Previous studies of the stable atmospheric boundary layer using techniques of non-linear dynamical systems (McNider et al. J.Atmos.Sci, 1995) have shown that the equations support multiple solutions in certain parameter spaces. For example, for geostrophic speeds as a bifurcation parameter two stable equilibria are found. One a warm solution corresponding to the high wind regime where the surface layer of the atmosphere stays coupled to the outer layer and a second corresponding to the low wind cold solution de-coupled case. In between the two stable equilibria is a unstable region where multiple solutions exist. The bifurcation diagram is a classic "s" shape with the fold back 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 three ordinary differential equations. The present paper extends this work by analyzing a partial differential equation set with full vertical dependence. In classic problems in dynamical analysis, interesting behaviors such as multiple solutions are often found for simple mathematical representations of physical phenomena, but, this behavior does not carry forward in more complete mathematical representations. In the present case, the basic "s" shaped bifurcation diagram remains with only slight variations from the two layer model.

In addition to the dynamical analysis of the solutions as a function of external parameters - roughness, windspeed etc., preliminary studies of the role of initial conditions are examined. The evolution of the boundary may be dramatically affected by the initial conditions at sunset. For example, a less stable initial profile may send the boundary to the coupled warm solution rather than the cold de-coupled solution at least temporarily. Thus, physical processes such as the entrainment of cool air from the surface by decaying large eddies may be important in setting the initial conditions (initial stability) for the subsequent evolution of the stable boundary layer. Sensitivity of the boundary solutions to initial conditions are explored for various external parameter spaces.