Finite time singularities in a class of hydrodynamic models

Abstract

Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form $\mathcal{L}\sim \int k^{\alpha }|{\mathbf{v}}_{\mathbf{k}}{|}^2{d}^3\mathbf{k}$ in 3D Fourier representation, where $\alpha$ is a constant, $0<\alpha <1$. Unlike the case $\alpha =0$ (the usual Eulerian hydrodynamics), a finite value of $\alpha$ 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. The 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 $(t^{\ast }-t{)}^{1/(2-\alpha )}$, where $t^{\ast }$ is the singularity time.

LaTeX, 17 pages, 3 eps figures. This version is close to the journal paper

References in corpus (4)

• Slow inviscid flows of a compressible fluid in spatially inhomogeneous systems
• A Nonlinear Analysis of the Averaged Euler Equations
• Interaction of a vortex ring with the free surface of ideal fluid
• The Euler-Poincare Equations and Semidirect Products with Applications to Continuum Theories

Cited by in corpus (5)

• Analytic solution of the dynamics of quantum vortex reconnection
• Slow inviscid flows of a compressible fluid in spatially inhomogeneous systems
• Current-sheet formation in incompressible electron magnetohydrodynamics
• Singularities in Fully Developed Turbulence
• The hodograph method applicability in the problem of long-scale nonlinear dynamics of a thin vortex filament near a flat boundary