Reynolds number scaling and energy spectra in geostrophic convection

Abstract

We report flow measurements in rotating Rayleigh--B\'enard convection in the rotationally-constrained geostrophic regime. We apply stereoscopic particle image velocimetry to measure the three components of velocity in a horizontal cross-section of a water-filled cylindrical convection vessel. At a constant, small Ekman number \(Ek=5\times 10^{-8}\) we vary the Rayleigh number \(Ra\) between \(10^{11}\) and \(4\times 10^{12}\) to cover various subregimes observed in geostrophic convection. We also include one nonrotating experiment. The scaling of the velocity fluctuations (expressed as the Reynolds number \(Re\)) is compared to theoretical relations expressing balances of viscous--Archimedean--Coriolis (VAC) and Coriolis--inertial--Archimedean (CIA) forces. Based on our results we cannot decide which balance is most applicable here; both scaling relations match equally well. A comparison of the current data with several other literature datasets indicates a convergence towards diffusion-free scaling of velocity as \(Ek\) decreases. However, the use of confined domains leads at lower \(Ra\) to prominent convection in the wall mode near the sidewall. Kinetic energy spectra point at an overall flow organisation into a quadrupolar vortex filling the cross-section. This quadrupolar vortex is a quasi-two-dimensional feature as it only manifests in energy spectra based on the horizontal velocity components. At larger \(Ra\) the spectra reveal the development of a scaling range with exponent close to \(-5/3\), the classical exponent for inertial-range scaling in three-dimensional turbulence. The steeper \(Re(Ra)\) scaling at low \(Ek\) and development of a scaling range in the energy spectra are distinct indicators that a fully developed, diffusion-free turbulent flow state is approached, sketching clear perspectives for further investigation.