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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 $l_d$. It is demonstrated that the stability of the flow with respect to growth of fluctuations with scale smaller than $l_d$ leads to a non-trivial constraint. That involves the dimensionless strength $Fl$ of fluctuations of the gradients of the scalar in the direction of gravity and the Rayleigh scale $L$ depending on the Rayleigh number $Ra$, the Nusselt number $Nu$ and $l_d$. 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)]. 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 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 swimmers.