Tracking pollutants using Lagrangian Coherent Structures.

ABSTRACT

In steady ows, the notion of boundaries separating dynamically distinct regions is

not ambiguous. This is because the invariant manifolds of time-independent ows and

the critical points of time-periodic ows provide adequate information to determine

the behaviour of the solutions of these systems. However, for time dependent systems,

it is strenuous to determine the nature of their solutions due to their dependence on

time. Nevertheless, it was observed that just like steady ows, most time-dependent

systems have boundaries that prevent cross-mixing of dynamically distinct regions.

They are known as Lagrangian Coherent Structures(LCSs) and they are embedded

in time-dependent ows as robust structures that determine the ow pattern of uid

particles.

This project investigates LCSs and also employs a numerical method to compute the

Finite Time Lyapunov Exponent to detect these structures. Initially, the coherent

structures are dened as hyperbolic material lines that separate dynamically distinct

regions in an unsteady ow. Then, the LCSs are classied into attracting and repelling

structures based on their inuence on the time-dependent ow. Subsequently,

the LCSs are also dened as a second derivative ridges of the FTLE elds. This de-

nition is perceptible from the numerical computations of the double-gyre model where

the coherent structures are extracted as ridges of the computed FTLE elds. Furthermore,

we employ the Finite time Lyapunov Exponent model to carry out numerical

simulations on satellite observed surface velocities along the coast of Ghana. The aim

of this realistic application is to determine the Lagrangian Coherent Structures that

are formed in geophysical ows. Finally, based on these results, we hypothesize the

implications of a crude oil spill along the coast of Ghana. It was realized that in the

event of a spill, the oil is likely to be conned to the coast temporarily due to the

concentration of repelling LCSs. Also, for a longer time interval the oil spill is likely

to be advected from the coastline.