Low Ekman Number Research Articles

Quasi-geostrophic kinematic dynamos at low magnetic Prandtl number

Rapidly rotating spherical kinematic dynamos at very low Ekman and Prandtl numbers are computed using the combination of a quasi-geostrophic (QG) model for the velocity field and a classical spectral 3D code for the magnetic field. The QG flow is computed in the equatorial plane of the sphere; it corresponds to Rossby wave instabilities of a geostrophic internal shear layer produced by differential rotation. The induction equation is computed in the whole sphere after the QG flow has been expanded along the rotation axis. Differential rotation and Rossby wave propagation are the key ingredients of this dynamo which can be interpreted in terms of Parker-Ω dynamo. Taking into account the quasi-geostrophy of the velocity field enables us to increase time and space resolution to compute the dynamics. For the first time, we report on numerical dynamos with very low Ekman numbers (10−8). Because the magnetic and velocity fields are computed on different grids, we compute dynamos for very low magnetic Prandtl numbers exhibiting a scale separation between magnetic and velocity field. These dynamos are asymptotically close to rapidly rotating, metallic planetary cores.

Observations of asymmetrical flow behaviour in transitional pipe flow of yield-stress and other shear-thinning liquids

The purpose of this brief paper is to report mean velocity profile data for fully developed pipe flow of a wide range of shear-thinning liquids together with two Newtonian control liquids. Although most of the data reported are for the laminar–turbulent transition regime, data are also included for laminar and turbulent flow. The experimental data were obtained in unrelated research programmes in UK, France and Australia, all using laser Doppler anemometry (LDA) as the measurement technique. In the majority of cases, axisymmetric flow is observed for the laminar and turbulent flow conditions, although asymmetry due to the Earth's rotation is evident for the laminar flow of a Newtonian fluid of low viscosity (i.e. low Ekman number). The key point, however, is that for certain fluids, both yield-stress and viscoelastic (all fluids in this study are shear-thinning), asymmetry to varying degrees is apparent at all stages of transition from laminar to turbulent flow, i.e. from the first indications to almost fully developed turbulence. The fact that symmetrical velocity profiles are obtained for both laminar and turbulent flow of all the non-Newtonian fluids in all three laboratories leads to the conclusion that the asymmetry must be a consequence of a fluid-dynamic mechanism, as yet not identified, rather than imperfections in the flow facilities.

Precession driven dynamos

The dynamo effect is demonstrated numerically in precession driven flow in a spherical container. Both laminar as well as unstable flows act as dynamos. At low Ekman numbers in unstable flows, the dynamo mechanism relies predominantly on the components of the flow excited by instabilities. All calculations are performed in a frame of reference attached to the boundaries. In this frame, the rotation axis of the fluid executes a periodic motion with a period equal to the rotation period of the boundaries, whereas the magnetic dipole moment undergoes slower variations interrupted by reversals.

A geostrophic-like model for large-Hartmann-number flows

A flow of electrically conducting fluid in the presence of a steady magnetic field has a tendency to become quasi-two-dimensional, i.e. uniform in the direction of the magnetic field, except in thin so-called Hartmann boundary layers. The condition for this tendency is that of a strong magnetic field, corresponding to large values of the dimensionless Hartmann number ( explains why the flow expected at large Hartmann number was not observed in experiments. The second problem exemplifies a situation where an analytical solution had been obtained in the past Walker & Ludford (J. Fluid Mech. vol. 56, 1972, p. 481) for the so-called ‘M-shaped’ velocity profile, which is here understood as an aspect of general quasi-two-dimensional magnetohydrodynamics.

Cartesian convection driven dynamos at low Ekman number

The Cartesian dynamo model of Childress and Soward [Phys. Rev. Lett. 29, 837 (1972)] is studied numerically in the regime of low viscosity. Dynamos with Ekman numbers E in the range 10(-4) > or =E> or =5 x 10(-7) are discussed and compared with the corresponding nonmagnetic states and with results obtained for imposed magnetic fields. We find that in the range of investigated Ekman numbers, a transition occurs from a flow regime where the planform of convection is only weakly affected by the dynamo generated field to a regime where the typical length scales of the flow are largely controlled by the Lorentz forces. The magnetic field acts to facilitate convection and leads to an increase in both the heat transport and in the amplitude of the flow. We demonstrate that this convection promoting effect allows for dynamo action even for Rayleigh numbers below the critical Rayleigh number for the onset of nonmagnetic convection.

Wind-induced exchange at the entrance to Concepción Bay, an equatorward facing embayment in central Chile

Abstract Hydrographic and underway water velocity profiles were measured at the entrance to Concepcion Bay, an equatorward facing embayment in central Chile, in order to illustrate the wind-induced exchange flow. Observations took place over a 25-h period (5–6 January 2002) during the austral summer, which is characterized by upwelling winds in the region. One portion of the observational transect was directly influenced by the wind and the other was protected from such influence by an island. The portion of the transect influenced by wind showed a surface layer, delimited by a pycnocline, moving downwind owing to the direct action of wind stress. This surface transport should have caused an onshore decrease of surface elevation that would drive upwind flow below the surface layer. The portion of the transect protected by the island was influenced by its wake and perhaps forced by a semidiurnal tide. This study is one of the few observational examples of wind-induced flow that documents the wind-induced exchange pattern under weak internal friction (low Ekman number). This pattern consists of downwind flow in an upper layer delimited by the pycnocline and upwind flow below.

The Magnetic Field in Giant Extrasolar Planets

We study the internal structure and the convective motions of giant extrasolar planets in order to calculate their dynamo-generated surface magnetic field and dipolar magnetic moment. Using evolutionary models, we investigate the existence of magnetism in planets with masses ranging from 0.3MJ to 10MJ and with rotation periods ranging from synchronism in hot Jupiters (with periods of 3-4 days) to the breakdown centrifugal limit (with periods of 2-5 hr). We find that the high Rayleigh and low Ekman numbers in the interior metallic-hydrogen region guarantee convective motions even for low-mass and evolved (aged) planets. The convective velocities estimated from mixing-length theory and from magnetostrophic balances yield high Reynolds and magnetic Reynolds numbers (~103-106), permitting the self-excited dynamo mechanism. Strong magnetism occurs in young massive and rapidly rotating extrasolar planets (with surface magnetic fields of ~30-60 G), but older or orbitally synchronized planets should have fields of ~1 G. We discuss the implications of these results for the detection of magnetic fields in extrasolar planets.

Elliptical instability in a rotating spheroid

This paper concerns the elliptical instability of a flow in a rotating deformed sphere. The aim of our work is to analyse the characteristics of this instability experimentally and to provide a theoretical model which accounts for the observations. For this purpose, an elastic and transparent hollow sphere has been moulded in a silicone gel block. The flow is visualized using Kalliroscope flakes illuminated with a laser sheet as the sphere is set into rotation and compressed by two rollers. The elliptical instability occurs by the appearance of the so-called ‘spin-over’ mode whose growth rate and saturation amplitude are measured by video image analysis at different Ekman numbers. Growth rates are predicted well by the linear stability analysis. A nonlinear model is developed and is shown to describe correctly the saturated regimes observed in the experiments. At low Ekman numbers, a secondary instability leading to an intermittent regime is also discovered.

Rotating convection-driven dynamos at low Ekman number.

We present a fully 3D self-consistent convection-driven dynamo model with reference to the geodynamo. A relatively low Ekman number regime is reached, with the aim of investigating the dynamical behavior at low viscosity. This regime is computationally very demanding, which has prompted us to adopt a plane layer model with an inclined rotation vector, and to make use of efficiently parallelized code. No hyperdiffusion is used, all diffusive operators are in the classical form. Our model has infinite Prandtl number, a Rayleigh number that scales as E(-1/3) (E being the Ekman number), and a constant Roberts number. The optimized model allows us to study dynamos with Ekman numbers in the range [10(-5),10(-4)]. In this regime we find strong-field dynamos where the induced magnetic fields satisfy Taylor's constraint to good accuracy. The solutions are characterized by (i) a MAC balance within the bulk, i.e., Coriolis, pressure, Lorentz, and buoyancy forces are of comparable magnitude, while viscous forces are only significant in thin boundary layers, (ii) the Elsasser number is O(10), (iii) the strong magnetic fields cannot prevent small-scale structures from becoming dominant over the large-scale components, (iv) the Taylor-Proudman effect is detectable, (v) the Taylorization decreases as the Ekman number is lowered, and (vi) the ageostrophic velocity component makes up 80% of the flow.

Zonal flow driven by strongly supercritical convection in rotating spherical shells

Thermal convection in a rotating spherical shell with free-slip boundaries can excite a dominant mean zonal flow in the form of differentially rotating cylinders concentric to the principal rotation axis. This process is studied numerically for Prandtl numbers of order 1, Ekman numbers in the range E = 3 × 10−4−10−5, and Rayleigh numbers up to 100× critical. Small-scale convection transfers kinetic energy into the mean zonal flow via Reynolds stresses. For low Ekman number and high Rayleigh number, the force balance is predominantly among the Coriolis, inertial and buoyancy forces, and viscosity plays a minor role. A modified Rayleigh number Ra* is introduced, which does not depend on viscosity or thermal diffusivity, and asymptotic scaling laws for the dependence of various properties on Ra* in the limit of negligible viscosity (E → 0) are estimated from the numerical results. The ratio of kinetic energy in the zonal flow to that in the non-zonal (convective) flow increases strongly with Ra* at low supercritical Rayleigh number, but drops at high values of Ra*. This is probably caused by the gradual loss of geostrophy of the convective columns and a corresponding decorrelation of Reynolds stresses. Applying the scaling laws to convection in the molecular hydrogen envelopes of the large gas planets predicts the observed magnitude of the zonal winds at their surfaces.

Zonal flow driven by deep convection in the major planets

The cause of the fast zonal winds on the large gas planets is disputed. To investigate their possible origin by convection in the planets' molecular hydrogen layer, I study simple numerical models of thermal convection in a rotating sphere at low Ekman numbers for Rayleigh numbers up to 100 times critical. With stress‐free boundaries and high Rayleigh number it is possible to reproduce the observed magnitudes of the zonal and non‐zonal flow components at the surfaces of Jupiter and Saturn. While only three flow bands are found for a thick convecting shell, the number increases to seven for a shell with an inner‐to‐outer radius of 0.6.

Numerical study of the motions within a slowly precessing sphere at low Ekman number

A geostrophic circulation and a pair of oblique oscillating shear layers arise in a spherical uid cavity contained in a slowly precessing rigid body. Both are caused by the breakdown of the Ekman boundary layer at two critical circles. We rely on numerical modelling to characterize these motions for very small Ekman numbers. Both the O(E1/5) amplitude of the velocity in the oscillating shear layer and the width (also O(E1/5)) of these oblique layers are the result of in ux into the interior from the regions where the Ekman layer breaks down. The oscillating motions are confined to narrow shear layers and their amplitude decays exponentially away from the characteristic surfaces. Nonlinear interactions inside the boundary layer drive the geostrophic shear layer attached to the critical circles. This steady layer, again of O(E1/5) thickness, contains O(E−3/10) velocities. Our results are in good agreement with the experimental measurement by Malkus of the geostrophic velocity arising in a slowly precessing spheroid.

Convection–driven geodynamo models

There has been significant progress in the development of numerical geodynamo models over the last five years. Advances in computer technology have made it possible to perform three–dimensional simulations, with thermal or compositional convection as the driving mechanism. These numerical simulations give reasonable results for the morphology and strength of the field at the core–mantle boundary, and the models are also capable of giving reversals and excursions which can be compared with palaeomagnetic observations; they also predict differential rotation between the inner core and the mantle. However, there are still a number of fundamental problems associated with the simulations, which are proving hard to overcome. Despite the advances in computing power, the models are still expensive and take a long time to run. This problem may diminish as faster machines become available, and new numerical methods exploit parallelization effectively, but currently there are no practical schemes available which work at low Ekman number. Even with turbulent values of the diffusivities (and the question of whether isotropic diffusivities are appropriate is still unresolved), the appropriate dynamical regime has not yet been reached. In consequence, modelling assumptions about the nature of the flow near the boundaries have to be made, and different choices can have profound effects on the dynamics. The nature of large–scale magnetoconvection at small E is still not well understood, and until we have more understanding of this issue, it will be difficult to have a great deal of confidence in the predictions of the numerical models.

Convection-driven dynamos in a rotating plane layer

The plane layer Childress–Soward dynamo model, consisting of a rotating layer of electrically conducting fluid between horizontal planes heated from below, is studied. Solutions periodic in the horizontal directions are sought, with electrically insulating boundary conditions applied. The large Prandtl number limit is used.Fully three-dimensional convection-driven dynamos have been studied numerically for this problem. Both the kinematic and the magnetically saturated regimes are studied, and a simple model of the dynamo mechanism is proposed. The dependence of the dynamo on the Rayleigh number, Ekman number and diffusivity ratio is studied, and the role of Taylor's constraint in low Ekman number convection-driven dynamos is considered.

Convection driven geodynamo models of varying Ekman number

We investigate the dynamo action arising from convection in a rapidly-rotating spherical shell. A single mode of the non-axisymmetric field is solved for, in addition to the axi-symmetric (mean) field. This allows dynamo action to be obtained without any imposed parameterisation, yet results in a system tractable enough that a range of physical regimes can be investigated. We describe the different types of dynamo obtained for varying Ekman and Roberts numbers. For the smaller values of these parameters, hyper-diffusivities have been used to model the effects of small lengthscale turbulent diffusion. At relatively high Ekman numbers (c. 10-3) dynamo action is obtained for moderately supercritical convective flows, with the flow in the form of travelling-wave convective rolls. At lower (hyperdiffusive) Ekman numbers, magnetic field maintenance occurs only for strongly supercritical flows. The resultant dynamos are temporally chaotic and dominated by strong fields and flows in the vicinity of the inner core, resembling the fully three-dimensional solutions of Glatzmaier and Roberts (1995b, 1996a). The inner core plays a critical role in these solutions, and rotates progradely at of order 1° per year. Calculations are conducted with ‘dipolar’ equatorial symmetry imposed and with no imposed symmetry. The imposed symmetry constraint proves an unphysical restriction for all our solutions. At lower Ekman numbers, the equatorially symmetric magnetic field is only intermittently significant; fluctuations in this component remain a plausible ‘trigger’ for reversals of the dominant dipole field, however, and so potentially important for the geodynamo. Taylor torque ‘integrals are calculated to quantify the adjustment of our solutions to the low viscosity regime. Our low Ekman number solutions satisfy the Taylor constraint appropriate to this regime more poorly than those at high Ekman number, however. Here the Lorentz torque on coaxial cylinders is balanced by the viscous torque associated with the hyperdiffusively-affected short wavelength velocities. Thus these solutions also remain viscously controlled, and might be expected to depend somewhat on the form of hyperviscosity assumed. At neither high nor low Ekman numbers does removing the imposed symmetry constraint reliably assist towards satisfying Taylor's constraint.

Inertial modes in the liquid core of the Earth

Using the simple model of an incompressible fluid, we have computed the eigenfrequencies of the lowest-order inertial modes (azimuthal wavenumber m = 0,1,2) in a spherical shell with the same aspect ratio as the liquid core of the Earth. The computed eigenfunctions show that all inertial modes have strong oscillating shear layers. For the very low Ekman number appropriate to the core, these layers might be the origin of some small-scale turbulence through shear instabilities. We have also studied the effect of a thin stable layer lying just below the core-mantle boundary, with the remainder of the core being neutrally stratified, as suggested by recent work. For plausible Nusselt numbers (0.8–0.9), the frequencies of the large-scale modes are only slightly increased (at best by 10 −4).

Numerical investigation of viscous gas secondary flows in a rotating cylinder with sources and sinks

A source-sink model of secondary flow excitation in a rotating cylinder, which describes the interaction between a circulator and a rotating gas, is proposed for a nonlinear system of Navier-Stokes equations, and the results of a numerical calculation of the resulting circulating flows are presented. The modified Newton's method employed in the numerical solution makes use of regularizing perturbations to ensure its stability and convergence at low Ekman numbers and high rates of rotation of the cylinder. The combined effect of mechanical and thermal means of flow excitation and the influence of viscous energy dissipation are considered.

A laboratory study of the lift forces on a moving solid obstacle in a rotating fluid

Abstract A series of experiments is described in which a novel technique is used to measure directly the lift force on a right cylynder circular moving through a homogeneous rotating fluid. These experiments have been conducted on the large rotating table facility at the Institut de Mecanique de Grenoble, in order to investigate simultaneously flows at high Reynolds number ( Re ), low Rossby ( Ro ), and low Ekman ( Ek ) number. The measurements have shown that within the range of parameters 0.25 Ro −6 Ek −6 ; 1.50 × 10 4 Re 5 , the cylinder experiences a mean lift due to (i) the structural asymmetry between cyclonic and anticyclonic eddies in its wake, and (ii) the Coriolis force on its excess mass. The total mean lift coefficient C ∗ shows no significant dependence upon Ek , but exhibits a linear relationship with Ro −1 . Similarly, the contribution of the mean lift coefficient C p due to the wake asymmetry alone is shown to vary approximately linearly with Ro −1 . In addition to the net mean lift, the cylinder experiences a fluctuating lift force generated by periodic vortex shedding. The amplitude of the fluctuating lift coefficient also increases linearly with increasing values of Ro −1 , though the Strouhal frequency shows no variation with either Ro or Ek . Measured values of the Strouhal number do not differ significantly from values typical of non-rotating flow past circular cylinders.

Laminar Compressible Flow Over a Stationary Disk in a Rotating Cylinder

The laminar isothermal compressible flow in a rotating cylinder with a stationary bottom is treated both numerically and by boundary layer matching. The numerical solution of the Navier-Stokes equations is based on a modified Cheng-Allen finite difference scheme. An approximate solution is obtained by matching boundary layers to an interior core. For sufficiently low Ekman numbers, the approximate and numerical solutions are in close agreement. The compressibility is shown to increase the angular velocity in the core and to decrease the meridional circulation. When the aspect ratio of the cylinder is increased, both the angular velocity in the core and the meridional circulation increase.

Discussion: “Flow Past a Right Circular Cylinder in a Rotating Frame” (Boyer, D. L., 1970, ASME J. Basic Eng., 92, pp. 430–435)

An analysis of low Rossby number and low Ekman number flow between parallel plates shows that the interior flow is irrotational and two-dimensional for all Rossby numbers and Ekman numbers in the range R0≪1, E≪1. The extra condition R0≪E1/2 is not needed for this proof.