1. Normal Mode Analysis of the Chesapeake Bay

2. USNA 13FEB05 Trident: Grant I. Gillary Advisers: Professor Reza Malek-Madani Assistant Professor Kevin McIlhany Company Officer LT Wilson, USN

3. Overview Purpose: Solve for a basis set of surface currents in the Chesapeake Bay using the eigenmodes of the potentials. Purpose: Solve for a basis set of surface currents in the Chesapeake Bay using the eigenmodes of the potentials. The basis set corresponds to a set of vector fields which can be derived from the eigenmodes of the potentials. The basis set corresponds to a set of vector fields which can be derived from the eigenmodes of the potentials. This basis set provides: This basis set provides: –Language necessary for complete description of any surface current. –Method to extrapolate empirical data from a small portion of the body of water to the rest of the domain.

4. The Representation of a Vector Field by Two Scalar Potentials Non-divergent vector fields can be represented by two scalar potentials. Non-divergent vector fields can be represented by two scalar potentials. is the stream potential is the stream potential is the velocity potential. is the velocity potential. The potentials can be separated into two eigenvalue equations. The potentials can be separated into two eigenvalue equations.

5. Eigenvalue-Eigenfunction Equations for the Basis Set The solutions to these eigenvalue-eigenfunction equations are the basis set for that domain. The solutions to these eigenvalue-eigenfunction equations are the basis set for that domain. – – – The potential takes into account any forcing functions at the boundary such as tidal forces.

6. Finite Difference Method

7. FEMLAB Throughout these calculations FEMLAB was used as a comparison for the solutions obtained using the finite differencing method. Throughout these calculations FEMLAB was used as a comparison for the solutions obtained using the finite differencing method. FEMLAB is an off-the-shelf finite element program which is generally used to solve multi-physics problems. FEMLAB is an off-the-shelf finite element program which is generally used to solve multi-physics problems.

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

14. Neumann Toy Problems The eigenvalue problem with Neumann boundary conditions was then solved on the same toy geometries. The eigenvalue problem with Neumann boundary conditions was then solved on the same toy geometries. Using Using The centered finite difference approximation was used to apply the boundary condition. The centered finite difference approximation was used to apply the boundary condition. At corners on the boundary both the x and y derivatives were set to zero to approximate the normal derivative at the corner. At corners on the boundary both the x and y derivatives were set to zero to approximate the normal derivative at the corner.

21. The Chesapeake Bay A picture of the bathymetry of the Chesapeake Bay obtained from the USCG website was used to extract the boundary of the Chesapeake Bay. A picture of the bathymetry of the Chesapeake Bay obtained from the USCG website was used to extract the boundary of the Chesapeake Bay. Using this boundary the eigenvalue problem with Dirichlet and Neumann boundary conditions was solved for the Chesapeake Bay. Using this boundary the eigenvalue problem with Dirichlet and Neumann boundary conditions was solved for the Chesapeake Bay.

22. Image Processing of the Chesapeake

23. Approximated Boundaries

32. Quoddy Boundary for the Chesapeake Bay Quoddy is a finite element solution to the Navier Stokes equation for the Chesapeake Bay. Quoddy is a finite element solution to the Navier Stokes equation for the Chesapeake Bay. The boundary used in Quoddy was extracted and solved using both the finite difference method and FEMLAB. The boundary used in Quoddy was extracted and solved using both the finite difference method and FEMLAB. Quoddy data was also used to create the source terms for the inhomogenous solution. Quoddy data was also used to create the source terms for the inhomogenous solution.

42. The 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, James, Patuxent and Susquehanna. Quoddy data was used to provide source terms for the four rivers that were removed from the Western side of the Chesapeake Bay: Potomac, James, Patuxent and Susquehanna. Using Using The same discrete approximations were used as for the Neumann boundary conditions. The same discrete approximations were used as for the Neumann boundary conditions.

51. Conclusion Toy problems were solved using both finite difference and FEMLAB solutions. Toy problems were solved using both finite difference and FEMLAB solutions. The Chesapeake Bay geometry was solved for the Dirichlet, Neumann and Inhomogenous solutions. The Chesapeake Bay geometry was solved for the Dirichlet, Neumann and Inhomogenous solutions. The Neumann solutions show significant errors and can be considered an open problem. The Neumann solutions show significant errors and can be considered an open problem.

52. Application The inner product of the basis set with empirical data gives the amplitude of each mode at that time slice. The inner product of the basis set with empirical data gives the amplitude of each mode at that time slice. This method is analogous to the Galerkin method. This method is analogous to the Galerkin method.

53. Derivation of Central Differences Taylor series expansion Taylor series expansion Adding these four equations: Adding these four equations:

54. Derivation Continued For : For : Rearranging Rearranging

56. The Discrete Square