Subgrid-scale modeling with memory using the Mori—Zwanzig formalism and variational multiscale method


Friday, 7 June, 2024 - 16:00 to 17:00

Speaker : Eric Parish, Sandia National Laboratories, USA

Time : 16:00 - 17.00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

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Organizers : Pavan Pranjivan Mehta* ( and Arran Fernandez** (

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: Mori-Zwanzig, turbulence, memory

Abstract: This talk discusses the development of subgrid-scale models for turbulent flows using the Mori—Zwanzig (MZ) formalism [1,2]. Originating from irreversible statistical mechanics and re-formulated by Chorin et al. in the early 2000s [3], the MZ formalism provides a mathematical procedure for the development of coarse-grained models of complex systems, such as turbulence, that lack scale separation. The formalism operates by partitioning the state of a dynamical system into a resolved “coarse-grained” component and an unresolved component. The impact of the unresolved scales on the resolved scales is then re-cast as a convolutional “memory” integral that depends only on the resolved scales. While exact computation of this memory integral is intractable, it provides a starting point for the systematic construction of subgrid-scale models. In this talk, we outline the development of MZ-based subgrid-scale models for coarse-grained simulations of turbulence. We introduce the Mori—Zwanzig formalism, and then provide a formulation for developing MZ-based subgrid-scale models for coarse-grained PDEs by combining MZ with the variational multiscale method (VMS) [4]. We introduce several approximations to the memory term and compare them with established techniques [4,5,6]. In doing so, we establish links between existing stabilization techniques and memory effects. Lastly, numerical results are shown for a variety of canonical turbulent flows

Biography: Eric Parish is a member of technical staff at Sandia National Laboratories. He received his PhD from the University of Michigan in 2018 and was the John von Neumann postdoctoral fellow at Sandia from 2018-2020. At Sandia, he works broadly in the fields of turbulence modeling, model reduction, and scientific machine learning. His present work is primarily targeted at developing an improved modeling capability for hypersonic turbulent flows.


[1] R. Zwanzig, “Nonlinear generalized Langevin equations”. In: Journal of Statistical Physics, 1973

[2] H. Mori. “Transport, Collective Motion, and Brownian Motion”. In: Progress of Theoretical Physics, 1965

[3] A. Chorin, O. Hald, and R. Kupferman. “Optimal prediction with memory”. In: Physica D: Nonlinear Phenomena, 2022.

[4] E. Parish, K. Duraisamy. “A unified framework for multiscale modeling using the Mori—Zwanzig formalism and the Variational Multiscale Method”. In: arXiv:1712:09669

[5] E. Parish, K. Duraisamy. “A dynamic subgrid scale model for Large Eddy Simulations based on the Mori-Zwanzig formalism”. In: Journal of Computational Physics, 2017.

[6] E. Parish, K. Duraisamy. “Non-Markovian Closure Models for Large Eddy Sim- ulation using the Mori-Zwanzig Formalism”. In: Physical Review Fluids, 2017.