On the Behavior of the Stable Boundary Layer and the Role of Initial Conditions

Abstract

Previous studies of the stable atmospheric boundary layer using techniques of nonlinear dynamical systems (MCNIDER et al., 1995) have shown that the equations support multiple solutions in certain parameter spaces. When geostrophic speed is used as a bifurcation parameter, two stable equilibria are found—a warm solution corresponding to the high-wind regime where the surface layer of the atmosphere stays coupled to the outer layer, and a cold solution corresponding to the low-wind, decoupled case. Between the stable equilibria is an unstable region where multiple solutions exist. The bifurcation diagram is a classic S shape with the foldback region showing the multiple solutions. These studies were carried out using a simple two-layer model of the atmosphere with a fairly complete surface energy budget. This allowed the dynamical analysis to be carried out on a coupled set of four ordinary differential equations. The present paper extends this work by examining additional bifurcation parameters and, more importantly, analyzing a set of partial differential equations with full vertical dependence. Simple mathematical representations of classical problems in dynamical analysis often exhibit interesting behavior, such as multiple solutions, that is not retained in the behavior of more complete representations. In the present case the S-shaped bifurcation diagram remains with only slight variations from the two-layer model. For the parameter space in the foldback region, the evolution of the boundary layer may be dramatically affected by the initial conditions at sunset. An eigenvalue analysis carried out to determine whether the system might support pure limit-cycle behavior showed that purely complex eigenvalues are not found. Thus, any cyclic behavior is likely to be transient.