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11/14/02 Colloquium
DR. EVGENII A. KUZNETSOV
Professor of Theoretical Physics, Deputy Director of Landau Institute for Theoretical Physics
Professor of Theoretical Physics, Deputy Director of Landau Institute for Theoretical Physics
Abstract: It is shown that the Euler hydrodynamics describing vortex flow of an ideal fluid coincide with the equations of motion for a charged compressible liquid flowing due to the Lorentz force in a self-consistent electromagnetic field. Transformation to the Lagrangian description in a new hydrodynamics is equivalent to the original Euler equations in a mixed Lagrangian-Eulerian description - the vortex line representation (VLR). Hydrodynamics collapse in the original Euler equations, i.e., formation of a singularity of vorticity in a finite time, can be considered as a process of the breaking of vortex lines. At the breaking point, the value of the vorticity blows up as where is a collapse time. The spatial structure of the collapsing distribution approaches a pancake form: contraction occurs by the law along the "soft" direction, the characteristic scales vanish like along two other ("hard") directions.
Within the 3D Euler equations, which are resolved relative to the infinite set of Cauchy invariants, the emergence of a singularity of vorticity at a single point [1], not related to any symmetry of the initial distribution, has been demonstrated numerically for the first time. Behavior of the maximum of vorticity near the point of collapse closely follows the dependence . This agrees with the interpretation of collapse in an ideal incompressible fluid as the process of vortex lines breaking. Sequences of such type of collapse are discussed for fully developed hydrodynamic turbulence. In particular, it is demonstrated that structure of vorticity near the breaking point has the Kolmogorov behavior. REFERENCES: [1] V.A. Zheligovsky, E.A. Kuznetsov, O.M. Podvigina, Numerical Modeling of Collapse in Ideal Incompressible Hydrodynamics, JETP Letters, 74, 367-370 (2001); e-Print physics/0110046.
