Theoretical analysis of convective instability of a growing horizontal thermal boundary layer

Abstract

An analytic description of the development of boundary layer and interface instabilities past the initial linear stage is given. A problem with an analytic approach is that whereas the quasi-static approximation, implying the fast growth of perturbations, is made in order to freeze in the time-dependent basic temperature profile, the nonlinear analysis of Stuart and Watson is valid only for small growth rates. A model in which these two assumptions are combined does, however, successfully describe a number of features of the nonlinear behavior of the system. The dominant role of the fastest growing periodic perturbation, including a damping of other disturbances and a short-lived equilibration, together with a breakaway of plumes from the outer boundaries of the active region is indicated. The analysis is made possible by the use of a simple approximation to the velocity for the linear problem. This approximation is shown to give a good estimate of the linear growth rates, as well as permitting a straightforward treatment of nonlinear effects.