  We perform linear and nonlinear stability analysis for thermal convection in
a fluid overlying a saturated porous medium. We use a coupled system, with the
Navier-Stokes equations and Darcy's equation governing the free-flow and the
porous regions respectively. Incorporating a dynamic pressure term in the Lions
interface condition (which specifies the normal force balance across the
fluid-medium interface) permits an energy bound on the typically uncooperative
nonlinear advection term, enabling new nonlinear stability results. Within
certain regimes, the nonlinear stability thresholds agree closely with the
linear ones, and we quantify the differences that exist. We then compare
stability thresholds produced by several common variants of the tangential
interface conditions, using both numerics and asymptotics in the small Darcy
number limit. Finally, we investigate the transition between full convection
and fluid-dominated convection using both numerics and a heuristic theory. This
heuristic theory is based on comparing the ratio of the Rayleigh number in each
domain to its corresponding critical value, and it is shown to agree reasonably
well with the numerics regarding how the transition depends on the depth ratio,
the Darcy number, and the thermal-diffusivity ratio.
