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 3Dimensional
 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 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 nonautonomous dynamical
systems
x  x
y  x 2  y
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
Wu
j 
j 
{x1 , x2
x j  x j 1
xj
2
xN }
x
x
j
j 1
w j 1 j 1  w j j
w j 1  w j
 j   j   j 1  / 2
 x j 1    x j 1  x j 
,
 x j   x j 1  x j
wj 
2
x
j
 x j 1 
x j 1  x j
x j 1  x j
2
 2
Redistribution
 j    j x j 1  x j
 nold 
nnew     j   2
 j

j 1
nold
l  j p 
 i  1

nnew
k 1

Redistribution algorithm due to Dritschel
[1989]
Chesapeake Results



Hyperbolicity
appears connected
to behavior near
boundaries
Manifolds observed
in few locations
Interesting finescale structure
observed
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

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 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 machinehours = 3 ½ MONTHS!!!

Accomplishments

Improvement of SLM Algorithm


Implementation of algorithms in
MATLAB


Weekend run on a single-processor
workstation
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, 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