To describe the vortical flows of ideal fluids including collapse we suggest a mixed Lagrangian-Eulerian description - the vortex line representation [1],
according to which each vortex line is labeled by two-dimensional Lagrangian marker and another coordinate coincides with a curve parameter given the
line. It is shown that the Euler hydrodynamics in a new representation coincides with equations of motion for charged compressible liquid flowing due to
the Lorents force in electromagnetic field. Both electric and magnetic fields satisfy the Maxwell equations [2]. As a sequence of compressibility of new
hydrodynamics breaking of continuously distributed vortex lines is possible which results in formation of the point singularities of the vorticity field
that is in correspondence with many performed numerics for the Euler equations. This scenario of the collapse is shown to take place in the integrable
three-dimensional hydrodynamics [3].
Within the 3D Euler equations, which are resolved relative to the infinite set of the Cauchy invariants, emergence of singularity of vorticity at a
single point [4], not related to any symmetry of the initial distribution, is demonstrated numerically. Behavior of the maximum of vorticity near the
point of collapse closely follows the dependence (t0 − t)−1, where t0 is the time of collapse. This agrees with the interpretation of collapse in an ideal
incompressible fluid as of the process of vortex lines breaking. Sequences of such type of collapse are discussed for fully developed hydrodynamic turbu-
lence. In particular, it is demonstrated that structure of vorticity near the breaking point has the Kolmogorov behavior that allows one to interpret the
Kolmogorov spectrum like the Phillips spectrum.
References
[1] E.A.Kuznetsov and V.P.Ruban, JETP Letters, 67, 1076 (1998); Phys. Rev. E, 61, 831 (2000).
[2] E.A. Kuznetsov, JETP Letters, 76, 346 (2002); Journal of Fluid Mechanics, 600, 167-180 (2008).
[3] E.A.Kuznetsov and V.P.Ruban, JETP, 91, 775 (2000).
[4] E.A.Kuznetsov, O.M.Podvigina and V.A.Zheligovsky, FLUID MECHANICS AND ITS APPLICATIONS, Volume 71: Tubes, Sheets and Singularities in
Fluid Dynamics. eds. K. Bajer, H.K. Moffatt, Kluwer, (2003) pp. 305-316; E.A.Kuznetsov, AIP Conf. Proc. 703, 16-25 (2004).