Uncertainty in finite-time Lyapunov exponent computations

Abstract

The Finite-Time Lyapunov Exponent (FTLE) is a well-established numerical tool for assessing stretching rates of initial parcels of fluid, which are advected according to a given time-varying velocity field (which is often available only as data). When viewed as a field over initial conditions, the FTLE's spatial structure is often used to infer the nonhomogeneous transport. Given the measurement and resolution errors inevitably present in the unsteady velocity data, the computed FTLE field should in reality be treated only as an approximation. A method which, for the first time, is able for attribute spatially-varying errors to the FTLE field is developed. The formulation is, however, confined to two-dimensional flows. Knowledge of the errors prevent reaching erroneous conclusions based only on the FTLE field. Moreover, it is established that increasing the spatial resolution does not improve the accuracy of the FTLE field in the presence of velocity uncertainties, and indeed has the opposite effect. Stochastic simulations are used to validate and exemplify these results, and demonstrate the computability of the error field.