The Cascade of Circulations in Fluid Turbulence

Abstract

Kelvin's Theorem on conservation of circulations is an essential ingredient of G. I. Taylor's theory of turbulent energy dissipation by the process of vortex-line stretching. In previous work, we have proposed a nonlinear mechanism for the breakdown of Kelvin's Theorem in ideal turbulence at infinite Reynolds number. We develop here a detailed physical theory of this ``cascade of circulations''. Our analysis is based upon an effective equation for large-scale ``coarse-grained'' velocity, which contains a turbulent-induced ``vortex-force'' that can violate Kelvin's Theorem. We show that singularities of sufficient strength, which are observed to exist in turbulent flow, can lead to non-vanishing dissipation of circulation for an arbitrarily small filtering length in the effective equations. This result is an analogue for circulation of Onsager's theorem on energy dissipation for singular Euler solutions. The physical mechanism of the breakdown of Kelvin's Theorem is diffusion of lines of large-scale vorticity out of the advected loop. This phenomenon can be viewed as a classical analogue of the Josephson-Anderson phase-slip phenomenon in superfluids due to quantized vortex lines. We show that the circulation cascade is local in scale and use this locality to develop concrete expressions for the turbulent vortex-force by a multi-scale gradient-expansion. We discuss implications for Taylor's theory of turbulent dissipation and we point out some related cascade phenomena, in particular for magnetic-flux in magnetohydrodynamic (MHD) turbulence.