A geometrical study of 3D incompressible Euler flows with Clebsch potentials — a long-lived Euler flow and its power-law energy spectrum

A simple initial condition for vorticity ω = [ sin ( y − z ) , sin ( z − x ) , sin ( x − y ) ] , which has Clebsch potentials, has been identified to lead to a flow evolution with a very weak energy transfer. This allows us to integrate the Euler equations in time longer than commonly expected, to reach a stage at which the total enstrophy attains its peak for the corresponding Navier–Stokes flow. It thereby enables us to study the relationship between the inviscid-limit and totally inviscid behaviours numerically. In spite of small energy dissipation rate, the Navier–Stokes flow shows a power-law spectrum whose exponent is around − 5 / 3 and − 2 . A similar behaviour is also observed for the Euler flow. In physical space, this flow has groups of vorticity layers, which hesitate to roll up.