Dresden 2006 – scientific programme

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DY: Dynamik und Statistische Physik

DY 38: Fluid Dynamics

DY 38.2: Talk

Wednesday, March 29, 2006, 17:00–17:15, SCH 251

Complex Singularities of the Euler Equation — •Walter Pauls — Observatoire de la Cote d’Azur, BP 4229, 06304 Nice Cedex 4, France — Fakultät für Physik, Universität Bielefeld, Universitätsstraße 25, 33615 Bielefeld

A detailed study of complex-space singularities of the two-dimensional incompressible Euler equation is performed in the short-time asymptotic regime when such singularities are very far from the real domain; this allows an exact recursive determination of arbitrarily many spatial Fourier coefficients. Using 35 to 100-digit high-precision arithmetic we find that the Fourier coefficients of the stream function are given over more than two decades of wavenumbers by F (k ) = C (θ ) k^−α e^{−δ (θ ) k} , where k = k (cosθ , sinθ ). The prefactor exponent α , typically between 5/2 and 8/3, is determined with an accuracy better than 0.01. It depends on the initial condition but not on θ . The vorticity diverges as s^−β , where α + β = 7/2 and s is the distance to the (complex) singular manifold. This new type of non-universal singularity is permitted by the strong reduction of nonlinearity (depletion) which is associated to incompressibility. Spectral calculations show that the scaling reported above persists well beyond the time of validity of the short-time asymptotics. A simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent α = 5/2 .