Abstract: We present several recent results regarding thermal convection in a fluid layer overlying a saturated porous media using the Navier-Stokes-Darcy-Boussinesq (NSDB) model with appropriate interface boundary conditions.

  First, we reconcile several seemingly contradictory interfacial boundary conditions utilized by different researchers, employing an asymptotic method  at the small Darcy number regime.

  Next, we establish the stability of the pure conduction state at small Rayleigh numbers using classical methods. We then investigate the loss of stability of the pure conduction state as the Rayleigh number surpasses a threshold value through a hybrid approach that combines analysis with numerical computation.

  Particularly, we discover that the transition between shallow and deep convection could be related to changes in system parameters. To predict the transition between deep and shallow convections, we propose a coarse-grained method.  Direct numerical simulations based on decoupled and energy-stable methods indicate that the coarse-grained method performs well in the small Darcy number regime.