Mathematical test models for superparametrization in anisotropic turbulence

The complexity of anisotropic turbulent processes over a wide range of spatiotemporal scales in engineering turbulence and climate atmosphere ocean science requires novel computational strategies with the current and next generations of supercomputers. In these applications the smaller-scale fluctuations do not statistically equilibrate as assumed in traditional closure modeling and intermittently send significant energy to the large-scale fluctuations. Superparametrization is a novel class of seamless multi-scale algorithms that reduce computational labor by imposing an artificial scale gap between the energetic smaller-scale fluctuations and the large-scale fluctuations. The main result here is the systematic development of simple test models that are mathematically tractable yet capture key features of anisotropic turbulence in applications involving statistically intermittent fluctuations without local statistical equilibration, with moderate scale separation and significant impact on the large-scale dynamics. The properties of the simplest scalar test model are developed here and utilized to test the statistical performance of superparametrization algorithms with an imposed spectral gap in a system with an energetic -5/3 turbulent spectrum for the fluctuations.