Using numerical simulations of rapidly rotating Boussinesq convection in a Cartesian box, we study the formation of long-lived, large-scale, depth-invariant coherent structures. These structures, which consist of concentrated cyclones, grow to the horizontal scale of the box, with velocities significantly larger than the convective motions. We vary the rotation rate, the thermal driving and the aspect ratio in order to determine the domain of existence of these large-scale vortices (LSV). We find that two conditions are required for their formation. First, the Rayleigh number, a measure of the thermal driving, must be several times its value at the linear onset of convection; this corresponds to Reynolds numbers, based on the convective velocity and the box depth, 100. Second, the rotational constraint on the convective structures must be strong. This requires that the local Rossby number, based on the convective velocity and the horizontal convective scale, 0.15. Simulations in which certain wavenumbers are artificially suppressed in spectral space suggest that the LSV are produced by the interactions of small-scale, depth-dependent convective motions. The presence of LSV significantly reduces the efficiency of the convective heat transport. arXiv:1403.7442v3 [physics.flu-dyn]

Convection is a fundamental physical process in the fluid cores of planets because it is the primary transport mechanism for heat and chemical species and the primary energy source for planetary magnetic fields. Key properties of convection, such as the characteristic flow velocity and lengthscale, are poorly quantified in planetary cores due to their strong dependence on planetary rotation, buoyancy driving and magnetic fields, which are all difficult to model under realistic conditions. In the absence of strong magnetic fields, the core convective flows are expected to be in a regime of rapidly-rotating turbulence, 1 which remains largely unexplored to date. Here we use a combination of numerical models designed to explore this low-viscosity regime to show that the convective lengthscale becomes independent of the viscosity and is entirely determined by the flow velocity and planetary rotation. For the Earth's core, we find that the characteristic convective lengthscale is approximately 30km and below this scale, motions are very weak. The 30-km cut-off scale rules out small-scale dynamo action and supports large-eddy simulations of core dynamics. Furthermore, it implies that our understanding of magnetic reversals from numerical geodynamo models does not relate to the Earth, because they require too intense flows. 2, 3 Our results also indicate that the liquid core of the Moon might still be in an active convective state despite the absence of a present-day dynamo. 4 Core convection is strongly affected by the rapid planetary rotation through the Proudman-Taylor constraint, 5 which obliges the fluid to move in columns with little variation along the rotation axis compared with the orthogonal directions. The very low fluid viscosity in liquid cores implies that the convective flows are turbulent, but this turbulence differs from both 3D turbulence due to the anisotropy imposed by the rotation and 2D turbulence due to the presence of Rossby waves. 6 Conditions in planetary cores correspond to small Ekman numbers (Ek = ν/ΩR 2 with viscosity ν, rotation rate Ω and core radius R), large Reynolds numbers (Re = UR/ν with flow speed U), and small Rossby numbers (Ro = U/ΩR = ReEk ), with, for instance, Ek ≈ 10 −15 , Re ≈ 10 9 and Ro ≈ 10 −6 in the Earth's core. 7 Numerical models of planetary cores must employ a fluid viscosity that is orders of magnitude larger than realistic values to keep the range of time and length scales involved in the dynamics manageable, with typically Ek ≥ 10 −7 and Re ≤ 10 4 . 8 Unfortunately this has the undesirable effect that convection properties are still controlled by the viscosity. 9, 10 To go beyond this range of parameters and into the rapidly-rotating turbulent convection regime, in which the fluid viscosity plays a sub-dominant role, we use a combination of a state-of-the-art 3D model 11 down to Ek = 10 −8 supplemented by a simplified model of rotating convection 12 down to Ek = 10 −11 using a quasi-geostrophic (QG) approximation. Here the QG approximation means that the axial vor...

We numerically investigate the efficiency of a spherical Couette flow at generating a self-sustained magnetic field. No dynamo action occurs for axisymmetric flow while we always found a dynamo when non-axisymmetric hydrodynamical instabilities are excited. Without rotation of the outer sphere, typical critical magnetic Reynolds numbers Rm c are of the order of a few thousands. They increase as the mechanical forcing imposed by the inner core on the flow increases (Reynolds number Re). Namely, no dynamo is found if the magnetic Prandtl number P m = Rm/Re is less than a critical value P m c ∼ 1. Oscillating quadrupolar dynamos are present in the vicinity of the dynamo onset. Saturated magnetic fields obtained in supercritical regimes (either Re > 2Re c or P m > 2P m c ) correspond to the equipartition between magnetic and kinetic energies. A global rotation of the system (Ekman numbers E = 10 −3 , 10 −4 ) yields to a slight decrease (factor 2) of the critical magnetic Prandtl number, but we find a peculiar regime where dynamo action may be obtained for relatively low magnetic Reynolds numbers (Rm c ∼ 300). In this dynamical regime (Rossby number Ro ∼ −1, spheres in opposite direction) at a moderate Ekman number (E = 10 −3 ), a enhanced shear layer around the inner core might explain the decrease of the dynamo threshold. For lower E (E = 10 −4 ) this internal shear layer becomes unstable, leading to small scales fluctuations, and the favorable dynamo regime is lost. We also model the effect of ferromagnetic boundary conditions. Their presence have only a small impact on the dynamo onset but clearly enhance the saturated magnetic field in the ferromagnetic parts. Implications for experimental studies are discussed.

We propose a self-consistent dynamo mechanism for the generation of large-scale magnetic fields in natural objects. Recent computational studies have described the formation of large-scale vortices in rotating turbulent convection. Here we demonstrate that for magnetic Reynolds numbers below the threshold for small-scale dynamo action, such turbulent flows can sustain large-scale magnetic fields, i.e., fields with a significant component on the scale of the system.

We study nonlinear convection in a rapidly rotating sphere with internal heating for values of the Prandtl number relevant for liquid metals (Pr ∈ [10 −2 , 10 −1 ]). We use a numerical model based on the quasi-geostrophic approximation, in which variations of the axial vorticity along the rotation axis are neglected, whereas the temperature field is fully three-dimensional. We identify two separate branches of convection close to onset: (i) a well-known weak branch for Ekman numbers greater than 10 −6 , which is continuous at the onset (supercritical bifurcation) and consists of thermal Rossby waves, and (ii) a novel strong branch at lower Ekman numbers, which is discontinuous at the onset. The strong branch becomes subcritical for Ekman numbers of the order of 10 −8 . On the strong branch, the Reynolds number of the flow is greater than 10 3 , and a strong zonal flow with multiple jets develops, even close to the nonlinear onset of convection. We find that the subcriticality is amplified by decreasing the Prandtl number. The two branches can co-exist for intermediate Ekman numbers, leading to hysteresis (Ek = 10 −6 , Pr = 10 −2 ). Non-linear oscillations are observed near the onset of convection for Ek = 10 −7 and Pr = 10 −1 .