How waves shape salt marshesNicoletta Leonardi and Sergio FagherazziDepartment of Earth and Environment, Boston University, 675 Commonwealth Avenue, Boston, Massachusetts 02215, USA
ABSTRACTWe present high-resolution field measurements of five sites along the United States Atlantic
Coast, and cellular automata simulations, to investigate the erosion of marsh boundaries by wave action. According to our analysis, when salt marshes are exposed to high wave energy conditions their boundaries erode uniformly. The resulting erosion events follow a Gauss-ian distribution, yielding a relatively smooth shoreline. On the contrary, when wind waves are weak and the local marsh resistance is strong, jagged marsh boundaries form. In this case, erosion episodes have a long-tailed frequency magnitude distribution with numerous low-magnitude events, but also high-magnitude episodes. The logarithmic frequency mag-nitude distribution suggests the emergence of a critical state for marsh boundaries, which would make the prediction of failure events impossible. Internal physical processes allowing salt marshes to reach this critical state are geotechnical and biological, and related to the non-homogeneity of salt marshes whose material discontinuities act as stress raisers.
INTRODUCTIONLarge salt marsh losses have been documented
in the United States, Asia, and Europe (e.g., Kirwan et al., 2010; Fagherazzi et al., 2013). If salt marshes continue declining, we risk los-ing their valuable ecosystem services. Among other benefits, salt marshes mitigate the impact of hurricanes and tsunamis, provide habitats for a variety of aquatic animal species, and medi-ate the exchange of sediments and contaminants between the marine and terrestrial environments (e.g., Zedler and Kercher, 2005).
Salt marshes are very resilient with respect to vertical dynamics because feedbacks among inun-dation, organic matter accretion, plant growth, and sediment deposition allow the marsh to keep pace with sea-level rise (e.g., Kirwan et al., 2010). However, recent results indicate that salt marshes are inherently weak with respect to horizontal erosion and that waves can trigger irreversible salt marsh deterioration even in the absence of sea-level rise (Fagherazzi et al., 2013). As a result, salt marshes do not display lateral equilibrium, but are always contracting or expanding at rates of meters per year (e.g., Fagherazzi, 2013).
Edge erosion has been also described as a repetitive stage, within an autocyclic mode, of marsh growth (e.g., Chauhan 2009; Van der Wal et al., 2008).
An understanding of the mechanisms control-ling salt marsh erosion is thus of crucial impor-tance for the correct management and preser-vation of coastal environments. Although salt marsh dynamics have been widely investigated, many processes are still poorly understood. Some studies adopt process-based or empirical models to estimate the location and size of ero-sion events (e.g., Fagherazzi et al., 2012; Ganju et al., 2013). An alternative conceptual model for marsh erosion can be based on simple sto-chastic models that could help extract universal features of the processes. Simple models, having few rules governing the interaction among indi-
vidual components, can simulate complex sys-tems, displaying structures with variability (e.g., Drossel et al., 1994; Goldenfeld and Kadanoff, 1999; Murray, 2007; Fagherazzi, 2008).
Here we present a cellular automata model and high-resolution field measurements of marsh boundaries to explain erosional trends under different wind-wave exposures.
According to our results, salt marshes that are very exposed to wind-wave power are retreating uniformly. However, low wave power conditions correspond to long-tailed distribution of erosion events, which create rougher marsh fronts.
STUDY SITESWe focus on five sites, located in Plum Island
Sound, Massachusetts, and Virginia Coast Reserve, Virginia (USA; Fig. DR1 in the GSA Data Repository1). The Plum Island Sound estuary is a coastal plain estuary characterized by extensive marshes. Tides are semidiurnal and the average tidal range is 2.9 m. Prevailing winds come from the westerly quadrant; winds having greatest frequency are from the north, while winds with the greatest duration are from southwest (e.g., McIntire and Morgan, 1964). The site Refuge North is the most sheltered from wave action and the slowest-eroding site, fol-lowed in order by Stackyard Road and Refuge South. The Virginia Coast Reserve encompasses more than 100 km of a dynamic system of barrier islands, shallow lagoons, and salt marshes sepa-rated by deep tidal inlets. Tides are semidiurnal with a mean tidal range of 1.2 m. Mean higher high water is 0.68 m while lower low water is -0.70 m with respect to mean sea level. Prevail-ing winds come from the southwest and north-
east quadrants. The northeast and southwest directions, parallel to the barrier island, also cor-respond to the highest fetch values (Fagherazzi and Wiberg, 2009). The two locations taken into account are Chimney Pole and Hog Island. Field data (McLoughlin et al., 2014) and model results (Mariotti et al., 2010) indicate that Chimney Pole is the site most susceptible to wind-wave erosion. Plum Island Sound and Virginia Coast Reserve salt marshes are both characterized by prominent scarps at their seaward edges, typi-cally 1.5 m or more above adjacent tidal flats.
METHODSThe stochastic model consists of a two-dimen-
sional (2-D) square lattice (Fig. 1) whose ele-ments, i, have randomly distributed resistance, ri, between 0 and 1. Each cell has erosion rate Ei:
E Pr
Pexpi
i= α −
β , (1)
1GSA Data Repository item 2014314, Figure DR1 (study areas) and Figure DR2 (marsh bound-ary profiles), is available online at www.geosociety .org /pubs/ft2014.htm, or on request from [email protected] or Documents Secretary, GSA, P.O. Box 9140, Boulder, CO 80301, USA.
GEOLOGY, October 2014; v. 42; no. 10; p. 1–4; Data Repository item 2014314 | doi:10.1130/G35751.1 | Published online XX Month 2014
Figure 1. Sketch of the cellular automata model. A: Possible domain configuration af-ter removal of two domain elements. Each cell has erosion rate Ei; gray cells represent erodible elements and have erosion probabil-ity pi . B: Possible domain configuration after removal of another cell (E9). C: Another cell is removed from the domain (E10). The crossed cell remains isolated from the rest of the boundary, and is thus automatically removed.
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where P is the wave power (kW/m) and a and b are nondimensional constants equal to 0.35 and 1.1, respectively (Schwimmer, 2001). The first part of Equation 1 is in agreement with classical theoretical and empirical investigations on salt marsh boundary erosion. According to these, the retreat rate is proportional to wave power and follows a power-law relationship, having an exponent close to 1 (e.g., Schwimmer, 2001; Marani et al., 2011). The model does not take into account wind-wave directionality and tem-poral heterogeneity of wind-wave conditions. The second part of Equation 1 is meant to take into account the variety of biological and geo-morphologic processes affecting each portion of the marsh. Among others, seepage erosion, crab burrowing, vegetation, and sediment cohe-sion make it difficult to predict which portion of the marsh will collapse first. Equation 1 is such that the local variability of marsh resistance is particularly relevant when the wave power is low (e.g., in this case the presence of vegeta-tion could actively prevent the failure of a cer-tain marsh portion). However, when wind-wave power is very high (e.g., during storms), local marsh characteristics play a secondary role and different marsh elements are eroded at a simi-lar rate, because their resistance is small com-pared to the main external driver. In this case, the exponent in Equation 1 goes to 1 and every cell has the same erosion rate. Soil tensile and shear strength, as well as vegetation root tensile strength (kPa) could be considered first indica-tors for marsh resistance (e.g., Francalanci et al., 2013; Bendoni et al., 2014). At each time step, only the neighbors of previously eroded cells are susceptible to erosion (herein, neigh-bors means having one side in common; Fig. 1). Each neighbor is eroded with a probability
p
E
Ei
i
i∑= ,
where the sum refers to all cells that can poten-tially be eroded for a given time step. A cell is also automatically eroded if it remains isolated from the rest of the domain (Fig. 1C; crossed cell). In fact, isolated cells would represent iso-lated marsh stacks that are disintegrated fast as they are attacked by waves from several direc-tions (Schwimmer, 2001). Similar models have already been adopted, in a different context, for the chemical etching of disordered solids (Štěpánek, 2008; Kolwankar et al., 2003). How-ever, our model uses a different formulation for the erosion rate Ei, and the sum at the denomi-nator of the erosion probability pi only runs over the boundary elements, rather than over the entire domain. Moreover, we automatically remove isolated cells. To our knowledge, there are no other applications of this kind of model to the marsh environment.
Marsh contours have been tracked using a real time kinematic global positioning system
and an electronic total station. Data were col-lected with an average resolution of 1 m. When marsh contours were characterized by signifi-cant variations in boundary geometry, measure-ments were taken as much as 20 cm apart. Marsh boundaries have been monitored every Septem-ber from A.D. 2008 to 2013 for the three sites in Plum Island Sound. For the two sites in the Vir-ginia Coast Reserve, measurements were taken in March 2008 and August 2010. We define as magnitude of an erosion event, for a given year and for a certain point along the marsh bound-ary, the shortest distance of that point from the marsh boundary of the subsequent year.
MODEL RESULTSFigure 2 shows results for two simulations
run for small and high wave power. In case of high wave power, marsh erosion proceeds uni-formly along the marsh shoreline and generates a profile that is rough at the scale of the single
cell, but smooth at a larger scale. This is because each cell has similar resistance if compared to the main external driver. Low wave power con-ditions, however, correspond to the development of a jagged boundary. Indentations are produced by different erosion rates of individual cells that affect the global system behavior.
From a statistical viewpoint, the system behaves differently for the two extreme condi-tions of very low and very high wave power. The frequency magnitude distribution of erosion events in a time interval Dt approaches a Gauss-ian distribution in the case of high wave power (Fig. 3A). In the model, the magnitude of an ero-sion event is the number of eroded cells. In the case of low wave power, the frequency distribu-tion is characterized by a long-tailed power-law distribution (Fig. 3B). For the low wave power case, a long time is required to erode very resis-tant cells. However, once the most resistant cells are eliminated, several weak sites remain exposed and can be rapidly removed, with conse-quent generation of large-scale failures. Similar results were shown by Kolwankar et al. (2003), who demonstrated that when the etching power of the solution approaches zero, their model is identical to classical invasion percolation, with reaction rate limited by the invasion percolation threshold (e.g., Wilkinson and Willemsen, 1983; Roux and Guyon, 1989).
Despite the occurrence of large-scale events, in the case of low wave power exposure, the remaining domain cells are very resistant due to the selected removal of the weakest sites. This differential removal is allowed by resistance variability among different cells. For a given low wave power condition, if cell resistance (r
i) values maintain the same mean but their range of variability (Dri) is reduced [e.g., ri ∈ (0.3; 0.7) instead of ri ∈ (0; 1)], the domain starts eroding
100 101 102100
101
102
B
(n)
N(n
)A
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0
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0 20 40 60
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75
Figure 2. Results of the stochastic model: marsh boundary under low and high wave power conditions.
Figure 3. Frequency magnitude distributions of erosion events; n is number of eroded cells, N(n) is number of times n cells are eroded within a time interval. Points are model results. A: High wave power conditions. Lines are the interpola-tion of model results using a Gaussian frequency magnitude distribution. B: Low wave power conditions. Lines are the interpolation of model results using a loga-rithmic frequency magnitude distribution.
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uniformly, as for the high wave power condition. This uniform erosion rate leads to an accelerated erosion (Fig. 4) and is thus unfavorable for the maintenance of the domain elements.
FIELD DATA ANALYSISIn Plum Island Sound, average erosion rates
for the period of record (September 2008 through September 2013) are 0.2 m/yr at Refuge North, 0.35 m/yr at Stackyard Road, and 0.45 m/yr at Refuge South. For the two sites in the Virginia Coast reserve, average erosion rates (March 2008 through August 2010) are 0.75 m/yr at Hog Island and 1 m/yr at Chimney Pole. The frequency mag-nitude distribution of biennial erosion events for each point along marsh shorelines is presented in Figure 5. The most sheltered sites in Plum Island Sound (Refuge North and Stackyard Road) have a logarithmic frequency magnitude distribution. Moreover, the lower the exposure to waves, the longer the tail of the power law (Fig. 5; the mag-nitude of the slope coefficient for Refuge North is lower than the one for Stackyard Road). For
the Refuge South site, the power-law distribu-tion starts to approach the Gaussian distribution and an intermediate condition arises, character-ized by a shorter tail and maximum number of erosion events no longer corresponding to mini-mum magnitudes.
For the two sites along the Virginia Coast reserve, Chimney Pole is the most exposed and its erosion events follow a Gaussian distribution. The frequency magnitude distribution of erosion events at Hog Island is intermediate between the Gaussian and the logarithmic distribution.
Thus, at both Plum Island and the Virginia Coast Reserve, the lower the site exposure to wave action, the longer the power-law tail of the erosion events distribution. The frequency magnitude distribution of erosion events is also clearly recognizable from marsh boundary pro-files (Fig. DR2).
DICUSSION AND CONCLUSIONSOur simple model appears to capture impor-
tant marsh boundary features and gives some new insight into salt marsh erosional processes. High-resolution field measurements at five loca-tions along the United State Atlantic Coast con-firm the numerical results. Our investigations have been related to high or low wave power exposure, as well as to weaker or more resistant marsh platforms (given a fixed wind-wave expo-sure). Very exposed sites are characterized by a uniform rate of marsh retreat along the shore-line, with erosion events following a Gaussian frequency magnitude distribution. Less exposed sites show a long-tailed frequency magnitude distribution with numerous small events and few (but not negligible) bigger events, which are unpredictable and can happen despite reduced wave action. The fact that sites at Plum Island have longer tailed distributions than the study sites in Virginia can be related to the increased likelihood of marsh slumping in macrotidal envi-ronments with respect to microtidal marshes; in the former case, higher marsh scarps promote undercutting and tensional break development.
We maintain that, in the case of low wave exposure, the system could reach self-organized
criticality (SOC). Self-organization refers to the ability of certain nonequilibrium systems to develop structures and patterns in the absence of any fine tuning from external agents. Criticality refers to the fact that all the members of the sys-tem influence each other and that local instabili-ties generate broader scale order disturbances (e.g., Bak et al., 1987; Bak, 1996).
The power-law relationship is a necessary condition for SOC and it is frequently used to test whether SOC is present. According to Bak et al. (1987, 1989), if frequency data fit a power-law distribution over a range of event magnitudes, the system is likely self-organized and could be at a critical state. However, the power-law relationship is a necessary but not sufficient condition for SOC, as it could also happen in a range of non-SOC systems. Other necessary conditions (but not sufficient, as sufficient conditions are unknown) for the development of SOC are (1) the existence of a quasi-steady critical state at which the system self-organizes; (2) an internal mechanism by which the system can reach this critical state; (3) the response of the system to perturbations varies in magnitude independently of perturba-tion size; (4) the presence of mechanisms for the system energy dissipation; and (5) the pres-ence of many degrees of freedom within which internal processes can operate.
Conditions 4 and 5 are easily verified, as salt marshes are inherently dissipative systems char-acterized by a continuous loss in both potential and kinetic wave energy, consequent sediment removal, and further energy dissipation. Condi-tion 3 is confirmed by the logarithmic distribu-tion of erosion events that spans multiple length scales. For conditions 1 and 2, we propose that the critical state for marsh boundaries is the one promoting the removal of weak sites and con-sequent exposure of more resistant and uniform marsh portions. The critical state would thus be the one maximizing salt marsh resistance to wave action. In our simplified model, this con-dition corresponds to the contour approaching the percolation cluster made of the slowest-eroding sites and surrounded by easily erodible
0 0.2 0.4 0.6 0.8−8
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/τ
)1
i
0 1 2 3 40
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D=-1.77D=-1.55
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CPHIRSSRRN
Increasing wave exposure
(n)
N(n
)
N(n
)
N(n
)
N(n
)
(n) (n) (n) (n)
N(n
)
Figure 4. Time (τi) required to erode half of the domain particles as a function of Dri and for a wave power P (see text) equal to 0.1; Dri is the amplitude of the range within which the cell resistance, ri, can vary. The time required to erode half of the domain cells for a certain Dri range has been normalized by the time τi corre-sponding to a Dri equal to 1 [i.e., ri ∈ (0; 1)].
Figure 5. Frequency-magnitude distributions of erosion events for three sites in Plum Island Sound and two sites in the Virginia Coast Reserve; n is the erosion event (m) occurring N(n) times. RN—Refuge North, SR—Stackyard Road, RS—Refuge South, HI—Hog Island, CP—Chimney Pole.
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ones. Field data confirm this assumption, con-sidering that the slope of the logarithmic distri-bution of the less exposed sites (Refuge North) is close to 1.53 and thus in agreement with clas-sical invasion percolation problems. Internal processes allowing the system to reach its criti-cal state are geotechnical and biological mecha-nisms connected to system discontinuities. Dis-continuities enhance wave stresses and lead to crack development. In the presence of cracks, the system approaches the minimum energy state independently from external agents (e.g., Roylance, 2001). Failures of marsh portions become possible and independent from any fine external tuning. Tension crack development is also promoted by drying and shrinkage of the soil, or by cyclic oscillations of mean and effec-tive stresses (Francalanci et al., 2013; Bendoni et al., 2014). Particular attention is required when trying to explain the behavior of a system through SOC (e.g., Frigg, 2003), and further efforts are required to definitely recognize it. In our simplified model, discontinuities corre-spond to contact area between clusters having relatively high or low erosional rates. In a natu-ral system, discontinuities could correspond to contact surfaces between marsh blocks weak-ened by groundwater seepage. Stress concen-tration along these discontinuities favors the failure of weak marsh clusters of different sizes. For example, marsh slumping, triggered by cracks and seepage, can lead to large-scale events, which in turn promote long-tailed dis-tribution. Once the weakest sites are removed, more resistant marsh portions are uncovered, which are difficult to erode. Thus, variability in marsh properties allows marsh boundaries to be resistant to wind-wave action by means of selected removal of weak elements.
ACKNOWLEDGMENTSWe thank BUMP (Boston University Marine Pro-
gram) students, G. Mariotti and A. Priestas for contri-bution to field data collection; D.H. Rothman, whose lectures inspired this work; editor R. Cox; and an anonymous reviewer. This research was supported by National Science Foundation awards OCE-0924287, DEB-0621014 (VCR-LTER, program), and OCE-1238212 (PIE-LTER, program) and U.S. Department of the Interior grant G14AC00045.
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Manuscript received 4 April 2014 Revised manuscript received 12 July 2014 Manuscript accepted 25 July 2014
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Nicoletta Leonardi and Sergio Fagherazzi How waves shape salt marshes
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