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Center for Atmosphere Ocean ScienceCourant Institute of Mathematical Sciences, New York University
Efficient modeling of unresolved small-scale turbulence is of primary importance in simulations of large-scale geophysical fluid dynamics. In many geophysical and astrophysical settings the unresolved turbulence is not homogeneous/isotropic, being affected by rotation, stratification, moist processes, magnetism, etc., and the multiscale interactions with the resolved large scales are complex and consist of more than inertial-range energy transfer. Furthermore, small-scale feedback to the resolved scales is not completely determined by the resolved large scales. The random nature of the small scales requires stochastic models, which in turn can improve the robustness of ensemble-based prediction and state estimation algorithms.
Superparameterization is a multiscale framework that models unresolved scales by PDEs evolving on pseudo-physical domains embedded into the coarse grid of a general circulation model. Although the small-scale PDEs are deterministic, their chaotic/turbulent dynamics generate an effectively stochastic feedback to the large scales. Though successful in modeling tropical atmospheric moist convection, superparameterization remains computationally costly, and of limited generality.
We develop an improved framework for superparameterization that models the small-scale turbulent dynamics by stochastic, quasilinear PDEs rather than nonlinear, deterministic ones. This greatly improves the efficiency of the algorithm, and our mathematical framework for developing the large- and small-scale PDEs increases the generality of superparameterization. The resulting algorithm is developed and tested in two idealized turbulent models: the one-dimensional complex-scalar MMT equation, and two-layer quasigeostrophic turbulence. In both settings the algorithm achieves several orders of magnitude of reduction in computational cost compared to direct simulation of all scales, and produces qualitatively accurate results. This is particularly impressive in the quasigeostrophic tests where the algorithm successfully parameterizes the inverse cascade of kinetic energy from unresolved to resolved scales. Future directions include more realistic applications, and optimization of the numerical algorithm.