  We develop time-series machine learning (ML) methods for closure modeling of
the Unsteady Reynolds Averaged Navier Stokes (URANS) equations applied to
stably stratified turbulence (SST). SST is strongly affected by fine balances
between forces and becomes more anisotropic in time for decaying cases.
Moreover, there is a limited understanding of the physical phenomena described
by some of the terms in the URANS equations. Rather than attempting to model
each term separately, it is attractive to explore the capability of machine
learning to model groups of terms, i.e., to directly model the force balances.
We consider decaying SST which are homogeneous and stably stratified by a
uniform density gradient, enabling dimensionality reduction. We consider two
time-series ML models: Long Short-Term Memory (LSTM) and Neural Ordinary
Differential Equation (NODE). Both models perform accurately and are
numerically stable in a posteriori tests. Furthermore, we explore the data
requirements of the ML models by extracting physically relevant timescales of
the complex system. We find that the ratio of the timescales of the minimum
information required by the ML models to accurately capture the dynamics of the
SST corresponds to the Reynolds number of the flow. The current framework
provides the backbone to explore the capability of such models to capture the
dynamics of higher-dimensional complex SST flows.
