Turbulent convection in liquid metal with and without rotation

Abstract

The magnetic fields of Earth and other planets are generated by turbulent, rotating convection in liquid metal. Liquid metals are peculiar in that they diffuse heat more readily than momentum, quantified by their small Prandtl numbers, Pr = 1. Most analog models of planetary dynamos, however, use moderate Pr fluids, and the systematic influence of reducing Pr is not well understood. We perform rotating Rayleigh-Bénard convection experiments in the liquid metal gallium (Pr = 0:025) over a range of nondimensional buoyancy forcing (Ra) and rotation periods (E). Our primary diagnostic is the efficiency of convective heat transfer (Nu). In general, we find that the convective behavior of liquid metal differs substantially from that of moderate Pr fluids, such as water. In particular, a transition between rotationally constrained and weakly rotating turbulent states is identified, and this transition differs substantially from that observed in moderate Pr fluids. This difference, we hypothesize, may explain the different classes of magnetic fields observed on the Gas and Ice Giant planets, whose dynamo regions consist of Pr < 1 and Pr > 1 fluids, respectively.T he interiors of Earth and other terrestrial bodies, as well as the Gas Giant planets, contain vast oceans of flowing metals. Mixed by turbulent convection as these planets cool, the flowing conductors produce electric currents that maintain planetary magnetic fields. These bodies also rotate, and it is expected that the resulting Coriolis forces strongly influence convective dynamo processes. Beyond linear theory (1), however, not much is known about the dynamics of rotating convection in liquid metal that underlie planetary magnetic field generation.Theory and experiments often focus on a simplified analog of geophysical and astrophysical systems, Rayleigh-Bénard convection (RBC). The RBC system consists of a fluid layer contained between flat, horizontal plates separated by distance h, the bottom of which is warmer than the top by ΔT, and with downward pointing gravity, g. Near the bottom boundary, the fluid warms and expands by a volumetric factor α (per degree kelvin) and similarly cools and contracts near the top boundary, such that the fluid is unstably stratified.Rotating convection is explored by spinning the RBC system about a vertical axis at an angular rate Ω, which is adopted as the reference frame for the model. The fluid dynamics of the system are characterized by three dimensionless parameters. The Prandtl number, Pr ≡ ν=κ, defines the diffusive transport properties of the fluid, where ν and κ are its viscous and thermal diffusivities. The Rayleigh number, Ra ≡ αgΔTh 3 =ðνκÞ, prescribes the magnitude of the buoyancy force. Finally, the rotation period is specified by the Ekman number, E ≡ ν=ð2Ωh 2 Þ. The primary diagnostic used in many convection studies, including this one, is the Nusselt number, which characterizes the efficiency of heat transport by convection as Nu ≡ qh=ðkΔTÞ, where q is total heat flux and k is the fluid's thermal conductivi...

Enhanced enstrophy generation for turbulent convection in low-Prandtl-number fluids

“…The Reynolds number is Re = u rms H=ν and the Nusselt number is Nu = 1 + ðRa PrÞ 1=2 hwT i V,t . The Kolmogorov and Bolgiano lengths follow from [6] and thermal boundary layer thickness from [2]. implies that the majority of kinetic energy is injected in the range r > δ T , i.e., at the larger scales, and that the low-Prandtl flow is thus closer to the classical Kolmogorov turbulence, at least for ℓ K δ T .…”

“…Turbulent convection in the Sun is present at Prandtl numbers Pr < 10 −3 (3)(4)(5). The Prandtl number in the liquid metal core of the Earth is Pr ∼ 10 −2 (6). Convection in material processing (7), nuclear engineering (8), or liquid metal batteries (9) has Prandtl numbers between 3 × 10 −2 and 10 −3 .…”

Inertial convection in a rotating narrow annulus: Asymptotic theory and numerical simulation

“…In comparison to the recent laboratory experiment 18 focusing on the scaling law of strongly turbulent states for (  Ra)/(  Ra) c ≫ O(10), the primary aim of our nonlinear study is at understanding weakly nonlinear states and the transition from the laminar to weakly turbulent states for the liquid metal gallium in the regime of moderately supercritical Rayleigh numbers 1 < (  Ra)/(  Ra) c < O(10). Since the thin viscous boundary layer plays an essential role in actively controlling the dynamics of inertial convection, numerical simulation must be performed longer than t = O(1/ √ E), which is computationally expensive for small Ekman numbers.…”

“…But the dynamics of strongly turbulent states for (Ra)/(Ra) c ≫ O(10) would be much more complicated, as revealed by the recent laboratory experiment, 18 since in that case the stronger nonlinear effects in connection with the term u · ∇u in (6) …”

“…5 In contrast to viscous convection, the problem of inertial convection has received relatively less attention with a relatively small existing literature: Zhang 4,5 first revealed the physical preference of inertial convection in the fluids of small Prandtl number and derived an asymptotic analytical solution for the onset of inertial convection in spherical geometry; Zhang and Roberts 16 extended the spherical analysis to that for a rotating Rayleigh-Bénard layer; Busse and Simitev 6 carried out a numerical study of spherical inertial convection; Aurnou and Olson 17 performed an experimental study of inertial convection in a rotating Rayleigh-Bénard layer using the liquid gallium; and the recent laboratory experiment 18,19 concentrates on the scaling law of strongly turbulent states with (Ra)/(Ra) c ≫ O(10) for the liquid gallium.…”