Reynolds number scaling and energy spectra in geostrophic convection

Abstract

We report flow measurements in rotating Rayleigh-Bénard 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 × 10^-8 we vary the Rayleigh number Ra between 10¹¹ and 4 × 10¹² 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; 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 bulk flow state is approached, sketching clear perspectives for further investigation.

Figure 1

Instantaneous velocity snapshots at Ra = 6.48 × 10¹¹: (a) with rotation, Ek = 5.00 × 10⁻⁸, (b) no rotation, Ek = ∞. The arrows display the horizontal velocity components; for clarity only one ninth of the total number of vectors is displayed. The background colour indicates the vertical velocity component. Velocities are normalised with the viscous velocity ν/H = 3.87 × 10⁻⁷ m/s.

Figure 2

(a) Radial dependence of Re_u (dashed lines) and Re_w (solid lines) for different Ra. Black crosses indicate the beginning of the sidewall boundary layer following 4.1. (b) Radial dependence of ${\omega }_z^{\text{rms}}$ for different Ra, normalised using the viscous time scale τ_ν = H²/ν. The legend entry ‘NR’ refers to the nonrotating experiment. All the quantities displayed in this figure are only shown from half the cylinder radius onward for clarity.

Figure 3

(a) Dependence on Ra of Reynolds numbers based on horizontal (Re_u) and vertical (Re_w) velocity components (red, left ordinate) and vertical vorticities (${\omega }_z^{\text{rms}}$; blue, right ordinate). Vorticity is normalised with the viscous timescale τ_ν = H²/ν. The solid lines represent power-law fits Re_u ∼ Ra^0.65±0.07, Re_w ∼ Ra^0.70±0.06 and ${\omega }_z^{\text{rms}}\sim R{a}^{0.63\pm 0.05}$. The dashed black diagonal line indicates the scaling Re ∼ Ra for reference. (b) Kinetic energy anisotropy $A=W^2/\left(U_h^2+W^2\right)$ versus Ra. For isotropy $A=\frac{1}{3}$ (red dashed line). In both panels the dotted and the dash-dotted lines represent the transitional Ra between CTCs and plumes and between GT and RIT, respectively, while open symbols represent the values for the nonrotating case.

Figure 4

Test of Re(Ra) scaling with VAC (eq. (2.11)) and CIA (eq. (2.14)) force balance arguments, for (a,b) Re_u and (c,d) Re_w. Three different Nu(Ra) relations are invoked: Nu ∼ Ra^0.52 (green; for RIT) and Nu ∼ Ra^0.64 (red; for plumes/GT) based on our heat transfer measurements in the same setup (Cheng et al. 2020); and Nu ∼ Ra^3/2 (blue) following eq. (2.16).

Figure 5

Comparison of current results for Re_w with published results. Plusses: data from Maffei et al. (2021), colour coded by Pr, results from asymptotically reduced model simulations on a horizontally periodic domain. Other symbols colour coded by Ek (see colour bar). Diamonds: current results. Small filled circles: experiments of Hawkins et al. (2023). Up triangles: DNS in a cylinder (Kunnen et al. 2010). Down triangles: experiments of Rajaei et al. (2018). Circles: DNS in a horizontally periodic domain (Aguirre Guzmán et al. 2020).

Figure 6

Kinetic energy spectra E (k) plotted as a function of normalised wavenumber kH, including total kinetic enery (E_tot) as well as the contributions from horizontal (E_hor) and vertical (E_ver) velocity components. Reference power-law slopes k^−5/3 and k⁻³ are also included. Graphs (e–h) are also plotted in compensated form k^5/3 E(k).