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Modeling Longshore Transport and Coastal Erosion due to Storms at Barrow, Alaska

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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

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

storm wind

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

Wave Refraction due to Shoaling

Snell’s Law Approximation

where

Solve for wave angle

where fis the contour angle.

Wave vector angle

Wave Heights and Wave Breaking

Wave height

Refraction effect

Shoaling effect

Breaking wave conditions

US Army Corps of Engineers

Komar & Gaughan (1972)

LeMehaute (1962)

Miche criterion

Deep water

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

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 )

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 = 212.4 miles

Required Duration = 17.07 hours

For both storms, fetch and

duration are sufficient for FDS.

Fully-Developed Sea for 35 mph Wind

Required Fetch = 342.6 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

Recasting the Longshore TransportEquation 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 1.308.

Predicted Longshore TransportRate 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)

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).

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.

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. 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-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.

Derivation of Wave Height Equation

Figure 5-42 from

Komar (1976)

Straight beach

with parallel

contour lines

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

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.

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

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

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