The combined effects of horizontal and vertical shear instabilities at low Prandtl number

Recent work has demonstrated the importance of horizontal shear in driving vertical transport in stratified fluids, especially at low Prandtl number (which is relevant in the interiors of stars and planets). Various regions of parameter space have been identified, each with its own set of scaling laws governing the dominant balance of term in the momentum and buoyancy equations, respectively. An important outstanding question of interest, however, is that of the combined effects of large-scale horizontal and vertical shear, which are often present at the same time in stars. In this paper, we demonstrate that vertical shear only influences the outcome provided it is stronger than the emergent vertical shear developing between unstable meanders of the horizontal flow. As such, it can be shown that the latter is not likely to play a role in most stellar interior conditions, and that theories that only incorporate transport by large-scale horizontal shear flows are sufficient when modeling stellar mixing.