Approaching the Asymptotic Regime of Rapidly Rotating Convection: Boundary Layers versus Interior Dynamics

Rapidly rotating Rayleigh-Bénard convection is studied by combining results from direct numerical simulations (DNS), laboratory experiments, and asymptotic modeling. The asymptotic theory is shown to provide a good description of the bulk dynamics at low, but finite Rossby number. However, large deviations from the asymptotically predicted heat transfer scaling are found, with laboratory experiments and DNS consistently yielding much larger Nusselt numbers than expected. These deviations are traced down to dynamically active Ekman boundary layers, which are shown to play an integral part in controlling heat transfer even for Ekman numbers as small as $10^{-7}$. By adding an analytical parametrization of the Ekman transport to simulations using stress-free boundary conditions, we demonstrate that the heat transfer jumps from values broadly compatible with the asymptotic theory to states of strongly increased heat transfer, in good quantitative agreement with no-slip DNS and compatible with the experimental data. Finally, similarly to nonrotating convection, we find no single scaling behavior, but instead that multiple well-defined dynamical regimes exist in rapidly rotating convection systems.

Figure 1

Nusselt number and interior temperature gradient at the midplane as a function of $\widetilde{\mathrm{Ra}}$. The filled symbols show DNS results at $E=10^{-7}$, while the open symbols are asymptotic predictions [7]. The stars show laboratory values obtained in a cylindrical rotating water tank with $5.79\le \mathrm{Pr}\le 6.2$ and $10^{-7}\le E\le 1.1\times 10^{-7}$.

Figure 2

Thermal anomaly $\theta =T-\overline{T}$ in DNS at $E=10^{-7}$ with no-slip boundary conditions. (a) Cellular regime, $\widetilde{\mathrm{Ra}}=10$, $\mathrm{Pr}=1$, (b) convective Taylor columns, $\widetilde{\mathrm{Ra}}=25$, $\mathrm{Pr}=7$, (c) plumes, $\widetilde{\mathrm{Ra}}=70$, $\mathrm{Pr}=7$, and (d) geostrophic turbulence, $\widetilde{\mathrm{Ra}}=90$, $\mathrm{Pr}=1$. For better visibility, the domain has been stretched horizontally by a factor of 4.5. $\theta$ is scaled with $\mathrm{\Delta }T$ in all cases.

Figure 3

Formation of large-scale, barotropic vortices of both signs of vorticity in the GT regime for stress-free boundaries at $E=10^{-7}$, $\widetilde{\mathrm{Ra}}=90$, $\mathrm{Pr}=1$. Shown is (a) vertical vorticity, (b) horizontal kinetic energy, (c) temperature anomaly at $z=0.99$, and (d) vertically averaged vertical vorticity. Units are $\kappa /H^2$, ${\kappa }^2/H^2$, $\mathrm{\Delta }T$, and $\kappa /H^2$, respectively.

Figure 4

Results obtained with parametrized Ekman pumping. Filled symbols denote no-slip cases, while open symbols show results obtained by applying the pumping boundary conditions (1) to the full equations. Vertical profiles for $E=10^{-7}$, $\mathrm{Pr}=7$, $\widetilde{\mathrm{Ra}}=20$ are shown in (c), with a boundary layer blowup in (d) for no-slip (solid) and Ekman pumping (dashed) boundary conditions. ($u_{\perp }$: horizontal velocity, $w$: vertical velocity, $T$: temperature, given in units of $\kappa /H$ and $\mathrm{\Delta }T$, respectively).

Figure 5

Ekman number dependence of Nu and pumping efficiency. Colored symbols represent no-slip DNS results (full symbols: $\mathrm{Pr}=1$, open symbols: $\mathrm{Pr}=7$), gray symbols show the asymptotic results. The panel in the lower left also contains experimental data (+: $2.56\times 10^{-8}\le E\le 2.77\times 10^{-8}$, *: $0.93\times 10^{-7}\le E\le 1.15\times 10^{-7}$, ×: $0.8\times 10^{-5}\le E\le 1.5\times 10^{-5}$, with $5\le \mathrm{Pr}\le 11$ in all cases.). The right panels show the rms vertical velocity close to the edge of the Ekman layer, measured at $z_E=3/4\pi \sqrt{2E}$, normalized by the rms vertical velocity at midlayer, a ratio we call $S$ here. Note that at $z=z_E$, classical Ekman layer theory predicts the pumping velocity to be identical to its far-field value, such that values at this location provide a good measure of Ekman pumping.