Multiple Transitions in Rotating Turbulent Rayleigh-Bénard Convection

Sometimes it is thought that sharp transitions between potentially different turbulent states should be washed out by the prevailing intense fluctuations and short coherence lengths and times. Contrary to this expectation, we found a sequence of such transitions in turbulent rotating Rayleigh-Bénard convection as the rotation rate was increased. This phenomenon was observed in cylindrical samples with aspect ratios (diameter/height) $\mathrm{\Gamma }=1.00$ and 0.50. It became most prominent at very large Rayleigh numbers up to $2\times {10}^{12}$, where the fluctuations are extremely vigorous, and was manifested most clearly for $\mathrm{\Gamma }=1.00$. It was found in the heat transport as well as in the temperature gradient near the sample center. We conjecture that the transitions are between different large-scale structures which involve changes of symmetry and thus cannot be gradual [L. Landau, Zh. Eksp. Teor. Fiz. 7, 19 (1937); L. D. Landau, Phys. Z. Sowjetunion 11, 26 (1937); L. D. Landau, in Collected Papers of L. D. Landau, (Oxford University Press, Oxford, 1965), pp. 193–216].

Figure 1

(a) ${\mathrm{Nu}}_r$, and (b) the temperature gradient at the sample center, both as a function of $1/\mathrm{Ro}$. Here and in subsequent figures the dashed lines are straight-line fits to various segments of data and serve as guides to the eye. Black vertical dotted lines, from left to right: the location of $1/{\mathrm{Ro}}_c=0.280$, $1/{\mathrm{Ro}}_{c2}=0.492$, and $1/{\mathrm{Ro}}_{c,3}\simeq 1.55$, respectively. Solid black circles: $\mathrm{Ra}=2.07\times {10}^{11}$ [$\mathrm{Nu}(0)=344$]. Open red circles: $\mathrm{Ra}=1.00\times {10}^{11}$ [$\mathrm{Nu}(0)=273$].

Figure 2

The reduced Nusselt numbers ${\mathrm{Nu}}_r$ as a function of $1/\mathrm{Ro}$ for water ($\mathrm{Pr}=4.38$), $\mathrm{\Gamma }=1.00$, $\mathrm{Ra}=2.3\times {10}^9$ (open purple diamonds), $\mathrm{Ra}=9.0\times {10}^9$ (open red circles), and $\mathrm{Ra}=1.8\times {10}^{10}$ (solid circles). Based on data from Ref. [30].

Figure 3

An example of the reduced Nusselt number ${\mathrm{Nu}}_r$ as a function of $1/\mathrm{Ro}$ for water over a wider range of $1/\mathrm{Ro}$ than in Fig. 2. For this case, $\mathrm{Pr}=4.38$ and $\mathrm{Ra}=1.8\times {10}^{10}$. Based on data from Ref. [30].

Figure 4

Results for $1/{\mathrm{Ro}}_c$ (solid symbols), $1/{\mathrm{Ro}}_{c,2}$ (open symbols), and $1/{\mathrm{Ro}}_{c,3}$ (open symbols with stars) as a function of Ra. Blue triangles: $\mathrm{Pr}=3.62$. Black circles: $\mathrm{Pr}=4.38$. Red squares: $\mathrm{Pr}=6.26$. Purple diamonds: $\mathrm{Pr}=12.3$.

Figure 5

The reduced Nusselt number ${\mathrm{Nu}}_r$ as a function of $1/\mathrm{Ro}$ for $\mathrm{\Gamma }=0.50$, $\mathrm{Pr}=12.3$, $\mathrm{Ra}=1.66\times {10}^{12}$ [solid black circles, $\mathrm{Nu}(0)=680$], and $\mathrm{Ra}=8.0\times {10}^{11}$ [open red circles, $\mathrm{Nu}(0)=540$]. The vertical dotted lines indicate the locations of $1/{\mathrm{Ro}}_c=0.612$ and $1/{\mathrm{Ro}}_{c,2}=2.10$.

Figure 6

The reduced Nusselt number ${\mathrm{Nu}}_r$ as a function of $1/\mathrm{Ro}$ on two different scales for $\mathrm{\Gamma }=0.50$, $\mathrm{Pr}=4.38$, and $\mathrm{Ra}=2.3\times {10}^9$. The vertical dotted lines, from left to right, indicate the location of $1/{\mathrm{Ro}}_c=0.85$, $1/{\mathrm{Ro}}_{c2}=2.38$, $1/{\mathrm{Ro}}_{c3}=4.53$, and $1/{\mathrm{Ro}}_{c4}=16.5$. The data are from Ref. [8].