Rigorous ultimate scaling in rapidly rotating steady convection

TL;DR

This paper rigorously analyzes steady solutions in rapidly rotating convection, deriving explicit scaling laws for heat and flow transport that approach ultimate theoretical limits through columnar structures.

Contribution

It provides a rigorous asymptotic characterization of steady convection solutions, revealing how they attain ultimate heat transport scalings with specific horizontal scales.

Findings

Derives explicit scaling laws for Nusselt and Reynolds numbers.

Shows solutions can reach diffusivity-free ultimate scalings with logarithmic corrections.

Identifies mechanisms for rapid rotation convection to approach ultimate heat transport.

Abstract

Rapidly rotating Rayleigh-B\'enard convection admits a class of exact steady single-mode solutions describing high-amplitude convection cells. Using a matched asymptotic analysis in the high-Rayleigh-number limit, we obtain a rigorous characterization of their bulk and boundary-layer structure, yielding explicit scaling laws for the Nusselt and Reynolds numbers, including their dependence on the horizontal wavenumber. We show that, for suitable wavenumbers, these solutions attain the diffusivity-free ultimate scalings frequently assumed for geophysical and astrophysical convection, with additional enhancing logarithmic corrections. This reveals a specific mechanism through which rapidly rotating convection can approach ultimate heat transport via coherent columnar structures with well-defined horizontal scales.