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The formation of strong and potentially singular fronts in a two-dimensional quasigeostrophic active scalar is studied through the symbiotic interaction of mathematical theory and numerical experiments. This active scalar represents the temperature evolving on the two dimensional boundary of a rapidly rotating half space with small Rossby and Ekman numbers and constant potential vorticity. The possibility of frontogenesis within this approximation is an important issue in the context of geophysical flows. A striking mathematical and physical analogy is developed between the structure and formation of singular solutions of this quasi-geostrophic active scalar in two dimensions and the potential formation of finite time singular solutions for the 3-D Euler equations. Detailed mathematical criteria are developed as diagnostics for self-consistent numerical calculations indicating strong front formation. These self-consistent numerical calculations demonstrate the necessity of nontrivial topology involving hyperbolic saddle points in the level sets of the active scalar in order to have singular behaviour; this numerical evidence is strongly supported by mathematical theorems which utilize the nonlinear structure of specific singular integrals in special geometric configurations to demonstrate the important role of nontrivial topology in the formation of singular solutions.
Constantin et al. (Tue,) studied this question.