reynolds number, R_m , based on U is small of order q and on exterior region in which R_m is of order unity.     

Slow hydromagnetic oscillations in a rotating spheroidal shell

Abstract

The ray method is used to study slow hydromagnetic waves in an incompressible, inviscid, perfectly conducting fluid of constant density in the presence of a constant toroidal magnetic field. The fluid is bounded below by a rigid sphere and above by a rigid spheroidal surface, and the mean fluid layer thickness is assumed to be small. Both the general time-dependent and time-harmonic (free oscillation) problems are studied and dispersion relations and conservation laws are derived. These results are applied to free oscillations with constant azimuthal wave number in a spherical shell and then compared to those of previous authors. Such oscillations propagate to the east and are trapped between circles of constant latitude. Wave propagation in axisymmetric shells is then studied with emphasis on the relationship between shell shape and direction of propagation, and it is found that such shells can sustain westward propagating modes wherever the shell thickness decreases sufficiently rapidly from a maximum at the poles to zero at the equator; no shells exist which can sustain westward propagation at the equator.     

The effect of anisotropic diffusivity on rotating magnetoconvection in the Earth's core

A thermal diffusive process in the Earth's core is principally enhanced by small-scale flows that are highly anisotropic because of the Earth's rapid rotation and a strong magnetic field. This means that a thermal eddy diffusivity should not be a scalar but a tensor. The effect of such anisotropic tensor diffusivity, which is to be prescribed, on dynamics in the Earth's core is investigated through numerical simulations of magnetoconvection in a rapidly rotating system. A certain degree of anisotropy has an insignificant effect on the character, like kinetic and magnetic energies, of magnetoconvection in a small region with periodic boundaries in the three directions. However, in a region with top and bottom rigid boundary surfaces, kinetic and magnetic energies of magnetoconvection can be altered by the same degree of anisotropy. This implies that anisotropic tensor diffusivity affects on dynamics in the core, in particular near the boundary surfaces.     

On the validity of a model for the reversals of the Earth's magnetic field

Abstract

We have used techniques of nonlinear dynamics to compare a special model for the reversals of the Earth's magnetic field with the observational data. Although this model is rather simple, there is no essential difference to the data by means of well-known characteristics, such as correlation function and probability distribution. Applying methods of symbolic dynamics we have found that the considered model is not able to describe the dynamical properties of the observed process. These significant differences are expressed by algorithmic complexity and Renyi information.     

Resistive wave breaking in the Earth's outer core

The equations for an electrically conducting fluid in cylindrical coordinates are linearized assuming that the inertial terms in the momentum equation can be ignored (small Rossby number), and that the ratio of the Elsasser number and magnetic Reynolds number is one. After these assumptions, the governing equations are linearized about an ambient solution which vanishes at the the equator. Upon assuming large Elsasser and magnetic Reynolds number, the solutions to the linearized equations are approximated by wave trains having very short wave length (relative to the core radius) but which vary slowly (on a scale of the core radius). The period of the waves is much longer than a day but much shorter than the period of the slow hydromagnetic oscillations. These waves are found to be trapped in a region about the equator and away from the axis of rotation. The waves break at a latitudinal wave region boundary, in the sense that the waves become exponentially large in a boundary layer, having as an exponent some positive power of the large azimuthal wave number. This behavior is amplified as the Elsasser number becomes smaller while still remaining relatively large. Waves in more Earth-like parameter regimes are discussed briefly.     

Magnetic instability in a rapidly rotating cylindrical annulus with a finitely conducting inner core

Abstract

We consider the stability of a toroidal magnetic field B = B(s*) (where (s*,φ,z*) are cylindrical polar coordinates) in a cylindrical annulus of conducting fluid with inner and outer radii s_i and s _o rotating rapidly about its axis. The outer boundary is taken to be either insulating or perfectly conducting, and the effect of a finite magnetic diffusivity in the inner core is investigated. The ratio of magnetic diffusivity in the inner core to that of the outer core is denoted by ηη→0 corresponding to a perfectly conducting inner core and η→∞ to an insulating one. Comparisons with the results of Fearn (1983b, 1988) were made and a good match found in the limits η→0 and η→∞ with his perfectly conducting and insulating results, respectively. In addition a new mode of instability was found in the eta;→0 regime. Features of this new mode are low frequency (both the frequency and growth rate →0 as η→0) and penetration deep into the inner core. Typically it is unstable at lower magnetic field strengths than the previously known instabilities.     

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.     

Numerical simulations of thermal convection in a rapidly rotating spherical shell cooled inhomogeneously from above

Abstract

Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell with and without inhomogeneous temperature anomalies on the top boundary have been carried out using a three-dimensional, time-dependent, spectral-transform code. The spherical shell of Boussinesq fluid has inner and outer radii the same as those of the Earth's liquid outer core. The Taylor number is 10⁷, the Prandtl number is 1, and the Rayleigh number R is 5R_c (R_c is the critical value of R for the onset of convection when the top boundary is isothermal and R is based on the spherically averaged temperature difference across the shell). The shell is heated from below and cooled from above; there is no internal heating. The lower boundary of the shell is isothermal and both boundaries are rigid and impermeable. Three cases are considered. In one, the upper boundary is isothermal while in the others, temperature anomalies with (l,m) = (3,2) and (6,4) are imposed on the top boundary. The spherically averaged temperature difference across the shell is the same in all three cases. The amplitudes of the imposed temperature anomalies are equal to one-half of the spherically averaged temperature difference across the shell. Convective structures are strongly controlled by both rotation and the imposed temperature anomalies suggesting that thermal inhomogeneities imposed by the mantle on the core have a significant influence on the motions inside the core. The imposed temperature anomaly locks the thermal perturbation structure in the outer part of the spherical shell onto the upper boundary and significantly modifies the velocity structure in the same region. However, the radial velocity structure in the outer part of the shell is different from the temperature perturbation structure. The influence of the imposed temperature anomaly decreases with depth in the shell. Thermal structure and velocity structure are similar and convective rolls are more columnar in the inner part of the shell where the effects of rotation are most dominant.     

The instability of barotropic circular vortices

Abstract

The linear, normal mode instability of barotropic circular vortices with zero circulation is examined in the f-plane quasigeostrophic equations. Equivalents of Rayleigh's and Fjortoft's criteria and the semicircle theorem for parallel shear flow are given, and the energy equation shows the instability to be barotropic. A new result is that the fastest growing perturbation is often an internal instability, having a finite vertical scale, but may also be an external instability, having no vertical structure. For parallel shear flow the fastest growing perturbation is always an external instability; this is Squire's theorem. Whether the fastest growing perturbation is internal or external depends upon the profile: for mean flow streamfunction profiles which monotonically decrease with radius, the instability is internal for less steep profiles with a broad velocity extremum and external for steep profiles with a narrow velocity extremum. Finite amplitude, numerical model calculations show that this linear instability analysis is not valid very far into the finite amplitude range, and that a barotropic vortex, whose fastest growing perturbation is internal, is vertically fragmented by the instability.     

Magnetic Rossby waves in a stably stratified layer near the surface of the Earth's outer core

Abstract

The stratification profile of the Earth's magnetofluid outer core is unknown, but there have been suggestions that its upper part may be stably stratified. Braginsky (1984) suggested that the magnetic analog of Rossby (planetary) waves in this stable layer (the ‘H’ layer) may be responsible for a portion of the short-period secular variation. In this study, we adopt a thin shell model to examine the dynamics of the H layer. The stable stratification justifies the thin-layer approximations, which greatly simplify the analysis. The governing equations are then the Laplace's tidal equations modified by the Lorentz force terms, and the magnetic induction equation. We linearize the Lorentz force in the Laplace's tidal equations and the advection term in the magnetic induction equation, assuming a zeroth order dipole field as representative of the magnetic field near the insulating core-mantle boundary. An analytical β-plane solution shows that a magnetic field can release the equatorial trapping that non-magnetic Rossby waves exhibit. A numerical solution to the full spherical equations confirms that a sufficiently strong magnetic field can break the equatorial waveguide. Both solutions are highly dissipative, which is a consequence of our necessary neglect of the induction term in comparison with the advection and diffusion terms in the magnetic induction equation in the thin-layer limit. However, were one to relax the thin-layer approximations and allow a radial dependence of the solutions, one would find magnetic Rossby waves less damped (through the inclusion of the induction term). For the magnetic field strength appropriate for the H layer, the real parts of the eigenfrequencies do not change appreciably from their non-magnetic values. We estimate a phase velocity of the lowest modes that is rather rapid compared with the core fluid speed typically presumed from the secular variation.     

The instability of precessing flow

Abstract

An explanation is put forward for the instability observed within a precessing, rotating spheroidal container. The constant vorticity solution for the flow suggested by Poincaré is found to be inertially unstable through the parametric coupling of two inertial waves by the underlying constant strain field. Such resonant couplings are due either to the elliptical or shearing strains present which elliptically distort the circular streamlines and shear their centres respectively. For the precessing Earth's outer core, the shearing of the streamlines and the ensuing shearing instability are the dominant features. The instability of some exact, linear solutions for finite precessional rates is established and used to corroborate the asymptotic analysis. A complementary unbounded analysis of a precessing, rotating fluid is also presented and used to deduce a likely upperbound on the growth rate of a small disturbance. Connection is made with past experimental studies.     

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.     

The effect of anisotropic heat transport in the earth's core on the geodynamo

Intermediate dynamos are axisymmetric, spherical models that evade Cowling's theorem by invoking an α-effect to create the meridional magnetic field from the zonal. Usually the energy source maintaining the motions is a specified thermal wind, but here the dynamo is driven by the buoyancy created by a uniform distribution of heat sources. It has been argued by Braginsky and Meytlis (this journal, vol. 55, 1990) that, in a rapidly rotating, strongly magnetic system such as the Earth's core, heat is transported principally by a small-scale turbulence that is highly anisotropic. They conclude that the diffusion of heat parallel to the rotation axis is then significantly greater than it is in directions away from that axis. A preliminary study of the consequences of this idea is reported here. Solutions are derived numerically using both isotropic and non-isotropic thermal diffusivity tensors, and the results are compared. It is shown that even a small degree of anisotropy can materially alter the character of the dynamo.     

Thermal and magnetically driven instabilities in a non-constantly stratified fluid layer

Abstract

The linear hydromagnetic stability of a non-constantly stratified horizontal fluid layer permeated by an azimuthal non-homogeneous magnetic field is investigated for various widths of the stably stratified part of the layer in the geophysical limit q→0 (q is the ratio of thermal and magnetic diffusivities). The choice of the strength of the magnetic field B_o is as in Soward (1979) (see also Soward and Skinner, 1988) and the equations for the disturbances are treated as in Fearn and Proctor (1983). It was found that convection is developed in the whole layer regardless of the width of its stably stratified part. The thermal instability penetrates essentially from the unstably stratified part of the layer into the stably stratified part for A ～ 1 (A characterises the ratio of the Lorentz and Coriolis forces). When the magnetic field is strong (A>1) the thermal convection is suppressed in the stably stratified part of the layer. However, in this case, it is replaced by the magnetically driven instability; which is fully developed in the whole layer. The thermal instabilities always propagate westward and exist for all the modes m. The magnetically driven instabilities propagate either westward or eastward according to the width of the stably and unstably stratified parts and exist only for the mode m=1.     

菲律宾海板块西缘强震活动与地球自转角加速度变化关系研究

统计分析了1962~2016年菲律宾海板块与欧亚板块挤压带俯冲带强地震活动与地球自转角加速度变化间的关系。结果显示,研究时段菲律宾海板块西边缘带的25次强地震活动(MS≥7.5)有21次都发生在地球自转角加速度上升期,表明该地区强地震活动与地球自转角加速度之间存在相关性,最后讨论了这种相关性可能的力学机理。