We consider the dynamics of a vertically stratified, horizontally-forced
Kolmogorov flow. Motivated by astrophysical systems where the Prandtl number is
often asymptotically small, our focus is the little-studied limit of high
Reynolds number but low P\'eclet number (which is defined to be the product of
the Reynolds number and the Prandtl number). Through a linear stability
analysis, we demonstrate that the stability of two-dimensional modes to
infinitesimal perturbations is independent of the stratification, whilst
three-dimensional modes are always unstable in the limit of strong
stratification and strong thermal diffusion. The subsequent nonlinear evolution
and transition to turbulence is studied numerically using direct numerical
simulations. For sufficiently large Reynolds numbers, four distinct dynamical
regimes naturally emerge, depending upon the strength of the background
stratification. By considering dominant balances in the governing equations, we
derive scaling laws for each regime which explain the numerical data
