Finite time singularities in a class of hydrodynamic models

V. P. Ruban, D. I. Podolsky, J. J. Rasmussen
2001 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form L∼∫ k^α| v_k|^2d^3 k in 3D Fourier representation, where α is a constant, 0<α< 1. Unlike the case α=0 (the usual Eulerian hydrodynamics), a finite value of α results in a finite energy for a singular, frozen-in vortex filament. This property allows us to study the dynamics of such filaments without the necessity of a regularization procedure for short length scales.
more » ... he linear analysis of small symmetrical deviations from a stationary solution is performed for a pair of anti-parallel vortex filaments and an analog of the Crow instability is found at small wave-numbers. A local approximate Hamiltonian is obtained for the nonlinear long-scale dynamics of this system. Self-similar solutions of the corresponding equations are found analytically. They describe the formation of a finite time singularity, with all length scales decreasing like (t^*-t)^1/(2-α), where t^* is the singularity time.
doi:10.1103/physreve.63.056306 pmid:11415005 fatcat:wjarxefqofavtfkhryojlpt2ra