publications - university of...

Unraveling the dynamics that scale cross-shoreheadland relief on rocky coastlines: Part 1.Model developmentPatrick W. Limber1,2, A. Brad Murray1, Peter N. Adams2, and Evan B. Goldstein1

1Division of Earth and Ocean Sciences, Nicholas School of the Environment, Center for Nonlinear and Complex Systems,Duke University, Durham, North Carolina, USA, 2Department of Geological Sciences, University of Florida, Gainesville,Florida, USA

Abstract We have developed an exploratory model of plan view, millennial-scale headland and bayevolution on rocky coastlines. Cross-shore coastline relief, or amplitude, arises from alongshore differences insea cliff lithology, where durable, erosion-resistant rocks protrude seaward as headlands and weaker rocksretreat landward as bays. The model is built around two concurrent negative feedbacks that control headlandamplitude: (1) wave energy convergence and divergence at headlands and bays, respectively, that increases inintensity as cross-shore amplitude grows and (2) the combined processes of beach sediment production by seacliff erosion, distribution of sediment to bays by waves, and beach accumulation that buffers sea cliffs fromwave attack and limits further sea cliff retreat. Paired with the coastline relief model is a numerical wavetransformation model that explores how wave energy is distributed along an embayed coastline. The twomodels are linked through genetic programming, a machine learning technique that parses wave modelresults into a tractable input for the coastline model. Using a pool of 4800 wave model simulations, geneticprogramming yields a function that relates breaking wave power density to cross-shore headland amplitude,offshore wave height, approach angle, and period. The goal of the coastline model is to make simple, butfundamental, scaling arguments on how different variables (such as sea cliff height and composition) affect theequilibrium cross-shore relief of headland and bays. Themodel’s generality highlights the key feedbacks involvedin coastline evolution and allows its equations (and model behaviors) to be easily modified by future users.

1. Introduction1.1. Overview

The diverse plan view shapes of rocky coastlines (broadly defined here as a cliff-backed coast consisting ofrock or well-consolidated sediments) have long been a topic of scientific and general interest [Gilbert andBrigham, 1908; Johnson, 1919]. A quick look at Google Earth reveals rocky coastline shapes ranging fromcrenulated to straight, with headlands interspersed with smooth sediment-filled (and sediment-free)embayments (Figure 1); cross-shore relief, or amplitude, of rocky coastlines varies from nearly zero to severalthousandmeters. While very large headlands, such as the Palos Verdes peninsula near Los Angeles, California,USA (Figure 1), are likely maintained by current or past regional tectonic uplift [Ward and Valensise, 1994], theapparent persistence of smaller headlands (approximately < 5 km) has traditionally been explained as adynamic balance between differing rock strength and wave energy focusing. Assuming that the coastlineretreat rate increases with wave energy and decreases with rock strength, headlands that are composed ofmore durable rock receive more wave energy (through refraction of waves and convergence of energy) tothe detriment of bays backed by soft rock as the coastline amplitude increases. Presumably, this wave-forcednegative feedback eventually results in equal landward erosion rates of the softer (bay) and harder(headland) rocks [Muir-Wood, 1971; Trenhaile, 1987; Trenhaile, 2002] (Figure 2) and controls rocky coastlineamplitude: headlands persist in time because of rock strength differences, or headlands are transient becausethere exists no meaningful rock strength difference between bay and headland. Beyond this qualitativedescription, no process-based quantitative model has been proposed and little observational data areavailable to support it [Carter et al., 1990]. As such several basic questions remain unanswered about thephysical characteristics that control cross-shore amplitude and the time scales over which rocky coasts adjust.

The wave energymodel is also missing a key process operating onmany rocky coastlines: the production anddistribution of beach sediment. New sediment is brought to the coast by sea cliff erosion and coastal rivers,

LIMBER ET AL. ©2014. American Geophysical Union. All Rights Reserved. 1

PUBLICATIONSJournal of Geophysical Research: Earth Surface

RESEARCH ARTICLE10.1002/2013JF002950

This article is a companion to Limberand Murray [2014] doi:10.1002/2013JF002978.

Key Points:• A new model of cross-shore headlandrelief on rocky coastlines is presented

• Wave energy delivery and beach/seacliff dynamics control headland relief

• A genetic programming routinerelates wave power to coastline shape

Correspondence to:P. W. Limber,[email protected]

Citation:Limber, P. W., A. Brad Murray, P. N.Adams, and E. B. Goldstein (2014),Unraveling the dynamics that scalecross-shore headland relief on rockycoastlines: Part 1. Model development,J. Geophys. Res. Earth Surf., 119,doi:10.1002/2013JF002950.

Received 13 AUG 2013Accepted 11 MAR 2014Accepted article online 13 MAR 2014

http://publications.agu.org/journals/

http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2169-9011

http://dx.doi.org/10.1002/2013JF002950

http://dx.doi.org/10.1002/2013JF002950

and the wave climate (i.e., wave height and angle) and coastline shape control its alongshore distribution[Ashton and Murray, 2006b]. Additionally, beach width exerts a fundamental control on the rate of sea clifferosion by acting either as a protective cover that buffers the cliff from wave attack or as an abrasive tool thatenhances wave attack [e.g., Sunamura, 1976, 1982; Robinson, 1977; Carter and Guy, 1988; Kamphuis, 1990;Ruggiero et al., 2001; Sallenger et al., 2002; Trenhaile, 2005;Walkden and Hall, 2005; Lee, 2008; Trenhaile, 2009;Trenhaile, 2010; Limber and Murray, 2011; Castedo et al., 2013; Young et al., 2014]. The feedback process, wherebeach sediment is produced by sea cliff erosion, distributed by alongshore sediment transport, and controlsfuture sea cliff erosion rates, can affect embayment sea cliff retreat rates and coastline morphology onsediment-rich rocky coasts over kilometer scales and decadal to millennial time scales.

Focusing on abrasion, recent modeling by Limber and Murray [2011] suggests that such internal sedimentdynamics can create persistent headlands and bays in the absence of rock heterogeneity. But what controlsheadland and bay evolution on rocky coastlines where abrasion is not an important geomorphic process and/orlithologic heterogeneities dominate coastline morphology? In this contribution we complement the originalapproach of Limber and Murray [2011] by examining other processes and configurations that occur on rockycoastlines: alongshore variations in rock type, wave energy convergences (headlands) and divergences (bays), abeach that acts only as a protective cover [e.g., Sallenger et al., 2002; Ruggiero et al., 2001; Lee, 2008], and fluvialsediment delivery to the coastline (see companion paper, Limber and Murray, in revision).

Thus, we present a simple model where two concurrent dynamic processes, internal sediment dynamics andwave energy delivery, can control the cross-shore amplitude of headlands and bays. The goal is to make basicscaling arguments on how different length scales (sea cliff height, headland spacing, and alongshoreheadland width, for example) and physical properties (rock composition and retreat rate) adjust the cross-

Figure 1. Headlands and embayments on rocky coastlines. Moving clockwise from upper left panel: (a) The Palos Verdes peninsula, Los Angeles, California, is an exampleof a large rocky promontory that is controlled by large-scale processes such as tectonics and is beyond the applicable scale of our model. (b) Within the peninsula,however, are kilometer-scale headlands and bays that are representative of the headlands and bays simulated by our model. (c) The Gaviota, California coastlinefeatures a near-continuous cliff-backed beach and few rocky headlands. (d–f) Further examples of headlands and bays of the type and scale applicable to our model.Note Montara Beach in Figure 1f that will be discussed in the text (and in the companion paper, Limber and Murray, in revision). All images from Google Earth.

Journal of Geophysical Research: Earth Surface 10.1002/2013JF002950


shore amplitude of kilometer-scale headland and bay systems on high-energy rocky coastlines. In thiscontribution, we develop and outline the coastline and wave models and discuss processes that can beadded to the model framework (e.g., sea level rise). In a companion paper (Part 2, Limber and Murray, inrevision) we will apply the model to address (1) how, and over what time scales, a lithologically diverse rockycoastline can reach equilibrium; (2) what fundamental physical properties (i.e., sea cliff height andcomposition) scale the eventual cross-shore equilibrium amplitude, if it exists; (3) under what conditions willheadlands be persistent or transient landscape features; (4) in what geologic settings will one morphologicalfeedback (wave focusing or sediment production) dominate over the other in shaping headland and bayamplitude; and (5) how models results can be compared to actual rocky coastlines using field observations.

1.2. Background

Early observers of rocky coastlines suggested that exposed headlands, if rock strength was constantalongshore, would be continually reduced in cross-shore amplitude by wave impacts until the coastlinereached a straight, featureless equilibrium configuration [Gilbert and Brigham, 1908; Johnson, 1919]. It wasalso recognized early on that alongshore variations in rock strength, or resistance to incoming wave energy,

Figure 2. A schematic of themodel feedbacks that control coastline amplitude through time. Alongshore differences in lithologycreate headlands and bays. Wave energy redistribution combined with bay sediment dynamics control the retreat ratesof headlands and bays and, eventually, the steady state amplitude at time te when retreat rates (shown by the vectors) areequal alongshore.



could allow higher curvature coastline shapes to persist [Gulliver, 1899; Johnson, 1919; Trenhaile, 1987], andthis view is still predominant as many workers have attributed rocky coast morphology over a range of spatialscales, from shore platforms to headlands, to variations in rock strength [Trenhaile, 1987; Benumof et al., 2000;Trenhaile, 2002; Dickson et al., 2004; Davies et al., 2004; Kennedy and Dickson, 2006; Thornton and Stephenson,2006; Naylor and Stephenson, 2010; Stephenson and Naylor, 2011]. However, there has been a relatively smallbody of work addressing long-term planform rocky coastline evolution and the shape and size of headlands.In Nova Scotia, Canada, Carter et al. [1990] made several valuable observations. First, they measured waveheights that were often 2–3 times greater along the headland apex relative to the flanks, which is consistentwith wave energy convergence and divergence. Second, they noted the coexistence of narrow “needle”-shaped headlands and “stubby” headlands with wider bases and found that the stubby types tended to belocated in exposed higher-energy wave zones, while needle types were found in more sheltered areas. Thissuggests that headland shape is related to the availability of wave energy. Finally, along with Muir-Wood[1971],May and Tanner [1973], and Komar [1985], Carter et al. [1990] also noted the importance of headlandsas an alongshore sediment transport boundary: beach sediment is distributed away from the headland toadjacent embayments, and headland spacing thus can affect coastline evolution. Similarly, Trenhaile [2002]related bay amplitude to headland spacing, as did Short [1999] and Bowman et al. [2009].

Only a few authors have modeled the plan view evolution of rocky coastlines. Prior models of rocky coastlineevolution have focused on the cross-shore development of features such as shore platforms [Trenhaile, 2000,2005, 2011], sea cliffs [Sunamura, 1982; Castedo et al., 2012, 2013; Young et al., 2014], marine terraces [Andersonet al., 1999; Trenhaile, 2002], and erosional continental shelves [Sunamura, 1978a; Trenhaile, 2001; Qartal et al.,2010]. Muir-Wood [1971] made an initial analytical scaling argument for the equilibrium cross-shoreembayment length of a rocky coastline, where headlands and bays result from varying rock strength. Thelengthwas estimatedmainly by assuming that wave energy dissipation is proportional to wave path length andthat as the cross-shore coastline length develops, the wave path length will increase in the bay (relative toheadlands). Wave energy is progressively dissipated in the bays as coastline relief increases, thereby slowingbay retreat until the coast reaches a steady state. Adams [2004] built a numericalmodel using similar conceptualarguments asMuir-Wood [1971] where shoaling waves drive sea cliffs of different rock types landward to forman embayment. Valvo et al. [2006] modeled the evolution of sandy coasts underlain by a geological framework,where beach sediment was derived from shoreface weathering. If shoreface lithology varied alongshore, thecoastline developed equilibrium undulations similar to headlands and bays. Other authors have used analongshore-extended profile-based model, Soft Cliff and Platform Erosion, or SCAPE, to simulate planformcoastline development during sea level rise [Walkden and Hall, 2005; Dickson et al., 2007; Dawson et al., 2009;Walkden and Hall, 2011]. The model includes beach and sea cliff dynamics similar to those presented here buthas not yet been used to model the development of coastline relief.

The equilibrium shape of log-spiral or headland-bay beaches on kilometer scales has received considerableattention [e.g., Silvester, 1960; Yasso, 1965; Littlewood et al., 2007; Weesakul et al., 2010; Jackson and Cooper,2010; Hsu et al., 2010]. These studies are mostly concerned with the curved, crenulate shape of the sandy baythat results from wave interactions with a stationary headland (either refraction/diffraction using a singlewave condition or wave-shadowing effects with a mix of wave directions) [Littlewood et al., 2007] rather thanthe understanding of how headland amplitude evolves. The equilibrium amplitude of log-spiral coasts hasbeen addressed [Hsu et al., 1989], but it is found empirically [Hsu et al., 2010] rather than through a dynamicalprocess-based approach as is presented here. Previous efforts also do not include the geomorphic feedbacksbetween beaches and sea cliffs, or between coastline length and sediment production.

2. The Rocky Coastline Model2.1. Conceptual Summary

Conceptually, the model and its feedbacks can be summarized as follows. Consider an initially flat, cliff-backedcoastline consisting of two different rock types, L1 and L2 (Figure 2). The rock type L1 is less durable than L2 andtherefore retreats faster. The difference in retreat rates between L1 and L2 causes the coastline to increase incross-shore amplitude and lengthen, as the stronger L2 emerges as headland. If unrestrained, the coastlineamplitude (and length) would perpetually increase (Figure 3a). But, the shape change from flat to embayedtriggers two concurrent feedbacks that stabilize the coastline shape.



First, waves are continuously delivering energy to the coastline. The increase in amplitude as a result of theretreat rate difference between L1 and L2 causes waves to refract and focus energy on the emergingheadland [Carter et al., 1990; Trenhaile, 2002]. The headland is able to capture more of the incoming waveenergy as it grows in cross-shore amplitude, which increases its retreat rate. In turn, wave energy fluxdivergence causes cliff retreat rates in the bays to decrease. This pushes the headland retreat rate toconverge with the embayment retreat rate (Figure 3b), thus stopping amplitude growth so that thecoastline reaches a steady state (Figures 2 and 3b).

At the same time, on rocky coasts experiencing a low-angle wave climate (which tends to smooth coastlines)[Ashton et al., 2001; Ashton and Murray, 2006b], the developing coastline curvature will set up a gradient inalongshore sediment flux that will transport and deposit sediment (sourced from sea cliff erosion and fluvialdelivery) into embayments, forming pocket beaches while leaving the emerging headland largely sedimentfree [May and Tanner, 1973; Komar, 1985; Carter et al., 1990; Valvo et al., 2006]. As sediment accumulates in baysand pocket beach width grows, the beach then controls the frequency and efficacy of wave attack on theembayed sea cliffs [Sunamura, 1982; Ruggiero et al., 2001; Sallenger et al., 2002]. Thus, by producing sediment asit retreats, the coastline can effectively manage its own morphological fate: if rocky coastline amplitudechanges through time, so does the total alongshore length, or sinuosity, of the coastline and, therefore, thelength scale over which sea cliff retreat (and subsequent sediment production) operates (Figure 2). So, largeramplitude headlands will produce more beach sediment than smaller amplitude headlands and will likewiseshed more sediment to surrounding bays. If amplitude is too large, total sediment production from cliff retreatwill be high, and pocket beaches will grow wide enough to stop cliff retreat in bays. Then, because headlandsare now retreating faster than the sediment-filled bays, amplitude will decrease. The opposite happens whencoastline amplitude is too small: sediment production is too low to sufficiently slow cliff retreat in the bays, andamplitude increases. Amplitude growth spurs more sediment production, which increases pocket beach widthand equalizes sea cliff retreat between the headlands and bays. This dynamic process progressively slows

Figure 3. The coastline reaches a stable steady state amplitude (Ae) when the bay and headland cliff retreat rates converge.The white and gray circles on each plot show the baseline (i.e., when A= 0 and w= 0, if applicable) bay and headland cliffretreat rates, respectively. (a) If there are no negative feedbacks operating, the differential retreat rate between the bay andheadland exists in perpetuity and amplitude grows to infinity. (b) If wave energy redistribution accelerates initially slowerheadland retreat and decelerates initially faster bay retreat, then the rates converge to a steady state. (c) Another way toreach steady state is through the accumulation of beach sediment in bays, which slows bay retreat until it converges with aconstant headland retreat. In that case, a steady state beach width also exists (we). (d) When the feedbacks work together,amplitude is limited more effectively and Ae and we are smaller. Also shown for reference is a cartoon of cliff retreat as afunction of beach width and abrasion, where a small amount of sediment can accelerate retreat. As argued by Limber andMurray [2011], a steady state exists in homogeneous lithology when beach width drives bay retreat at the same rate assediment-free headlands.



embayment retreat and moves toward a stabilizing convergence of the initially faster embayment retreat rate(L1) and the initially slower headland retreat rate (L2; Figures 2 and 3c). The beach sediment production andprotection feedback is most effective on sediment-rich rocky coasts. On sediment-poor coasts, where the cliffsaremade ofmudstone, for example, the coastlinewill grow in length until there is enough sea cliff area that canerode to produce a stabilizing beach width. The wave energy feedback operates on rocky coasts regardless ofthe amount of sediment input, and when beach sediment is present, both feedbacks work together to moreeffectively limit steady state amplitude (Figure 3d).

2.2. Modeling Approach, Domain Set-Up, and Key Variable Definitions

Our ultimate modeling goal is to make basic physical scaling arguments and clear testable predictions, and inthe process shed light on the broadly applicable large-scale dynamics that scale plan view rocky coastlinemorphology. Our goal is not to model rocky coastline evolution in detail, or model a particular location. As aresult, the modeling approach is purposefully simple [Murray, 2007]. We will focus on large-scale (tens ofmeters to kilometers) and long-term (centuries to millennia) emergent behavior, within which many small-scale and short-term processes are only implicitly represented, as will be explained below. The model isintended to be a starting point for exploring large-scale rocky coastline evolution, or an easily customizableand transparent analytical platform that can be built upon in the future. Several processes that can be addedto the model framework, including sea level rise and shore platform dynamics, are discussed in section 4.

Figure 4. Plan view and cross-sectional model schematic showing variable definitions.



The model domain setup, with variable definitions, is shown schematically in Figure 4. Variables are listed inTable 1 (these variables will be explained in subsequent sections). Landward-directed rates of change arenegative, and seaward-directed rates are positive. In the model, rock type varies alongshore and moreresistant lithology juts seaward as a headland and less resistant lithology retreats landward as an embayment(e.g., Figure 2). Sea cliff lithology variations are assumed to extend landward to infinity so that there is no limitto cross-shore coastline retreat. The headland has a cross-shore amplitude, A, that evolves through time andis measured relative to the embayment cliff position. Primary model assumptions are that sediment is sweptfrom headlands to bays (i.e., headlands must remain sediment free) and the beach cannot extend seaward ofthe headland (see section 2.6 for further explanation). Sea cliff height and composition determine howmuchbeach sediment is produced per unit length of cliff retreat.

Sea cliff retreat is the long-term landward movement of the entire cliff profile, while erosion refers to theshorter-term weathering and removal of cliff material from the cliff base that, over time, causes cliff retreat.The model cannot resolve short-term cliff erosion and localized changes to the cliff profile (e.g., notchdevelopment and local rockfalls), and the sea cliff retreat rate and profile shape are averaged over seasonalchanges, storms, tides, and episodic failure events [Young et al., 2009; Rosser et al., 2013]. Over the millennialtime scales considered here, erosion and retreat are considered synonymous. Sea cliff retreat in the model isdriven only by wave impacts for simplicity and tractability; however, many other processes such asgroundwater percolation and surface runoff [Emery and Kuhn, 1982; Hampton, 2002], human impacts [Runyanand Griggs, 2003], and bioerosion [Naylor et al., 2012] can also affect sea cliff stability. There are two sea cliffretreat notations (Table 1). The rates E0H and E0B are respective bare-rock, or baseline, retreat rates for theheadland and bay when the coastline is flat and not fronted by a beach (Figure 3) [Limber and Murray, 2011].Baseline retreat rates can be estimated using long-term (decadal) observations of sea cliff retreat [e.g., Kirk,1977; Moore and Griggs, 2002; Hapke and Reid, 2007]. The variables ηH and ηB are the comprehensiveheadland and bay sea cliff retreat rates, respectively. Measured relative to the baseline rates E0H and E0B, theydescribe the total effect of wave power delivery and beach sediment on sea cliff retreat rates.

Alongshore sediment distribution is treated implicitly. The model assumes a low-angle wave climate [Ashtonand Murray, 2006b] so that headlands act as divergent sediment transport boundaries [May and Tanner, 1973;Carter et al., 1990] (Figure 4). Over long time scales (centuries to millennia), all sediment produced along theirperimeter is swept into bays to form pocket beaches of cross-shore widthw that are evenly distributed over thealongshore embayment length [Limber and Murray, 2011]. Resolving alongshore transport over shorter timescales is not possible using our simplified analytic approach. Beach sediment cannot be transported alongshoreout of bays, which is not always the case in nature as sediment can be shared between bays during stormevents [e.g., Storlazzi and Field, 2000; Bowman et al., 2014]. Within bays, there is a sediment budget where

Table 1. List of Central Variables

Variable Description Units

A Cross-shore headland amplitude mAe Steady state amplitude mα0 Offshore wave angle degBB,H Headland or embayment alongshore length mcB,H Constant that relates wave power to sea cliff retreat m3�s2�J�1

CB,H Proportion of coarse sediment in bay or headland sea cliffsE0B,H Baseline headland or bay retreat rates m�yr�1

HB,H Headland or bay sea cliff height divided by beach depth, dHCB,H Effective dimensionless sea cliff heightHw0 Offshore wave height m�ηH;B Headland or bay comprehensive sea cliff retreat rate m�yr�1

P Að Þ Mean wave power density as a function of headland amplitude averagedover headland width BH

J�m�1�s�1

S Beach sediment loss rate m�yr�1

T Wave period sw Beach width mwmax Beach width that causes bay cliff retreat rate to reach zero mΩ1 Constant that relates wave power to headland amplitude J�m�2�s�1

Dw Cumulative forcing by waves on sea cliff retreat J�m�1�s�1



additions from cliff retreat across the entire domain are balanced with losses (see section 2.4). A time-varying beach width, w(t), results from the embayment sediment budget with periodic alongshoreboundary conditions. The beach acts as a protective cover for embayment sea cliffs by mitigating thefrequency that the cliffs are subjected to wave impacts [e.g., Sumamura, 1982; Komar and Shih, 1993; Lee,2008]. The beach rests on a flat shore platform surface [Trenhaile, 2004] that widens as the sea cliffs retreat(Figure 4). Although not fully presented here, the dynamics between shore platforms (flat and sloped), seacliff retreat, and beach accumulation [Trenhaile, 2005; Marshall and Stephenson, 2011; Ogawa et al., 2011]could be included in future work. An approach to including basic dynamics between shore platforms andsea cliffs is presented in section 4.

2.3. Wave Energy Focusing on Headlands

The transmission of wave energy from the deep ocean, across the continental shelf, and ultimately to thecoastline is modulated by the deep-water wave angle and height, bathymetry, as well as coastline shape[Adams et al., 2002; Adams et al., 2005; Lim et al., 2011]. Bathymetric contours in the model are assumed toreflect the presence of headlands and bays. Variations in alongshore bathymetry result in wave energyconvergence and divergence caused by wave refraction [Komar, 1997]. Refraction tends to focus wave energyand locally increase wave heights on protruding rocky headlands while dispersing wave energy (stretchingwave crests and lowering wave heights) in neighboring embayments [Komar, 1985; Komar, 1997; Trenhaile,1987]. The rate of sea cliff erosion increases with breaking wave height because wave impacts are a maindriver of sea cliff erosion and retreat [Sunamura, 1976, 1977, 1982; Trenhaile, 2000]. As coastline amplitudeincreases, headlands will capture a greater amount of incoming wave energy at the expense of the adjacent,sheltered bays. This amplitude-dependent energy redistribution increases and decreases wave heights,respectively, and causes headland retreat to accelerate relative to bay retreat [Trenhaile, 2002].

We can use the concept of alongshore wave energy redistribution (as a function of amplitude) to beginbuilding our analytical model. Starting with a flat coastline with two different rock types (Figure 2), thechange in coastline amplitude (A) through time (t) is taken as the difference between the mean embayment

and headland sea cliff retreat rates ( �ηB and �ηH, Figure 2):

dAdt

∝ �ηB � �ηH: (1)

If the embayment sea cliffs retreat faster than the headland cliffs, amplitude grows; if bay sea cliffs retreatslower than headlands, amplitude decays. The greater the cross-shore amplitude is relative to the initially flatcoastline, the more wave energy is captured by the headland and lost to the embayment. Because energyflux is conserved along our initial coastline, the amount of energy that is dispersed is simply the remainder ofthe total energy left over after focusing on the headland. Therefore, the rate that energy is captured by theheadland equals the rate that is dispersed in the bay. If sea cliff retreat rates are a function of wave energy

focusing and dispersal, �ηB and �ηH are a function of coastline amplitude. Although these functions will beestablished below using a wave transformation model, let us assume momentarily, as a thought experiment,

that the forms of �ηB and �ηH are linear so that

�ηB ¼ E0B �ΩA tð Þ (2)

and

�ηH ¼ E0H þΩA tð Þ (3)

where E0B and E0H are time-averaged baseline retreat rates of the bay and headland, respectively, whencoastline amplitude is zero and Ω is a constant in units of inverse time that describes the rate at which waveenergy is captured on headlands and lost in embayments as amplitude changes. Note that E0B> E0H.Equation (1) now becomes

dAdt

∝ E0B �ΩAð Þ � E0H þΩAð Þ (4)

and when the coastline is in steady state and dA/dt= 0, we can solve for equilibrium amplitude, Ae:

Ae∝E0B � E0H

2Ω: (5)



Although �ηB and �ηH are yet to bedetermined and Ω is unconstrained,useful qualitative insights can bemade from equation (5). First,steady state amplitude isproportional to the difference inretreat rates between the headlandand bay. This makes physical sense,as the greater the difference is, thegreater the coastline amplitudemust be for wave energyredistribution to equalize the retreatrates. Second, we see that whenE0B> E0H, amplitude goes to zeroand headlands cannot persist. Thisis because wave energyconvergence or divergence when

A> 0 will always make headlands retreat faster than bays, and headlands will be transient (neglectingbeach-cliff interactions; see below).2.3.1. The Wave Transformation ModelWe use a numerical wave transformation model developed and detailed by Adams et al. [2002] and Adams[2004] to understand how wave energy is redistributed alongshore as coastline amplitude changes. In themodel, wave energy propagating toward shore is simulated using discretized parallel wave rays of highspatial resolution (Figure 5). The rays originate in deep water (>50m) and are prescribed with wave height(Hw0), wave period (T), and wave angle relative the mean coastline orientation (αο). A bathymetry grid is alsoinitially set, over which the wave transformation is calculated using linear wave theory [Komar, 1997] for eachray along its particular refracted path toward the coast. The wave modeling goals are to (1) calculate thechange in breaking wave energy flux, or power (P), per length of coastline as the cross-shore amplitudechanges and (2) relate that information to long-term rates of sea cliff retreat.

To reach these goals, we begin in deep water and follow the wave rays toward the coast as they shoal andrefract. Wave speed, Cw, is found from the dispersion equation [Komar, 1997]:

Cw ¼ gT2π

tanh d2πCwT

� �(6)

where g is gravitational acceleration and d is local water depth. Wave height, Hw, evolves concurrently alongthe wave path as

Hw ¼ Hw0KsKr : (7)

The variable Ks is the shoaling coefficient,

Ks ¼ 12n

Cw0

Cw(8)

where C0 is the initial deep-water wave celerity and n is a shoaling parameter defined as

n ¼ 12

1þ 2kdsinh 2kdð Þ

� �(9)

in which k is the wavenumber. The final term in equation (7), Kr, is the refraction coefficient:

Kr ¼ffiffiffiffiffiffiffiffiffiffiffifficosα0cosα

r(10)

where α is local wave direction found using Snell’s Law.

Figure 5. Schematic of the numerical wave transformationmodel, wherewaverays transport energy onshore and distribute it via refraction according tocoastline shape.



The wave ray is tracked along its path until a criticaldepth is reached, at which point the wave breaks.Theoretically, solitary waves oversteepen and breakat a critical ratio of wave height to water depth thathas been empirically estimated by Kaminsky andKraus [1993] as

Hwb

db¼ 1:2

βffiffiffiffiffiffiffiffiffiffiffiffiffiffiHw0=L0

p !0:27

(11)

where Hwb is breaking wave height, db is breaking wave depth, β is local seabed slope, and L0 is deep-water wavelength. At this ratio, the wave breaks and releases energy upon the sea cliff to do geomorphicwork (i.e., mechanical and hydraulic cliff erosion). How that energy is translated to a sea cliff retreat rate isdiscussed below. The wavelength-integrated energy flux per unit length of wave crest, or power density(P; W/m), is given as

P ¼ 18ρgH2

wbCwn (12)

where ρ is the density of seawater.2.3.2. Relating Wave Power, Coastline Amplitude, and Sea Cliff RetreatWith the wave ray tracing model, we next find out how coastline amplitude alters wave power deliveryalongshore. Conservation of energy flux causes seaward-growing headlands to capture wave power at thesame rate that landward-receding bays disperse it, so we can explore wave convergence on a singleheadland and subsequently extrapolate to find wave divergence. We will focus on a handful of key variablesthat affect wave power delivery for a given headland amplitude including the offshore wave characteristics(height, period, and angle) and alongshore headland width, BH. The chosen values of each variable used inthe wave ray tracing model are listed in Table 2. Wave height and period are coupled, but other variables areindependent of one another. A model iteration consists of a single wave transformation event, from deepwater to breaking, using a set of variables. Bathymetry contours reflect the coastline shape in shallow waterbut gradually decay (using a Gaussian function) in the offshore direction (Figure 5) toward flat contours thatrepresent a planar, seaward-sloping shelf by 50m depth. The bathymetry is an idealization intended tocapture the basic qualitative effects of wave energy convergence and divergence.

For each wave front, the delivered power density, P, is averaged alongshore across headland width BH(Figure 4) to yield a mean wave power density, P, that is a function of the prescribed wave height, period,angle, headland width, and headland amplitude. The spatial averaging effectively obscures any headland-scale (i.e.,<BH) patterns of power delivery. However, for our simple large-scale model, an aggregated signal isapt because we are interested in alongshore scales greater than BH to resolve spatial wave power patternsbetween headlands and bays, rather than within them.

We perform a suite of model experiments using all possible combinations of variable values. Iterating the

model for every combination yields a large database of P values (n= 4,800; Table 2). Using this data set, we

develop a predictor for P as a function of input variables using machine learning. Common machine learningtechniques such as artificial neural networks and boosted regression trees have been used previously instudying coastal processes [e.g., Pape et al., 2010; Oehler et al., 2011; van Maanen et al., 2010]. In this work weuse genetic programming [Koza, 1992; Poli et al., 2008], a population based machine learning routine: arandomly generated population of equations (a population of predictors) is manipulated based onevolutionary processes (e.g., mutation and crossover) to develop a smooth, optimized, physically reasonable,

and general model for how P varies as a function of input variables (headland amplitude, wave properties,and alongshore headland width). We use the proven, well-performing genetic programming packagedeveloped by Schmidt and Lipson [2009, 2013] that has been deployed successfully to study other coastalprocesses [Goldstein et al., 2013]. In the results section we demonstrate how the genetic programming-

derived equation for P, based on single-wave events, can be recast to represent the long-term time-averaged

wave power delivery as a function of headland amplitude (P Að Þ) for a given long-term wave climate [Ashtonand Murray, 2006b].

Table 2. List of Variable Values Used in the Wave RayTracing Model

Variable Values n

A (m) 0–1200 20α0 0–89° 20BH (km) 0.8, 1.6, 2.3 3Hw0 (m), T (s) {0.5, 8} {1, 10} {2, 12} {4, 15} 4

Total 4800



Using the long-term P Að Þ relationship, the final step is to translate wave power density into rates of sea cliffretreat (i.e., equations (2) and (3)). Unfortunately, there is no straightforward method to link wave forcing andsea cliff response, although many studies have addressed it along a range of coastline types from soft rockand hard rock [e.g., Sunamura, 1982; Kamphuis, 1987; Trenhaile, 2000; Adams, 2004; Walkden and Hall, 2005;Adams et al., 2005; Trenhaile, 2009; Trenhaile, 2011] to “cliffed” marsh edges [e.g., Schwimmer, 2001; Mariottiand Fagherazzi, 2010;Marani et al., 2011; Silliman et al., 2012]. Most approaches rely on site-specific empiricalcalibration and are limited by their lack of generality. Marani et al. [2011], however, found a general,transferable solution based only on dimensional analysis where the rate of retreat on cliffed marsh edges islinearly related to incident wave power. Using the linear relationship, sea cliff retreat along bays (subscript B)and headlands (subscript H) as a function of wave energy convergence and divergence and cross-shoreheadland amplitude is

�ηB;H ¼ cB;HPB;H Að Þ (13)

where cB,H is a constant (c≥ 0) that relates wave power to sea cliff retreat. Although Marani et al. [2011]derived the above relationship in the context of marsh edge retreat, the authors treated the marsh edge as asmall sea cliff in the dimensional analysis and included only variables that sea cliffs and marsh edges have incommon (i.e., cliff height, cliff strength, and still water depth). Thus, the wave power/erosion relationship canbe tuned to rocky sea cliff retreat using c. For our minimalist model, the linear style of equation (13) is asuitable fit because it is easy to manipulate and it is derived solely from dimensional analysis, whichmaximizes its generality. The constant cB,H is different for bays and headlands as a result of variations in rockstrength (and the resultant retreat rates). Similarly, because wave energy converges on headlands anddiverges in bays, the power density function P Að Þ will be different (inverse) for each and is noted using thesame subscripts.

2.4. Beach and Sea Cliff Dynamics

As incoming wave power drives cliffs landward, beach sediment is produced through rock erosion alongthe entire coastline that is transported alongshore and ultimately accumulates in bays. Sediment-drivensea cliff retreat (not including wave power delivery) in our model is a monotonically decaying function ofbeach width:

�ηB ¼ E0B e� w tð Þ

wscale (14)

where wscale is a characteristic length scale determined by E0B and wmax, and wmax is the beach width thatprovides near complete protection to sea cliffs from wave attack and causes the sea cliff retreat rate todecrease to zero. The width can be approximated using field observations [Everts, 1991; Lee, 2008] or usingwave runup models [e.g., Sallenger et al., 2002]. We use values of between 75 and 100m and a discussion ofmodel sensitivity to the specific value of wmax is included in Limber and Murray (in revision).

Because embayment sea cliff retreat is a function of time-varying beach width, we must add another centralequation to the analytical model to describe beach width change. Beach width change is simply a balance ofsediment gains from sea cliff retreat and any sediment losses [Limber and Murray, 2011]:

dwdt

¼ HBCB�ηB

� �þ HHCH�ηH ℓHð Þ 1

BB

� �� S (15)

where HB,H is sea cliff height (where B and H denote embayment or headland, respectively) divided by acharacteristic beach depth (Figure 4), CB,H reflects sea cliff composition and is the proportion of cliff-derivedsediment that is coarse enough to remain in the surf zone and contribute to beach width [Eittreim et al., 2002;Perg et al., 2003; Limber et al., 2008], ℓH is the total alongshore length of the headland (Figure 4); BB is thealongshore length of the embayment (Figure 4), and S is the time invariant rate of beach sediment loss. HB,H

and CB,H can be considered together as an effective dimensionless cliff height. For example, a small value ofHC does not necessarily mean that the cliff is small; it may alsomean that the cliff is quite tall but supplies littlesediment to the beach per unit length of retreat (i.e., CB,H is small), or the beach deposit is relatively thick.Small values of CB,H correspond to sand-poor cliffs made of mudstone, for example, and higher values couldimply a sandstone lithology, or other sediment-rich cliff material. The length BB is the alongshore distancethat beach sediment is produced by bay cliff retreat, and the distance that the total amount of beach



sediment (during a given time step) is distributed across as a pocket beach. The alongshore length of theheadland (ℓH, constrained by A and the alongshore headland lengthmeasured at the headland base, BH (Figure 4))determines the length over which beach sediment is produced as the headland sea cliffs retreat. Becauseamplitude changes through time, the headland length also changes, causing headland sediment production tochange through time. Headland length is the bare headland length that protrudes seaward of the beach and ismeasured relative to beach width. We use a simple Euclidean approximation of headland length (Figure 4),

ℓH ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA� wð Þ2 þ 1

2BH

� �2s

; (16)

so that the headland maintains a triangular shape of basal width BH as the coastline translates landward alongthe fixed-width lithologic heterogeneities. Many headlands are more rounded or rectangular in shape (e.g.,Figures 1 and 4), and so a Euclidean length is likely an underestimation of headland length that does not affectqualitative model dynamics. In future model experiments, a more complex arc length could be substituted.

To clarify the broad physical meaning of equation (15), we will interpret it term by term. The first term on theright-hand side represents sediment production from embayment sea cliff retreat. Sea cliff height (relative tobeach depth, HB,H) and the proportion of coarse material that is eroded from the cliff (CB,H) determine theamount of sediment produced per unit length of cliff retreat. The second term is sediment production fromheadland sea cliff retreat. Like in the embayment, the headland’s effective sea cliff height (HHCH) determinesthe amount of sediment released to the beach per unit length of retreat. The total amount of beach sedimentproduced as the headland sea cliff retreats is multiplied by the alongshore headland length scale ℓH and thendistributed across the embayment length BB, implicitly simulating alongshore sediment transport from theheadlands to the bays [Limber and Murray, 2011]. So, for a given input of beach sediment, beach widthchanges more for smaller embayment lengths than for larger ones because sediment is not as spread out.Finally, the third term, S, is a bulk sediment loss rate that can be broadly interpreted as, for example, surf zonesediment attrition by abrasion [Gibb and Adams, 1982; Dornbusch et al., 2002], losses associated with sea levelrise and the landward translation of the beach profile [Wolinsky and Murray, 2009], or human interventionsuch as sand mining [Thornton et al., 2006].

2.5. The Combined Model: Qualitative Dynamics and Equilibrium

Having described the wave energy redistribution and beach sediment dynamics, we can now better define

the comprehensive embayment and headland retreat rates, �ηB and �ηH. The headland rate is a function of localwave power convergence as amplitude changes and is written as

�ηH ¼ cHPH Að Þ: (17)

The bay retreat rate, �ηB, is more complex because it is a function of wave power divergence as well as beachsediment control:

�ηB ¼ cBPB Að Þ e�

w tð Þwscale : (18)

Note that the unknown functionP Að Þ is negative because wave energy is dispersed in bays at the same rate asit is captured by headlands.

With the comprehensive sea cliff retreat rates (nearly) defined, equations (1) (dA/dt) and 15 (dw/dt) are now acoupled dynamical system as the dependent variables A and w appear on the right-hand side of bothequations. These are the governing equations of our exploratory model:

dAdt

¼ cBPB Að Þ� �e�

w tð Þwscale � cHPH Að Þ (19)

dwdt

¼ HBCB cBPB Að Þ� �e�

w tð Þwscale

� �þ HHCHcHPH Að Þ ℓHð Þ 1

BB

� �� S: (20)

The functions that scale wave power delivery, sea cliff retreat, and coastline amplitude will be determined inthe following section using the results of the numerical wave transformation model.

The system will reach a stable steady state when equations (19) and (20) equal zero. This happens when wavedispersion and accumulating cliff-derived beach sediment reduce the (initially faster) embayment retreat rate so



that it is equal to the (initially slower) headland retreat rate that is hastened by wave energy focusing. The retreatrates converge (Figure 3) at an equilibrium amplitude and beach width where these geomorphic processes arebalanced. The equilibrium is stable (rather than unstable) because of the negative feedbacks between coastlineamplitude (or length) and sediment production, and between coastline amplitude and wave energyredistribution (Figures 2 and 3). For example, if coastline amplitude is increased from steady state, (1) embaymentcliff retreat will slow as a result of wave energy dispersion and the buildup of extra beach sediment shed from thenow larger headland, (2) headland retreat will quicken because wave energy focusing becomes more intense,and then (3) the imbalance will cause amplitude to decay to its original steady state. Sea cliff sedimentproduction will return to equilibrium rates as well, and the pocket beach will return to its equilibrium width.

2.6. A Note on Unphysical Beach Widths and Amplitudes, and Measuring Conventions

During sediment-poor conditions (i.e., when dw/dt< 0 for extended periods of time), beachwidth can decreasebelow zero in time-dependent numerical simulations. This unphysical result has consequences for modelstability and interpretation. When w< 0, embayment cliff retreat speeds up exponentially, causing cross-shoreamplitude growth to quickly (and incorrectly) accelerate. The amplitude and coastline length will consequentlygrow above the steady state solution and overproduce beach sediment. The sediment-poor conditions are thenrapidly reversed (dw/dt> 0), and the beach width swings from below zero to well above zero (e.g., 100m).Then, the wide beach buffers the embayment and causes coastline amplitude to decrease. In turn, sedimentproduction and beach width will decrease, too, until finally the system recovers to a physically realistic space.

An intuitive way of fixing this problem is to impose a constraint that does not allow beach width to drop belowzero. But, this creates a sharp discontinuity in the continuous solution that is incompatible with our choice ofnumerical methods (Runge-Kutta). There are two accessible ways to resolve the issue: recast the equations tooperate completelywithin a nonnegative parameter space [Shampine et al., 2005] or disregard the unphysical partsof a time-dependent solution.We choose the latter option, given that (1) a negative beachwidth arises only duringextended sediment-poor conditions and is not an issue otherwise and (2) themodel result of primary interest hereis a basic steady state scaling argument that can be found by setting dA/dt =dw/dt =0 and not iteratively solvingthe model equations. Still, the time-dependent model solutions are key to understanding how the systemprogresses to steady state, and we will not disregard them completely. A companion manuscript (Limber andMurray, in revision) shows model results and addresses only physical time-dependent results (w(t)> 0).

Conversely, during sediment-rich conditions (i.e., sea cliffs are tall and/or sediment rich, or the sediment lossrate is low), the equilibrium headland amplitude can drop below beach position or become negative.Fortunately, this result is easier to control. When A<w or A< 0, a fundamental assumption of the model isviolated: beach sediment must be confined to the embayments. So, if A(t) violates either condition, themodelsimulation is terminated and the equilibrium headland amplitude is taken as zero. Setting dA/dt =dw/dt = 0to find steady state solutions can also yield negative or unphysical results when conditions are sediment rich.These values are morphologically equivalent to A= 0.

Because beach position cannot move seaward of the headland position, headland amplitude could bemeasured relative to beach position. In this case, amplitude of zero does not necessarily mean that the coastlineis flat but instead that the beach is wider in cross-shore extent than the headland so that the coastline is likely tobe fronted by a continuous sandy beach. Gaviota, California, USA, is an example (Figure 1) where the cliffedcoastline exhibits subtle planform bumps but is fronted by a continuous subaerial beach with few exposedrocky headlands. This measurement convention means that changes in headland amplitude can be caused bychanges in beach width because beach width also varies through time. Defining headland amplitude usingdifferent conventions does not change model dynamics but alters the modeled quantitative headlandamplitude values and the physical interpretation of amplitude (especially when A= 0). In the companion paperto this contribution (Limber and Murray, in revision), we use both conventions to explore model results.

3. Results3.1. Wave Model and Genetic Programming

Each modeled value of P (wave power density averaged across headland width) is plotted in Figure 6. Thereare several nested data clusters, each corresponding to a wave model parameter (Table 2). The mostobvious clusters are the four determined by the wave height (Hw0) and period (T) pairings. Increasing the



magnitude of Hw0 and T increases mean wave power density, as should be expected. Within each of thosefour main groups, there are three additional groups, each representing one of the three chosen alongshoreheadland widths. Headland width has a negligible effect on wave power density. Within each headlandwidth cluster, the wave power decreases with increasing initial wave angle (light gray to black colors). Thisis because high-angle waves refract more than low-angle waves as they shoal, which causes wave crests tostretch and results in smaller breaking wave heights. Zooming in further shows the effects of headlandamplitude on mean wave power density. For each wave angle (within a given headland width cluster andwave height/period pairing), mean power density increases with headland amplitude (Figure 6 inset).Headlands that protrude further offshore can capture more incoming wave energy flux by inducing localwave refraction and wave ray convergence.

The genetic programming algorithm uses the results from the wave model to find a predictor of P. Thecomplexity of the predictors ranges from very simple (with large residuals) to very complex (with lowresiduals). We choose a smooth, physically interpretable predictor that best balances error and simplicity andis tractable in our rocky coastline model:

P Að Þ ¼ Ω2H2w0T cosα0 ±Ω1A tð Þ (21)

where Ω1 = 1.61 with units of J · m�2 s�1 and Ω2 = 930 with units of Jm�3 s�2 relate wave characteristics towave power density. To represent long-term forcing, equation (21) can be recast as

P Að Þ ¼ Dw ±Ω1A tð Þ (22)

where Dw (with units of power density) is a long-term time-averaged wave climate that represents thecumulative forcing by waves on sea cliff retreat, similar to the long-term effective diffusivity forced by waveson sandy coastlines [Ashton and Murray, 2006b]:

Dw ¼Σn

i¼1Ω2H 2

w0T cosα0�Δt

Σn

i¼1Δt

: (23)

Equation (22) can be calculated using wave buoy data or wave hindcasts (e.g., Wave Information Studies,http://wis.usace.army.mil/) as is demonstrated below.

The function is linear with respect to changes in headland amplitude, and Ω1 defines the rate of the

increase in P with A, and the first term defines the y axis intercept when A = 0 and shifts the functionvertically. When the second term is positive, (21) is the power delivery to a headland (power increases withamplitude; denoted by subscript “H”); when it is negative, it is the power delivery to a bay (power decreaseswith amplitude; denoted by subscript “B”). Values of mean power density from the wave model are plotted

Figure 6. Plot of all wavemodel runs. Each data point is breaking wave power density averaged over a headland width for agiven headland amplitude, wave height, wave angle, and wave period. Inset shows how each data cluster is a function ofheadland amplitude.



http://wis.usace.army.mil/

against values predicted by (21)(Figure 7). The model tends tooverpredict wave power densityfor large waves, but the overalltrend shows that mean wavepower is predicted reasonably wellacross all wave height/period pairs(different data marker shades inFigure 7). It is encouraging that thewave power predictor isproportional to wave heightsquared, as it follows the definitionof wave power (equation (12)). Thepredictor is also proportional tothe cosine of the offshore waveangle and wave period, which is inagreement with observations andwave power models [Adams et al.,2005; Marani et al., 2011].

We can now substituteP Að Þ into the headland and embayment sea cliff retreat rates (equations (17) and (18)).The final task is to estimate the constant that relates wave power to cliff retreat, cB,H. For the rocky coastlinemodel, when A=0, we can use the assigned baseline headland and bay retreat rates (E0H and E0B) to define c:

E0 B;Hð Þ ¼ cB;HPB;H 0ð Þ ¼ cB;HDw : (24)

Rearranging,

cB;H ¼ E0 B;Hð ÞDw

(25)

where the subscripts H and B refer to headlands and bays. The constant c can be physically interpreted as an“efficiency factor,” or a balance between long-termwave forcing (Dw) and rock response (E0) that determines howeffectively wave power is manifest as a sea cliff retreat rate. In terms of coastline dynamics, a higher value of cmeans that the coastline will evolve more rapidly. Inherent in the definition of c is rock strength, such that lessdurable rocks provide less resistance to incoming wave energy. Therefore, increasing E0 (and c) for a given waveclimate increases the efficiency of wave power in driving sea cliff retreat by implicitlymaking the rock less durable.Making thewave climatemore energetic (increasingHw0 or T, or decreasing α0) for a given rate of E0 decreases theefficiency and increases rock strength because the rock is “responding” less to more powerful forcing.

To understand the possible real-world range and magnitude of Dw, sample calculations are presented for twocoastlines: Montara and Gaviota, California (Figure 1), using National Data Buoy Center stations 46026 and46053. Wave characteristics were available for spans of 5 and 2 years, respectively, although a longer (decadal)

data set is desirable. The variableDw must be found using the local coastline orientation, which is about 275° forMontara (facing almost due west) and nearly 180° for Gaviota (facing almost due south). Gaviota is shelteredfrom wave energy by several islands and a large headland (Point Conception) [Adams et al., 2011]. Montara, onthe other hand, is on open-ocean coast and faces in the general direction of dominant incoming swell

[Wingfield and Storlazzi, 2007]. The approximate value of Dw for Montara is 4.3 × 104 Jm�1 s�1 as compared to0.3 × 104 Jm�1 s�1 for Gaviota. The power difference is a result of the abundance of high-angle waves and

sheltering effects (leading to smaller wave heights) in Gaviota. Thus, we have a range ofDw estimates for higherand lower energy coastlines to add context to the values used in the model results (Part 2, Limber and Murray,

in revision). The Dw values can also be used to calculate the efficiency factor, c, using (25).

4. Discussion

Themodel for rocky coastline amplitude was built using results from awavemodel that were further interpretedby genetic programming. With such a large data set of wave model results (n=4800), deriving an appropriate,

Figure 7. Modeled mean power density versus mean power density predictedby the genetic programming algorithm.



general predictor for wave power delivery (equation (21)) would otherwise be more challenging because of theinherent difficulty in visualizing relationships in more than three dimensions (i.e., the relationship between wavepower, headland amplitude, wave period, wave height, and deepwater wave angle). The choice of equation (21)also prescribed the value of a key constant, Ω1, which was partly the result of the idealized bathymetry used inthe wave model. A bathymetry that reflected the coastline shape to a much shallower depth, or hadobstructions like an offshore reef, may yield a smaller value of Ω1. Smaller values of Ω1 will lead to largerheadland amplitudes, and vice versa. Model results using different values of Ω1 are included in Part 2 of thisresearch (Limber and Murray, in revision).

In the Limber and Murray [2011] model, rocky coastline morphology (defined as “rockiness,” or the relativealongshore proportion of rocky headlands and sediment-filled bays) emerged as a result of abrasion-driveninternal sediment dynamics. In this paper, the alongshore variations in rock type dictate the coastline’srockiness, and it is not dynamic. Instead, we investigate the cross-shore coastline amplitude of headlands andbays whose positions are static through time. The effect of abrasion (initially increasing cliff retreat rates asbeach width increases from zero) is not included in our model, but using the model framework, we canconsider how it would affect the results. With abrasion and alongshore-homogeneous lithology, a stablesteady state exists when the pocket beach reaches an equilibrium width that drives embayment sea cliffretreat at the same rate as (sediment-free) headland retreat, equalizing sea cliff retreat rates alongshore[Limber and Murray, 2011] (Figure 3c). If we add a more erosion-resistant rock type to that steady statedepiction (and assume that the more resistant rock will protrude seaward as a sediment-free headland), thesteady state disappears because the cliff retreat rates are no longer equalized alongshore (Figure 3c). Now,the more durable headland is retreating more slowly than the rest of the coastline (i.e., any headlands andbays that had developed in the homogeneous lithology), and cross-shore amplitude grows. With theamplitude growth comes (1) coastline lengthening and the production of additional beach sediment, as seacliff retreat operates over a longer length scale, and (2) the destruction of any plan viewmorphology that haddeveloped through abrasion-induced feedbacks in the previously homogeneous coastline lithology, as bayretreat rates slow relative to headlands of the softer lithology. The geologic framework externally forcesheadlands and bays to exist along harder and softer lithology, respectively, and will override the abrasion-induced nonlinear dynamics and self-organization of headlands and bays. If the bay cliff retreat rate as afunction of beach width (equation (14)) includes the nonlinear effects of abrasion in our model, the modeledheadland amplitudes would not be affected: the steady state would still be the same (Figure 3c). While it isshown that abrasion does not theoretically play a significant role in setting cross-shore headland amplitudealong lithologically diverse rocky coastlines, it is important for short-term sea cliff dynamics that our modeldoes not resolve [Sunamura, 1982; Kamphuis, 1990; Trenhaile, 2005].

The goal of the current model is to generate a fundamental scaling argument. Treating a complex process likerocky coastline evolution in such a broad, simplified style allows us to clearly derive this argument becauseour approach highlights the key feedbacks leading to the scaling of cross-shore headland amplitude [Murray,2007]. In addition, large-scale behavior, as is explored here, may be better predicted by the use of large-scaleparameterizations rather than parameterizations of much smaller scale (and faster) processes [Murray, 2007].Still, many of the simplifications in our modeling framework could be relaxed for more in-depthinvestigations if the model equations were discretized over horizontally extended domain. The crudegeometry assumptions (Figure 4) could be improved and include a beach width that is not averagedalongshore, headlands that do not completely prevent sediment leakage between embayments, and anembayment shape that is not assumed to be flat alongshore. A more complex alongshore sediment transportscheme could be used to track beach position, such as that described in detail by Ashton and Murray [2006a,2006b]. Discretizing the equations would also lead to exploring how headlands of different amplitudeinteract with one another and possibly compete for incoming wave energy. Future work will address suchchanges and add further model complexity, with possible additions discussed below.

Sea level rise is not included in this initial work. But, Walkden and Dickson [2008], Ashton et al. [2011], andTrenhaile [2011] derived several general and quantitative predictions of rocky coastline recession from sealevel rise. These predictions can be appended to equations (17) and (18) that describe headland and bay seacliff retreat. As sea level rise accelerates, sea cliff retreat rates will speed up and the coastline amplitude willadjust more rapidly [Trenhaile, 2011]. In our case, sea level rise is likely to adjust the time scale over which thecoastline amplitude evolves rather than the amplitude itself. But, as sea level rises on a rocky coast, the sea



cliff height may change depending on the landscape slope landward of the cliff edge. If the landscape slopeis greater than zero, eventually the landward and upward trajectory of coastline retreat (i.e., as predicted bythe generalized Bruun Rule, adjusted for coastal lithology effects and gradients in alongshore flux) will matchthe landscape slope, and sea cliff height will remain constant through time [Wolinsky andMurray, 2009]. Usingthis perspective we could consider our cliff retreat model to be in equilibrium with rising sea level, such thatcliff height is constant and some sediment released by cliff retreat is used to build the beach profile verticallyand lost from the nearshore system (as represented by the constant S).

An implicit assumption in our model is that waves can consistently reach across the shore platform and erodethe sea cliff [Marshall and Stephenson, 2011; Ogawa et al., 2011; Dickson et al., 2013]. Implicitly, we couldconsider that the outer shore platform (seaward of the beach deposit) evolves through downcutting [Kirk,1977; Stephenson and Kirk, 1996, 1998; Porter et al., 2010; Stephenson et al., 2010] and allows this to occur.Some rocky coastline models include wave energy dissipation across the shore platform, so that as theplatform widens less wave energy is available to drive sea cliff retreat [e.g., Anderson et al., 1999; Trenhaile,2000]. Like sea level rise, this is an effect that can be easily added to equations (17) and (18). Previous researchhas treated wave energy dissipation as an exponentially decaying function dependent on platform width:

η̇c∝E0e�kb (26)

where k is the decay constant and b is shore platform width defined as the sum of previous sea cliff retreatthrough time:

b ¼ ∑∞

t¼0η̇c�Δt: (27)

This is another negative feedback that can control coastline evolution: sea cliff retreat discourages future seacliff retreat by lengthening the shore platform. If sea level is constant, it is possible that the platform can reacha width that completely dissipates wave energy and causes cliff retreat to cease [Edwards, 1941; Trenhaile,1972; Sunamura, 1978b; Trenhaile, 2000]. If the shore platform reaches such a width in front of both headlandsand bays, this could be a mechanism that ‘locks’ coastline shape into place by halting its evolution until sealevel adjusts and allows coastline retreat to resume. The addition of a shore platform and associated wavetransformation (along with sea level change) is another facet of this model that could be included (using theequations above) in future work.

ReferencesAdams, P. N. (2004), Assessing coastal wave energy and the geomorphic evolution of rocky coasts, PhD dissertation, University of California,

Santa Cruz, 175 p.Adams, P. N., R. S. Anderson, and J. Revenaugh (2002), Microseismic measurement of wave energy delivery to a rocky coast, Geology, 30(10),

895–898, doi:10.1130/0091-7613(2002)030<0895:MMOWED>2.0.CO;2.Adams, P. N., C. D. Storlazzi, and R. S. Anderson (2005), Nearshore wave-induced cyclical strain of sea cliffs: A possible fatigue mechanism,

J. Geophys. Res., 110, F02002, doi:10.1029/2004JF000217.Adams, P. N., D. L. Inman, and J. L. Lovering (2011), Effects of climate change and wave direction on hotspots of coastal erosion in Southern

California, Clim. Change, SI109, S211–S228, doi:10.1007/s10584-011-0317-0.Anderson, R. S., A. L. Densmore, and M. A. Ellis (1999), The generation and degradation of marine terraces, Basin Res., 11, 7–19, doi:10.1046/

j.1365-2117.1999.00085.x.Ashton, A., A. B. Murray, and O. Arnault (2001), Formation of coastlines features by large-scale instabilities induced by high-angle waves,

Nature, 414, 296–300.Ashton, A., and A. B. Murray (2006a), High-angle wave instability and emergent shoreline shapes: 1. Modeling of sand waves, flying spits, and

capes, J. Geophys. Res., 111, F04011, doi:10.1029/2005JF000422.Ashton, A., and A. B. Murray (2006b), High-angle wave instability and emergent shoreline shapes: 2. Wave climate analysis and comparisons

to nature, J. Geophys. Res., 111, F04012, doi:10.1029/2005JF000423.Ashton, A., M. J. A. Walkden, and M. E. Dickson (2011), Equilibrium responses of cliffed coasts to changes in the rate of sea level rise,Mar. Geo.,

284, 217–229, doi:10.1016/j.margeo.2011.01.007.Benumof, B. T., C. D. Storlazzi, R. J. Seymour, and G. B. Griggs (2000), The relationship between incident wave energy and seacliff erosion rates:

San Diego County, California, J. Coastal Res., 16(4), 1162–1178.Bowman, D., J. Guillén, L. López, and V. Pellegrino (2009), Planview geometry and morphological characteristics of pocket beaches on the

Catalan coast (Spain), Geomorphology, 108, 191–199, doi:10.1016/j.geomorph.2009.01.005.Bowman, D., V. Rosas, and E. Pranzini (2014), Pocket beaches of Elba Island (Italy)—Planview geometry, depth of closure, and sediment

dispersal, Estuarine Coastal Shelf Sci., doi:10.1016/j.ecss.2013.12.005.Carter, R. W. G., S. C. Jennings, and J. D. Orford (1990), Headland erosion by waves, J. Coast. Res., 6(3), 519–529.Carter, C. H., and D. E. Guy (1988), Coastal erosion: Processes, timing and magnitudes at the bluff toe, Mar. Geol., 84, 1–17.Castedo, R., W. Murphy, J. Lawrence, and C. Paredes (2012), A new process-response coastal recession model of soft rock cliffs,

Geomorphology, 177–178, 128–143, doi:10.1016/j.geomorph.2012.07.020.



AcknowledgmentsThis work is supported by NationalScience Foundation grants EAR-1024815 and EAR-1053033. We aregrateful to Alan Trenhaile and twoanonymous reviewers for detailed sug-gestions and expertise that greatlyimproved this paper.

http://dx.doi.org/10.1130/0091&hyphen;7613(2002)030<0895:MMOWED>2.0.CO;2



http://dx.doi.org/10.1029/2004JF000217

http://dx.doi.org/10.1007/s10584&hyphen;011&hyphen;0317&hyphen;0

http://dx.doi.org/10.1046/j.1365&hyphen;2117.1999.00085.x

http://dx.doi.org/10.1046/j.1365&hyphen;2117.1999.00085.x

http://dx.doi.org/10.1029/2005JF000422

http://dx.doi.org/10.1029/2005JF000423

http://dx.doi.org/10.1016/j.margeo.2011.01.007

http://dx.doi.org/10.1016/j.geomorph.2009.01.005

http://dx.doi.org/10.1016/j.ecss.2013.12.005


Castedo, R., M. Fernandez, A. S. Trenhaile, and C. Paredes (2013), Modelling cyclic recession of cohesive clay coasts: Effects of wave erosionand bluff stability, Mar. Geol., 335, 162–176, doi:10.1016/j.margeo.2012.11.001.

Davies, P., T. Sunamura, I. Takeda, H. Tsujimoto, and A. T. Williams (2004), Controls of shore platform width: The role of rock resistance factorsat selected sites in Japan and Wales, U. K., J. Coast. Res., SI39, 160–164.

Dawson, R. J., et al. (2009), Integrated analysis of risks of coastal flooding and cliff erosion under scenarios of long term change, Clim. Change,95(1–2), 249–288, doi:10.1007/s10584-008-9532-8.

Dickson, M. E., D. M. Kennedy, and C. D. Woodroffe (2004), The influence of rock resistance on coastal morphology around Lord Howe Island,southwest Pacific, Earth Surf. Processes Landforms, 29, 629–643, doi:10.1002/esp.1058.

Dickson, M. E., M. J. A. Walkden, and J. W. Hall (2007), Systemic impacts of climate change on an eroding region over the twenty-first century,Clim. Change, 84, 141–166, doi:10.1007/s10584-006-9200-9.

Dickson, M. E., H. Ogawa, P. S. Kench, and A. Hutchinson (2013), Sea-cliff retreat and shore platform widening: Steady-state equilibrium?,Earth Surf. Processes Landforms, 38, 1046–1048, doi:10.1002/esp.3422.

Dornbusch, U., R. B. G. Williams, C. Moses, and D. A. Robinson (2002), Life expectancy of shingle beaches: measuring in situ abrasion, J. Coast.Res., SI36, 249–255.

Edwards, A. B. (1941), Storm wave platforms, J. Geomorph., 4, 223–236.Eittreim, S. L., J. P. Xu, M. Noble, and B. D. Edwards (2002), Towards a sediment budget for the Santa Cruz shelf, Mar. Geol., 181, 235–248.Emery, K. O., and G. G. Kuhn (1982), Sea cliffs: Their processes, profiles, and classification, Geol. Soc. of Am. Bull., 93, 644–654.Everts, C. H. (1991), Seacliff Retreat and Coarse Sediment Yields in Southern California, Proceedings of Coastal Sediments ’91, vol. 2, pp. 1586–1598,

Amer. Soc. of Civil Eng., Seattle, Wash.Gibb, J. G., and J. Adams (1982), A sediment budget for the east coast between Oamaru and Banks Peninsula, South Island, New Zealand,

N. Z. J. Geol. Geophys., 25(3), 335–352, doi:10.1080/00288306.1982.10421497.Gilbert, G. K., and A. P. Brigham (1908), An Introduction to Physical Geography, pp. 410, D. Appleton and Co., New York.Goldstein, E. B., G. Coco, and A. B. Murray (2013), Prediction of wave ripple characteristics using genetic programming, Cont. Shelf Res., 71,

1–15, doi:10.1016/j.csr.2013.09.020.Gulliver, F. P. (1899), Shoreline topography, Proc. Am. Acad. Arts Sci., 34(8), 151–258.Hampton, M. A. (2002), Gravitational failure of sea cliffs in weakly lithified sediment, Env. and Eng. Geosci., 8(3), 175–191.Hapke, C. and D. Reid (2007), National assessment of shoreline change, Part 4: Historical coastal cliff retreat along the California coast, U.S.

Geological Survey Open-File Report, 2007–1133, 51 p.Hsu, J. R.-C., M.-J. Yu, F. C. Lee, and L. Benedet (2010), Static bay beach concept for scientists and engineers: A review, Coastal Eng., 57, 76–91.Hsu, J. R.-C., R. Silvester, and Y. M. Xia (1989), Generalities on static equilibrium bays, Coastal Eng., 12, 353–369.Jackson, D. W. T., and J. A. G. Cooper (2010), Application of the planform equilibrium concept to natural beaches in North Ireland, Coastal

Eng., 57, 112–123.Johnson, D. W. (1919), Shore Processes and Shoreline Development, pp. 584, Wiley, New York.Kaminsky, G. M., and N. C. Kraus (1993), Evaluation of depth-limited wave breaking criteria, in Proc. of 2nd Int. Symposium on Ocean Wave

Measurement and Analysis, pp. 180–193, Amer. Soc. of Civil Eng., New Orleans.Kamphuis, J. (1987), Recession rate of glacial till bluffs, J. Waterways, Ports, Coastal, and Ocean Eng., 113, 60–73.Kamphuis, J. W. (1990), Influence of sand or gravel on the erosion of cohesive sediment, J. Hydraulic Res., 28(1), 43–53.Kennedy, D. M., and M. E. Dickson (2006), Lithological control on the elevation of shore platforms in a microtidal setting, Earth Surf. Processes

Landforms, 31, 1575–1584.Kirk, R. M. (1977), Rates and forms of erosion on intertidal platforms at Kaikoura Peninsula, South Island, New Zealand, N. Z. J. Geol. Geophys.,

20, 571–613.Komar, P. D. (1985), Computer models of shoreline configuration: Headland erosion and the graded beach revisited, in Models in

Geomophology, edited by M. J. Woldenberg, pp. 155–170, Allen and Unwin, London.Komar, P. D. (1997), Beach Processes and Sedimentation, pp. 544, Prentice Hall, Upper Saddle River, N. J., U.S.A.Komar, P. D., and S. Shih (1993), Cliff erosion along the Oregon coast: A tectonic-sea level imprint plus local controls by beach processes,

J. Coastal Res., 9(3), 747–765.Koza, J. R. (1992), Genetic Programming: On the Programming of Computers by Means of Natural Selection, MIT Press, Cambridge, Mass., U.S.A.Lee, E. M. (2008), Coastal cliff behavior: Observations on the relationship between beach levels and recession rates, Geomorphology, 101,

558–571.Lim, M., N. J. Rosser, D. N. Petley, and M. Keen (2011), Quantifying the controls and influence of tide and wave impacts on coastal rock cliff

erosion, J. Coastal Res., 27(1), 46–56.Limber, P. W., and A. B. Murray (2011), Beach and sea cliff dynamics as a driver of rocky coastline evolution and stability, Geology, 39(12),

1149–1152.Limber, P. W., K. B. Patsch, and G. B. Griggs (2008), Coastal sediment budgets and the littoral cut-off diameter: A grain-size threshold for

quantifying active sediment inputs, J. Coastal Res., 24(supplement 2), 122–133, doi:10.2112/06-0675.1.Littlewood, R., A. B. Murray, and A. D. Ashton (2007), An alternative explanation for the shape of ‘log-spiral’ bays, in Proceedings of Coastal

Sediments ’07, pp. 341–350, Amer. Soc. of Civil Eng., New Orleans, Louisiana.Marani, M., A. D’Alpos, S. Lanzoni, and M. Santalucia (2011), Understanding and predicting wave erosion of marsh edges, Geophys. Res. Lett.,

38, L21401, doi:10.1029/2011GL048995.Mariotti, G., and S. Fagherazzi (2010), A numerical model for the coupled long-term evolution of salt marshes and tidal flats, J. Geophys. Res.,

115, F01004, doi:10.1029/2009JF001326.Marshall, R. J. E., and W. J. Stephenson (2011), The morphodynamics of shore platforms in a micro-tidal setting: Interactions between waves

and morphology, Mar. Geol., 288, 18–31, doi:10.1016/j.margeo.2011.06.007.May, J. P., and W. F. Tanner (1973), The littoral power gradient and shoreline changes, in Coastal Geomorphology, edited by D. R. Coates,

Binghampton State University, New York.Moore, L. J., and G. B. Griggs (2002), Long-term cliff retreat and erosion hotspots along the central shores of Monterey Bay National Marine

Sanctuary, Marine Geol., 181, 265–283.Muir-Wood, A. (1971), Engineering aspects of coastal landslides, Proc. Inst. Civ. Eng., 50(3), 257–276.Murray, A. B. (2007), Reducing model complexity for explanation and prediction, Geomorphology, 90, 178–191, doi:10.1016/j.

geomorph.2006.10.020.Naylor, L. A., and W. J. Stephenson (2010), On the role of discontinuities in mediating shore platform erosion, Geomorphology, 114(1–2),

89–100.





http://dx.doi.org/10.1002/esp.1058



http://dx.doi.org/10.1080/00288306.1982.10421497

http://dx.doi.org/10.1016/j.csr.2013.09.020

http://dx.doi.org/10.2112/06&hyphen;0675.1

http://dx.doi.org/10.1029/2011GL048995

http://dx.doi.org/10.1029/2009JF001326




Naylor, L. A., M. A. Coombes, and H. A. Viles (2012), Reconceptualising the role of organisms in the erosion of rock coasts: A new model,Geomorphology, 157, 17–30.

Oehler, F., G. Coco, M. O. Green, and K. R. Bryan (2011), A data-driven approach to predict suspended-sediment reference concentrationunder non-breaking waves, Continental Shelf Res., 46, 96–106.

Ogawa, H., M. E. Dickson, and P. S. Kench (2011), Wave transformation on a sub-horizontal shore platform, Tatapouri, North Island, NewZealand, Continental Shelf Res., 31, 1409–1419, doi:10.1016/j.csr.2011.05.006.

Pape, L., Y. Kuriyama, and B. G. Ruessink (2010), Models and scales for cross-shore sandbar migration, J. Geophys. Res., 115, F03043,doi:10.1029/2009JF001644.

Perg, L. A., R. S. Anderson, and R. C. Finkel (2003), Use of cosmogenic radionuclides as a sediment tracer in the Santa Cruz littoral cell,California, United States, Geology, 31, 299–302, doi:10.1130/0091-7613(2003)031<0299:UOCRAA>2.0.CO;2.

Poli, R., W. B. Langdon, and N. F. McPhee (2008), A Field Guide to Genetic Programming, Lulu Enterprises, U. K., Limited.Porter, N. J., A. S. Trenhaile, K. J. Prestanski, and J. I. Kanyaya (2010), Patterns of surface downwearing on shore platforms in eastern Canada,

Earth Surf. Processes Landforms, 35, 1793–1810.Qartal, R., A. S. Trenhaile, N. C. Mitchell, and F. Tempera (2010), Development of volcanic insular shelves: Insights from observations and

modelling of Faial Island in the Azores Archipelago, Mar. Geol., 275(1–4), 66–83, doi:10.1016/j.margeo.2010.04.008.Robinson, L. A. (1977), Marine erosive processes at the cliff foot, Mar. Geol., 23, 257–271, doi:10.1016/0025-3227(77)90022-6.Rosser, N. J., M. J. Brain, D. N. Petley, M. Lim, and E. Norman (2013), Coastline retreat via progressive failure of rocky coastal cliffs, Geology,

41(8), 939–942.Ruggiero, P., P. D. Komar, W. G. McDougal, J. J. Marra, and A. Beach (2001), Wave runup, extreme water levels and the erosion of properties

backing beaches, J. Coastal Res., 117, 407–419.Runyan, K., and G. B. Griggs (2003), The effects of armoring seacliffs on the natural sand supply to the beaches of California, J. Coastal Res.,

19(2), 336–347.Sallenger, A. H., W. Krabill, J. Brock, R. Swift, S. Manizade, and H. Stockdon (2002), Sea cliff erosion as a function of beach changes and extreme

wave runup during the 1997–1998 El Nino, Marine Geol., 187, 279–297, doi:10.1016/S0025-3227(02)00316-X.Schmidt, M., and H. Lipson (2009), Distilling free-form natural laws from experimental data, Science, 324, 81–85.Schmidt, M., and H. Lipson (2013), Eureqa (Version 0.98 beta) [Software]. [Available at http://www.eureqa.com/.]Schwimmer, R. A. (2001), Rates and processes of marsh shoreline erosion in Rehoboth Bay, Delaware, U.S.A, J. Coastal Res., 17(3), 672–683.Shampine, L. F., S. Thompson, J. A. Kierzenka, and G. D. Byrne (2005), Non-negative solution to ODEs, Appl. Math. Comput., 170, 556–569.Short, A. D. (1999), Beach and Shoreface Morphodynamics, pp. 379, John Wiley, Chichester.Silliman, B. R., J. van de Koppel, M. W. McCoy, J. Diller, G. Kasozi, K. Earl, P. N. Adams, and A. R. Zimmerman (2012), Degradation and resilience in

Louisiana salt marshes after the BP–Deepwater Horizon oil spill, Proc. Nat. Acad. Sci., 109(28), 11,234–11,239, doi:10.1073/pnas.1204922109.Silvester, R. (1960), Stabilization of sedimentary coastlines, Nature, 188, 467–469.Stephenson, W. J., and R. M. Kirk (1996), Measuring erosion rates using the micro-erosion meter: 20 years of data from shore platforms,

Kaikoura Peninsula, South Island New Zealand, Mar. Geol., 131, 209–218.Stephenson, W. J., and R. M. Kirk (1998), Rates and patterns of erosion on shore platforms, Kaikoura, South Island, New Zealand, Earth Surf.

Processes Landforms, 23, 1071–1085.Stephenson, W. J., and L. A. Naylor (2011), Within site geological contingency and its effect on rock coast erosion, J. Coast. Res., SI64, 831–835.Stephenson, W. J., R. M. Kirk, S. A. Hemmingsen, and M. A. Hemmingsen (2010), Decadal scale micro erosion rates on shore platforms,

Geomorphology, 114, 22–29, doi:10.1016/j.geomorph.2008.10.013.Storlazzi, C. D., and M. E. Field (2000), Sediment distribution and transport along a rocky, embayed coast: Monterey Peninsula and Carmel

Bay, California, Marine Geol., 170, 289–316.Sunamura, T. (1976), Feedback relationship in wave erosion of laboratory rocky coast, J. Geol., 84, 427–437, doi:10.1086/628209.Sunamura, T. (1977), A relationship between wave-induced cliff erosion and erosive force of waves, J. Geol., 85, 613–618.Sunamura, T. (1978a), A model of the development of continental shelves having erosional origin, Geol. Soc. Am. Bull., 89, 504–510.Sunamura, T. (1978b), A mathematical model of submarine platform development, Math. Geol., 10, 53–58.Sunamura, T. (1982), A wave tank experiment on the erosional mechanism at a cliff base, Earth Surf. Processes Landforms, 7, 333–343,

doi:10.1002/esp.3290070405.Thornton, E. B., A. Sallenger, J. Conforto Sesto, L. Egley, T. McGee, and R. Parsons (2006), Sand mining impacts on long-term dune erosion in

southern Monterey Bay, Mar. Geol., 229, 45–58, doi:10.1016/j.margeo.2006.02.005.Thornton, L., andW. J. Stephenson (2006), Rock strength: A control of shore platform elevation, J. Coast. Res., 22, 224–231, doi:10.2112/05A-0017.1.Trenhaile, A. S. (1972), The shore platforms of the Vale of Glamorgan, Wales, Trans. Inst. Br. Geogr., 56, 127–144.Trenhaile, A. S. (1987), The Geomorphology of Rock Coasts, Clarendon Press, Oxford.Trenhaile, A. S. (2000), Modeling the development of wave-cut shore platforms, Marine Geol., 166, 163–178.Trenhaile, A. S. (2001), Modeling the Quaternary evolution of shore platforms and erosional continental shelves, Earth Surf. Processes

Landforms, 26, 1103–1128.Trenhaile, A. S. (2002), Rock coasts, with particular emphasis on shore platforms, Geomorphology, 48, 7–22.Trenhaile, A. S. (2004), Modeling the accumulation and dynamics of beaches on shore platforms, Mar. Geol., 206, 55–72.Trenhaile, A. S. (2005), Modeling the effect of waves, weathering, and beach development on shore platform development, Earth Surf.

Processes Landforms, 30, 613–634.Trenhaile, A. S. (2009), Modeling the erosion of cohesive clay coasts, Coastal Eng., 56, 59–72, doi:10.1016/j.coastaleng.2008.07.001.Trenhaile, A. S. (2010), Modeling cohesive clay coast evolution and response to climate change, Mar. Geol., 277, 11–20, doi:10.1016/j.

margeo.2010.08.002.Trenhaile, A. S. (2011), Predicting the response of hard and soft rock coasts to changes in sea level and wave height, Clim. Change, 109,

599–615, doi:10.1007/s10584-011-0035-7.Valvo, L., A. B. Murray, and A. Ashton (2006), How does underlying geology affect coastline change? An initial modeling investigation,

J. Geophys. Res., 111, F02025, doi:10.1029/2005JF000340.van Maanen, B., G. Coco, K. R. Bryan, and B. G. Ruessink (2010), The use of artificial neural networks to analyze and predict alongshore

sediment transport, Nonlin, Processes Geophys., 17, 395–404, doi:10.5194/npg-17-395-2010.Walkden, M. J. A., and J. W. Hall (2005), A predictive mesoscale model of the erosion and profile development of soft rock shores, Coastal Eng.,

52, 535–563.Walkden, M. J. A., and M. E. Dickson (2008), Equilibrium erosion of soft rock shores with a shallow or absent beach under increased sea level

rise, Mar. Geol., 251, 75–84.



http://dx.doi.org/10.1016/j.csr.2011.05.006

http://dx.doi.org/10.1029/2009JF001644

http://dx.doi.org/10.1130/0091&hyphen;7613(2003)031<0299:UOCRAA>2.0.CO;2




http://dx.doi.org/10.1016/0025&hyphen;3227(77)90022&hyphen;6

http://dx.doi.org/10.1016/S0025&hyphen;3227(02)00316&hyphen;X

http://www.eureqa.com/

http://dx.doi.org/10.1073/pnas.1204922109


http://dx.doi.org/10.1086/628209



http://dx.doi.org/10.2112/05A&hyphen;0017.1

http://dx.doi.org/10.1016/j.coastaleng.2008.07.001




http://dx.doi.org/10.1029/2005JF000340

http://dx.doi.org/10.5194/npg&hyphen;17&hyphen;395&hyphen;2010

Walkden, M. J. A., and J. W. Hall (2011), A mesoscale predictive model of the evolution and management of a soft-rock coast, J. Coast. Res.,27(3), 529–543, doi:10.2112/JCOASTRES-D-10-00099.1.

Ward, S. M., and G. Valensise (1994), The Palos Verdes terraces, California: Bathtub rings from a buried reverse fault, J. Geophys. Res., 99(B3),4485–4494, doi:10.1029/93JB03362.

Weesakul, S., T. Rasmeemasmuang, S. Tasaduak, and C. Thaicharoen (2010), Numerical modeling of crenulate bay shape, Coastal Eng., 57,184–193.

Wingfield, D. K., and C. D. Storlazzi (2007), Spatial and temporal variability in oceangraphic and meteorlogic forcing along Central Californiaand its implications on nearshore processes, J. Mar. Syst., 68, 457–472.

Wolinsky, M. A., and A. B. Murray (2009), A unifying framework for shoreline migration: 2. Application to wave-dominated coasts, J. Geophys.Res., 114, F01009, doi:10.1029/2007JF000856.

Yasso, W. E. (1965), Plan geometry of headland bay beaches, J. Geol., 73, 702–714.Young, A. P., R. E. Flick, R. Gutierrez, and R. T. Guza (2009), Comparison of short-term seacliff retreat measurement methods in Del Mar,

California, Geomorphology, 112, 318–323.Young, A. P., R. E. Flick, W. C. O’Reilly, D. B. Chadwick, W. C. Crampton, and J. J. Helly (2014), Estimating cliff retreat in southern California

considering sea level rise using a sand balance approach, Mar. Geol., 348, 15–26, doi:10.1016/j.margeo.2013.11.007.



http://dx.doi.org/10.2112/JCOASTRES&hyphen;D&hyphen;10&hyphen;00099.1

http://dx.doi.org/10.1029/93JB03362

http://dx.doi.org/10.1029/2007JF000856


publications - university of...

Documents