a b s t r a c tAn accurate description of turbulent core convection is necessary in order to build robust models of planetary core processes. Towards this end, we focus here on the physics of rapidly rotating convection. In particular, we present a closely coupled suite of advanced asymptotically-reduced theoretical models, efficient Cartesian direct numerical simulations (DNS) and laboratory experiments. Good convergence is demonstrated between these three approaches, showing that a comprehensive understanding of the dynamics appears to be within reach in our simplified rotating convection system. The goal of this paper is to review these findings, and to discuss their possible implications for planetary cores dynamics.

A convection-driven multiscale dynamo model is developed in the limit of low Rossby number for the plane layer geometry in which the gravity and rotation vectors are aligned. The small-scale fluctuating dynamics are described by a magnetically-modified quasigeostrophic equation set, and the large-scale mean dynamics are governed by a diagnostic thermal wind balance. The model utilizes three timescales that respectively characterize the convective timescale, the large-scale magnetic evolution timescale, and the large-scale thermal evolution timescale. Distinct equations are derived for the cases of order one and low magnetic Prandtl number. It is shown that the low magnetic Prandtl number model is characterized by a magnetic to kinetic energy ratio that is asymptotically large, with ohmic dissipation dominating viscous dissipation on the large-scales. For the order one magnetic Prandtl number model the magnetic and kinetic energies are equipartitioned and both ohmic and viscous dissipation are weak on the large-scales; large-scale ohmic dissipation occurs in thin magnetic boundary layers adjacent to the solid boundaries. For both magnetic Prandtl number cases the Elsasser number is small since the Lorentz force does not enter the leading order force balance. The new models can be considered fully nonlinear, generalized versions of the dynamo model originally developed by Childress and Soward [Phys. Rev. Lett., 29, p.837, 1972]. These models may be useful for understanding the dynamics of convection-driven dynamos in regimes that are only just becoming accessible to direct numerical simulations.

We report on axisymmetric numerical simulations of rapidly rotating spherical shells in which the axial rotation rate of the outer shell is modulated in time. This allows us to model planetary bodies undergoing forced longitudinal libration. In this study we systematically vary the Ekman number, 10 −7 Յ E Շ 10 −4 , which characterizes the ratio of viscous to Coriolis forces in the fluid, and the libration amplitude, ⌬. For libration amplitudes above a certain threshold, Taylor-Görtler vortices form near the outer librating boundary, in agreement with the previous laboratory experiments of Noir et al. ͓Phys. Earth Planet. Inter. 173, 141 ͑2009͔͒. At the lowest Ekman numbers investigated, we find that the instabilities remain spatially localized at onset in the equatorial region. In addition, nonzero time-averaged azimuthal ͑zonal͒ velocities are observed for all parameters studied. The zonal flow is characterized by predominantly retrograde flow in the interior, with a stronger prograde jet in the outer equatorial region. The magnitude of the zonal flow scales as the square of the librational forcing, ⑀ 2 , where ⑀ = ⌬f and f is the dimensionless libration frequency defined as the ratio between the libration frequency and the mean angular rotation rate. In addition, the zonal flow is primarily independent of the Ekman number, implying that the zonal flow does not depend on the viscosity of the fluid. The simulations show that the zonal flow is driven by nonlinearities in the Ekman boundary layer; it is not driven by Taylor-Görtler vortices or by inertial waves in the fluid interior. Application of our results suggests that many librating bodies in the solar system are above the onset for centrifugal instabilities, with values up to ϳ30 times supercritical. However, the spatial localization of the instabilities at onset in our simulations suggests that their effects are limited on the global dynamics of librating bodies. We find that the zonal flows driven by libration in axisymmetric spherical shells are unlikely to produce significant planetary magnetic fields, but will likely generate nonzero mean torques on the bounding surfaces.

It is a well established result of linear theory that the influence of differing mechanical boundary conditions, i.e., stress-free or no-slip, on the primary instability in rotating convection becomes asymptotically small in the limit of rapid rotation (Chandrasekhar 1961). This is accounted for by the diminishing impact of the viscous stresses exerted within Ekman boundary layers and the associated vertical momentum transport by Ekman pumping (Niiler & Bisshopp 1965;Heard & Veronis 1971). By contrast, in the nonlinear regime recent laboratory experiments and supporting numerical simulations are now providing evidence that the efficiency of heat transport remains strongly influenced by Ekman pumping in the rapidly rotating limit (Stellmach et al. 2014;Cheng et al. 2015). In this paper, a reduced model is developed for the case of low Rossby number convection in a plane layer geometry with no-slip upper and lower boundaries held at fixed temperatures. A complete description of the dynamics requires the existence of three distinct regions within the fluid layer: a geostrophically balanced interior where fluid motions are predominately aligned with the axis of rotation, Ekman boundary layers immediately adjacent to the bounding plates, and thermal wind layers driven by Ekman pumping in between. The reduced model uses a classical Ekman pumping parameterization to alleviate the need for spatially resolving the Ekman boundary layers. Results are presented for both linear stability theory and a special class of nonlinear solutions described by a single horizontal spatial wavenumber. It is shown that Ekman pumping (which correlates positively with interior convection) allows for significant enhancement in the heat transport relative to that observed in simulations with stress-free boundaries. Without the intermediate thermal wind layer the nonlinear feedback from Ekman pumping would be able to generate a heat transport that diverges to infinity. This layer arrests this blowup resulting in finite heat transport at a significantly enhanced value. With increasing buoyancy forcing the heat transport transitions to a more efficient regime, a transition that is always achieved within the regime of asymptotic validity of the theory, suggesting this behavior may be prevalent in geophysical and astrophysical settings. As the rotation rate increases the slope of the heat transport curve below this transition steepens, a result that is in agreement with observations from laboratory experiments and direct numerical simulations.

Numerical simulations of quasi-static magnetoconvection with a vertical magnetic field are carried out up to a Chandrasekhar number of Q = 10 8 over a broad range of Rayleigh numbers Ra. Three magnetoconvection regimes are identified: two of the regimes are magnetically-constrained in the sense that a leading-order balance exists between the Lorentz and buoyancy forces, whereas the third regime is characterized by unbalanced dynamics that is similar to non-magnetic convection. Each regime is distinguished by flow morphology, momentum and heat equation balances, and heat transport behavior. One of the magnetically-constrained regimes is believed to represent an 'ultimate' magnetoconvection regime in the dual limit of asymptotically-large buoyancy forcing and magnetic field strength; this regime is characterized by an interconnected network of anisotropic, spatially-localized fluid columns aligned with the direction of the imposed magnetic field that remain quasi-laminar despite having large flow speeds. As for nonmagnetic convection, heat transport is controlled primarily by the thermal boundary layer. The heat transport and dynamics in the magnetically-constrained, high-Q regimes appear to be independent of the thermal Prandtl number.

International audienceForced longitudinal librations are oscillatory perturbations of the rotation rate of a planet resulting from a gravitational coupling with orbital partners. In the present study we report the first experimental evidence that a librating cylindrical container can viscously drive mean azimuthal flows in the liquid interior, hereafter referred to as zonal flows. Using a novel laser Doppler velocimetry system, the current work extends upon the study of libration-driven flows by Noir et al. (2009). We investigate the different mechanisms underlying the zonal flow generation. It is found that zonal flows in the interior result primarily from non-linearities in the Ekman boundary layer. Furthermore, the zonal flow scales as the square of the libration amplitude and is independent of the Ekman number. This scaling implies that forced longitudinal libration in an axisymmetric container (purely viscous coupling) will drive unobservably small zonal flows at planetary conditions. Thus, purely viscous librational coupling will not generate significant energy dissipation in a planetary fluid layer. It follows that any observed phase lag between the gravitational forcing and the orbital response of a planet requires non-viscous coupling mechanisms to account for the energy dissipatio

Abstract

The rotating cylindrical annulus geometry was first developed by Busse (J. Fluid Mech., vol. 44, 1970, pp. 441-460) as a simplified analogue for studying convection in rapidly rotating spherical geometries. Although it has provided a more tractable two-dimensional model than the sphere, it is formally limited to asymptotically small slopes and thus weak velocities in the direction parallel to the rotation axis. We present an asymptotically reduced three-dimensional equation set to model quasi-geostrophic convection in the annulus geometry where order-one slopes are permissible; this model provides a closer analogue to quasi-geostrophic convection in spheres and spherical shells where steeply sloping boundaries are present. A linear stability analysis of the reduced equations shows that a new class of three-dimensional, convectively driven Rossby waves is present in this system. The gravest modes exhibit strong axial variations as the slope of the boundaries becomes large. In addition, higher-order eigenmodes showing increasingly complex axial dependence are found that possess critical Rayleigh numbers close to that of the gravest mode.