There is growing interest in discovering interpretable, closed-form equations for subgrid-scale (SGS) closures/parameterizations of complex processes in Earth system. Here, we apply a common equation-discovery technique with expansive libraries to learn closures from filtered direct numerical simulations of 2D forced turbulence and Rayleigh-Benard convection (RBC). Across common filters, we robustly discover closures of the same form for momentum and heat fluxes. These closures depend on nonlinear combinations of gradients of filtered variables (velocity, temperature), with constants that are independent of the fluid/flow properties and only depend on filter type/size. We show that these closures are the nonlinear gradient model (NGM), which is derivable analytically using Taylor-series expansions. In fact, we suggest that with common (physics-free) equation-discovery algorithms, regardless of the system/physics, discovered closures are always consistent with the Taylor-series. Like previous studies, we find that large-eddy simulations with NGM closures are unstable, despite significant similarities between the true and NGM-predicted fluxes (pattern correlations > 0.95). We identify two shortcomings as reasons for these instabilities: in 2D, NGM produces zero kinetic energy transfer between resolved and subgrid scales, lacking both diffusion and backscattering. In RBC, backscattering of potential energy is poorly predicted. Moreover, we show that SGS fluxes diagnosed from data, presumed the ‘truth’ for discovery, depend on filtering procedures and are not unique. Accordingly, to learn accurate, stable closures from high-fidelity data in future work, we propose several ideas around using physics-informed libraries, loss functions, and metrics. These findings are relevant beyond turbulence to closure modeling of any multi-scale system.

Convection-generated gravity waves (CGWs) transport momentum and energy, and this momentum is a dominant driver of global features of Earth’s atmosphere’s general circulation (e.g. the quasi-biennial oscillation, the pole-to-pole mesospheric circulation). As CGWs are not generally resolved by global weather and climate models, their effects on the circulation need to be parameterized. However, quality observations of GWs are spatiotemporally sparse, limiting understanding and preventing constraints on parameterizations. Convection-permitting or -resolving simulations do generate CGWs, but validation is not possible as these simulations cannot reproduce the forcing convection at correct times, locations, and intensities. Here, realistic convective diabatic heating, learned from full-physics convection-permitting Weather Research and Forecasting (WRF) simulations, is predicted from weather radar observations using neural networks and a previously developed look-up table. These heating rates are then used to force an idealized GW-resolving dynamical model. Simulated CGWs forced in this way did closely resemble those observed by the Atmospheric InfraRed Sounder in the upper stratosphere. CGW drag in these validated simulations extends 100s of kilometers away from the convective sources, highlighting errors in current gravity wave drag parameterizations due to the use of the ubiquitous single-column approximation. Such validatable simulations have significant potential to be used to further basic understanding of CGWs, improve their parameterizations physically, and provide more restrictive constraints on tuning \textit{with confidence}.

In this work, we develop a data-driven subgrid-scale (SGS) model using a fully convolutional neural network (CNN) for large eddy simulation of forced 2D turbulence. Forced 2D turbulence is a fitting prototype for many large-scale geophysical and environmental flows (where rotation and/or stratification dominate) and has been widely used as a testbed for novel techniques, including machine-learning-based SGS modeling. We first conduct direct numerical simulation (DNS) and obtain training, validation, and testing data sets by applying a Gaussian spatial filter to the DNS solution. With the filtered DNS (FDNS) data in hand, we train the CNN with the filtered state variables. A priori analysis shows that the CNN-predicted SGS term accurately captures the inter-scale energy transfer. A posteriori analysis indicates that the LES-CNN outperforms the physics-based models in both short-term prediction and long-term statistics. Although the CNN-based model is promising in predicting the SGS term, it requires big data to perform satisfactorily. In the small-data limit, the LES-CNN generates artificial instabilities and thus leads to unphysical results. We propose three remedies for the CNN to work in the small-data limit, i.e., data augmentation and group convolution neural network (GCNN), leveraging the rotational equivariance of the SGS term and incorporating a physical constraint on the SGS enstrophy transfer. The SGS term is both translational and rotational equivariant in a square periodic flow field. While primitive CNN can capture the translational equivariance, the rotational equivariance can be accounted for by either including rotated snapshots in the training data set or by a GCNN that enforces rotational equivariance as a hard constraint. Additionally, The SGS enstrophy transfer constraint can be implemented in the loss function of the CNN. A priori and a posteriori analyses show that the CNN/GCNN with knowledge/constraints of rotational equivariance and SGS enstrophy transfer enhances the SGS model and allows the data-driven model to work stably and accurately in a small-data limit. These findings can potentially help the ongoing efforts in using machine-learning for SGS modeling in weather/climate models, where high-quality training data are scarce and instabilities have been reported in many past studies.