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Trajectory and Invariant Manifold Computation for Flows in the Chesapeake Bay

This study focuses on the computation of trajectories and invariant manifolds for oceanographic flows in the Chesapeake Bay. The algorithm used linearizes the vector field, evolves the segments in time, and interpolates new nodes. Results show the observation of interesting fine-scale structures near boundaries.

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Trajectory and Invariant Manifold Computation for Flows in the Chesapeake Bay

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

  2. Acknowledgements • Advisers • 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 • 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 80-100 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 = 50 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

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