Complex manifolds for the Euler equations: a hierarchy of ODEs and the case of vanishing angle in two dimensions

Abstract

This paper considers the two-dimensional Euler equation for complex spatial variables and two complex modes in the initial condition. A hierarchy of third-order ordinary differential equations (ODEs) is used to study the location and structure of the complex singular manifold for short times. The system has two key parameters, the ratio η of the wave numbers of the two modes, and the angle ϕ between the two wave vectors. Using this hierarchy for the case ϕ=π/2 the results of earlier authors (Pauls et al 2006 Physica D 219 40–59) are reproduced numerically. To make analytical progress, the paper considers the limit ϕ→ 0 in which the wave vectors become parallel, rescaling time also. By considering the limiting behaviour of the ODE hierarchy, an asymptotic framework is set up that describes the geometry of the singular manifold and local behaviour of vorticity in this limiting case ϕ=0 of parallel modes. In addition, the hierarchy of ODEs can be solved analytically, order by order, in the parallel case using computer algebra. This is used to confirm the asymptotic theory and to give evidence for a scaling exponent β=1 for the blow-up of vorticity on the singular manifold, ω=O(s− β) in this case of vanishing angle ϕ.