Jai Sukhatme and Leslie Smith
Mathematics Dept, University of Wisconsin-Madison.
Abstract :
We study the evolution of large scale density perturbations in the stably stratified two-dimensional Boussinesq equations. As is known, density (or temperature) fronts that form in the early stages of evolution become unstable and spontaneously develop into severely distorted sheets that possess structure at very fine scales. We show here that implied evolution of the spectral energy distribution culminates in the establishment of a self-similar state. In particular, this self-similarity is reflected in the invariant nature of the probability density function associated with the normalized vorticity field. Further, this state is characterized by a frontally dominated $k^{-1}$ potential energy spectrum, the signature of these fronts is also apparent in the saturation of higher order structure functions. The kinetic energy on the other hand is consistent with a dimensionally anticipated $k^{-5/3}$ spectrum although its scaling extends over a shorter range as compared to the potential energy. Ofcourse, given the decaying nature of the problem, this self-similar state is transient and terminates with a transition into a large scale slowly decaying vertically sheared (almost) horizontal field --- i.e. the Pearson-Linden regime. Finally, a conjecture regarding the inviscid limit along with implications for unbalanced components in the fully three-dimensional problem are put forth.