  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.
