An Analytical and AI-discovered Stable, Accurate, and Generalizable Subgrid-scale Closure for Geophysical Turbulence

Abstract

By combining AI and fluid physics, we discover a closed-form closure for 2D turbulence from small direct numerical simulation (DNS) data. Large-eddy simulation (LES) with this closure is accurate and stable, reproducing DNS statistics including those of extremes. We also show that the new closure could be derived from a 4th-order truncated Taylor expansion. Prior analytical and AI-based work only found the 2nd-order expansion, which led to unstable LES. The additional terms emerge only when inter-scale energy transfer is considered alongside standard reconstruction criterion in the sparse-equation discovery.

Figures (4)

• Figure 1: Representative examples of the effects of increasing the sparsity-level hyperparameter, $\alpha$, on the CC in the discovered closure. (a)-(b) The CC-$\alpha$ relationship. In (a), the shading represents the max-min spread of all three elements of $\tau_{ij}$. (a) uses the common metric, CC of $\tau_{ij}$, while (b) uses a physics-informed metric that accounts for the CC of total ($P_{\tau}$), diffusion ($P_{\tau} > 0$), and backscattering ($P_{\tau} < 0$) inter-scale energy transfers as well.
• Figure 2: Comparison of the a priori (offline) performance of different closures. The table shows the ratio of domain-averaged $P_{\tau}$ and $P_Z$ to that of FDNS. The bar plots present the CC for $\tau_{12}$, $P_{\tau}$, and $P_Z$ of each closure and FDNS. All values are the mean and standard deviations calculated for 4 setups together (Cases 1-2 at $N_{\text{LES}}=128$ and Cases 3-4 at $N_{\text{LES}}=512$); Table S1 shows the values for each case.
• Figure 3: Comparison of the a posteriori performance of different closures for Cases 3-4 at $N_{\text{LES}}=512$. (a)-(b) The PDF of vorticity normalized by its standard deviation, $\overline{\omega}/\sigma_{\omega}$. The upper-right insets of (a)-(b) compares the accuracy of short-term forecasting of each LES against FDNS. The $x$-axis represents the eddy-turnover time, $t_{\eta} = 1/\sqrt{\langle\overline{\omega}^2\rangle}$. (c)-(d) Comparison of the enstrophy spectra, $\hat{Z}\left(k\right)$, of LES with different closures. A similar figure for (Cases 1-2 at $N_{\text{LES}}=128$ ) is shown in the supporting information.
• Figure 4: The four cases considered in this study. These cases represent a broad range of dynamics in 2D turbulence and mimic the diversity of flow regimes (jets and vortices) in the atmosphere and ocean. $(k_x,k_y)$ are the wavenumbers of the time-invariant forcing in the $x$ and $y$ directions and $\beta$ is the gradient of the Coriolis force (see jakhar2024learning for details). $N_\mathrm{DNS}$ is the resolution of the pseudo-spectral solver (in each direction) used for DNS (py2d). The first row presents snapshots of the DNS vorticity, $\omega$. The second row shows snapshots of the FDNS vorticity, $\overline{\omega}$. The third row displays a snapshot of an element of the subgrid-scale (SGS) term, $\tau_{12}$. The last row depicts the energy (red line) and enstrophy (blue line) spectra for DNS (dashed line) and FDNS (solid line). FDNS is at $N_{\text{LES}}=128$ for Cases 1-2 and $N_{\text{LES}}=512$ for Cases 3-4.