Touching the classical scaling in penetrative convection

The Cassini missions have identified the tiger stripes on Enceladus as the source of both thermal emission and plume jets. The hot spots in the tiger stripes are highly localized, and the plumes suggest active hydrothermal processes within the subglacial ocean of Enceladus. However, understanding the mechanism responsible for the heat anomalies in the tiger stripes remains a challenge. About 60 y ago, geoscientist George Veronis proposed a model for cold water oceans, along with the classical notion of a 1/3 scaling relationship between vertical heat transfer and the Rayleigh number (Ra), a dimensionless number representing the strength of buoyancy driving the convection within the fluid body. In this study, by delving into the steady coherent rolls arising in Veronis' model, we first confirm the existence of the classical scaling as proposed. We identify two distinct pathways, characterized by different flow patterns, that converge to the unified classical scaling. In scenarios where stratification resulting from the density's nonlinear temperature dependence is insignificant, convection rolls with fixed (width-to-height) aspect ratios achieve the classical scaling as Ra approaches infinity. Conversely, under high stratification levels, convection rolls with a heat-transport-maximizing aspect ratio achieve the classical scaling. This investigation also reveals a notable correspondence between optimal coherent rolls and turbulent flow patterns. Based on these coherent rolls, we offer insights into the formation of heat anomalies in the tiger stripes. The predicted heat flux through the ocean and the lateral flow length, are consistent with the measurements obtained by the Cassini spacecraft.