Variational Multiscale Modeling and Memory Effects in Turbulent Flow Simulations

Effective computational models of multiscale problems have to account for the impact of unresolved physics on the resolved scales. This dissertation advances our fundamental understanding of multiscale models and develops a mathematically rigorous closure modeling framework by combining the Mori-Zwanzig (MZ) formalism of Statistical Mechanics with the variational multiscale (VMS) method. This approach leverages scale-separation projectors as well as phase-space projectors to provide a systematic modeling approach that is applicable to complex non-linear partial differential equations. %The MZ-VMS framework is investigated in the context of turbulent flows. Spectral as well as continuous and discontinuous finite element methods are considered. The MZ-VMS framework leads to a closure term that is non-local in time and appears as a convolution or memory integral. The resulting non-Markovian system is used as a starting point for model development. Several new insights are uncovered: It is shown that unresolved scales lead to memory effects that are driven by an orthogonal projection of the coarse-scale residual and, in the case of finite elements, inter-element jumps. Connections between MZ-based methods, artificial viscosity, and VMS models are explored. The MZ-VMS framework is investigated in the context of turbulent flows. Large eddy simulations of Burgers' equation, turbulent flows, and magnetohydrodynamic turbulence using spectral and discontinuous Galerkin methods are explored. In the spectral method case, we show that MZ-VMS models lead to substantial improvements in the prediction of coarse-grained quantities of interest. Applications to discontinuous Galerkin methods show that modern flux schemes can inherently capture memory effects, and that it is possible to guarantee non-linear stability and conservation via the MZ-VMS approach. We conclude by demonstrating how ideas from MZ-VMS can be adapted for shock-capturing and filtering methods.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145847/1/parish_1.pd

Stochastic climate theory and modeling

Stochastic methods are a crucial area in contemporary climate research and are increasingly being used in comprehensive weather and climate prediction models as well as reduced order climate models. Stochastic methods are used as subgrid-scale parameterizations (SSPs) as well as for model error representation, uncertainty quantification, data assimilation, and ensemble prediction. The need to use stochastic approaches in weather and climate models arises because we still cannot resolve all necessary processes and scales in comprehensive numerical weather and climate prediction models. In many practical applications one is mainly interested in the largest and potentially predictable scales and not necessarily in the small and fast scales. For instance, reduced order models can simulate and predict large-scale modes. Statistical mechanics and dynamical systems theory suggest that in reduced order models the impact of unresolved degrees of freedom can be represented by suitable combinations of deterministic and stochastic components and non-Markovian (memory) terms. Stochastic approaches in numerical weather and climate prediction models also lead to the reduction of model biases. Hence, there is a clear need for systematic stochastic approaches in weather and climate modeling. In this review, we present evidence for stochastic effects in laboratory experiments. Then we provide an overview of stochastic climate theory from an applied mathematics perspective. We also survey the current use of stochastic methods in comprehensive weather and climate prediction models and show that stochastic parameterizations have the potential to remedy many of the current biases in these comprehensive models