Dynamical stability and flow regimes in a stably stratified valley-shaped cavity heated from below

We investigate the three-dimensional stability of a stably stratified fluid in a valley-shaped cavity heated from below using linear stability analysis and direct numerical simulations. We first describe the pure-conduction flow state and derive a dimensionless criterion that provides a lower bound for the onset of instability, valid for any slope angle. We then examine the sequence of flow regimes for a slope angle of $\alpha = 30^{\circ}$ and Prandtl number $Pr = 7$, including two-dimensional steady states, the emergence of a Hopf bifurcation, and the formation of steady and oscillatory three-dimensional structures preceding the transition to fully unsteady, chaotic flow. Although the nonlinear governing equations depend on two dimensionless parameters, we find that the flow dynamics across a wide parameter range collapse to depend on a single parameter--the composite stratification parameter $\Pi_c$. However, as the system becomes more unstable, sensitivity to the second parameter, $\Pi_h$, increases. We construct a regime map of all observed flow states as a function of $\Pi_c$ and $\Pi_h$, and confirm the onset of chaos using Lyapunov exponents. Across all regimes, asymmetric circulation remains the dominant flow structure, persisting even in time-averaged fields of chaotic states. Finally, we characterize heat transfer in the cavity using the Nusselt number, which scales as $Nu \sim \Pi_c^{0.43}$ or equivalently $Nu \sim Ra^{0.275}$. This result further establishes $\Pi_c$ as a key dimensionless parameter governing the flow dynamics preceding the chaotic regime.