Data-driven stochastic modeling for coarsened computational geophysical fluid dynamics
Sagy Ephrati is a PhD student in the department Multiscale Modeling and Simulation. (Co)Promotors are prof.dr.ir. B.J. Geurts and dr.ir. P. Cifani from the faculty of Electrical Engineering, Mathematics and Computer Science.
This thesis deals with data-driven stochastic modeling of coarsened computational geophysical fluid dynamics. Geophysical fluid dynamics concerns the study of fluid flows in large-scale geophysical systems, such as the Earth's atmosphere or oceans. Physical phenomena in these flows consist of motions described by a vast range of scales. Fully resolving all these scales in a numerical simulation requires tremendous computational resources, making accurate high-resolution numerical simulations computationally expensive and time-consuming. As a result, coarse numerical simulations can be used, which employ lower resolutions to accommodate for the large computational costs. However, the use of coarse computational grids introduces uncertainty and error to the numerical solution. Uncertainty is introduced via the loss of small-scale flow features on coarse grids. That is, the small-scale flow features often cannot be determined with absolute certainty for a given large-scale flow configuration. Errors arise in the coarsening process due to poorly resolved spatial derivatives, generally leading to a deterioration of the numerical solution. To address these challenges, stochastic modeling can serve both to compensate for the errors due to coarsening and to quantify the inherent uncertainty. The aim of the work presented in this thesis is to study stochastic modeling for coarsened flows when data of the fully resolved system is available.
In this thesis, the models enter the governing equations as a space- and time-dependent forcing term. This term is decomposed into fixed spatial basis functions and corresponding time series, of which only the latter are modeled based on high-resolution numerical data. Including a forcing term of this form immediately leads to modeling choices such as where the forcing enters the governing equations, what the spatial basis functions are, how the time series are modeled, and which high-resolution data is used. The work presented in this thesis explores these aspects of modeling. More specifically, we first investigate how to measure coarsening effects exactly. These measurements are inserted into a coarse numerical simulation as a reduced-order correction with prescribed time series to get an exact agreement with a pre-computed reference solution. The quality of the reconstruction depends strongly on the employed discretization method. Subsequently, Stochastic Advection by Lie Transport is used to quantify uncertainty due to unresolved small-scale processes in coarse numerical simulations. Here, the time series are modeled as stochastic processes with statistics matching those of the measurements. Stochastic processes with similar temporal correlation as the measurements were found to lead to a reduced ensemble spread without a loss of accuracy when compared to uncorrelated noise, indicating a closer adherence of the former to the reference solution. Finally, a data-driven subgrid-scale model is proposed. The model is derived from a data assimilation algorithm and acts on the spectral coefficients of the solution with the aim of reconstructing a reference kinetic energy spectrum. This method is found to perform well in terms of flow statistics for two different fluid dynamical systems.
