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Finite-time Lagrangian Transport Analysis: Stable and Unstable Manifolds of Hyperbolic Trajectories and Finite-time Lyapunov Exponents : Volume 17, Issue 1 (05/01/2010)

By Branicki, M.

Book Id:WPLBN0003981905 Format Type:PDF Article : File Size:Pages 36 Reproduction Date:2015

Wiggins, S., & Branicki, M. (2010). Finite-time Lagrangian Transport Analysis: Stable and Unstable Manifolds of Hyperbolic Trajectories and Finite-time Lyapunov Exponents : Volume 17, Issue 1 (05/01/2010). Retrieved from http://hawaiilibrary.net/

Description
Description: School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK. 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 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 area of dynamical systems theory.

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

Excerpt
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