The boundary layer near a cooled inclined plate, which is immersed in a stably stratified fluid rotating about an axis parallel to the direction of gravity, is a model for katabatic flows at high latitudes. In this paper the base flow of such an inclined buoyancy layer is solved analytically for arbitrary Prandtl numbers. By applying linear stability analyses, five unstable modes are identified for both the fixed temperature and the isoflux boundary conditions, i.e., the stationary longitudinal roll (LR) mode, the oblique roll with low streamwise wave-number (OR-1) and high streamwise wave-number (OR-2) modes, and the Tolmien-Schlichting (TS) wave with low streamwise wave-number (TS-1) and high streamwise wave-number (TS-2) modes. It is indicated that the Coriolis effect induced by the rotation leads the critical modes to be three dimensional, and a larger tilt angle of the plate and stronger Coriolis effect cause both TS wave modes to be more unstable for both thermal boundary conditions. When the Coriolis effect is considered, the OR-1 and OR-2 modes are the most unstable mode at low and high tilt angles, respectively, but the TS-1 wave mode may be the most unstable one when the plate is nearly vertical. In addition, the spanwise phase velocities of the TS wave modes change directions as the tilt angle passes some threshold values for both thermal boundary conditions except for the TS-1 wave mode with a fixed temperature boundary condition, which propagates in the same spanwise direction for all explored tilt angles.

The analysis of the Taylor-Couette problem in the small gap limit is extended to the cases of tertiary and quaternary solutions. The theoretical results are compared with experimental observations. Although in the latter the small-gap approximation is not always well approximated, the comparison of theoretical results and observations yields reasonable agreements. The absence of the wavy twist mode in the observed patterns is explained by the presence of no-slip boundary conditions in the axial direction of the experimental apparatus, which differ from the periodic conditions imposed in the theoretical analysis. Quaternary solutions bifurcating from the tertiary ones through subharmonic instabilities are presented and compared with experimental observations. Reasonable agreement has been found.

Convection in a homogeneously cooled layer heated by a fixed temperature at the lower boundary is investigated numerically for various Rayleigh numbers Ra and Prandtl numbers Pr. The upper boundary is kept insulating. Large aspect ratios up to 20 are used in order to study the large scale pattern of convection. The problem is mathematically identical with the problem of a homogeneously heated convection layer cooled from above. The mean temperature difference between the two boundaries decreases in proportion to Ra- 0.216.

The onset of convection in a horizontal layer of an electrically conducting fluid heated from below is studied in the presence of a horizontal magnetic field rotating about a vertical axis. Two different ranges of the rotation frequency are considered: (1) the case of high frequencies of rotation for which the averaging approach is used and (2) the case of finite frequencies for which the averaging approach might be implemented. It is shown that in general the magnetic field stabilizes the static state of pure thermal conduction. However, there exists a parameter range where the inhomogeneity of the magnetic field is strong enough to provide energy for the growth of disturbances even if the fluid layer is stably stratified.

Numerical simulations of the geodynamo have successfully represented many observable characteristics of the geomagnetic field, yielding insight into the fundamental processes that generate magnetic fields in the Earth's core. Because of limited spatial resolution, however, the diffusivities in numerical dynamo models are much larger than those in the Earth's core, and consequently, questions remain about how realistic these models are. The typical strategy used to address this issue has been to continue to increase the resolution of these quasi-laminar models with increasing computational resources, thus pushing them toward more realistic parameter regimes. We assess which methods are most promising for the next generation of supercomputers, which will offer access to O(10(6)) processor cores for large problems. Here we report performance and accuracy benchmarks from 15 dynamo codes that employ a range of numerical and parallelization methods. Computational performance is assessed on the basis of weak and strong scaling behavior up to 16,384 processor cores. Extrapolations of our weak-scaling results indicate that dynamo codes that employ two-dimensional or three-dimensional domain decompositions can perform efficiently on up to approximate to 10(6) processor cores, paving the way for more realistic simulations in the next model generation.

The theory of planetary dynamos and its applications to observed phenomena of planetary magnetism are outlined. It is generally accepted that convection flows driven by thermal or compositional buoyancy are the most likely source for the sustenance of global planetary magnetic fields. While the existence of dynamos in electrically conducting fluid planetary cores provides constraints on properties of the latter, the lack of knowledge about time dependences of the magnetic fields and about their toroidal components together with the restricted parameter regions accessible to theory have prevented so far a full understanding of the phenomena of planetary magnetism.

We present a set of convective dynamo simulations in rotating spherical fluid shells based on an anelastic approximation of compressible fluids. The simulations extend into a regime where the buoyancy forcing is dominant while the Coriolis force is no longer balanced by pressure gradients and strong anti-solar differential rotation develops as a result. Dynamos in this regime are strongly dominated by dipole components but at the same time their magnetic energies are relatively small compared to the corresponding kinetic energies of the flow. Despite being relatively weak the self-sustained magnetic fields are able to reverse the direction of differential rotation to solar-like. We find that the convection in the buoyancy-dominated regime is significantly stronger near the pole than in the equatorial region, leading to non-oscillatory dipolar dynamo solutions. The results are obtained with a new simulation code for modelling of convection and MHD dynamo generation in rotating spherical shells under the anelastic approximation. The model equations and the new code are also presented here along with code validation results based on benchmark dynamo models.

Numerical MHD simulations play an increasingly important role for understanding the mechanisms of stellar magnetism. We present simulations of convection and dynamos in density-stratified rotating spherical fluid shells. We employ a new 3D simulation code for obtaining the solution of a physically consistent anelastic model of the process with a minimum number of parameters. The reported dynamo simulations extend into a “buoyancydominated” regime where the buoyancy forcing is dominant while the Coriolis force is no longer balanced by pressure gradients, and strong anti-solar differential rotation develops as a result. We find that the self-generated magnetic fields, despite being relatively weak, are able to reverse the direction of differential rotation from antisolar to solar-like. We also find that convection flows in this regime are significantly stronger in the polar regions than in the equatorial region, leading to non-oscillatory dipole-dominated dynamo solutions, and to a concentration of magnetic field in the polar regions. We observe that convection has a different morphology in the inner and the outer part of the convection zone simultaneously such that organized geostrophic convection columns are hidden below a near-surface layer of well-mixed highly chaotic convection. While we focus our attention on the buoyancy-dominated regime, we also demonstrate that conical differential rotation profiles and persistent regular dynamo oscillations can be obtained in the parameter space of the rotation-dominated regime even within this minimal model.

Convection in planetary cores can generate fluid flow and magnetic fields, and a number of sophisticated codes exist to simulate the dynamic behaviour of such systems. We report on the first community activity to compare numerical results of computer codes designed to calculate fluid flow within a whole sphere. The flows are incompressible and rapidly rotating and the forcing of the flow is either due to thermal convection or due to moving boundaries. All problems defined have solutions that allow easy comparison, since they are either steady, slowly drifting or perfectly periodic. The first two benchmarks are defined based on uniform internal heating within the sphere under the Boussinesq approximation with boundary conditions that are uniform in temperature and stress-free for the flow. Benchmark 1 is purely hydrodynamic, and has a drifting solution. Benchmark 2 is a magnetohydrodynamic benchmark that can generate oscillatory, purely periodic, flows and magnetic fields. In contrast, Benchmark 3 is a hydrodynamic rotating bubble benchmark using no slip boundary conditions that has a stationary solution. Results from a variety of types of code are reported, including codes that are fully spectral (based on spherical harmonic expansions in angular coordinates and polynomial expansions in radius), mixed spectral and finite difference, finite volume, finite element and also a mixed Fourier–finite element code. There is good agreement between codes. It is found that in Benchmarks 1 and 2, the approximation of a whole sphere problem by a domain that is a spherical shell (a sphere possessing an inner core) does not represent an adequate approximation to the system, since the results differ from whole sphere results.

Rayleigh-Bénard convection is investigated in rotating annular cavities at a moderate dimensionless rotation rate Ω=60. The onset of convection is in the form of azimuthal traveling waves that set in at the sidewalls and at values of the Rayleigh number significantly below the value of the onset of convection in an infinitely extended layer. The present study addresses the effects of curvature and confinement on the onset of sidewall convection by using three-dimensional spectral solutions of the Oberbeck-Boussinesq equations. Such solutions demonstrate that the curvature of the outer boundary promotes the onset of the wall mode, while the opposite curvature of the inner boundary tends to delay the onset of the wall mode. An inner sidewall with a radius as low as one tenth of its height is sufficient, however, to support the onset of a sidewall mode. When radial confinement is increased the two independent traveling waves interact and eventually merge to form a nearly steady pattern of convection.

Significant effects of compression on the dynamics of convection in gases can be obtained in centrifuges. Because of the dominant role of the Coriolis force, convection in the form of propagating thermal Rossby waves will be generated at onset. At finite amplitudes transitions to more complex forms of convection interacting with mean zonal flows must be expected which have been studied only numerically until now.

In 1976 Morten Tveitereid and Enok Palm established that the preference for down-hexagons in high Prandtl number fluids in an internally heated layer is changed to a preference for up-hexagons for Prandtl numbers below P c = 0.2506 . In this note it is demonstrated analytically that in a slightly modified problem the critical Prandtl number P c = 1 / 4 holds exactly.

Abstract The onset of convection in a rotating cylindrical annulus with parallel ends filled with a compressible fluid is studied in the anelastic approximation. Thermal Rossby waves propagating in the azimuthal direction are found as solutions. The analogy to the case of Boussinesq convection in the presence of conical end surfaces of the annular region is emphasised. As in the latter case, the results can be applied as an approximation for the description of the onset of anelastic convection in rotating spherical fluid shells. Reasonable agreement with three-dimensional numerical results published by Jones, Kuzanyan & Mitchell (J. Fluid Mech., vol. 634, 2009, pp. 291–319) for the latter problem is found. As in those results, the location of the onset of convection shifts outwards from the tangent cylinder with increasing number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}N_{\rho }$ of density scale heights until it reaches the equatorial boundary. A new result is that at a much higher number $N_{\rho }$ the onset location returns to the interior of the fluid shell.

It is demonstrated theoretically that the nonlinear stage of the Rayleigh-Taylor instability can be retarded at arbitrary Atwood numbers in a rotating system with the axis of rotation normal to the acceleration of the interface between two uniform inviscid fluids. The Coriolis force provides an effective restoring force on the perturbed interface, and the uniform rotation will always decrease the nonlinear saturation amplitude of the interface at any disturbance wavelength.

It is frequently considered that many planetary magnetic fields originate as a result of convection within planetary cores. Buoyancy forces responsible for driving the convection generate a fluid flow that is able to induce magnetic fields; numerous sophisticated computer codes are able to simulate the dynamic behaviour of such systems. This paper reports the results of a community activity aimed at comparing numerical results of several different types of computer codes that are capable of solving the equations of momentum transfer, magnetic field generation and heat transfer in the setting of a spherical shell, namely a sphere containing an inner core. The electrically conducting fluid is incompressible and rapidly rotating and the forcing of the flow is thermal convection under the Boussinesq approximation. We follow the original specifications and results reported in Harder & Hansen to construct a specific benchmark in which the boundaries of the fluid are taken to be impenetrable, non-slip and isothermal, with the added boundary condition for the magnetic field B that the field must be entirely radial there; this type of boundary condition for B is frequently referred to as 'pseudovacuum'. This latter condition should be compared with the more frequently used insulating boundary condition. This benchmark is so-defined in order that computer codes based on local methods, such as finite element, finite volume or finite differences, can handle the boundary condition with ease. The defined benchmark, governed by specific choices of the Roberts, magnetic Rossby, Rayleigh and Ekman numbers, possesses a simple solution that is steady in an azimuthally drifting frame of reference, thus allowing easy comparison among results. Results from a variety of types of code are reported, including codes that are fully spectral (based on spherical harmonic expansions in angular coordinates and polynomial expansions in radius), mixed spectral and finite difference, finite volume, finite element and also a mixed Fourier-finite element code. There is good agreement among codes.

Current ideas related to the problem of the generation of the geomagnetic field by motions in the liquid iron core of the Earth are reviewed and the difficulties of theoretical approaches are outlined. The importance of theoretical models starting from basic principles is emphasized and recent results from computations by Zhang and Busse [1988] of the magnetohydrodynamic problem of magnetic field generation by buoyancy driven convection in rotating spherical shells are described. This particular approach follows the sequence of bifurcating solutions of the full set of nonlinear equations. Future extensions and possible limitations of this numerical approach are discussed in a concluding section.

Two new experiments on fluid turbulence have attained conditions needed to establish asymptotic scalings for turbulent transports of heat and angular momentum.

Simple models of magnetic field generation by convection in rotating spherical shells exhibit properties resembling those observed on the sun. The assumption of the Boussinesq approximation made in these models prevents a realistic description of the solar cycle, but through a physically motivated change in the boundary condition for the differential rotation, the propagation of dynamo waves towards higher latitudes can be reversed, at least at low latitudes.

We report the results of three-dimensional numerical simulations of convection-driven dynamos in relatively thin rotating spherical shells that show a transition from an strong non-oscillatory dipolar magnetic field to a weaker regularly oscillating dipolar field. The transition is induced primarily by the effects a stress-free boundary condition. The variation of the inner to outer radius ratio is found to have a less important effect.