Patrick W. Limber

Beach and sea-cliff dynamics as a driver of long-term rocky coastline evolution and stability

We investigate rocky coastline evolution over millennial time scales using exploratory analytical and numerical models based on interactions between beaches and sea cliffs. In the models, wave-driven sea-cliff retreat is a nonlinear function of beach width, where cliff retreat is maximized by sediment abrasion and minimized by either a lack of beach sediment or too much sediment (which prevents waves from reaching the sea cliff). As sea cliffs retreat, beach sediment is produced and distributed alongshore by wave-driven sediment transport, and local beach widths determine future cliff retreat rates. Numerical experiments indicate that through such interactions, rocky coastlines can reach an equilibrium configuration where headlands and embayments remain stable through time, even in the absence of alongshore variations in sea-cliff lithology. Furthermore, the equilibrium coastline configuration, or the alongshore proportion of rocky headland to cliff-backed pocket beach, can be predicted analytically. Initial tests suggest that predictions match well qualitatively with actual landscapes.

Coastal Sediment Budgets and the Littoral Cutoff Diameter: A Grain Size Threshold for Quantifying Active Sediment Inputs

Abstract The use of coastal sediment budgets has garnered wide acceptance since its inception nearly 40 years ago. Since then, many researchers have used sediment budgets to quantify littoral transport rates and understand coastal processes on diverse coastlines including the high-energy Pacific coast of North America, the Black Sea, the Nile Delta and beyond. Here, we suggest further improvement on an already successful conceptual tool by questioning the broad definition of sand set forth by the classic Wentworth grain size scale (63–2000 microns) that is often used in quantifying coastal sediment budget inputs from sources such as coastal-draining rivers and eroding sea cliffs. A smaller range of sediment sizes is found on many beaches in California. This range is defined by a minimum grain size threshold, termed the littoral cutoff diameter. Sediment contributed to the littoral system that is smaller than this threshold, even if defined as sand by the Wentworth scale, may not remain on the beach in any significant quantity. The littoral cutoff diameter ranges from 88 to 180 microns on the California beaches studied herein, and results from a variety of locations show that yearly littoral sediment flux from coastal-draining rivers and eroding sea cliffs can be overestimated by 16–300% percent if the littoral cutoff diameter is not considered. The presence of the littoral cutoff diameter suggests that quantifying sediment inputs within the context of preexisting littoral sediments is of first-order importance when constructing sediment budgets in California and in other analogous coastal environments.

Unraveling the dynamics that scale cross‐shore headland relief on rocky coastlines: 1. Model development

We have developed an exploratory model of plan view, millennial-scale headland and bay evolution on rocky coastlines. Cross-shore coastline relief, or amplitude, arises from alongshore differences in sea cliff lithology, where durable, erosion-resistant rocks protrude seaward as headlands and weaker rocks retreat landward as bays. The model is built around two concurrent negative feedbacks that control headland amplitude: (1) wave energy convergence and divergence at headlands and bays, respectively, that increases in intensity as cross-shore amplitude grows and (2) the combined processes of beach sediment production by sea cliff erosion, distribution of sediment to bays by waves, and beach accumulation that buffers sea cliffs from wave attack and limits further sea cliff retreat. Paired with the coastline relief model is a numerical wave transformation model that explores how wave energy is distributed along an embayed coastline. The two models are linked through genetic programming, a machine learning technique that parses wave model results into a tractable input for the coastline model. Using a pool of 4800 wave model simulations, genetic programming 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, but fundamental, scaling arguments on how different variables (such as sea cliff height and composition) affect the equilibrium cross-shore relief of headland and bays. The models generality highlights the key feedbacks involved in coastline evolution and allows its equations (and model behaviors) to be easily modified by future users.

Unraveling the dynamics that scale cross‐shore headland relief on rocky coastlines: 2. Model predictions and initial tests

We explore the behavior of a theoretical model of cross-shore headland relief caused by alongshore differences in lithology and rock strength on rocky coastlines. Results address the question of why some rocky coasts exhibit frequent headland and embayment sequences while others evolve to a flat, smooth, and sandy configuration. Main model predictions are that cross-shore headland amplitude is inversely proportional to beach sediment supply and the strength of wave energy convergence and divergence along the headland and bay, and proportional to the alongshore embayment length (or distance between headlands) and the difference between headland and bay rock strength. The coastlines initial physical properties (sea cliff height, composition, etc.) largely determine whether headlands will be persistent or transient landscape features. Model timescales over which the headland and bay reach steady state amplitude, or disappear to a flat coastline, range from 120 to 175,000u2009years depending on how close the initial amplitude is to steady state. In many cases, the coastline must evolve over several sea level highstands in order to reach equilibrium. A characteristic timescale (independent of initial conditions) shows that the coastline evolves most rapidly when: wave focusing is stronger; sea cliff rock is weaker or retreats faster in a given wave climate; the sea cliff retreat rate decreases rapidly as a function of beach width (i.e., the beach is very effective at dampening wave energy); and the coastline is sediment rich. Comparisons to nature suggest that our model is qualitatively capturing general rocky coastline dynamics and that modeled headland amplitudes are consistent with observed amplitudes.

A model integrating longshore and cross‐shore processes for predicting long‐term shoreline response to climate change

We present a shoreline change model for coastal hazard assessment and management planning. The model, CoSMoS-COAST (Coastal One-line Assimilated Simulation Tool), is a transect-based, one-line model that predicts short-term and long-term shoreline response to climate change in the 21st century. The proposed model represents a novel, modular synthesis of process-based models of coastline evolution due to longshore and cross-shore transport by waves and sea level rise. Additionally, the model uses an extended Kalman filter for data assimilation of historical shoreline positions to improve estimates of model parameters and thereby improve confidence in long-term predictions. We apply CoSMoS-COAST to simulate sandy shoreline evolution along 500u2009km of coastline in Southern California, which hosts complex mixtures of beach settings variably backed by dunes, bluffs, cliffs, estuaries, river mouths, and urban infrastructure, providing applicability of the model to virtually any coastal setting. Aided by data assimilation, the model is able to reproduce the observed signal of seasonal shoreline change for the hindcast period of 1995–2010, showing excellent agreement between modeled and observed beach states. The skill of the model during the hindcast period improves confidence in the models predictive capability when applied to the forecast period (2010–2100) driven by GCM-projected wave and sea level conditions. Predictions of shoreline change with limited human intervention indicate that 31% to 67% of Southern California beaches may become completely eroded by 2100 under sea level rise scenarios of 0.93 to 2.0u2009m.

New Techniques to Measure Cliff Change from Historical Oblique Aerial Photographs and Structure-from-Motion Photogrammetry

ABSTRACT Warrick, J.A.; Ritchie, A.C.; Adelman, G.; Adelman, K., and Limber, P.W., 2017. New techniques to measure cliff change from historical oblique aerial photographs and Structure-from-Motion photogrammetry. Oblique aerial photograph surveys are commonly used to document coastal landscapes. Here it is shown that adequate overlap may exist in these photographic records to develop topographic models with Structure-from-Motion (SfM) photogrammetric techniques. Using photographs of Fort Funston, California, from the California Coastal Records Project, imagery were combined with ground control points in a four-dimensional analysis that produced topographic point clouds of the study areas cliffs for 5 years spanning 2002 to 2010. Uncertainty was assessed by comparing point clouds with airborne LIDAR data, and these uncertainties were related to the number and spatial distribution of ground control points used in the SfM analyses. With six or more ground control points, the root mean squared errors between the SfM and LIDAR data were less than 0.30 m (minimum = 0.18 m), and the mean systematic error was less than 0.10 m. The SfM results had several benefits over traditional airborne LIDAR in that they included point coverage on vertical-to-overhanging sections of the cliff and resulted in 10–100 times greater point densities. Time series of the SfM results revealed topographic changes, including landslides, rock falls, and the erosion of landslide talus along the Fort Funston beach. Thus, it was concluded that SfM photogrammetric techniques with historical oblique photographs allow for the extraction of useful quantitative information for mapping coastal topography and measuring coastal change. The new techniques presented here are likely applicable to many photograph collections and problems in the earth sciences.

Can beaches survive climate change

Anthropogenic climate change is driving sea level rise, leading to numerous impacts on the coastal zone, such as increased coastal flooding, beach erosion, cliff failure, saltwater intrusion in aquifers, and groundwater inundation. Many beaches around the world are currently experiencing chronic erosion as a result of gradual, present-day rates of sea level rise (about 3xa0mm/year) and human-driven restrictions in sand supply (e.g., harbor dredging and river damming). Accelerated sea level rise threatens to worsen coastal erosion and challenge the very existence of natural beaches throughout the world. Understanding and predicting the rates of sea level rise and coastal erosion depends on integrating data on natural systems with computer simulations. Although many computer modeling approaches are available to simulate shoreline change, few are capable of making reliable long-term predictions needed for full adaption or to enhance resilience. Recent advancements have allowed convincing decadal to centennial-scale predictions of shoreline evolution. For example, along 500xa0km of the Southern California coast, a new model featuring data assimilation predicts that up to 67% of beaches may completely erode by 2100 without large-scale human interventions. In spite of recent advancements, coastal evolution models must continue to improve in their theoretical framework, quantification of accuracy and uncertainty, computational efficiency, predictive capability, and integration with observed data, in order to meet the scientific and engineering challenges produced by a changing climate.

Indications of a positive feedback between coastal development and beach nourishment

Beach nourishment, a method for mitigating coastal storm damage or chronic erosion by deliberately replacing sand on an eroded beach, has been the leading form of coastal protection in the United States for four decades. However, investment in hazard protection can have the unintended consequence of encouraging development in places especially vulnerable to damage. In a comprehensive, parcel-scale analysis of all shorefront single-family homes in the state of Florida, we find that houses in nourishing zones are significantly larger and more numerous than in non-nourishing zones. The predominance of larger homes in nourishing zones suggests a positive feedback between nourishment and development that is compounding coastal risk in zones already characterized by high vulnerability.

Coastal Storm Modeling System (CoSMoS)

Projected coastal squeeze derived from CoSMoS Phase 2 shoreline change and cliff retreat projections. Projected coastal squeeze extents illustrate the available area between shoreline (mean high water; MHW) positions and man-made structures and barriers (referred to as non-erodible structures) or cliff-top retreat, as applicable, for a range of sea-level rise scenarios. The coastal squeeze polygons include results from the Coastal Storm Modeling System (CoSMoS) shoreline change (CoSMoS-COAST; Vitousek and others, 2017; available at https://www.sciencebase.gov/catalog/item/57f426b9e4b0bc0bec033fad) and cliff retreat models (Limber and others, 2015; available at https://www.sciencebase.gov/catalog/item/57f4234de4b0bc0bec033f90) using future wave-climate conditions derived from Global Climate Models (GCMs). Coastal squeeze areas are identified and defined from combined model projections, using model scenarios where erosion was limited by non-erodible structures (for shoreline change models) and armoring (for cliff retreat models; both cliff and shoreline cases referred to as hold the line) and where no beach-nourishment was included. Coastal squeeze projections are defined for each sea-level rise scenario. Shoreline change and cliff retreat model details and data sources are outlined in CoSMoS_3.0_Phase_2_Southern_California_Bight:_Summary_of_data_and_methods (available at https://www.sciencebase.gov/catalog/file/get/57f1d4f3e4b0bc0bebfee139?name=CoSMoS_SoCalv3_Phase2_summary_of_methods.pdf). Phase 2 data for Southern California include information for the coast from the border of Mexico to Pt. Conception. Please read the Summary of methods and inspect output carefully. Data are complete for the information presented.This dataset contains projections of coastal cliff-retreat rates and positions for future scenarios of sea-level rise (SLR). Present-day cliff-edge positions used as the baseline for projections are also included. Projections were made using numerical and statistical models based on field observations such as historical cliff retreat rate, nearshore slope, coastal cliff height, and mean annual wave power, as part of Coastal Storm Modeling System (CoSMoS) v.3.0 Phase 2 in Southern California. Details: Cliff-retreat position projections and associated uncertainties are for scenarios of 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, and 5 meters of SLR. Projections were made at CoSMoS cross-shore transects (CST) spaced 100 m alongshore using a baseline sea-cliff edge from 2010 (included in the dataset). Within each zip file, there are two separate datasets available: one that ignores coastal armoring, such as seawalls and revetments, and allows the cliff to retreat unimpeded (“Do Not Hold the Line”); and another that assumes that current coastal armoring will be maintained and 100% effective at stopping future cliff erosion (Hold the Line). Eight numerical models synthesized from literature (Trenhaile, 2000; Walkden and Hall, 2005; Trenhaile, 2009; Trenhaile, 2011; Ruggiero and others, 2011; Hackney and others, 2013) were used to make projections. All models relate breaking-wave height and period to cliff rock or unconsolidated sediment erosion. Models range in complexity from 2-D models in which the entire profile evolves, from below water to the cliff edge, to simple 1-D empirical or statistical models in which only the cliff edge evolves as a function of wave impact intensity and frequency. The projections are a robust average of all models, and the uncertainties are proportional to 1) underlying uncertainties in the model input data, such as historical cliff retreat rates, and 2) the differences between individual model forecasts at each CST so that uncertainty is larger when the models do not agree. As sea level rises, waves break closer to the sea cliff, more wave energy impacts the cliffs, cliff erosion rates accelerate. Model behavior also includes wave run-up (Stockdon and others, 2006), wave set-up that raises the water level during big-wave events, and tidal levels. The more complex 2-D models were run on idealized cliff profiles extending from about 10 m water depth to 1 kilometer inland from the cliff edge. Profiles were extracted by overlaying the cross-shore transects on a high-resolution digital elevation model (DEM) covering the Southern California study area. For all models, the presence of a beach was recorded (yes or no) for all transects using aerial photography, and the cliff toe elevation (or beach/cliff junction) was digitized from the DEM profiles. Using historic cliff edge retreat rates by Hapke and Reid (2007), unknown coefficients within the cliff-profile models were calibrated using a Monte Carlo simulation (in other words, coefficients were tuned until the modeled mean retreat rate equaled the observed mean retreat rate for a given transect). Uncertainty was tallied using a root mean squared error (RMSE) approach. The RMSE represents cumulative uncertainty from multiple sources and assumes that different sources of error will, at times, cancel each other out. It is therefore not a worst-case uncertainty (in other words, a straight sum of errors) but instead an average uncertainty. Total RMSE increased with SLR rate and varied between +/- 2-3 m to a maximum of +/- 50 m for the extreme 5 m SLR scenario. For more information on model details, data sources, and integration with other parts of the CoSMoS framework, see CoSMoS_3.0_Phase_2_Southern_California_Bight:_Summary_of_data_and_methods (available at https://www.sciencebase.gov/catalog/file/get/5633fea2e4b048076347f1cf?name=CoSMoS_SoCalv3_Phase2_summary_of_methods.pdf).Maximum depth of flooding surface (in cm) in the region landward of the present day shoreline that is inundated for the storm condition and sea-level rise (SLR) scenario indicated. Note: Duration datasets may have occasional gaps in open-coast sections. The Coastal Storm Modeling System (CoSMoS) makes detailed predictions (meter-scale) over large geographic scales (100s of kilometers) of storm-induced coastal flooding and erosion for both current and future sea-level rise (SLR) scenarios. CoSMoS v3.0 for Southern California shows projections for future climate scenarios (sea-level rise and storms) to provide emergency responders and coastal planners with critical storm-hazards information that can be used to increase public safety, mitigate physical damages, and more effectively manage and allocate resources within complex coastal settings. Model details and data sources are outlined in CoSMoS_3.0_Phase_2_Southern_California_Bight:_Summary_of_data_and_methods (available at https://www.sciencebase.gov/catalog/file/get/57f1d4f3e4b0bc0bebfee139?name=CoSMoS_SoCalv3_Phase2_summary_of_methods.pdf). Phase 2 data for Southern California include flood-hazard information for the coast from the border of Mexico to Pt. Conception. Several changes from Phase 1 projections are reflected in many areas; please read the Summary of methods and inspect output carefully. Data are complete for the information presented.Model-derived significant wave height (in meters) for the given storm condition and sea-level rise (SLR) scenario. The Coastal Storm Modeling System (CoSMoS) makes detailed predictions (meter-scale) over large geographic scales (100s of kilometers) of storm-induced coastal flooding and erosion for both current and future sea-level rise (SLR) scenarios. CoSMoS v3.0 for Southern California shows projections for future climate scenarios (sea-level rise and storms) to provide emergency responders and coastal planners with critical storm-hazards information that can be used to increase public safety, mitigate physical damages, and more effectively manage and allocate resources within complex coastal settings. Model details and data sources are outlined in CoSMoS_3.0_Phase_2_Southern_California_Bight:_Summary_of_data_and_methods (available at https://www.sciencebase.gov/catalog/file/get/57f1d4f3e4b0bc0bebfee139?name=CoSMoS_SoCalv3_Phase2_summary_of_methods.pdf). Phase 2 data for Southern California include flood-hazard information for the coast from the border of Mexico to Pt. Conception. Several changes from Phase 1 projections are reflected in many areas; please read the Summary of methods and inspect output carefully. Data are complete for the information presented.Maximum depth of flooding surface (in cm) in the region landward of the present day shoreline that is inundated for the storm condition and sea-level rise (SLR) scenario indicated. Note: Duration datasets may have occasional gaps in open-coast sections. The Coastal Storm Modeling System (CoSMoS) makes detailed predictions (meter-scale) over large geographic scales (100s of kilometers) of storm-induced coastal flooding and erosion for both current and future sea-level rise (SLR) scenarios. CoSMoS v3.0 for Southern California shows projections for future climate scenarios (sea-level rise and storms) to provide emergency responders and coastal planners with critical storm-hazards information that can be used to increase public safety, mitigate physical damages, and more effectively manage and allocate resources within complex coastal settings. Model details and data sources are outlined in CoSMoS_3.0_Phase_2_Southern_California_Bight:_Summary_of_data_and_methods (available at https://www.sciencebase.gov/catalog/file/get/57f1d4f3e4b0bc0bebfee139?name=CoSMoS_SoCalv3_Phase2_summary_of_methods.pdf). Phase 2 data for Southern California include flood-hazard information for the coast from the border of Mexico to Pt. Conception to include the Channel Islands. Please read the Summary of Methods and inspect output carefully. Data are complete for the information presented.