Nonlinear convection in the boundary layer above a sinusoidally heated flat plate

The thermal boundary layer induced within a horizontal semi-infinite layer of Boussinesq fluid by a sinusoidally heated bounding plate is known to be susceptible to vortex instability. The structure of short wavelength convection is considered here and the analysis taken from linear, through weakly nonlinear and onto the fully nonlinear stage. At this point the activity is sufficiently strong that it induces large changes in the underlying thermal profile and the overall flow characteristics are determined by the solution of a free boundary-value problem. This solution demonstrates that the highly nonlinear vortices tend to settle into a periodic form in which intervals of intense activity alternate with times of relative quiescence.