Spatial Covariance Modeling for Stochastic Subgrid-Scale Parameterizations Using Dynamic Mode Decomposition

Abstract

Stochastic parameterizations are increasingly being used in climate modeling to represent subgrid-scale processes. While different parameterizations are being developed considering different aspects of the physical phenomena, less attention is given to technical and numerical aspects. In particular, empirical orthogonal functions (EOFs) are employed when a spatial structure is required. Here, we provide evidence they might not be the most suitable choice. By applying an energy-consistent parameterization to the two-layer quasi-geostrophic (QG) model, we investigate the model sensitivity to a priori assumptions made on the parameterization. In particular, we consider here two methods to prescribe the spatial covariance of the noise: first, by using climatological variability patterns provided by EOFs, and second, by using time-varying dynamics-adapted Koopman modes, approximated by dynamic mode decomposition (DMD). The performance of the two methods are analyzed through numerical simulations of the stochastic system on a coarse spatial resolution and the outcomes compared to a high-resolution simulation of the original deterministic system. The comparison reveals that the DMD-based noise covariance scheme outperforms the EOF-based one. The use of EOFs leads to a significant increase of the ensemble spread and to a meridional misplacement of the bimodal eddy kinetic energy (EKE) distribution. Conversely, using DMDs, the ensemble spread is confined, the meridional propagation of the zonal jet stream is accurately captured, and the total variance of the system is improved. Our results highlight the importance of the systematic design of stochastic parameterizations with dynamically adapted spatial correlations, rather than relying on statistical spatial patterns.Plain Language Summary Exact and accurate representations of the climate system would require enormous amounts of computational resources and data storage. Hence, to circumvent this problem, climate models resolve explicitly only the large slow scales, while the fast small modes are represented inside climate models via parameterizations. Due to the different evolution times of the resolved and unresolved scales, the latter can be represented by means of a stochastic process. While different parameterizations are being developed considering different aspects of the physical phenomena, less attention is given to the technical and numerical aspects. In particular, the use of a constant in time noise covariance for the noise is very common. In the framework of a simplified model for the large-scale dynamics, we propose an alternative method to define the noise covariance, which allows it to be regularly updated during the simulation. This might be of crucial importance in the context of climate change. The results show that a dynamically adapted spatial correlation leads to a reduced growth of the uncertainties and better captures the system behavior.

Correlation-based flow decomposition and statistical analysis of the eddy forcing

“…1995; Zanna et al. 2017; Berloff 2018) and stochastic (Frederiksen 1999; Berloff 2005; Crommelin & Vanden-Eijnden 2008; Mana & Zanna 2014; Gugole & Franzke 2020) parametrization approaches for the oceanic mesoscale eddies demonstrate the spectrum of ideas and vigorous intensity of the ongoing research (reviewed by Hewitt et al. 2020).…”