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Fetch-limited self-organization of elongate water bodies

¹ Coastal Systems Group, Department of Geology and Geophysics, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543, USA
² Department of Earth and Ocean Sciences, Nicholas School of the Earth and Environment/Center for Complex and Nonlinear Processes, Duke University, Durham, North Carolina 27707, USA

	ABSTRACT

TOP ABSTRACT INTRODUCTION WAVE-INFLUENCED SEGMENTED LAKES HIGH-ANGLE-WAVE SHORELINE... METHODS: NUMERICAL MODEL RESULTS: MODELED BEHAVIOR RESULTS: LOCAL WAVE CLIMATES DISCUSSION REFERENCES CITED

 
Naturally occurring elongate bodies that are segmented or appear to be in the process of segmentation occur in a variety of environments and scales. A simple, process-based numerical model of planform shoreline evolution demonstrates that fetch controls on alongshore sediment transport can result in the segmentation of an elongate water body into smaller, rounded lakes or ponds. The shape of elongate water bodies leaves their long coasts prone to a high-angle-wave instability in shoreline shape that results in the formation of capes that grow through interactions with one another along the same coast. In a numerical model, as capes extend farther offshore, a new behavior emerges, whereby capes on opposing coasts attract one another laterally as they grow, suggesting a novel mechanism for large-scale shoreline self-organization through fetch-limiting interactions. We demonstrate these interactions through analysis of local net sediment flux and coastline stability. Ensemble model runs suggest that, for a symmetric wind distribution, the initial segmentation of a water body requires four lengths per initial width, yet water bodies with higher initial aspect ratios segment to one final round water body per factor of two of the initial aspect ratio. Wave-dominated elongate water bodies with coasts consisting of clastic sediment (and a lack of vegetation) are most likely to undergo this predicted segmentation.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION WAVE-INFLUENCED SEGMENTED LAKES HIGH-ANGLE-WAVE SHORELINE... METHODS: NUMERICAL MODEL RESULTS: MODELED BEHAVIOR RESULTS: LOCAL WAVE CLIMATES DISCUSSION REFERENCES CITED

 
The long-axis shores of some elongate water bodies exhibit waveformed features such as sandy spits and capes. Appearing at many scales, these features often attain a size commensurate with that of the width of the water body itself (Fig. 1; for geographical locations and data sources, see the GSA Data Repository¹). In many cases, similar cuspate features are also found on the opposing coast, sometimes opposite one another (Fisher, 1955; Fig. 1E), sometimes not (Fig. 1B). In other settings, series of rounded lakes outline a larger basin, suggesting that opposing cuspate shoreline features have joined, segmenting the lake. Although qualitative models of this segmentation have long been suggested, this behavior has neither been directly observed in nature nor reproduced quantitatively (Zenkovich, 1967). Explaining the origin of these enigmatic series of rounded lakes not only deepens our understanding of the evolution of fetch-limited environments (Cooper et al., 2007), but can allow us to constrain and accurately interpret the environmental conditions during their formation as recorded in the sedimentary record (e.g., Wright et al., 2000).

Zenkovich (1959, 1967) suggested a qualitative model whereby the appearance of long-axis cuspate forms and the eventual segmentation of elongate water bodies could be attributable to waves generated by winds blowing across the long fetch parallel to the main axis, arriving with crests at angles >45° relative to the long coastlines. Recent theoretical and numerical studies (Ashton et al., 2001; Ashton and Murray, 2006a) have investigated how such high-angle waves lead to the initial formation and subsequent self-organization of cuspate features. Here we present results from a model of coastline evolution modified to represent the generation of waves within an enclosed water body. As simulated cuspate shoreline features extend across a water body, a fascinating new dynamic emerges: by changing the wave fetch fields on opposing coasts, these growing capes and spits attract one another across the water body, eventually segmenting it into smaller, round water bodies. This process modeling confirms many aspects of Zenkovich's phenomenological model (1959 (1967), but suggests significantly different mechanisms for final water body segmentation.

	WAVE-INFLUENCED SEGMENTED LAKES

TOP ABSTRACT INTRODUCTION WAVE-INFLUENCED SEGMENTED LAKES HIGH-ANGLE-WAVE SHORELINE... METHODS: NUMERICAL MODEL RESULTS: MODELED BEHAVIOR RESULTS: LOCAL WAVE CLIMATES DISCUSSION REFERENCES CITED

 
The presence of segmented water bodies at a variety of scales and (modern) environments, from arctic to tropical, suggests a universal formation mechanism. (Accordingly, although we use the term "lake" in the following discussion, we imply any general enclosed or semi-enclosed water body, including ponds, estuaries, or lagoons.) Although we have not performed a comprehensive global survey, many examples of these depositional features exist at high latitudes (Fig. 1), so we must clearly distinguish the bodies of water we are interested in from thermo-karst and other oriented lakes also found in these settings (Livingstone, 1954). While thermo-karst lakes are erosional and expand due to permafrost thawing of cohesive banks (West and Plug, 2008, and references therein), the examples presented in Figure 1 share a geomorphic expression that suggests accretion of the cuspate shoreline forms, implying that clastic material has been reworked and deposited by waves.

	HIGH-ANGLE-WAVE SHORELINE INSTABILITY

TOP ABSTRACT INTRODUCTION WAVE-INFLUENCED SEGMENTED LAKES HIGH-ANGLE-WAVE SHORELINE... METHODS: NUMERICAL MODEL RESULTS: MODELED BEHAVIOR RESULTS: LOCAL WAVE CLIMATES DISCUSSION REFERENCES CITED

 
Recent research has illuminated the importance of deep-water (unrefracted) wave-approach angle on shoreline evolution, and the instability in the shoreline shape when waves approach the coast with high angles (Ashton et al., 2001; Ashton and Murray, 2006a). Numerical modeling demonstrates how, along an open ocean coast, a predominance of high-angle waves leads to the self-organization of quasi-periodic shoreline features, including alongshore sandwaves, cuspate flying spits, and capes. In enclosed elongate water bodies, even if winds do not predominantly blow along the long axis, the increased fetch in that direction will generate waves that are larger than those generated by winds blowing across the short axis, tending to produce a dominance of high-angle waves along the long coast. This situation should cause the long coast to self-organize into capes and cuspate spits (Zenkovich, 1959; Ashton et al., 2001). However, this formation mechanism for shoreline undulations does not explain how cuspate features on opposing sides of a water body can attract one another, eventually joining.

	METHODS: NUMERICAL MODEL

TOP ABSTRACT INTRODUCTION WAVE-INFLUENCED SEGMENTED LAKES HIGH-ANGLE-WAVE SHORELINE... METHODS: NUMERICAL MODEL RESULTS: MODELED BEHAVIOR RESULTS: LOCAL WAVE CLIMATES DISCUSSION REFERENCES CITED

 
We use a model similar to the one described in Ashton and Murray (2006a), that evolves a discretized plan-view shoreline based upon gradients in alongshore sediment transport, determined using the CERC (Coastal Engineering Research Center) equation (Komar, 1971):

where Q_s is the alongshore sediment transport (m³/s, deposited volume), the generally empirical parameter K is related to sediment characteristics (typically ~0.4 m^1/2s^–1), H_b is the breaking wave height (m), _b is the orientation of breaking wave crests, and is the shoreline trend. As in other one-contour-line models (Hanson and Kraus, 1989), gradients in alongshore sediment transport are spread across the shoreface depth D. The model has been improved from previous generations to allow the evolution of more than one simply connected shoreline domain, including islands not attached to the main coast.

The major difference between the model used here and the one presented in Ashton and Murray (2006a) is that rather than forcing shoreline changes from an assumed external wave source, waves are generated locally as a function of the length of water, or fetch, that winds from a given direction blow across. We use a basic equation to determine wave height assuming no depth limitation (using relationships from Komar, 1998):

where H₀ is the deep-water wave height (m), U_w is the 10-m-elevation wind speed (m/s), and F is the fetch (m). Wave period (T, s) is also computed (U.S. Army Corps of Engineers, 1984):

To determine the breaking values used in Equation 1, the computed local deep-water waves are iteratively refracted onshore using linear approximations and assuming shore-parallel contours until depth-limited breaking. All of the simulations here use the simplest wind climate scenario possible, an isotropically distributed wind climate, with a new random wind direction chosen each simulated high wind day.

The crucial relationship in Equation 2 is the wave height dependence on the square root of the fetch. As long as the wind distribution remains unchanged, altering the wind speed only affects the scaling of simulated time to actual time (e.g., Ashton and Murray 2006a). Although the simulations presented here are for a set of fixed parameters, as with a physical experiments, the results can be rigorously rescaled for different characteristic values of U_w and D and different-sized water bodies as long as alongshore sediment transport processes remain dominant.

In keeping with the general exploratory modeling approach focusing on the fundamental system behaviors and nonlinear feedbacks (Murray, 2007), the treatment of waves is simple. More detailed treatment of waves, including refraction over non-shore-parallel contours (Falqués and Calvete, 2005) or accounting for depth constraints upon wave generation (e.g., Carniello et al., 2005), should have a quantitative, rather than qualitative effect on modeled behavior. The model is also idealized as it assumes that the coast consists of mobile, noncohesive sediment whose long-term evolution is unaffected by the underlying geological framework.

	RESULTS: MODELED BEHAVIOR

TOP ABSTRACT INTRODUCTION WAVE-INFLUENCED SEGMENTED LAKES HIGH-ANGLE-WAVE SHORELINE... METHODS: NUMERICAL MODEL RESULTS: MODELED BEHAVIOR RESULTS: LOCAL WAVE CLIMATES DISCUSSION REFERENCES CITED

 
Simulations begin from a hypothetical initial configuration of a rectangular basin with white-noise initial perturbations to the shoreline (Fig. 2). Although the wind field is isotropic, the large fetch along the long axis results in high-angle waves along these coasts. The predominance of high-angle waves results in shoreline instability, causing the coastlines to self-organize into cuspate shapes that appear along the long axis and grow and increase wavelength through merging (Fig. 2). Whereas shoreline undulations in the middle of the water body have symmetrical cuspate bump shapes, asymmetrical fetch distributions closer to the ends result in flying spits that migrate away from the center of the lake. This initial evolution resembles that of a single coast exposed to high-angle waves (Ashton et al., 2001; Ashton and Murray, 2006a).

Once these shapes coarsen in wavelength and cross-shore amplitude such that they extend significantly offshore (approximately half-way across the water body), a new dynamic emerges: regardless of whether they are aligned directly across the water body, the cusps grow together. Offset cuspate shapes migrate toward one another and eventually merge, segmenting the domain. After segmentation, the new, smaller water bodies are not sufficiently oblong to cause shoreline instability, and the lakes eventually tend toward a circular shape, reflecting the characteristics of the wave climate (in other simulations, not shown, asymmetrical wave distributions lead to more elliptical bodies of water).

	RESULTS: LOCAL WAVE CLIMATES

TOP ABSTRACT INTRODUCTION WAVE-INFLUENCED SEGMENTED LAKES HIGH-ANGLE-WAVE SHORELINE... METHODS: NUMERICAL MODEL RESULTS: MODELED BEHAVIOR RESULTS: LOCAL WAVE CLIMATES DISCUSSION REFERENCES CITED

 
The manner by which opposing capes affect one another can be better understood by examining how the evolving shapes of the water body affect wave climates along other coasts. The first way to examine these fetch-controlling feedbacks is to look directly at local wave energy roses (Fig. 3), computed by freezing the model domains and then summing the deep-water wave contribution to alongshore sediment transport in the directional wave climate. Further techniques for analyzing wave climates were presented in Ashton and Murray (2006b). Two key metrics express the locally normalized net alongshore sediment transport (where positive values indicate flux to the right, looking offshore) and locally normalized shoreline-shape diffusivity (, with –1.0 indicating a fully unstable wave climate and 1.0 indicating a fully stable wave climate) (Fig. 3).

Similar to an open coast (Ashton and Murray, 2006b), the generation of shoreline undulations results in increased local stability ( > 0), with unstable wave climates only at the cape tips ( < 0) (Fig. 3). The wave roses also illustrate how the growth of shoreline capes and segmentation change the fetch fields, leading to reduced local proportions of high-angle waves.

Net alongshore sediment transport is directed toward the tips of the cuspate features and they continue to grow. If a shoreline feature extends across the water body and it is not directly opposite another cape or spit, it begins to block waves from its side of the lake or lagoon (Figs. 2 and 3). The resultant asymmetry in the wave climate experienced by a cape or spit on the opposing shoreline causes it to migrate alongshore toward the feature creating the fetch limitation. This fetch-limiting feedback acts symmetrically across the water body. Thus, growing features attract one another and eventually merge. This new dynamic revealed by the model contrasts with the hypothesis that features are either inclined to initially form across from each other, or that the "laws of wave refraction" set the scale for segmentation (Zenkovich, 1967, p. 518).

	DISCUSSION

TOP ABSTRACT INTRODUCTION WAVE-INFLUENCED SEGMENTED LAKES HIGH-ANGLE-WAVE SHORELINE... METHODS: NUMERICAL MODEL RESULTS: MODELED BEHAVIOR RESULTS: LOCAL WAVE CLIMATES DISCUSSION REFERENCES CITED

 
Shoreline protuberances are expected to form in any wave-dominated elongate water bodies that have shoreline reaches dominated by high-angle waves. For the simple case of an isotropic wind field, the presence of any unstable coastal reach should require an aspect ratio >~2:1 in the initial water body. Model results, however, suggest that for a water body to divide even once, it needs an initial aspect ratio of 4:1 (Fig. 4). For smaller initial aspect ratios, even though nascent capes form along the long coast, the rectangular lake becomes progressively rounder (as sediment collects at the ends) at a rate sufficiently rapid to reduce the predominance of high-angle waves along the long axes. Features eventually stop extending across the body and disappear as the water body becomes round; the opposing features never manage to connect. As the initial aspect ratio is increased beyond this threshold to segment a water body once, final lakes develop at a rate of one per factor of two of the initial aspect ratio (Fig. 4).

Where can this segmentation process be expected to occur? The phenomenon discussed here should not be expected to occur in all elongate water bodies, only those with noncohesive (e.g., sand or gravel) banks actively reworked by waves. In many small fetch environments, development of bank (shoreline) vegetation and the potential subsequent deposition of cohesive sediment could prevent wave transportation of sediments. Segmentation would therefore be favored where vegetation growth is slow or frequently interrupted, such as in cold and arid environments where even small waves can keep the shore free of vegetation, or on the backs of frequently overwashing barriers. Elsewhere, larger fetches and stronger winds would be required for waves to be sufficiently strong to prevent vegetation from stabilizing banks.

In some wave-influenced locations, cuspate shoreline features exist, but cross-body connection does not appear imminent. Significant tidal flows, which would become faster as flow is constricted, appear to often prevent segmentation. Also, if the water body is too deep and sedimentation rates are low, segmentation through cuspate spit growth may take too long to occur compared to other long-term environmental changes.

Here we have, for the first time, used a process-based approach to study the fascinating behavior arising from the interactions of opposing shores, delivering additional insight through analysis of modeled wave climates. The presence of cuspate shoreline features along many of the long coasts of natural elongate water bodies serves as an additional test of the high-angle shoreline instability hypothesis presented in Ashton et al. (2001). The segmentation of water bodies reproduced here is observed in many locations and environments, which include backbarrier lagoons (Figs. 1A, 1B, 1E, and 1G), reworked kettle ponds (Figs. 1D and 1H), flooded drainages (Fig. 1I), and even Carolina Bays (i.e., elliptical depressions found along the mid-Atlantic coastal plain of the United States) (Fig. 1F) (Johnson, 1942). Whatever the formation mechanism of an elongate water body, the process of fetch-limiting self-organization as a means of cross-water-body connection should be considered as a probable reason for the appearance of chains of round lakes in many clastic settings.

View larger version (92K): [in this window] [in a new window]   Figure 1. Natural examples of enclosed water bodies with cuspate features and segmented water bodies. A: Laguna Val'karkynmangkak, Russia. B: Inset, A. C: Lagoa Dos Patos, Brazil. D: Gull-Higgins-Williams Ponds, Massachusetts, USA. E: St. Lawrence Island, Alaska, USA. F: Near Rains, South Carolina, USA. G: Western Alaska, USA. H: Near Maine, Minnesota, USA. I: Falmouth, Massachusetts, USA. Geographical locations and data sources are included in GSA Data Repository (see footnote 1).

View larger version (46K): [in this window] [in a new window]   Figure 2. Plan views of simulation domains spaced at equal intervals of 1875 days of high wind events (10-m-elevation wind speed, Uw = 15 m/s = 54 km/h). Cells are 100 m wide; shoreface depth, D = 3 m. Note that comparing model time with a natural example requires re-scaling both the spatial scale and wind climate of the prototype.

View larger version (23K): [in this window] [in a new window]   Figure 3. Wave climate metrics along model shoreline at time steps during different phases of segmentation. A: After 0 days. B: After 15,000 days. C: After 30,000 days. Also shown are inset roses of weighted wave "energy" (height contributions to sediment transport). Qs—along-shore sediment transport;—normalized shoreline-shape diffusivity.

View larger version (23K): [in this window] [in a new window]   Figure 4. Contributing length per width of final water body () versus aspect ratio of initial water body. A: Graphical demonstration of . As example, the simulation shown in Figure 3 has initial aspect ratio of 6.65 (4.6 km wide/30.6 km long). This body segments into three final water bodies, yielding = 6.65/3 = 2.22. B: derived from ensemble simulations with differing initial aspect ratios. Data labels indicate number of final water bodies. (Simulation details are in the GSA Data Repository; see footnote 1.)

	ACKNOWLEDGMENTS

	FOOTNOTES

 
¹ GSA Data Repository item 2009046, geographical locations, data sources, and simulation details, is available online at www.geosociety.org/pubs/ft2009.htm, or on request from editing{at}geosociety.org or Documents Secretary, GSA, P.O. Box 9140, Boulder, CO 80301, USA.

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TOP ABSTRACT INTRODUCTION WAVE-INFLUENCED SEGMENTED LAKES HIGH-ANGLE-WAVE SHORELINE... METHODS: NUMERICAL MODEL RESULTS: MODELED BEHAVIOR RESULTS: LOCAL WAVE CLIMATES DISCUSSION REFERENCES CITED

 

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Received for publication 27 June 2008

Revised manuscript received 13 October 2008

Manuscript accepted 15 October 2008

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