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

Abstract

Abstract.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 characterising 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 finitetime 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 area of dynamical systems theory.Correspondence to: M. Branicki

Hyperbolicity in temperature and flow fields during the formation of a Loop Current ring

“…Strong FTLE ridges with a persistent intersection are generally good surrogates for these manifolds, with the critical trajectory coinciding with the intersection (Peacock and Dabiri, 2010;Branicki and Wiggins, 2010). Before analyzing the relationship between the FTLE, velocity, and temperature fields (see Sect.…”

“…The temperature, velocity, and SSHa fields used in the analyses are from a Gulf of Mexico forecast system based on the HYbrid Coordinate Ocean Model (HYCOM) (Bleck, 2002;Chassignet et al, 2007). The implementation used here has ≈ 4 km horizontal resolution and 20 layers in the vertical.…”

“…As such they capture the magnitude of dispersion along the main axis of separation. Ridges (curves of local maxima) of the FTLE field can, under certain conditions (Branicki and Wiggins, 2010), be used as surrogates for repelling and attracting manifolds that form distinguished material transport barriers. It is in this sense that we apply FTLEs here in the context of LCR formation.…”

“…If the FTLEs are faithful surrogates of distinguished material boundaries, then these LCS will intersect at a distinguished hyperbolic trajectory. Otherwise, FTLEs merely indicate high strain regions and do not necessarily delineate material advective transport boundaries (Branicki and Wiggins, 2010).…”

Path-integrated Lagrangian measures from the velocity gradient tensor

“…These structures govern the stretching and folding mechanisms that control flow mixing. A variety of definitions of LCS in aperiodic flows have been developed and discussed during the last years (Haller, 2000(Haller, , 2011Shadden et al, 2005;Branicki and Wiggins, 2010). However, the identification of hyperbolic LCS with the FTLE method exhibits some drawbacks.…”