  Developing reduced-order models for turbulent flows, which contain dynamics
over a wide range of scales, is an extremely challenging problem. In
statistical mechanics, the Mori-Zwanzig (MZ) formalism provides a
mathematically formal procedure for constructing reduced-order representations
of high-dimensional dynamical systems, where the effect due to the unresolved
dynamics are captured in the memory kernel and orthogonal dynamics. Turbulence
models based on MZ formalism have been scarce due to the limited knowledge of
the MZ operators, which originates from the difficulty in deriving MZ kernels
for complex nonlinear dynamical systems. In this work, we apply a recently
developed data-driven learning algorithm, which is based on Koopman's
description of dynamical systems and Mori's linear projection operator, on a
set of fully-resolved isotropic turbulence datasets to extract the Mori-Zwanzig
operators. With data augmentation using known turbulence symmetries, the
extracted Markov term, memory kernel, and orthogonal dynamics are statistically
converged and the Generalized Fluctuation-Dissipation Relation can be verified.
The properties of the memory kernel and orthogonal dynamics, and their
dependence on the choices of observables are investigated to address the
modeling assumptions that are commonly used in MZ-based models. A series of
numerical experiments are then constructed using the extracted kernels to
evaluate the memory effects on predictions. Results show that the prediction
errors are strongly affected by the choice of observables and can be further
reduced by including the past history of the observables in the memory kernel.
