Convective stability of turbulent Boussinesq flow in the dissipative range and flow around small particles

We consider arbitrary, possibly turbulent, Boussinesq flow which is smooth below a dissipative scale ld. It is demonstrated that the stability of the flow with respect to growth of fluctuations with scale smaller than ld leads to a nontrivial constraint. That involves the dimensionless strength of fluctuations of the gradients of the scalar in the direction of gravity Fl and the Rayleigh scale L depending on the Rayleigh number Ra, the Nusselt number Nu, and ld. The constraint implies that the stratified fluid at rest, which is linearly stable, develops instability in the limit of large Ra. This limits observability of solution for the flow around small swimmer in quiescent stratified fluid that has closed streamlines at scale L [A. M. Ardekani and R. Stocker, Phys. Rev. Lett. 105, 084502 (2010)PRLTAO0031-900710.1103/PhysRevLett.105.084502]. Correspondingly, to study the flow at scale L one has to take turbulence into account. We demonstrate that the resulting turbulent flow around small particles or swimmers can be described by a scalar integro-differential advection-diffusion equation. Describing the solutions, we show that closed streamlines persist with finite probability. Our results seem to be the necessary basis in understanding flows around small particles and swimmers in natural marine environments.