GSNP Dissertation Award: Critical Transitions in Turbulence

Turbulence is ubiquitous in the universe, and comprises a wide range of space and time scales. In three dimensions (3D), energy is transferred from large to small scales (direct cascade). In two dimensions (2D), the opposite is true (inverse cascade). Here I present results from direct numerical simulations (DNS) of the Navier-Stokes equations, modelling, asymptotics and methods from statistical physics pertaining to different scenarios where the largest scales in a turbulent flow change their properties abruptly at a critical parameter value. In each case, the physics close to the critical point is characterised in detail. First, rotating turbulence in an elongated domain is studied using an asymptotic expansion, with a single parameter combining layer height and rotation rate. At a critical parameter value, energy begins to be transferred inversely. Density stratification is found to impact the energy cascades nontrivially. The second scenario concerns turbulence in a thin layer of variable depth, forming large-scale vortices (LSV) below a critical height. The flow is studied numerically and a mean field model is proposed, explaining observed scaling laws. The transition to LSVs is shown to be subcritical. Third, a simplified model of 3D perturbations on 2D flow is presented. The model facilitates a stability analysis of LSV at a reduced cost, reproducing intermittent growth in perturbation amplitude recently observed in DNS. The perturbation growth rate fluctuates following a heavy-tailed distribution. The mathematical structure of the model is studied in detail using a Langevin equation with Lévy noise. Finally, exact results on the microcanonical statistical mechanics of truncated 2D Euler flows are presented. By evaluating phase space integrals, we compute the reversal statistics for the largest-scale mode in a square domain with free-slip boundaries. We validate the microcanonical results numerically by using a minimal model, in contrast with the canonical ensemble, which is shown to fail in this example.