Reduced Data-Driven Parametrizations for Turbulent Flow

It is well known that the wide range of spatial and temporal scales present in geophysical flow problems represents a (currently) insurmountable computational bottleneck, which must be circumvented by a coarse-graining procedure. The effect of the unresolved fluid motions enters the coarse-grained equations as an unclosed forcing term, denoted as the `eddy forcing. Traditionally, the system is closed by approximate deterministic closure models, i.e. so-called parametrizations. Instead of creating a deterministic parametrization, some recent efforts have focused on creating a stochastic, data-driven surrogate model for the eddy forcing from a (limited) set of reference data, with the goal of accurately capturing the long-term flow statistics. Since the eddy forcing is a dynamically evolving field, a surrogate should be able to mimic the complex spatial patterns displayed by the eddy forcing. Rather than creating such a (fully data-driven) surrogate, we propose to precede the surrogate construction step by a procedure that replaces the eddy forcing with a new source term which: i) is tailor-made to capture spatially-integrated quantities of interest, ii) strikes a balance between physical insight and data-driven modelling , and iii) significantly reduces the amount of training data that is needed. Instead of creating a surrogate model for an evolving field, we now only require a surrogate model for one scalar time series per quantity-of-interest. We derive the new source terms for a simplified an ocean model of two-dimensional turbulence in a doubly periodic square domain, and show that the time-series training data produces the same statistics for our quantities of interest as the exact (full-field) eddy-forcing term, which is extracted from a high-resolution reference model. Additionally, we discuss the challenges of training data-driven surrogates models which are coupled to physical partial-differential equations of the (macroscopic) flow variables, and the role that uncertainty quantification plays in the prediction of such coupled systems.