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We present a study using a class of post-hoc local explanation methods i.e., feature importance methods for "understanding" a deep learning (DL) emulator of climate. Specifically, we consider a multiple-input-single-output emulator that uses a DenseNet encoder-decoder architecture and is trained to predict interannual variations of sea surface temperature (SST) at 1, 6, and 9 month lead times using the preceding 36 months of (appropriately filtered) SST data. First, feature importance methods are employed for individual predictions to spatio-temporally identify input features that are important for model prediction at chosen geographical regions and chosen prediction lead times. In a second step, we also examine the behavior of feature importance in a generalized sense by considering an aggregation of the importance heatmaps over training samples. We find that: 1) the climate emulator's prediction at any geographical location depends dominantly on a small neighborhood around it; 2) the longer the prediction lead time, the further back the "importance" extends; and 3) to leading order, the temporal decay of "importance" is independent of geographical location. An ablation experiment is adopted to verify the findings. From the perspective of climate dynamics, these findings suggest a dominant role for local processes and a negligible role for remote teleconnections at the spatial and temporal scales we consider. From the perspective of network architecture, the spatio-temporal relations between the inputs and outputs we find suggest potential model refinements. We discuss further extensions of our methods, some of which we are considering in ongoing work.

Fluid turbulence is characterized by strong coupling across a broad range of scales. Furthermore, besides the usual local cascades, such coupling may extend to interactions that are non-local in scale-space. As such the computational demands associated with explicitly resolving the full set of scales and their interactions, as in the Direct Numerical Simulation (DNS) of the Navier-Stokes equations, in most problems of practical interest are so high that reduced modeling of scales and interactions is required before further progress can be made. While popular reduced models are typically based on phenomenological modeling of relevant turbulent processes, recent advances in machine learning techniques have energized efforts to further improve the accuracy of such reduced models. In contrast to such efforts that seek to improve an existing turbulence model, we propose a machine learning(ML) methodology that captures, de novo, underlying turbulence phenomenology without a pre-specified model form. To illustrate the approach, we consider transient modeling of the dissipation of turbulent kinetic energy, a fundamental turbulent process that is central to a wide range of turbulence models using a Neural ODE approach. After presenting details of the methodology, we show that this approach outperforms state-of-the-art approaches.

In the Large Eddy Simulation (LES) framework for modeling a turbulent flow, when the large scale velocity field is defined by low-pass filtering the full velocity field, a Taylor series expansion of the full velocity field in terms of the large scale velocity field leads (at the leading order) to the nonlinear gradient model for the subfilter stresses. Motivated by the fact that while the nonlinear gradient model shows excellent a priori agreement in resolved simulations, the use of this model by itself is problematic, we consider two models that are related, but better behaved: The Rational LES model that uses a sub-diagonal Pade approximation instead of a Taylor series expansion and the Lagrangian Averaged Navier-Stokes-�� model that uses a regulariza- tion approach to modeling turbulence. In this article, we show that these two latter models are identical in two dimensions.

The applicability of multi-speed discrete-velocity gases to compressible flow situations is considered. First, the equation of state, the anisotropies and the advection velocities for any multi-speed model on the square and triangular lattices are derived. The dependence on the model of any of these to leading order in the flow velocity is shown to be only through a fourth moment of the stationary equilibrium speed distribution. Next, a computation scheme is introduced, wherein adjacent cells in a cell network interact through an exchange of particles, commensurate with the equilibrium fluxes of mass, momentum, and energy. This corresponds to the infinite collision rate limit of the model gas, resulting in very low viscosities. Finally, a simple multi-speed model, the nine-velocity model is studied in detail: Solving the shock tube flow with the model yields almost all phenomenology associated with a perfect gas. An exact shock profile is computed for the model and is compared to a Navier-Stokes shock profile. An adiabatic channel flow is simulated with the model and the results compared to an integral solution of the Navier-Stokes equation. The comparisons in both the cases are excellent. It is also shown that the nine-velocity gas does not permit steady supersonic flow.