Sufficient condition for finite-time singularity and tendency towards self-similarity in a high-symmetry flow

Abstract

A highly symmetric Euler flow, first proposed by Kida (1985), and recently simulated by Boratav and Pelz (1994) is considered. It is found that the fourth order spatial derivative of the pressure (p _xxxx) at the origin is most probably positive. It is demonstrated that if p _xxxx grows fast enough, there must be a finite-time singularity (FTS). For a random energy spectrum E(k) ∞ k ^−v, a FTS can occur if the spectral index v<3. Furthermore, a positive p _xxxx has the dynamical consequence of reducing the third derivative of the velocity u _xxx at the origin. Since the expectation value of uxxx is zero for a random distribution of energy, an ever decreasing uxxx means that the Kida flow has an intrinsic tendency to deviate from a random state. By assuming that u _xxx reaches the minimum value for a given spectral profile, the velocity and pressure are found to have locally self-similar forms similar in shape to what are found in numerical simulations. Such a quasi self-similar solution relaxes the requirement for FTS to v<6. A special self-similar solution that satisfies Kelvin’s circulation theorem and exhibits a FTS is found for v=2.