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 −5/3−5/3 and −2−2. 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.