Subgrid Model with Scaling Laws for Atmospheric Simulations

Abstract

Subgrid-scale parameterizations with self-similar scaling laws are developed for large-eddy simulations (LESs) of atmospheric flows. The key new contribution is the development of scaling laws that govern how these parameterizations depend on the LES resolution and flow strength. Both stochastic and deterministic representations of the effects of subgrid-scale eddies on the retained scales are considered. The stochastic subgrid model consists of a backscatter noise term and a drain eddy viscosity, while in the deterministic subgrid model the net effect of these two terms is represented by a net eddy viscosity. In both cases the subgrid transfers are calculated self-consistently from the statistics of higher-resolution-reference direct numerical simulations (DNSs). The dependence of the subgrid parameterizations on the resolution of the LESs is determined for DNSs having resolutions up to triangular 504 wavenumber truncations. The subgrid parameterizations are developed for typical large-scale atmospheric flows and for different strengths and spectra of kinetic energy within a quasigeostrophic spectral model. LESs using the stochastic and deterministic subgrid parameterizations are shown to replicate the kinetic energy spectra of the reference DNS at the scales of the LESs. It is found that the maximum strengths of the drain, net, and backscatter viscosities satisfy scaling laws dependent on the LES truncation wavenumber and that the dependence of these eddy viscosities on total wavenumber can also be written as essentially universal functions that depend on flow strength and resolution. The scaling laws make the subgrid-scale parameterizations more generally applicable in LESs and remove the need to generate them from reference DNSs.

Statistical Dynamical Closures and Subgrid Modeling for Inhomogeneous QG and 3D Turbulence

“…The multi-field QDIA and its underpinning Langevin equation also provide further support for the direct stochastic modeling approach to subgrid-scale parameterizations developed by Frederiksen and Kepert [2] and applied also to the baroclinic QG equations for atmospheric and oceanic flows by Zidikheri and Frederiksen [32][33][34] and Kitsios, Frederiksen and Zidikheri [35]. In some of these studies, particularly for higher resolution LESs of atmospheric flows [32,33,35], where the baroclinic instability is resolved, the renormalized mean forcing is essentially just the bare forcing: ) , ( ) [34], the subgrid contribution to the mean field forcing is in general significant and the renormalized forcing is given by ) ,…”

“…Baroclinic oceanic flows were considered by Zidikheri and Frederiksen [33,34]; they applied the stochastic modeling approach and showed that it can be successfully used to maintain the correct LES spectra for both simple flows and more complex flows characteristic of the Antarctic circumpolar current. In very recent work, Kitsios, Frederiksen and Zidikheri [35] derived universal scaling laws for subgrid models of eddy-eddy interactions applicable to baroclinic atmospheric flows.…”

“…In some of these studies, particularly for higher resolution LESs of atmospheric flows [32,33,35], where the baroclinic instability is resolved, the renormalized mean forcing is essentially just the bare forcing: ) , ( ) [34], the subgrid contribution to the mean field forcing is in general significant and the renormalized forcing is given by ) ,…”

“…Frederiksen and Kepert [2] have found that their approach gave very similar subgrid-scale parameterizations, and successful LES, to the closure based method of Frederiksen and Davies [31]. Unlike the case for the QDIA, we know of no general proof of the realizability of the direct stochastic approach with memory effects and stochastic backscatter [2] but all subsequent applications have yielded realizable positive definite energy spectra [2,[32][33][34][35]. We begin by expanding a k ζˆ in Equation (15b) in a perturbation series:…”

Subgrid modelling for geophysical flows

“…Furthermore, at sufficiently high resolutions of the LESs, the off-diagonal elements of the aforementioned matrices became less important in magnitude compared with the diagonal elements. Recently, Kitsios et al [50,51] showed that universal scaling properties can be obtained for these parametrizations, making them applicable to a wide range of flows.…”