Recent acceleration in coastal cliff retreat rates on the south coast of Great Britain

Significance

Cliffed coasts erode when attacked by energetic waves. Cliff retreat threatens coastal assets and livelihoods. Understanding rates of past erosion is vital to quantifying these risks, particularly when confronted with expected increases in storminess and sea level, and given continued human occupation and engineering of coastal regions. Historical observations of cliff retreat span 150 y at most. We derived past cliff retreat rates over millennial time scales for chalk cliffs on the south coast of Great Britain by interpreting measured cosmogenic nuclides with numerical models. Our results provide evidence for accelerated erosion in recent centuries, which we suggest is driven by reduced sediment supply and thinning of beaches in the face of environmental and anthropogenic changes.

Abstract

Rising sea levels and increased storminess are expected to accelerate the erosion of soft-cliff coastlines, threatening coastal infrastructure and livelihoods. To develop predictive models of future coastal change we need fundamentally to know how rapidly coasts have been eroding in the past, and to understand the driving mechanisms of coastal change. Direct observations of cliff retreat rarely extend beyond 150 y, during which humans have significantly modified the coastal system. Cliff retreat rates are unknown in prior centuries and millennia. In this study, we derived retreat rates of chalk cliffs on the south coast of Great Britain over millennial time scales by coupling high-precision cosmogenic radionuclide geochronology and rigorous numerical modeling. Measured ¹⁰Be concentrations on rocky coastal platforms were compared with simulations of coastal evolution using a Monte Carlo approach to determine the most likely history of cliff retreat. The ¹⁰Be concentrations are consistent with retreat rates of chalk cliffs that were relatively slow (2–6 cm⋅y⁻¹) until a few hundred years ago. Historical observations reveal that retreat rates have subsequently accelerated by an order of magnitude (22–32 cm⋅y⁻¹). We suggest that acceleration is the result of thinning of cliff-front beaches, exacerbated by regional storminess and anthropogenic modification of the coast.

Rocky coasts are “erosional environments formed as a result of the landward retreat of bedrock at the shoreline” (1). They leave scant evidence of any previous state, making it difficult to interpret their history. Cliff retreat is driven by a combination of wave-driven cliff-base erosion, subaerial weathering, and mass wasting processes, whose efficiencies are dependent on lithology and climate. Sediment generated by mass wasting processes such as abrasion, plucking, landslides, and rockfalls tends to be reworked and transported away by waves and currents, particularly for softer rock types.

Cliff erosion due to mass wasting threatens livelihoods and both public and private clifftop infrastructure and development; quantitative estimates of the rate of cliff retreat are necessary to assess the associated risk. Rising sea levels and increased storminess may lead to accelerated coastal erosion rates in the future, potentially increasing hazard exposure (2–5). To accurately assess coastal hazards in the face of future climate and land-use changes, it is necessary to understand the dynamics of cliff erosion over length and time scales relevant to the processes that drive change. To establish the context for modern change, we must quantify the natural variability and the long-term behavior of cliff retreat. Historical records are too short to allow us to do this: they typically span no longer than ∼150 y (6, 7), which can be less than the characteristic return period of significant coastal failures (8), and they coincide with the period over which humans have significantly modified the coast. It is therefore vital that we obtain longer, reliable records of coastal change to compare with historical observations to understand how coastal erosion may have changed through time, what the drivers are, and how coasts may evolve into the future (5).

Measurement of in situ concentrations of ¹⁰Be provide a versatile geochronometer for geomorphic studies, which facilitates dating of surface exposure and the deposition and burial of sediments and estimation of weathering and erosion rates (9). The technique has recently been applied to rocky coasts to estimate rates of cliff retreat (10, 11) and to understand the Quaternary history of exposure, inheritance, and reoccupation of shore platforms (12). Here we report a long-term record of cliff retreat in the relatively soft chalk cliffs of East Sussex, United Kingdom, which have been observed to be eroding at rates of 10–80 cm⋅y⁻¹ over the last 150 y (7). Our long-term record was generated by coupling high-precision measurement of concentrations of ¹⁰Be on a coastal platform with a numerical and statistical model that inverts these data for rates of cliff retreat at millennial time scales.

The model assumes that the coastal profile evolves through equilibrium retreat such that cliff height, platform gradient, and beach width are constant through time (Fig. 1A). In nature, stable beaches play an important role in mediating cliff erosion by providing protective cover to dissipate wave energy; however, mobile beaches may provide abrasive tools to erode the cliff toe (13). Beach cover on a shore platform will also shield the platform, at least in part, from the incoming cosmic ray flux that produces ¹⁰Be (10). The model presented here assumes beach width and cover is constant through time and of sufficient thickness to completely shield the underlying platform from the production of ¹⁰Be. As the cliff recedes, the rocky platform is exposed to the production of ¹⁰Be. Exposure is mediated, however, by a number of variables, including the rate of cliff retreat and the cover of water (10–12). The local water depth is dictated by tides, relative sea-level history, and vertical downwearing of the platform. This generates a theoretical “humped” pattern of ¹⁰Be concentration with distance offshore (10). We extend this model to account for beach cover, the intrinsic variability of ¹⁰Be production (14), and the influence of cliff height (topographic shielding) (15) and use an established glacial isostatic adjustment model (16) to provide relative sea-level history for the past 7,000 y covered by the simulations. We develop a rigorous statistical analysis to compare the resulting predictions with measured ¹⁰Be concentrations to generate quantitative estimates of cliff retreat histories (Fig. 1B) (see Materials and Methods for a full description of the numerical and statistical model).

Fig. 1.

Setup for modeling the accumulation of ¹⁰Be on a coastal platform. (A) The model assumes equilibrium retreat such that as the coast evolves the cross-section morphology remains steady while translating shoreward according to the prescribed retreat rate. Beach width was held constant during each model run, and the elevation of the coastal profile tracks relative sea level change. (B) Schematic illustration of a rocky coast and platform showing the expected “humped” relationship between distance from the cliff and ¹⁰Be concentration.

Our study site was the Cretaceous chalk cliffs in East Sussex, United Kindgom (Fig. 2), where cliff retreat has generated wide coastal platforms characterized by abundant bands of chemically inert and erosionally resistant flint (Fig. 2 A and B). Both the lithology and structure of the chalk are relatively uniform along the examined coast, although there are known subtle variations in jointing pattern, in the orientation of gentle fold axes, and the associated dip of subhorizontal bedding of the chalk and flint bands (17). Our modeling assumes that the geological properties of the cliff and platform have been constant as retreat has occurred. Waves approach predominantly from the open Atlantic Ocean into the relatively narrow English Channel (Fig. 2C). Previous studies suggest the wave directions have been consistent during the midlate Holocene (18), although storminess may have varied (19, 20). The coastline is managed as part of the South Downs National Park and is designated a Site of Special Scientific Interest, a Marine Conservation Zone, an Area of Outstanding Natural Beauty, and a Heritage Coast by the UK government. There has been little direct human intervention; the chalk cliffs therefore evolve without any attempts to control erosion (21).

Fig. 2.

Location and observed historical cliff retreat rates. (A) Photograph of platform and Seven Sisters chalk cliffs. (B) Location map showing study area in Cretaceous Chalk in East Sussex, United Kingdom. (C) Shaded relief map derived from stitched LiDAR topography and multibeam bathymetry [data courtesy of the Channel Coast Observatory (CCO); www.channelcoast.org]. Mapped 1870s and 2001 cliff lines and associated observed cliff retreat rates are plotted along the coast after Dornbusch et al. (7). The box plot shows the 5th, 25th, 50th, 75th, and 95th percentile of these historic retreat rates above the legend. The wave rose diagram shows wave conditions during 2014 with dominant wave approach from the southwest (data courtesy of CCO). (D and E) Shaded relief draped with 2008 aerial photographs (data courtesy of CCO) for field sites at (D) HG and (E) BH, respectively. Black triangles show the locations of flint samples collected for ¹⁰Be analysis. Average 20th-century retreat rates are 0.32 and 0.22 m⋅y⁻¹, respectively.

Chalk cliff heights range from 12 m near Cuckmere Haven up to 150 m at Beachy Head (BH). The cliffs are nearly vertical along the length of the coastline and are connected to a low gradient rock platform extending several hundred meters offshore (Fig. 2 D and E). At the junction between cliff and platform there are intermittent fringing beaches composed of flint pebbles and cobbles mixed with sand. These are known to have been more continuous and of larger volume during the 19th century (7). Frequent cliff failures result in aprons of chalk debris that are subsequently reworked by wave action. A variety of cliff failure mechanisms have been observed, including vertical collapses, wedge collapses, rockfalls, rotational failures, and toppling (17); all of these processes can result in several meters of clifftop retreat in a single event. Erosion of platforms seems to occur through a combination of vertical downwearing due to frost action, mechanical and biological abrasion (22), and subhorizontal step retreat (23).

Mapped clifftop positions from 1873 to 2001 historical maps and aerial photographs reveal that cliff retreat rates vary between 0.05 and 0.8 m⋅y⁻¹ (Fig. 2C) (7). Extrapolating this range of historical retreat rates back in time, a ∼350-m platform (widest observed subaerially exposed platform at the study site) can form in between 450 and 7,000 y, and therefore certainly within the Holocene. The model and ¹⁰Be data presented here allowed us to constrain more precisely the platform age and cliff retreat rates.

Samples of in situ flint exposed on the rock platform were collected along transects roughly perpendicular to the cliff face at Hope Gap (HG; Fig. 2D) and Beachy Head (BH; Fig. 2E) during spring tides on July 24–25, 2013. Cliff heights at HG and BH are 15 m and 50 m, respectively. These transects were chosen to maximize platform width (minimizing platform gradient) to sample as far offshore as possible. We collected samples from local topographic highs on sections of the platform away from areas that exhibited significant roughness due to runneling or block removal (Fig. 3). Distance to a fixed position on the cliff and the height of the cliff were measured with a laser range finder. In addition, we sampled rock from inside a sea cave near HG to estimate inherited ¹⁰Be concentration before platform exposure.

Fig. 3.

Measured ¹⁰Be concentrations and 1σ uncertainties (open circles and whiskers, respectively) and most likely retreat scenarios (colored lines and shaded regions showing median and 95% confidence interval) for (A) HG and (B) BH transects. Concentrations of ¹⁰Be generally increase and then decrease offshore. The sample highlighted in red on the HG transect (A) was treated as an outlier (Discussion). The minimum measured concentration in each transect was assumed to represent the inherited concentration of ¹⁰Be (see text for further discussion). The most likely retreat scenarios in both cases were a recent step change in retreat rate, with (A) a reduction from 5.7 (+0.3/−0.3) to 1.3 (+1.1/−0.3) cm⋅y⁻¹, 308 (+135/−100) years ago at HG and (B) an increase in retreat rate from 2.6 (+0.2/−0.2) to 30.4 (+8.3/−106.) cm⋅y⁻¹, 293 (+170/−80) years ago at BH. (C) Steady-state ¹⁰Be concentrations as a function of depth generated by deep-penetrating muons for surface lowering rates of up to 0.1 mm⋅y⁻¹. Red symbols show measured concentrations with depth taken as the local cliff height for each site. Measured inheritance is consistent with surface lowering rates of 0.01–0.04 mm⋅y⁻¹.

The ¹⁰Be sample preparation was carried out at the Scottish Universities Environmental Research Centre (SUERC) using isotope dilution chemistry. The ¹⁰Be/⁹Be analyses by accelerator mass spectrometry (AMS) were conducted at Lawrence Livermore National Laboratory (LLNL) to determine ¹⁰Be concentrations (Materials and Methods).

To interpret Holocene cliff retreat rate, we compared the measured distributions of ¹⁰Be concentrations across the coastal platform to predicted concentrations from numerical modeling of coastal retreat and ¹⁰Be accumulation. We searched for the most likely cliff retreat rate histories by comparing observed ¹⁰Be concentrations to modeling results via maximum likelihood estimation (MLE) using Markov chain Monte Carlo (MCMC) (24) ensembles (each with 200,000 iterations). We modeled three possible scenarios for the history of cliff retreat: (i) steady rate of cliff retreat for the entire Holocene, (ii) linear change in erosion rate throughout the Holocene (either acceleration or deceleration), and (iii) step change in erosion rate at an unknown time (acceleration or deceleration). The presence of a beach was incorporated assuming that no ¹⁰Be production occurs beneath the beach (i.e., that the beach thickness is sufficient to diminish ¹⁰Be production entirely). Beach width was treated as a free parameter in the MCMC procedure but is held constant throughout any single cliff retreat model run, because there is little information about beach width change during the Holocene. Estimates and confidence intervals of cliff retreat rates and beach width for each scenario were obtained from the MCMC-derived posterior probability distributions as the median and 95% confidence limits (SI Appendix).

Results

Broadly, concentrations of ¹⁰Be across the coastal transects show a “humped” profile (10) (Fig. 3 A and B). One sample (HG-12) showed anomalously high ¹⁰Be concentration and we therefore treated it as an outlier. Despite taking care to sample only in situ flint nodules, it is possible that this HG-12 sample was not in situ and had been transported for a significant period at the surface, allowing high exposure to cosmic rays. We collected sample HG-15 from an inward-directed face 8 m deep inside a cave in the 30-m-high cliff, adjacent to the HG transect. This sample contained an appreciable concentration of ¹⁰Be, suggesting that any newly exposed platform may contain an inherited contribution of ¹⁰Be (up to 30–50% of the measured concentrations). This inherited contribution is likely due to production by the deep penetration of the energetic muons (25) into the landscape. The inherited concentration measured here is similar to concentrations measured on a similar platform at Mesnil-Val on the opposite side of the English Channel (10). This highlights that future studies on coastal platforms should be careful to assess potential inheritance or risk significantly underestimating retreat rates. We modeled the production of muogenic ¹⁰Be as a function of depth and surface lowering rates (26) (Materials and Methods) to compare with the measured inherited ¹⁰Be concentrations (Fig. 3C). Concentrations are consistent with muogenic production for slow surface lowering rates in the range 0.01–0.04 mm⋅y⁻¹.

Before MCMC inversion used to determine most likely retreat scenario and rates we corrected concentrations for inherited ¹⁰Be using the measured concentrations at both HG-15 and BH-13 for the HG and BH transects, respectively (shaded gray area labeled “inheritance” in Fig. 4 A and B). Note that site HG-10 was sampled twice (HG-10a and HG-10b) (i.e., from two different adjacent flint nodules on the rock platform). The concentrations returned from these two were within measurement error of one another (Fig. 3A and SI Appendix, Table S1).

Fig. 4.

Example probability density (Top) and cumulative probability (Bottom) of the two retreat rates, the timing of change, and beach width for the step-change scenario MCMC ensemble at HG. Values and uncertainties were taken as the median (solid line) and 95% confidence range (dashed lines and gray shading) from the cumulative density plots on the bottom row.

The most likely retreat scenarios were determined by MLE using MCMC ensembles, resulting in likelihood-weighted probability distributions (Fig. 4 and SI Appendix). At both transects the best-fit scenario included a recent step change in retreat rate, with a reduction from 5.7 (+0.3/−0.3) to 1.3 (+1.1/−0.3) cm⋅y⁻¹, 308 (+135/−100) years ago at HG and an increase in retreat rate from 2.6 (+0.2/−0.2) to 30.4 (+8.3/−106.) cm⋅y⁻¹, 293 (+170/−80) years ago at BH (SI Appendix, Tables S2 and S3). However, both sites have experienced a recent acceleration in erosion rates as evidenced by observed rates of ∼32 cm⋅y⁻¹ and ∼22 cm⋅y⁻¹ since 1870 at HG and BH, respectively (7).

Discussion

To date, application of ¹⁰Be to quantify long-term coastal process rates have been few (10–12), but these techniques provide a new opportunity to integrate observations with long-term rates and antecedent coastal conditions. Observed rates of cliff retreat at HG (∼32 cm⋅y⁻¹) and BH (∼22 cm⋅y⁻¹) imply that the 250- to 350-m width of platform that we have sampled is young, forming in the last 1,500 y. Such recent retreat and young platform age would result in negligible ¹⁰Be accumulation on the platform, which is inconsistent with the measured ¹⁰Be concentrations. Thus, the rates suggested by historical observations cannot be extrapolated back in time; instead, cliff retreat rates must have recently accelerated to their observed values.

The ¹⁰Be concentrations at HG demonstrate that slower cliff retreat (∼5.7 cm⋅y⁻¹) persisted for much of the Holocene and do not match the historically observed higher rates (Fig. 3A). On the contrary, our modeling results suggest a recent slowdown to ∼1.3 cm⋅y⁻¹ over the last 300 y. This slowdown is principally allowing better fit to HG-13 and HG-14, the samples nearest the cliff. These sites may have elevated ¹⁰Be concentrations due to minimal platform downwear in this zone, sampled at ∼1-m elevation above mean sea level in the upper intertidal zone. Nevertheless, the most landward platform sample (HG-14) is 50 m from the modern cliff; at 32 cm⋅y⁻¹ (the observed retreat rate since 1870s), this 50 m would have occurred in the last 156 y. Hence, we may not have sampled close enough to the cliff to detect an acceleration in cliff retreat rates that must have occurred during this time. Future studies could focus on higher-resolution sampling nearer the cliff to resolve the historical signal.

Measured ¹⁰Be concentrations at BH indicate long-term average retreat rates that are much slower than historical rates for most of the Holocene. In contrast with nearshore samples at HG, low concentrations in the nearshore region of BH are consistent with recent, rapid retreat, as corroborated by historical observations. Low concentrations persist to 145 m out from the modern cliff (Fig. 3B); at historical retreat rates of 22 cm⋅y⁻¹ this cliff would have retreated 145 m in the last 650 y, implying acceleration must have occurred within this timeframe. Our modeling results suggest a significant increase in retreat rates in the last 200–500 y. The large uncertainty estimates with respect to the timing of this change result from a tradeoff between the timing of acceleration in retreat rates and the increased retreat rate itself. More rapid retreat rates require the acceleration to have occurred more recently to expose the 145 m of platform with consistently low ¹⁰Be concentrations.

At both sites, ¹⁰Be concentrations demonstrate that cliff retreat was slow for much of the Holocene, in contrast to substantially higher observed retreat rates. We conclude that the coast of East Sussex, previously a relatively stable, slowly eroding coastline, has undergone a recent increase in rates of cliff retreat.

We assumed that equilibrium retreat is an appropriate model for the morphological evolution of the studied shorelines. Alternative morphological models include shore platforms that widen and shallow through time, potentially causing deceleration in cliff retreat rates due to increased wave energy dissipation (27, 28). The studied platforms are relatively steep (gradient 1:60 m), suggesting that equilibrium retreat is appropriate over the millennial time scales studied. Moreover, our modeling concludes that platforms that were widening and shallowing through time have distributions of ¹⁰Be concentrations that are distinct from those predicted assuming equilibrium retreat (29). The distribution of concentrations measured in the shore platforms for this study are consistent with equilibrium retreat. Nevertheless, differences in lithological resistance or susceptibility perhaps related to jointing (17) between our two studied transects may account for the 45% differences in retreat rates, with HG recording more rapid retreat over both long time scales as revealed by ¹⁰Be concentrations, and historical time scales, compared with the equivalent time periods at BH.

In addition, our modeling assumes that beach width has not changed during the Holocene. If beach widths had in fact been wider and thicker in the midlate Holocene, less ¹⁰Be would have accumulated on the coastal platform because the platform would have been shielded by sedimentary cover (11). The influence of additional cover would require even slower long-term retreat rates to match the observed ¹⁰Be concentrations and would increase the difference between long-term and historic cliff retreat rates. Beaches play a dual role in affecting cliff erosion: They provide the abrasive tools to achieve erosion but also provide protective cover to dissipate wave energy before it reaches the cliff toe (13, 30). Our modeling demonstrates that the presence or absence and variability of beach cover exerts only minor control on the distribution of ¹⁰Be across the shore platform (29). If beaches were wider and thicker in the past, then measured ¹⁰Be concentrations would be lower than if no beaches were present; lower concentrations would suggest faster apparent erosion rates than had actually occurred. In this sense, our estimates of long-term cliff retreat rates may be maxima.

Acceleration of chalk cliff erosion is likely related to an increase in wave energy delivered to the cliff face, and we offer two potential explanations for this increase. The first is related to climate change during the Little Ice Age (LIA, ∼600–150 y B.P.). A growing body of proxy-based evidence supports increased storminess in the north Atlantic ca. 600–250 y B.P. (19) associated with the negative phase of the North Atlantic Oscillation that resulted in a drier, colder climate in northern Europe (20). General circulation climate model simulations have shown that during the LIA the paths and the intensity of cyclones, and associated extremes of precipitation and wind speed, may have shifted southward below 50°N. Such conditions may have increased the delivery of wave energy to the coast due to both the number of energetic events and their severity. The second explanation is related to the availability and role of beach sediment. Sediment protects the platform against vertical downwearing and serves to dissipate wave energy otherwise available to drive cliff erosion. Beaches within the study area are known to have been thinning during the Holocene (7), in part supplying the wider beaches to the east (downdrift) (31–33).

Sediment supply to the beaches may also be related to human intervention at the coast. Although there are no active interventions protecting the studied coastline, engineering activities since the late 19th century, designed to protect several kilometers of the coastline 2–15 km to the west (updrift), have reduced the supply of littoral sediment along the studied coastline; beach widths have been observed to be declining or been lost along the length East Sussex coastline (7). Numerical modeling has demonstrated that shoreline interventions can result in significant nonlocal impact many kilometers downdrift from the protected sites (3, 34).

Our methods do not allow us to attribute the recent acceleration in cliff retreat rates in East Sussex to anthropogenic activity, to a response to progressive thinning of beach material, or to increased storminess during the LIA. However, these results would suggest that beaches play an important role in regulating coastal erosion along the East Sussex coast of southern Great Britain. The dynamics and fate of beaches on shore platforms and how they link to long-term coastal evolution remains an outstanding research area within coastal geomorphology (35).

Conclusions

Forecasting future coastal change at cliffed coasts, faced with rising sea level and increased storminess, requires understanding of past cliff retreat rates in response to environmental conditions over long time scales. Measuring cosmogenic ¹⁰Be in shore platforms offers a promising approach to obtaining such records (35). Here, ¹⁰Be concentrations from two shore platforms in East Sussex in southern Great Britain reveal that retreat rates between 2–6 cm⋅y⁻¹ prevailed for most of the Holocene and contrast dramatically with historical records of rapid retreat at 22–32 cm⋅y⁻¹ at the same sites during the last 150 y (7). Our measurements demonstrate that acquisition of long-term records of coastal change can reveal marked changes in coastal dynamics in the relatively recent past. At our study site, these changes likely reflect beach dynamics that has led to thinning of beach sediment, which in turn has increased cliff retreat rates.

Materials and Methods

We processed samples at SUERC according to modified protocols developed for this study. We crushed and sieved flint nodule samples to 0.25- to 0.50-mm size fraction and performed magnetic separation to remove magnetically susceptible particles. To purify flint (amorphous SiO₂ with the same chemical formula as quartz, but a different structure) and remove atmospherically derived ¹⁰Be adhered to the outer parts of the grains (36), samples were washed and leached in subboiling 2% (vol/vol) nitric acid. Samples were dried and etched in 35% (vol/vol) hexaflorosilicic acid, followed by repeated 16% (vol/vol) hydrofluoric acid etches. The samples were then dried and aliquots assayed to determine their elemental abundances by inductively coupled plasma optical emission spectrometry (ICP-OES). Samples contained high levels of impurities, including Al, Ca, Na, K, Mg, Ti, and/or Fe, and were additionally etched; upon reassay, elemental concentrations remained constant, and we therefore judged that observed concentrations were inherent to the samples.

Samples were transferred to a cleanroom, rinsed in 18.2 MΩ water, and dried. Samples were then massed (∼50–60 g of flint) and ∼200 µg low-background beryl-derived Be carrier was added by mass. The samples were dissolved in subboiling hydrofluoric acid. The hydrofluoric acid was evaporated and the resulting digestion cakes were fumed to dryness at least three times to convert to chloride form then taken up in hydrochloric acid (37). Insoluble residues were removed by centrifugation. To reduce the high concentrations of cations and anions in the solution, samples were first precipitated at pH 8 as hydroxides (38). Postprecipitation, ∼30 mg of anions and cations were still present in each sample. Because the vast majority of the ions in solution were cations, the samples were passed through anion exchange columns using 2 mL of AG 1-X8 (200–400 dry mesh) resin to remove iron, using standard protocols. After conversion to sulfate form with sulfuric acid, samples were passed through large (20 mL) cation exchange AG 50W-X8 (20–50 dry mesh size) resin columns to remove impurities (39) and to isolate Be. Elution curves for these large columns with high cation loads were developed before sample processing and milliequivalent calculations were made based on postprecipitation ICP-OES data to ensure that cation loads were below ∼50% of the available column capacity. After elution, yield test samples were collected from the Be fractions to determine their purity and to ensure that sufficient material was available for high-quality isotopic analyses; Be fractions from large columns were ∼75% (∼150 µg) with a few hundred micrograms each of Al, Mg, and K. Nearly all of the missing Be was lost during the first pH 8 hydroxide precipitation, rather than during subsequent ion exchange chromatography. To further purify the Be fractions, these solutions were dried down, dissolved in sulfuric acid, and passed through an additional 2-mL cation column using standard procedures. After the second cation column, Be fractions were free of impurities and no additional Be was lost during the second elution. The final Be fractions were precipitated at pH 8 as hydroxides, centrifuged, washed with 18.2 MΩ water, centrifuged, decanted, and dried. The dried material was ignited in a furnace to convert to Be oxide, mixed with Nb in a 1:1 molar ratio, and packed into stainless steel cathodes for isotopic analysis at LLNL by AMS (40).

At the LLNL AMS facility, each cathode was measured at least three times. Initial sample ⁹Be³⁺ beam currents averaged ∼18 μA, ∼75% of standard cathodes. The data were normalized to the 07KNSTD3110 standard with a reported ¹⁰Be/⁹Be ratio of 2.85 × 10⁻¹², which is consistent with the revised ¹⁰Be decay constant (41). Secondary standards produced by K. Nishiizumi were run as unknowns to confirm the linearity of the isotopic measurements. Two full-process blanks (Be carrier only) were processed with each batch of samples. The average measured blank isotopic ratio for each batch was subtracted from the measured isotopic ratios of the samples in that batch with uncertainties (i.e., SD samples and blanks) propagated in quadrature (SI Appendix, Table S1). The ¹⁰Be/⁹Be blank ratios for two blanks run with the samples in one batch (HG samples) averaged 2.1 ± 0.07 × 10⁻¹⁵, whereas two blanks in the second batch (BH samples) averaged 6.3 ± 2.0 × 10⁻¹⁵, both representing a relatively small portion (∼3–11% and ∼11–35%, respectively) of the measured sample isotopic ratios of samples in each batch.

The concentration of ¹⁰Be in rock, N (atoms⋅gram⁻¹), at depth below the rock platform surface, z (meters), evolves through time, t, according to (29)

$$\frac{dN}{dt}={\displaystyle \sum _i{S}_T{S}_G{S}_W{P}_i}{e}^{-(z/z_i^{*})}-\lambda N.$$

The first term on the right-hand side reflects production of radionuclides and the second term their decay. The subscript i refers to different production pathways; for ¹⁰Be this is dominated by spallation (26), with a minor contribution from muogenic production. Production due to muons is modeled with a single exponential term (25). S_T is a topographic shielding scaling factor that adjusts the incoming cosmic ray flux depending on the proportion of the sky blocked by the presence of the cliff and is modeled following established procedures (15). S_T varies with distance from the cliff, and the model assumes a vertical cliff of constant height in space and time. S_G is a scaling factor reflecting temporal variation in incoming cosmic ray flux due to solar activity and deviation in the strength of Earth’s magnetic field, calculated following Lifton et al. (14). S_W is a scaling factor reflecting shielding of the platform due to water cover, averaged over a single tidal cycle, calculated following Regard et al. (10). We used a glacio-isostatic adjustment model for the United Kingdom to predict relative sea level change at the field sites (16). P_i is the surface production rate specific to the production pathway. For spallation, the value of P = 4.008 at⋅g⁻¹⋅y⁻¹ was obtained for the field site from the Lifton et al. (14) scaling scheme. For muogenic production a single median value of P = 0.028 at⋅g⁻¹⋅ y⁻¹ was used to integrate both fast muon interactions and negative muon capture reactions (25). $z_i^{\ast }=\hspace{0.17em}{\rho }_r/{\mathrm{\Lambda }}_i$ is a production pathway-specific attenuation length scale, where ρ_r is rock density (1,800 kg/m³ used here for chalk) (17) and Λ_i is the attenuation factor. For spallation, Λ = 1,600 kg⋅m⁻², and for muons, Λ = 42,000 kg⋅m⁻² were used. λ = 4.99 × 10⁻⁷ is the ¹⁰Be radioactive decay constant (42, 43).

Prediction of the expected ¹⁰Be concentration inherited (Fig. 3C) due to deep penetration of energetic muons N_μ (atoms⋅gram⁻¹), where the subscript μ refers to the muogenic production pathway, were calculated assuming steady-state surface lowering rate ε (millimeters per year) (26) according to

$$N_{\mu }\left(z\right)=\frac{P_{\mu }}{\lambda +\epsilon /z_{\mu }^{\ast }}{e}^{-(z/z_{\mu }^{*})}.$$

To find the retreat rate histories that best replicate the observed ¹⁰Be concentrations, we performed an MCMC analysis (24) to produce posterior probability density functions for cliff retreat rates [similar to Hurst et al. (44)]. A Metropolis–Hastings algorithm was used to vary parameters (45). We calculate and maximize the likelihood L for a given set of parameters:

$$L={\displaystyle \prod _{j=1}^n\frac{1}{\sqrt{2\pi }{\sigma }_j}\exp \left[-\frac{{\left(N_j^{meas}-N_j^{\mathit{mod}}\right)}^2}{2{{\sigma }_j}^2}\right]},$$

where n is the number of observations of ¹⁰Be concentration N, the superscripts ^meas and ^mod refer to corresponding measured and modeled ¹⁰Be concentrations, and σ is the confidence range of measured ¹⁰Be concentrations.

Three scenarios of cliff retreat were run for comparison with measured ¹⁰Be concentrations: (i) a single retreat rate ε₁ applied through the entire Holocene, (ii) a step change in retreat rate from ε₁ to ε₂ at time t, and (iii) a gradual change in retreat rate from ε₁ to ε₂ throughout the Holocene (7 ka to present). A fixed beach width W was assumed throughout each model run. After each run in the MCMC, new values for ε₁, ε₂, t, and W were randomly selected from a Gaussian probability distribution centered on the previous accepted values, with SDs tailored to a target acceptance rate of 23% (46). The likelihood of each iteration is compared with that of the last accepted parameter set such that if the ratio of the current to the last accepted iteration >1 then the new parameter set is accepted. If the ratio <1, then the new parameters may be accepted with a probability of acceptance equal to the likelihood ratio (to allow the chain to fully explore the parameter space). The “burn-in” period was less than 1,000 iterations in all cases, and each MCMC was run for 200,000 iterations (45). The posterior probability distribution of each parameter was generated as a likelihood-weighted frequency distribution from the MCMC iterations. Parameter values and confidence intervals were then determined as the median and 95% limits on the probability distribution (SI Appendix).