Convection, Stability, and Low Dimensional Dynamics

Abs t r ac t . Recent developments concerning the connection between notions of hydrodynamic stability usually associated with stationary laminar flows-and dynamics, most notably turbulent fluid flows, are reviewed. Based on a technical device originally introduced by Hopf in 1941, a rigorous mathematical relationship between criteria for nonlinear energy stability and bounds on global t ransport by steady, unsteady, or even turbulent flows, has been established. The optimal "marginal stability" criteria for the best bound leads to a novel variational problem, and the differential operator associated with the stability condition generates an adapted basis in which turbulent flow fields may naturally be decomposed. The application and implications of Galerkin truncations in these bases to produce low dimensional dynamical systems models is discussed in the context of thermal convection in a saturated porous layer.