Stochastic superparameterization and multiscale filtering of turbulent tracers

Data assimilation or filtering combines a numerical forecast model and observations to provide accurate statistical estimation of the state of interest. In this paper we are concerned with accurate data assimilation of a sparsely observed passive tracer advected in turbulent flows using a reduced-order forecast model. The turbulent flows which contain anisotropic and inhomogeneous structures such as jets are typical in geophysical turbulent flows in atmosphere and ocean science, and passive tracers with a mean gradient can exhibit anisotropic transport with intermittent extreme events, as shown below. Stochastic superparameterization, which is a seamless multiscale method developed for large-scale models of atmosphere and ocean models without scale-gap between the resolved and unresolved scales, generates large-scale turbulent velocity fields using a significantly smaller degree of freedom compared to a direct fine resolution numerical simulation. In a large-scale model of the tracer transport, the tracer is advected by the large-scale velocity field generated by superparameterization with a parameterization of eddies, an additional eddy diffusion given by an anisotropic biharmonic diffusion. The turbulent tracer is sparsely observed in space in only the upper surface layer. These observations naturally mix the resolved and unresolved scales, and so we develop an ensemble multiscale data assimilation algorithm which provides estimates of the resolved scales using the mixed observations. The reduced-order model is 200 times cheaper than the fine resolution solution and thus enables us to increase the number of ensemble members for accurate predictions of state distributions. Numerical experiments show positive results in the estimation of the resolved scales of the tracer as well as in capturing anisotropic intermittent extreme events for the unresolved portions of the tracer field.