Heat Transport in the Geostrophic Regime of Rotating Rayleigh-Bénard Convection

Abstract

We report experimental measurements of heat transport in rotating Rayleigh-Bénard convection in a cylindrical convection cell with aspect ratio Γ = 1/2. The fluid was helium gas with Prandtl number Pr = 0.7. The range of control parameters was Rayleigh number 4 × 10 9 < Ra < 4 × 10 11 and Ekman number 2 × 10 −7 < Ek < 3 × 10 −5 (corresponding to Taylor number 4 × 10 9 < Ta < 1 × 10 14 and convective Rossby number 0.07 < Ro < 5). We determine the crossover from weakly rotating turbulent convection to rotation dominated geostrophic convection through experimental measurements of the normalized heat transport Nu. The heat transport for the rotating state in the geostrophic regime, normalized by the zero-rotation heat transport, is consistent with scaling of (RaEk −7/4 ) β with β ≈ 1. A phase diagram is presented that encapsulates measurements on the potential geostrophic turbulence regime of rotating thermal convection. PACS numbers: 47.20.Bp, 47.54.+r Thermal convection in the presence of rotation occurs in many geophysical contexts, including the Earth's mantle [1], oceans [2], planetary atmospheres such as Jupiter [3], and solar interiors [4]. It also remains a fundamental problem in fluid dynamics, balancing rotation and buoyancy in a simple system that can be studied theoretically [5], experimentally [6][7][8][9][10][11] and numerically [12,13] with high precision. Thus, the problem of rotating thermal convection is of interest across a wide spectrum of scientific disciplines.The parameters of rotating convection are Ra = gα∆Td 3 /νκ which measures the buoyant forcing of the flow, Ek = ν/(2d 2 Ω) which represents an inverse dimensionless rotation rate, and Pr = ν/κ, where g is acceleration of gravity, ∆ T is the temperature difference between top and bottom plates separated by distance d, ν and κ are the fluid kinematic viscosity and thermal diffusivity, respectively, and Ω = 2πf is the angular rotation about an axis parallel to gravity for rotation frequency f . Rotation can also be represented by the Taylor number Ta = 1/Ek 2 or by the convective Rossby number Ro = Ek Ra/Pr which reflects the ratio of rotational time to buoyancy time. Here we use the representation of Ek or Ro such that high dimensionless rotation rates correspond to small values of the rotational control parameter in the spirit of the asymptotic equation approach of expanding in a small variable [14]. The measured response of the system in this space of buoyant and rotational forcing is the Nusselt number, Nu =Q/(λ∆T) whereQ is the applied heater power through the fluid and λ is the thermal conductance of the fluid.Much of the experimental work on rotating convection at high dimensionless rotation rates has focused on either the transition to convection where rotation-induced wall modes play an important role [7,15] or the turbulent state far from onset where thermal boundary layers control heat transport [6,[8][9][10][11]. Recently, the numerical simulation [16,17] of the appropriate equations of motion [14] in the asymptotic limit of h...

Approaching the Asymptotic Regime of Rapidly Rotating Convection: Boundary Layers versus Interior Dynamics

“…Beyond this transition, a much slower growth of Nu withR is observed. Unfortunately, laboratory experiments have yet to reach small enoughR at low E to investigate this regime change [2][3][4]. The experimental data presented here are no exception (cf.…”

“…While experiments easily reach high levels of turbulence, they struggle to ensure that Coriolis forces remain dominant in the force balance [2][3][4]. Numerical simulations suffer from the enormous range of spatial and temporal scales that need to be resolved.…”

Exploring the geostrophic regime of rapidly rotating convection with experiments

“…Up to now, most of the experiments [13][14][15][16][17][18][19][20][21][22][23] and DNSs 16,18,[24][25][26][27] did not reach deep into the rapidly rotating convection regime. The few experimental and numerical studies that entered decisively into this regime 9,[28][29][30] primarily use the overall heat transfer to characterize rapidly rotating Rayleigh-Bénard convection and identify transitions between flow regimes from changes in the heat-flux scaling. These findings suggest a phase diagram as sketched in Figure 1, with the Ekman number Ek indicating the ratio of viscous forces over Coriolis forces and the strength of thermal forcing is quantified by the Rayleigh number Ra and normalized by its critical value Ra cr for the onset of convective motion.…”

Laboratory-numerical models of rapidly rotating convection in planetary cores

“…In our E ≥ 3 × 10 −8 data, we can unambiguously detect a steep and a shallow Nu-Ra scaling, but even lower values of E are required to differentiate the independent scalings for intermediate regimes (cf. Ecke & Niemela 2014). …”