Data‐Driven Stochastic Lie Transport Modeling of the 2D Euler Equations

Abstract

A major challenge in geophysical and observational sciences is the representation and quantification of uncertainty in numerical predictions. Uncertainty stems from various sources, most relevantly from incomplete inclusion of all relevant physical mechanisms in the models and uncertainty in the initial and boundary conditions (T. N. Palmer, 2000). Important models for geophysical fluid dynamics, such as the two-dimensional Euler equations, quasi-geostrophic equations or rotating shallow water equations are derived from the three-dimensional Navier-Stokes equations. A sequence of simplifying assumptions is applied in order to reduce the complexity of the model to a more manageable level, while retaining main flow physics (Zeitlin, 2018). Stochastic extensions to these models have also been derived (Holm & Luesink, 2021). These approximate models are nevertheless rich in dynamics and contain a wide range of spatial and temporal scales. Numerically resolving the entire spectrum of scales is often not computationally feasible, meaning that either the complexity of the model should be reduced even further such that the resulting model is simple enough the be solvable numerically, or the complex model is represented on a coarse computational grid and unresolved scales are replaced by a sub-grid model. The latter option may be combined with stochastic forcing, which provides an effective way to represent unresolved scales in numerical simulations (Buizza et al., 1999;Frederiksen & Davies, 1997;Majda et al., 2001). The use of stochasticity as a means to represent the unresolved scales serves to restore some of the missing small-scale

Computational Modeling for High-Fidelity Coarsening of Shallow Water Equations Based on Subgrid Data

“…Randomness is introduced via the term dB t lm in which B t lm is a general random process, defined for each pair l, m separately. The random process can be tailored to fit the measurement data [18], though the common choice is to let dB t lm be normally distributed with a variance depending on the time step size [29]. We choose the latter in what follows and include the variance scaling in σ lm .…”

“…These approaches have also been applied successfully to more complete geophysical models. Examples include the modeling of uncertainty through Casimir-preserving stochastic forcing for the two-dimensional Euler equations [15,18,13] and energy-preserving stochastic forcing in the quasi-geostrophic equations [43]. An alternative approach is based on statistics of subgrid data that lead to a stochastic forcing and eddy viscosity, which has been applied to the barotropic vorticity equation on the sphere [20].…”

Sparse-stochastic model reduction for 2D Euler equations

“…Indeed, it is shown that a suitable model for large scales is given by the so called SALT equations [12], in which a transport noise term models the infinitesimal action of the small scales on the large ones. Several numerical tests have shown the usefulness of the SALT equations as a powerful tool for model reduction [6,8].…”

Subgrid Parameterization of Eddy, Meanfield and Topographic Interactions in Simulations of an Idealized Antarctic Circumpolar Current

“…(2021) and Ephrati et al. (2023) also implement stochastic forcing by exploiting the statistics from a high resolution reference case. In these studies, the full covariance matrix of the unresolved field is sampled, which is then represented using a truncated number of typically Empirical Orthogonal Functions (EOFs).…”

“…A similar approach was applied to a multi-layer double gyre primitive equation simulation in Cooper (2017). Cotter et al (2020), Gugole and Franzke (2020), Brecht et al (2021) and Ephrati et al (2023) also implement stochastic forcing by exploiting the statistics from a high resolution reference case. In these studies, the full covariance matrix of the unresolved field is sampled, which is then represented using a truncated number of typically Empirical Orthogonal Functions (EOFs).…”