# Rare transitions to thin-layer turbulent condensates

## TL;DR

This study investigates rare transitions between 2-D condensate and 3-D flow states in thin-layer turbulence, revealing exponential distributions of transition times and modeling the dynamics with a stochastic Langevin equation near the critical layer height.

## Contribution

It provides a detailed statistical analysis of bistable transitions in thin-layer turbulence and introduces a stochastic model to describe the large-scale energy dynamics near the critical threshold.

## Key findings

- Transition times follow exponential distributions with mean values increasing rapidly near the threshold.
- The large-scale kinetic energy dynamics can be modeled by a stochastic Langevin equation.
- Effective potential analysis reveals two minima corresponding to 2-D and 3-D states.

## Abstract

Turbulent flows in a thin layer can develop an inverse energy cascade leading to spectral condensation of energy when the layer height is smaller than a certain threshold. These spectral condensates take the form of large-scale vortices in physical space. Recently, evidence for bistability was found in this system close to the critical height: depending on the initial conditions, the flow is either in a condensate state with most of the energy in the two-dimensional (2-D) large-scale modes, or in a three-dimensional (3-D) flow state with most of the energy in the small-scale modes. This bistable regime is characterised by the statistical properties of random and rare transitions between these two locally stable states. Here, we examine these statistical properties in thin-layer turbulent flows, where the energy is injected by either stochastic or deterministic forcing. To this end, by using a large number of direct numerical simulations (DNS), we measure the decay time $\tau_d$ of the 2-D condensate to 3-D flow state and the build-up time $\tau_b$ of the 2-D condensate. We show that both of these times $\tau_d,\tau_b$ follow an exponential distribution with mean values increasing faster than exponentially as the layer height approaches the threshold. We further show that the dynamics of large-scale kinetic energy may be modeled by a stochastic Langevin equation. From time-series analysis of DNS data, we determine the effective potential that shows two minima corresponding to the 2-D and 3-D states when the layer height is close to the threshold.