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