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We consider arbitrary, possibly turbulent, Boussinesq flow which is smoothbelow a dissipative scale $l_d$. It is demonstrated that the stability of theflow with respect to growth of fluctuations with scale smaller than $l_d$ leadsto a non-trivial constraint. That involves the dimensionless strength $Fl$ offluctuations of the gradients of the scalar in the direction of gravity and theRayleigh scale $L$ depending on the Rayleigh number $Ra$, the Nusselt number$Nu$ and $l_d$. The constraint implies that the stratified fluid at rest, whichis linearly stable, develops instability in the limit of large $Ra$. Thislimits observability of solution for the flow around small swimmer in quiescentstratified fluid that has closed streamlines at scale $L$ [A. M. Ardekani andR. Stocker, Phys. Rev. Lett. 105, 084502 (2010)]. Correspondingly to study theflow at scale $L$ one has to take turbulence into account. We demonstrate thatthe resulting turbulent flow around small particles or swimmers can bedescribed by scalar integro-differential advection-diffusion equation.Describing the solutions we show that closed streamlines persist with finiteprobability. Our results seem to be the necessary basis in understanding flowsaround small swimmers.

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