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1.
It has been suggested that there exists a stably stratified electrically conducting layer at the top of the Earth's outer fluid core and that lateral temperature gradients in the lower mantle is capable of a driving thermal-wind-type flow near the core–mantle boundary. We investigate how such a flow in a stable layer could influence the geomagnetic field and the geodynamo using a very simple two-dimensional kinematic dynamo model in Cartesian geometry. The dynamo has four layers representing the inner core, convecting lower outer core, stable upper core, and insulating mantle. An α2 dynamo operates in the convecting outer core and a horizontal shear flow is imposed in the stable layer. Exact dynamo solutions are obtained for a range of parameters, including different conductivities for the stable layer and inner core. This allows us to connect our solutions with known, simpler solutions of a single-layer α2 dynamo, and thereby assess the effects of the extra layers. We confirm earlier results that a stable, static layer can enhance dynamo action. We find that shear flows produce dynamo wave solutions with a different spatial structure from the steady α2 dynamos solutions. The stable layer controls the behavior of the dynamo system through the interface conditions, providing a new means whereby lateral variations on the boundary can influence the geomagnetic field.  相似文献   

2.
A key non-linear mechanism in a strong-field geodynamo is that a finite amplitude magnetic field drives a flow through the Lorentz force in the momentum equation and this flow feeds back on the field-generation process in the magnetic induction equation, equilibrating the field. We make use of a simpler non-linear?α?2-dynamo to investigate this mechanism in a rapidly rotating fluid spherical shell. Neglecting inertia, we use a pseudo-spectral time-stepping procedure to solve the induction equation and the momentum equation with no-slip velocity boundary conditions for a finitely conducting inner core and an insulating mantle. We present calculations for Ekman numbers (E) in the range 2.5× 10?3 to 5.0× 10?5, for?α?=α 0cos?θ?sin?π?(r?ri ) (which vanishes on both inner and outer boundaries). Solutions are steady except at lower E and higher values of?α?0. Then they are periodic with a reversing field and a characteristic rapid increase then equally rapid decrease in magnetic energy. We have investigated the mechanism for this and shown the influence of Taylor's constraint. We comment on the application of our findings to numerical hydrodynamic dynamos.  相似文献   

3.
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 5Rc (Rc 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.  相似文献   

4.
This article commences by surveying the basic dynamics of Earth's core and their impact on various mechanisms of core-mantle coupling. The physics governing core convection and magnetic field production in the Earth is briefly reviewed. Convection is taken to be a small perturbation from a hydrostatic, “adiabatic reference state” of uniform composition and specific entropy, in which thermodynamic variables depend only on the gravitational potential. The four principal processes coupling the rotation of the mantle to the rotations of the inner and outer cores are analyzed: viscosity, topography, gravity and magnetic field. The gravitational potential of density anomalies in the mantle and inner core creates density differences in the fluid core that greatly exceed those associated with convection. The implications of the resulting “adiabatic torques” on topographic and gravitational coupling are considered. A new approach to the gravitational interaction between the inner core and the mantle, and the associated gravitational oscillations, is presented. Magnetic coupling through torsional waves is studied. A fresh analysis of torsional waves identifies new terms previously overlooked. The magnetic boundary layer on the core-mantle boundary is studied and shown to attenuate the waves significantly. It also hosts relatively high speed flows that influence the angular momentum budget. The magnetic coupling of the solid core to fluid in the tangent cylinder is investigated. Four technical appendices derive, and present solutions of, the torsional wave equation, analyze the associated magnetic boundary layers at the top and bottom of the fluid core, and consider gravitational and magnetic coupling from a more general standpoint. A fifth presents a simple model of the adiabatic reference state.  相似文献   

5.
We consider the effect of compressibility on mixed Ekman–Hartmann boundary layers on an infinite plane (z = 0), in the presence of an external magnetic field oblique to the boundary. The aim is to investigate the influence of the magnetic pressure on the fluid density, and hence, via mass conservation, on the mass flow into or out of the boundary layer. We find that if the z-component of vorticity in the main flow, immediately above the boundary layer, is negative, then there is a competition between Ekman suction and the magnetic pressure effect. Indeed, as the magnetic field strength is increased, the magnetic pumping may overcome the Ekman suction produced by anti-cyclonic main flow vortices. Such a mechanism, based on the competition between these effects, may be of importance for understanding the dynamics of the magnetic field in stellar (or planetary) interiors. For the solar tachocline, we find that the analysed magnetic pressure effect is unlikely to play a significant role; however, we give examples of what changes in the assumed scalings would be necessary for the effect to become important.  相似文献   

6.
Convection in the Earth's core is driven much harder at the bottom than the top. This is partly because the adiabatic gradient steepens towards the top, partly because the spherical geometry means the area involved increases towards the top, and partly because compositional convection is driven by light material released at the lower boundary and remixed uniformly throughout the outer core, providing a volumetric sink of buoyancy. We have therefore investigated dynamo action of thermal convection in a Boussinesq fluid contained within a rotating spherical shell driven by a combination of bottom and internal heating or cooling. We first apply a homogeneous temperature on the outer boundary in order to explore the effects of heat sinks on dynamo action; we then impose an inhomogeneous temperature proportional to a single spherical harmonic Y 2² in order to explore core-mantle interactions. With homogeneous boundary conditions and moderate Rayleigh numbers, a heat sink reduces the generated magnetic field appreciably; the magnetic Reynolds number remains high because the dominant toroidal component of flow is not reduced significantly. The dipolar structure of the field becomes more pronounced as found by other authors. Increasing the Rayleigh number yields a regime in which convection inside the tangent cylinder is strongly affected by the magnetic field. With inhomogeneous boundary conditions, a heat sink promotes boundary effects and locking of the magnetic field to boundary anomalies. We show that boundary locking is inhibited by advection of heat in the outer regions. With uniform heating, the boundary effects are only significant at low Rayleigh numbers, when dynamo action is only possible for artificially low magnetic diffusivity. With heat sinks, the boundary effects remain significant at higher Rayleigh numbers provided the convection remains weak or the fluid is stably stratified at the top. Dynamo action is driven by vigorous convection at depth while boundary thermal anomalies dominate in the upper regions. This is a likely regime for the Earth's core.  相似文献   

7.
A recent dynamo model for Mercury assumes that the upper part of the planet's fluid core is thermally stably stratified because the temperature gradient at the core–mantle boundary is subadiabatic. Vigorous convection driven by a superadiabatic temperature gradient at the boundary of a growing solid inner core and by the associated release of light constituents takes place in a deep sub-layer and powers a dynamo. These models have been successful at explaining the observed weak global magnetic field at Mercury's surface. They have been based on the concept of codensity, which combines thermal and compositional sources of buoyancy into a single variable by assuming the same diffusivity for both components. Actual diffusivities in planetary cores differ by a large factor. To overcome the limitation of the codensity model, we solve two separate transport equations with different diffusivities in a double diffusive dynamo model for Mercury. When temperature and composition contribute comparable amounts to the buoyancy force, we find significant differences to the codensity model. In the double diffusive case convection penetrates the upper layer with a net stable density stratification in the form of finger convection. Compared to the codensity model, this enhances the poloidal magnetic field in the nominally stable layer and outside the core, where it becomes too strong compared to observation. Intense azimuthal flow in the stable layer generates a strong axisymmetric toroidal field. We find in double diffusive models a surface magnetic field of the observed strength when compositional buoyancy plays an inferior role for driving the dynamo, which is the case when the sulphur concentration in Mercury's core is only a fraction of a percent.  相似文献   

8.
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 si 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.  相似文献   

9.
Summary An exact solution of electrically conducting viscous incompressible flow in an annulus with porous walls under an external radial magnetic field is obtained when the motion is due to both longitudinal motion of the inner boundary and a constant axial pressure gradient, and the fluid injection rate at one wall is equal to the fluid withdrawal rate at the other. The fluid may be injected at the outer wall and sucked at the inner or vice versa. The solution for the hydromagnetic flow between two flat plates has also been obtained as a limiting case of the annulus problem.  相似文献   

10.
The mathematical formulation of an iterative procedure for the numerical implementation of an ionosphere-magnetosphere (IM) anisotropic Ohm’s law boundary condition is presented. The procedure may be used in global magnetohydrodynamic (MHD) simulations of the magnetosphere. The basic form of the boundary condition is well known, but a well-defined, simple, explicit method for implementing it in an MHD code has not been presented previously. The boundary condition relates the ionospheric electric field to the magnetic field-aligned current density driven through the ionosphere by the magnetospheric convection electric field, which is orthogonal to the magnetic field B, and maps down into the ionosphere along equipotential magnetic field lines. The source of this electric field is the flow of the solar wind orthogonal to B. The electric field and current density in the ionosphere are connected through an anisotropic conductivity tensor which involves the Hall, Pedersen, and parallel conductivities. Only the height-integrated Hall and Pedersen conductivities (conductances) appear in the final form of the boundary condition, and are assumed to be known functions of position on the spherical surface R=R1 representing the boundary between the ionosphere and magnetosphere. The implementation presented consists of an iterative mapping of the electrostatic potential , the gradient of which gives the electric field, and the field-aligned current density between the IM boundary at R=R1 and the inner boundary of an MHD code which is taken to be at R2>R1. Given the field-aligned current density on R=R2, as computed by the MHD simulation, it is mapped down to R=R1 where it is used to compute by solving the equation that is the IM Ohm’s law boundary condition. Then is mapped out to R=R2, where it is used to update the electric field and the component of velocity perpendicular to B. The updated electric field and perpendicular velocity serve as new boundary conditions for the MHD simulation which is then used to compute a new field-aligned current density. This process is iterated at each time step. The required Hall and Pedersen conductances may be determined by any method of choice, and may be specified anew at each time step. In this sense the coupling between the ionosphere and magnetosphere may be taken into account in a self-consistent manner.  相似文献   

11.
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 Tm ? 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.  相似文献   

12.
Lower mantle heterogeneity could cause deviations from axial symmetry in geodynamo properties. Global tomography models are commonly used to infer the pattern of core–mantle boundary heat flux via a linear relation that corresponds to a purely thermal interpretation of lower mantle seismic anomalies, ignoring both non-thermal origins and non-resolved small scales. Here we study the possible impact on the geodynamo of narrow thermal anomalies in the base of the mantle, originating from either compositional heterogeneity or sharp margins of large-scale features. A heat flux boundary condition composed of a large-scale pattern and narrow ridges separating the large-scale positive and negative features is imposed on numerical dynamos. We find that hot ridges located to the west of a positive large-scale core–mantle boundary heat flux anomaly produce a time-average narrow elongated upwelling, a flow barrier at the top of the core and intensified low-latitudes magnetic flux patches. When the ridge is located to the east of a positive core–mantle boundary heat flux anomaly, the associated upwelling is weaker and the homogeneous dynamo westward drift leaks, precluding persistent intense low-latitudes magnetic flux patches. These signatures of the core–mantle boundary heat flux ridge are evident in the north–south component of the thermal wind balance. Based on the pattern of lower mantle seismic tomography (Masters et al., 2000), we hypothesize that hot narrow thermal ridges below central Asia and the Indian Ocean and below the American Pacific coast produce time-average fluid upwelling and a barrier for azimuthal flow at the top of the core. East of these ridges, below east Asia and Oceania and below the Americas, time-average intense geomagnetic flux patches are expected.  相似文献   

13.
A 3D kinematic geodynamo model in a sphere with the conductive solid inner core is considered. The 3D magnetic field and velocity field are resolved in the physical space for r- and -coordinates, whereas the sin- and cos-decomposition is applied to the -coordinate. The additional boundary conditions for the case of non-zero velocity field on the boundaries of the liquid spherical shell and for different magnetic diffusivities of the inner and outer core are applied. The computer code was tested by free decay mode solutions and comparisons were made also with results reported by other authors. This work is a part of a project to study 3D inviscid geodynamo models.  相似文献   

14.
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.  相似文献   

15.
The inner core has long been recognized as a part of the process by which fluid core convection is maintained and as an influence upon the magnetic field. Evidence from several seismological studies in recent years has mounted to indicate that the inner core has anisotropic velocities with large-scale (degree one) variation in strength of the differences from isotropy, and fine-scale (a few kilometres) inhomogeneities of structure. Indications have also been found of systematic changes in the travel time of seismic waves passing through the inner core – which have been interpreted as evidence that the inner core is rotating in an easterly direction, relative to the mantle, at a rate fast enough to be perceived on human timescales. The rate has been dismissed as too slow or physically impossible, but additional evidence has accumulated in support of what may be an emerging consensus that the inner core has a super-rotation of a few tenths of a degree per year.  相似文献   

16.
Oversampling techniques are often used in porous media simulations to achieve high accuracy in multiscale simulations. These methods reduce the effect of artificial boundary conditions that are imposed in computing local quantities, such as upscaled permeabilities or basis functions. In the problems without scale separation and strong non-local effects, the oversampling region is taken to be the entire domain. The basis functions are computed using single-phase flow solutions which are further used in dynamic two-phase simulations. The standard oversampling approaches employ generic global boundary conditions which are not associated with actual flow boundary conditions. In this paper, we propose a flow based oversampling method where the actual two-phase flow boundary conditions are used in constructing oversampling auxiliary functions. Our numerical results show that the flow based oversampling approach is several times more accurate than the standard oversampling method. We provide partial theoretical explanation for these numerical observations.  相似文献   

17.
A new nonlinear boundary layer in rotating hydromagnetic flows is presented. The purpose of this layer is to provide a smooth transition between the magnetic field at a rigid, electrically insulating boundary and that far from the boundary. The boundary layer problem is solved in a cylindrical geometry, assuming that the variables obey the hydromagnetic analog of the von Karman similarity. The primary momentum balance within this boundary layer is between pressure, Coriolis and hydromagnetic forces; the viscous and nonlinear inertia forces are unimportant. Diffusion of the magnetic field is balanced by advection where the advecting fluid flow is induced by the action of the hydromagnetic body forces within the boundary layer. The structure of the layer depends upon a single parameter which measures the strength of the normal magnetic field. Steady solutions are presented and their uniqueness and temporal stability are analyzed. The relevance of this layer to the core of the Earth is discussed. It is estimated that this boundary layer would be at least as thick as one tenth of the core radius if it exists within the core.  相似文献   

18.
We address issues associated with non-local magnetic boundary conditions for non-spectral dynamo simulations. We introduce an integro-differential formulation for a domain bounded by an insulating outer domain. We show how to combine the flexibility of a local discretisation with a rigorous formulation of magnetic boundary conditions in arbitrary geometries. This formulation substantiates from mathematical point of view a new method for numerical solution of magnetohydrodynamic problems with non-local boundary conditions based on coupling finite volumes and boundary elements. Finally, we discuss practical efficiency of this new method.  相似文献   

19.
We study magnetic field variations in numerical models of the geodynamo, with convection driven by nonuniform heat flow imposed at the outer boundary. We concentrate on cases with a boundary heat flow pattern derived from seismic anomalies in the lower mantle. At a Rayleigh number of about 100 times critical with respect to the onset of convection, the magnetic field is dominated by the axial dipole component and has a similar spectral distribution as Earth’s historical magnetic field on the core-mantle boundary (CMB). The time scales of variation of the low-order Gauss coefficients in the model agree within a factor of two with observed values. We have determined the averaging time interval needed to delineate deviations from the axial dipole field caused by the boundary heterogeneity. An average over 2000 years (the archeomagnetic time scale) is barely sufficient to reveal the long-term nondipole field. The model shows reduced scatter in virtual geomagnetic pole positions (VGPs) in the central Pacific, consistent with the weak secular variation observed in the historical field. Longitudinal drift of magnetic field structures is episodic and differs between regions. Westward magnetic drift is most pronounced beneath the Atlantic in our model. Although frozen flux advection by the large-scale flow is generally insufficient to explain the magnetic drift rates, there are some exceptions. In particular, equatorial flux spot pairs produced by expulsion of toroidal magnetic field are rapidly advected westward in localized equatorial jets which we interpret as thermal winds.  相似文献   

20.
We establish a general theory that describes the rotational motion of a layered, oblate, elastic Earth under the influence of tidal forces when account is taken of the liquid outer core. We obtain a linearized version of the Navier-Stokes equation; within it not only have we retained the Coriolis and centrifugal acceleration terms, but also have included the nutational terms. We also make use of the Euler equation for angular momentum to analytically relate the nutational motion of the rotational axis with the oscillations of the liquid core and obtain a constraint for the nutational amplitude. Consideration of the Poisson equation for density variation completes our analytical model.We primarily discuss the equations of motion for the liquid core and present the solution as the sum of two terms: one being a component of the spheroidal displacement field, the other of the toroidal field. We also formulate the equations valid for the solid mantle when rotational effects are included, and establish the boundary conditions that must hold at the various interfaces in order that a complete integration of the differential system of equations be accomplished.We assume that the outer core consists of an inviscid fluid and ignore the existence of any boundary layer. We do not impose, however, any restriction on the stratification of the fluid. The dynamical coupling between liquid core and solid mantle is represented by a torque which is generated by the forced oscillations within the liquid core; these oscillations are in turn triggered by the diurnal tides.The expected influence of the liquid core/solid mantle boundary on the nutational motion is discussed in view of Poincare's results concerning a liquid core surrounded by a rigid shell. Comparison is finally made of our model with Molodenskii's 1961 theory for a neutral core and the 1976 Shen-Mansinha nutational theory for an unrestricted core.  相似文献   

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