An unusual bifurcation scenario in a stably stratified, valley-shaped enclosure heated from below

Abstract

We delineate the structure of steady laminar flows within a stably stratified, valley-shaped triangular cavity heated from below through linear stability analysis and Navier-Stokes simulations. We derive an exact solution to the quiescent conduction state, and characterize the flow via the stratification perturbation parameter, $\Pi_s$, which is a measure of the strength of the surface heat flux relative to the background stable stratification. Beyond a threshold value of $\Pi_s$, two unstable eigenmodes appear, one marked by a dominant central circulation, and the other one exhibiting dual circulations of equal strength. Through Navier-Stokes simulations, we confirm that the central-circulation eigenmode generates a pair of asymmetric steady states, whereas the dual-circulation eigenmode leads to distinct upslope and downslope symmetric steady states. Linear stability analysis and Navier-Stokes simulations jointly confirm the instability of the two symmetric steady states, both of which transition to the asymmetric steady state under a perturbation. Thus, for a given set of dimensionless parameters, the Navier-Stokes equations admit at least five possible steady-state solutions. Two of these solutions, namely the quiescent, pure conduction state and the counter-intuitive symmetric downslope state, have previously been overlooked in heated, stably stratified, valley-shaped enclosures. These five flow solutions reveal an intriguing bifurcation structure, including both a perfect pitchfork bifurcation and a nested bifurcation that gives rise to two distinct states. The inner bifurcation, while resembling a pitchfork in some respects, does not break any symmetry of the valley due to the lack of any possible horizontal axis of symmetry. The categorization of this inner bifurcation remains an unresolved matter, as it does not conform to any established descriptions of canonical bifurcations.