Sufficient condition for a finite-time singularity in a high-symmetry Euler flow: Analysis and statistics.

Abstract

A sufficient condition is obtained for the development of a finite-time singularity in a highly symmetric Euler flow, first proposed by Kida [J. Phys. Soc. Jpn. 54, 2132 (1995)] and recently simulated by Boratav and Pelz [Phys. Fluids 6, 2757 (1994)]. It is shown that if the second-order spatial derivative of the pressure (${\mathit{p}}_{\mathit{xx}}$) is positive following a Lagrangian element (on the x axis), then a finite-time singularity must occur. Under some assumptions, this Lagrangian sufficient condition can be reduced to an Eulerian sufficient condition which requires that the fourth-order spatial derivative of the pressure (${\mathit{p}}_{\mathit{xxxx}}$) at the origin be positive for all times leading up to the singularity. Analytical as well as direct numerical evaluation over a large ensemble of initial conditions demonstrate that for fixed total energy, ${\mathit{p}}_{\mathit{xxxx}}$ is predominantly positive with the average value growing with the numbers of modes. \textcopyright{} 1996 The American Physical Society.