Normal Mode Analysis of the Chesapeake Bay

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Normal Mode Analysis of the Chesapeake Bay
Oct/24/05
FEMLAB Conference 2005
Kevin McIlhany, Physics Dept. USNA Lt. Grant I.
Gillary, USMC Reza Malek-Madani, Math Dept. USNA
http//web.usna.navy.mil/rmm/
ONR N0001405WR20243
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Outline

• Problem Statement / History
• Motivation / Applications
• Methods Applied Toy Problems
• Chesapeake Bay Results
• Analysis / Conclusions

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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

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Methods of solution

• Zeldovich (1985) Velocity vectors fields can
  be extracted from two scalar potentials
• 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.

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Putting It All Together

• is the stream potential (vorticity mode).
• is the velocity potential (divergent mode).
• Situation analogous to (E,B) fields from EM.
• The vector field representation can be separated
  into two eigenvalue equations.
• Source term solved via Poissons equation.
• The total vector field is written as a sum over
  all states for each representation.

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Outline

• Problem Statement / History
• Motivation / Applications
• Methods Applied Toy Problems
• Chesapeake Bay Results
• Analysis / Conclusions

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Motivation
Collection of Velocity Data
Incomplete Velocity Data
Normal Mode Analysis

• Waves
• Inadequate
• Infrastructure
• Covert Access
• Processing Errors

• HF Radar
• Langrangian Drifters
• Naval Ships

• Building Blocks
• Completion of Velocity Field

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This project was motivated by the seminal work
of Eremeev, Kirwan, Lipphardt and Wiggins.
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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

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Outline

• Problem Statement / History
• Motivation / Applications
• Methods Applied Toy Problems
• Chesapeake Bay Results
• Analysis / Conclusions

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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.

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Dirichlet Test Problems

• The eigenvalue problem was solved using Dirichlet
  boundary conditions on the square, circle and
  equilateral triangle where
  and .
• The zero value Dirichlet boundary condition was
  applied by removing rows and columns in the
  differentiation matrix corresponding to the
  boundary nodes.

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Neumann Test Problems

• The eigenvalue problem with Neumann boundary
  conditions was solved on the same test geometries
  where
• and
  .
• The centered finite difference approximation was
  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.

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Outline

• Problem Statement / History
• Motivation / Applications
• Methods Applied Toy Problems
• Chesapeake Bay Results
• Analysis / Conclusions

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Stream and Velocity Potentials in the Chesapeake
Bay

• The QUODDY boundary for the Chesapeake Bay 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.
• Amess method with 5-point configuration used.

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Image Processing of the Chesapeake
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Approximated Boundaries
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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
  .
• 
  
• The same discrete approximations were used as for
  the Neumann boundary conditions.

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Outline

• Problem Statement / History
• Motivation / Applications
• Methods Applied Toy Problems
• Chesapeake Bay Results
• Analysis / Conclusions

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Analysis

• Checks for orthogonality of all modes show most
  modes had inner products (vs. lowest eigenmode)
  less then 10 .
• Compare eigenvalues between FEMLAB and Finite
  Difference schemes.
• Study convergence of eigenvalue of lowest mode
  vs. changes in resolution.

-9

• Concurrent analysis modeling fluid packets from
  integration of Navier-Stokes equation.

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Future Work

• Further analysis of FEMLAB solutions
• Extend calculation into 3D
• Multi-layered 2D vs full 3D
• Lagrangian drifters are beginning to collect data
  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.

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Monterey Bay

• Lipphardt et al. at Univ. of Deleware have
  continued work on Monterey Bay, developing near
  real-time Nowcasts.
• http//newark.cms.udel.edu/brucel/slmaps/
• http//newark.cms.udel.edu/brucel/realtimemaps

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Conclusion

• FEMLAB is a suitable environment to study the
  surface 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/

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Acknowledgements

• USNA Trident Scholar program gave opportunity
  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 DArchangelo
• Professors Denny Kirwan and Bruce Lipphardt - UD
• Mrs. Lisa Becktold and the CADIG staff
• Dr. Tom Gross - NOAA
• ONR grant N0001405WR20243