scholarly journals Finite-Time Lagrangian Transport Analysis: Stable and Unstable Manifolds of Hyperbolic Trajectories and Finite-Time Lyapunov Exponents

2009 ◽  
Author(s):  
Michal Branicki ◽  
Stephen Wiggins
Keyword(s):  
Lyapunov Exponents ◽  
Finite Time ◽  
Stable And Unstable Manifolds ◽  
Lagrangian Transport ◽  
Finite Time Lyapunov Exponents ◽  
Transport Analysis ◽  
Unstable Manifolds ◽  
Hyperbolic Trajectories

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

2010 ◽  
Vol 17 (1) ◽  
pp. 1-36 ◽  
Author(s):  
M. Branicki ◽  
S. Wiggins
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
Vector Fields ◽  
Transient Flow ◽  
Time Interval ◽  
Stable And Unstable Manifolds ◽  
Lagrangian Transport ◽  
Transport Barriers ◽  
Unstable Manifolds ◽  
Hyperbolic Trajectories

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 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.

Hyperbolic neighbourhoods as organizers of finite-time exponential stretching

2016 ◽  
Vol 807 ◽  
pp. 509-545 ◽  
Author(s):  
Sanjeeva Balasuriya ◽  
Rahul Kalampattel ◽  
Nicholas T. Ouellette
Keyword(s):  
Finite Time ◽  
Time Interval ◽  
Stable And Unstable Manifolds ◽  
Periodic Flows ◽  
Chaotic Flow ◽  
Finite Time Lyapunov Exponent ◽  
Exponential Stretching ◽  
Unstable Manifolds ◽  
Hyperbolic Trajectories ◽  
Specific Sense

Hyperbolic points and their unsteady generalization – hyperbolic trajectories – drive the exponential stretching that is the hallmark of nonlinear and chaotic flow. In infinite-time steady or periodic flows, the stable and unstable manifolds attached to each hyperbolic trajectory mark fluid elements that asymptote either towards or away from the hyperbolic trajectory, and which will therefore eventually experience exponential stretching. But typical experimental and observational velocity data are unsteady and available only over a finite time interval, and in such situations hyperbolic trajectories will move around in the flow, and may lose their hyperbolicity at times. Here we introduce a way to determine their region of influence, which we term a hyperbolic neighbourhood, that marks the portion of the domain that is instantaneously dominated by the hyperbolic trajectory. We establish, using both theoretical arguments and empirical verification from model and experimental data, that the hyperbolic neighbourhoods profoundly impact the Lagrangian stretching experienced by fluid elements. In particular, we show that fluid elements traversing a flow experience exponential boosts in stretching while within these time-varying regions, that greater residence time within hyperbolic neighbourhoods is directly correlated to larger finite-time Lyapunov exponent (FTLE) values, and that FTLE diagnostics are reliable only when the hyperbolic neighbourhoods have a geometrical structure that is ‘regular’ in a specific sense.

scholarly journals Surface Ocean Mixing Inferred from Different Multisatellite Altimetry Measurements

2010 ◽  
Vol 40 (11) ◽  
pp. 2466-2480 ◽  
Author(s):  
Francisco J. Beron-Vera ◽  
María J. Olascoaga ◽  
Gustavo J. Goni
Keyword(s):  
Lyapunov Exponents ◽  
Coherent Structures ◽  
Diagnostic Tool ◽  
Turbulent Mixing ◽  
Finite Time ◽  
The Other ◽  
Sea Surface ◽  
Anomaly Field ◽  
Finite Time Lyapunov Exponents ◽  
Surface Ocean

Abstract Two sea surface height (SSH) anomaly fields distributed by Archiving, Validation, and Interpretation of Satellite Oceanographic (AVISO) Altimetry are evaluated in terms of the effects that they produce on mixing. One SSH anomaly field, tagged REF, is constructed using measurements made by two satellite altimeters; the other SSH anomaly field, tagged UPD, is constructed using measurements made by up to four satellite altimeters. Advection is supplied by surface geostrophic currents derived from the total SSH fields resulting from the addition of these SSH anomaly fields to a mean SSH field. Emphasis is placed on the extraction from the currents of Lagrangian coherent structures (LCSs), which, acting as skeletons for patterns formed by passively advected tracers, entirely control mixing. The diagnostic tool employed to detect LCSs is provided by the computation of finite-time Lyapunov exponents. It is found that currents inferred using UPD SSH anomalies support mixing with characteristics similar to those of mixing produced by currents inferred using REF SSH anomalies. This result mainly follows from the fact that, being more easily characterized as chaotic than turbulent, mixing as sustained by currents derived using UPD SSH anomalies is quite insensitive to spatiotemporal truncations of the advection field.

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