Abstract
Turbulent convection processes in nature are often found to be organized in a hierarchy of plume structures and flow patterns. The gradual aggregation of convection cells or granules to a supergranule which eventually fills the whole horizontal layer is reported and analysed in spectral element direct numerical simulations of three-dimensional turbulent Rayleigh-B\'{e}nard convection at an aspect ratio of $60$. The formation proceeds over a time span of more than $10^4$ convective time units for the largest accessible Rayleigh number and occurs only when the turbulence is driven by a constant heat flux which is imposed at the bottom and top planes enclosing the convection layer. The resulting gradual inverse cascade process is observed for both temperature variance and turbulent kinetic energy. An additional analysis of the leading Lyapunov vector field for the full turbulent flow trajectory in its high-dimensional phase space demonstrates that turbulent flow modes at a certain scale continue to give rise locally to modes with longer wavelength in the turbulent case. As a consequence successively larger convection patterns grow until the horizontal extension of the layer is reached. This instability mechanism, which is known to exist near the onset of constant heat flux-driven convection, is shown here to persist into the fully developed turbulent flow regime thus connecting weakly nonlinear pattern formation with the one in fully developed turbulence. We discuss possible implications of our study for observed, but not yet consistently numerically reproducible, solar supergranulation which could lead to improved simulation models of surface convection in the Sun.
Highlights
Turbulent convection, the essential mechanism by which heat is transported in natural flows, manifests often in a hierarchy of structures and flow patterns
We show that the supergranule is absent in direct numerical simulations (DNS) with constant temperature boundary conditions at the top and bottom planes of the Rayleigh-Bénard convection (RBC) layer
Our study suggests that the mechanisms of supergranule formation in a simple convection flow are related to linear instabilities in the turbulent flow that give rise to longerwavelength structures
Summary
The essential mechanism by which heat is transported in natural flows, manifests often in a hierarchy of structures and flow patterns. We show that the supergranule is absent in DNS with constant temperature boundary conditions at the top and bottom planes of the RBC layer For these cases, the formation of the recently comprehensively investigated turbulent superstructures [28,29,30,31,32,33,34,35,36,37,38]—well-ordered patterns of temperature and velocity with characteristic convection roll widths up to /2 ∼ 3 − 4H [36,37,38]—takes place. Instabilities of finite-amplitude convection rolls for Rayleigh numbers slightly above Rac showed that each mode is unstable to one longer wavelength [22] This gradual aggregation process has not been observed in previous turbulent RBC simulations with constant flux boundary conditions [44,45,46], most probably because they were conducted in smaller aspect ratio domains and for shorter total integration times. Our investigation can shed light on the fundamentals of solar granulation processes
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