Collapse of vortex lines in hydrodynamics

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Abstract
A new mechanism is proposed for collapse in hydrodynamics associated with the “breaking” of vortex lines. The collapse results in the formation of point singularities of the vorticity field, i.e., a generalized momentum curl. At the point of collapse the vorticity |Ω| increases as ((t0 − t)−1 and its spatial distribution for t → t0 approaches quasi-two-dimensional: in the “soft” direction contraction obeys the law l1 → (t0 − t)3/2 whereas in the other two “hard” directions it obeys l2 → (t0 − t)1/2. It has been shown that this collapse scenario takes place in the general case for three-dimensional integrable hydrodynamics with the Hamiltonian ℋ = ∫|Ω| dr.