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Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. (English) Zbl 0809.35057

Summary: The formation of strong and potentially singular fronts in a two- dimensional quasi-geostrophic active scalar is studied here 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 here between the structure and formation of singular solutions of this quasigeostrophic 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.

MSC:

35L67 Shocks and singularities for hyperbolic equations
76L05 Shock waves and blast waves in fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
76E20 Stability and instability of geophysical and astrophysical flows
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99 Numerical methods for partial differential equations, boundary value problems
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