Nonlinear Processes in Oceanic and Atmospheric Flows


An adaptive method for computing invariant manifolds in unsteady, three-dimensional flows.

Author: Michal Branicki, Department of Mathematics, University of Bristol.

Names and affiliation of other authors:
Stephen Wiggins, Department of Mathematics, University of Bristol.

Oral or poster: Poster

Invariant manifolds have been widely used within the past two decades to study and visualise the processes of Lagrangian transport in two-dimensional, unsteady, advection-dominated fluid flows. In the 2D setting, the (invariant) stable and unstable manifolds of hyperbolic flow trajectories are given by material curves and the powerful (topological in nature) technique of `lobe dynamics' can be implemented relatively easily. In this work we discuss a non-trivial extension of the `invariant manifold' approach to the Lagrangian transport and present a method for computing two-dimensional invariant manifolds in 3D, aperiodically time-dependent flows. The presented method is adaptive in space which allows for detailed, computationally efficient determination of highly convoluted geometry of time-dependent invariant manifolds in unsteady 3D flows. We apply this procedure to compute the stable and unstable manifolds of relevant hyperbolic trajectories in a number of examples and study the associated Lagrangian transport processes using the 3D lobe dynamics. We show that the method is capable of providing detailed information on the evolving Lagrangian flow structure for long periods of time. The developed tools can be potentially very useful in providing a much needed insight into transport processes in 3D oceanic flows, obtained from 3D ocean models or from 3D data assimilation, including transport mechanisms due to the vertical coupling of the upper-and-deep ocean eddies.

Formatted version of the abstract or additional information

*Satellite images from NASA and ESA

Nonlinear Processes in Oceanic and Atmospheric Flows. July 2-4, 2008. Castro Urdiales, Cantabria, Spain.