Universality class of turbulent transitions in stably stratified flows

In geophysical shear flows, ocean thermoclines, atmospheric boundary layers, stable density stratification suppresses vertical motion and competes with shear-driven turbulence, controlling whether turbulence is sustained or collapses to a laminar state. The laminar-turbulent transition in unstratified shear flow is now understood as a directed percolation (DP) phase transition: the turbulent fraction vanishes continuously at a critical Reynolds number $Re_c$ with universal scaling exponents. Whether stratification, by selectively damping the vertical fluctuations that sustain turbulent patches, shifts $Re_c$, modifies these exponents, or drives the transition into a different universality class entirely is an open question with both fundamental and geophysical significance.

We investigate this in stratified Waleffe flow, a minimal model of geophysically relevant stratified shear turbulence. A key challenge is that stratification breaks the wall-normal mode truncation that made large-domain DP simulations feasible in the unstratified case (Chantry, Tuckerman & Barkley 2017), forcing computations into a finite-size regime where the critical point is shifted and clean scaling is difficult to resolve. We develop finite-size scaling and Binder cumulant methods to disentangle finite-domain artefacts from genuine stratification-induced modifications to the critical behaviour, and thereby determine whether stratification is a relevant perturbation to DP universality at the turbulent onset.