Stochastic uncertainty of advected curves in finite-time unsteady flows

Identifying coherent structures in unsteady flows acting over a finite-time is a well-established research area, in part due to the applicability to realistic velocities obtained from experimental, observational, or numerically generated data. More recently, there is an emerging need to understand the impact of small-scale uncertainties on larger scale structures; for example, the "stochastic parametrization" problem in climate models. This article establishes a rigorous tool in this direction, specifically quantifying the uncertainty of advected curves in the presence of small stochasticity. Explicit expressions are derived for the expectation and the variance of the curves' location. The velocity field may be unsteady and compressible, and the Wiener process driving the stochasticity can have general spatiotemporal dependence. Monte Carlo simulations are used to verify the uncertainty expressions.