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

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 (pxx) 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 (pxxxx) 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, pxxxx is predominantly positive with the average value growing with the numbers of modes. @S1063-651X~96!13008-7#