Spectral reduction: a statistical description of turbulence

A method is described for predicting statistical properties of turbulence. Collections of Fourier amplitudes are represented by nonuniformly spaced modes with enhanced coupling coefficients. The statistics of the full dynamics can be recovered from the time-averaged predictions of the reduced model. A Liouville theorem leads to inviscid equipartition solutions. Excellent agreement is obtained with two-dimensional forced-dissipative pseudospectral simulations. For the two-dimensional enstrophy cascade, logarithmic corrections to the high-order structure functions are observed. Typeset using REVTEX 1 Many practical applications for spectral simulations of turbulence exist where it would be desirable to evolve modes that are distributed nonuniformly in Fourier space, devoting most of the computational resources to the length scales of greatest physical interest. This idea has led to the development of a new reduced statistical description of turbulence, called spectral reduction [1], which dramatically reduces the number of spectral modes that must be retained in simulations of turbulent phenomena. It exploits the fact that statistical moments are much smoother functions of wave number than are the underlying stochastic amplitudes. The concept of wave-number reduction is not new. In the method of constrained decimation [2–4], a stochastic forcing term is added to model the effect of the deleted modes on the retained modes. She and Jackson have proposed a reduction scheme in which the linear (viscous) term is modified [5]. In spectral reduction, a third alternative is chosen: the nonlinear coefficients are enhanced to account for the effect of the discarded modes on the explicitly evolved modes. There have been other more heuristic attempts at wave-number reduction [6–9]; these methods typically neglect nonlocal wave-number triad interactions (which play a particularly important role in two-dimensional turbulence). Unlike the renormalization group [10] method, which retains only large-scale modes and attempts to express the effect of the small-scale modes using a self-similarity Ansatz , spectral reduction retains certain modes from all scales, while discarding other modes associated with these same scales. The generality of the formulation allows one to refine the partition wherever the physics dictates. In this Letter we restrict our attention to homogeneous and isotropic incompressible turbulence in two dimensions. The appropriate spectral transform in this limit is the integral Fourier transform, under which the two-dimensional Navier–Stokes vorticity equation becomes ∂ωk ∂t + νkωk = ∫