Trajectory and Invariant Manifold Computation for Flows in the Chesapeake Bay

Nathan Brasher

February 13, 2005

Acknowledgements

AdvisersProf. Reza Malek-MadaniAssoc. Prof. Gary Fowler

CAD-Interactive Graphics Lab Staff

Chesapeake Bay Analysis

QUODDY Computer ModelFinite-Element

ModelFully 3-

Dimensional9700 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 4th 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

yxy

xx

2

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

1 2

1 1

2 2

1 1 1 1

11 1

2 21 1

1

{ , }

,

/ 2

uj N

j j j j

j

j j j j j j j j

j jj j j jj j

j j j j

j j j

W x x x x

x x x x

x x x x x x x x

x xw ww

w w x x

Redistribution

1

1

1

2

1

old

j j j j

n

new jj

jold

l jk new

x x

n

np i

n

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 AlgorithmUses data from previous time-sliceImproves efficiency and resolutionNeeds residence time computation for

80-100 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

New Method

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 cluster15 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 AlgorithmWeekend run on a single-processor

workstation Implementation of algorithms in

MATLABPlatform 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, 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