On the onset of convection in rotating spherical shells

Abstract

The linear problem of the onset of convection in rotating spherical shells is analysed numerically in dependence on the Prandtl number. The radius ratio η=r _i/r _o of the inner and outer radii is generally assumed to be 0.4. But other values of η are also considered. The goal of the analysis has been the clarification of the transition between modes drifting in the retrograde azimuthal direction in the low Taylor number regime and modes traveling in the prograde direction at high Taylor numbers. It is shown that for a given value m of the azimuthal wavenumber a single mode describes the onset of convection of fluids of moderate or high Prandtl number. At low Prandtl numbers, however, three different modes for a given m may describe the onset of convection in dependence on the Taylor number. The characteristic properties of the modes are described and the singularities leading to the separation with decreasing Prandtl number are elucidated. Related results for the problem of finite amplitude convection are also reported.     

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.     

Finite amplitude convection and magnetic field generation in a rotating spherical shell

Abstract

Finite amplitude solutions for convection in a rotating spherical fluid shell with a radius ratio of η=0.4 are obtained numerically by the Galerkin method. The case of the azimuthal wavenumber m=2 is emphasized, but solutions with m=4 are also considered. The pronounced distinction between different modes at low Prandtl numbers found in a preceding linear analysis (Zhang and Busse, 1987) is also found with respect to nonlinear properties. Only the positive-ω-mode exhibits subcritical finite amplitude convection. The stability of the stationary drifting solutions with respect to hydrodynamic disturbances is analyzed and regions of stability are presented. A major part of the paper is concerned with the growth of magnetic disturbances. The critical magnetic Prandtl number for the onset of dynamo action has been determined as function of the Rayleigh and Taylor numbers for the Prandtl numbers P=0.1 and P=1.0. Stationary and oscillatory dynamos with both, dipolar and quadrupolar, symmetries are close competitors in the parameter space of the problem.     

Nonlinear dynamics of a stellar dynamo in a spherical shell

Abstract

A simple mean-field model of a nonlinear stellar dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the only nonlinearity retained is the influence of the Lorentz forces on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.'s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.     

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.     

Nonlinear free thermal convection in a spherical shell:A variable viscosity model

Introduction The velocity field of surface plate motion can be split into a poloidal and a toroidal parts.At the Earth′s surface,the toroidal component is manifested by the existence of transform faults,and the poloidal component by the presence of convergence and divergence,i.e.spreading and subduc-tion zones.They have coupled each other and completely depicted the characteristics of plate tec-tonic motions.The mechanism of poloidal field has been studied fairly clearly which is related to …     

Travelling wave convection in a rotating layer

Abstract

Small amplitude two-dimensional Boussinesq convection in a plane layer with stress-free boundaries rotating uniformly about the vertical is studied. A horizontally unbounded layer is modelled by periodic boundary conditions. When the centrifugal force is balanced by an appropriate pressure gradient the resulting equations are translation invariant, and overstable convection can take the form of travelling waves. In the Prandtl number regime 0.53 < [sgrave] < 0.68 such solutions are preferred over the more usual standing waves. For [sgrave] < 0.53, travelling waves are stable provided the Taylor number is sufficiently large.     

Steady-state convection in a rotating long cavity

Abstract

Convection experiments in a rotating long cavity were conducted to investigate the changes introduced by Coriolis-force to the well understood convection in the (nonrotating) long cavity. In the far convective regime, the results indicated that large-scale features of the convection were dominated by geostrophically balanced confined primary currents. Close to the transition regime, where the intruding currents gain a thickness close to the vertical dimension of the tank, secondary features became predominant. In the transition regime, these secondary features were not present and the simple picture of two cavity filling counter currents with laterally tilted isotherms could be confirmed.     

Bénard convection in a rotating fluid

Abstract

The steady nonlinear regime of Bénard convection in a uniformly rotating fluid is treated using a two-dimensional primitive-equation numerical model with rigid boundaries. Quantitative comparisons with laboratory heat transport data for water are made in the parameter ranges for which the experimental flows are approximately two-dimensional and steady. When an experimentally realistic spatial periodicity is imposed upon the numerical solution, the model simulates the experimental determinations of Nusselt number fairly accurately. In particular, it predicts the observed non-monotonic dependence on Taylor number. When spatial periodicities corresponding to those of the linear stability problem are specified, however, the accuracy of the simulation is less and the Taylor number dependence is monotonic.     

Implementation of a high-order combined compact difference scheme in problems of thermally driven convection and dynamo in rotating spherical shells

We present an improved solution method for modeling thermally driven convection and dynamo in a rotating spherical shell. In this method, we introduce a high-order three-point combined compact difference scheme (CCDS) on non-uniform grid points in radius, while spherical harmonic expansion is conventionally performed in the angular direction. The governing equations in the spectral form are time-stepped together with the implicit CCDS up to the second derivative. To improve stability of the scheme, a boundary closure scheme is developed on non-uniform mesh. Numerical comparison with a published benchmark solution at moderate Ekman and Rayleigh numbers demonstrates that accuracy and convergence of the CCDS is fairly good and superior to the existing finite difference scheme using more stencil. With this scheme, we could more accurately solve problems of convection and also dynamo action in planetary core with less grid points.     

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.     

Two-and three-dimensional linear convection in a rotating annulus

Abstract

An exceptional case to the model-independent theory of Knobloch (1995) is presented, by investigating a rotating cylindrical annulus of height H and side wall radii r _o and r _i, with non-slip, perfectly thermally conducting side walls and thermally insulating stress-free ends. Radial heating permits the possibility of either two- or three-dimensional convective solutions being the preferred mode. An analytical solution is obtained for the two-dimensional case and a numerical solution for the three-dimensional solution, which is also applied to the two-dimensional solution. It is shown that both two- and three-dimensional solutions can be realized depending on the aspect ratio, γ = H/d, where d = r _o-r _i is the thickness of the annulus, the radii ratio λ = r _i/r _o and the rotation rate of the model. For γ = O(1) and λ = 0.4, the preferred convective solution is three-dimensional when the Taylor number, T < 10² and two-dimensional for T > 10². For small aspect ratios, γ ? 1, the preferred mode is two-dimensional for all rotation rates.     

Taylor cylinder and convection in a spherical shell

The successive development of thermal convection in a rotating spherical shell heated from below has been studied. It has been indicated that the convection properties strongly differ within and outside a Taylor cylinder. The distributions of the kinetic energy, helicity, temperature, and differential rotation in these regions are presented. The consequences for the geodynamo theory and interpretation of geomagnetic observations are considered.     

Three-dimensional convection in spherical shells

Abstract

In this study, the equations of the three-dimensional convective motion of an infinite Prandtl number fluid are solved in spherical geometry, for Rayleigh numbers up to 15 times the critical number. An iterative method is used to find stationary solutions. The spherical parts of the operators are treated using a Galerkin collocation method while the radial and time dependences are expressed using finite difference methods. A systematic search for stationary solutions has led to eight different stream patterns for a low Rayleigh number (1.28 times the critical number). They can be classified as:

I) Axisymmetrical solutions, analogous to rolls in plane geometry.

II) Solutions which have several ascending plumes within a large area of ascending current, and also several descending plumes within an area of descending current. This type of flow is analogous to bimodal circulation in plane geometry.

III) Solutions characterized by isolated ascending (or descending) plumes separated from each other by a closed polyhedral network of descending (or ascending) currents. This type of circulation is called ‘polygonal’ in analogy with hexagonal circulation in plane geometry.

The behaviour of each of the eight solutions has been studied by increasing the Rayleigh number up to 15 times the critical number. A trend towards transitions from type (I) and type (II) solutions to type (III) solutions is observed. It is inferred that only the “polygonal” solutions are stable for a Rayleigh number greater than 15 times the critical number.     

Hydromagnetic waves in a thin rotating spherical shell

We consider an electrically conducting fluid confined to a thin rotating spherical shell 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 a possible stable layer at the top of the Earth's outer core. It may also be a model for the thin shells which are thought to be a source of the magnetic fields of some planets such as Mercury or Uranus. Linear hydromagnetic waves are studied using a multiple scale asymptotic scheme in which boundary layers and the associated boundary conditions determine the structure of the waves. These waves are assumed to be of the form of an asymptotic series expanded about an ambient magnetic field which vanishes on the equatorial plane and velocity and pressure fields which do not. They take the form of short wave, slowly varying wave trains. The results are compared to the author's previous work on such waves in cylindrical geometry in which the boundary conditions play no role. The approximation obtained is significantly different from that obtained in the previous work in that an essential singularity appears at the equator and nonequatorial wave regions appear.     

Wave propagation in a rotating fluid of spherical configuration

Abstract

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

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.