We study the onset of magnetoconvection between two infinite horizontal planes subject to a vertical magnetic field aligned with background rotation. In order to gain insight into the convection taking place in the Earth's tangent cylinder (TC), we target regimes of asymptotically strong rotation. The critical Rayleigh number Ra c and critical wavenumber are computed numerically by solving the linear stability problem in a systematic way. A parametric study is conducted, varying the Ekman number, E (ratio of viscous to Coriolis forces) and the Elsasser number, Λ (ratio of the Lorentz force to the Coriolis force). E is varied from 10 −9 to 10 −2 and Λ from 10 −3 to 1. Apply to arbitrary thermal and magnetic Prandtl numbers, our results verify and confirm previous experimental and theoretical results showing the existence of two distinct unstable modes at low values of E -one being controlled by the magnetic field, the other being controlled by viscosity (often called the viscous mode).Asymptotic scalings for the onset of these modes have been numerically confirmed and precisely quantified. We show that with no-slip boundary conditions, the asymptotic behaviour is reached for E < 10 −6 and establish a map in the (E, Λ) plane. We distinguish regions where convection sets in either in the magnetic mode or in the viscous mode. Our analysis gives the regime in which the transition between magnetic and viscous modes may be observed. We also show that within the asymptotic regime, the role played by the kinematic boundary conditions is minimal.

This paper experimentally investigates the convection in a rapidly rotating Tangent Cylinder (TC), for Ekman numbers down to E = 3.36 × 10 −6 . The apparatus consists of a hemispherical fluid vessel heated in its centre by a protruding heating element of cylindrical shape. The resulting convection that develops above the heater, i.e. within the TC, is shown to set in for critical Rayleigh numbers and wavenumbers respectively scaling as Ra c ∼ E −4/3 and a c ∼ E −1/3 with the Ekman number E. Although exhibiting the same exponents as for plane rotating convection, these laws reflect much larger convective plumes at onset. The structure and dynamics of supercritical plumes are in fact closer to those found in solid rotating cylinders heated from below, suggesting that the confinement within the TC induced by the Taylor-Proudman constraint influences convection in a similar way as solid walls would do. There is a further similarity in that the critical modes in the TC all exhibit a slow retrograde precession at onset. In supercritical regimes, the precession evolves into a thermal wind with a complex structure featuring retrograde rotation at high latitude and either prograde or retrograde rotation at low latitude (close to the heater), depending on the criticality and the Ekman number. The intensity of the thermal wind measured by the Rossby number Ro scales as Ro 5.33(Ra * q ) 0.51 with the Rayleigh number based on the heat flux Ra * q ∈ [10 −9 , 10 −6 ]. This scaling is in agreement with heuristic predictions and previous experiments where the thermal wind is determined by the azimuthal curl of the balance between the Coriolis force and buoyancy.Within the range Ra ∈ [2 × 10 7 , 10 9 ] which we explored, we also observe a transition in the heat transfer through the TC from a diffusivity-free regime where N u 0.38E 2 Ra 1.58 to a rotation-independent regime where N u 0.2Ra 0.33 .

In this paper, we present a new experimental facility, Little Earth Experiment, designed to study the hydrodynamics of liquid planetary cores. The main novelty of this apparatus is that a transparent electrically conducting electrolyte is subject to extremely high magnetic fields (up to 10 T) to produce electromagnetic effects comparable to those produced by moderate magnetic fields in planetary cores. This technique makes it possible to visualise for the first time the coupling between the principal forces in a convection-driven dynamo by means of Particle Image Velocimetry (PIV) in a geometry relevant to planets. We first present the technology that enables us to generate these forces and implement PIV in a high magnetic field environment. We then show that the magnetic field drastically changes the structure of convective plumes in a configuration relevant to the tangent cylinder region of the Earth's core.

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