Estimating stable and unstable sets and their role as transport barriers in stochastic flows

Abstract

We consider the situation of a large-scale stationary flow subjected to small-scale fluctuations. Assuming that the stable and unstable manifolds of the large-scale flow are known, we quantify the mean behavior and stochastic fluctuations of particles close to the unperturbed stable and unstable manifolds and their evolution in time. The mean defines a smooth curve in physical space, while the variance provides a time- and space-dependent quantitative estimate where particles are likely to be found. This allows us to quantify transport properties such as the expected volume of mixing as the result of the stochastic fluctuations of the transport barriers. We corroborate our analytical findings with numerical simulations in both compressible and incompressible flow situations. We moreover demonstrate the intimate connection of our results with finite-time Lyapunov exponent fields, and with spatial mixing regions.

Stochastic Sensitivity: A Computable Lagrangian Uncertainty Measure for Unsteady Flows

“…The third example, when using an explicit diffusion, may be viewed as possessing a stochastic δx, and so can be considered to fall within the type (i) uncertainty quantification realm. It is highlighted that uncertainties of type (ii)-the impact of uncertainties in the evolving dynamics on eventual Lagrangian locations-do not seem to have been explicitly addressed until the present article (though hints of this appear in some recent diffusive approaches [26,50,6,9]). Thus, a fundamental contribution to uncertainties in Lagrangian trajectories has been made.…”

“…Interest in the impact of uncertainties in the Eulerian velocity field is only recently emerging: finite-time Lyapunov [44,8] and Lagrangian diagnostics [13] calculations using ensembles of stochastic realizations, fattening of material curves [6], fuzziness imparted on stable/unstable manifolds and consequent mixing [9], surfaces across which diffusive flux is minimal [50], and a numerical method for transfer operator computation which supplants the standard initial-and/or end-time set diffusion by continuous-time diffusion [26]. The current article presents a particular framework for quantifying resulting Lagrangian uncertainties as a physically interpretable field which is easily computable using velocity data.…”

“…While such equations have been studied extensively within contexts of financial mathematics, and in theoretical analyses of noise-induced bifurcation and related phenomena [16,15,21,23,17], escape/passage times from sets [33,60,32], and controlling such systems [61,1,2], they have seen little usage in the Lagrangian coherent structure community. However, there is emerging interest in this modelling approach [26,6,9,50,20]. The drift term would be the deterministic velocity, and the form of the additional diffusion term could be chosen based on any information available from the application of interest, and would relate to the uncertainty of the Eulerian velocity.…”

“…Moreover, this circumvents brute force Monte-Carlo numerics on stochastic systems which are well-known to be computationally expensive [31]. There are closely related analyses to this paper [6,14,9] which address uncertainties in advected curves and stable/unstable manifolds using formal calculations. In this article, however, an uncertainty measure across the global flow domain, as well as a description of its anisotropy, are developed and rigorously justified.…”

Uncertainty in finite-time Lyapunov exponent computations

“…Another slight issue is that, even if FTLE ridges are relevant and exist in the flow, averaged stochastic simulations will make them less sharp, and hence more difficult to identify. (This is as it should be: uncertainties do mean that the ridge is less certain [8].) Having a theoretical error field-computable directly from the deterministic data, and incorporating the spatial resolution L r and Eulerian velocity uncertainty scale v r -is a more desirable approach.…”