Breaking of vortex lines - a new mechanism of collapse in hydrodynamics

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A new mechanism of the collapse in hydrodynamics is suggested, due to breaking of continuously distributed vortex lines. Collapse results in formation of the point singularities of the vorticity field |Ω|. At the collapse point, the value of the vorticity blows up as (t₀-t)⁻¹ where t₀ is a collapse time. The spatial structure of the collapsing distribution approaches a pancake form: contraction occurs by the law l₁ (t₀-t)³/² along the "soft" direction, the characteristic scales vanish like l₂ (t₀-t)¹/² along two other ("hard") directions. This scenario of the collapse is shown to take place in the integrable three-dimensional hydrodynamics with the Hamiltonian H=∫|Ω|d r. Most numerical studies of collapse in the Euler equation are in a good agreement with the proposed theory.