Experimental observation of the geostrophic turbulence regime of rapidly rotating convection

Abstract

The competition between turbulent convection and global rotation in planetary and stellar interiors governs the transport of heat and tracers, as well as magnetic field generation. These objects operate in dynamical regimes ranging from weakly rotating convection to the "geostrophic turbulence" regime of rapidly rotating convection. However, the latter regime has remained elusive in the laboratory, despite a worldwide effort to design ever-taller rotating convection cells over the last decade. Building on a recent experimental approach where convection is driven radiatively, we report heat transport measurements in quantitative agreement with this scaling regime, the experimental scaling law being validated against direct numerical simulations (DNS) of the idealized setup. The scaling exponent from both experiments and DNS agrees well with the geostrophic turbulence prediction. The prefactor of the scaling law is greater than the one diagnosed in previous idealized numerical studies, pointing to an unexpected sensitivity of the heat transport efficiency to the precise distribution of heat sources and sinks, which greatly varies from planets to stars.

Keywords: geophysical and astrophysical fluid dynamics; rotating flows; turbulent convection.

Fig. 1

Radiatively driven rotating convection. A powerful spotlight shines from below at a mixture of water and dye. The resulting internal heat source decreases exponentially with height over the absorption length $\ell$, delivering a total heat flux P. The cylindrical tank is attached from above to a DC motor that imposes global rotation at a rate Ω (slight curvature of the top free surface not represented). Two thermocouples T₁ and T₂ measure the vertical temperature drop in the rotating frame, the data being communicated through WIFI to a remote Arduino microcontroller. On the right-hand side is a DNS snapshot of the temperature field in horizontally periodic geometry devoid of centrifugal and sidewall effects, highlighting the vertically elongated structures of rotating convection (${\text{Ra}}_{\text{P}}={10}^{12}$, $\text{E}=2\times {10}^{-6},\hspace{0.17em}\text{Pr}=7,\hspace{0.17em}\ell /H=0.048$, arbitrary color scale ranging from blue for cool fluid to red for warm fluid).

Fig. 2

Suppression of heat transport by global rotation. Heat transport efficiency Nu as a function of the Ekman number E, for various fluid heights: blue, H = 10 cm, ${\text{Ra}}_{\text{P}}\simeq 2.5\times {10}^{10}$; green, H = 15 cm, ${\text{Ra}}_{\text{P}}\simeq 1.3\times {10}^{11}$; red, H = 20 cm, ${\text{Ra}}_{\text{P}}\simeq 3.5\times {10}^{11}$; black, H = 25 cm, ${\text{Ra}}_{\text{P}}\simeq 9\times {10}^{11}$. The dimensionless absorption length is $\ell /H=0.024$ (filled circles) or $\ell /H=0.048$ (open squares). For fixed H and $\ell$, the mixing efficiency dramatically decreases with increasing rotation rate (decreasing E). Error bars are estimated from the values obtained for the first and second halves of the measurement interval; see Materials and Methods and SI Appendix.

Fig. 3

Observation of the geostrophic turbulence regime. (Top) In terms of the diffusivity-independent parameters $\mathcal{N}$ and $\mathcal{R}$, the data gathered for a given value of $\ell /H$ collapse onto a master curve, which validates the “fully turbulent” assumption. In the rapidly rotating regime $\mathcal{R}\lesssim 3\times {10}^{-7}$, the master curve displays a power-law behavior over one and a half decades in $\mathcal{R}$, in excellent agreement with the prediction $\mathcal{N}\approx {\mathcal{R}}^{3/5}$ associated with the geostrophic turbulence scaling regime of rapidly rotating convection (shown as an eye guide; see Table 1 for best-fit exponents). Same symbols as in Fig. 2 for the experimental data. The triangles are DNS data for ${\text{Ra}}_{\text{P}}={10}^{12},\hspace{0.17em}\text{Pr}=7$, and $\ell /H=0.048$. Experimental and numerical error bars are visible when larger than the symbol size. (Bottom) Same data compensated by the geostrophic turbulence scaling prediction. An approximate plateau is observed for $\mathcal{R}\lesssim 3\times {10}^{-7}$.

Fig. 4

Raw signals from thermocouples T₁ (red, z = 0) and T₂ (blue, $z=3H/4$) as a function of time t for H = 25 cm, Ω = 30 rpm, and $\ell /H=0.048$. Also shown are the instantaneous temperature drop between the two probes (black), and room temperature (green). The solid box indicates the total measurement interval, separated by a dashed line into two subintervals, I and II.