Trajectory and Invariant Manifold Computation for Flows in the Chesapeake Bay Nathan Brasher February 13, 2005

Acknowledgements Advisers Prof. Reza Malek-Madani Assoc. Prof. Gary Fowler CAD-Interactive Graphics Lab Staff

Chesapeake Bay Analysis QUODDY Computer Model Finite-Element Model Fully 3- Dimensional 9700 nodes

QUODDY Boussinesq Equations Temperature Salinity Sigma Coordinates No normal flow Winds, tides and river inflow included in model

Bathymetry

Trajectory Computation Surface Flow Computation Radial Basis Function Interpolation Runge-Kutta 4 th order method Residence Time Calculations Synoptic Lagrangian Maps Method of displaying large amounts of trajectory data

Trajectory Computation

Invariant Manifolds Application of dynamical systems structures to oceanographic flows Create invariant regions and direct mass transport Manifolds move with the flow in non- autonomous dynamical systems

Algorithm Linearize vector field about hyperbolic trajectory 5-node initial segment along eigenvectors Evolve segment in time, interpolate and insert new nodes Algorithm due to Wiggins et. al.

Algorithm

Redistribution Redistribution algorithm due to Dritschel [1989]

Chesapeake Results Hyperbolicity appears connected to behavior near boundaries Manifolds observed in few locations Interesting fine- scale structure observed

Synoptic Lagrangian Maps Improved Algorithm Uses data from previous time-slice Improves efficiency and resolution Needs residence time computation for particles to maintain ~10,000 total data points

Old Method Square Grid Each data point recomputed for each time-slice

New Method Initial hex-mesh Advect points to next time-slice Insert new points to fill gaps Compute residence time for new points only

New Method

Final Result Scattered Data Interpolated to square grid in MATLAB for plotting purposes

Day 1

Day 3

Day 5

Day 7

Computational Improvement SLM Computation no longer requires a supercomputing cluster 15 Hrs for initial time-slice + 35 Hrs to extend the SLM for a one-week computation = 50 total machine – hours Old Method 15*169 = 2185 machine- hours = 3 ½ MONTHS!!!

Accomplishments Improvement of SLM Algorithm Weekend run on a single-processor workstation Implementation of algorithms in MATLAB Platform independent for the scientific community Investigation of hyperbolicity and invariant manifolds in complex geometry

References Dritschel, D.: Contour dynamics and contour surgery: Numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows, Comp. Phys. Rep., 10, 77–146, Mancho, A., Small, D., Wiggins, S., and Ide, K.: Computation of stable and unstable manifolds of hyperbolic trajectories in two-dimensional, aperiodically time-dependent vector fields, Physica D, 182, 188–222, Mancho A., Small D., and Wiggins S. : Computation of hyperbolic trajectories and their stable and unstable manifolds for oceanographic flows represented as data sets, Nonlinear Processes in Geophysics (2004) 11: 17–33