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

We report experimental measurements of heat transport in rotating Rayleigh-Bénard convection in a cylindrical convection cell with an aspect ratio of $\mathrm{\Gamma }=1/2$. The fluid is helium gas with a Prandtl number $\mathrm{Pr}=0.7$. The range of control parameters for Rayleigh numbers $4\times {10}^9<\text{Ra}<4\times {10}^{11}$ and for Ekman numbers $2\times {10}^{-7}<\text{Ek}<3\times {10}^{-5}$ (corresponding to Taylor numbers $4\times {10}^9<\mathrm{Ta}<1\times {10}^{14}$ and convective Rossby numbers $0.07<\text{Ro}<5$). We determine the transition from weakly rotating turbulent convection to rotation dominated geostrophic convection through experimental measurements of the heat transport Nu. The heat transport, best collapsed using a parameter ${\mathrm{RaEk}}^{\beta }$ with $1.65<\beta <1.8$, defines two boundaries in the phase diagram of ${\text{Ra}/\text{Ra}}_c$ versus Ek and elucidates properties of the geostrophic turbulence regime of rotating thermal convection. We find $\mathrm{Nu}\sim {(\text{Ra}/{\text{Ra}}_c)}^{\gamma }$ with $\gamma \approx 1$ from direct measurement and $1.2<\gamma <1.6$ inferred from scaling arguments.

Figure 1

$\mathrm{Nu}/{\mathrm{Nu}}_0$ vs Ra for constant Ek: $1.1\times {10}^{-6}$ (solid diamond, gray), $5.9\times {10}^{-7}$ (solid square, blue), $3.1\times {10}^{-7}$ (solid circle, black), $2.1\times {10}^{-7}$ (solid up triangle, red), and for constant $\text{Ra}=6.2\times {10}^9$ (solid down triangle, gray). Dashed lines are guides to the eye.

Figure 2

Log-Log plot of $\mathrm{Nu}/{\mathrm{Nu}}_0$ vs Ro for constant Ek (${\mathrm{Nu}}_0=0.121{\text{Ra}}^{0.309}$): $1.1\times {10}^{-6}$ (solid diamond, gray), $5.9\times {10}^{-7}$ (solid square, blue), $3.1\times {10}^{-7}$ (solid circle, black), $2.1\times {10}^{-7}$ (solid up triangle, red), and for constant Ra: $6.2\times {10}^9$ (solid down triangle, gray), DNS [9]—$1\times {10}^8$ (open square, black). The solid lines show approximate power law variations in the range $0.8<{\mathrm{Nu}/\mathrm{Nu}}_0<1$ with ${\mathrm{Nu}\sim \text{Ra}}^{0.45}$ (top, blue) and for ${\mathrm{Nu}/\mathrm{Nu}}_0<0.8$ with $\mathrm{Nu}\sim {\text{Ra}}^{1.1}$—(bottom, red), respectively. Vertical arrows indicate approximate transition values.

Figure 3

Log-Log plot of $\mathrm{Nu}/{\mathrm{Nu}}_0$ vs ${\mathrm{RaEk}}^{7/4}$ for constant Ek: $1.1\times {10}^{-6}$ (solid diamond, gray), $5.9\times {10}^{-7}$ (solid square, blue), $3.1\times {10}^{-7}$ (solid circle, black), $2.1\times {10}^{-7}$ (solid up triangle, red), and for constant Ra: $6.2\times {10}^9$ (solid down triangle, gray), DNS [9]—$1\times {10}^8$ (open square, black). The solid lines show approximate power law variations of the region $0.8<{\mathrm{Nu}/\mathrm{Nu}}_0<1$ with $({\mathrm{RaEk}}^{7/4}{)}^{1/7}$ (top, blue) and for ${\mathrm{Nu}/\mathrm{Nu}}_0<0.8$ with $({\mathrm{RaEk}}^{7/4}{)}^{2/3}$—$\mathrm{Nu}\sim \text{Ra}$—(bottom, red), respectively. Vertical arrows indicates approximate transition values.

Figure 4

Phase diagram of rotating convection in parameters $\text{Ra}/{\text{Ra}}_c$ and Ek. Rotation first affects turbulent convection below line $A$ for $\mathrm{Pr}=6$ (${\text{Ro}}_T=2$) and below line $B$ for $\mathrm{Pr}=0.7$ (${\text{Ro}}_T=0.35$). The crossover to geostrophic turbulence for $\mathrm{Pr}=0.7$ occurs along line $C$ where ${\text{Ra}}_t=0.25{\text{Ek}}^{-1.8}$ (the dotted line below $C$ is the lower limit of data presented here) whereas line $D$ describes the higher Pr data with ${\text{Ra}}_t=1.3{\text{Ek}}^{-1.65}$. Line $E$ indicates $\text{Ra}/{\text{Ra}}_c=3$ which is the rough upper bound to a regime of weakly nonlinear convection near onset ($\text{Ra}/{\text{Ra}}_c=1$). Data are from Ref. [8] (solid square—red), Refs. [9, 11] [open (blue) and solid (black) up triangles], Ref. [10] (solid down triangles, black), Ref. [25] (open diamond, blue), and this Letter (open and solid circles, red). The shaded region corresponds to states where geostrophic turbulence may be accessible.