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Keywords:

  • permafrost;
  • thermokarst;
  • thaw lakes

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Results and Discussion
  6. 4. Conclusions
  7. Acknowledgments
  8. References

[1] Thaw lakes, widespread in permafrost lowlands, expand their basins by conduction of heat from warm lake water into adjacent permafrost, subsidence of icy permafrost on thawing, and movement of thawed sediment from lake margins into basins by diffusive and advective mass wasting. We describe a cross-sectional numerical model with thermal processes and mass wasting. To test the model and provide an initial investigation of its utility, the model is driven using historical daily temperatures and permafrost conditions for the northern Seward Peninsula, Alaska (NSP; thick syngenetic ice, mean annual air temperature (MAAT) −6°C) and Yukon coastal plain (YCP; thin epigenetic ice, MAAT −10°C). In the model, lakes develop dynamic equilibrium profiles that are independent of initial morphology. These profiles migrate outward episodically and match the morphology of profiles from lakes that were measured at each site. Modeled NSP lakes expand more rapidly than YCP lakes (0.26 versus 0.10 m a−1) under respective modern climates. When identical climates are imposed, NSP lakes still grow more rapidly because their deeper basins and steeper bathymetric slopes move thawed insulating sediment away from the lake margin. In sensitivity tests, an increase of 3°C in MAAT causes 2.5× (NSP) and 1.6× (YCP) faster expansion of lakes. An 8°C decrease essentially halts expansion for both sites, consistent with paleostudies which attribute basins to postglacial warming. In the model, basins expand monotonically but lakes do not. The 1σ interannual variability of lake expansion is 0.51 (NSP) and 0.44 m a−1 (YCP), with single year rates of up to ±8 m occurring because of instabilities from thermal/mass movement coupling even under a stationary climate. This variability is likely a minimum estimate, compared to natural variability, and suggests that long measurement time series, of basins not lake surfaces, would best detect thermokarst acceleration resulting from a climate change.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Results and Discussion
  6. 4. Conclusions
  7. Acknowledgments
  8. References

[2] Thaw lakes are dynamic features of high-latitude lowlands underlain by ice-rich permafrost. The lakes may expand radially and downward by coupled thermal processes (heat conduction and thawing of permafrost) and geomorphic processes (mass wasting, such as slumps, on the thawing margins of basins) [Hopkins, 1949; French and Egginton, 1973; Burn and Smith, 1988; Hopkins and Kidd, 1988; Rampton, 1988; Murton, 1996, 2001]. Active thaw lakes presently cover up to 40% of lowland regions in northwestern Canada [Mackay, 1997; Burn, 2002] (principally the Tuktoyaktuk lowlands and Mackenzie Delta), northern and western Alaska [Hopkins et al., 1955; Sellman et al., 1975; Hinkel et al., 2005], and one million km2 of Siberia [Tomirdiaro, 1982; Zimov et al., 1997]. Relic drained basins formed by partial or complete drainage are even more widespread, and sometimes are superimposed into palimpsests because the basin floors may refreeze and reaccumulate excess ground ice, starting another cycle of thaw lake development [Hopkins, 1949; French, 1975; Hunter et al., 1981; Mackay, 1981; Brewer et al., 1993; Marsh and Neumann, 2001; Hinkel et al., 2003]. Thaw lakes and other forms of permafrost melting are normal processes that result from a variety of drivers including a wide range of surface disturbances. Although climate warming is not an exclusive driver of thaw lake initiation and expansion, recent attention has been focused on how the extent and expansion rate of lakes may be sensitive to modern and anticipated climate change over coming decades. Notably, the lakes might act as a globally significant positive feedback to high-latitude warming, possibly ∼5°C by the end of the 21st century (the mean increase in mean annual surface temperatures across model ensemble MMD-A1B [Christensen et al., 2007]). Faster thawing and expansion might accelerate the anaerobic decomposition in lakes of the Pleistocene age organic material sequestered in some permafrost, and hence increase the rate of release of CH4 to the atmosphere [Zimov et al., 1997; Phelps et al., 1998; Walter et al., 2006]. Individual lakes and the total areal coverage by lakes may have grown in continuous permafrost regions of N. America and Siberia during recent decades [Yoshikawa and Hinzman, 2003; Smith et al., 2005; Walter et al., 2006], evidence that lakes might be sensitive to climate over decadal time scales as well as to interannual variations in precipitation [Plug et al., 2008].

[3] There is therefore a clear need for predictive models of thaw lake dynamics. However, most existing numerical and analytical models for thaw lakes treat their morphology as unchanging and, under this assumption, have focused solely on the affect of a lake on ground temperature. Finite element models of heat conduction and phase change in frozen ground have been used to examine thaw bulb (talik) dynamics and their dependence on lake water temperature [Ling and Zhang, 2003; Zhou and Huang, 2004]. Hydrodynamic calculations [Rex, 1961] have been used to explain the preferred orientation of lakes observed in some regions [Hinkel et al., 2005] but erosion is not treated. Another approach investigates lake orientation by combining ground temperature change with a migrating shoreline in cross section, using a wave equation to approximate bank retreat [Pelletier, 2005]. Although a dependence of shoreline retreat on bank height is found, this approach has not been tested against lake bathymetry and sensitivity to climate is unexplored. Dynamic morphology has been partly treated by coupling a numerical model for heat transfer and phase change with thaw subsidence (in our earlier paper [West and Plug, 2008]). In that model, however, bank retreat is imposed as an external condition, not a dynamical variable, and there are no geomorphic processes of slumping and sediment redistribution. The consequences of coupled geomorphic and thermal processes therefore could not be explored. One approach to a fully coupled model would be to combine thermal models with models of mass movement on hill slopes. The latter are widely used to explore landscape change across many substrates and scales, including linear and nonlinear diffusion models for deg3radation of scarps [Hanks et al., 1984], moraines [Hallet and Putkonen, 1994], badlands [Howard, 1997] and the response of hillslopes to tectonic and climatic forcing [e.g., Roering et al., 2001]. Although some of these other geomorphic systems are similar in scale to the ≈1–50 m relief of thaw lake basins, an adapted version of these models would need to treat processes unique to thaw lake landscapes, such as thaw subsidence, advective mass wasting both above and below the waterline, and rates of bluff retreat of 0–10 m a−1 [French and Egginton, 1973; Burn and Smith, 1988; Rampton, 1988; Zimov et al., 1997; Walter et al., 2006; West and Plug, 2008], much more rapid than for most hillslopes.

[4] In this paper we (1) describe a two-dimensional numerical model of thaw lake expansion in cross section which combines hillslope processes with heat conduction and thaw-driven subsidence; (2) compare dynamics and morphology of modeled lakes against available measurements from the northern Seward Peninsula (NSP), Alaska, and the Yukon Coastal Plain (YCP), northwestern Canada, regions with thick and thin ice-rich layers respectively (“thick” and “thin”, as used here, are synonymous with “deep” and “shallow” as in the work by West and Plug [2008]); (3) conduct sensitivity tests using colder-than-present climate (8°C below current MAAT) to test if the model is consistent with paleoenvironmental reconstructions that attribute paleobasins in NW North America solely to early Holocene warming, not Pleistocene glacial periods; and (4) use a warmer-than-present climate scenario (MAAT +3°C) to provide an initial probing of the response of thaw lakes to the warming predicted for thaw lake regions over the next 100 years.

2. Model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Results and Discussion
  6. 4. Conclusions
  7. Acknowledgments
  8. References

2.1. Aims and Limitations

[5] The construction of any model for a complicated natural system requires selection of relevant variables and processes, guided by intended use and time scale. This has relevance to our model, and its limitations, in several ways.

[6] First, because our aim is to address the fundamental behaviors of thaw lake expansion in general, we have included classes of processes which are common to all thaw lakes: heat transfer and thawing in permafrost, thaw-driven subsidence, and mass movement. Other geomorphic processes have been described to influence particular and/or an unknown number of lakes, including erosion by the expansion and rafting of lake ice (“ice push”) and lake shallowing by the growth of ice wedge polygons beneath already shallow lakes [Mackay, 1963]. Such processes are intentionally not included because they are not universal; hence, as with most models other than those that are highly “tuned” to match a particular natural instance of a system, our results are intended to be applicable to the average behavior of a large number of thaw lakes, rather than one in particular. Also, because our model is two-dimensional we do not attempt to include long-shore transport of sediment by water, which has been hypothesized to influence sediment budgets on some lake margins where there is a strongly dominant wind direction [Rex, 1961; Hinkel et al., 2005].

[7] Second, substrates around natural lakes can be highly heterogeneous, especially in terms of ice content. Unlike most spatially extended geomorphic models, we do explicitly include, and test the sensitivity to, spatially heterogenous material properties (sections 2.7.2 and 3.3). However, our treatment of heterogeneity is at a spatial scale of ≈0.5 m, the scale of discretization in the model (ie. variability is imposed by adding normally distributed random perturbations to ice content in individual cells). In nature, ice wedges and massive buried ice bodies [e.g., Mackay, 1971] may cause variability over 10s to 1000s of meters (the scale of lakes), which can cause neighboring lakes to exhibit variable dynamics and morphology.

[8] Third, the construction of a mass movement submodel requires the representation of processes over appropriate scales of space and time. On the one hand, a reductionist approach which uses mechanistic parameterization of processes might be attempted. For example, detailed calculations [e.g., Chandler, 1972; McRoberts and Morgenstern, 1974] and measurements [Harris and Lewkowicz, 2000] of pore fluid pressure and effective stress on thawing slopes have been used to predict when and where a particular planar slope is unstable to shear failure. A reductionist model of the evolution of subaerial and subaqueous thaw lake slopes over years to centuries is unwieldy, however, because it would require simultaneous treatment of processes of toppling, skin flows, mud flows, active layer detachment slides, slumps, and underlying processes including pore fluid migration, thaw consolidation and growth of binding roots. On the other hand, highly generalized methods are used by most landscape evolution models, such as linear diffusion, to simulate slope evolution by multiple processes over thousands of years or longer [e.g., Hanks et al., 1984; Hallet and Putkonen, 1994; Kooi and Beaumont, 1996; Howard, 1997; Dietrich et al., 2003]. These approaches may reproduce the essential character of a landscape over millennial time scales [Martin and Church, 1997], but the time scale of interest for thaw lakes is shorter. Because of coupling between mass movement and thawing, infrequent fast events may have a significant effect which is difficult to capture by a long-term-averaging approach. Hence, our approach is a middle ground between short-time-scale mechanistic models and landscape evolution models. Our model is not an attempt to reproduce a particular slope process (for example, fast retrogressive thaw slumps [Burn, 2000]) in detail, although geomorphic phenomena of this type do emerge in the model. We reserve discussion of the model's “geomorphic” behavior to section 3.1.

2.2. Overview

[9] The model treats a single thaw lake in cross section, using a lattice of square cells which may be air, lake water, lake ice or ground. Ground cells have properties of phase and volumetric ice content. The heat capacity and thermal conductivity of ground cells vary with phase and ice content. The upper boundary is air with time-varying temperature driven by daily averages from historical climate data. The lower boundary is deep in the ground, with a geothermal heat flux applied. Vertical boundaries are no-flux (heat and mass) owing to symmetry at the right boundary and a modeled infinite surface at the left boundary. The initial condition is a small half lake symmetric across the right boundary, chosen because our goal is to model expansion, not initiation of lakes. The temperature of lake water and maximum thickness of winter lake ice depend on climate. The initial volumetric ice content of each ground cell is set according to a simulation-specific, depth-dependent relation plus a superimposed heterogeneity, the latter representing the spatial variability arising in nature from ice wedges, lenses and veins. The nonice component of ground is spatially uniform, unconsolidated soil. In time steps of 1 day, the morphology evolves first by thaw subsidence, followed by slope failure and redeposition of sediment, though we have found that results do not depend on the order used. Ground temperatures are updated daily by conductive heat transfer as morphology and surface temperature change.

2.3. Mass Movement

[10] For natural thaw lakes, momentum-dominated processes including small debris flows and skin flows redistribute sediment from bluffs to the lake margin and into the basin [Hopkins, 1949; Hopkins and Kidd, 1988; McRoberts and Morgenstern, 1974]. Active layer detachment slides and retrogressive thaw slumps on low-angle slopes may cause long-range (1–100 m) transport of material to the lake shore [Rampton, 1982; Murton, 2001]. These have been described for other periglacial slopes, with headwalls migrating up to 10 m a−1 [Harris and Lewkowicz, 2000; Burn, 2000]. Intact blocks of peat and frozen soil may raft on the lake surface until thawed and waterlogged, then settle on the basin floor. The lake floor can be smoothed by wave- and gravity-driven disturbances [Hopkins, 1949; Hopkins and Kidd, 1988; Murton, 1996]. Turbidity currents, initiated by subaerial slope failures or below the water line, can cause long-range subaqueous transport [Hopkins and Kidd, 1988]. Superimposed on the episodic but fast advective long-range processes is creep by freeze-thaw cycles, shrink-swell, and faunal disturbances, processes treated as linear diffusion in a broad range of climates and substrates [e.g., Dietrich et al., 2003].

[11] To capture the net transport of sediment from the broad range of advective and diffusive processes, we seek a minimal model which (1) exhibits accelerated erosion as slopes approach a critical threshold gradient; (2) transports eroded material distances ranging from centimeters (creep) to meters (debris and mud flows) to hundreds of meters (turbidity currents and rafting) per year, weighted to the former but with increasing probability of long-range transport from higher and steeper lake bluffs; and (3) deposits sediment primarily in topographic lows nearest the erosion site but, because of rafting events and flows with significant momentum, also occasionally in more distant depressions.

[12] The time-averaged cumulative effect of diffusive slope transport (hillslope creep) and advective mass movement has been treated using a nonlinear sediment transport relation [Howard, 1998; Roering et al., 2001]

  • equation image

where z is elevation above an arbitrary datum (L), Sc is the critical slope angle of the ground material (which may be identified as the angle of internal friction), and K a rate constant (L2T−1). This produces near-pure creep behavior where ∣∇z∣ ≪ Sc (linear diffusion: equation image = Kz), and rapid increase in sediment flux with increasing ∣∇z∣, becoming infinite where ∣∇z∣ = Sc, to effectively capture small slope failures that do not travel long distances downslope [Roering et al., 1999, 2001]. One limitation of this approach, important for our purposes, is that sediment released from a hillslope must fill the nearest downslope depression before deposition in more distant depressions can occur. It does not treat advective movements such as toppling, rolling, and rafting in water which can transport material over one or more topographic depressions.

[13] In our discrete model, sediment flux from a position i on the ground surface has a similar nonlinear dependence on local slope Si, treated by setting the probability of slope failure PE,i to

  • equation image

For frozen material, the critical slope angle Sc is set to 90° because ice-cemented permafrost can maintain vertical faces for short periods, as shown by the near-vertical angles of retreating permafrost cliffs along the eroding coast of the NSP field site, and the headwalls of some slumps [Burn, 2000]. Sc is set to 45° for unfrozen sediment, because this value is intermediate between estimates of 31° to 50° from calibration of nonlinear transport models for unfrozen, unconsolidated soil against topographic measurements [Howard, 1997; Roering et al., 1999]. In surveys of the subaerial bluffs around active thaw lakes in the YCP and NSP, the mean slope of surveyed subaerial margins (measured from water line to bluff crest) was ≈33° and the maximum slope angle, over 1 m long distances, was 43° [Plug, 2003b] (section 3.2). Sc must be significantly greater than the mean gradient and ≥ the steepest gradient. In clay-rich muds, headwalls of recently stabilized thaw slumps may have slopes as low as 12.5° [Burn, 2000], suggesting that 45° is an overestimate of Sc for clay-rich soils during a short period following thaw (because their slow drainage causes positive pore water pressure and hence liquification). Fast retrogressive thaw slumping that can occur in thawing clay-rich soils [e.g., Burn, 2000; Harris and Lewkowicz, 2000] is probably insignificant or absent in most thaw lakes (section 3.1). A modification of the model to extend its application to this process is considered in section 3.6.

[14] As with K in (1), CE in (2) acts as a rate constant encapsulating substrate properties which determine shear strength (pore pressure, angle of internal friction ϕ, and density ρ), with weaker soil having higher CE. In general, barring pure clays where cohesion is important, the shear strength of a potential failure plane at depth h beneath a fully submerged soil column is

  • equation image

where θ is slope angle. The reduction in shear strength by pore pressure is proportional to the density contrast of dry soil solids, ρb, and water. In our model, the effect of pore pressure on slope stability is treated by making CE proportional to the density contrast of soil solids and water. For an unfrozen soil with porosity 30% and a solid component that is 80% mineral (2700 kg m−3) and 20% organic (400 kg m−3) (section 2.7.2), for example, CE is ∼40% greater below the water line than above. Subaqueous slopes relax more quickly than subaerial slopes, all other factors being equal. The rate of modeled bank retreat in our model generally increases with CE because more sediment detaches in a time step. However, interactions between erosion, thawing, and insulation by sediment accumulation at the base of lake bluffs preclude a straightforward dependence of bank retreat rate on CE.

[15] Once detached by slope failure, eroded material moves downslope from i until deposited at position k with a probability of %

  • equation image

where Δxi,k and Δyi,k are the horizontal and vertical distances between i and k respectively. The exponent n sets sediment transport distance, possibly expressing the rate of dissipation of flow energy as a function of transit distance.

[16] As with CE, n might, arguably, vary for transport through air versus water. The higher viscosity of water would increase dissipative losses and so might shorten travel distances. However, the greater density of water can maintain sediment in suspension, particularly silt-sized grains common in thaw lake terrain, and allow rafting of peat blocks into the basin. Both would counteract dissipative losses in other mass movement types. In the absence of a quantitative argument for varying n between subaqueous and subaerial settings, we use constant values. This assumption may be tested by comparing modeled to measured lake basin morphology: if subaqueous transport distances in natural lakes are significantly longer than subaerial transport, we expect that our use of constant values would produce spurious build-ups of sediment at the water line of modeled lake basins. The selection of n and CE are considered in section 2.7.3 and sensitivity of results to these parameters in section 3.3.

[17] For round natural lakes, sediment transported from a unit length along the margin bluff converges with proximity to a lake's center. To treat this three-dimensional effect in our cross-sectional model, the deposition rate at position k in the basin is

  • equation image

where Dkt is the rate of deposition at k in the cross-sectional mass movement model during the time step, rlt is the lake radius, and rk is the distance of k from the lake center.

2.4. Thawing and Subsidence

[18] Temperatures are updated as morphology evolves using a time-marching, Dufort-Frankel finite difference scheme to solve the two-dimensional heat equation [West and Plug, 2008; L. N. Trefethen, Finite difference and spectral methods for ordinary and partial differential equations, 1996, available at http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/pdetext.html]. The effect of latent heat on phase change is addressed by reducing thermal diffusivity across a -1 to 0°C envelope appropriate for silty soil [Comini et al., 1974; Ling and Zhang, 2003]. Ground cells have a thermal diffusivity α = k/C. k is thermal conductivity and C is volumetric heat capacity, which depends on the cell's phase (frozen or thawed) and ice or water content. In nature, thawing permafrost with a volumetric ice content in excess of its unfrozen porosity subsides to partly or fully recover its native porosity, owing to the phase transition and to drainage of excess water (the latter may be slow or partial). In the model, permafrost cells are removed when passing through the phase change envelope, with a probability that depends on ice volume. For example, a permafrost cell with 20% excess ice (in excess of native porosity) has a 21.8% chance of removal from the simulation (the 20% ice content, plus 0.09 of the 20% to account for the conversion of ice to water). Overlying sediment, if there is any, is shifted downward as cells beneath compact from thaw subsidence. Thaw subsidence occurs at the phase change interface of the talik and at the lake margin. Subsidence in the model is instantaneous on thawing, an appropriate treatment for the slow 1–2 cm a−1 rate of thaw front advance in the talik for lakes ≥100 years old (e.g., section 3.1) [Ling and Zhang, 2003]. Before this, and during annual thawing in the active layer, the more rapid advance of the thaw front may exceed the rate at which excess water can migrate away from the thaw front through low-permeability sediment. Under these conditions, thaw subsidence would occur over time after thawing, and may have greater spatial variability than is modeled owing to pore water escape processes of liquifaction and fluidization that rearrange sediment. Most water escape structures are submeter in scale [Lowe, 1975], so any morphology they generate is likely below the scale of thaw lake basins relevant here.

2.5. Lake Ice

[19] In measurements and thermodynamic models for lake ice in shallow subarctic lakes, ice thickness increases monotonically and approximately linearly over the course of a winter (with the most rapid thickening in early winter), reaching a maximum at winter's end that is followed by rapid thawing [Duguay et al., 2003]. In our model, lake ice forms when mean daily temperature is <−1°C and linearly increases to a maximum value reached at the close of the winter season. Lake ice is removed when the daily surface temperature exceeds 0°C. This might underestimate ice cover duration for centers of natural lakes, where ice can persist 30–60 days into the thaw season, but not for the shore lines relevant to our model where ice thins and disappears first [Duguay et al., 2003]. This method for treating ice is not intended to be a comprehensive model of lake ice, but instead allows a reasonable approximation for the time when the lake margin is frozen at the water line. The model is insensitive to details of the rate of ice thickening because the section of the profile that is affected is minor (<1 m) and has a consistently low slope angle hence is rarely a site for initiation of mass movements. In a simulation in which ice thickness increased with equation image during winter, the difference in mean retreat rate of the lake bluff, compared to the reference model, was within the interrealization variability of the model.

2.6. Initial Morphology and Ground Ice

[20] The simulation begins with a small half basin of radius ∼34 m, depth 4 m, and basin slope 40° in a 300 × 300 m lattice. The basin floor is overall flat but with ±0.3 m cells-to-cell irregularity. These dimensions were chosen because they are appropriate for an already developed but small natural thaw lake [e.g., West and Plug, 2008, Figure 7E].

[21] In the model, the initial ground ice content decreases with depth from a maximum value at the permafrost table. The volume of ground ice in i depends on its depth, di:

  • equation image

where υ is the unfrozen porosity of the soil and υE the maximum volumetric excess ice. Both υE and dE (the depth of the layer with excess ice) are simulation-specific parameters that depend on the geographic region modeled. ε is a random perturbation representing ground ice heterogeneity owing to ice lenses, veins, and wedges. Because our goal is to examine the sensitivity of lake dynamics to ground ice and climate, we use this generalized representation of ground ice. A measured depth profile of ice content might be used if available and if the goal were simulation of a specific lake, but would complicate comparisons between simulations and analysis of the intrinsic variability of lake expansion. We use the linear dependence in (6) because ground ice enrichment occurs over 1000s of years within the upper few meters of permafrost by formation of massive ice (e.g., ice wedges), lenses and pore ice by infiltration of unfrozen water under thermal gradients that cause a matric potential [Williams and Smith, 1991]: Hence, the slowly aggrading or static surfaces during deglaciation and the Holocene would accumulate greater ground ice within the few meters below their paleosurfaces, compared to faster aggrading deeper loess. Ice volume above the permafrost table depth, dp, is set to υ. This suppresses thaw subsidence in the active layer during seasonal thawing. Other than a slight effect on modeled active layer depths owing to latent heat effects, our results are not sensitive to this assumption.

2.7. Parameters

2.7.1. Alaskan and Yukon Thaw Lake Terrains

[22] In our reference simulations, we drove the model using modern climate and approximations of ground ice conditions for the northern Seward Peninsula, Alaska (NSP) and the Arctic coastal plain of the Yukon Territory, NW Canada (YCP). Characterized by markedly different depositional history and ground ice thickness, these provide an opportunity to test model behavior against near end-member types of thaw lake terrain. The northern Seward Peninsula is a coastal plain along the Chukchi Sea, a large embayment of the Arctic Ocean above Bering Strait. Mean annual precipitation is 250 mm and mean annual air temperature (MAAT) is −6°C (at nearby Kotzebue (Western Regional Climate Center, Historical climate for western region of the United States, http://www.wrcc.dri.edu)). Permafrost is >90 m thick and active layers up to 0.6 m deep [Hopkins, 1949; Hopkins et al., 1955]. During Pleistocene sea level lowstands, silt transported by glacially charged rivers to what is now Kotzebue Sound [Goetcheus and Birks, 2001] was mobilized by wind and deposited as loess on the NSP. Ground ice formed in step with the rising surface and permafrost table, producing a thick layer of syngenetic ground ice in the form of wedges, lenses and pore ice [Hopkins, 1949; Hopkins et al., 1955; Hopkins and Kidd, 1988]. The Yukon site (YCP) is colder and with a thinner layer in which permafrost is ice-rich. The site is located on the coastal plain bordering the Beaufort Sea, near Shingle Point, MAAT is −10°C and mean annual precipitation 250 mm (Environment Canada, Historical climate normals, available at http://www.climate.weatheroffice.ec.gc.ca/). Permafrost thickness is >300 m, and the active layer is ∼0.3 m deep. Late Pleistocene glacial advances deposited more than 60 m of unconsolidated gravel, sand and silt [Rampton, 1982], with up to 90% excess ice concentrated within the upper several meters below the modern permafrost table [Rampton, 1982]. Buried beds of massive ice some 10s of meters thick do occur in permafrost along the western Arctic coast [Mackay, 1971], a region which includes the YCP study area, and have been described for Sabine Point [Harry et al., 1988] and Herschel Island [Pollard, 1990], but these beds are localized and do not appear to have effected lakes in the particular YCP study area [West and Plug, 2008].

2.7.2. Temperatures and Substrates

[23] Daily air temperatures in the reference models were set using historical 30 year mean air temperatures from Kotzebue (Western Regional Climate Center, Historical climate for western region of the United States, http://www.wrcc.dri.edu) (NSP) and Shingle Point (Environment Canada, Historical climate normals, available at http://www.climate.weatheroffice.ec.gc.ca/) (YCP). A layer of snow with thermal conductivity 0.2 W m−1 K−1 increases from a depth of 0 m at the end of the thaw season, to the historical mean end-of-winter depth. We assume the snow's depth is spatially uniform, and so do not account for drifts or blowouts owing to topography or vegetation, which may change the thermal regime at lake margins. Lake bottom temperatures were set to 4°C, as used in an earlier thermal model for thaw lake taliks in N. Alaska [Ling and Zhang, 2003] and the measured value for thaw lakes in NW Canada that are sufficiently deep that summer water depth exceeds winter lake ice thickness [Burn, 2002]. Temperatures are fixed at this 4°C value all year, for the part of the lake that remains unfrozen. In the layer that freezes (ie. in the lake ice), temperatures are dynamic during winter because of conductive heat transfer from the surface. The dependence of lake bottom temperature on surface air temperatures also was investigated using a validated 1-D physically based eddy diffusion model for lake water temperature and evaporation [Hostetler and Bartlein, 1990], which we modified for thaw lakes by changing the lower boundary of the lake from a no-heat-flux condition to flux condition (equal to the spatially averaged heat flux in the upper 0.5 m of the modeled talik). The water temperature model has no tuning parameters and includes the effects of ice cover and convective mixing. Using physical parameters from Hostetler and Bartlein [1990] and the daily averaged meteorological data and latitude from Shingle Point, a lake of 100 m diameter and 3.5 m depth (comparable to the size of modeled lakes) has mean annual bottom water temperature of 2–5°C, depending on cloudiness (fraction of the sky obscured by clouds) and lake turbidity [Hostetler and Bartlein, 1990], confirming the reasonableness of the 4°C water temperature value used.

[24] The geothermal heat flux measured in deep permafrost in N. Alaska, 0.055 W m−2 [Lachenbruch et al., 1982; Ling and Zhang, 2003], is used for both simulations. Nonice components of soil are assumed to be 20% organic and 80% mineral by volume, and the thermal conductivity therefore set to 0.4–2.24 W m−1K−1 for frozen ground cells (range is due to varying ground ice content) and 0.31 W m−1K−1 for unfrozen ground cells; heat capacity is 5.6–4.6 × 108 J kg−1K−1 (frozen), and 1.4 × 109 J kg−1K−1 for unfrozen ground cells [Williams and Smith, 1991; West and Plug, 2008]. To simulate the effects of surface peat, the organic and mineral fractions are changed to 80% and 20% respectively in the upper 0.6 m. This approach is probably a reasonable approximation for homogeneous beds of Pleistocene loess topped by thin peat, as on upland surfaces at the field areas, but probably not for the 1–1.5 m thick Holocene peat that blankets lower ground at both sites. The effect of assuming a uniform substrate is considered in section 3.2 and the dependence of thaw bulb deepening on substrate in the work by West and Plug [2008].

[25] The thick ice-rich layer on the Seward Peninsula is treated in NSP simulations by setting excess ground ice at the top of permafrost, υE, to 50%, and depth of excess ice, dE, to 80 m. For the YCP simulations, dE is 10 m and υE is 50%, from observations by Rampton [1982] that the upper 5–10 m are ice-rich. Our results are therefore not relevant to lakes, if any, that have developed on those uplands which are underlain by massive bodies of pure ice. Porosity υ is 30% for both simulations, giving a mean volumetric ice content at the permafrost table of 80%. For both simulations, the normally distributed spatial variability in ice content, ε, is 10%.

[26] Lake ice thickness is primarily controlled by surface air temperature, but water temperature and the depth, density and albedo of snow also play a role [Duguay et al., 2003]. Winter lake ice thickness is set to 1.5 m in the model, consistent with measured lake ice in northern Alaska and northwestern Canada where MAAT is −6°C [Allen, 1977; Jeffries et al., 1996; Zhou and Huang, 2004]. The interannual variability of lake ice in natural lakes is sufficiently small in magnitude (the 1σ variability was 8.2 cm over 12 years of measurement for a lake inland of Shingle Point, in the YCP field area [Allen, 1977]) that the annual maximum ice thickness is held constant in each model scenario.

2.7.3. Erosion and Discretization

[27] Exponent n in (4) controls the length scale of sediment transport. Transport mechanisms are dissipative of momentum and energy. Therefore a priori n > 0 and probably > 1. Increasing n causes transport to be limited to ever shorter distances, analogous to processes that are more dissipative. To better constrain n, we used transport distances predicted in an advective flow simulation along a slope comparable to a thaw lake margin.

[28] For avalanche-style movements of snow, rock, and debris, the resistance to flow depends linearly on a coefficient of sliding friction between the flow and underlying bed, μ, and on flow velocity squared (V2) owing to “turbulent” friction arising from air drag, internal viscous dissipation, and dislodging of underlying surface material [e.g., Perla et al., 1980; Cannon and Savage, 1988; McEwen and Malin, 1989; Howard, 1998]. The change in momentum of the advective movement is the difference between gravitational acceleration and flow resistance [Howard, 1998]:

  • equation image

Ct is the coefficient of “turbulent” friction, θ is slope angle. Assuming the density, ρ, and depth, h, of the layer that fails does not change appreciably downslope, velocity varies with downslope distance s, according to

  • equation image

Deposition occurs where V = 0. An initial condition of a linear slope profile and a shear-stress-dependent failure criterion (which decreases through time owing to weathering), results in a backwearing upper slope profile and deposition on an apron with a linear slope [Howard, 1998].

[29] We incorporated (8) into a finite difference simulation for a longitudinal profile with dimensions approximating a lake margin into a basin (an 8 m high slope of mean θ = 30°), and imposed downslope variability in θ and μ because natural lake margin profiles are not smooth and sliding friction will vary with ground ice, water content, and substrate texture (caption of Figure 1). Slope failures initiate in this simulation according to (2). The resultant transport distances are widely distributed between 0 and 30 m (Figure 1). Morphologic variability along the margin profile causes local decelerations that can cause midprofile deposition, depending on the prior path because this determines V. The travel distances in this transport model do not follow a single power law frequency-magnitude relationship over the full range of transport distances, but instead decline rapidly at long distances (Figure 1). The relationship is approximately s−2 over the full range of transport distances, so we set n to 2 in our reference simulations. Comparisons between modeled and natural lake profiles are an additional method of testing the appropriate value of n (section 3.2).

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Figure 1. Distribution of transport distances in a finite difference simulation of advective movement on a slope with dimensions comparable to a thaw lake bluff: an 8 m high bluff with mean gradient θ = 25° into a flat basin. Sliding friction angle, μ, has mean 25°, and Ct is 500 [Howard, 1998]. θ and μ are normally distributed (σ = 0.5), representing heterogeneity in morphology, soil moisture, and substrate on a natural slope. Depth of failures (d in equation (8)) are uniformly distributed between 0 and 0.6 m, the depth of maximum seasonal thawing. Resulting advective transport distances are not power law distributed (points form a curve), but the overall decline in frequency has slope −2 (solid line) matching the PD ∼ Δx−2 dependence in the transport relation (equation (4)) used in our thaw lake model.

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[30] Rate constant CE in (4) encapsulates properties of substrates which control slope stability such as density, internal friction, cohesion, and binding by roots (if any). CE is not, to the best of our knowledge, constrained by available process measurements. We ran the model using parameter sets for the NSP and YCP, varying CE between simulations, and quantified the difference between modeled profiles for each CE and profiles measured using differential GPS and sonar at each site. CE is set to 5 × 10−10 m2 s−1. The successful use of one parameter set for both regions (section 3.2) indicates that this value probably is a robust selection.

[31] The model has potential for cell size to affect dynamics, particularly slope failures, because erosion is coupled to heat transfer. For example, a cell must be completely thawed before it is assigned the lower, thawed, value for critical slope Sc, and, until it erodes, it insulates underlying permafrost. Discretization is most appropriate for processes that involve movement of consolidated blocks and for flows whose volume exceeds the volume of cells, but probably less so for individual, small grain-by-grain continuous movements. In our reference simulations we chose a cell size of 0.3 × 0.3 m because in our qualitative observations this is a typical value for peat and sediment dislodged from thaw lake margins. The sensitivity of results to cell size, n, CE, and ɛ, is considered in section 3.3.

3. Results and Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Results and Discussion
  6. 4. Conclusions
  7. Acknowledgments
  8. References

3.1. Geomorphology and Dynamics

[32] In modeled lakes, a profile that is in dynamic equilibrium develops out of the initial morphology after approximately 60 years. This profile, which includes the subaerial lake bluff and the outer ≈40 m of the lake bottom, is variable through time, within bounds, and migrates outward as the basin expands (Figure 2).

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Figure 2. Thermal and morphological profiles of a simulated lake after (a and b) 300 years and (c and d) 310 years in one realization using NSP model parameters. Time of year is the end of winter; lake ice lies between indicated air and lake. Over the 10 year period shown, the basin (defined by the top of the bluff) has expanded 2.4 m compared to ≈0.5 m for the lake, the latter less because of sediment deposited at the lake margin prior to year 300 (which has partly diffused into the basin by year 310) (Figures 2b and 2d). Water depth and talik thickness at the lake center remained constant because basin subsidence approximately balanced sediment deposition.

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[33] In the NSP simulations, lakes expand at 0.26 ± 0.51 m a−1 (Figure 3a; ± is 1σ of interannual variability). The maximum single-year lake expansion is 4.80 m, and maximum single-year shrinkage is −3.80 m. This episodic, nonmonotonic expansion occurs despite stationary water temperatures and MAAT for two reasons. First, advective mass movements (which cause short-term rapid expansion) are triggered where banks are vertical, with resultant movement also destabilizing overlying slopes. Second, the rapid retreat of a subaerial bluff can occur by a form of retrogressive thaw slumping where permafrost is particularly rich in ice (≥80% ice over several meters in a horizontal direction). In these settings, 1–2 m high vertical bluffs (“headwalls”) can form by thaw subsidence (i.e., thawed cells are removed, forming a “step”), which then migrates up to ≈3 m in one thaw season because they are unstable, and can be reactivated for one or several following years depending on ice content. This behavior is qualitatively similar to retrogressive thaw slumps that have been described for other shorelines [e.g., Burn, 2000]. The model does not reproduce the extreme rates and duration of headwall retreat measured or estimated for exceptional natural examples (for example, Burn [2000] reports a headwall migration of 450 m in ≈44 years, measured back from a river shoreline). However, such sustained rapid events probably are rare in natural thaw lake margins. Large slump scars 10–100 m across are not evident around most lakes, and the rounded, smoothed margins of many lakes indicate that fast localized retreat can only be a minor factor in the expansion of the majority of lakes.

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Figure 3. Distributions of annual expansion rates for (a) lake radius (lake center to water edge) and (b) basin radius (to crest of the bluff) in NSP and YCP simulations. Mean lake expansion rates are 0.26 m a−1 (NSP) and 0.09 m a−1 (YCP), but both have modal rates of 0 m a−1. n = 3500 model years for all distributions.

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[34] Compared to the episodic expand/shrink behavior of modeled lakes, basins (their radius defined by the top of the subaerial bluff) expand monotonically and with less interannual variability (Figure 3b). Basin expansion is only infrequently synchronized with rapid lake expansion. When the bluff is steep (approaching the critical angle) and the near-shore of the lake relatively deep (>0.3 m), sediment released by slope failures near the water line can be deposited into the lake and hence synchronized bluff retreat and lake expansion may occur. In most cases, sediment from failures on the bluff is retained at or near the water line, in the shallow near-shore where the gradient is gentle. Slumped sediment ultimately relaxes into the lake basin, but in the meantime insulates permafrost and so delays thaw subsidence and hence lake expansion. Temporary lake shrinkage occurs by large-scale mass movements from the subaerial margin to near the water line, and where thaw subsidence on shallow subaerial shelves generates water-filled depressions into which slumped sediment is redeposited. This temporarily diminishes the radius of the lake.

[35] Below the water line, the modeled lake floor has a gentle gradient and is generally smooth (within the range of ±1–2 model cells), except where locally roughened by deposits of recent large advective events. Subaqueous topography is chiefly smoothed in the model by diffusive mass movement because the slope angle is far below Sc (Figure 2, for example, illustrates lake floor smoothing over 10 years, at 97–100 m from lake center), behavior consistent with natural lakes where large blocks of material are diffused by wave- and gravity-driven disturbances [e.g., Hopkins and Kidd, 1988, Figure 2]. The small-scale topography that persists for days to months on the model lake floor is similar in scale to the <1 m amplitude of measured relief of lake basins on the Seward Peninsula [West and Plug, 2008].

[36] YCP simulations also develop margin profiles at a dynamic equilibrium, but rates and morphology differ from the NSP simulations. Whereas NSP lakes deepen though time, YCP lakes decrease in depth from the 4 m initial condition to reach an approximately steady state depth of ≈2 m after 500 years (Figures 4 and 5) . YCP taliks deepen more slowly because the faster radial expansion of NSP lakes causes an areally larger thermal disturbance at the ground surface, i.e., a warm lake of radius 160 m (NSP) versus 50 m (YCP) after 500 years. Also, compaction of the talik by thaw subsidence causes a thinner layer through which heat is conducted from the lake to the thaw front. With lesser thaw subsidence in the talik, and the addition of sediment eroded from the expanding margin, YCP basins remain shallow.

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Figure 4. Cross sections of (a) an NSP modeled basin, (b) measured profile of Pear Lake basin, (c) a YCP modeled basin, and (d) measured profile of Doe Lake basin (YCP). The morphology of modeled basins is shown at winter's end at 100 year intervals. Isotherms are for year 500. Vertical exaggeration is 3×. Lake measurements, from West and Plug [2008], are for Pear Lake (66°32′30″N, 164°27′30″W, area 9.5 × 105 m2, water depth 17 m), Lake Claudi (66°33′00″N, 164°27′00″W, area 2.0 × 105 m2, water depth ∼9 m), and Doe Lake (68°54′45″N, 137°16′05″W, area 1.2 × 105 m2, water depth 2 m.)

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image

Figure 5. Modeled talik thickness, lake depth, and radius of lakes during 500 years for (a) NSP and (b) YCP simulations. In a thick ice-rich layer (NSP), lake expansion is 0.26 ± 0.009 m a−1 (± refers to 1σ over seven models), lake depth reaches 5.2 ± 0.34 m, and talik thickness is 12.7 ± 0.40 m. The distance between the lake margin and the basin margin (subaerial margin) is 5.2 ± 1.3 m. In a thin ice-rich layer (YCP), banks retreat at 0.09 m a−1, lake depth decreases to 2.1 m, and talik thickness reaches 8.7 m. The distance between the lake margin and the basin margin, the subaerial shelf of the thaw basin, is 7.8 ± 1.8 m. Data are from single model realizations.

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[37] In YCP simulations, lakes display significantly more variability in year-to-year expansion (Figure 3). Because YCP lakes are shallow and have gentler subaqueous slopes, slumped material from above the water line is more likely to temporarily accumulate near the water line. In contrast, NSP basins draw slumped material into the basin more rapidly because they are deeper and have steeper marginal gradients.

[38] In both NSP and YCP simulations the talik extends downward from the lake margin, beginning at the position of deepest winter lake ice. The part of the talik beneath the outer part of the lake is thinner and with gentler inclination in NSP simulations (compared to YCP simulations) because the lake expands more rapidly. Active layer depths are 0.3 to 0.6 m on undisturbed terrain away from the lake. On the actively retreating lake margin, active layer depths vary from as little as 0 m where material has been eroded, to as much as >2 m where material has been freshly deposited earlier during the thaw season. In places of shallow water where there has been neither erosion or deposition, active layer depths depend on thaw season length but are typically 0.6–1 m.

3.2. Tests

3.2.1. Morphology

[39] We compared modeled morphology to measurements of bathymetry [West and Plug, 2008] and subaerial bluffs [Plug, 2003b] of natural lakes Claudi and Pear (NSP) and Doe (YCP) (Figure 6 and geographic coordinates in caption of Figure 4). Lakes Pear and Claudi are deeper than modeled NSP lakes because they are larger in diameter and much older than the 500 year duration of our simulations [West and Plug, 2008]; modeled NSP basins reach a similar depth, ∼25 m, after 8000 years [West and Plug, 2008] (this depth is for a model including thaw subsidence but not mass movement, but the effect of deposition on maximum depth at lake center is minor for a 500 m diameter lake). Our morphologic comparisons are valid because they use the subaerial bluff and the outer part of the subaqueous profile, which are at dynamic equilibrium and so have morphologies that are independent of lake age and maximum depth.

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Figure 6. (a) High-resolution DEM of Lake Claudi's (NSP) east facing subaerial bluff. (b) Modeled versus measured profiles from Lake Claudi. (c) Comparisons of gradients across the profiles. Distance and elevation are normalized by bluff height in Figures 6b and 6c. Modeled profiles are the mean and 1σ variation of 40 randomly sampled years, excluding the first 100 years because the equilibrium profile has not been achieved. Modeled bluffs reproduce the cross-section shape of measured bluffs except near the crest. Fibrous peat probably reinforces crests of natural bluffs, an effect not included in the model. The Lake Claudi DEMs were constructed by linear interpolation of ≈600 DGPS point measurements each, collected at ≈2 m cross-slope and ≈1 m slope-parallel spacing, with closer spacing used where necessary to capture steep relief [Plug, 2003b].

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[40] In comparisons of modeled and the two natural bluffs from Lake Claudi (Figure 6b), the slope of all three profiles is overall approximately linear. The morphology of modeled bluffs match measured bluffs across most of the profile, within the range of variability in both model and nature. Modeled bluffs lie outside the 1σ range of measured bluffs only near the upper 15% of the profile, which corresponds to the upper 1 m for the ≈8 m high bluffs of Lake Claudi. Unlike measured bluffs, modeled bluffs lack well-defined crests, probably because the effect of fibrous modern peat on enhancing the stability of the upper 1–2 m of the slope is not included in the model. For both modeled and measured bluffs, the steepest gradients (0.6–0.8, or 31–39°) occur in the lower half of the profile (Figure 6c). In the model, this location corresponds to the average position of the lower edge of slump material that preferentially accumulates immediately above the water line, and a similar slump block “landing zone” mechanism might explain the steep lower profile of natural bluffs.

[41] In comparisons of bathymetric profiles, the simulated NSP basins are inclined 3° over an ≈50 m wide ramp starting near the water line (Figure 4a). In comparison, the outer ≈50 m of Pear Lake's subaqueous slopes also average 3°. A flat central basin occurs in both simulated and measured NSP lakes: in the former, and probably the latter, where the bottom of the underlying talik is approximately flat. Simulated YCP basins have maximum depths of 2–2.5 m and subaqueous gradients of 6° over ≈20 m (Figure 4c), similar to the 2–2.5 m depths and marginal slopes inclined 5–7° of lakes on the YCP [West and Plug, 2008]. Second-generation lakes in the central Seward Peninsula, also developed in thin ground ice, have been noted to have maximum slope angles of 5° [Hopkins, 1949], also comparable to the shallow ice simulations.

[42] Beneath modeled lake margins, the upper limit of the talik extends downward from the position of deepest winter lake ice, with a tangent to this portion of the talik inclined at 20–30°. We do not have measurements of talik shape for the NSP or YCP, but the talik of a natural lake of similar size in northwestern Canada (Lake Illisarvik), was inclined 23° at this position (220–260 m from the lake center on a north-south transect [Hunter et al., 1981]).

3.2.2. Dynamics

[43] In NSP simulations, the long-term mean rate of lake expansion is 0.26 m a−1 (Figure 5a), compared to ∼0.4 m a−1 for Lake Claudi over 400+ years as inferred from overlapping basins [West and Plug, 2008]. In comparison, lakes in the central Yukon probably grow at rates of 0.2–0.9 m a−1, sustained over decades or longer [Burn and Smith, 1988], and lakes in Siberian yedoma reportedly grow at 0.5 m a−1, presumably averaged over many lakes and a long duration [Zimov et al., 1997; Walter et al., 2006]. Both rates are faster than that of simulated lakes, but are probably generally consistent with the model, given that summer temperatures at both these continental locations are warmer than either the YCP or NSP, and so experience deeper subaerial thawing. In general, most other published expansion rates seem to be anecdotal, probably focusing on extreme rates sustained over a short time because these are most notable, or span durations of only one or several years. Reports include up to 4.5 m a−1 on the Tuktoyaktuk Peninsula [Rampton, 1988] and 7–10 m a−1 on Banks Island, NWT [French and Egginton, 1973]. A rate of 5 m a−1 over 2 years has been reported for a NW Alaskan lake [Hopkins, 1949]. These rates are consistent with extreme rates that occur in the model over one to several years (Figure 3, section 3.5). The mean expansion rates of NSP simulations would generate a 2 km radius lake over the duration of the Holocene, assuming a constant growth rate. This is similar to the size of the largest modern lakes that can be identified in aerial photographs of the area; although not a precise constraint on long-term expansion rates, this does suggest that modeled mean rates are more accurate than the 1–10 m a−1 reported single year extreme rates.

[44] Model active layer depths in the undisturbed flat ground distant from the lake range from 0.3 to 0.6 m. These bounds are set by the 0.3 m cell size in reference simulations. The average depth is 0.45 m when a 0.05 m cell size is used, similar to those of peaty, wet soils characteristic of many lowland thaw lake areas [Hopkins et al., 1955; Plug, 2003a; Jorgenson et al., 2006]. For the NSP, the equilibrium temperature profile (away from lake) has 180 m deep permafrost, consistent with reported depths for NW Alaska [Hopkins et al., 1955; Brown and Pewe, 1973]. For YCP simulations, the permafrost depth (without the influence of a lake) is ∼400 m, comparable to estimates for the YCP and elsewhere in the coastal western Arctic [Rampton, 1982; Judge and Bawden, 1987]. The -1 m ground temperature away from the lake reaches a minimum in March of ∼−15°C and a September maximum of ∼−1°C (both temperatures vary somewhat across the simulation, owing the effects of heterogeneous ground ice content on thermal diffusivity). Both are comparable to measurements from Richards Island to the east of the YCP and with a comparable climate and substrate [Burn, 2002]. Thermal properties of modeled soils and heat flux across the ground surface are therefore likely in an appropriate range.

3.3. Sensitivity to Parameters and Discretization

[45] We used NSP parameters to generate a reference simulation, and then varied each parameter one at a time. Mean annual expansion of the modeled lake, and interannual variability in expansion, were measured for each parameter set.

[46] The erosion constant CE encapsulates properties of substrates that control slope stability by setting the probability of erosion. Increasing CE from the reference value of 5 × 10−10 m2 s−1 results in more rapid lake expansion (Figure 7a) but sensitivity to CE is not great: varying CE by two orders of magnitude (0.1 to 10×10−10 m2 s−1) causes <2× increase in lake expansion rate (0.18 to 0.32 m a−1), all within the range of expansion rates reported for thaw lakes. Interannual variability in lake expansion is insensitive to CE. When erosion is turned off (CE = 0), expansion occurs only by thaw subsidence and is 0.02 m a−1.

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Figure 7. Sensitivity of mean modeled lake expansion rate (solid lines) and interannual variability (dashed lines) to (a) coefficient of erosion CE; (b) exponent n, which sets transport length scale; (c) spatial heterogeneity of ground ice, ε (1σ for normally distributed random perturbations); and (d) cell size. Arrows indicate reference model values.

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[47] The rate and interannual variability of lake expansion both decrease with increases in n (Figure 7b). For n = 3, slow diffusive-type behavior dominates. Sediment accumulates at lake margins, insulating ice-rich frozen ground from further thawing, and margin profiles are smoother and expansion rates significantly less variable through time. For n = 1, lake margins remain relatively free of thick redeposited sediment, enhancing thawing. Despite the power dependence of transport length on n, modeled expansion rates of lakes are not highly sensitive over the 1 ≤ n ≤ 3 range explored, varying from ∼0.1 m a−1 to ∼1.3 m a−1, again within the range of natural lake expansion. Even for n = 3, transport is sufficiently rapid that thawing of permafrost and lake growth is sustained.

[48] For cell sizes of 0.05–1 m, modeled expansion rates are within the range of natural lakes (Figure 7d), decreasing with increasing cell size, probably because the slower removal of cells by erosion slows thawing. Interannual variability in expansion is insensitive to cell size within the range of 0.05 to 0.4 m, decreasing thereafter with increasing cell size. The maintenance of interannual variability indicates that this is a robust phenomenon in the model, not arising from discretization. Likewise, interannual variability is independent of heterogeneity in ground ice concentration (Figure 7c), indicating that variable retreat arises from dynamics, not from model discretization.

[49] The long-term average behavior of modeled lakes is reproducible across different realizations of the model (each realization having a different initial morphology, ground ice distribution, and seed for random number generation). For ten realizations of the 500 year long NSP simulation, the mean lake expansion varied by only ±0.01 m a−1 and the 1σ range of interannual variability by only ±0.02 m a−1. Results are therefore not highly sensitive to initial conditions or particular ground ice distribution.

3.4. Lake Dynamics Under Different MAAT Scenarios

[50] To provide an initial probing of the sensitivity of modeled lakes to climate, we ran two scenarios with surface temperatures shifted from modern values. By the end of the 21st century, mean annual air temperature may increase across the Arctic by 5°C (the mean of General Circulation Model (GCM) ensemble MMD-A1B), roughly twice the projected global mean warming [Christensen et al., 2007]. There is considerable variability within the region and between GCM models, however, with a minimum projected cross-model increase being 2.8°C [Christensen et al., 2007; Chapman and Walsh, 2007]. Most predictions are for warming to be greatest in winter, but changes in seasonality are variable cross-model and cross-ensemble. Given this uncertainty, and since our goal is a first-order test of model dynamics, we conservatively increased mean air temperature by +3°C in our warm scenarios by uniformly shifting daily air temperatures from their modern values. Similarly, for cold scenarios, we decreased mean annual surface temperature by 8°C, the magnitude of cooling during the Last Glacial Maximum at high latitudes (for example, in a CCSM3 simulation, the mean annual surface temperature at ≈21 ka BP is −4 to −12°C for central Siberia and NW Canada [Otto-Bleisner et al., 2006]). Mean annual lake water temperature was adjusted by +1°C (warm scenarios) and −2.5°C (cold scenarios) using the water temperature model (section 2.7.2). Because we do not know how the secondary controls over lake ice would vary for scenarios, we set ice thickness in the warm scenario to −0.3 m from reference, so it matches the 1.2 m mean of measured maximum values for Alaskan lakes which have MAAT of the warm scenario (National Weather Service (Alaska Pacific) River Forecast Center, available at http://aprfc.arh.noaa.gov/). Lacking a subarctic analog for lake ice under a −8°C climate, we use the 30 year mean maximum for a lake at Resolute in the Canadian High Arctic [Lenormand et al., 2002] (MAAT −16.4°C), 2 m. Given the relative insensitivity of the model to lake ice thickness, the uncertainties in ice thickness have little effect on results.

[51] In the warm (+3°C shift) scenarios, deeper seasonal thawing of subaerial margins causes larger and more frequent mass movements over a longer thaw season. Lake expansion increases by 2.5× for the NSP, to 0.63 m a−1, and by 1.5× for the YCP (Figure 8). Warmer water causes faster thaw subsidence beneath the lake, particularly near margins because the talik there is shallow. Faster thaw subsidence provides accommodation space for sediment from slope failures. Taliks are deeper in the warm scenario, owing to warmer water and larger lake size, but the relative change is less than that of radial expansion of lakes: for the NSP, taliks are 36% thicker (to 17 m) and for the YCP 45% (12.6 m) after 500 years. YCP lakes are shallower under the warmer scenario, 1.8 m after 500 years, a decrease of 15% from the reference model. Conversely, the warmer climate causes NSP lakes to be deeper: 6.9 m versus 5.7 m after 500 years. In NSP simulations, the increased rate of sediment deposition in the basin is approximately matched by faster thaw subsidence. Conversely, for the YCP warming causes the rate of sediment infilling to exceed the increase in thaw-driven basin subsidence (because ground ice is thin), resulting in reduced lake depth after 500 years.

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Figure 8. Sensitivity of (a) NSP and (b) YCP modeled lakes and taliks to cold (MAAT 8°C below present climate) and warm (MAAT 3°C above present climate) climate scenarios.

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[52] In the cold (−8°C shift) scenarios, expansion is dramatically slowed to 0.03 m a−1 for the NSP (Figure 8a) and 0.02 m a−1 for the YCP (Figure 8b). The slope of subaerial margins increases from 8° (reference) to 17° for the YCP, and from 20° to 30° for the NSP in the cold scenario. Colder, more stable substrates maintain steeper slopes and reduced lateral talik development beneath lake margins contributes to bank stability. Paleoenvironmental reconstructions have attributed paleobasins to Holocene warming in northwest Canada between 10 and 9 ka [Rampton, 1982; Michel et al., 1989], and in northern Alaska between 8 and 4 ka [Carson, 1968], and thermokarst cessation during cooler full-glacial climate [Rampton, 1988; Hopkins and Kidd, 1988]. Cessation of lake expansion in the model cold scenario is consistent with this and therefore provides additional support for the realism of modeled lake dynamics.

[53] We emphasize that these results are initial sensitivity tests using simple MAAT scenarios, with uncertainty particularly arising from the complicated effect of climate on water temperature and level in lakes, as well as snow, vegetation and albedo feedbacks. However, several general implications can be drawn. First, expansion rates of lakes may vary over more than an order of magnitude and do not depend linearly on MAAT. For the NSP, an increase in MAAT of 3°C from the present value increases the expansion rate by ≈0.4 m a−1, compared to a 0.25 m a−1 slowing for 8°C cooling. Second, lakes of different morphology can display different expansion rates under a similar climate. For example, a deep lake in thick ground ice would expand more rapidly than a shallow lake under similar climate, because of its greater accommodation volume and margin gradients. Expansion of the shallower lakes would be (relatively) slowed by redeposited sediment that obstructs and insulates the lake margin. Therefore, in regions with deep syngenetic ground ice (such as the NSP and Siberian yedoma), modern active first generation lakes, which are deeper, might expand more rapidly than those modern lakes that are expanding in the refrozen drained basins of earlier lakes. Likewise, the average expansion rates of lakes during the early Holocene may have exceeded that of modern lakes, even if climate then and now were similar. This is because many modern lakes are of the later generation, shallower, and slower-expanding type. Finally, given a modern climate warming, lakes which developed deep basins but small surface areas under a sustained cool climate (when mean annual water temperature was >0°C but low summer air temperatures slowed thawing of subaerial margins) might show rapid sustained expansion under a warming climate but neighboring, younger, shallower lakes might not. Predictions of this type, which illustrate the richness of the model, might be tested by long-term observations and/or from aerial photograph change studies. We emphasize that other feedbacks not presently in our model, including vegetation colonization and snow accumulation at margins of lakes, may influence bank stability and hence the dependence of expansion rates on climate.

3.5. Interannual Variability in Margin Retreat and Implications for Lake Measurements

[54] In the model, lake expansion is broadly variable and shrinkage occurs in some years, even under a stationary climate (Figure 3). This variability occurs because of the coupling between thawing and mass movement, and because advective slope processes are nonlinear: mass movements change slope morphology, and in so doing influence subsequent mass movements. This variability has implications for measurements of natural lake expansion, which are necessarily acquired over short periods, 1–100 years at most. To quantify the possible measurement error of finite numbers of observations, we used a Monte Carlo approach [Plug et al., 2007]. 3500 independent observations of annual lake expansion in simulations were randomly resampled to form 30 synthetic measurement series. The variation between the 30 synthetic series, measured as the standard deviation of their means, was then assessed as a function of the number of observations in the series. These measured means are compared to the “true” mean of all 3500 observations (Figure 9). The procedure was repeated for NSP and YCP simulations.

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Figure 9. (a and b) Convergence of measured annual expansion rates toward the population mean rate (i.e., “true mean”) (1σ envelope of measurements shown) and (c) the maximum and minimum mean rates (max/min envelopes shown), as a function of number of observations. Each observation is a measurement at a different position, or different year for a single position. In NSP simulations, approximately 50 years (or independent measurements) are required for convergence to within ±0.1 m of the true mean rate; for the YCP, 50 observations converge to within 0.1 m of the mean rate, but because the mean rate is low the predicted value may still misestimate the lake as stable or even shrinking. The broader variability of YCP annual expansion rates causes larger max/min rates, as much as ±∼8 m in a single year.

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[55] For 10 measurements of lake expansion in NSP simulations, the mean of measurements is within 0.20 m of the true mean, 0.26 m, and likely to be slightly below the true mean because rapid expansion years have not been sampled (Figure 9a). Single-year extreme rates are 4.40 and −3.80 m a−1 (Figure 9c). Significant misestimates of the long-term expansion rate can therefore occur from one or a small number of measurements (i.e., years). For 100 measurements, the measured mean is not biased and has converged to within 0.05 m of actual expansion rate. In comparison, a single measurement of YCP expansion is likely to be as low as −8 or as high as 0.9 m a−1, and possibly −7.3 m to 10 m, compared to the true long-term mean of 0.05 m a−1. For 10 measurements, the 1σ error decreases to a ± range of 0.15 m a−1, and for 50 measurements to 0.1 m a−1. A negative expansion rate (lake shrinkage) might therefore be erroneously inferred from as many as 50 years of measurement.

[56] These results show that measurements of lake expansion over years to decades may significantly underestimate or overestimate the true mean rate. Amalgamating measurements of bank retreat from multiple (50+) lakes might overcome the large uncertainty inherent to using measurements from the cross section of a single lake. Alternatively, areal measurements of a single or small number of lakes might reduce uncertainty because this effectively samples more than one cross section. However, to the best of our knowledge, the typical cross-slope width of slope failures around thaw lakes has not been measured. This width(s) will set the cross-slope correlation distance for the rate of bank retreat. Since it is not known, the area and number of lakes that would be required to accurately measure expansion rates also is presently unknown. These uncertainties in measurement are caused by intrinsic variability that arises from processes of thawing and mass movement in the model, even under a relatively uniform volumetric ground ice content and a substrate that has homogeneous erodibility. In nature, coarse-grained heterogeneity in ground ice (such as ice wedges, which are meters wide and deep) and surface vegetation (such as shrubs) presumably induces even greater accelerations and decelerations in lake expansion from year to year. Water level fluctuations would add additional variability, particularly for remote sensing measurements. Our model-based results for the uncertainty of measurements are therefore probably minimum estimates for natural lakes.

3.6. Future Work

[57] Our model is a first attempt at numerical modeling of thaw lake development by combining geomorphic and thermal processes. It provides realistic rates of lake expansion and bathymetric development when compared with a small set of field observations. As already noted in section 2.1, however, our model does not include the full complexity of processes and substrates that can influence individual natural lakes in the wide range of different settings and climates in which thaw lakes can occur. Future work therefore should include adding more complex and site-specific processes to the model. In step with this, additional field observations are needed to rigorously test the model.

[58] Several types of measurements might be conducted to further test and extend the model. First, bathymetric measurements from more lakes in more locations could be used to test the universality of mass movement parameters n and CE. The effect of wind-driven circulation (waves and longshore transport) on sediment transfer might be inferred by comparing the model to bathymetric measurements of lakes of a wide range of diameters and depths, and/or from the bathymetric cross sections at different angles across oriented lakes. Second, more morphologic comparisons (section 3.2) might be conducted between the model and the profiles of subaerial bluffs around lakes, focusing in particular on the possible role of vegetation feedbacks on stabilizing lake margins. For example, fibrous peat may stabilize bluffs and locally increase the critical angle SC. Vegetation that is able to colonize bluffs of very slowly expanding lakes might retard expansion even more, imposing an additional nonlinearity and possibly a threshold effect (in that a single expansion event that removes vegetation might allow sustained growth, even when the temporary warm conditions that caused the short-term expansion cease). Third, in addition to comparisons of modeled and natural morphology, the model's treatment of mass movement might be tested by comparing the distribution of transport distances in the model to measured distances on bluffs of natural lakes. Our comparisons here (section 3.2) focus on morphology because process measurements collected over a suitably long time frame, >10 years, are not available. Given the magnitude of interannual variability, we anticipate that 20+ years of measurements from multiple lake cross sections would be required.

[59] Presently the model does not treat the time dependence in soil shear strength, which might occur by transient pore pressure increases in melting clay-rich permafrost. To address these substrates, Sc of thawed sediment could be made time-dependent, i.e., low immediately after thawing but increasing to a high value after days to weeks. Likewise, the model treats surface boundary conditions as spatially uniform, but there might be significant differences between stable vegetated surfaces and recently disturbed surfaces that have exposed mineral soil. In the context of an eroding thaw lake margin, the rates of disturbance can be highly variable and peat and organic fragments can be transported from upslope but still retain an insulating effect, complicating parameterization of heat flow differences. A time- and rate-dependent treatment of the thermal diffusivity could be incorporated in future versions of the model, given calibration measurements.

[60] Over time scales of 103 years and longer, thaw lake landscapes are modified by lake coalescence, by partial or catastrophic drainage of lakes, and by an evolving channel network that affects drainage pathways and where surface water pools and initiates lakes. To treat the evolution of landscapes over long time scales, the single, fast events which modify single lakes, such as individual mass movement events and seasonal thawing, probably are unimportant. Our two-dimensional model for a single lake might be used to calibrate abstractions of landscape-scale processes, such as the dependence of mean rates of lake expansion deepening on climate parameters, which can be used in a three-dimensional model for thaw lake landscape evolution over 103 years and longer.

4. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Results and Discussion
  6. 4. Conclusions
  7. Acknowledgments
  8. References

[61] We have presented a model for thaw lake expansion which combines heat transfer with mass movement and thaw-driven subsidence. The model produces morphology which is broadly consistent with the subaerial and subaqueous parts of measured profiles from natural lakes, and its dynamics (including expansion rate) are consistent with reported observations. The mean modeled rates of thaw lake expansion in a landscape with a thick ice-rich layer (the Northern Seward Peninsula, MAAT −6°C), 0.26 m a−1, is nearly 3× higher than modeled rates in a thin ice-rich layer (Yukon Coastal Plain, MAAT −10°C). Most of this faster rate is due to the greater ground ice thickness permitting deeper lakes that better accommodate slump material released from thawing banks. In sensitivity tests, a climate with MAAT +3°C compared to present causes lake expansion to accelerate by 1.5–2.5X, the larger value for thick ground ice, suggesting natural lakes might be highly sensitive to projected warming at high latitudes over the next ≈100 years. A −8°C cooling of MAAT from present, roughly comparable to the magnitude of cooling during the Last Glacial Maximum, causes modeled lake expansion to essentially halt, consistent with paleoenvironmental studies that attribute natural thaw lakes only to interglacial warming. Modeled lakes show highly variable rates of year-to-year expansion, of order ±5 m, indicating long-term measurements of expansion over 20+ years are likely to be reliable indicators of a sustained change in expansion rates.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model
  5. 3. Results and Discussion
  6. 4. Conclusions
  7. Acknowledgments
  8. References

[62] We thank T. Jorgenson, M. Church, and an anonymous reviewer for valuable reviews and B. T. Werner, J. C. Gosse, and H. Beltrami for comments and discussions. Research was supported by the Natural Science and Engineering Research Council of Canada, the Polar Continental Shelf Project, and the Northern Scientific Training Programme. Equipment was supported by the Canadian Foundation for Innovation, IBM Canada, and Leica Geosystems.

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  5. 3. Results and Discussion
  6. 4. Conclusions
  7. Acknowledgments
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