Normal Modes Analysis can be used to estimate full 2D velocity vectors from radial CODAR measurements in tidal channels. It models flow as a series of eigenfunctions satisfying the boundary conditions. By fitting the radial radar data to the Normal Modes solutions, coefficients for each mode can be determined via least squares. This allows reconstruction of the full velocity field and visualization of particle trajectories over time, providing insight into tidal flow dynamics. Future work aims to validate the method against direct measurements during flow reversals and determine the optimal number of modes.

1. Normal Modes Analysis
of RiverSonde Data in a
Tidal Channel
Calvin C. Teague, Donald E. Barrick
CODAR Ocean Sensors, Ltd.
Mountain View, CA
David Honegger
Oregon State University
Corvallis, OR

2. RiverSonde Description
• UHF (435 MHz, 70-cm wavelength) radar
• Bragg scattering from 35-cm wavelength waves
• Based on SeaSonde hardware
• 1 W transmit power
• MUSIC direction finding using 3-yagi antenna array
• 5–15 m range bins, 1° angle bins
• Typically installed on a river bank

3. Antennas
Center array used for Transmit & Receive
Side arrays used for receive only

4. Typical Geometry
River Mean Flow
Radar Field of View
1 2 3
Radar

5. Newport Experiment
• CODAR student grant program
• Installed in September 2010 at Newport, Oregon
• Channel connecting Yaquina Bay to Pacific Ocean
• Tidal flow between parallel jetties

6. Experiment Location

7. Radial Vectors
2010-12-07-13:00 UTC

8. Velocity Vector Estimation
• From a single site, find along- and cross-channel
components from least-squares fit to radials
• If 2 sites are available, find full vectors by combining
radial measurements from both sites
• From a single site, use Normal Modes Analysis of radial
measurements to infer full vectors
• For arbitrary boundary, numerical solution required
• For rectangular boundary, closed-form solution possible
• Useful for dynamic flow conditions like tidal reversals

9. Normal Modes Analysis
• Assume water incompressible
• Express horizontal flow in terms of velocity potentials and stream func-
tions
• Boundary conditions
– Zero normal flow at banks
– No impedance to tangential flow at banks
– Periodic boundary at ends of analysis region
−
→
• Horizontal surface velocity vector U
−
→
U = ∇ × [z (−Ψ) + ∇ × (ˆΦ)]
ˆ z
where z is the vertical unit vector, Ψ is the stream function, and Φ is
ˆ
the velocity potential
• Allow up to 20 modes across river, only 2 along river
• Closed-form solutions in terms of sines and cosines

10. Homogeneous Equations
• Stream function satisfying Dirichlet condition at bank
∇2ψn + νnψn = 0, where ψn|Γ = 0
D , v D = −∂ψn , ∂ψn
un n
∂y ∂x
where ψn is the n-th eigenfunction of the stream function Ψ, νn is the
corresponding n-th eigenvalue and uD and vn are the velocity compo-
n
D
nents in the x and y directions, respectively.
• Velocity function satisfying Neumann condition at bank
∂φn
∇2φn + µnφn = 0, where (ˆ · ∇φn) =
λ =0
Γ ∂λ Γ
N , v N ] = ∂φn , ∂φn
[un n
∂x ∂y
where φn is the n-th eigenfunction of the velocity potential Φ, µn is the
corresponding n-th eigenvalue and ˆ is the direction perpendicular to
λ
the boundary.

11. Normal Modes Solutions
Velocity potential modes with a periodic boundary at x = ±L/2 and bank at y = ±W/2

cos(j2πx/L) cos(mπy/W )



 for j = 0, 1, 2, 3, . . . ; m = 0, 2, 4, 6, . . .



cos(j2πx/L) sin(mπy/W )




for j = 0, 1, 2, 3, . . . ; m = 1, 3, 5, 7, . . .
φn(x, y) =
sin(j2πx/L) cos(mπy/W )


 for j = 0, 1, 2, 3, . . . ; m = 0, 2, 4, 6, . . .




sin(j2πx/L) sin(mπy/W )



 for j = 0, 1, 2, 3, . . . ; m = 1, 3, 5, 7, . . .
Corresponding stream modes

cos(j2πx/L) cos(mπy/W )



 for j = 0, 1, 2, 3, . . . ; m = 1, 3, 5, 7, . . .



cos(j2πx/L) sin(mπy/W )




for j = 0, 1, 2, 3, . . . ; m = 0, 2, 4, 6, . . .
ψn(x, y) =
sin(j2πx/L) cos(mπy/W )


 for j = 0, 1, 2, 3, . . . ; m = 1, 3, 5, 7, . . .




sin(j2πx/L) sin(mπy/W )



 for j = 0, 1, 2, 3, . . . ; m = 0, 2, 4, 6, . . .

12. Velocity Mode Examples 1
150
150
100
100
50
50
0
0
-400
-400 -200
-200 0
0 200
200 400
400
83, 0, 1, Null, Null
85, 1, 0, Null, Null
350
350
300
300
250
250
j=1 200
200
m=0 150
150
100
100
50
50
0
0
-400
-400 -200
-200 0
0 200
200 400
400
86, 1, 1, Null, Null
350
300
250
j=1 200
150
m=1 100
50
0
-400 -200 0 200 400
87, 2, -1, Null, Null
350
300
250
200

13. Velocity Mode Examples 2
150
100
50
0
-400 -200 0 200 400
85, 1, 0, Null, Null
350
300
250
j=0 200
150
m=1 100
50
FitModes6-Mod
0
-400 -200 0 200 400
86, 1, 1, Null, Null
350 88, 2, 0, Null, Null
350
300
300
250
250
200
j=0 200
150
150
m=2 100
100
50
50
0
0
-400 -200 0 200 400
-400 -200 0 200 400
87, 2, -1, Null, Null
350 89, 2, 1, Null, Null
350
300
300
250
250
200
200
150

14. 2
2 2
2
Velocity Mode Examples 3
n § Length@uPsiD, n ++, title = n = ToString@nD; title = modeIDPLength@uPhiD +
ToString@nD; title = modeIDPLength@uPhiD
plt = HVectorPlot@8uPsiPnT, vPsiPnT, 8x, x1, x2, 8y, y1, y2, Frame Ø True,
x2, 8y, y1, y2, Frame Ø True,
PlotLabel Ø title, AspectRatio Ø Automatic, DisplayFunction Ø IdentityDL;
DisplayFunction Ø IdentityDL;
Print@Show@plt, Graphics@8RGBColor@1, 0, 0D, Disk@80, 0, 82, 2DD,
Disk@80, 0, 82, 2DD,
PlotRange Ø 88x1 - 10 - 0.01`, x2 + 10, 8yp1, yp2,
yp2,
ImageSize Ø modesPlotWidth, DisplayFunction Ø $DisplayFunctionDDD;F;F;
$DisplayFunctionDDD;F;F;
864, Null, Null, 1, 0
350
300
250
j=0 200
150
m=1 100
50
0
-400 -200 0 200 400
400
865, Null, Null, 2, 0
350
300
250
j=0 200
150
m=2 100
50
0
-400 -200 0 200 400
400

15. Mode Coefficients
Determination
• Evaluate model in terms of unknown mode coefficients
at each point where radar data are available
• At each point, equate sum of radial components of
model to radial radar measurement
• Repeat over all available radar measurements
• Solve overdetermined set of equations for mode
coefficients (~5000 equations in ~50 unknowns) using
least-squares
• Allow up to 20 modes across river for along-river
component (mmax), only 2 along river for both along- and
cross-river components (jmax)

16. Streamline Examples
NWPT_2010_12_07_0600 NWPT_2010_12_07_0945
2.0 2.0
300 300
1.5 1.5
250 250
y m
y m
200 200
1.0 ms 1.0 ms
150 150
100 0.5 100 0.5
50 50
200 100 0 100 200 200 100 0 100 200
0.0 0.0
x m x m
NWPT_2010_12_07_1300 NWPT_2010_12_07_1530
2.0 2.0
300 300
1.5 1.5
250 250
y m
y m
200 200
1.0 ms 1.0 ms
150 150
100 0.5 100 0.5
50 50
200 100 0 100 200 200 100 0 100 200
0.0 0.0
x m x m

17. Mode Limits
NWPT_2010_12_07_1530
2.0
300
1.5
250
u: jmax = 1, mmax = 5
y m
200
1.0 ms
v: jmax = 0, mmax = 2 150
100 0.5
50
200 100 0 100 200
0.0
x m
NWPT_2010_12_07_1530
2.0
300
1.5
250
u: jmax = 1, mmax = 20
y m
200
1.0 ms
v: jmax = 0, mmax = 2 150
100 0.5
50
200 100 0 100 200
0.0
x m

18. Lagrangian Particle Trajectories
• Compute velocity vectors at 5-minute intervals
• Seed study area with 100 particles randomly placed
every 2 minutes
• Integrate particle velocity in 10-second steps
• Display 10 locations of particles with lighter color for
older positions
• Movie covers 2.5 hours around a tidal reversal

19. Particle Trajectory Example
2010 12 07 08:30:00 0000
350
300
250
y m
200
150
100
50
200 100 0 100 200
x m

20. Summary
• For an arbitrary boundary, Normal Modes solution must be
found numerically
• For the special case of a rectangular boundary, with no normal
flow across banks and periodic continuation at open
boundaries, a closed-form solution can be found as a series of
products of sines and cosines
• Least-squares fit of radial components of Normal Modes to
radar radial velocity vectors gives coefficients
• Lagrangian visualization of particle trajectories may be useful in
dynamic conditions like tidal reversals
• Future studies
• Compare this 2D fitting to 1D radial data with ADCP or other
in-situ measurements, especially during flow reversals
• Determine how many modes are meaningful