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We report flow measurements in rotating Rayleigh--B\'enard convection in therotationally-constrained geostrophic regime. We apply stereoscopic particleimage velocimetry to measure the three components of velocity in a horizontalcross-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 ingeostrophic convection. We also include one nonrotating experiment. The scalingof the velocity fluctuations (expressed as the Reynolds number $Re$) iscompared to theoretical relations expressing balances ofviscous--Archimedean--Coriolis (VAC) and Coriolis--inertial--Archimedean (CIA)forces. Based on our results we cannot decide which balance is most applicablehere; both scaling relations match equally well. A comparison of the currentdata with several other literature datasets indicates a convergence towardsdiffusion-free scaling of velocity as $Ek$ decreases. However, the use ofconfined domains leads at lower $Ra$ to prominent convection in the wall modenear the sidewall. Kinetic energy spectra point at an overall flow organisationinto a quadrupolar vortex filling the cross-section. This quadrupolar vortex isa quasi-two-dimensional feature as it only manifests in energy spectra based onthe horizontal velocity components. At larger $Ra$ the spectra reveal thedevelopment of a scaling range with exponent close to $-5/3$, the classicalexponent for inertial-range scaling in three-dimensional turbulence. Thesteeper $Re(Ra)$ scaling at low $Ek$ and development of a scaling range in theenergy spectra are distinct indicators that a fully developed, diffusion-freeturbulent flow state is approached, sketching clear perspectives for furtherinvestigation.

Reynolds number scaling and energy spectra in geostrophic convection