LAUSR

Abstract Turbulent flows driven by a vertically invariant body force were proven to become exactly two-dimensional (2-D) above a critical rotation rate, using upper bound theory. This transition in dimensionality of a turbulent flow has key consequences for the energy dissipation rate. However, its location in parameter space is not provided by the bounding procedure. To determine this precise threshold between exactly two-dimensional and partially three-dimensional (3-D) flows, we perform a linear stability analysis over a fully turbulent 2-D base state. This requires integrating numerically a quasi-2-D set of equations over thousands of turnover times, to accurately average the growth rate of the 3-D perturbations over the statistics of the turbulent 2-D base flow. We leverage the capabilities of modern graphics processing units to achieve this task, which allows us to investigate the parameter space up to $Re=10^5$. At the Reynolds numbers typical of 3-D direct numerical simulations and laboratory experiments, $Re\in [10^2, 5\times 10^3]$, the turbulent 2-D flow becomes unstable to 3-D motion through a centrifugal-type instability. However, at even higher Reynolds numbers, another instability takes over. A candidate mechanism for the latter instability is the parametric excitation of inertial waves by the modulated 2-D flow, a phenomenon that we illustrate with an oscillatory 2-D Kolmogorov flow.