Boundary layers in thermal convection are fluctuation-dominated*

We study the dynamics of thermal and momentum boundary regions in three-dimensional direct numerical simulations of Rayleigh-B\'{e}nard convection for the Rayleigh number range $10^5\le Ra \le 10^{11}$ and $Pr=0.7$. Using a Cartesian slab with horizontal periodic boundary conditions and an aspect ratio of 4, we obtain statistical homogeneity in the horizontal $x$- and $y$-directions, thus approximating an infinitely extended system. We also supplement these results by similar simulations for aspect ratios of 2 and 8 at $Ra = 10^9$. We observe upon canonical use of long-time and area averages, with averaging periods of at least 100 free-fall times, that a coherent mean flow is practically absent and that the magnitude of the velocity fluctuations is larger than the mean by up to 2 orders of magnitude. The velocity field close to the wall is a collection of differently oriented local shear-dominated patches interspersed with shear-free incoherent flow regions. The incoherent regions occupy a 60 % area fraction for all Rayleigh numbers. Rather than resulting in a pronounced mean with small fluctuations about this mean, the velocity field is dominated by strong fluctuations of all three components around a non-existent or weak mean. This feature is particularly pronounced for $Ra \ge 10^9$, which underlines the necessity for large aspect ratios and high Rayleigh numbers. We discuss the consequences of these observations for convection layers with larger aspect ratios, including boundary layer instabilities and the resulting turbulent heat transport.