Trajectory and Invariant Manifold Computation for Flows in the Chesapeake Bay Nathan Brasher...
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![Page 1: Trajectory and Invariant Manifold Computation for Flows in the Chesapeake Bay Nathan Brasher February 13, 2005.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d385503460f94a11301/html5/thumbnails/1.jpg)
Trajectory and Invariant Manifold Computation for Flows in the Chesapeake Bay
Nathan Brasher
February 13, 2005
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Acknowledgements
AdvisersProf. Reza Malek-MadaniAssoc. Prof. Gary Fowler
CAD-Interactive Graphics Lab Staff
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Chesapeake Bay Analysis
QUODDY Computer ModelFinite-Element
ModelFully 3-
Dimensional9700 nodes
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QUODDY
Boussinesq Equations Temperature Salinity
Sigma Coordinates No normal flow Winds, tides and river
inflow included in model
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Bathymetry
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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
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Trajectory Computation
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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
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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.
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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
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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]
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Chesapeake Results
Hyperbolicity appears connected to behavior near boundaries
Manifolds observed in few locations
Interesting fine-scale structure observed
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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
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Old Method
Square Grid Each data point
recomputed for each time-slice
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New Method
Initial hex-mesh Advect points to
next time-slice Insert new points
to fill gaps Compute
residence time for new points only
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New Method
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New Method
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New Method
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Final Result
Scattered Data Interpolated to square grid in MATLAB for plotting purposes
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Day 1
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Day 3
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Day 5
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Day 7
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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!!!
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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
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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