Normal Mode Analysis of the
Chesapeake Bay
USNA
13FEB05
Trident:
Grant I. Gillary
Advisers:
Professor Reza Malek-Madani
Assistant Professor Kevin McIlhany
Company Officer
LT Wilson, USN
Overview
• 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.
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.
The Representation of a Vector
Field by Two Scalar Potentials
u     nˆ      nˆ  
• Non-divergent vector fields can be represented
•
•
•
by two scalar potentials.
 is the stream potential
 is the velocity potential.
The potentials can be separated into two
eigenvalue equations.
Eigenvalue-Eigenfunction Equations
for the Basis Set
• The solutions to these eigenvalue-eigenfunction
equations are the basis set for that domain.
–
2  Dn  n  Dn
–
2 Nm  m Nm (nˆ  N ) |boundary  0
–
 D |boundary  0
 2 ( x, y , 0, t )  S  (t )
(mˆ ) |boundary  (mˆ  umod el ) |boundary
The  potential takes into account any forcing functions
at the boundary such as tidal forces.
Finite Difference Method
   
2

-4   i 1   i 1   j 1   j 1
h
2
 
 1 
 4 1 0 1 0 0 0 0 0   1 
 
 1 4 1 0 1 0 0 0 0    
 2

 2
3 
 0 1 4 0 0 1 0 0 0    3 
 

 
4 
1 0 0 4 1 0 1 0 0    4 


1 
0 1 0 1 4 1 0 1 0    5      5 
2
h 
 
 
 6 
 0 0 1 0 1 4 0 0 1    6 
 
 0 0 0 1 0 0 4 1 0    
 7

 7
 8 
 0 0 0 0 1 0 1 4 1    8 
 
 0 0 0 0 0 1 0 1 4    

 9
 9
FEMLAB
• 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.
Dirichlet Toy Problems
• The eigenvalue problem was solved using
Dirichlet boundary conditions on the
square, circle and equilateral triangle.
D
2
D
D







|boundary  0
n
• Using
• The zero boundary condition was applied
by removing rows and columns in the
differentiation matrix corresponding to
boundary nodes.
n
n
Neumann Toy Problems
• The eigenvalue problem with Neumann
•
•
•
boundary conditions was then solved on the
same toy geometries.
2
N
N
N
ˆ






(
n

) |boundary  0
Using
m
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.
m
m
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.
Image Processing of the Chesapeake
Approximated Boundaries
Quoddy Boundary 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.
• Quoddy data was also used to create the
source terms for the inhomogenous solution.
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.
• Using
 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.
Conclusion
• 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 Neumann solutions show significant errors and
can be considered an open problem.
Application
• The inner product of the basis set with
empirical data gives the amplitude of each
mode at that time slice.
N
M
u ( x, y, 0, t )   An (0, t )u ( x, y )   Bm (0, t )umN ( x, y ) u i ( x, y, 0, t )
n 1
D
n
N
m 1
M
v( x, y, 0, t )   An (0, t )v ( x, y )   Bm (0, t )vmN ( x, y ) v i ( x, y, 0, t )
n 1
D
n
m 1
• This method is analogous to the Galerkin
method.
Derivation of Central Differences
• Taylor series expansion
1
1
 (a  x, b)   (a, b)  x x (a, b)  x 2  xx (a, b)  x 3 xxx (a, b)  ...
2
6
1
1
 (a  x, b)   (a, b)  x x (a, b)  x 2  xx (a, b)  x 3 xxx (a, b)  ...
2
6
1
1
 (a  y, b)   (a, b)  y y (a, b)  y 2  yy (a, b)  y 3 yyy (a, b)  ...
2
6
1
1
 (a  y, b)   (a, b)  y y (a, b)  y 2  yy (a, b)  y 3 yyy (a, b)  ...
2
6
• Adding these four equations:
 (a  x, b)   (a  x, b)   (a  y, b)   (a  y, b)  4 (a, b) 
1
y 2  yy (a, b)  x 2 xx (a, b)   ...

2
Derivation Continued
• For x  y  h :
1
 (a  h, b)   (a  h, b)   (a  h, b)   (a  h, b)  4 (a, b)  h 2 2   ...
2
• Rearranging
1
   2  4 (a, b)   (a  h, b)   (a  h, b)   (a, b  h)   (a, b  h) 
h
2
The Discrete Square

1  4  7
 2 5 8
3 6 9