FEMLAB Conference 2005
Oct/24/05
Normal Mode Analysis of the
Chesapeake Bay
Kevin McIlhany, Physics Dept. USNA
Lt. Grant I. Gillary, USMC
Reza Malek-Madani, Math Dept. USNA
ONR# N0001405WR20243
http://web.usna.navy.mil/~rmm/
Outline
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Problem Statement / History
Motivation / Applications
Methods Applied – Toy Problems
Chesapeake Bay Results
Analysis / Conclusions
Problem: Obtain surface current
vector fields for a coastal region
• Oceanography: long history of various
schemes – packet tracking (local) vs.
systemic (global) schemes
• Eremeev et al. (1992) solved the lowest
eigenmodes of the Black Sea.
• Lipphardt et al. (2000) extends the
calculation to include forcing terms
Methods of solution
• Zel’dovich (1985): Velocity vectors fields can be
extracted from two scalar potentials
u    nˆ     nˆ  
  n  n  n
2
D
D
2 Nm  m Nm
 D |boundary  0
(nˆ  N ) |boundary  0
• Lipphardt et al. (2000): Addition of forcing terms
allows for non-conservation of mass through a
boundary – ie. Water from rivers or the ocean is
accounted.
 2 ( x, y, 0, t )  S (t )
(mˆ ) |boundary  (mˆ  umod el ) |boundary
Putting It All Together
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 is the stream potential (vorticity mode).
 is the velocity potential (divergent mode).
Situation analogous to (E,B) fields from E&M.
The vector field representation can be separated
into two eigenvalue equations.
Source term solved via Poisson’s equation.
The total vector field is written as a sum over all
states for each representation.
N
(u, v)   an (un , vn ) D 
n 1
M
b
m1
m
(um , vm ) N  (u (t ), v(t )) S
Outline
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Problem Statement / History
Motivation / Applications
Methods Applied – Toy Problems
Chesapeake Bay Results
Analysis / Conclusions
Motivation
Collection of
Velocity Data
•HF Radar
•Langrangian
Drifters
•Naval Ships
Incomplete
Velocity Data
Normal Mode
Analysis
•Waves
•Inadequate
•Infrastructure
•Covert Access
•Processing Errors
•Building Blocks
•Completion of
Velocity Field
This project was motivated by the
seminal work of:
Eremeev, Kirwan, Lipphardt and Wiggins.
Applications
•Provide surface current data for
Military Operations
•Study spread of wildlife in a body of water
•Study spread of pollution in a body of water
•Computation of particle trajectories
•Unmanned underwater vehicles
Outline
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Problem Statement / History
Motivation / Applications
Methods Applied – Toy Problems
Chesapeake Bay Results
Analysis / Conclusions
Methodology
• FEMLAB
– Using standard PDE module with default settings for
Dirichlet boundaries, solve for 100 modes.
– Varied resolution from very coarse to very fine.
– Under plot parameters, use either (ux,ux) for velocity or
(-uy,ux) for stream potential.
• Finite Differences – provided a check for FEMLAB
results – given that no analytic solutions exist.
• Created toy models of the square, circle and
triangle in order to track / understand error
propogation.
Dirichlet Test Problems
• The eigenvalue problem was solved using
Dirichlet boundary conditions on the
square, circle and equilateral triangle
where 2 D  n  D and  D |boundary  0 .
n
n
• The zero value Dirichlet boundary
condition was applied by removing rows
and columns in the differentiation matrix
corresponding to the boundary nodes.
Neumann Test Problems
• The eigenvalue problem with Neumann
boundary conditions was solved on the same
test geometries where
N
2 N  m N and (nˆ  ) |boundary  0 .
m
m
• The centered finite difference approximation was
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used to apply the boundary condition.
On the corners of the grid both the x and y
derivatives were set to zero to approximate the
normal derivative at the boundary.
Outline
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Problem Statement / History
Motivation / Applications
Methods Applied – Toy Problems
Chesapeake Bay Results
Analysis / Conclusions
Stream and Velocity Potentials
in the Chesapeake Bay
• The QUODDY boundary for the Chesapeake Bay
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was used.
Both Neumann and Dirichlet modes were solved on
the QUODDY geometry.
The finite difference method in MATLAB (up to
400x1400 nodes) and the FEMLAB (up to 140,000
elements) were used to test the consistency of the
solutions.
Ames’s method with 5-point configuration used.
Image Processing of the Chesapeake
Approximated Boundaries
Inhomogenous Modes
• Quoddy data was used to provide source terms
for the four rivers that were removed from the
Western side of the Chesapeake Bay: Potomac,
Rappahannock, York and James plus the Atlantic
Ocean
where
 2 ( x, y , 0, t )  S  (t )
(mˆ ) |boundary  (mˆ  umod el ) |boundary
.
• The same discrete approximations were used as
for the Neumann boundary conditions.
Outline
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Problem Statement / History
Motivation / Applications
Methods Applied – Toy Problems
Chesapeake Bay Results
Analysis / Conclusions
Analysis
• Checks for orthogonality of all modes
show most modes had inner products (vs.
-9
lowest eigenmode) less then 10 .
• Compare eigenvalues between FEMLAB
and Finite Difference schemes.
• Study convergence of eigenvalue of lowest
mode vs. changes in resolution.
• Concurrent analysis modeling fluid packets
from integration of Navier-Stokes equation.
Future Work
• Further analysis of FEMLAB solutions
– Extend calculation into 3D
– Multi-layered 2D vs full 3D
• Lagrangian drifters are beginning to collect data
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for the Chesapeake starting in Fall 2005.
Apply the basis set using Normal Mode Analysis
of the Chesapeake Bay.
Compare integrated (u,v) set vs. packets.
Development of a database of waterways.
Through Galerkin method, develop Nowcasts for
Chespeake Bay.
Monterey Bay
• Lipphardt et al. at Univ. of Deleware have
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continued work on Monterey Bay, developing near
real-time “Nowcasts”.
http://newark.cms.udel.edu/~brucel/slmaps/
http://newark.cms.udel.edu/~brucel/realtimemaps
Conclusion
• FEMLAB is a suitable environment to study the surface
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currents of the Chesapeake Bay.
FEMLAB – well documented (with tutorials and training
seminars) ideal for student invovlement.
Three bodies of water have successfully had eigenmodes
calculated – ranging from simple to complex boundaries
(Black Sea, Monterey Bay, Chesapeake Bay).
The Neumann problem was not validated by the analysis in
this research.
• Visit us at: http://web.usna.navy.mil/~rmm/
Acknowledgements
• USNA Trident Scholar program – gave opportunity
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to Lt. Gillary to study this problem.
USNA Chesapeake Bay Group: Reza Malek-Madani,
Kevin McIlhany, Gary Fowler, John Pierce, Irina
Popovici, Sonia Garcia, Tas Liakos, Louise
Wallendorf, Bob Bruninga, Jim D’Archangelo
Professors Denny Kirwan and Bruce Lipphardt - UD
Mrs. Lisa Becktold and the CADIG staff
Dr. Tom Gross - NOAA
ONR grant #N0001405WR20243