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

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

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

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

5. Bathymetry

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

7. Trajectory Computation

8. 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

9. 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.

10. Algorithm

11. Redistribution
Redistribution algorithm due to Dritschel [1989]

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

13. 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

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

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

16. New Method

17. New Method

18. New Method

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

20. Day 1

21. Day 3

22. Day 5

23. Day 7

24. 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 = total machine – hours
Old Method 15*169 = 2185 machine-hours = 3 ½ MONTHS!!!

25. 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

26. 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, 1989.
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, 2003.
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