XII Congreso Geológico Chileno
Santiago, 22-26 Noviembre, 2009
S1_017
Wave-sediment interaction in the nearshore zone
Le Roux, J.P.1, Demirbilek, Z.2, Brodalka, M.3, B.W. Flemming4
(1) Departamento de Geología, Facultad de Ciencias Físicas y Matemáticas, Universidad
de Chile, Casilla 13518, Correo 21, Santiago, Chile.
(2) U.S. Army Engineer R&D Centre, Coastal and Hydraulics Laboratory, Vicksburg,
MS 31980
(3) University of Pretoria, Pretoria, South Africa.
(4) Senckenberg Institute, Department of Marine Science, Suedstrand 40, 26382
Wilhelmshaven, Germany.
[email protected]
Introduction
Coastal processes involve waves, currents and bottom sediments, which interact to cause
either erosion or progradation. Erosion constitutes a hazard to coastal constructions and
may also destroy valuable assets such as natural beaches, whereas progradation may silt
up harbors or clog navigation channels. It is therefore essential to understand how these
processes interact from a coastal engineering point of view. These processes are also of
considerable interest to sedimentologists, oceanographers, and marine biologists or
ecologists. In this paper we focus on wave-sediment interaction in the nearshore zone,
and demonstrate an Excel program to model some of these processes.
Methodology and use of spreadsheet
Table 1 shows the equations used in the spreadsheet, of which 60% are standard and have
been applied for decades [1], [2], [3]. The remainder have only been published since 2007
[4], [5], [6], [7] and are presented here for the first time in an integrated form, providing a
seamless transition from deep water to breaking conditions. The spreadsheet requires
relatively few input data, including the sustained wind speed Ua, bottom slope α, and the
median sediment grain size D, although the water density ρ and dynamic viscosity μ can
be adapted to refine the calculations for specific conditions.
After providing the wind speed, an initial value is entered into the water depth cell and
the “Iterate” button is clicked, which calculates the breaking depth db and height Hb.
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XII Congreso Geológico Chileno
Santiago, 22-26 Noviembre, 2009
These values are used subsequently to calculate other wave parameters, and it is only
required to change the depth d to obtain the local conditions.
Changes in wave parameters with depth
In deep water, wind waves forming on the sea surface have a sinusoidal shape, but as
they begin to shoal there is an initial decrease and then an increase in the wave height Hw,
together with a decrease in wavelength Lw, wave celerity Cw, and shape expressed by
η(x), the water surface elevation at any distance x from the wave crest. Seaward of the
breaker zone, the waves change their profile from sinusoidal to trochoidal and cnoidal,
i.e. the crests as defined by the median crest diameter MCD [6] become shorter, steeper
and sharper, whereas the troughs flatten out. The MCD reaches a minimum value equal to
1/6th of the deepwater wavelength Lo, from where it remains constant up to breaking over
any bottom slope. The median trough diameter MTD remains constant until the MCD
reaches its minimum value and then decreases according to Lw - MCD. Before breaking,
the mean water surface elevation also rises relative to the still water level SWL, which is
directly related to the thickness of the boundary layer δ at the bottom.
Sediment entrainment
The changes in wave shape described above have a profound influence on the horizontal
water particle velocity at the top of the boundary layer Uδ, which determines whether
sediment will be entrained or not. First, a critical boundary velocity Uδcr is calculated
using the wave period Tw, the sediment size D and density ρs, as well as the water density
ρ and kinematic viscosity ν. This is based on the method of You and Yin [8] as modified
by Le Roux [9]. The actual boundary velocity is then calculated using the horizontal
water particle displacement or semi-excursion A at the top of the boundary layer for the
specific wave conditions and water depth, which is then compared with the critical
boundary velocity. If it exceeds the latter, sediment entrainment will take place. Here the
bottom slope α is also taken into account, as it increases the critical velocity under the
wave crest and decreases it under the trough. By varying the water depth d for any
specific wave conditions, it can be determined at which depth landward or seaward
entrainment will commence.
An example of fair-weather and stormy conditions
Consider a shoreface with a seaward bottom slope of 1º, consisting of well-sorted sand
with a median grain size of 0.5 mm. During prolonged fair-weather conditions with a
sustained wind speed of 3 m s-1, fully developed waves will be generated with a
deepwater height Ho of 0.2 m and a wavelength Lo of 5.76 m. Such waves will break at a
depth of 0.27 m and distance of 16 m from the shore, with a breaker height of 0.25 m.
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XII Congreso Geológico Chileno
Santiago, 22-26 Noviembre, 2009
Gradually increasing the water depth indicates that landward entrainment of the 0.5 mm
sediment will commence at a depth of 1.32 m, i.e. 75.6 m from the shore. However,
seaward transport will only take place from a depth of 0.31 m, 17.8 m from the shore,
practically within the breaker zone. The result of prolonged fair-weather conditions will
therefore be a net landward transport of sediments or coastal progradation.
In the case of a major storm with a sustained wind speed of 17 m s-1, the deepwater wave
height and length will be 6.55 and 185.1 m, respectively. These waves will break 499 m
from the shore with a breaker depth and height of 8.71 m and 7.97 m, respectively. Both
landward and seaward entrainment will commence at a depth of about 72 m, but because
the duration of seaward transport under the wave troughs is longer than under the crests,
net erosion of the shoreface will take place.
References
[1] Airy, G.B. (1845) Tides and Waves. Encyclop. Metrop., Article 192, 241-396.
[2] Boussinesq, J. (1871) Theorie de l’intumescence liquide appelee onde solitaire ou de
translation se propageant dans un canal rectangulaire. Comptes Rendus Acad. Sci. Paris,
vol. 72, 755-759.
[3] Korteweg, D.J., De Vries, G. (1895) On the change in form of long waves advancing
in a rectangular channel, and on a new type of stationary waves. Phil. Mag., 5th Series 39,
422-443.
[4] Le Roux, J.P. (2007) A simple method to determine breaker height and depth for
different deepwater height/length ratios and sea floor slopes. Coastal Engineering, vol.
54, 271-277.
[5] Le Roux, J.P. (2007) A function to determine wavelength from deep into shallow
water based on the length of the cnoidal wave at breaking. Coastal Engineering, vol. 54,
770-774.
[6] Le Roux, J.P. (2008) An extension of the Airy theory for linear waves into shallow
water. Coastal Engineering, vol. 55, 295-301.
[7] Le Roux, J.P. (2008) Profiles of fully developed (Airy) waves in different water
depths. Coastal Engineering, vol. 55, 701-703.
[8] You, Z.J. Yin, B. (2006) A unified criterion for initiation of sediment motion and
inception of sheet flow under water waves. Sedimentology, vol. 53, 1181-1190.
[9] Le Roux, J.P. (2007) A unified criterion for initiation of sediment motion and
inception of sheet flow under water waves - Discussion. Sedimentology, vol. 54, 14471448.
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XII Congreso Geológico Chileno
Santiago, 22-26 Noviembre, 2009
Table 1. Equations used in WAVECALC program
f = {[2 + cosh(4πd/Lw)][cosh(2πd/Lw)]/2[sinh(2πd/Lw)]3}3
Acceleration of gravity: g = {978.049[1+5.2884 x 10-3
Distance crest from bottom: yc = yt + Hw
2
-6
2
(sin πΩ) ]-5.9 x 10 (sin 2πΩ) -0,011}/100, where Ω is
the latitude
Horizontal water particle semi-excursion under wave crest:
Ahwz = (Ho/2){cosh[π(d-z)/MCDw]/cosh
Radian frequency: ω = 2π/Tw
[πd/MCDw]}
Wave number: k = 2π/Lo
Vertical water particle semi-excursion under wave crest:
Ahwz = (Ho/2){sinh[π(d-z)/MCDw]/sinh
Wave phase: θ = abs[-360(kx – ωt)/2π]
[πd/MCDw]}
Wave period: Tw = 2πUa/g
Horizontal water particle velocity in wave crest at distance
z from SWL: Uchwz = 2AhwzgTwLw/8MCDw2
Wave celerity: Co = Lw/Tw
Wavelength: Lo = gTw2/2π
Wave height: Ho = Lo/9π = gTw2/18π2
Horizontal water particle velocity in wave trough at
distance z from SWL: Uthwz = 2AhwzgTwLw/8MTDw2
Orbital diameter at depth z from SWL: 2Az = (Ho)
exp(2πz/Lo) sin θ
Vertical water particle velocity under wave trough at depth
z from SWL: Uhtwz = (2AvwzgTwLw/8MTDw2)
Water surface elevation: η = (Ho/2) cos [2πx/Lo-2πt/Tw]
Distance of breaker from shoreline: Xbα = dbα/tan α
Median crest and trough diameter: MCDo = MTDo = Lo/2
Thickness of boundary layer: δ = (Ho/2) + ηt
Wave celerity: Co = gTw/2π
where ηt = Lw(-0.017683 + 9.64 x 10-7f)
Horizontal and vertical water particle velocity under
wave crest and trough at depth z from SWL: Uchoz = Uthoz
= Uchoz = Utvoz = πHo/Tw exp(-2πz/Lo)
Subsurface gauge pressure: P = ρgη(exp 2πz/Lo)-ρgη
Total wave energy: E = ρgHo2Lo/8
Breaker height and depth: Iterate d in:
Slope-adjusted critical boundary velocity: Uδcrα =
Uδcr[sin(φ + α)/sin φ] where φ = 31º
Hbα = dbα(-0.0036α2 + 0.0843α + 0.835) and
Hw = Ho{A exp[(Ho/Lo)B]} where A = 0.5875(d/Lo)-0.18
when d/Lo ≤0.0844; A = 0.9672(d/Lo)2 - 0.5013(d/Lo) +
0.9521 when 0.0844 ≤ (d/Lo) ≤ 0.6; A = 1 when (d/Lo) >
0.6; B = 0.0042(d/Lo)-2.3211 until = Hbα = Hw
1/2
Boundary velocity under wave crest: Uδc =
(HogTwLw)/[8MCDw2cosh(πd/MCDw]
Boundary velocity under wave trought: Uδt =
(HogTwLw)/[8MTDw2cosh(πd/MTDw]
Critical boundary velocity under wave crest:
Uδcr = 2πJ[1 + 5(TR/Tw)2]-0.25
s* = D[(ρs - ρ)gD]0.5/4ν
J = 2.53s*0.92ν/D
Breaker length: Lbα = Tw[g(0.5Hbα + dbα)]
Wavelength: Lw = {LbαTw [g(0.5Hbα +d)]1/2}1/2 (Max.
value of d = Lo/2.965)
TR = 159s*-1.3D2/ν
Median crest diameter: MCDw = Lw - Lo/2; Min. Value
of MCD = Lo/6
Ursell number: UR = Lw2Hw/d3
Median trough diameter: MTDb = Lo/2; Lw - MCDw
when MCDw = Lo/6.
Distance of trough from bottom: yt = d + Lw(-0.017683 +
9.64 x 10-7f)
When Uδcr > 0.8 m s-1, 1.5Uδcr - 0.4.
Wave profile: η(x) = Lw[G cos(2πx/Lw) + I cos(4πx/Lw)]
Corrections to wave profile: η = ηp/(ηpc/ηc)h where h = 0.8,
0.6, 0.455 and 0.2 at increment distances of Lw/24 from the
wave crest
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