Spherical Fluid Shells Research Articles

Numerical simulations of thermal convection in a rotating spherical fluid shell at high Taylor and Rayleigh numbers

In this study, we carry out numerical simulations of thermal convection in a rapidly rotating spherical fluid shell at high Taylor number Ta and Rayleigh number R with a nonlinear, three-dimensional, time-dependent, spectral-transform code. The parameters used in the simulations are chosen to be in a range which allows us to study two different types of convection, i.e., single column and multi-layered types, and the transition between them. Numerical solutions feature highly time-dependent north–south open columnar convective cells. The cells occur irregularly in longitude, are quasi-layered in cylindrical radius, and maintain alternating bands of mean zonal flow. The complex convective structure and the banded mean zonal flow are results of the high Taylor and Rayleigh numbers. The transition between the two types of convection appears to occur gradually with increasing Rayleigh and Taylor numbers. At a Taylor number of 107 the differential rotation pattern consists of an inner cylindrical region of subrotation and an outer cylindrical shell of superrotation manifest at the outer boundary as an equatorial superrotation and a high latitude subrotation. The differential rotation pattern is similar at Ta=108 and low Rayleigh number. Cylindrical shells of alternately directed mean zonal flow begin to develop at Ta=108 and R=50Rc and at Ta=109 and R=25Rc. This pattern is seen on the outer surface as a latitudinally-banded zonal flow consisting of an equatorial superrotation, a middle and high latitude subrotation, and a polar superrotation. At Ta=109 and R=50Rc the differential rotation appears at the surface as a broad eastward flow in the equatorial region with alternating bands of westward and eastward flow at high latitudes.

Onset of convection in a rapidly rotating compressible fluid spherical shell

Abstract We consider a rapidly rotating compressible fluid spherical shell, and study linear perturbations of a polytropic equilibrium state. Instead of considering the fully compressible problem, we make the anelastic approximation. We start from Boussinesq solutions and study the effect of introducing compressibility for different choices of the Prandtl number, Pr, comparing our results with those for a similar model of Glatzmaier and Gilman (1981). For Pr = 1 and 10, the results are similar. As compressibility is increased, convection becomes localised near to the inner boundary, an effect which is magnified by increasing the rotation rate. When we consider Pr = 0.1 we find different results. As compressibility is introduced, the critical Rayleigh number, R c decreases sharply and becomes negative. This behaviour was not found by Glatzmaier and Gilman.

A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle

We present the first three-dimensional (3D), time-dependent, self-consistent numerical solution of the magneto-hydrodynamic (MHD) equations that describe thermal convection and magnetic field generation in a rapidly rotating spherical fluid shell with a solid conducting inner core. This solution, which serves as a crude analog for the geodynamo, is a self-sustaining supercritical dynamo that has maintained a magnetic field for three magnetic diffusion times, roughly 40 000 years. Fluid velocity in the outer core reaches a maximum of 0.4 cm s −1, and at times the magnetic field can be as large as 560 gauss. Magnetic energy is usually about 4000 times greater than the kinetic energy of the convection that maintains it. Viscous and magnetic coupling to both the inner core below and the mantle above cause time-dependent variations in their respective rotation rates; the inner core usually rotates faster than the mantle and decadal variations in the length of the day of the mantle are similar to those observed for the Earth. The pattern and amplitude of the radial magnetic field at the core-mantle boundary (CMB) and its secular variation are also similar to the Earth's. The maximum amplitudes of the longitudinally averaged temperature gradient, shear flow, helicity, and magnetic field oscillate between the northern and southern hemispheres on a time scale of a few thousand years. However, only once in many attempts does the field succeed in reversing its polarity because the field in the inner core, which has the opposite polarity to the field in most of the outer core, usually does not have enough time to reverse before the field in the outer core changes again. One successful magnetic field reversal occurs near the end of our simulation.

Stationary and chaotic dynamos in rotating spherical shells

Convection-driven dynamos in rotating spherical fluid shells are analyzed numerically. A Galerkin method is used for the expansion of the dependent variables in spherical harmonics and in complete systems of functions of the radial coordinate satisfying the respective boundary conditions at the inner and outer spherical boundaries. Forward integration in time is done with a Crank-Nicolson scheme. It appears that time-dependent chaotic solutions tend to exist at lower magnetic Reynolds numbers than stationary dynamos. The magnetic field typically exhibits an oscillatory component with maximum strength at low latitudes. The oscillation period is related to the circulation time in the convection rolls which are nearly stationary in contrast to the motion in the non-magnetic case at the same parameter values. The axisymmetric components of the magnetic field are relatively small and exhibit aperiodic variations on a long time-scale. Although applications to the problem of the origin of geomagnetism have not been the primary goal of the analysis, some tentative conclusions about the geodynamo can be drawn.

Spherical shell rotating convection in the presence of a toroidal magnetic field

Convective instabilities of a self-gravitating, rapidly rotating fluid spherical shell are investigated in the presence of an imposed azimuthal axisymmetric magnetic field in the form of the toroidal decay mode that satisfies electrically insulating boundary conditions and has dipole symmetry. Concentration is on two major questions: how purely thermal convection of the different forms (Zhang 1992, 1994) is affected by the Lorentz force, the strength of which is measured by the Elsasser number ∧, and in what manner purely magnetic instabilities in a spherical shell (Zhang & Fearn 1993, 1994) are associated with magnetic convection. It is found that the two-dimensionality of purely thermal convection (Busse 1970) survives under the influence of a strong Lorentz force. Convective motions always attempt to satisfy the Proudman–Taylor constraint and remain predominantly two-dimensional in the whole range of ∧, 0 ≤ ∧ ≤ ∧ c , where ∧ c ═ O (10) is the critical Elsasser number for purely magnetic instabilities. Though the optimum azimuthal wave number m of convection rolls decreases drastically, from m ~ O ( T 1/6 ) at ∧ ═ 0 to m ═ O (5) at ∧ ═ O (1). We show that there exist no optimum values of ∧ that can give rise to an overall minimum of the (modified) Rayleigh number R *; the optimum value of R * is a monotonically, smoothly decreasing function of ∧, from R * ═ O ( T 1/6 ) at ∧ < O ( T -1/6 ) to R * ═ O (–10) at ∧ ═ 20. We also show that the influence of the magnetic field on thermal convection is crucially dependent on the size of the Prandtl number. At sufficiently small Prandtl number, the Poincaré convection mode (Zhang 1994) is preferred in the region 0 ≤ ∧ < ∧ c , and is only slightly affected by the presence of the toroidal magnetic field. Analytical solutions of the magnetic convection problem are then obtained based on a perturbation analysis, showing a good agreement with the numerical solution.

Hydromagnetic waves in rapidly rotating spherical shells generated by magnetic toroidal decay modes

Abstract Instabilities in the form of slow azimuthally travelling hydrodynamic waves in a rapidly rotating, stress-free, electrically conducting spherical fluid shell are investigated. The instabilities are generated by the toroidal decay mode of the lowest order or a combination of toroidal decay modes. It is found that the Elsasser number Λc at the onset of instability is always Λc = O(10) for various profiles of the basic magnetic field. It is also found that the hydromagnetic waves of the preferred instability propagate eastward [i.e. for solutions proportional to exp i(mφ + ωt), ω < 0] and are characterized by nearly two-dimensional columnar fluid motions attempting to satisfy the Proudman-Taylor theorem, indicating that the most rapidly growing magnetic disturbance arranges itself in such a way that the corresponding magnetic forces balance only the ageostrophic component of the Coriolis force. Except for the Stewartson-type velocity boundary layer at the equator of the inner core, the velocity and ...

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 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 te...

Transitions to chaotic thermal convection in a rapidly rotating spherical fluid shell

Abstract Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell heated from below and within have been carried out with a nonlinear, three-dimensional, time-dependent pseudospectral code. The investigated phenomena include the sequence of transitions to chaos and the differential mean zonal rotation. At the fixed Taylor number T a =106 and Prandtl number Pr=1 and with increasing Rayleigh number R, convection undergoes a series of bifurcations from onset of steadily propagating motions SP at R=R c = 13050, to a periodic state P, and thence to a quasi-periodic state QP and a non-periodic or chaotic state NP. Examples of SP, P, QP, and NP solutions are obtained at R = 1.3R c , R = 1.7 R c , R = 2R c , and R = 5 R c , respectively. In the SP state, convection rolls propagate at a constant longitudinal phase velocity that is slower than that obtained from the linear calculation at the onset of instability. The P state, characterized by a single frequency and its harmonics, has a two-layer cellular structure in radius. Convection rolls near the upper and lower surfaces of the spherical shell both propagate in a prograde sense with respect to the rotation of the reference frame. The outer convection rolls propagate faster than those near the inner shell. The physical mechanism responsible for the time-periodic oscillations is the differential shear of the convection cells due to the mean zonal flow. Meridional transport of zonal momentum by the convection cells in turn supports the mean zonal differential rotation. In the QP state, the longitudinal wave number m of the convection pattern oscillates among m = 3,4,5, and 6; the convection pattern near the outer shell has larger m than that near the inner shell. Radial motions are very weak in the polar regions. The convection pattern also shifts in m for the NP state at R = 5R c , whose power spectrum is characterized by broadened peaks and broadband background noise. The convection pattern near the outer shell propagates prograde, while the pattern near the inner shell propagates retrograde with respect to the basic rotation. Convection cells exist in polar regions. There is a large variation in the vigor of individual convection cells. An example of a more vigorously convecting chaotic state is obtained at R = 50R c . At this Rayleigh number some of the convection rolls have axes perpendicular to the axis of the basic rotation, indicating a partial relaxation of the rotational constraint. There are strong convective motions in the polar regions. The longitudinally averaged mean zonal flow has an equatorial superrotation and a high latitude subrotation for all cases except R = 50R c , at this highest Rayleigh number, the mean zonal flow pattern is completely reversed, opposite to the solar differential rotation pattern.

Steady, finite-amplitude, rotating magnetoconvection and the mean poloidal circulation of the Earth's core

Certain features of the Earth's magnetic field remain stationary for at least a sizeable fraction of the presumed convective time-scale of the outer core. These features may result from forced convection (laterally variable boundary buoyancy conditions) in the core as a result of core-mantle thermal coupling and the longer convective time-scale of the mantle. Although time-dependent motions in the core are certainly present, the observation of stationary features has motivated us to develop a method to find steady, finite-amplitude solutions to the non-linear convective equations. In this paper, we develop the iterative numerical method, and apply it to free convection (laterally uniform boundary buoyancy conditions), which is better understood than forced convection. Although we cannot yet be sure of the stability with respect to time perturbations of our solutions, our method has computational advantages over time-stepping that allow us to examine a large parameter space. We provide destabilization in our model by uniformly fluxing buoyancy across the bottom boundary with a concominant buoyancy sink at the top boundary. The Rayleigh number Ra is a measure of the destabilization. All calculations are at infinite Prandtl number Pr and magnetic Prandtl number q = 1. We first apply our method to a non-rotating, electrically insulating, spherical fluid shell, to demonstrate the method's viability. We then look for solutions in a rapidly rotating spherical fluid shell (high Taylor number; Ta = O(10 6)). We obtain axisymmetric polar modes, but not the non-axisymmetric equatorial modes (columns) predicted by linear theory, which require an azimuthal drift in spherical geometry. We next impose a poloidal magnetic field, and look for the magnetic analog of the polar modes. When the Lorentz force is comparable with the Coriolis force (El (Elsasser number) = O(10 0)), the modes fill the shell and most efficiently transport buoyancy. Thus, g z , the component of gravity parallel to the rotation axis, may play an important dynamical role for supercritical rotating convection, especially for an electrically conducting fluid in which the polar modes are more efficient than for an insulating fluid. In conjunction with the greater amplitude of convection near the inner spherical boundary than near the outer one, we find that the non-linear interaction between the advectively created toroidal magnetic field and the associated radial electrical current leads to a consistent bias towards equatorial upwelling flow. Until we determine the growth rates of instabilities, we cannot be sure of the stability of any particular converged solution, but we nevertheless believe that this non-linear result is a general feature of the ω-effect, with possible relevance to the poloidal circulation in the Earth's core.

Convection in a rotating spherical fluid shell with an inhomogeneous temperature boundary condition at infinite Prandtl number

We examine thermal convection in a rotating spherical shell with a spatially non-uniformly heated outer surface, concentrating on three distinct heating modes: first, with wavelength and symmetry corresponding to the most unstable mode of the uniformly heated problem; secondly, with the critical wavelength but opposite equatorial symmetry; and thirdly, with wavelength much larger than that of the most unstable mode. Analysis is focused on boundary-locked convection, the associated spatial resonance phenomena, the stability properties of the resonance solution, and time-dependent secondary convection. A number of new forms of instability and convection are found: the most interesting is perhaps the saddle-node bifurcation, which is the first to be found for realistic fluid systems governed by partial differential equations. An analogous Landau amplitude equation is also analysed, providing an important mathematical framework for understanding the complicated numerical solutions.

On equatorially trapped boundary inertial waves

Solutions of the Poincaré equation describing equatorially trapped three-dimensional boundary travelling waves in rotating spherical systems are discussed. It is shown that the combined effects of Coriolis forces and spherical curvature enable the equatorial region to form an equatorial waveguide tube with characteristic latitudinal radius (2/m)1/2 and radial radius (1/m), where m is azimuthal wavenumber. Inertial waves with sufficiently simple structure along the axis of rotation and sufficiently small azimuthal wavelength must be trapped in the equatorial waveguide tube. The structure and frequency of the inertial waves are thus hardly affected by the presence of an inner sphere or by the condition of higher latitudes. Further calculations on rotating spherical fluid shells of finite internal viscosity and stressfree boundaries are also discussed.

The propagation and stability of linear wave motions in rapidly rotating spherical shells: Weak magnetic fields

+The propagation properties and stability of wave motions in a spherical fluid shell, rotating uniformly in a co-rotating zonal magnetic field, are studied. Various profils of magnetic field and temperature gradient are examined. The fluid is viscous and electrically and thermally conducting. The analysis is applicable when the rotation rate is very high and the Elsasser number A 1. (Λ measures the ratio of Lorentz to Coriolis forces). The wave motions occur in the form of annular cylindrical cells whose thickness is determined by the rotation rate and magnetic field amplitude (Eltayeb and Kumar, 1977). It is shown that an infinity of modes exists and both westward and eastward phase propagation is possible.

Convection in a rapidly rotating spherical shell at infinite Prandtl number: transition to vacillating flows

Stability of steadily drifting three-dimensional finite-amplitude columnar rolls at infinite Prandtl number, in a rapidly rotating spherical fluid shell of radius ratio η = r i r 0 = 0.4 , is examined. Two types of instability, modulational and amplitude-vacillating, are found. Both instabilities result in the simultaneous breaking of temporal and spatial symmetry of the drifting columnar rolls. Amplitude-vacillating instability leads to a periodic change in amplitude without substantially changing the pattern or scale of convection. Modulational instability gives rise to the transition from the regularly placed columnar rolls to ones which are strongly spatially modulated on a large scale. A dramatic enhancement in heat transfer is produced in both cases through the combined effects of time-dependence and spatial modulations.

The effect of a stable layer at the core-mantle boundary on thermal convection

Abstract As a model for convection in the Earth's core we study the linear stability of a rapidly rotating electrically conducting spherical fluid shell, permeated by a toroidal magnetic field B. We look at the effect of introducing a stably stratified layer into the fluid adjacent to the core-mantle boundary (CMB). For values of the Elsasser number (a non-dimensional measure of the magnetic field strength), Λ ≪ 0(1), convection can penetrate significantly into the stable layer. In this weak field case, the constraints of the Taylor-Proudman theorem cause convection to become columnar in structure. As Λ is decreased the azimuthal wave number m, corresponding to the most unstable mode, increases. For Λ = O(1), convection is still unaffected by the introduction of a stable layer, but is no longer columnar. For Λ ≫ O(1), we find that convection becomes concentrated in the unstably stratified region. Except in the low magnetic field strength regime, our findings agree with previous work of Boda (1988) and Sev...

Three-dimensional convection of an infinite-Prandtl-number compressible fluid in a basally heated spherical shell

A numerical investigation is made of the effects of compressibility on threedimensional thermal convection in a basally heated, highly viscous fluid spherical shell with an inner to outer radius ratio of approximately 0.55, characteristic of the Earth’s whole mantle. Compressibility is implemented with the anelastic approximation and a hydrostatic adiabatic reference state whose bulk modulus is a linear function of pressure. The compressibilities studied range from Boussinesq cases to compressibilities typical of the Earth’s whole mantle. Compressibility has little effect on the spatial structure of steady convection when the superadiabatic temperature drop across the shell AT,, is comparable to a characteristic adiabatic temperature. When AT,& is approximately an order of magnitude smaller than the adiabatic temperature, compressibility is significant. For all the non-Boussinesq cases, the regular polyhedral convective patterns that exist at large AZ, break down at small AT, into highly irregular patterns ; as AZ, decreases convection becomes penetrative in the upper portion of the shell and is strongly time dependent at Rayleigh numbers only ten times the critical Rayleigh number, (Ra),,. Viscous heating in the compressible solutions is concentrated around the upwelling plumes and is greatest near the top and bottom of the shell. Solutions with regular patterns (and large AEJ remain steady up to fairly high Rayleigh numbers (100(Ra),,), while solutions with irregular convective patterns are time dependent at similar Rayleigh numbers. Compressibility affects the pattern evolution of the irregular solutions, producing fewer upwelling plumes with increasing compressibility.

Spiralling columnar convection in rapidly rotating spherical fluid shells

It is shown that the fundamental features of both thermal instabilities and the corresponding nonlinear convection in rapidly rotating spherical systems (in the range of the Taylor number 109 &lt; T &lt; 1012) are determined by the fluid properties characterized by the size of the Prandtl number. Coefficients of the asymptotic power law for the onset of convection at large Taylor number are estimated in the range of the Prandtl number 0.1 ≤ Pr ≤ 100. For fluids of moderately small Prandtl number, a new type of convective instability in the form of prograde spiralling drifting columnar rolls is discovered. The linear columnar rolls extend spirally from near latitude 60° to the equatorial region, and each spans azimuthally approximately five wavelengths with the inclination angle between a spirally elongated roll and the radial direction exceeding 45°. As a consequence, the radial lengthscale of the linear roll becomes comparable with the azimuthal lengthscale. A particularly significant finding is the connection between the new instability and the predominantly axisymmetric convection. Though non-axisymmetric motions are preferred at the onset of convection, the nonlinear convection (at the Rayleigh number of the order of (R—Rc)/Rc = O(0.1)) bifurcating supercritically from the spiralling mode is primarily dominated by the component of the axisymmetric zonal flow, which contains nearly 90% of the total kinetic energy. For fluids of moderately large Prandtl numbers, thermal instabilities at the onset of convection are concentrated in a cylindrical annulus coaxial with the axis of rotation; the position of the convection cylinder is strongly dependent on the size of the Prandtl number. The associated nonlinear convection consists of predominantly non-axisymmetric columnar rolls together with a superimposed weak mean flow that contains less than 10% of the total kinetic energy at (R—Rc)/Rc = O(0.1). A double-layer structure of the temperature field (with respect to the basic state) forms as a result of strong nonlinear interactions between the nonlinear flow and the temperature field. It is also demonstrated that the aspect ratio of the spherical shell does not substantially influence the fundamental properties of convection.

Convection in a rapidly rotating spherical shell at infinite Prandtl number: steadily drifting rolls

Three-dimensional linear and non-linear convection at infinite Prandtl number in a rapidly rotating spherical fluid shell of radius ratio η = r i r o = 0.4 is investigated numerically. An asymptotic criterion for the onset of convective instability (at large Taylor number) is derived, and the asymptotic relationship describing the typical finite amplitude of steadily drifting rolls, U, the Rayleigh number, R, and the Taylor number, T, U = 20.1(R/T 2 3 − 1.63) 1 2 = 25.6[ ( R − R c ) R c ) ] 1 2 is obtained from non-linear solutions of the full equations. A particularly interesting feature of the finite amplitude convection is the formation of the double-layer structure of the temperature field (with respect to the basic state); the spherically symmetric basic distribution of the buoyancy force is therefore strongly modified. As a consequence, the linear convection columnar rolls positioned at mid-latitudes are moved to much lower latitudes at finite amplitudes. Instead of transporting heat into the narrow belt of the top and bottom of the convection cylinder at small amplitudes, the non-linear convection transfers heat much more effectively in the large equatorial area, reflecting the way in which the non-linear effects overcome the constraints of rotation.

Vacillatory convection in a rotating spherical fluid shell at infinite Prandtl number

The temporal and spatial behaviour of three-dimensional convection for a rotating spherical fluid shell in the limit of infinite Prandtl number is investigated. The main features of the instabilities and the bifurcation solutions at two representative Taylor numbers are determined

Longitudinal axial resonances in the form function for backscattering from a fluid spherical shell

Reverberations of longitudinal waves between the inner and outer surfaces of a hollow shell can strongly affect the backscattering amplitudes at frequencies associated with a thickness resonance. The phenomena are studied for the idealized case of vanishing shear stresses by letting the material of the shell be an inviscid fluid; the surrounding fluid is water. An exact partial-wave series (PWS) gives the form function f (θ,ka) and plots of |f(θ = π,ka)| display a structure from resonances. Here, ka = 2πa/λ is the size parameter, where a is the outer radius of the shell and λ is the wavelength in water of the incident acoustic wave. The resonant structure is also recovered in a geometrical calculation of f (π,ka), which sums the amplitudes associated with rays multiply reflected within the curved shell. The geometric synthesis demonstrates that the effects of curvature are essential to modeling f (π,ka). The analysis gives the geometric divergence factors of successive internal reflections. In addition to numerical comparisons with the PWS, the geometrical synthesis is tested by considering several limiting cases. These limiting cases correctly give results anticipated from elementary considerations. [Worked supported by ONR.]

Marangoni convection in a rotating spherical geometry

New results concerning the problem of surface-tension-driven instability in a rotating spherical fluid shell are presented. The fluid layer is assumed to be Boussinesq-like, heated from the inner sphere and placed in a microgravity environment (Marangoni problem). Small-amplitude disturbances are considered and the normal mode problem is solved numerically by means of a finite difference technique. Marginal stability curves are computed and surstability is exhibited. Onset of convection is subordinated to several parameters like the thickness of the layer, its angular velocity, and its viscous and diffusive properties; all these effects receive a special treatment.