Steady Rayleigh–Bénard convection between no-slip boundaries

The central open question about Rayleigh–Bénard convection – buoyancy-driven flow in a fluid layer heated from below and cooled from above – is how vertical heat flux depends on the imposed temperature gradient in the strongly nonlinear regime where the flows are typically turbulent. The quantitative challenge is to determine how the Nusselt number $Nu$ depends on the Rayleigh number $Ra$ in the $Ra\to \infty$ limit for fluids of fixed finite Prandtl number $Pr$ in fixed spatial domains. Laboratory experiments, numerical simulations and analysis of Rayleigh's mathematical model have yet to rule out either of the proposed ‘classical’ $Nu \sim Ra^{1/3}$ or ‘ultimate’ $Nu \sim Ra^{1/2}$ asymptotic scaling theories. Among the many solutions of the equations of motion at high $Ra$ are steady convection rolls that are dynamically unstable but share features of the turbulent attractor. We have computed these steady solutions for $Ra$ up to $10^{14}$ with $Pr=1$ and various horizontal periods. By choosing the horizontal period of these rolls at each $Ra$ to maximize $Nu$ , we find that steady convection rolls achieve classical asymptotic scaling. Moreover, they transport more heat than turbulent convection in experiments or simulations at comparable parameters. If heat transport in turbulent convection continues to be dominated by heat transport in steady rolls as $Ra\to \infty$ , it cannot achieve the ultimate scaling.

Stellar Turbulent Convection: The Multiscale Nature of the Solar Magnetic Signature

The multiscale dynamics associated with turbulent convection present in physical systems governed by very high Rayleigh numbers still remains a vividly disputed topic in the community of astrophysicists, and in general, among physicists dealing with heat transport by convection. The Sun is a very close star for which detailed observations and estimations of physical properties on the surface, connected to the processes of the underlying convection zone, are possible. This makes the Sun a unique natural laboratory in which to investigate turbulent convection in the hard turbulence regime, a regime typical of systems characterized by high values of the Rayleigh number. In particular, it is possible to study the geometry of convection using the photospheric magnetic voids (or simply voids), the quasi-polygonal quiet regions nearly devoid of magnetic elements, which cover the whole solar surface and which form the solar magnetic network. This work presents the most extensive statistics, both in the spatial scales studied (1–80 Mm) and in the temporal duration (SC 23 and SC 24), to investigate the multiscale nature of solar magnetic patterns associated with the turbulent convection of our star. We show that the size distribution of the voids, in the 1–80 Mm range, for the 317,870 voids found in the 692 analyzed magnetograms, is basically described by an exponential function.