Modeling erosion of ice-rich permafrost bluffs along the Alaskan Beaufort Sea coast

Authors


Abstract

The Arctic climate is changing, inducing accelerating retreat of ice-rich permafrost coastal bluffs. Along Alaska's Beaufort Sea coast, erosion rates have increased roughly threefold from 6.8 to 19 m yr−1 since 1955 while the sea ice-free season has increased roughly twofold from 45 to 100 days since 1979. We develop a numerical model of bluff retreat to assess the relative roles of the length of sea ice-free season, sea level, water temperature, nearshore wavefield, and permafrost temperature in controlling erosion rates in this setting. The model captures the processes of erosion observed in short-term monitoring experiments along the Beaufort Sea coast, including evolution of melt notches, topple of ice wedge-bounded blocks, and degradation of these blocks. Model results agree with time-lapse imagery of bluff evolution and time series of ocean-based instrumentation. Erosion is highly episodic with 40% of erosion is accomplished during less than 5% of the sea ice-free season. Among the formulations of the submarine erosion rate we assessed, we advocate those that employ both water temperature and nearshore wavefield. As high water levels are a prerequisite for erosion, any future changes that increase the frequency with which water levels exceed the base of the bluffs will increase rates of coastal erosion. The certain increases in sea level and potential changes in storminess will both contribute to this effect. As water temperature also influences erosion rates, any further expansion of the sea ice-free season into the midsummer period of greatest insolation is likely to result in an additional increase in coastal retreat rates.

1 Introduction

The Arctic environment is responding to global climate change [Arctic Climate Impact Assessment, 2004; Richter-Menge et al., 2006; Serreze and Barry, 2011; Intergovernmental Panel on Climate Change (IPCC), 2013]. Along the ice-rich permafrost coastal bluffs of the Alaskan Beaufort Sea, this change manifests as rapid and accelerating coastal retreat [Reimnitz and Maurer, 1979; Are, 1988; Solomon et al., 1993; Jones et al., 2008, 2009; Lantuit and Pollard, 2008; Mars and Houseknecht, 2009; Lantuit et al., 2012; Overeem et al., 2011; Wobus et al., 2011]. The presence of ground ice and the control exerted on erosional processes by sea ice distinguishes Arctic coastlines from their temperate counterparts and renders them particularly sensitive to climate changes.

Along the central-western Beaufort Sea region, 3–5 m tall permafrost bluffs with high ice content and fine sediment abut the shallow Beaufort Sea shelf. The coastal bluffs retreat by the incision of a notch into the base of the bluff, which subsequently causes bluff collapse. The failed portion of the bluff is cohesive and rotates toward the sea as it degrades. The composition and morphology of the coast allows efficient coastal retreat.

This coastline bounds the Beaufort Sea, which along with the Bering and Chukchi Seas has already experienced significant change and is expected to experience some of the greatest future changes in the Arctic [Chapman and Walsh, 2007; Walsh, 2008]. Nearshore sea ice now disappears earlier in the beginning of the open-water season, and open-water conditions persist longer into the autumn [Markus et al., 2009; Overeem et al., 2011]; the Arctic Ocean has warmed substantially, with sea surface temperature anomalies up to 5°C [Steele et al., 2008]; and sea levels in the Russian Arctic are rising [Proshutinsky et al., 2004]. Arctic storminess is characterized by large interannual variability [Atkinson, 2005]. Increased storminess in the winter and autumn is anticipated based on current coupled climate models [Clow et al., 2011].

In the face of an expanding sea ice-free season, erosion rates along a 60 km stretch of the Beaufort Sea coast from Drew Point to Cape Halkett doubled from 6.8 m yr−1 (1955–1979) to 13.6 m yr−1 (2002–2007) [Jones et al., 2008, 2009; Mars and Houseknecht, 2009], with local observations as high as 30 m yr−1 [Wobus et al., 2011]. Our own observations yield rates of 19 year−1 for 2011–2012. Existing petroleum and military infrastructure along the Alaskan Beaufort Sea coast, such as oil exploration wells and abandoned Defense Early Warning line sites, are increasingly in jeopardy, and several structures have already been destroyed by land loss [Jones et al., 2008]. In addition, these coastal bluffs contribute increasing amounts of carbon to the ocean [Ping et al., 2011].

The long-term coastal retreat rate integrates many episodes of notch incision that occur when water rises to the base of the bluff, typically during a storm. Submarine erosion of the bluff material incises the notch during each storm. While detailed treatment of notch incision during a single event exist [e.g., Kobayashi, 1985], the relative influences of the length of the sea ice-free season, water and permafrost temperatures, and storm-associated set up and wavefield on the long-term coastal retreat rate remain to be evaluated.

As the anticipated climate changes should enhance Arctic coastal erosion, this motivates a detailed understanding of notch incision, block degradation, and the details of the submarine erosion process aimed at answering two questions: 1. What is the simplest relationship between environmental conditions and the short-term rate of submarine notch incision? and 2. How sensitive are long-term rates of coast retreat to individual environmental conditions?

We describe the geologic and environmental setting at Drew Point and our field methods. We then outline the process of coastal retreat, observations of coastal retreat erosion, and a numerical model designed to capture the essence of the observed processes. We test this model and three rules for the rate of submarine notch incision over two time periods: 1 week in summer 2010 and the period of sea ice observation, 1979–2011. We compare the results of the short-term model runs with the detailed time-lapse record of coastal erosion and validate the results of the long-term model runs against remotely sensed and field-based observations of coastal position. Finally, we perform numerical experiments in which we modify the environmental conditions in order to explore the sensitivity of the system to the individual environmental conditions.

2 Description of the Beaufort Sea Coast System

2.1 Geologic Setting

Drew Point is located between Barrow and Prudhoe Bay on the Alaskan Beaufort Sea coast. The coast is characterized by 3–5 m high coastal bluffs with high ice content (typically greater than 30% and as high as 70%) and fine-grained sediment (silt and clay) (Figure 1). This type of coast comprises ∼480 km of the nearly 2000 km long American Beaufort Sea coast [Ping et al., 2011]. Inland of the coast, the coastal plain rises toward the Brooks Range in the south and is lightly incised with small streams and dotted with thaw lakes. The coastal plain is underlain by the shallow, nearshore marine sediments of the Quaternary Gubik Formation [Black, 1964].

Figure 1.

Regional map and field photographs.

2.2 Nearshore Environment

The shelf bathymetry slopes offshore at ∼0.001 m m−1 [Rickets, 1953; Greenberg et al., 1981] and is roughly uniform along the shore. A narrow beach with a steeper slope is present in places. Bathymetric surveys at Drew Point conducted in 2010 show a 10 m wide zone with a slope of 0.035 m m−1 that gives way to a slope of 0.002 m m−1 until a water depth of 5 m, at which point the slope shallows further to 0.0005–0.001 m m−1. We consider the nearshore zone the region adjacent to the coast that is well mixed during storms.

The region is microtidal. The mean tidal range is 15 cm for daily to monthly cycles. These are superimposed on a yearly tidal cycle with a range of 66 cm that peaks in late July (around ordinal day 210; Prudhoe Bay, NOAA Tide Gauge Station 9497645).

2.3 Sea Level

Relative sea level change along the Beaufort Sea coast results from a combination of changing global mean sea level and local components. The best current estimate for end of 21st century global sea level rise is 0.18–0.59 (relative to 1980–1999 [IPCC, 2013]). These estimates should be considered as the lower bound for the Beaufort Sea region as sea level rise is not expected to be globally uniform [Yin et al., 2010], and the Arctic Ocean is projected to experience one of the greatest magnitudes of sea level rise [Yin et al., 2010; Clow et al., 2011].

Documentation of local sea level change requires a tide gauge record of at least 30 years. This is not available at the nearest station (Prudhoe Bay). Modeled glacial isostatic adjustment suggests rates of relative sea level rise for the past 500 years has been 0.61 mm yr−1 at Drew Point (0.78 mm yr−1 at Barrow and 0.74 mm yr−1 at Prudhoe Bay) [Mitrovica and Milne, 2003; Peltier, 2004; Kendall et al., 2005], much smaller than the global rate of ∼3 mm yr−1 in the last decades.

2.4 Climatology and Permafrost

Clouds influence the radiative forcing and the nearshore water temperature, particularly during the cloudy open water season. The average cloud fraction is 0.76 from May to October with a summertime peak of 0.9, and maximum cloud radiative forcing (63 W m−2) occurs in August [Dong et al., 2010]. Mean annual air temperature at this the U.S. Geological Survey (USGS) Drew Point station in 2012 is −11°C, with temperatures reaching as low as −47°C in the winter and as high as 25°C in the summer (http://data.usgs.gov/climateMonitoring/station/show/4).

Long-term mean annual ground temperature at a depth of ∼1 m was −11.1°C during much of the twentieth century [Lachenbruch and Marshall, 1986]. Shallow ground-temperature observations from this station and others in the USGS Permafrost and Climate Monitoring Network indicate that the permafrost temperature has been progressively increasing; the mean annual temperature in recent years ranges from −8.5°C to −6.5°C at the USGS Drew Point station (5 cm to 120 cm depth) and the summertime active layer ranges in thickness from 30 to 50 cm (for 1998–2013).

2.5 Sea Ice, Storms, and Wind Field

In this region the sea ice-free season begins in roughly middle July to early August, and ends in late September to early October. The sea ice-free season is expanding asymmetrically; the end of open-water season for the period 1979–2009, extended by 0.92 d yr−1 whereas the onset of open water started earlier at a rate of 0.71 d yr−1 [Overeem et al., 2011]. Autumn typically experiences more storms than summer in the Beaufort Sea region. For the period 1950–2000, an average of 8.5 storms (wind speeds greater than 10 m s−1 over a 48 h period) occurred in September and October, whereas an average of 6.5 storms occurred in July and August [Atkinson, 2005].

To the east of Drew Point, the coast is oriented roughly east-west. Surface winds from the west result in coastward Ekman transport that raises water levels and allows the ocean water to reach the base of the bluff. Conversely, winds from the east set down water levels, increasing the width of the exposed beach. Winds are most commonly from the east and northeast and show a secondary maximum from the west [Urban and Clow, 2013]. In the spring and autumn, this is consistent with winds coming off the Beaufort Sea High, a persistent atmospheric feature characterized by high sea level pressure [Serreze and Barrett, 2011]. Winds from the west that develop water level setup are typically associated with the passage of synoptic-scale storms.

3 Field Methods

Measurements of water level, wavefield, water temperature, and atmospheric conditions were collected at Drew Point during the summers of 2009 and 2010 (Figure 2). The hourly wavefield, recording water level, water temperature, significant wave height, and wave period were measured with a wave logger custom built by Tim Stanton at the Naval Postgraduate School. Four wave loggers were deployed in 2009 in water depths ranging from 1.9 to 6.8 m (up to 9 km from the shore). Water temperature was measured in 2009 at different depths using Onset Tidbits at 30 min intervals. One wave logger was deployed in 2010 at a water depth of 88 cm. Water level and temperature were measured in 2010 at 5 min intervals using a Solinst Levelogger placed at an average water depth of 33 cm. A Campbell Scientific meteorological station installed in 2008 measured and recorded air temperature and other atmospheric conditions (sensors monitored every 30 s and averaged and recorded once per hour).

Figure 2.

Map of the Drew Point site including the position of deployed instruments (circles are meteorological stations, squares are wave loggers, and the hexagon is the summer 2010 camera locations), and the three coast surveys (purple, yellow, and green lines) taken between 2008 and 2012.

We documented the process of coastal erosion at Drew Point using time-lapse imagery (Movies S1 and S2 in the supporting information, and others in Overeem et al. [2011] and Wobus et al. [2011]). The first time lapse was taken from just offshore during the period 13 to 23 August 2010 and records the degradation of a toppled block (Movie S1). The second time lapse shows a bluff failure during the period 12 August to 11 September 2010 (Movie S2).

We surveyed the position of the top of the coastal bluff for 7 km to the east of Drew Point using a handheld GPS (Garmin Vista H) set to record position every 10 m (along-track accuracy of 1–2 m) on 18–21 June 2008, 5 August 2011, and 8 August 2012 (Figure 2).

4 Process of Erosion

Inspired by field and time-lapse observations, we outline a conceptual model for retreat of ice-rich coastal bluffs bordering flat permafrost terrain. This conceptual model is similar to that of prior work by Kobayashi [1985], Kobayashi et al. [1999], Hoque and Pollard [2009], and Ravens et al. [2012]. First, a notch is carved at the base of the coastal bluff by submarine erosion [Kobayashi, 1985; Kobayashi et al., 1999]. Once the notch has incised enough to destabilize the bluff, it fails, generally separating along an ice wedge [Hoque and Pollard, 2009; Wobus et al., 2011]. The failed portion of the bluff (here called a toppled block) rotates toward the sea, rather than sliding down the failure plane. The bluff material is cohesive, and the block maintains its shape while rotating. Finally, the toppled block degrades; ocean water melts material at the base of the block, which causes it to destabilize and roll toward the ocean. As the bluff material is ice-cemented fine-grained sediment, the sediment released by ice melt is quickly transported away from the system in suspension. Little occurs when the water level is below the base of the coastal bluffs or the toppled blocks.

Subaerial erosion occurs over the entire summer season. Sediment is liberated from the coastal bluff or toppled block as the interstitial ice melts and sloughs off the block and bluff faces. During times of low water, this material remains at the base of the bluff face and around the block as a muddy plinth. When water level is set up to the level of this sloughed sediment, it is washed away. Vegetation present in the active layer, the uppermost portion of the permafrost that freezes and thaws on an annual basis, binds the upper few tens of centimeters. This cohesive mat hangs over the top of the bluffs with increasing length as material is removed from below until it rips under its own weight.

The time-lapse record of block degradation (Movie S1) shows that as a block degrades, it rotates at least an additional 180° toward the ocean. This process is important as it moves the block down the beach to lower elevations where the nearshore water has more frequent access to the block. Block degradation is rapid, occurring over a few days, and takes place at lower elevation than the notch at the base of the bluff. If the water level is high enough to incise a notch, it likely already destroyed or substantially degraded any blocks that previously broke away from the bluff. Thus, the toppled blocks do not serve as protection for the shoreline for any appreciable length of time. In the example of the time-lapse movie, the block completely erodes within 10 days.

5 Observed Erosion Rates

In section 6, we outline a numerical model of coastal erosion. In order to test this model, we use short-term and long-term observations as a benchmark against which we compare our model results.

5.1 Short-Term Erosion Rates: Proxy Derived From Time-Lapse Imagery

We determine the apparent cross-sectional area of the degrading block in the time-lapse imagery collected in summer 2010 every 1.25 h (every fifth image). Red dots placed manually around the edges of the block provide estimates of the cross-sectional block size (in pixels) from each image (Figure 3). The apparent cross-sectional area is scaled using the observed cross-sectional area measured at time-lapse camera deployment (trapezoidal in cross section with a width of ∼4 m and a height of ∼3.5 m).

Figure 3.

Construction of the short-term block degradation record used as a proxy for short-term erosion rate.

This is not a direct measurement of mass or volume loss. However, we believe that comparing the rate of change of the apparent cross-sectional area of a degrading block to potential erosion drivers provides insight into the physical processes that dominate coastal erosion in this landscape. We know of no other observations that better constrain short-term erosion rates in this landscape.

5.2 Long-Term Erosion

Prior studies indicate that coastal retreat rates along the segment of the Alaskan Beaufort Sea coast between Drew Point and Cape Halkett have increased over the past 60 years. Mars and Houseknecht [2009] document a doubling of the rate of area loss from 0.48 km2 yr−1 to 1.08 km2 yr−1 (1955–1985 compared with 1985–2005). Estimates of coast-perpendicular retreat rates show an increase from 6.8 m yr−1 to 8.7 m yr−1, and again to 13.6 m yr−1 (rates for 1955–1979, 1979–2002, and 2002–2007)—again, essentially doubling over the period of observation [Jones et al., 2009]. Surveys of the coast as part of this study indicate that the mean erosion rate over the 7 km just east of Drew Point was 15 m yr−1 for the period 2008–2011 and 19 m yr−1 for the period 2011–2012 (Figure 2).

6 Ice-Rich Coastal Bluff Erosion Model

We outline a numerical model that captures the process of erosion and mechanics of the system observed in the time-lapse imagery. It includes both subaerial and submarine erosion, allows for a notch to form at the base of a coastal bluff, evaluates the force balance on the bluff, and forms and tracks a toppled block as it degrades and rotates away from the coast.

We consider the absence of sea ice and water that exceeds the base of the coastal bluffs as prerequisites to submarine erosion (Figure 4). These two environmental conditions act like “switches” that allow water to erode the base of the bluff. The absence of sea ice occurs seasonally, while water level, controlled by the combination of setup and wave height, is controlled by passing storms. This makes erosion in this environment a highly episodic process. While other environmental conditions, such as the water temperature, influence the instantaneous rate of submarine erosion, these environmental conditions are only important at times when the sea ice is absent and nearshore water is high enough to contact the coastal bluffs.

Figure 4.

Conceptual sketch of the environmental conditions that influence the rate of coastal erosion in ice-rich permafrost environments. The absence of nearshore sea ice and water levels setup to the base of the bluffs are prerequisites to erosion. The wavefield and water temperature influence only the rate of submarine erosion. The sea ice cover, insolation, and wind field interact to produce variations in water level, wavefield, and water temperature.

This work expands upon prior work modeling this type of Arctic coast. Kobayashi [1985] laid a theoretical framework for the incision of a thermoerosional notch in contact with warm water at a constant level above the base of the bluff. As his formulation does not consider how the development of an erosional notch changes with time-variable water level, we reformulate it to apply it with time-variable conditions. Kobayashi et al. [1999] considered the overall horizontal cliff retreat in the face of a single storm event without explicitly treating the thermoerosional notch or toppled block. Ravens et al. [2012] combined the original thermoerosional notch rule developed by Kobayashi [1985] with a simple parameterization for block degradation, and addresses two periods (1979–2002 and 2002-2007) using a hindcast of environmental conditions.

We improve upon these contributions by the following: (1) using a model for block degradation that honors field and time-lapse imagery observations, (2) using shorter time steps (1 h instead of the 12 h used by Ravens et al. [2012]), (3) evaluating three different possible formulations of the physics that set the submarine erosion rate, and (4) evaluating the model against both short-term (1 week) and long-term (1979–2011) erosion rate observations. Short time steps are important because storms are often short lived, and significant erosion can occur within a 12 h time period. Our 1 h time step matches the finest temporal resolution of observations of the local wind field at Barrow for the entire 32 years period.

Our two modeling exercises are fundamentally different. The short-term model period focuses on block degradation alone, whereas the long-term time period includes both block degradation and notch incision. The short-term model runs are driven entirely by in situ observations of environmental conditions whereas the long-term model runs are driven by a hindcast of environmental conditions. In the application of the different submarine erosion rate formulations, we do not apply any “tuning” factors. Each of the three formulations relates the environmental conditions to instantaneous submarine erosion rate based on either physical or empirical relationships, and thus, we are not able to calibrate these rules to fit model results to observations.

Instead, we evaluate both long- and short-term model time periods as robust test of the model and the three formulations—a successful formulation for submarine notch incision should perform well over both time periods. The process of submarine erosion should be the same for the erosional niche at the base of the bluff and the toppled block, as both elements of the eroding coast are composed of the same material. Based on the model and formulation performance, we interpret the sensitivity of the model results to changing environmental conditions.

6.1 Model Domain

The model environment is a cross section of coastal bluff that spans from inland of the coastal bluff to the nearshore portion of the continental shelf (Figure 5). The bluff topography and block outline are discretized with either horizontal or vertical spatial steps as the independent variable. This allows for horizontally directed erosion, such as the observed cutting of a notch at the base of the bluff. As the eroded bluff sediment is fine and is transported out of the model domain as suspended sediment, the model does not track sediment.

Figure 5.

Schematic of the numerical model for coastal erosion showing discretization of the (a) coastal bluff and (b) toppled block.

Model inputs include the material properties and geometry of the coastal bluff, the nearshore bathymetry, the ice-wedge distribution, the timing of sea ice-free conditions, and time series of environmental conditions needed to calculate instantaneous submarine and subaerial erosion rates (Table 1). Based on field observations, the height of the bluff is set to 4 m and the ice-wedge depth to 4 m. At the beginning of each model run, a set of ice-wedge spacing values is randomly generated based on field measurements of ice-wedge spacing [Wobus et al., 2011].

Table 1. Long-Term Numerical Experiment Results
ParameterDefinitionTypical ValueUnitsReference
Model Geometry
ZbBluff Height4mfield measurements
inline imageMean Water Level Relative to Base of Bluffs−45cmfield measurements
DIWIce Wedge Depth4mfield measurements
LbBeach Length10mfield measurements
SbBeach Slope−0.035m m−1field measurements
SsShelf Slope−0.001m m−1Rickets [1953] and Greenberg et al. [1981]
ΔtModel Time Step0.25–1h 
Ice, Water, and Soil Material Properties
WIce Fraction65%field measurements
ρiDensity of Ice917kg m−3
ρwDensity of Water1,000kg m−3
ρsDensity of Dry Soil1,200kg m−3
CiHeat Capacity of Ice2,108J kg−1 K−1
CwHeat Capacity of Water4,210J kg−1 K−1
CsHeat Capacity of Dry Soil1,000J kg−1 K−1
LpLatent Heat of Water334,000J kg−1
τiTensile Strength of Ice1 × 104Panot well constrained, is less than τpf
τpfTensile Strength of Permafrost2 × 105PaHoque and Pollard [2009]
TpfTemperature of Bluff Materialvaries, see text°Cfield measurements
δwFreezing Temperature of Water−1.8°C
δpfFreezing Temperature of Permafrost0°C
Erosion Rules
αAreSubaerial Erosion Rate0.01C−1 d−1Are [1988]
αRHSubmarine Erosion Rate Constant2.08 × 10−7C−1 s−1Russell-Head [1980]
βRHSubmarine Erosion Rate Constant1.5unitlessRussell-Head [1980]
RWater Roughness Height0.01mWhite et al. [1980]
χconstant in equation (12)0.000146unitlessWhite et al. [1980]
κMomentum DiffusivitySee equation (14)m2 s−1Longuet Higgins [1970]
AκMomentum Diffusivity Coefficient0.4unitlessOstendorf [1982] and Inman et al. [1971]
λMaterial Property Adjustment FactorSee equation (10)unitless 
Storm Surge Model
UWind Speedvariablem s−1Barrow Observations
θWind DirectionvariableazimuthBarrow Observations
τwxWind Shear StressSee equation (3) in Text S1kg m−1 s−2Dean and Dalrymple [1991]
kFriction Factororder 10−6, see equation (1) in Text S1s2 m−1Dean and Dalrymple [1991]
fFriction Factor0.01–0.08m s−2Dean and Dalrymple [1991]
nFriction Factor1.15–1.3unitlessDean and Dalrymple [1991]
VAlong Shore Water VelocitySee equation (5) in Text S1m s−1Dean and Dalrymple [1991]
ΩCoriolis Parameter7.29 × 105rad s−1
ΦLatitude70°N 
Water Temperature Model
hrReference Water Depth3m 
RTOATop of Atmosphere RadiationSee equation (20)kg s−3 
RSWShort Wave Radiation (net)See equation (21)kg s−3 
RLWDownward Longwave Radiationvariablekg s−3Barrow Observations
ϵEmissivity of Water0.95unitless 
σStefan-Boltzman Constant5.67 × 10−8W m−2 K−4
CSea IceSea Ice Concentration0–100unitlessNSIDC
CFCloud Fraction0.7–0.9unitlessDong et al. [2010]
fcloudTransmittance of Clouds0.5unitlessSerreze and Bradley [1987]
αAlbedovariableunitless 
I0Solar Constant1368W m−2Frohlich [1993]
θiSolar Incidence Anglevariableradians 
ζLocal Zenith Anglevariableradians 
Rm/REccentricity correctionvariableunitlessHock [1999]
ψaMean Atmospheric Clear-Sky Transmissivity0.7unitlessWendler and Eaton [1990].

The water level indicates what portion of the bluff and toppled block, if present, is eroded. We use h to refer to the time series of water level, inline image to refer to the mean water level relative to the base of the bluffs, and η to refer to the water level setup or set down relative to the mean water level (inline image, see Figure 5). The switching on and off of erosion is controlled by the water level and wave height. We use a Rayleigh distribution to construct the wave height probability density function [Dean and Dalrymple, 1991, equation (7.18)]. This yields the distribution of water levels around the mean, and thus the portion of bluff and block that experience erosion at each time step.

Based on the instantaneous erosion rate and distribution of water levels for a specific time step, the model removes material from the bluff and block and then evaluates the rotational stability of both the bluff and the block. If the bluff is unstable, it will fail and rotate seaward, forming a new block. A new bluff edge will form at the location of the ice wedge along which failure occurred. If a block is unstable, the model will rotate it. Similar to observations from the time-lapse imagery, toppled blocks tend to roll toward the sea as they are undercut to a greater extent on the seaward side. However, the block is allowed to roll in whichever direction it is unstable.

6.2 Force Balance

The stability of the bluff and the toppled block are evaluated each time step (Figure 5) based on the sum of resisting and driving torques. In the case of the bluff, the failure criterion (equation (1)) is met when the driving torque per unit length along the coast about a pivot point at the apex of the notch, TD, is greater than the sum of the resisting torque, TR, and the torques needed to overcome the cohesive strength on the ice-wedge face, TIW, and the permafrost, Tpf,

display math(1)

The subscripts a, w, i, pf, and IW denote air, water, ice, permafrost, and the ice wedge, respectively. The seaward-facing portion of the bluff geometry is represented by x(z) in Figure 5a and is defined between z = 0 and z = Zb where Zb is the height of the bluff. The torque around the pivot point is the product of the weight of each element (ρbg dx dz) with the lever arm (the horizontal distance between the element and the pivot point). The driving and resisting torques from the weight of the block about a pivot point at the apex of the notch (located at x = xp and z = 0) are given by,

display math(2)

and

display math(3)

The cohesion at the ice wedge located at x = xIW and along the horizontal permafrost surface resists bluff failure:

display math(4)
display math(5)

where τi and τpf are the tensile strength of ice and permafrost, respectively.

The toppled block is represented as the right and left sides xright(z) and xleft(z) which are defined between z = zp and z = zt. The top of the block is located at (xt, zt). The failure criterion for rotation of a toppled block about a pivot point located at (xp, zp) is met when the driving torque exceeds the resisting torque,

display math(6)

where,

display math(7)
display math(8)

6.3 Formulations of Subaerial Erosion

The rate of subaerial erosion on the exposed coastal bluff face is calculated using the empirical formulation of Are [1988] based on the temperature difference between the permafrost and the air, ΔTa=(TaTpf).

display math(9)

where αAre is a constant with a value of 0.01 m °C−1 d−1 [Are, 1988; Wobus et al., 2011].

6.4 Formulations of Submarine Erosion Rate

A number of processes contribute to the rate of submarine notch incision. The nearshore wavefield controls how well water is mixed with respect to sediment, salinity, and temperature, as well as how effective the water is at suspending and transporting sediment. The water temperature and wavefield control the rate at which the heat that melts interstitial ice is delivered to the coast. Waves provide energy to suspend sediment which, if there is rock to abrade, could result in mechanical erosion at the base of the coastal bluffs. In our study region, however, we focus on thermal erosion because the sediment is very fine and the ice content is high.

The temperature difference between the water and permafrost is clearly important to the submarine erosion rate. However, the relative roles of the temperature, water depth, and wavefield are unknown. Thus, we evaluate multiple formulations for submarine erosion to determine the minimum information necessary to predict erosion rates.

Wobus et al. [2011] evaluated two empirical formulations developed for iceberg melting that, to first order, match observations of coastal erosion at Drew Point; however, the observation period over which they compare the two models was not long enough to distinguish between the formulations. The first model evaluated by Wobus et al. [2011] is the Russell-Head formulation which considers only the temperature difference between the water and the melting substrate [Russell-Head, 1980]. The second model evaluated by Wobus et al. [2011]is the White formulation which considers this temperature difference and the effect of the wavefield on heat transport [White et al., 1980; Kubat et al., 2007]. We evaluate these models and the comprehensive physically based model developed by Kobayashi [1985]. In addition to the water temperature and wavefield, the Kobayashi formulation considers a variable length scale for the diffusion of heat from the water at the front of the bluff to the back of the notch.

6.4.1 Russell-Head Formulation

The Russell-Head formulation is an empirical relationship for the melting of an iceberg based solely on the temperature difference between the water and ice [Russell-Head, 1980]. The ice-rich permafrost bluffs near Drew Point, AK, are at temperatures near or below 0°C, and are a composite material containing ice, silt, and peat. In order to apply the Russell-Head relationship to ice-rich permafrost, the original formulation is adjusted using a factor, λ, that accounts for the difference between the heat needed to bring the bulk material to its melting point, δpf= 0°C, and the heat required to cross the phase barrier in material with a lower water content. This dimensionless factor is the ratio of the energy needed to melt a unit volume of pure ice at 0°C and the energy needed to bring a unit volume of composite material to 0°C and then melt it:

display math(10)

where the permafrost bluffs are considered as a composite material with a water fraction W, a bulk density of ρb and an ice density of ρi, a bulk specific heat capacity Cb, and a temperature of Tpf. The latent heat of fusion of water is given as Lw. The bulk density and specific heat capacity are calculated by summing the individual material properties weighted by the volume fraction of each.

The modified Russell-Head erosion rate is thus given by,

display math(11)

where inline image is the instantaneous submarine erosion rate of the permafrost bluff, αRH and βRH are empirically derived constants, Tw is the temperature of the water, and δw is the freezing point of the water. In this setting, the value (Twδw) is the thermal driving parameter [e.g., Josberger and Martin, 1981; Josberger, 1983; Kobayashi, 1985; Ravens et al., 2012] that represents the sensible heat available in the water to melt ice. Russell-Head [1980] found αRH=2.08 × 10−7 m s−1°C−1, β=1.5, and δ= −1.8°C.

6.4.2 White Formulation

The White formulation is an empirical formulation based on laboratory experiments to describe the melting of an iceberg under wave action in the open ocean, and accounts for both the temperature differential and the characteristics of the wavefield [White et al., 1980; Kubat et al., 2007]. White et al. [1980] found that the dimensionless waterline melt rate in iceberg melting experiments (inline image) correlated exactly to the ratio of the roughness and wave height (RH−1). Again adjusting the formulation using λ, the White formulation is given by,

display math(12)

where inline image is the instantaneous submarine erosion rate of the permafrost bluff given in meters per second, H is the wave height, τ is the wave period, and R is the surface roughness (taken as 10 cm after White et al. [1980]). χ is an empirical constant with a value of 0.000146 [White et al., 1980, their equation (77)].

6.4.3 Kobayashi Formulation

Kobayashi [1985] developed a physically based formulation for the rate of thermoerosional notch development in ice-rich permafrost from time-averaged mass conservation equations in the cross-shore direction for salinity, suspended sediment, water, and heat.

Kobayashi [1985] assumes that during a storm, the water temperature, salinity, and suspended sediment concentration will remain constant within the surf zone. Near the shore on the Beaufort Sea shelf, a well-mixed surf zone is reasonable: the shelf slopes shallowly, resulting in a wide surf zone during storms [Weingartner et al., 2009]. Kobayashi [1985] assumes the cross-shore surf zone diffusivities for the seawater, suspended sediment, salt, and heat are all constant and equal to the cross-shore-directed momentum diffusivity associated with the longshore current induced by breaking waves, κ [Longuet Higgins, 1970].

The width of a thermoerosional notch, xm, after a given time, t, is given by Kobayashi [1985, his equation (25)]

display math(13)

and κ is given by Kobayashi [1985, his equation (15), after Ostendorf [1982]],

display math(14)

with Aκ∼ 0.4 [following Kobayashi [1985] after Ostendorf [1982] and Inman et al., 1971]. This formulation assumes that in the surf zone, wave breaking is depth limited, and thus, the water depth, h, and wave height are proportional. ζm is a nondimensional constant that is a function of the temperature, salinity, and sediment concentration of the water offshore, the physical properties of water, ice, and sediment, and the ice content of the frozen bluff material (given by equations (29)–(37) in Kobayashi [1985]). The calculation of ζm is straightforward, with the exception of Kobayashi [1985, equation (30)], which we solve using the standard Newton-Raphson numerical method.

Equation (13) can be rearranged and differentiated to yield the instantaneous rate of notch incision as a function of notch width,

display math(15)

Note that in the Kobayashi formulation, the submarine erosion rate, inline image, goes as inline image such that the erosion rate decreases as the notch width increases as the diffusion of heat to the notch face becomes less efficient. For bluff incision, we calculate the notch width at each time step based on the bluff geometry. Determining the notch width for the toppled block is harder. We examined values of 10 cm to 10 m for the block notch width and chose to use a constant notch width of 1 m for a toppled block.

7 Short-Term Coastal Changes: Comparing Observations and Models

The data set collected in summer 2010 allows us to test the model over a short time period. The primary goal of this modeling effort is to identify which of the three formulations of instantaneous submarine erosion correctly capture the observed block degradation.

The model is initialized with a toppled block of the same cross-sectional area as the block observed in the time-lapse imagery. Observations of the environmental conditions (Figures 6a–6d) set the water level and the submarine and subaerial erosion rates. The permafrost temperature is set to a value of −7.5°C based on local USGS observations. Figure 7 shows the model domain for the experiment (animation given in Movie S3).

Figure 6.

(a–d) Observed environmental conditions at Drew Point, Alaska, during time-lapse imagery collection and results of model experiments from August 2010. (e) In the model run, the Russell-Head formulation underpredicts the rate of block degradation. The White formulation correctly predicts, and the Kobayashi formulation overpredicts the rate of block degradation.

Figure 7.

Example model output from the short-term model experiments. Animation available in Movie S3.

The time-lapse observations indicate that the degrading block does little to protect the coast from erosion and thus is not a rate-limiting factor for long-term erosion. This may seem to provide an argument to ignore the degrading block in the model; however, this model component allows us to evaluate the process of erosion over a short time period. The physical process that governs submarine erosion rate should be the same for both the notch at the base of the bluff and the base of the block. Although we have observations of block degradation, making short-term observations of notch incision is difficult. For the short time period, we have a complete data set of environmental conditions and what we feel is an appropriate proxy for erosion rate.

The observed rate of block degradation and the rate predicted by the White formulation show good agreement (Figure 6e). The Russell-Head formulation predicts a much slower rate of block degradation. The Kobayashi formulation is initially similar to the White formulation but predicts the rapid disappearance of the block nearly 5 days too early. A potential source of this discrepancy is a block notch width that is too small. However, we examined model output for the Kobayashi formulation for block notch widths of 10 cm to 10 m. Notch widths of 5 m, which we consider too large result in complete block degradation by 19 August, still earlier than observations.

These results indicate that the Russell-Head formulation, which does not include the influence of the wavefield on the submarine erosion rate, is not sufficient. While the White formulation preforms best, we cannot rule out the most complex Kobayashi formulation. This formulation yields rapid erosion rates at small notch depths, and because this model time period focuses exclusively on block degradation and not notch incision, even though we explored a range of block notch widths, we are not able to rule it out.

8 Reconstructing the Coastal Erosion History for the Period 1979–2011

Having tested our coastal erosion model to a short time period, we apply it over the period 1979–2011, when satellite-derived sea ice concentration observations exist to constrain oceanographic and environmental conditions. The goal of this modeling effort is both to evaluate the performance of the three submarine erosion rate formulations and to determine the sensitivity of the long-term erosion rate to variations in environmental conditions.

8.1 Hindcasts of Environmental Conditions

Here we outline the construction of time series of air and water temperature, water level, and wavefield for the time period 1979–2011. Table 1 lists the parameters and typical values used to construct the environmental conditions. In addition to measurements at Drew Point, we use three observational data sets.

8.1.1 Observational Data Sets

Barrow Meteorology: We employ the hourly wind speed and direction measured at Barrow (located ∼110 km northwest of Drew Point, http://www.esrl.noaa.gov/gmd/obop/brw/) to calculate wind setup and wavefield. This station also records ambient pressure, which is used for storm surge calculations. We adjust the wind speed and direction measured at Barrow for Drew Point using a transfer function optimized for winds greater than 5 m s−1 during sea ice-free conditions ([Overeem et al., 2011] calibrated over June 2008–September 2010, R2 = 0.8). The calculated wind speed, in m s−1, at 10 m height at Drew Point, U10m,DrewPoint, is given as a function of the 10 m wind speed measured at Barrow, U10m,Barrow:

display math(16)

The wind direction at Drew Point was found to correlate best with that at Barrow when the latter was rotated 24° counterclockwise.

We develop a transfer function for positive air temperature from Barrow to Drew Point based on comparison of air temperature measured at Barrow and at the Drew Point UCB station (http://data.usgs.gov/climateMonitoring/station/show/5) over the time period June 2008 to September 2013 (R2 = 0.73). The calculated surface air temperature, in°C at Drew Point, Ta, is given as a function of the 2 m air temperature measured at Barrow, Ta,Barrow :

display math(17)

Barrow Radiation: We use radiation observations at Barrow as input for the water temperature model. This data set includes measurements of upward and downward total and longwave radiation, and direct and diffuse radiation (ftp://ftp.cmdl.noaa.gov/data/radiation/baseline/brw/). Using the Barrow radiation record as a proxy for that at Drew Point is reasonable as both sites are located in the maritime zone of the Beaufort Sea coast and experience a similar climate.

Arctic Sea Ice: The concentration of sea ice is given by the Nimbus7 SMMR/SSM/I and DMSP SSMI Passive Microwave Sea Ice Concentrations (SIC) derived from brightness temperature [Cavalieri et al., 1996, http://nsidc.org/data/nsidc-0051.html]. We employ this data set to locate the sea ice edge and model nearshore water temperatures.

8.1.2 Sea Ice Edge-Based Fetch

The fetch, F, is the length of open water over which wind setup and wave generation occurs. Fetch varies in space and time with the wind field, the geometry of the shoreline, and the location of the sea ice edge. For wind blowing from the open ocean, we set the fetch to the distance to the sea ice edge in the direction from which the wind blows, calculated using the method outlined in Overeem et al. [2011]. The fetch is taken as 1 km for winds blowing offshore (from azimuths ranging from 085° to 200°) because we observe nearshore waves at these times. The fetch for wave generation is limited to 15 km for winds blowing from the direction of the interior of Smith Bay (to the west; 200° to 260°, Figure 1). These parameters were determined based on the geometry of the shoreline in the Drew Point region and on optimization of the correlation between observed and predicted water level and wavefield characteristics during the summer 2010 field season.

8.1.3 Storm Surge and Wavefield Models

We model wind-driven setup using a bathystrophic storm surge model [Dean and Dalrymple, 1991, section 5.9.1]. This model calculates the water level setup or set down, η, as a function of wind speed measured at 10 m height, U; wind direction, θ; fetch, F; and bathymetry, h(x). The wavefield is calculated using the method outlined in the Shore Protection Manual [Coastal Engineering Research Center, 1984, equations (3)–(39) and (3)–(40)]. A full description of the storm surge and wavefield models is available in the supporting information Text S1. Measured and modeled waves from summers 2009 and 2010 (Figure 8) reveal that measurements and modeled water level setup are generally in good agreement. While much scatter exits in the comparison between the observed and modeled wave heights for low values, the modeled and observed wave heights converge at higher values, when most erosion occurs.

Figure 8.

(a) Modeled water level setup and (b) wave height for summers 2009 and 2010. (c and d) Comparison of the measured and modeled values, respectively.

8.1.4 Water Temperature

Water temperature at a point is a function of incoming solar radiation, advection of heat by ocean currents, and transfer of heat from the subsurface and air into the water column. Field observations (Figure 9) reveal highly dynamic nearshore water temperatures. In the early sea ice-free season of 2009, the season in which we collected the longest record of water temperature, the water column is initially thermally stratified. It becomes mixed after a storm event on 9 August 2009, after which water temperatures collected at a variety of depths and distances from the coast are more uniform. This supports the assumption that the nearshore water is well mixed. The deepest sensor at the most nearshore location, Wave Logger 2 Tidbit Lower, shows cooling when winds are from the east and water levels are set down (Figure 9). Winds from the west that set up the water level mix the stratified early-season water column, but do not result in cooling events.

Figure 9.

Measured and modeled water temperatures from summers 2009 and 2010. Wave loggers had Onset Tidbits attached to record temperature in the upper and lower portions of the water column. (top) The modeled top of the atmosphere and station-measured radiation. The difference is due to the effect of clouds.

We model water temperature with a 1-D radiation balance on a column of water. This method captures the seasonal cycle of water temperature variations during the sea ice-free period. As it is difficult to constrain other elements of the water column energy balance and we find that this method is sufficient, we do not add additional complexity to the model (e.g., advection of water). Water temperature, Tw, is calculated using a forward difference method. The rate of change in water temperature with time during the sea ice-free season, dTw/dt, is governed by the radiation balance at the top of a water column:

display math(18)

where Rnet is the radiation balance at the surface,

display math(19)

and hr is a reference water depth.

The net radiation at the water surface is given as the sum of the downwelling and upwelling shortwave radiation, RSW (equation (21)), the downwelling longwave, RLW, and the upwelling longwave radiation, inline image, where σ is the Stefan-Boltzman constant and ϵ is the emissivity of water.

The Arctic is cloudy, particularly in the summer, and clouds influence downwelling longwave radiation. We use the annual cycle of RLW based on day of year averages of the radiation data set from Barrow as model input. While day of year average of radiation observations does not account for interyear variations in the downwelling longwave radiation history, we find that this parameterization captures the seasonal cycle of water temperatures measured in two summers (2009 and 2010).

Incoming shortwave radiation at the top of the atmosphere, RTOA, is calculated after Hock [1999, equation (3)]:

display math(20)

where I0 is the solar constant (Frohlich [1993], 1368 W m−2), R is the instantaneous Sun-Earth distance, Rm is the mean Earth-Sun distance, ψa is the mean atmospheric clear-sky transmissivity, P is the atmospheric pressure, P0 is the mean atmospheric pressure at sea level, ζ is the local zenith angle, and θi is the angle of incidence including the effect of local slope. Summer clear-sky transmissivity at Barrow is set at a constant value of 0.7 [Wendler and Eaton, 1990]. In this formulation (Rm/R)2 corrects for the eccentricity in the Earth's orbit, P/P0 corrects for the effect of altitude and cosζ introduces variation in the path length through the atmosphere throughout the day. For both the North Slope of Alaska and the Beaufort Sea, the local topographic slope is nearly zero.

The top of the atmosphere radiation, RTOA, is modified by the observed sea ice concentration, CSea Ice, and a factor to account for absorption of incoming radiation by clouds, Acloud. This factor incorporates the seasonal cloud fraction, CF [Dong et al., 2010], and average transmission of incoming radiation by clouds, fcloud (Serreze and Bradley [1987], taken as 0.5). This results in the following:

display math(21)
display math(22)

In this simplified model, we consider neither the latent heat necessary to melt sea ice nor the albedo of sea ice. Changes in the sea ice cover, including the latent heat and albedo, are built into the nearshore water temperature model by the time series of sea ice concentration, CSeaIce in equation (21). Albedo of the ocean surface, α, is calculated as a function of the solar incidence angle using the Fresnel equations [e.g., Cogley, 1979, equation (5)].

A reference water thickness, hr, of 3 m captures the observed seasonal amplitude of nearshore water temperature, which appears reasonable considering the shallow shelf near the study site (Figure 9). We note that we should not expect this model to capture the cooling events associated with storms because it lacks a parameterization of the effect of storms on water temperature. While the simplified model for the evolution of water temperature does not capture the day- to several-day-long cooling events (∼3°C), but as these occur during times of water level set down, they should not affect the overall rate of coastal erosion.

8.2 Design of Long-Term Experiments

The constructed environmental conditions serve as inputs for both a base case model run that targets matching the long-term observations of coastal retreat (Figure 10) and five numerical experiments meant to explore the effects of changing environmental conditions on the long-term rate of coastal retreat (Figures 11 and 12). As in the short-term numerical experiments, we consider all three submarine erosion formulations and only change one environmental condition at a time. We present the time series of coast position for each experiment (Figure 11), a sensitivity analysis (Figure 12), and the average erosion rate for each model run (Table 2).

Figure 10.

Results of standard long-term numerical model runs. (a) The location of the coast relative to 1979 is shown for each of the three submarine erosion rate formulations and the observed location of the coast documented by Jones et al. [2009] and the results of our coastal surveys (Figure 2). The Russell-Head formulation underpredicts the coastal erosion rate, the White formulation does better, and the Kobayashi formulation is too fast. (b) Comparison of the coast position relative to 1979 with two metrics for exposure of the base of the bluff to water: the cumulative time that the water is set up to the base of the bluff and the cumulative water level above the base of the bluff. To first order, the duration of time water level is adjacent to the coastal bluffs predicts the amount of erosion.

Figure 11.

Results of long-term numerical experiments. The location of the coast relative to 1979 is shown for each of the three submarine erosion rate formulations and the observed location of the coast documented by Jones et al. [2009] and the results of our coastal surveys (Figure 2). (top) An explanation of each numerical experiment. For clarity, only a selection of results are shown.

Figure 12.

Sensitivity of the average erosion rate to each of the variables explored in the numerical experiments. Each dot represents an individual numerical experiment.

Table 2. Long-Term Numerical Experiment Resultsa
    Average Coastal Erosion Rate (m yr−1)
  1. a

    Bold emphasis is used to distinguish model results from observations.

    1979–20111979–20022002–20072008–2011
  TemperatureMean Water
 Wave FactorOffset (°C)Level Offset (cm)RHWHIKOBRHWHIKOBRHWHIKOBRHWHIKOB
Observations            
        8.7  13.6  15 
No Changes            
 1001.311.435.30.87.021.32.523.579.43.831.190.3
Sea Ice Season            
 constant at 1979 projection 0.63.65.40.52.44.10.66.712.40.99.24.8
Offset to Mean Water Level            
 10−300.93.610.60.72.57.21.07.223.92.67.917.2
 10−251.04.312.90.73.08.71.08.328.22.88.527.3
 10−201.05.015.00.73.49.51.010.134.62.910.831.3
 10−151.06.118.40.73.910.61.012.337.41.716.057.4
 10−101.07.423.10.74.713.92.114.145.51.721.667.9
 10−51.29.027.10.85.615.82.314.456.52.930.682.4
 1051.414.945.30.89.529.83.831.093.22.138.2110.1
 10101.719.357.31.213.239.13.039.8129.33.443.3114.6
 10151.824.669.91.317.649.43.151.4156.81.748.5125.2
 10202.130.988.71.723.767.63.562.6193.84.452.9130.5
 10252.738.9106.01.931.085.05.174.8217.25.460.4147.2
 10302.949.8128.42.441.5106.76.590.7251.91.777.7171.3
Permafrost Temperature            
 constant at −4°C 1.311.937.00.87.722.72.523.584.94.031.991.0
 constant at −7.5°C 1.311.436.30.87.021.92.523.580.73.831.194.2
 linear increase: −11 to −7.5°C 1.311.435.30.87.021.32.523.579.43.831.190.3
 constant at −11°C 1.311.435.00.87.021.92.523.576.93.830.986.7
Water Temperature            
 1−500.53.25.90.41.93.40.57.216.50.99.111.0
 1−400.54.510.70.52.46.10.612.228.81.111.022.5
 1−300.66.114.00.63.49.30.714.235.51.316.523.6
 1−200.97.820.10.64.611.70.916.644.03.322.156.9
 1−101.09.727.20.75.716.22.420.760.51.627.772.6
 1101.414.144.90.88.728.12.428.288.22.239.7130.3
 1201.616.554.60.810.635.23.932.2114.34.445.4137.4
 1301.719.268.21.212.144.43.136.9136.03.455.4181.7
 1401.721.683.51.314.354.83.039.1159.33.661.6230.5
 1502.124.392.61.315.865.14.743.8155.54.969.5253.2
Wave Height Magnitude            
 0.25000.91.42.50.70.92.01.03.52.42.32.15.2
 0.5000.92.55.70.71.93.91.05.313.32.72.99.1
 2003.6110.8371.13.498.3356.15.9179.9735.03.6151.0102.7

Five numerical experiments explore the effect of changing (1) the length of the sea ice-free season, (2) the mean water level, (3) the wave height magnitude, (4) the permafrost temperature, and (5) the water temperature (Figure 11). The experiments on the length of the sea ice-free season, the mean water level, and the wave height change the prerequisites for erosion. The wave height, water temperature, and permafrost temperature influence the rate of submarine erosion once the prerequisites are met (Figure 4).

Water temperature and wave height magnitude are varied in the same manner as in the short-term modeling exercise, with the exception of the 4X wave height magnitude experiment which yields extremely high erosion rates and was omitted. A short sea ice-free-season run uses the 1979 values of the trends in the first and last day of open water calculated by Overeem et al. [2011]. Permafrost temperature in the standard model run linearly increases from −11.1°C in 1979 to −7.5°C in 2011. We examine the effect of constant permafrost temperature at −11.1°C, −7.5°C, and −4°C. In the water level experiments, the entire water level record is offset by a constant value.

In this environment, determining the magnitude of inline image, the mean water level relative to the base of the bluffs, is difficult. The water level changes due to tides on a daily and yearly basis, with a seasonal component of the tide that is high in the summer. Shore-fast sea ice is present for much of the year, and during the sea ice-free season winds from the east commonly set down the water level. Here the mean water level is set at 45 cm below the base of the coastal buffs (inline image= −45 cm) based on both field observations and matching observed and modeled long-term erosion rates for the best performing White formulation. In order to match the observed ∼300 m of erosion over the time period 1979–2011 using the Russell-Head method, we must set the water level to 0.5 m above the base of the bluffs at all times open water is present, whereas in order to match observed erosion with the Kobayashi method, we must lower the mean water level an additional 25 cm (to 70 cm below the base of the bluffs). We think neither of these is reasonable.

8.3 Results of Long-Term Model Runs

As in the short-term model experiment, the coastal erosion rate predicted by the Russell-Head formulation is much slower than both the observed rate of coastal erosion and the rates predicted by the other two formulations, whereas the coastal erosion rates predicted by the White formulation is reasonable and the rates predicted by the Kobayashi formulation are too high (Figure 10a). The White formulation underpredicts the rate of coastal erosion in the period before 2002 and overpredicts from 2002 onward.

The rate of long-term coastal erosion closely follows both the cumulative time that water is set up to the base of the bluffs and the cumulative water exposure, defined as the cumulative sum of water levels above the base of the bluff (Figure 10b). We note that both of these metrics for exposure to nearshore water display a similar history which indicates that while the amount of time water levels exceed the base of the bluff varies through time, the average positive water level does not change over the model time period.

The coastal erosion history predicted by all three of the submarine erosion rule formulations is episodic; much of the change in coast position occurs during short time intervals. This is consistent with the field observation that little erosion occurs until the water level contacts the base of the coastal bluffs, at which point erosion becomes rapid. This episodic nature is built into the model—submarine erosion only occurs when water contacts the base of the bluffs. The length of the sea ice-free season greatly affects the predicted rates. In the first experiment, we compare the standard model run with a shortened sea ice-free season run. The long-term erosion rate for the shortened sea ice-free season is drastically slower than the standard run.

The relationship between the erosion rates and both the mean water level and wave height is nonlinear, indicating that the rate of coastal erosion is especially sensitive to the mean water level. These results differ from those of Ravens et al. [2012] in which they found a nearly linear relationship between the beach elevation and the predicted rate of coastal erosion. Both the water level and wave height control the amount of time water interacts with the coastal bluffs, but in the White and Kobayashi formulations, the wave height also influences the instantaneous rate of submarine erosion. This explains the high sensitivity of the long-term erosion rate to this environmental condition.

The average rates of coastal erosion are sensitive to any offset in water temperature. The relationship between the long-term erosion rates predicted by the White formulation and the water temperature is nearly linear, whereas that of the Kobayashi formulation is nonlinear. The long-term rates for the White formulation track the linear dependence of the submarine erosion rate on the thermal driving parameter. The relationship between temperature and instantaneous erosion rate in the Kobayashi formulation is incorporated into ζm which goes as the thermal driving parameter to the 0.93 power. Thus, the Kobayasi formulation is more sensitive to the water temperature than we would expect based only on the equation for instantaneous submarine erosion rate.

The modeled seasonal water temperature shows warming through time. This reflects a longer sea ice-free season and thus a longer time over which nearshore water absorbs solar radiation. As the rate of submarine notch incision is sensitive to water temperature, this environmental condition is partially responsible for the acceleration in modeled coast erosion rate through time. That the model results under-predict coastal erosion rates for the early portion of the model time period and overpredict in the later portion of the record is discussed below.

Varying the permafrost temperature history has little effect on the predicted rate of coastal retreat. The predicted rate only increases slightly when the permafrost temperature is set to −4°C, a temperature that is warmer than the present-day Drew Point permafrost temperatures. We conclude that the erosion rate is not sensitive to the permafrost temperature because the heat required to increase the temperature of the bluff material by a few degrees is small compared with the amount of heat required to melt the interstitial ice.

9 Discussion

We set out two goals in section 1. First, to identify the simplest relationship between environmental conditions and the short-term rate of submarine notch incision, and second, to determine the sensitivity of long-term rates of coast retreat to environmental conditions. We also discuss implications for predicting coastal erosion and the impact of episodicity on the rate of coastal erosion in this environment.

9.1 Submarine Erosion Rate Formulations

The short-term model experiments indicate that the wavefield must be considered in order to predict accurately the rate of block degradation. The Russell-Head formulation, which only considers the temperature difference between the nearshore water and permafrost does not include this factor, and underpredicts the rate of block degradation. The White formulation, which does incorporate the influence of the wavefield on the rate of submarine erosion, matches the block degradation history well. We therefore suggest that the Russell-Head parameterization for submarine erosion is too simple, and that characteristics of the wavefield must be considered to predict the correct rate of notch incision.

The long-term coastal erosion model results also indicate that the formulations for submarine erosion that include the wavefield capture the essence of the long-term erosion history. The model results that employ the White formulation capture the correct erosion rates when averaged over the entire model time period. The rates predicted by the Kobayashi formulation, however, are much faster. This reflects the dependence of submarine erosion rate on notch width in the Kobayashi formulation, which enhances the submarine erosion rate for small notch widths and results in very rapid initial notch incision. While the Kobayashi formulation may do well over a sustained single storm event in which a submarine notch is initiated and propagates to a depth resulting in bluff failure, we find that it is not successful when applied with a time series of water level wave heights. Instead, we advocate for use of the White formulation, which incorporates both the wavefield and water temperature into the instantaneous rate of submarine erosion and preforms well over the long-term model time period.

9.2 Sensitivity of Long-Term Coastal Retreat to Environmental Conditions

The increase in the duration of the sea ice-free season at Drew Point over the satellite record has been roughly twofold, and to first order the change in coastal erosion rate over 1979–2009 tracks this expansion [Overeem et al., 2011]. The rate of long-term coastal erosion follows both the cumulative time that water is set up to the base of the bluffs and the cumulative water exposure (Figure 10b). However, this relationship is not linear, implying that other environmental conditions are important for the long-term rate of coastal retreat. Extending the sea ice-free season may result in other feedbacks that enhance the rate of coast retreat. One such feedback is the expansion of the sea ice-free season earlier into the summer when insolation is closer to its peak, promoting warmer sea surface temperatures.

The White formulation underpredicts the long-term erosion rates before 2002 and overpredicts rates after 2002 (Figure 10 and Table 2). For example, the White formulation predicts a rate of 23.5 m yr−1 for 2002–2007, whereas measurements indicate that the erosion rate for this period is 13.6 m yr−1. We suggest two main reasons for this discrepancy: (1) we may be incorrectly hindcasting water levels and wave heights and thus the amount of time that the base of the bluff is able to erode, and (2) we may be incorrectly hindcasting other variables that influence the submarine erosion rate when the water level is sufficiently elevated. The sea ice history is well documented over the model time period. However, small errors in the reconstructed water level record are likely to be greatly amplified in the calculated erosion history, particularly when employing the sensitive Kobayashi formulation. Although our algorithm for wind-driven setup and storm surge is based on a well-established model, it may be too simple to capture the details of the erosion record in this environment. Although the model results do not capture the full detail of the erosion rate history, we believe that they are sufficiently faithful to reality to interpret the results of the sensitivity analysis.

The length of the sea ice season, the mean water level, and the wave height, which all control the length of time over which erosion occurs, all profoundly impact the long-term rate of coastal erosion. The system is most sensitive to wave height, which influences both when water contacts the coastal bluffs and the rate of submarine notch incision.

Not surprisingly, of the environmental conditions that influence the rate of submarine erosion but not the timing of erosion, water temperature has the greatest effect. As with the mean water level, erosion rates increase with changes in water temperature nonlinearly with a power greater than one (Figures 11d and 11c). The mismatch between the measured and the modeled erosion history may therefore also reflect errors in our hindcast of the nearshore water temperature history. To first order, the seasonal cycle of nearshore water temperature can be modeled with a radiation balance of the sort we have employed (Figure 9). However, other details of the oceanic and atmospheric system that we have not explicitly modeled, such as the location of the sea ice edge, the passage of storms, and the connection between deep ocean circulation and the horizontal circulation across the broad shelf, may play an important role in governing the nearshore water temperature history and thus the long-term erosion history.

As the length of the sea ice-free season extends, nearshore water temperatures is likely to rise. However, the impact of the water temperature should not be considered in isolation, as water level in excess of the base of the bluffs is a prerequisite for erosion. If storminess increases during peak summer insolation and associated high water temperatures, coastal erosion rates may increase yet more rapidly.

These results strongly suggest that any environmental change that influences the amount of time the water contacts the base of the coastal bluffs will influence the rate of coastal erosion. An increase in the length of the sea ice-free season, sea level rise, increases in the frequency of storms, or changes to the morphology of the Beaufort Sea Shelf (which influence the magnitude of wind-driven setup and set down) all will influence the amount of time water contacts the base of the coastal bluffs.

9.3 Implications for Predicting Future Rates of Coastal Erosion

We show that the long-term rate of coastal erosion in regions with ice-rich permafrost bluffs is controlled by episodic events during which rapid coastal erosion occurs. The exposure of the base of the coastal bluffs to water exerts first-order control on the long-term rate of coastal retreat (Figure 11b). The frequency of events that set up water is controlled by both the duration of the sea ice-free season and the seasonal passage of synoptic scale storms. However, we have also shown that the details of the nearshore water temperature and wavefield play an important role in governing the rate of instantaneous coastal erosion, and thus the total amount of erosion in a single event.

Detailed knowledge of all of the environmental conditions is required to predict the magnitude of coastal erosion during an event. Although reconstructing all necessary input parameters may be feasible for the past, extrapolating into the future is much less certain, particularly in an environment in which a small set of short duration events so greatly impacts the longer term rate.

9.4 Episodicity

The time-lapse video observations indicate retreat occurs during relatively short lived discrete events in which the water level is at the base of the coastal bluffs. Using the results from the long-term model run for the preferred White formulation, we estimate that nearly 60% of the coastal erosion in the 1979–2011 period occurred during hourly time steps in which less than 0.5 m of total notch incision occurs (Figure 13a). These time steps, however, represent 95% of the total number of time steps in which erosion occurred. The remaining erosion occurred in a small number of large events (most with greater than 3 m of notch incision within a single time step).

Figure 13.

Episodicity of erosion events. (a) Proportion of total erosion and number of events as a function of erosion threshold for the White formulation. The timing within each year of the onset of (b) small and (c) large events is similar, but small events extend later into the fall.

Examination of the environmental conditions during erosion events reveals that the biggest difference between the environmental conditions during high erosion events is higher water temperature, particularly in the later time periods. None of the large events occurred with water temperatures between −2°C and 0°C. The timing of onset of large (>2.5 m of notch incision) and small events is similar (<0.5 m) (Figures 13b and 13c). The last date of most large and small events, however, occurs at different times of the year with small events ending later than large events (27 September instead of 12 September). Although there is not a large difference between these date ranges, the larger events occur closer to the peak in summertime insolation and associated warm water temperatures. We note that while the window for large erosion events ends in September, ice-free conditions persist into October. Over the historic record, the fall season was not an effective time for erosion as sea surface temperature has cooled by then. The opposing effects of increased storminess and cooler water temperatures make the fall a more difficult time period to predict future erosion.

10 Conclusions

Application of a numerical model of the erosion of ice-rich permafrost coastal bluffs on both short-term (∼1 week) and long-term (∼30 years) time periods indicates that the length of sea ice-free season, water exposure, and water temperature exert the greatest control over the rate of submarine notch incision and long-term coastal erosion. In this environment, coastal erosion is a highly episodic process. The nearshore water level, primarily controlled by set up and waves from the passage of storms, serves as a threshold for notch incision because erosion rates are negligible unless water levels reach the base of the coastal bluff. The nearshore water temperature exerts control on the rate of notch incision as it controls the melt rate of the interstitial ice that holds bluff sediments together. Very little coastal erosion occurs in the fall when water temperatures are cooler. These factors complicate the prediction of the response of ice-rich permafrost coastal bluffs to the future Arctic.

However, as most projections for the future Arctic include increased sea levels, warmer ocean waters, and lengthened sea ice-free seasons, it is best to anticipate that coastal erosion rates will continue to increase.

Acknowledgments

The authors acknowledge the U.S. Geological Survey and Bureau of Land Management for helicopter support, and thank Shane Walker for support. CPS Polarfield helped organize field logistics in 2008 and 2009. Adam LeWinter was critical to the time-lapse camera deployment in 2010. We thank Tim Stanton for the use of his wave loggers. Gene Ellis helped complete the coast position surveys. Andy Wickert extracted the relative sea level change rates and provided helpful discussion. We also thank our bear guards from 2008 and 2009, Robert Brower and Eben Brower from Barrow, for their support and companionship during the fieldwork. Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government. The authors kindly thank JGR Assistant Editor Andrew Ashton, Aart Kroon, two anonymous reviewers, and Gene Ellis of the U.S. Geological Survey for extensive comments that significantly improved the quality and clarity of the manuscript. This research was funded by ONR-1547946, ONR-1544301, NSF/OPP-1549620 to R.S.A., I.O., and C.W. K.R.B. is supported by a NASA Earth and Space Science Fellowship (award 1549613) and thanks the Alaska Geological Society for their support. We acknowledge computing time on the CU-CSDMS High-Performance Computing Cluster. The oceanographic data presented here is available through the CSDMS data repository (http://csdms.colorado.edu/wiki/Data_portal).

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