Strain, vortices, and the enstrophy inertial range in two-dimensional turbulence

Abstract

The properties of vortices in a strain field are used to construct a phenomenological theory of the enstrophy inertial range in two-dimensional incompressible turbulence. The theory, based in part on the results and behavior of numerical simulations, attempts to combine spectral inertial range theories of the Kolmogorov type with the dynamics of vortex interactions in physical space. It is based on the assumptions that coherent vortices can survive in a turbulent flow if of sufficient strength compared to the background straining field, and that coherent structures feel a mean strain field, independent of their scale. The first assumption is suggested by a result in the theory of uniform elliptic vortices, while the second comes from numerical simulations. The theory employs a single non-dimensional parameter, essentially the ratio between the enstrophy flux and the mean strain, which then characterizes flows from extremely intermittent decaying turbulence to nearly Gaussian passive scalar dynamics. The theory predicts that in forced two-dimensional turbulence, coherent structures reside in a “background” straining field. The coherent vortices will dominate the flow at a sufficiently large scale, with a fairly abrupt transition at a small scale to a flow in which the classical k−1 enstrophy spectrum holds. In this classical region small amplitude vortices do not survive because the (large-scale) straining field is of larger amplitude than the (small-scale) vorticity. The vorticity itself is passively advected in this regime. If the enstrophy flux is very small compared to the enstrophy itself, then the dynamics will be highly intermittent, with a spectrum determined by the spectrum of the vortices themselves, rather than by the dynamics of the enstrophy flux. The theory predicts that at small scales in forced-dissipative two-dimensional turbulence the energy spectrum will obey the classical enstrophy inertial range predictions even though the non-linear interactions remain spectrally non-local. Passive scalar dynamics are predicted to be similar to vortex dynamics, at small scales. Available numerical simulations are consistent with these suggestions.