The asymptotics of blow-up in inviscid Boussinesq flow and related systems

We consider the approach to blow-up in two-dimensional inviscid flows with stagnation-point similitude, in particular a buoyancy-driven flow resulting from a horizontally quadratic density variation in a horizontally unbounded slab. The blow-up, which is only possible because the flow has infinite energy, proceeds by intensification of the vorticity and density gradient in a layer adjacent to the upper boundary, while the remainder of the flow tends towards irrotationality. The governing Boussinesq flow equations are first solved numerically, and the results suggest scalings which are then used in an asymptotic analysis as tau --> 0, where tau is the time remaining until blow-up. The structure of the asymptotic solution, involving exponential orders as well as powers and logarithms of the small parameter, is suggested by the analysis of a simpler related problem for which an exact solution is available. The expansion is uniformly valid across the upper boundary layer and the outer region, but there is a layer adjacent to the lower boundary where the flow remains dependent on the initial conditions and is undetermined by the asymptotics.