1. Modeling Longshore Transport and Coastal Erosion due to
Storms at Barrow, Alaska
Scott D. Peckham and
James P.M. Syvitski
INSTAAR, University of Colorado, Boulder
March 11, 2004, Boulder, Colorado
34th Annual International Arctic Workshop

2. This coastal erosion modeling project is part of a larger NSF
Arctic System Science (ARCSS) Program project entitled:
An Integrated Assessment of the Impacts of Climate
Variability on the Alaskan North Slope Coastal Region
Principal Investigators:
Amanda Lynch, Ronald Brunner, Judith Curry, James
Maslanik, Linda Mearns, Anne Jensen, Glenn Sheehan
and James Syvitski
It is also part of HARC: Human Dimensions of the Arctic

3. Geography Near Barrow, Alaska
storm wind

4. Geography Near Barrow, AK

5. Beach Sediment at Barrow, AK

6. Characteristics of Ocean Waves
Wave number
L = wavelength (m)
Wave frequency
T = wave period (s)
Dispersion relation, Airy waves
shallow
( d < 0.07 L )
Phase velocity
or celerity
deep
( d > 0.28 L )
Group
velocity

7. Wave Refraction due to Shoaling
Snell’s Law Approximation
where
Solve for wave angle
where f is the contour angle.
Wave vector angle

8. Wave Heights and Wave Breaking
Refraction effect
Shoaling effect
Breaking wave conditions
US Army Corps of Engineers
Komar & Gaughan (1972)
LeMehaute (1962)
Miche criterion
Deep water

9. Nearshore Sediment Transport
Semi-empirical wave power formulation (Komar and Inman, 1970)
Each variable on right-hand side is evaluated at the breaker zone.
Longshore wave power
Immersed-weight transport rate
( K = 0.7 for sand and K is unitless )
Volumetric transport rate (m3 / sec)
( a’ = (1 - p), usually taken as 0.6 )
Bed erosion and deposition
Different grain sizes

10. Waves on a Fully-Developed Sea
Fetch (miles) required to
reach fully-developed sea
state vs. wind speed (mph)
Duration (hours) required to
reach fully-developed sea
state vs. wind speed (mph)
Wave characteristics
for peak frequency of
fully developed sea
( ft )
( sec )
Theoretical wave spectrum for a fully-developed sea
( ft )

11. Large Storms at Barrow, Alaska
August 2000 Storm
October 1963 Storm
Winds from west
Fetch of 500 miles
9 continuous hours over 35 mph
16 continuous hours over 30 mph
Winds from west
Fetch of 360 miles
14 continuous hours over 35 mph
18 continuous hours over 30 mph
Fully-Developed Sea for 30 mph Wind
Required Fetch = miles
Required Duration = hours
For both storms, fetch and
duration are sufficient for FDS.
Fully-Developed Sea for 35 mph Wind
Required Fetch = miles
Required Duration = 23.6 hours
For both storms, fetch is sufficient
for FDS, but duration is too short.
For a fully-developed sea in equilibrium with 30 mph wind, the dominant
deep-water wave characteristics are predicted by theory to be:
H = 9.6 feet, L = 189 feet, T = 7.4 seconds

12. Recasting the Longshore Transport Equation in Terms of FDS Values
Volumetric transport rate ( m3 / s )
Longshore wave power
Snell’s law
Wave speed, shallow
deep
Breaking wave height in terms of FDS values
Combining these equations with those that give H, L and T in terms of
wind speed, w, for a fully-developed sea, we can write Q as:
( m3 / sec )
To change units to ( m3 / hour ), multiply Q by 3600.
To change units to ( yard3 / hour ), multiply Q by

13. Predicted Longshore Transport Rate for Storms at Barrow, AK
For Barrow’s coastline, with a sustained wind from due west, we have
degrees, with a value of about 39 degrees at Barrow.
Beach material is black, pea-sized gravel (0.5 to 1 cm), so we will use
K = 0.45, rs = 3000 kg / m3, and a’ = 0.6 as initial estimates.
For a fully-developed sea with a wind speed of 30 mph, we get:
Q = 3934 (m3 / hour) or Q = 5145 (yd3 / hour)
as the estimated sediment transport rate during both the August 2000
and October 1963 storms.
For FDS conditions and a wind speed of 35 mph, we would get:
Q = 10,230 (m3 / hour) or Q = 13,381 (yd3 / hour)

14. Longshore Transport vs. Wind

15. Longshore Transport vs. Wind

16. Simplified Coastal Erosion Model
y = coastline position measured from a meridian line
x = distance along this meridian line
t = time
d = closure depth
Q = longshore sediment transport rate
a = incoming wave angle
Note that coastal erosion is predicted to be zero at points
where Q reaches a maximum with respect to alpha (45 degrees)
(as near Barrow) and for straight coastlines (via last term).

17. Conclusions
The physical cause of longshore currents due to waves is well-understood
and is well-described by mathematical models.
Sediment transport due to longshore currents is less well understood, but can
be described by an empirically validated formula due to Komar. This formula predicts maximum sediment transport for a breaking wave angle of 45 degrees, which is close to the value near Barrow, Alaska.
A “nodal point” is predicted to occur near the city of Barrow, Alaska with erosion south of this point and deposition north of this point. This agrees
with an analysis of remotely-sensed images.
Longshore transport is a highly nonlinear function of sustained wind speed,
so that relatively small changes in wind speed result in very large changes in sediment transport rates.
The results presented here provide a practical method for residents of Barrow,
Alaska to assess risk using real-time wind data and to understand why erosion
or accretion occurs at particular locations along the coast.

18. References
Dean, R.G. and Dalrymple, R.A. (19??) Coastal Processes: with Engineering Applications, Cambridge University Press.
Ebersole, B.A. and Dalrymple, R.A. (1979) A numerical model for nearshore circulation including convective accelerations and lateral mixing, Technical Report No. 4, Contract No. N C-0342, with ONR Research Geography Programs, Ocean Engineering Report No. 21, Dept. of Civil Engineering, Univ. of Delaware, Newark.
Komar, P.D. (1998) Beach Processes and Sedimentation, 2nd ed., Prentice-Hall Inc., New Jersey, 544 pp.
Lighthill, J. (1978) Waves in Fluids, Cambridge University Press, 504 pp.
Longuet-Higgins, M.S. (1970) Longshore currents generated by obliquely incident sea waves, J. Geophysical Research, 75,
Longuet-Higgins, M.S. and Stewart, R.W. (19 64) Radiation stress in water waves: A physical discussion with application, Deep Sea Research, 11,
Martinez, P.A. and Harbaugh, J.W. (1993) Simulating Nearshore Environments, Pergamon Press, New York.
U.S. Army Corps of Engineers (1984) Shore Protection Manual, Vol. 1, Coastal Engineering Research Center, Vicksburg, Mississippi.

20. Conclusions
The physical cause of longshore currents due to waves is well-understood
and is well-described by mathematical models.
The following physical effects can all be captured by this model:
(1) conservation of mass and momentum (and energy)
(2) refraction of waves due to shoaling
(3) the effects of waves interacting with mean currents
(4) the effect of breaking waves in the surf zone
(5) lateral mixing across the breaker zone due to turbulence
(6) wave set-up and set-down (surface displacement)
(7) rip currents and vortices
Sediment transport due to longshore currents is less well understood, but
can be described by an empirically validated formula due to Komar.
This last part of the model is not yet fully implemented.
Further work is need to fully understand the nature of numerical
instabilities in the model and the approach to steady state.

21. Derivation of Wave Height Equation
Figure 5-42 from
Komar (1976)
Straight beach
with parallel
contour lines

22. Mass and Momentum Conservation
Equations of motion are depth-integrated and time-averaged over one wave period.
These can be iterated until steady-state conditions are achieved.
Wave set-up and set-down

23. Radiation Stress Terms
Radiation stress includes excess momentum flux and pressure terms
due to waves. Original formulation due to Longuet-Higgins (1962).
Where q is the wave angle, E is the energy density of the wave per unit area, H is the wave height, d is the water depth, k is the wave number and Cg is the group velocity.

24. Frictional Loss via Bed Stress
Bed stress has a quadratic dependence on the total velocity, which is composed of
mean currents (U, V) and orbital velocities due to waves (u cos(q), v cos(q)).
Shear stress on the bed
where the total velocity is given by
Wave orbital velocity
Max orbital velocity

25. Equations that Govern Waves
Wave number vector field k is irrotational
( This implies that k = grad(f). )
Dispersion relation between frequency and wave number
Solution via Newton iteration
Wave-current interaction equation

26. Waves on a Fully-Developed Sea