Closed-form Discovery of Subgrid-Scale Closures: Promises and Challenges
Abstract
Resolving the entire range of relevant spatial and temporal scales in turbulent flows can be computationally prohibitive. As a result, in practice, turbulent atmospheric and oceanic flows are often resolved on a relatively coarse grid, and the subgrid-scale (SGS) processes equal to the grid scale are parameterized. Quality of parametrizations effect the accuracy of turbulent flow simulations; machine learning is opening new avenues to extract these parametrizations directly from the data. Here, we focus on discovering a closed-form equation of SGS term to have full interpretability. Following the pioneering work of Zanna and Bolton (2020, GRL), we use Bayesian sparse regression for this purpose. This is achieved by first forming a library of basis functions including products and/or derivatives of relevant flow variables. We then employ a relevance vector machine (RVM) to determine the fewest functions from the library required to accurately represent the SGS terms. The optimization is done using a mean-square-error loss function. The SGS momentum stress is discovered for 2D-Forced Homogeneous Isotropic turbulence (2D-FHIT) and Rayleigh-Bénard convection (RBC); SGS heat flux is also discovered for RBC. We use filtered data generated at various coarse grid sizes and three filter types. We find that the dominant terms of equations discovered for all SGS parameterizations of 2D-FHIT and RBC are nonlinear combination of velocity gradients, the so called nonlinear gradient model can also be derived from the Taylor series expansion of the SGS terms. Coefficients of discovered SGS terms are dependent on filter type and size, but the discovered structures are independent of grid size and filters. This happens because the first term of the Taylor expansion dominates the mean-square-error loss function. However, a posteriori (online) tests with the discovered parametrizations suggest that the coarse resolution models are always unstable, owing to the lack of representation of inter-scale energy transfer. Our work suggest that with this kind of approach to SGS parameterization discovery, the discovered model is always the gradient model, if its components are present in the library. Discovering stable SGS parameterizations requires alternative approaches to building the library and loss function.
- Publication:
-
AGU Fall Meeting Abstracts
- Pub Date:
- December 2022
- Bibcode:
- 2022AGUFMNG22B0356J