Test Models for Filtering with Superparameterization

Abstract

Superparameterization is a fast numerical algorithm to mitigate implicit scale separation of dynamical systems with large-scale, slowly varying “mean” and smaller-scale, rapidly fluctuating “eddy” term. The main idea of superparameterization is to embed parallel highly resolved simulations of small-scale eddies on each grid cell of coarsely resolved large-scale dynamics. In this paper, we study the effect of model errors in using superparameterization for filtering multiscale turbulent dynamical systems. In particular, we use a simple test model, designed to mimic typical multiscale turbulent dynamics with small-scale intermittencies without local statistical equilibriation conditional to the large-scale mean dynamics, and simultaneously force the large-scale dynamics through eddy flux terms. In this paper, we consider the Fourier domain Kalman filter for filtering regularly spaced sparse observations of the large-scale mean variables. We find high filtering and statistical prediction skill with superparameterization (identical to the skill with perfect model), beyond conventional approaches such as the “bare truncation model” that ignores completely the eddy fluxes and the “equilibrium closure” model that crudely approximates the eddy fluxes with classical averaging theory. We show that this high filtering skill is robust even for very sparse observation networks and turbulent signals with a very steep, $-6$, spectrum. This is a counterexample to naive thinking that the small-scale processes are not so important in multiscale turbulent dynamics with steep energy spectrum. We find that the high filtering skill with superparameterization is robust for small enough scale gap, provided that the filter prior model satisfies the classical linear controllability condition. We will demonstrate a spectacular failure of filtering deterministically forced true signals with the exactly perfect model that does not satisfy controllability condition and a dramatic improvement when the controllability condition is restored with additional stochastic forcings. This result reconfirms and justifies the counterintuitive viewpoint that judicious model errors (or noises) can help filtering turbulent signals.