Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of "transport barriers" in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterizing transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate, in a concrete manner, the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of "flow transition" which occurs when finite-time hyperbolicity is lost, or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing, and important, new area of dynamical systems theory

Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities

Here, we examine the suitability of truncated Polynomial Chaos Expansions (PCE) and truncated Gram-Charlier Expansions (GrChE) as possible methods for uncertainty quantification (UQ) in nonlinear systems with intermittency and positive Lyapunov exponents. These two methods rely on truncated Galerkin projections of either the system variables in a fixed polynomial basis spanning the ‘uncertain ’ subspace (PCE) or a suitable eigenfunction expansion of the joint probability distribution associated with the uncertain evolution of the system (GrChE). Based on a simple, statistically exactly solvable non-linear and non-Gaussian test model, we show in detail that methods exploiting truncated spectral expansions, be it PCE or GrChE, have significant limitations for uncertainty quantification in systems with intermittent instabilities or parametric uncertainties in the damping. Intermittency and fat-tailed probability densities are hallmark features of the inertial and dissipation ranges of turbulence and we show that in such important dynamical regimes PCE performs, at best, similarly to the vastly simpler Gaussian moment closure technique utilized earlier by the authors in a different context for UQ within a framework of Empirical Information Theory. Moreover, we show that the non-realizability of the GrChE approximations is linked to the onset of intermittency in the dynamics and it is frequently accompanied by an erroneou

Lagrangian structure of flows in the Chesapeake Bay:Challenges and perspectives on the analysis of estuarine flows

In this work we discuss applications of Lagrangian techniques to study transport properties of flows generated by shallow water models of estuarine flows. We focus on the flow in the Chesapeake Bay generated by Quoddy (see Lynch and Werner, 1991), a finite-element (shallow water) model adopted to the bay by Gross et al. (2001). The main goal of this analysis is to outline the potential benefits of using Lagrangian tools for both understanding transport properties of such flows, and for validating the model output and identifying model deficiencies. We argue that the currently available 2-D Lagrangian tools, including the stable and unstable manifolds of hyperbolic trajectories and techniques exploiting 2-D finite-time Lyapunov exponent fields, are of limited use in the case of partially mixed estuarine flows. A further development and efficient implementation of three-dimensional Lagrangian techniques, as well as improvements in the shallow-water modelling of 3-D velocity fields, are required for reliable transport analysis in such flows. Some aspects of the 3-D trajectory structure in the Chesapeake Bay, based on the Quoddy output, are also discussed