Modelling dispersion in stratified turbulent flows as a resetting process

In 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 (Kimura and Herring JFM Vol 328 1996). 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 such a stationary distribution for vertical displacement. 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 and Majumdar, PRL, Vol 106 2011) with memory. We assume that Lagrangian tracers in stratified turbulent flows 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. The model parameters are constrained by analysing the dynamics of an idealised density structure. Even though highly idealised, the model has the advantage of being analytically solvable. We show the emergence of a stationary distribution for the vertical displacement of Lagrangian tracers, as well as identify some instructive scalings.