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Strong turbulent convection: distributed chaos and large-scale circulation

Two types of spontaneous breaking of the space translational symmetry in distributed chaos have been considered for turbulent thermal convection at large values of Rayleigh number. First type is related to boundaries and second type is related to appearance of inertial range of scales. The first type is dominated by vorticity correlation integral: $\int_{V} \langle {\boldsymbol ω} ({\bf x},t) \cdot {\boldsymbol ω} ({\bf x} + {\bf r},t) \rangle_{V} d{\bf r}$ and is characterized by stretched exponential spectrum $\exp-(k/k_β)^{β}$ with $β=1/2$. The second type is dominated by energy correlation integral: $\int_{V} \langle {\bf u}^2 ({\bf x},t) ~ {\bf u}^2({\bf x} + {\bf r},t) \rangle_{V} d{\bf r}$ and is characterized by $β=3/5$. Good agreement has been established with laboratory experimental data obtained at large values of Rayleigh number $Ra \sim 10^{11}-10^{14}$ (the range relevant to solar photosphere) in upright cylinder cells. Taylor hypothesis transforms the wavenumber spectrum into frequency spectrum $\exp-(f/f_β)^{1/2}$. It is shown that turnover frequency of large-scale circulation (wind): $f_w = f_β/2$. Results of an experiment in horizontal cylinder are also briefly discussed. The analysis suggests that in this case the large-scale circulation can be considered as a natural (harmonic) part of the distributed chaos.