LARGE-SCALE MORPHODYNAMIC CONSEQUENCES OF AN INSTABILITY
IN ALONGSHORE TRANSPORT
A. Brad Murray1, Andrew Ashton1, Olivier Arnoult2
ABSTRACT: On a sandy shoreline, the alongshore current caused by waves breaking at
oblique angles transports sediment. When the angle between wave crests and the regional
trend of the shoreline is small enough, this sediment transport will tend to smooth the largescale shape of the coast. This case, in which a straight shoreline is the ultimate stable
configuration, has been explicitly assumed in models of large-scale shoreline morphology [13], and is commonly assumed implicitly. However, simple analysis shows that when the
angle between wave crests and the shoreline trend is large enough, perturbations on a straight
shoreline will grow. While the possibility of this instability has also been pointed out
historically [4,5], the implications for large-scale morphology have not been widely
appreciated or quantitatively investigated. Numerical modeling suggests that this
fundamental instability could lead to the self-organization of various kinds of enigmatic
large-scale shoreline features, including capes and cuspate forelands, which are exemplified
along the southeastern coast of the North America. The instability could also play a role in
coastal behavior on smaller spatial and temporal scales.
1. INTRODUCTION
The possibility of an instability in alongshore transport that could lead to an increasingly
rough, complex shoreline on large scales follows from two simple premises. The first is the
common approximation that the exchange of sediment between the nearshore region strongly
affected by waves and areas of deeper water is negligible compared with the alongshore flux.
This assumption of the conservation of nearshore sediment (with the assumption that the
cross-shore bed profile in the nearshore zone remains constant) can be expressed as
∂η/∂t = (1/D)∂Qs/∂x,
(1)
where η is the cross-shore position of the shoreline (e.g. the water line at high tide), x is the
alongshore coordinate, Qs is the volumetric alongshore flux of sediment, and D is depth to
which the accumulation or erosion extends.
The second premise needed to show the existence of the instability is the fact that the
relationship between Qs and the angle between the wave crests in deep water and the local
shoreline orientation is nonlinear, exhibiting a maximum. The existence of the maximum
___________________________________________________________________________
1
Division of Earth and Ocean Sciences/Center for Nonlinear and Complex Systems, Duke University, Box
90227, Durham, NC 27708-0227, USA
2
Present address: Ecole Normale Superieure, 45 Rue d’ULM, 75005 Paris, France
arises robustly from the mechanism that drives alongshore current and sediment transport.
The presence of waves creates a momentum flux in the water in excess of that which would
exist without the waves. This momentum flux, which arises largely because of wave
oscillatory motions, was termed the “radiation stress” by Longuet-Higgins [6]. The
oscillatory motions constitute a momentum flux across a plane with a normal parallel to the
wave-propagation direction. This flux is analogous to the molecular motions across any plane
in a fluid, which gives rise to the macroscopic variable of pressure. Analogous to the effect of
a pressure gradient, a gradient in the radiation stress can drive a flow. Therefore, in an area
where waves are becoming smaller in the direction of wave propagation, the gradient will
tend to drive a flow in that direction. When waves enter the nearshore zone of wave breaking
and dissipation (the ‘surf zone’) at oblique angles, the component of the momentum-flux
gradient in the alongshore direction produces an alongshore current (Fig. 1). This current
then advects in the alongshore direction sediment that is mobilized by a combination of the
current motion, wave motions, and turbulent motions resulting from the dissipation of wave
energy.
w
cr ave
es
ts
φ
nt
momentum−flux
gradient
g
reakin
wave b
curre
θ
Figure 1 Schematic plan view of waves breaking at an angle to shore.
The rate that alongshore momentum is transported toward shore, termed the alongshore
‘Thrust’ by Longuet-Higgins [7], can be expressed as
Thrust = Ecgsin(φ -θ)cos(φ -θ),
(2)
where E is wave energy density, cg is the group velocity (the speed at which wave energy
propagates), φ and θ are the orientation of wave crests and the shoreline, respectively,
relative to a global coordinate system. A semi-empirical relationship for alongshore sediment
transport, Qs, can be expressed as the Thrust multiplied by the breaking wave celerity [8,9].
Assuming both refraction over shore-parallel contours and the conservation of energy, the
thrust is a constant outside the surf zone [10]. Thus the formula for Qs can be written in
terms of either breaking-wave properties:
Qs = KHb5/2sin(φb-θ)cos(φb-θ),
(3)
or in terms of deep water quantities,
Qs = K2H012/5sin(φ0-θ)cos6/5(φ0-θ),
(4)
where K and K2 are constants, H is wave height, and the subscripts b and 0 refer to breaking
an deep water values. (We have used the approximation cos5(φb) ≈ 1 in deriving equation (4),
and in both expressions we have used the common approximations that that E ∝ H2, Hb ∝ Db,
and cb = (gDb)1/2, where g is the acceleration of gravity.)
To describe the instability (in the next section), we need to show how Qs depends on
shoreline orientation explicitly. Although equation (3) is the more familiar expression, φb is
implicitly a function of θ (assuming refraction over shore parallel contours). Equation (4)
shows that the relationship between Qs and φ0-θ exhibits a maximum. As the relative angle
between the waves and the shoreline approaches zero, the alongshore component of the wave
momentum, and therefore the thrust, approaches zero. As the relative angle approaches 90°,
the flux of wave momentum and energy toward shore, and therefore the thrust, approaches
zero. Between these extremes, the thrust—and therefore alongshore sediment transport—
must reach a maximum. Equation (4) predicts that the maximum occurs for φ0-θ ≈ 42°. An
explicit analysis of surf-zone sediment transport processes by Diegaard and others [11]
predicts a maximum for φ0-θ slightly greater than 45°.
(In equation (3), if Hb is held constant, Qs is maximized for φb-θ = 45°. In general, φb and φ0
have very different values because of nearshore refraction. This apparent conflict is resolved
by considering that, as φ0 and H0 are held constant (as is appropriate for examining how
sediment flux depends on local shoreline orientation), then φb-θ and Hb do not vary
independently along the shoreline. Higher values of φb-θ are associated with greater
nearshore refraction (assuming shore-parallel contours), which tends to reduce wave heights.
In this case, Qs is maximized for φb-θ << 45°. As an example, if H0 = 2 m and the wave
period = 10 s, then Qs is maximized for φb-θ ≈ 10°.)
2. THE INSTABILITY
The instability occurs when waves approach a shoreline at a relative angle greater than this
maximum (which we will call ‘high angle’ waves). In this case, moving along a nearly
straight shoreline in the transport direction, when first encountering an infinitesimalamplitude perturbation (Fig. 2), the relative angle between the waves and the local shoreline
orientation decreases, coming closer to the value that maximizes the transport. This creates a
divergence of sediment flux, and the shoreline will erode locally. However, moving between
the inflection points of the perturbation, the relative angle increases. Thus, along the crest of
the perturbation, the alongshore sediment flux converges, causing accretion of sediment; the
perturbation grows (Fig. 2).
Finite amplitude effects will eventually dictate the evolution of a growing feature. For
example, with a constant wave direction, as the amplitude of the perturbation in Fig. 2
increases, sediment flux just downdrift (in the transportation direction) of the crest will
approach 0, leading to the extension of a spit-like protuberance. The shoreline angle at the
updrift inflection point will increase until reaching the one that maximizes the sediment
transport. (This inflection-point angle depends on the deep-water wave angle; for waves with
an angle relative to the shoreline trend only slightly greater than the transport-maximizing
angle, the features will form subtle undulations of the shoreline.) Erosion updrift of the
inflection point will cause the inflection point to approach the crest; the feature will translate
alongshore while continuing to extend. However, the morphological evolution resulting from
this instability operating on an extended alongshore domain, with the possibility of multiple
features interacting, cannot be predicted analytically.
Wave
Crests
φ
Flux nce
erge
Conv
OCEAN
Y
X
θ
Shoreline at t0:
SHORE
Inflection
Points
Shoreline at t1:
Figure 2 The relative magnitude of the alongshore sediment flux, shown
by the length of the arrows, and the consequent zones of erosion and
accretion on a perturbation to a straight shoreline when the angle between
the wave crests and the shoreline trend is greater than that which
maximizes the flux.
3. NUMERICAL MODEL
To investigate the long-term effect of this instability, especially with varying wave
directions, we have developed a numerical model. This model is similar to others discretizing
equations (1) and (3) [12-14]. This model, however, treats larger spatial and temporal scales,
and employs an algorithm designed for arbitrarily shaped shorelines (Fig. 3).
55555555555555555555555555555
55555555555555555555555555555
Subaerial
A55555555555555555555555555555
D
Beach
55555555555555555555555555555
Continental
55555555555555555555555555555
Shelf
Shoreface
55555555555555555555555555555
55555555555555555555555555555
55555555555555555555555555555
55555555555555555555555555555
Incoming
Waves
Shadowed
Area
B
Sediment
Transport
555555555555
555555555555
Shoreline
555555555555
555555555555
555555555555
555555555555
555555555555
Figure 3 Schematic illustration of the model algorithm. A. Cross-shore profile. B. Plan view.
Figure 3 A shows a cross-shore profile of the model domain. D represents the depth below
which cross-shore transport is considered negligible. D is defined as the intersection of the
continental shelf (which has a slope Scs perpendicular to the global shoreline trend) with a
linear nearshore profile (representing the ‘shoreface’, which has a slope Ssf extending from
the subaerial portion of the beach in a direction perpendicular to the local shoreline
orientation). A minimum depth, Dmin, is imposed, to represent the erosion of the shoreface
by wave action where the shore moves landward. For all simulations shown, Scs = 0.001, Ssf
= 0.01, Dmin = 10 m, and the initial subaerial beach starts at an average of 8000 m seaward of
the intersection of the projection of the continental shelf slope and sea level.
In planview (Fig. 3 B), the domain is divided into cells that are filled with a fractional
amount of sediment, Fi. Cells with Fi = 1 form the subaerial coast, and cells adjacent to these
with 0 < Fi < 1 constitute the shoreline, with Fi representing the fractional position of the
shoreline within the cell. Boundary conditions are periodic in the alongshore direction. Local
sediment fluxes are determined using equation (3). (Deep-water wave heights and angles are
converted to breaking-wave quantities using linear-wave theory for shoaling and refraction
over shore parallel contours, and assuming depth-limited breaking).
For each boundary between shoreline cells, the sign of the angle between deep-water wave
crests and the line connecting the shoreline positions in adjacent shoreline cells determines
the direction of alongshore sediment transport between the neighboring cells. To avoid
discretization artifacts (for the case of high angle waves), we combine upwind and central
finite-difference techniques: If the magnitude of the local relative wave angle—defined
using the position of the shoreline in cells bounding the cell sediment transport is from—is
less than (greater than) that which maximizes the transport, we use central (upwind) finite
difference to determine the sediment flux. When moving from a boundary with centraldifferencing to one with upwind conditions, sediment transport for the latter boundary is set
to that occurring for φb-θ = 45°, with the local Hb).
Sediment is not transported out of cells that are ‘shadowed’ from incoming waves by other
portions of the shoreline. Sediment transport into shadowed cells is determined using an
upwind finite difference. Sediment contents of shoreline cells are adjusted according to the
net sediment flux into or out of the cell, Qs,net: ∆Fi = Qs,net ∆t/(W2Di), where W is the
horizontal cell width (∆t = 0.1 days and W = 100 m except where noted).
In most simulations, wave angles are distributed according to a probability distribution
function: Each simulated day, 1) a random number between 0 and 1, X, is chosen, and the
deep-water wave angle is determined by φ0 = (4π /9) Xα ; and 2) a different random number
between 0 and 1, Y is chosen, and sign of the wave angle is positive if Y > β and negative if
Y < β (positive = from the left, negative from the right).
4. MODEL RESULTS AND COMPARISON WITH NATURAL FEATURES
In the simplest model experiments, waves always approach an initially nearly straight
shoreline from deep water with the same high angle relative to the global shoreline trend.
With this steady, spatially uniform forcing, the simple, deterministic interactions in the
model produce progressively more complex shapes (Fig. 4 A). Such simulations, while
interesting, omit a factor that is important on most natural shorelines: varying wave angles.
A
t ~ 0 yrs
7 B
km
t ~14 yrs
D
t ~ 27 yrs
t ~1 yrs
t ~27 yrs
t ~ 55 yrs
t ~2 yrs
t ~55 yrs
t ~ 83 yrs
t ~3 yrs
C
t ~87 yrs
t ~110 yrs
20 km
Figure 4. Results of the numerical model, showing plan views of the shoreline. In each
simulation, the initial conditions consisted of a straight shoreline plus uncorrelated random
perturbations to the shoreline position at each alongshore location, with an amplitude of one
cell width. Except where noted, cell widths represent 100 m. A, Results using a single angle
(55°) between deep-water wave crests and the global shoreline orientation. B, Results when
deep-water wave angles are chosen from a distribution function weighted toward high
angles and toward waves approaching from the left. In this simulation, α = 0.8 and β = 0.8. C,
Results with α = 0.6, β = 0.7. D, Results with α = 0.6, but with waves symmetrically distributed
from left and right, β = 0.5.
As long as the wave probability distribution function is weighted toward high-angle waves,
features grow.
If the distribution is also weighted toward waves approaching from positive (or negative)
angles, so that there is a net sediment flux, features translate alongshore (Fig. 4 B and C). In
this case, features can overtake each other and merge, and the average wavelength increases
with time. These features resemble a class of shoreline phenomena called variously “cuspate
spits” [4,15] (e.g. Fig. 5 A), “sandwaves” [16-19], “ords” [20], and “humps” [16]. On some
open ocean coasts, these have been described as migrating, spatially correlated zones of
erosion and accretion [16,20]. In elongate water bodies, where the only significant waves are
those that propagate sub-parallel to the long axis, with wave crests at high angles to the
shore, more pronounced spit- or wave-shaped shoreline shapes develop [4,15,21].
If the wave distribution is symmetrical, with waves approaching equally often from positive
and negative angles, a cusp-shaped pattern develops (Fig. 4 D), resembling the shapes
observed on some elongated water bodies [4,15,22] (e.g. Fig. 5 B). Somewhat surprisingly,
the wavelength also continually increases in the symmetric-distribution case. Observations of
simulations have revealed that, even without net migration, neighboring features interact in a
way that leads to this perpetual increase; Slightly larger features shelter their neighbors from
the highest-angle waves, so that the larger neighbors grow in amplitude relative to the
smaller ones. As the sheltering effect increases, a smaller feature will eventually experience a
wave distribution weighted toward low-angle waves, and will be smoothed away.
Animations of simulations show that when one feature disappears, its neighbors undergo a
pulse of growth, which increases the sheltering on the adjacent neighbor, causing its
disappearance. This effect rapidly propagates through the pattern, causing a relatively sudden
doubling of the wavelength. This process tends to create a pattern exhibiting regularly spaced
features at most times, even though the wavelength increases over time. Simulations with a
slightly asymmetric wave-angle distribution exhibit similar behavior.
N
A
100 km
N
B
Regional
shoreline trend
42o
42o
100 km
Figure 5. Satellite images (Courtesy of SeaWIFS). A. Cuspate spits on the shore of
the Sea of Azov, Ukraine. The thick arrow shows the direction of the prevailing
winds [5]. B. The capes and cuspate foreland of the coast of south eastern coast of
the United States (North and South Carolina). Inset show sdirectional distribution of
wave energy in approximately 30 m water depth, according to wave hindcasts [23],
shoreline trend defined by joining the backs of the bays, and the directions dividing
high from low angle waves regionally.
After a simulated time on the order of 105 years (with a constant H0 of 2 m), a cuspate pattern
develops a length scale on the order of 100 km, commensurate with that of large cuspate
forelands in nature. The capes and cuspate coastline of the southeastern North America
provide the best-known example of this phenomenon. Estimates of the modern wave climate
off of the Outer Banks of North Carolina (in approximately 30 m water depth) [23] suggest
that the wave energy is distributed with a predominance of the energy coming from angles
greater than approximately 42° to the regional shoreline trend (Fig. 5 B). Many hypotheses
for the origin of these capes and cuspate pattern have been suggested, most involving
external templates such as paleo-river deltas [24]. While the wave data is not conclusive, and
does not necessarily reflect past wave conditions, it is consistent with the hypothesis that the
striking large-scale pattern of the open ocean shoreline in this region has arisen
spontaneously, driven by the instability inherent in the relationship between alongshore
transport and local shoreline orientation.
Sea level has stood near its current high elevation only thousands of years, rather than the 105
years required in the model for cape-scale features to develop. However, each time sea level
rises during an interglacial epoch, a horizontally undulating beach ridge left behind after the
previous high stand would likely be encountered, rather than an approximately straight
coastline.
5. DISCUSSION
This model is limited in several ways. It applies only to sediment-covered shorelines, not to
predominantly rocky coasts. In addition, we have intentionally omitted some processes that
occur in natural settings, including: 1) alongshore inhomogeneities, such as variations in
grain size and lithology; 2) wave diffraction; 3) the convergence and divergence of wave rays
that tends to concentrate wave energy on headlands and decrease it in bays, especially when
the curvature of the coastline is high; and 4) the residual motion that arises when tidal
currents pass a headland, which can transport sediment off of a cape onto subaqueous shoal
that act as sediment sinks that can influence cape dynamics and morphology [25]. In this
initial numerical exploration, our goal is not to reproduce any natural system in detail, but
rather to investigate the range of coastal landforms and behaviors that can result from the
fundamental instability in alongshore sediment transport that occurs in regions with
predominantly high-angle waves.
Other limitations prevent using the present model to explore some other shoreline patterns,
with scales on the order of 100 m to kilometers, which we conjecture to be related to the
instability. Shore attached bars, with length scales on the order of 100 m, have been modeled
as developing because of a feedback between flow and subaqueous morphology [26].
Development of those attached bars that are exposed subaerially during at least part of the
tidal cycle could also be influenced by shoreline sediment transport while the features are
exposed. When waves break at or very near the shoreline on the locally steep bar face,
sediment transport in the narrow breaking and swash zones will be a function of the local
angle between waves and shoreline, on the scale of meters to tens of meters. This relationship
will exhibit a maximum; locally high-angle waves could influence growth and extension of
the bars. However, the approximation in the present model that breaking-wave parameters
can be related to those in deep water by assuming refraction over contours parallel to the
shoreline is not appropriate for treating changes in shoreline orientation on such small scales
(except where wave period is very small).
Alongshore bars can exhibit a pattern in which a bar, on the order of kilometers long, is very
close to shore at one end and shifts farther offshore in the alongshore direction, possibly
becoming the outermost of two bars at the other end (for example, in the vicinity of Duck,
North Carolina; Bill Birkemeier, personal commumication, 2001). During large-wave events,
with waves breaking even over the farthest-offshore segments of the bars, a portion of the
wave energy and momentum will be expended there, causing alongshore transport along the
length of the bar. This transport will be concentrated in the cross-shore direction near the bar
crest. When large waves approach from high angles, these bars could behave much like
subaerial spits subjected to high-angle waves in the model, growing in a direction oblique to
shore. However, a model to explore this speculation will have to treat the partitioning of
energy, momentum and sediment transport in the cross-shore direction, and variations in this
partitioning with changing wave conditions.
The instability could also relate to the transient erosion “hot spots” observed recently along
the East Coast of the United States [27,28]. After storms, GPS and LIDAR surveys of long
stretches of coast show that some segments of shore experience considerable erosion
(landward moment of beach contours), while intervening areas remain stable. Post-storm
accretion patterns form a near mirror image of the erosion. The hot spots could represent the
erosional areas on the flanks of perturbations (subtle bends in the shoreline) that would occur
during storms with high wave angles. During subsequent lower-wave periods, this localized
sediment-transport divergence would not exist, and low-angle waves could then resmooth the
shoreline, causing the localized accretion in the same areas that suffered storm erosion.
Considerable work remains to explore the possible morphodynamics consequences of the
instability in alongshore transport.
ACKNOWLEDGEMENTS
The Andrew W. Mellon Foundation supported this work.
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