This paper examines the postulate of local isotropy in stratified
homogeneous turbulence from a theoretical point of view. The study is based on
a priori analysis of the evolution equations governing single-point turbulence
statistics that are formally consistent with the Navier--Stokes
equations. The Boussinesq approximation has been
utilized to account for the effect of buoyancy---a simplifying assumption which
constitutes an excellent approximation for the case considered here. The study
concludes that the hypothesis of local isotropy is formally inconsistent with
the Navier--Stokes equations in homogeneous
stratified turbulence. An estimate is provided that suggests that local
isotropy may constitute only a physically justifiable approximation in the limit
of a clear-cut separation between the time scales associated with the imposed
buoyancy and the turbulent eddy turnover time scale. This is unlikely to happen
in most flows, at least those not too far from equilibrium. The results also
suggest that the dynamical dependence of the small-scale turbulence on
large-scale anisotropies associated with imposed density stratification is
significantly stronger than that caused by an imposed mean straining.