Modeling Longshore Transportand Coastal Erosion due toStorms at Barrow, Alaska

Scott D. Peckham andJames P.M. Syvitski

INSTAAR, University of Colorado, Boulder

March 11, 2004, Boulder, Colorado34th Annual International Arctic Workshop

This coastal erosion modeling project is part of a larger NSFArctic System Science (ARCSS) Program project entitled:

An Integrated Assessment of the Impacts of ClimateVariability on the Alaskan North Slope Coastal Region

Principal Investigators:Amanda Lynch, Ronald Brunner, Judith Curry, JamesMaslanik, Linda Mearns, Anne Jensen, Glenn Sheehanand James Syvitski

It is also part of HARC: Human Dimensions of the Arctic

Geography Near Barrow, Alaska

storm wind

Geography Near Barrow, AK

Beach Sediment at Barrow, AK

Characteristics of Ocean Waves

shallow ( d < 0.07 L )

Wave number

deep( d > 0.28 L )

Phase velocityor celerity

Groupvelocity

L = wavelength (m)

T = wave period (s)Wave frequency

Dispersion relation, Airy waves

Wave Refraction due to Shoaling

Snell’s Law Approximation

Wave vector angle

where

Solve for wave angle

where is the contour angle.

Wave Heights and Wave Breaking

LeMehaute (1962)

Breaking wave conditions

Refraction effect

Shoaling effect

Deep water

US Army Corps of Engineers

Wave height

Miche criterion

Komar & Gaughan (1972)

Nearshore Sediment Transport

Semi-empirical wave power formulation (Komar and Inman, 1970)Each variable on right-hand side is evaluated at the breaker zone.

Bed erosion and deposition

Different grain sizes

Volumetric transport rate (m3 / sec)( a’ = (1 - p), usually taken as 0.6 )

Longshore wave power

Immersed-weight transport rate( K = 0.7 for sand and K is unitless )

Waves on a Fully-Developed SeaFetch (miles) required toreach fully-developed seastate vs. wind speed (mph)

Duration (hours) required toreach fully-developed seastate vs. wind speed (mph)

Wave characteristicsfor peak frequency offully developed sea

( ft )

( ft )

( sec )Theoretical wave spectrum for a fully-developed sea

Large Storms at Barrow, Alaska

August 2000 Storm

Winds from westFetch of 500 miles9 continuous hours over 35 mph16 continuous hours over 30 mph

October 1963 Storm

Winds from westFetch of 360 miles14 continuous hours over 35 mph18 continuous hours over 30 mph

Fully-Developed Sea for 30 mph Wind Required Fetch = 212.4 miles Required Duration = 17.07 hours

Fully-Developed Sea for 35 mph Wind Required Fetch = 342.6 miles Required Duration = 23.6 hours

For both storms, fetch andduration are sufficient for FDS.

For both storms, fetch is sufficientfor FDS, but duration is too short.

For a fully-developed sea in equilibrium with 30 mph wind, the dominantdeep-water wave characteristics are predicted by theory to be:

H = 9.6 feet, L = 189 feet, T = 7.4 seconds

Recasting the Longshore TransportEquation in Terms of FDS Values

Combining these equations with those that give H, L and T in terms ofwind speed, w, for a fully-developed sea, we can write Q as:

Wave speed, shallow

Snell’s law

Longshore wave power

Volumetric transport rate ( m3 / s )

deep

To change units to ( m3 / hour ), multiply Q by 3600.To change units to ( yard3 / hour ), multiply Q by 1.308.

( m3 / sec )

Breaking wave height in terms of FDS values

Predicted Longshore TransportRate for Storms at Barrow, AK

For Barrow’s coastline, with a sustained wind from due west, we havedegrees, with a value of about 39 degrees at Barrow.

Q = 3934 (m3 / hour) or Q = 5145 (yd3 / hour)

Beach material is black, pea-sized gravel (0.5 to 1 cm), so we will useK = 0.45, s = 3000 kg / m3, and a’ = 0.6 as initial estimates.

For a fully-developed sea with a wind speed of 30 mph, we get:

as the estimated sediment transport rate during both the August 2000and 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)

Longshore Transport vs. Wind

Longshore Transport vs. Wind

Simplified Coastal Erosion Model

y = coastline position measured from a meridian linex = distance along this meridian linet = timed = closure depthQ = longshore sediment transport rate = incoming wave angle

Note that coastal erosion is predicted to be zero at pointswhere Q reaches a maximum with respect to alpha (45 degrees)(as near Barrow) and for straight coastlines (via last term).

ConclusionsThe physical cause of longshore currents due to waves is well-understoodand is well-described by mathematical models.

Sediment transport due to longshore currents is less well understood, but canbe 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 agreeswith 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 erosionor accretion occurs at particular locations along the coast.

ReferencesDean, 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. N0014-76-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, 6778-6789.

Longuet-Higgins, M.S. and Stewart, R.W. (19 64) Radiation stress in water waves: A physical discussion with application, Deep Sea Research, 11, 529-563.

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.

Conclusions

The physical cause of longshore currents due to waves is well-understoodand 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, butcan 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 numericalinstabilities in the model and the approach to steady state.

Derivation of Wave Height Equation

Figure 5-42 fromKomar (1976)

Straight beachwith parallelcontour lines

Mass and Momentum ConservationEquations 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

Radiation Stress Terms

Radiation stress includes excess momentum flux and pressure termsdue to waves. Original formulation due to Longuet-Higgins (1962).

Where 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.

Frictional Loss via Bed Stress

Max orbital velocity

Wave orbital velocity

Bed stress has a quadratic dependence on the total velocity, which is composed ofmean currents (U, V) and orbital velocities due to waves (u cos(), v cos()).

Shear stress on the bed

where the total velocity is given by

Equations that Govern Waves

Wave number vector field k is irrotational

Dispersion relation between frequency and wave number

Wave-current interaction equation

( This implies that k = grad(f). )

Solution via Newton iteration

Waves on a Fully-Developed Sea