Modelling dispersion in stratified turbulent flows as a resetting process

In freely decaying stably stratified turbulent flows, numerical evidence shows that the horizontal displacement of Lagrangian tracers is diffusive while the vertical displacement converges towards a stationary distribution, as shown numerically by Kimura & Herring ( J. Fluid Mech. , vol. 328, 1996, pp. 253–269). Here, we develop a stochastic model for the vertical dispersion of Lagrangian tracers in stably stratified turbulent flows that aims to replicate and explain the emergence of a stationary probability distribution for the vertical displacement of such tracers. More precisely, our model is based on the assumption that the dynamical evolution of the tracers results from the competing effects of buoyancy forces that tend to bring a vertically perturbed fluid parcel (carrying tracers) to its equilibrium position and turbulent fluctuations that tend to disperse tracers. When the density of a fluid parcel is allowed to change due to molecular diffusion, a third effect needs to be taken into account: irreversible mixing. Indeed, ‘mixing’ dynamically and irreversibly changes the equilibrium position of the parcel and affects the buoyancy force that ‘stirs’ it on larger scales. These intricate couplings are modelled using a stochastic resetting process (Evans & Majumdar, Phys. Rev. Lett. , vol. 106, issue 16, 2011, 160601) with memory. More precisely, Lagrangian tracers in stratified turbulent flows are assumed to follow random trajectories that obey a Brownian process. In addition, their stochastic paths can be reset to a given position (corresponding to the dynamically changing equilibrium position of a density structure containing the tracers) at a given rate. Scalings for the model parameters as functions of the molecular properties of the fluid and the turbulent characteristics of the flow are obtained by analysing the dynamics of an idealised density structure. Even though highly idealised, the model has the advantage of being analytically solvable. In particular, we show the emergence of a stationary distribution for the vertical displacement of Lagrangian tracers. We compare the predictions of this model with direct numerical simulation data at various Prandtl numbers $Pr$ , the ratio of kinematic viscosity to molecular diffusion.