Inertial Ranges and Small-Scale Intermittency in One-Dimensional Turbulence Models

Abstract

A set of one-dimensional turbulence models can be constructed by applying severe D-1 directional Fourier truncation to the D-dimensional fluid equations, allowing for numerical calculation over an extremely wide range of scales. At low resolution a reduced 2D model displayed both inviscid energy-enstrophy equipartition and a spectrum consistent with the enstrophy cascade phenomenology in the decay problem. In the present note, these results are extended by using reduced models to examine both the inverse (D = 2) and direct (D = 3) energy cascades. In addition, higher numerical resolution (up to 4096 grid points) is employed to demonstrate unequivocal adherence to the 2D phenomenologies. Small-scale intermittency in the form of spatially intermittent vorticity gradients is also observed. Simulations based on the reduced 3D model were less successful. Although the truncated inviscid equilibrium agreed with an energy equipratition spectrum, forced-viscous simulations failed to show a clear Kolmogorov range. It is argued that the technique, although not justified for 3D problems, produces an interesting 1D turbulence model when applied to 2D flow.