  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.
