Number : Title : Nonlinear Laplacian Spectral Analysis of Rayleigh

The analysis of physical datasets using modern methods developed in machine learning presents unique challenges and opportunities. These datasets typically feature many degrees of freedom, which tends to increase the computational cost of statistical methods and complicate interpretation. In addition, physical systems frequently exhibit a high degree of symmetry that should be exploited by any data analysis technique. The classic problem of Rayleigh Benard convection in a periodic domain is an example of such a physical system with trivial symmetries. This article presents a technique for analyzing the time variability of numerical simulations of two-dimensional Rayleigh-Benard convection at large aspect ratio and intermediate Rayleigh number. The simulated dynamics are highly unsteady and consist of several convective rolls that are distributed across the domain and oscillate with a preferred frequency. Intermittent extreme events in the net heat transfer, as quantified by the time-weighted probability distribution function of the Nusselt number, are a hallmark of these simulations. Nonlinear Laplacian Spectral Analysis (NLSA) is a data-driven method which is ideally suited for the study of such highly nonlinear and intermittent dynamics, but the trivial symmetries of the Rayleigh-Benard problem such as horizontal shift-invariance can mask the interesting dynamics. To overcome this issue, the vertical velocity is averaged over parcels of similar temperature and height, which substantially compresses the size of the dataset and removes trivial horizontal symmetries. This isothermally averaged dataset, which is shown to preserve the net convective heat-flux across horizontal surfaces, is then used as an input to NLSA. The analysis generates a small number of orthogonal modes which describe the spatiotemporal variability of the heat transfer. A regression analysis shows that the extreme events of the net heat transfer are primarily associated with a family of modes with fat tailed probability distributions and low frequency temporal power spectra. On the other hand, the regular oscillation of the heat transfer is associated with a pair of modes with nearly uniform probability distributions. Physical mechanisms for the regular oscillation and the extreme heat transfer events are hypothesized based on an analysis of the spatio-temporal structure of these modes. Finally, proposals are made for this approach to be applied to the study of other problems in turbulent convection, including three-dimensional Rayleigh-Benard convection and moist atmospheric convection. Suggested Reviewers: Jorg Schumacher [email protected] Expert in Rayleigh Benard convection Nadine Aubry [email protected] Expert in the data analysis of dynamical systems Daan Crommelin [email protected] This manuscript presents an algorithm for the analysis of a numerical simulation of Rayleigh Benard convection using state of the art methods adapted from the machine learning community. There are two primary contributions: 1) The transient dynamics of two dimensional Rayleigh Benard convection are analyzed with a special focus on intermittency in the bulk heat transfer, which is quantified by the probability distribution of the Nusselt number. 2) An algorithm which combines nonlinear Laplacian spectral analysis (NLSA) and an isothermal averaging procedure is developed. This algorithm is specifically designed to take into account the trivial symmetries inherent to fluid dynamics simulations in idealized geometries. This algorithm generates mode families which successfully explain the intermittent aspects of heat transfer. The isothermal averaging procedure has the added benefit of substantially reducing the computational cost of the NLSA algorithm, which suggests that the algorithm is applicable to large three-dimensional simulations. Significance and Novelty of this paper