High Rayleigh number convection with double diffusive fingers

An electrodeposition cell is used to sustain a destabilizing concentration difference of copper ions in aqueous solution between the top and bottom boundaries of the cell. The resulting convecting motion is analogous to Rayleigh-B\'enard convection at high Prandtl numbers. In addition, a stabilizing temperature gradient is imposed across the cell. Even for thermal buoyancy two orders of magnitude smaller than chemical buoyancy, the presence of the weak stabilizing gradient has a profound effect on the convection pattern. Double diffusive fingers appear in all cases. The size of these fingers and the flow velocities are independent of the height of the cell, but they depend on the ion concentration difference between top and bottom boundaries as well as on the imposed temperature gradient. The scaling of the mass transport is compatible with previous results on double diffusive convection

Rapidly rotating plane layer convection with zonal flow

The onset of convection in a rapidly rotating layer in which a thermal wind is present is studied. Diffusive effects are included. The main motivation is from convection in planetary interiors, where thermal winds are expected due to temperature variations on the core-mantle boundary. The system admits both convective instability and baroclinic instability. We find a smooth transition between the two types of modes, and investigate where the transition region between the two types of instability occurs in parameter space. The thermal wind helps to destabilise the convective modes. Baroclinic instability can occur when the applied vertical temperature gradient is stable, and the critical Rayleigh number is then negative. Long wavelength modes are the first to become unstable. Asymptotic analysis is possible for the transition region and also for long wavelength instabilities, and the results agree well with our numerical solutions. We also investigate how the instabilities in this system relate to the classical baroclinic instability in the Eady problem. We conclude by noting that baroclinic instabilities in the Earth's core arising from heterogeneity in the lower mantle could possibly drive a dynamo even if the Earth's core were stably stratified and so not convecting.Comment: 20 pages, 7 figure

Tilt-over mode in a precessing triaxial ellipsoid

The tilt-over mode in a precessing triaxial ellipsoid is studied theoretically and numerically. Inviscid and viscous analytical models previously developed for the spheroidal geometry by Poincar\'e [Bull. Astr. 27, 321 (1910)] and Busse [J. Fluid Mech., 33, 739 (1968)] are extended to this more complex geometry, which corresponds to a tidally deformed spinning astrophysical body. As confirmed by three-dimensional numerical simulations, the proposed analytical model provides an accurate description of the stationary flow in an arbitrary triaxial ellipsoid, until the appearance at more vigorous forcing of time dependent flows driven by tidal and/or precessional instabilities.Comment: http://link.aip.org/link/doi/10.1063/1.350435

Towards an experimental von Karman dynamo: numerical studies for an optimized design

Numerical studies of a kinematic dynamo based on von Karman type flows between two counterrotating disks in a finite cylinder are reported. The flow has been optimized using a water model experiment, varying the driving impellers configuration. A solution leading to dynamo action for the mean flow has been found. This solution may be achieved in VKS2, the new sodium experiment to be performed in Cadarache, France. The optimization process is described and discussed, then the effects of adding a stationary conducting layer around the flow on the threshold, on the shape of the neutral mode and on the magnetic energy balance are studied. Finally, the possible processes involved into kinematic dynamo action in a von Karman flow are reviewed and discussed. Among the possible processes we highlight the joint effect of the boundary-layer radial velocity shear and of the Ohmic dissipation localized at the flow/outer-shell boundary

Dynamo action at low magnetic Prandtl numbers: mean flow vs. fully turbulent motion

We compute numerically the threshold for dynamo action in Taylor-Green swirling flows. Kinematic calculations, for which the flow field is fixed to its time averaged profile, are compared to dynamical runs for which both the Navier-Stokes and the induction equations are jointly solved. The kinematic instability is found to have two branches, for all explored Reynolds numbers. The dynamical dynamo threshold follows these branches: at low Reynolds number it lies within the low branch while at high kinetic Reynolds number it is close to the high branch.Comment: 4 pages, 4 figure

A spherical shell numerical dynamo benchmark with pseudo vacuum magnetic boundary conditions

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>B</b> that the field must be entirely radial there; this type of boundary condition for <b>B</b> is frequently referred to as ‘pseudo-vacuum’. 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

Full sphere hydrodynamic and dynamo benchmarks

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

The alpha effect with imposed and dynamo-generated magnetic fields

Estimates for the nonlinear alpha effect in helical turbulence with an applied magnetic field are presented using two different approaches: the imposed-field method where the electromotive force owing to the applied field is used, and the test-field method where separate evolution equations are solved for a set of different test fields. Both approaches agree for stronger fields, but there are apparent discrepancies for weaker fields that can be explained by the influence of dynamo-generated magnetic fields on the scale of the domain that are referred to as meso-scale magnetic fields. Examples are discussed where these meso-scale fields can lead to both drastically overestimated and underestimated values of alpha compared with the kinematic case. It is demonstrated that the kinematic value can be recovered by resetting the fluctuating magnetic field to zero in regular time intervals. It is concluded that this is the preferred technique both for the imposed-field and the test-field methods.Comment: 10 pages, 8 figures, published versio

A spherical shell numerical dynamo benchmark with pseudo-vacuum magnetic boundary conditions

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 ‘pseudo-vacuum'. 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 code

On the magnetic fields generated by experimental dynamos

We review the results obtained by three successful fluid dynamo experiments and discuss what has been learnt from them about the effect of turbulence on the dynamo threshold and saturation. We then discuss several questions that are still open and propose experiments that could be performed to answer some of them.Comment: 40 pages, 13 figure