Bounds to the Basset-Boussinesq force on particle laden stratified flows

Abstract

The Basset-Boussinesq force is often perfunctorily neglected when studying small inertial particles in turbulence. This force arises from the diffusion of vorticity from the particles and, since it depends on the particles' past history, complicates the dynamics by transforming their equations of motion into integro-differential equations. However, this force is of the same order as other viscous forces acting on the particles, and beyond convenience, the reasons for neglecting it are unclear. This study addresses the following question: Under what conditions can the Basset-Boussinesq force be neglected in light particles in geophysical flows? We derive strict bounds for the magnitude of the Basset-Boussinesq force in stably stratified flows, in contexts of interest for geophysical turbulence. The bounds are validated by direct numerical simulations. The Basset-Boussinesq force is negligible when a buoyancy Stokes number $\textrm{Sb} = N \tau_p$ is small, where $N$ is the flow Brunt-V\"ais\"al\"a frequency and $\tau_p$ is the particle's Stokes time. Interestingly, for most oceanic particles this force may be negligible. Only for very strong stratification, or for particles with very large inertia, this force must be considered in the dynamics.