Topic
Ekman number
About: Ekman number is a research topic. Over the lifetime, 764 publications have been published within this topic receiving 18521 citations.
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Papers
TL;DR: In this paper, an extensive set of dynamo models in rotating spherical shells, varying all relevant control parameters by at least two orders of magnitude, were studied and their scaling laws were established.
Abstract: SUMMARY We study numerically an extensive set of dynamo models in rotating spherical shells, varying all relevant control parameters by at least two orders of magnitude. Convection is driven by a fixed temperature contrast between rigid boundaries. There are two distinct classes of solutions with strong and weak dipole contributions to the magnetic field, respectively. Non-dipolar dynamos are found when inertia plays a significant role in the force balance. In the dipolar regime the critical magnetic Reynolds number for self-sustained dynamos is of order 50, independent of the magnetic Prandtl number Pm. However, dynamos at low Pm exist only at sufficiently low Ekman number E. For dynamos in the dipolar regime we attempt to establish scaling laws that fit our numerical results. Assuming that diffusive effects do not play a primary role, we introduce non-dimensional parameters that are independent of any diffusivity. These are a modified Rayleigh number based on heat (or buoyancy) flux Ra ∗ , the Rossby number Ro measuring the flow velocity, the Lorentz number Lo measuring magnetic field strength, and a modified Nusselt number Nu ∗ for the advected heat flow. To first approximation, all our dynamo results can be collapsed into simple power-law dependencies on the modified Rayleigh number, with approximate exponents of 2/5, 1/2 and 1/3 for the Rossby number, modified Nusselt number and Lorentz number, respectively. Residual dependencies on the parameters related to diffusion (E, Pm, Prandtl number Pr) are weak. Our scaling laws are in agreement with the assumption that the magnetic field strength is controlled by the available power and not necessarily by a force balance. The Elsasser number � , which is the conventional measure for the ratio of Lorentz force to Coriolis force, is found to vary widely. We try to assess the relative importance of the various forces by studying sources and sinks of enstrophy (squared vorticity). In general Coriolis and buoyancy forces are of the same order, inertia and viscous forces make smaller and variable contributions, and the Lorentz force is highly variable. Ignoring a possible weak dependence on the Prandtl numbers or the Ekman number, a surprising prediction is that the magnetic field strength is independent both of conductivity and of rotation rate and is basically controlled by the buoyancy flux. Estimating the buoyancy flux in the Earth’s core using our Rossby number scaling and a typical velocity inferred from geomagnetic secular variations, we predict a small growth rate and old age of the inner core and obtain a reasonable magnetic field strength of order 1 mT inside the core. From the observed heat flow in Jupiter, we predict an internal field of 8 mT, in agreement with Jupiter’s external field being 10 times stronger than that of the Earth.
TL;DR: In this paper, it was shown that if the perturbation velocity is a smooth function of r, the distance from the axis, then the angular velocity of the main body of fluid is determined by balancing the outflow from the boundary layer on one disc with the inflow to the boundary layers on the other at the same value of r.
Abstract: In order to answer some of Proudman's questions (1956) concerning shear layers in rotating fluids, a study is made of the flow between two coaxial rotating discs, each having an arbitrary small angular velocity superposed on a finite constant angular velocity. It is found that, if the perturbation velocity is a smooth function of r, the distance from the axis, then the angular velocity of the main body of fluid is determined by balancing the outflow from the boundary layer on one disc with the inflow to the boundary layer on the other at the same value of r. At a discontinuity in the angular velocity of either disc a shear layer parallel to the axis occurs. If the angular velocity of the main body of the fluid is continuous, according to the theory given below the purpose of this shear layer is solely to transfer fluid from the boundary layer on one disc to the boundary layer of the other. It has a thickness O(v1/3), where v is the kinematic viscosity, and in it the induced angular velocity is O(v1/6) of the perturbation angular velocity of the discs. On the other hand, if the angular velocity of the main body of fluid is discontinuous, according to the theory given below the thickness of the shear layer is O(v1/4). A secondary circulation is also set up in which fluid drifts parallel to the axis in this shear layer and is returned in an inner shear layer of thickness O(v1/3).The theory is also applied to the motion of fluid inside a closed circular cylinder of finite length rotating about its axis almost as if solid.
501 citations
TL;DR: In this paper, three mechanisms for self-induced Ekman pumping in the interiors of mesoscale ocean eddies are investigated, including the surface stress that occurs because of differences between surface wind and ocean velocities, resulting in Ekman upwelling and downwelling in the cores of anticyclones and cyclones, respectively.
Abstract: Three mechanisms for self-induced Ekman pumping in the interiors of mesoscale ocean eddies are investigated. The first arises from the surface stress that occurs because of differences between surface wind and ocean velocities, resulting in Ekman upwelling and downwelling in the cores of anticyclones and cyclones, respectively. The second mechanism arises from the interaction of the surface stress with the surface current vorticity gradient, resulting in dipoles of Ekman upwelling and downwelling. The third mechanism arises from eddy-induced spatial variability of sea surface temperature (SST), which generates a curl of the stress and therefore Ekman pumping in regions of crosswind SST gradients. The spatial structures and relative magnitudes of the three contributions to eddy-induced Ekman pumping are investigated by collocating satellite-based measurements of SST, geostrophic velocity, and surface winds to the interiors of eddies identified from their sea surface height signatures. On average, e...
347 citations
TL;DR: In this article, the authors analyse 50 3D numerical calculations of hydrodynamic dynamos driven by convection in a spherical shell, with rigid and stress-free boundaries, with Prandtl number 1, magnetic Prandttl numbers in the range 0.5-5, Ekman numbers E=10− 3−10−4 and Rayleigh numbers to 15 times critical.
Abstract: Summary
We analyse ~ 50 3-D numerical calculations of hydrodynamic dynamos driven by convection in a spherical shell. We examine rigid and stress-free boundaries, with Prandtl number 1, magnetic Prandtl numbers in the range 0.5–5, Ekman numbers E=10− 3–10− 4 and Rayleigh numbers to 15 times critical. No parametrizations such as hyperviscosities are used. Successful dynamos are compared with non-magnetic convection solutions. Results for various spectral truncations suggest that the calculations are well resolved when the kinetic and magnetic energy drops by more than two orders of magnitude from the spectral peak to the cut-off, although the basic features are still captured at lower resolution. With few exceptions we obtain dipole-dominated magnetic fields. The dynamos operate in the strong-field regime where Lorentz and Coriolis forces are of similar order. The critical magnetic Reynolds number for self-sustained dynamos is of order Rm=50. However, we also find that the field can die away when Rm is too large. The minimum magnetic Prandtl number at which we find dynamo action depends on the Ekman number to the 3/4 power. Dynamos at E=10− 3 are subcritical whereas those at E=10− 4 are generally supercritical. The presence of the magnetic field tends to break the equatorial symmetry of the flow and favours convection inside the inner core tangent cylinder. With stress-free boundaries, dynamo action suppresses the axisymmetric azimuthal wind that dominates in non-magnetic convection. The field morphology is broadly similar for both kinds of boundary conditions. When low-pass filtered, several models exhibit field structures that resemble the geomagnetic field at the core–mantle boundary to a surprising degree.
325 citations
TL;DR: In this paper, a sequence of three convection-driven simulations in a rapidly rotating spherical shell is used to reach realistic turbulent regime in direct numerical simulations of the geodynamo.
Abstract: We present an attempt to reach realistic turbulent regime in direct numerical simulations of the geodynamo. We rely on a sequence of three convection-driven simulations in a rapidly rotating spherical shell. The most extreme case reaches towards the Earth's core regime by lowering viscosity (magnetic Prandtl number Pm=0.1) while maintaining vigorous convection (magnetic Reynolds number Rm>500) and rapid rotation (Ekman number E=1e-7), at the limit of what is feasible on today's supercomputers. A detailed and comprehensive analysis highlights several key features matching geomagnetic observations or dynamo theory predictions – all present together in the same simulation – but it also unveils interesting insights relevant for Earth's core dynamics.
In this strong-field, dipole-dominated dynamo simulation, the magnetic energy is one order of magnitude larger than the kinetic energy. The spatial distribution of magnetic intensity is highly heterogeneous, and a stark dynamical contrast exists between the interior and the exterior of the tangent cylinder (the cylinder parallel to the axis of rotation that circumscribes the inner core).
In the interior, the magnetic field is strongest, and is associated with a vigorous twisted polar vortex, whose dynamics may occasionally lead to the formation of a reverse polar flux patch at the surface of the shell. Furthermore, the strong magnetic field also allows accumulation of light material within the tangent cylinder, leading to stable stratification there. Torsional Alfven waves are frequently triggered in the vicinity of the tangent cylinder and propagate towards the equator.
Outside the tangent cylinder, the magnetic field inhibits the growth of zonal winds and the kinetic energy is mostly non-zonal. Spatio-temporal analysis indicates that the low-frequency, non-zonal flow is quite geostrophic (columnar) and predominantly large-scale: an m=1 eddy spontaneously emerges in our most extreme simulations, without any heterogeneous boundary forcing.
Our spatio-temporal analysis further reveals that (i) the low-frequency, large-scale flow is governed by a balance between Coriolis and buoyancy forces – magnetic field and flow tend to align, minimizing the Lorentz force; (ii) the high-frequency flow obeys a balance between magnetic and Coriolis forces; (iii) the convective plumes mostly live at an intermediate scale, whose dynamics is driven by a 3-term 1 MAC balance – involving Coriolis, Lorentz and buoyancy forces. However, small-scale (E^{1/3}) quasi-geostrophic convection is still observed in the regions of low magnetic intensity.
255 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 3 |
| 2024 | 2 |
| 2023 | 20 |
| 2022 | 68 |
| 2021 | 10 |
| 2020 | 14 |