Finite-Size Effects Lead to Supercritical Bifurcations in Turbulent Rotating Rayleigh-Bénard Convection

In turbulent thermal convection in cylindrical samples with an aspect ratio $\Gamma \equiv D/L$ ($D$ is the diameter and $L$ the height), the Nusselt number Nu is enhanced when the sample is rotated about its vertical axis because of the formation of Ekman vortices that extract additional fluid out of thermal boundary layers at the top and bottom. We show from experiments and direct numerical simulations that the enhancement occurs only above a bifurcation point at a critical inverse Rossby number $1/{\mathrm{Ro}}_c$, with $1/{\mathrm{Ro}}_c\propto 1/\Gamma$. We present a Ginzburg-Landau–like model that explains the existence of a bifurcation at finite $1/{\mathrm{Ro}}_c$ as a finite-size effect. The model yields the proportionality between $1/{\mathrm{Ro}}_c$ and $1/\Gamma$ and is consistent with several other measured or computed system properties.

Figure 1

The Nusselt number $\mathrm{Nu}(1/\mathrm{Ro})$, normalized by $\mathrm{Nu}(0)$ without rotation, as a function of the inverse Rossby number $1/\mathrm{Ro}$. Data are from experiments unless mentioned otherwise. (a) $\Gamma =0.50$ and $\mathrm{Pr}=4.38$; Rayleigh numbers $\mathrm{Ra}=9.0\times {10}^9$ (solid circles, red online), $1.8\times {10}^{10}$ (open circles, purple online), and $3.6\times {10}^{10}$ (solid squares, blue online). (b) $\Gamma =1.00$. Main figure: $\mathrm{Pr}=4.38$, $\mathrm{Ra}=2.25\times {10}^9$ (solid circles, red online), $8.97\times {10}^9$ (open circles, purple online), and $1.79\times {10}^{10}$ (solid squares, blue online). Inset: $\mathrm{Pr}=6.26$ and $\mathrm{Ra}=2.73\times {10}^8$. DNS: open squares (black online). Experiment: solid circles (red online). (c) $\Gamma =2.00$ and $\mathrm{Pr}=4.38$; the data are for $\mathrm{Ra}=2.91\times {10}^8$ (open circles, red online, DNS; solid circles, purple online, experiment), $5.80\times {10}^8$ (open squares, blue online), and $1.16\times {10}^9$ (solid squares, black online). Note the different vertical scale for (c) compared to (a) and (b). The small vertical lines indicate the locations of the bifurcation points.

Figure 2

The critical inverse Rossby number $1/{\mathrm{Ro}}_c$ as a function of the inverse aspect ratio $1/\Gamma$ for $\mathrm{Pr}=4.38$ and different Ra (see Fig. 1). The line is a fit of a quadratic equation with the constant coefficient set to zero.

Figure 3

The vortices as identified by the $Q_{2D}$ criterion [3, 21] for $\mathrm{Ra}=2.73\times {10}^8$, $\mathrm{Pr}=6.26$, and $\Gamma =1$. (a) $1/\mathrm{Ro}=2/3$, (b) $1/\mathrm{Ro}=1$, (c) $1/\mathrm{Ro}=1.54$, and (d) $1/\mathrm{Ro}=3.33$. The vortex area increases with increasing $1/\mathrm{Ro}$. This trend is quantified in Fig. 4.

Figure 4

Fraction $\overline{A}$ of a horizontal slice covered with vortices at the edge of the kinetic BL (circles) and at a distance $0.023L$ (the kinetic BL thickness without rotation, squares) from the plates as a function of $1/\mathrm{Ro}$ for $\mathrm{Ra}=2.73\times {10}^8$, $\mathrm{Pr}=6.26$, and $\Gamma =1$. The vertical dashed line indicates the bifurcation point at $1/{\mathrm{Ro}}_c$, and the horizontal dash-dotted line is a background vorticity level ${\overline{A}}_0$ present even below $1/{\mathrm{Ro}}_c$.

Figure 5

Azimuthal average $⟨A{⟩}_{\varphi }(r/L)$ of the vortex density $A$ for $\mathrm{Ra}=2.73\times {10}^8$, $\mathrm{Pr}=6.26$, $\Gamma =1$, and $2.22<1/\mathrm{Ro}<3.33$. In total, the statistics is based on 8 snapshots. The dashed line is the uniform case. The density approaches zero close to the side wall.