  This work addresses the question of the stability of stratified, spatially
periodic shear flows at low P\'eclet number but high Reynolds number. This
little-studied limit is motivated by astrophysical systems, where the Prandtl
number is often very small. Furthermore, it can be studied using a reduced set
of "low-P\'eclet-number equations" proposed by Lignieres [Astronomy &
Astrophysics, 348, 933-939, 1999]. Through a linear stability analysis, we
first determine the conditions for instability to infinitesimal perturbations.
We formally extend Squire's theorem to the low-P\'eclet-number equations, which
shows that the first unstable mode is always two-dimensional. We then perform
an energy stability analysis of the low-P\'eclet-number equations and prove
that for a given value of the Reynolds number, above a critical strength of the
stratification, any smooth periodic shear flow is stable to perturbations of
arbitrary amplitude. In that parameter regime, the flow can only be laminar and
turbulent mixing does not take place. Finding that the conditions for linear
and energy stability are different, we thus identify a region in parameter
space where finite-amplitude instabilities could exist. Using direct numerical
simulations, we indeed find that the system is subject to such finite-amplitude
instabilities. We determine numerically how far into the linearly stable region
of parameter space turbulence can be sustained.
