Simulating the decadal- to millennial-scale dynamics of morphology and sequestered carbon mobilization of two thermokarst lakes in NW Alaska

Authors


Abstract

[1] Thermokarst lakes alter landscape topography and hydrology in widespread permafrost regions and mobilize significant permafrost carbon pools, including releasing methane (CH4) to the atmosphere. Despite this, the dynamics of lake evolution, permafrost thawing, and carbon mobilization are not well known. We present a 3-D numerical model of thermokarst lakes on organic-rich yedoma permafrost terrains with surface water flow and pooling naturally defining lakes that deepen, expand laterally, and drain due to talik formation, bank retreat, and both gradual and catastrophic drainage. We predict the 3-D pattern of microbial methane production within the talik over time. As a first model test and calibration, beginning with small protolakes, we simulated 10,000 years of evolution of Pear and Claudi lakes, two neighboring thermokarst features on the northern Seward Peninsula, Alaska. Simulated lakes approximated observed bathymetry, but results are sensitive to initial topography and soil ice content. Local topography caused markedly different dynamics for the two lakes. Pear expanded rapidly across low-relief topography, fully drained multiple times, and released little methane in later stages due to Pleistocene carbon depletion by the first and largest lake generation. Claudi grew slowly and continuously across high-relief topography, forming high subaerial banks; partial drainages left remnant horseshoe lakes that continued to expand into virgin yedoma, mobilizing carbon at roughly the same rate irrespective of lake drainage. The ∼2× discrepancy between simulated CH4 production and observed emission rates in Claudi likely results from misestimation of hot spot ebullition, labile carbon content, CH4:CO2 production ratio, or microbial CH4 oxidation.

1. Introduction

[2] A large quantity (375–500 Gt) of highly labile carbon (C) is stored in the Yedoma Ice Complex, an ice-supersaturated, silt-dominated permafrost deposit that formed in unglaciated lowlands in Siberia during the late Pleistocene [Zimov et al., 2006a; Schirrmeister et al., 2011]. Similar Late Pleistocene yedoma deposits have been identified in North America, and are common across formerly unglaciated regions of Alaska [Kanevskiy et al., 2011].

[3] In regions of ice rich permafrost, lakes can result from a well known positive feedback in which the lower albedo of pooled water promotes thawing of underlying permafrost, which consolidates, deepening the surface depression [Hopkins, 1949; Hopkins et al., 1955]. Originally known as thaw lakes, they are now commonly termed thermokarst lakes, or more generally thermokarst terrains, in reference to landscapes shaped by dissolution of rock (karst). Over hundreds of years, thermokarst lakes can form deep taliks [Ling and Zhang, 2003; West and Plug, 2008], which are regions of perennially thawed soil beneath the lakes. Carbon sequestered in permafrost is mobilized in taliks, even in regions where the mean annual surface temperature is well below 0°C [Hinkel et al., 2003; Burn, 2005; Walter et al., 2007a].

[4] Thermokarst lakes and drained lake basins covering 46% of the western Arctic Coastal Plain [Frohn et al., 2005; Hinkel et al., 2005, 2007] and 73% of the tip of Cape Espenberg on the northern Seward Peninsula [Charron, 1995; Jones et al., 2011] are a prominent feature shaping the landscape in these regions. With the ability of thermokarst lakes to produce methane (CH4) through anaerobic decomposition at depth, they represent a potentially large source of atmospheric CH4 that could drive a significant positive feedback to climate warming at high latitudes [Zimov et al., 1997; Bockheim et al., 2004; Zimov et al., 2006a; Walter et al., 2006, 2007a; Khvorostyanov et al., 2008a; Walter Anthony, 2009; Isaksen et al., 2011]. Thermokarst lakes may be an important consideration when estimating methane production in northern regions; however, quantification of the potential release rate of methane from thermokarst lakes is needed. An important question that has not been addressed in previous studies is how rapidly permafrost carbon is mobilized beneath lakes evolving on a thermokarst-impacted landscape.

[5] Recent 1-D and 2-D numerical models of thaw lakes and their taliks have elucidated the fundamental thermal and mechanical processes responsible for thaw lakes [e.g.,Ling and Zhang, 2003; Zhou and Huang, 2004; Pelletier, 2005; West and Plug, 2008; Plug and West, 2009; Taylor et al., 2008; Khvorostyanov et al., 2008b, 2008c]; however, they have not attempted to account for the interaction of thermokarst lakes with the topography that defines the lakes. The 2-D landscape model ofvan Huissteden et al. [2011] accounts for lake/lake and lake/river interactions through stochastic methods; however, it does not simulate the effect of topography on the growth and draining of lakes, or the role of lakes in modifying topography. Interaction with topography is a major factor in determining how large and how old thermokarst lakes can be, which dictates the depth of talik penetration and total carbon mobilization.

[6] In this paper we employ a 3-D numerical model of thermokarst lakes that includes topographic interactions, built upon the 2-D landscape model ofPlug and Werner [2000], to simulate the development of a specific thermokarst lake and a neighboring drained thermokarst lake basin in Northwest Alaska, known informally as Lake Claudi and Pear Basin (Figure 1). This two lake simulation is an important first step in predicting methane release from landscapes with many lakes responding to and creating topography over millennia and through multiple lake generations.

Figure 1.

Pan-sharpened, multispectral IKONOS image (GeoEye Satellite Image, IKONOS scene acquired 24 August 2006, Dulles, Virginia), with a resolution of 1 m showing Lake Claudi, Pear Basin, and the Kitluk River.

1.1. Study Area

[7] While the model presented here is relevant to all yedoma-dominated landscapes, including those in Siberia, this study focuses on a set of thermokarst features located on the northern Seward Peninsula, Alaska (66.6°N, 164.5°W). The northern Seward Peninsula, one of the major lake districts in Alaska [Arp and Jones, 2009], is underlain by continuous permafrost and covered by extant lakes and drained lake basins with relatively few intact upland yedoma permafrost deposits that have not thawed since their formation in the Pleistocene, hereafter called “virgin yedoma.” Yedoma deposits are composed of organic-rich Late Quaternary aeolian loess interspersed with tephra layers from nearby volcanic sources, and sandy coastal sediments [Höfle et al., 2000; Beget et al., 1996]. Alas, or drained lake basin, deposits consist of reworked silt, tephra and alternating peat-rich horizons from peat-forming wetlands that frequently succeed lakes after drainage [Jones et al., 2011]. The study area remained unglaciated during the Last Glacial Maximum [Manley and Kaufman, 2002] and was dominated by graminoid-herb tundra with a cold-dry climate [Colinvaux, 1964; Ager, 1982, 1983]. Modern climate in this region is subarctic, with mean summer temperatures of 7–10°C, mean winter temperatures of −12°C, and mean annual precipitation of <58 cm [Arp and Jones, 2009].

1.2. Field Work

[8] Topographical mapping of the study area was conducted in August 2008 using a centimeter-accurate RTK differential GPS (Leica Geosystems AG). We mapped the bathymetry of Lake Claudi using an outboard motor boat equipped with a hydroacoustic system (BioSonics DT-X Digital Echosounder) and Garmin 76CSx GPS with Wide Area Augmentation System differential correction. Bathymetry data were recorded along 24 E-W and 3 N-S transects across the lake surface in August 2008 (Figure 2). Similar setups show a depth error of <0.2 m [Valley and Drake, 2005], well below the error associated with the 60 m DEM used in the simulations shown here.

Figure 2.

Topography from NED DEM [Gesch et al., 2002; Gesch, 2007] and bathymetry from echosounding DGPS transects (white) of Lake Claudi. Contours indicate the predicted expanding Lake Claudi boundary at years 0 (black), 100 (red), 200 (red), and 300 (black) simulated into the future.

[9] We estimated whole lake annual CH4emissions from Lake Claudi as the sum of measured seep ebullition in November 2008, and assumed background ebullition and diffusive flux based on the relationship between modes of emissions observed by intensive high temporal resolution, year-round measurements in similar thermokarst lakes, Shuchi Lake and Tube Dispenser Lake, in Siberian yedoma deposits [Walter et al., 2006; Walter Anthony et al., 2010]. This follows the field survey method of K. M. Walter Anthony and P. Anthony (Constraining spatial variability of methane ebullition in thermokarst lakes using point process models, submitted to Journal of Geophysical Research, 2011) and Walter Anthony et al. [2010], which demonstrated November to be the optimal time for quantifying lake methane ebullition.

[10] We collected samples of ebullition bubbles from 8 seeps at the surface of Lake Claudi using submerged bubble traps or from bubbles frozen in lake ice in Lake Claudi following methods described by Walter et al. [2008]. Gases were collected into 60 ml glass serum vials, sealed with butyl rubber stoppers, and stored under refrigeration in the dark until analysis in the laboratory. We determined δ13CCH4 to inform 14C dating using a Finnegan Mat Delta V mass spectrometer at Florida State University. Samples of ebullition gas were then combusted to CO2, purified, and catalytically reduced to graphite [Stuiver and Polach, 1977], and the 14C/12C isotopic ratios were measured by accelerator mass spectrometry at the Woods Hole Oceanographic Institution's National Ocean Sciences AMS Facility.

2. Numerical Model

[11] The 3-D numerical model presented here simulates thermokarst terrains in which multiple lakes initiate, grow and drain over thousands of years. In this model, surface topography defines water flow and pooling into lakes that modify the topography through thaw driven subsidence, bank erosion and catastrophic lake drainage events. Simulation of the subsurface temperature and water phase is used to predict carbon mobilization through anaerobic methanogenesis.

[12] We hypothesize that the long-timescale dynamics of thermokarst lakes are largely determined by the topography they interact with and gross thermokarst processes, but are not sensitive to the details of the submeter- and subannual-scale processes. Bank retreat, for example, results from subgrid-scale processes, such as thaw undercutting, bank collapse and slumping, which are abstracted collectively in the model as a local lowering of the ground surface due to ice loss and a small amount of redistribution of sediment into the lake. This and other abstractions of small-scale processes to the 10 m scale, which will be described, enable us to simulate both individual lakes and landscapes with multiple lakes over thousands of years.

[13] This model is coded in the Matlab programming language. We used a quad core 2.8Ghz AMD desktop computer running Ubuntu Linux; however the code is single threaded and can run on any pc running the Matlab computing environment. Run times vary with simulation and cell size, topography, and lake properties (e.g., number of lakes, frequency of drainage events). The largest simulations shown here, which include both Pear Basin and Lake Claudi, are 178 by 160 cells and typically simulate between 1000 and 2000 years per hour. As the complexity of the terrain increases with thermokarsting, run times tend to increase; however, this only becomes substantial with a high frequency of lake drainage events, which does not occur in the simulations shown here. The 5 year time step used in looping through the physical processes was chosen to reduce the numerical cost of calculating the surface water flow, which does not in general change dramatically between iterations. Individual process algorithms can use sub-time-steps, for example, the thermal calculation employs a 1 year time step. Smaller or larger cell sizes and time steps (minimum time step is 1 year) may be used in this model; however, the validity of the process abstractions should be considered for much smaller or larger cell sizes. For details of the model code, please contact the authors.

2.1. Three-Dimensional Model Structure

[14] The numerical model structure is a two-dimensional array of square cells (10 m wide) defining the surface topography, each underlain by a column of sediment extending past the depth of ice-rich permafrost to a uniform elevation across the entire simulation (−47 m in the simulations presented here). Each column is divided into 50 layers, creating a 3-D array of cells in which the following state variables are tracked: soil thickness, excess ice thickness (ice in excess of the soil pore space), frozen fraction, and labile carbon content. In the simulations shown here, the mean layer thickness is 1.6 ± 0.3 m, varying with excess ice content, which is up to 50% of the layer thickness. Subsurface cells do not interact laterally with cells of neighboring columns as they are at disparate elevations, and of varying thickness depending on the columns surface elevation and the cells current excess ice content, thus this model is more accurately termed a 2 + 1-D model. In the simulations shown here, lateral simulation boundaries are defined to be static rivers: surface water sinks with fixed surface elevation.

2.2. Sequence of Operations

[15] During each time step (Δt = 5 years was used in the simulations shown here for numerical efficiency), the numerical model iterates a sequence of six algorithms, each simulating a physical process or processes (Figure 3). The first algorithm simulates surface water, including: precipitation, evaporation, and downslope surface water flow, producing a map of surface water thickness across the modeled space. The second algorithm simulates catastrophic lake drainage including the resulting water transfer and channel erosion. The third algorithm simulates talik formation and refreezing, taking into account surface temperature, winter lake ice, water temperature, permafrost temperature, phase change, and soil compaction due to the melting of excess ice. The fourth algorithm simulates bank retreat due to thawing and the resulting compaction. For the third and fourth algorithms, representing the fundamental thermokarsting processes, the model employs a 1 year time step cycling between these two algorithms. The fifth algorithm simulates both subaerial and subaqueous diffusive mass movement. The sixth algorithm simulates the sequestration of active layer carbon, the production of CH4 and the simultaneous reduction in carbon stock within talik cells. The details of each algorithm and their modification of state variables throughout the modeled space are described in this section.

Figure 3.

Sequence of submodels performed during one cycle of the numerical model. First, precipitation, evaporation, and surface water flow. Second, catastrophic lake drainage. Third, thaw and compaction, which iterate on a shorter (1 year) time step with the fourth submodel, bank retreat. Fifth, slope diffusion. Sixth, carbon sequestration and methane production.

2.2.1. Surface Water

[16] Modeled lakes are topographically defined by water pooling in depressions. Basins with a maximum water depth greater than 0.2 m are considered lakes, which is approximately the total summer lake evaporation in the simulations shown here (210 mm), thus lakes are perennial water bodies. No hydrological distinction is made between shallow lakes (<0.2) and standing water such as fens and bogs, as they are not treated as thermokarst lakes. This overly simplistic water depth definition of thermokarst lakes glosses over the wide range of shallow water behaviors seen on thermokarst terrains; however, no good model for the initiation of thermokarst lakes currently exists, and the processes involved are likely below the resolution of this model. In the simulations shown here, we assume thermokarst lakes initiate since we are modeling terrains that are known to have thermokarst lakes. While it is outside the scope of this study, neglecting lake initiation details may overpredict or underpredict the number and coverage of thermokarst lakes because it assumes all water greater and no water less than 0.2 m deep initiates a lake, when in fact nonthermokarst lake shallow water bodies are observed on thermokarst terrains. Claudi and Pear lakes are manually initiated by creating a small depression in the surface as described in section 4.1.

[17] Water depth in each cell is determined by precipitation, evaporation and topographically driven surface water flow using the following algorithm:

[18] 1. define lake basins by filling topographic depressions [Planchon and Darboux, 2002];

[19] 2. add total/net precipitation in lake/land cells;

[20] 3. route water downslope to basins or rivers: for each land cell in order of descending elevation transfer water to downslope neighbors in proportion to slope [Quinn et al., 1991];

[21] 4. evaporate Δt − 0.5 year water from lake cells;

[22] 5. for each lake, in order of descending basin sill elevation, either:

[23] (i) reduce lake size to the available water volume, or

[24] (ii) route excess water through the sill to a downslope lake or river;

[25] 6. evaporate 0.5 year water from lake cells and reduce lake area accordingly.

[26] This treatment of multiple years (5 year time steps) in one iteration is intended to simulate long-term average lake levels, but of course neglects the subannual lake expansion/shrinkage due to precipitation-driven water availability that occurs in some natural lakes [Plug et al., 2008]. Half of the annual evaporation is applied after the water routing to approximate the mean summer water level and subaerial bank heights. This assumes water is abundant and early summer lake size is determined by the sill elevation, which is typical with the climate parameters used in these simulations.

2.2.2. Catastrophic Lake Drainage Events

[27] In this part of the model, we simulate the gross progression of drainage events, including: erosion of the sill, shrinkage of the lake, and propagation of the eroding channel up into the lake. For each lake, in order of descending sill elevation, drainage events are tested for and proceed each model time step using a drainage event simulation of the simultaneous water discharge and channel erosion resulting from a random overfilling of the water level (between 0 m and 0.1 m above the sill, but not more than the annual lake discharge), conceptually resulting from the blockage of the sill by an ice or snow dam [Marsh and Neumann, 2001]. The magnitude of the random overfilling is chosen as the largest of N random numbers with N being the number of years in a single model time step (N = 5 herein). Multiple events in the same lake during a single time step are missed; however, the time span between drainage events is generally much larger than a single time step and drainage events change the topography such as to reduce the propensity for future events.

[28] Discharge and erosion during the drainage event are simulated by iterating the following algorithm for each cell in the lake or drainage path until the maximum erosion rate in all cells drops below 1/100 of the event maximum: (1) calculate the water surface elevation drop (ΔHw) to the neighbor with the lowest water surface (Ndrain), (2) transfer the mobilized water (water that entered from upstream on the previous time step) plus ΔHw/4 to Ndrain, and (3) erode the bed linearly proportional to the total transferred water volume (Vwater) and the bed slope (Sdrain): E = CerodeVwaterSdrain, in which Cerode is an erosion coefficient (= 0.01), which we calibrated to produce channel volume to lake volume ratios of about 1/100, within the range seen in natural thermokarst lakes on the Seward Peninsula (L. J. Plug and D. W. Gardner, Landscape modification by drainage of Arctic thermokarst, unpublished manuscript, 2008). Erosion is modeled as linearly proportional to discharge (volume/time step) because the event time is not simulated by the water routing algorithm; erosion proportional to some other power of the discharge, would make the erosion sensitive to the numerical details (e.g., water moved each iteration) by erroneously linking the numerical time step to a physical time interval.

[29] Modeled drainage events likely do not reproduce the dynamics of natural drainage events, but they do produce geomorphically similar simple branching drainage channels dominated by a main trunk as we observed on the northern Seward Peninsula during flights, on the ground, and in satellite imagery (Figure 1).

2.2.3. Subsurface Thermal Regime

[30] Talik formation and soil compaction below thermokarst lakes impact lake growth through their effect on lake depth, which determines shore profile and mean water temperature. Simulation of the talik depth is also essential to predicting the mobilization of ancient carbon below thermokarst lakes. Numerically solving the heat equation, as has been done in recent 2-D thermokarst lake models [Ling and Zhang, 2003; West and Plug, 2008], is not computationally feasible for our 3-Dmodel given a potential simulation space of many square kilometers and time frame of thousands of years. Hence, we simulate talik thawing and refreezing using an approximate solution to the Stefan problem that assumes temperature change in the talik is negligible compared to latent heat release at the phase change boundary (PCB) allowing a simple and widely used conduction equation to be employed [Osterkamp and Gosink, 1991; Burn, 2000; Woo et al., 2004; Hayashi et al., 2007]. Our thermal model tracks the PCB in each column by converting heat flux (Fourier's Law) into a thickness of ice melted/frozen (ΔHice, volume per unit area):

display math

in which ktalik is the mean thermal conductivity of the talik, Tbed is the mean lake bed temperature, Tpcb is the temperature at the PCB (= 0°C), Hthaw is the PCB depth or talik thickness, L is the latent heat of fusion of water, and ρice is the density of ice. Equation (1) could be applied directly; however, because the thawing front is descending, the temperature gradient decreases through a time step, thus requiring small time steps. We instead use a solution that is less sensitive to time step, which is derived (Appendix A) by substituting the mean thaw depth over a time step, Hthaw + ΔHthaw/2, for Hthaw in equation (1):

display math

in which Rice is the ratio of ice volume to dry soil volume. Tbed is a time weighted average of the mean summer and winter water temperatures. Below the depth of winter lake ice, we assume the mean winter water temperature increases linearly from 0°C at the ice up to a maximum of Twaterdeep reached at Hwaterdeep m (= 4 m). Lake ice thickens with the square root of time over the winter to a predefined maximum (1.5 m in these simulations [Plug and West, 2009]). The lake bottom temperature is assumed to be 0°C when the lake ice thickness exceeds the water depth, which overestimates the initial deepening rate because it neglects refreezing of the lake bed where the water is shallow. Accurate modeling of talik formation below shallow water is a factor in predicting lake deepening and expansion of shallow water lakes and littoral shelves [Burn, 2002], but it requires assumptions about snow depth beyond the scope of this model. However, calibration of the bank thermal diffusivity and the subaqueous diffusion constant should produce reasonable model behavior for the steady climate used here. This approximate solution of the Stefan problem (equation (2)) has the added advantage that it can be applied at zero thaw depth.

[31] Simple 1-D heat conduction has been used to approximate thaw progression in permafrost [e.g.,Burn, 2000; Hayashi et al., 2007]; however, it does not simulate lateral heat exchange with the surrounding permafrost, which could be important at lake edges. For this purpose, we augment this 1-D thermal model with an estimation of lateral heat flow using a depth averaged approach. We assume that ice volume melted at the PCB is reduced by lateral heat flow from the talik into the surrounding permafrost and by vertical heat flow into the underlying permafrost.

[32] A 1-D vertical three-point linear heat conduction calculation is used to calculate the mean permafrost temperatureTperm below talik and land cells, assuming an upper boundary temperature of 0°C and the mean annual surface temperature Tmean. A geothermal gradient of 0.025°C m−1 is applied at the lower boundary below land cells. However, since the entire depth of permafrost is not simulated a zero heat flow lower boundary is applied below the subtalik permafrost, thus producing a conservative estimate of the rise in permafrost temperature below a lake.

[33] Depth averaged lateral heat flow is simulated in two steps. First, lateral heat flow is calculated for an upper layer defined by the maximum talik depth within neighboring columns. Temperature changes within the talik are converted to ice volume changes at the PCB. Second, lateral heat flow is calculated for the entire soil column and the heat flow from the first step is subtracted. The remaining heat flow modifies the permafrost temperature. For both heat flow calculations a Dufort-Frankel scheme, with an initial forward time central space time step is used to solve a 2-D (lateral) heat diffusion equation (equation (3)) [Ling and Zhang, 2003, 2004; West and Plug, 2008; Plug and West, 2009]. The main consequence of heat exchange between the PCB and the permafrost is to limit the talik depth near the edges of slowly expanding lakes.

display math

in which kij is the thermal conductivity in column ij, ρij is the density, and Cij is the heat capacity. These three properties are calculated from the porosity, frozen fraction, excess ice content, and mineral and organic fractions using a geometric mean for kij [Johansen, 1975; Ling and Zhang, 2004] and a weighted average for ρij and Cij of standard property values of the constituents given by Williams and Smith [2008].

[34] We compared this simplified thermal model with a fully three dimensional numerical solution of the heat equation in the permafrost surrounding and beneath a 200 m wide lake with the same soil and thermal parameters used in the simulations described below, but imposed a condition of no bank retreat and no diffusive mass movement (Figure 4). The 3-D full numerical solution of the heat equation accounts for latent heat by modifying the thermal diffusivity over a 1°C thawing envelope as done byLing and Zhang [2003] and West and Plug [2008]. For the purpose of this modeling effort, our simplified 3-D model adequately captures the retarding effect that lateral heat transfer has on the growth of the talik while running ∼2000 times faster. It should be noted that this approximate solution is influenced by the depth of permafrost, and should be tested under the conditions being simulated.

Figure 4.

Downward progression of the talik, shown at years 0, 50, 200, 500, 1000, and 1500, for the thermal model used here (blue) and a 3-D finite difference solution of the heat equation (black).

[35] After each iteration of the thermal algorithm, compaction is simulated by removing all subsurface water in excess of the pore space, reducing layer thicknesses and shifting overlying layers downward. This water is added to the overlying lake, thus increasing lake depth locally and preserving lake water level. In nature compaction rates vary with the hydraulic conductivity and depth; however, we assume progression of the thawing front is much slower than the rate of soil consolidation. In tests where we assigned a 10 year timescale to compaction there was no qualitative change in model behavior; however, this should be revisited for fine grained soils with low hydraulic conductivity. Thaw consolidation rate would likely have its greatest impact on lake initiation which is not treated explicitly; the rate of soil consolidation relative to the timescale of the thermal disturbance that initiates a thermokarst lake warrants further investigation.

[36] In addition to computational efficiency, the simplified 3-D thermal model easily treats changing layer thickness due to thaw compaction because layers are not linked computationally to neighboring cells. In contrast, the finite difference solutions used in other models [Ling and Zhang, 2003; Khvorostyanov et al., 2008b] do not treat compaction, but instead assume a higher porosity. The limitation of not treating compaction is problematic for simulating thermokarst lakes in substrates with significant excess ice. In these settings, compaction of thawed sediments maintains high vertical thermal gradients and fast thaw front decent. The probabilistic method of West and Plug [2008], in which whole cells are removed to simulate compaction, does treat compaction and allows for a more accurate thermal simulation than the depth averaged approach used in our simplified thermal model, but at the cost of discretizing cell thickness and surface topography.

[37] Refreezing of taliks after lake drainage is calculated using a descending freezing front and the same assumption of linear temperature gradient between 0°C at the freezing front and the mean annual temperature at the ground surface.

2.2.4. Growth of Excess Ice

[38] This model also allows for the growth of excess ice in near-surface layers that have been depleted by thermokarst lake formation. We model this as a decay of the ice deficit toward a specified maximum excess ice fraction, which decreases linearly with depth:

display math

in which τ (= 1000 years) is the excess ice growth timescale, Fxsice is the volumetric fraction of excess ice, and Ficexsmax (= 0.5) is the specified maximum excess ice fraction, D is the layer depth, Dmax (10 m [Black, 1976]) is the maximum depth of excess ice growth, and Vsoil is the soil volume in the layer. The excess ice growth timescale is not well constrained, with ice growth deriving from both segregation and ice wedges formation. For comparison, ice wedges [Black, 1976] grow on the order of millimeters per year to an ultimate size of a few meters, while segregation ice [Konrad, 1990] under optimal conditions can grow tens of meters thick in 1000 years.

2.2.5. Thermal Bank Degradation/Retreat

[39] In natural thermokarst lakes, the long-term mean bank retreat rate is controlled by interactions between thermal and mechanical erosion processes, including: thaw, compaction, bank collapse, sediment shielding and near shore erosion. A typical sediment packet near a lake starts frozen in ice-rich permafrost. Subaerial or subaqueous thawing caused by the thermal disturbance of the lake removes the ice matrix in and around the sediment packet. The now thawed, and likely saturated, sediment forms a thawed layer near the surface that shields the bank, slowing further thermal and physical degradation of underlying permafrost. Either through low-energy processes such as consolidation and slumping, or through high-energy processes such as bank collapse, the shielding sediment is eventually transported from the bank to the shore or into the lake. High-energy processes may transport some sediment, especially that which is fine grained, large distances into the lake. Sediment deposited at the shore shields the bank until near shore processes transport it offshore.

[40] Although the cause of lake expansion is thermal erosion of the banks, the rate of retreat can be very sensitive or largely insensitive to temperature, depending on the amount of near shore energy available to remove the collapsed bank material. In this model we assume the rate of thermal degradation, ice melting, is proportional to the total bank height, including both subaerial and subaqueous portions; however, higher banks have lower long-term retreat rates because they produce more shielding sediment consistent with observations [Jones et al., 2009, 2011].

[41] The details of thermal bank degradation are assumed to be subgrid-scale processes that at the 10 m scale of a grid cell act to remove ice, cause compaction and redistribute sediment within the cell making it susceptible to near shore erosion. Thermal degradation of the bank is simulated as a lowering of a cell's surface elevation through the removal of excess ice and a transition of sediment from a frozen, nonerodible state to a state that is erodible by near shore processes. The collapsed bank material acts to thermally shield the bank from further thawing in the same way that the thickening talik slows the thaw front propagation in the talik. A similar set of equations are used:

display math

in which Δtshield is the time it takes to rethaw the sediment deposit shielding the bank (Hshield), which we assume has water filled pores that refreeze each winter. Tsumm is the summer surface temperature; however, we use the summer air temperature assuming the difference will be accounted for in the calibration of thermal diffusivity kshield. kshield is poorly constrained because a single thermal diffusivity simplifies a complex problem involving both subaerial and subaqueous heat transfer. We calibrate kshield by assuming that the annual thermal penetration with no excess ice under the reference climate conditions is 0.5 m yr−1, which is roughly the active layer thickness. This value will limit the maximum mean annual retreat rate, so we choose 0.5 m yr−1 because it is near the maximum retreat rates observed over 50 years in lake Claudi [Jones et al., 2009, 2011]. Δtsumm is the summer length, and Rice is the volumetric ratio of ice melted to dry soil thawed as given in equation (2). ΔXice is the volume melted of ice per unit area of bank; the ice volume melted per cell area, is obtained by multiplication of ΔXice by the bank height, division by the cell width, and summation over all subaqueous neighbors.

2.2.6. Diffusive Mass Movement

[42] In addition to the thermokarst lake specific transport processes, surface diffusion of soil is included in the model to simulate slope driven transport and nearshore erosion. Soil transport is simulated using a linear relation between soil flux and slope (qx = k* math formula), applied in two dimensions with separate subaerial and subaqueous diffusion steps and diffusion constants (kland = 0.01 m2 yr−1, klake = 1 m2 yr−1). kland was determined by fitting scarp profiles to a linear diffusion model [Plug, 2003]; klake was calibrated using Pear Basin, as described below. Since bank cells are partially subaerial and partially subaqueous, they are included in both diffusion steps, using the bank elevation in the subaerial step and the water surface elevation in the subaqueous step.

[43] Diffusive mass movement acts to remove collapsed bank material from the shore, and in doing so regulates the bank retreat rate. In the model, subaqueous bank erosion is limited to the available thawed/collapsed bank sediment, Hshield. This limit prevents the model from exaggerating shore erosion and lake expansion rates under conditions in which thermal erosion is low. In contrast, under conditions with high bank sediment production, either due to high subaerial banks or high thermal bank retreat rate, diffusive removal of sediment becomes the limiting factor in lake expansion.

[44] Longer spatial scale transport of fine sediment from the banks into the lake is also simulated in this slope diffusion algorithm. A fraction (= 0.5) of the soil and associated carbon removed from bank cells during the subaqueous diffusion step is distributed across the lake bed linearly proportional to the water depth, simulating fine suspended sediment that settles slowly from the water column. The crude fraction, 0.5, was used because this algorithm abstracts a variety of poorly known, soil-dependent processes that sort and transport sediment during both bank collapse and shore erosion.

2.2.7. Peatland Carbon Sequestration

[45] Modern peatlands in drained thermokarst lake basins are both sequestering carbon and releasing methane. Over thousands of years after lake drainage, basin floors experience topographic and plant community changes, often from highly productive fens to bogs and other less productive plant communities, with corresponding changes in the carbon sequestration rate [Zona et al., 2010; M. C. Jones et al., Peat accumulation in drained thermokarst lake basins in continuous ice-rich permafrost, Seward Peninsula, Alaska, submitted toJournal of Geophysical Research, 2011]. We simulate the sequestering of modern carbon in peatlands by increasing the carbon stock with time in former lake cells with low drainage. This stored modern carbon is incorporated into lakes by lake expansion and dispersal of sediments across the lake bed (sections 2.2.5 and 2.2.6).

[46] In the model, cells are classified as peatland by the relative magnitude of the annual discharge of water and the drainage capacity determined by the local slope. We identified poorly drained areas as cells conforming to this relation:

display math

in which Kpeat is the hydraulic conductivity, Hal is the active layer thickness, S is the local slope, and Δtsumm is the summer length, Δx is the cell width, and Hw is the thickness of water passing through the cell annually. We roughly calibrated Kpeat (25 m d−1) to produce peatland area in Pear basin similar to the observed peatland; this value is well within the wide range of observed peat hydraulic conductivities [Quinton et al., 2008].

[47] Within peatland cells we apply a carbon sequestration rate that varies with time since lake drainage. We fit the total carbon given by Jones et al. (submitted manuscript, 2011) for this study site to a power function of time since lake drainage (C = 265(ttdrain)0.62, R2 = 0.85), giving a rate of carbon sequestration (g C m−2 yr−1) that decreases over time as: dC/dt = 164(ttdrain)−0.38, in which ttdrain is the time since lake drainage. The sequestration rate given by this curve is lower than the mean rate calculated by dividing the total sequestered carbon by the basal soil age because this curve describes the instantaneous rate at a given age. The predicted rate at year 1000 is 11.9 g C m−2 yr−1 while the mean rate over the first 1000 years is 19.0 g C m−2 yr−1, consistent with reported rates [Clymo et al., 1998; Vitt et al., 2000; Turunen et al., 2002; Belyea and Malmer, 2004; Robinson, 2006; Sannel and Kuhry, 2009]. However, the mean rate over the Holocene of 8.0 g C m−2 yr−1 is at the lower end of reported rates. In the model, this sequestered carbon is added to the soil layer immediately below the active layer, thus progressively reducing the mean carbon age in that layer.

2.2.8. Methane Production

[48] Methane is produced within a talik by anaerobic decomposition of organic matter contained within taberal sediments (permafrost soils that thawed in situ) [Zimov et al., 1997; Walter et al., 2006]. The primary controls on methanogenesis are the size of the available labile carbon pool, which we assume to be 1/3 of the total carbon based on consistent observations of carbon mass loss from taberal sediments beneath yedoma lakes in different yedoma regions of Siberia [Kholodov et al., 2003; Zimov et al., 2006a; Walter et al., 2007b], and soil temperature [Walter and Heimann, 2000; van Huissteden et al., 2006]. We assumed a 1:1 stoichiometric relation between CO2 and CH4 produced from the labile fraction [Conrad et al., 2002], thus the methane production is half the carbon mobilization (ΔC). We express the carbon mobilization within a cell over a time step (Δt) using a formulation similar to Walter and Heimann [2000] and van Huissteden et al. [2006]:

display math

in which, RC is the fraction of carbon converted per year (0.02), C is the labile carbon content in the cell, T is temperature, Q10 (= 2.2) is the factor by which the rate increases with a 10°C rise in temperature, and TQ10 (= 3.5°C) is a reference temperature. We estimated RC, Q10 and TQ10from the incubation experiment with Pleistocene-aged yedoma sediments reported on byZimov et al. [1997]; the range of published Q10 values is 1.7 to 16 [Walter and Heimann, 2000]; although, most observations are in the bottom third of this range [Bridgham and Richardson, 1992; Valentine et al., 1994; Rivkina et al., 2004; Petrescu et al., 2008].

[49] Given the long timescale that the model is intended to simulate, with multiyear time steps, we assume the surface release rate is equal to the depth-integrated production rate without treating the relatively short timescale transport from depth. Since the majority of simulated methane production at a given time is from a narrow zone of very high production (Figures 5 and 8), ebullition will be favored as the primary means of methane transport and the opportunity for methane oxidation in transit will be small. Discreet seep ebullition accounted for 70% and background ebullition accounted for 24% of the total methane emission from two extensively studied thermokarst lakes in Siberia [Walter et al., 2006].

Figure 5.

Methane production versus time since incorporation into the lake for each layer (black) and depth integrated (red) in a single column. Blue line is the methane production from Holocene organics.

[50] It should be emphasized that ΔCH4 is the methane production and could differ substantially from the methane emission at the surface due to methane oxidation in transit through the soil layers or in the water column, neither of which are included in this model. We assume methane oxidation in surface sediments is negligible since Lake Claudi is stratified with anoxic conditions at the bed, as has been assumed in other thermokarst lake methane modeling efforts [Stepanenko et al., 2011]. However, methane diffuses out of bubbles trapped beneath winter lake ice; field observations indicate a 34% loss of methane from bubbles trapped beneath winter lake ice [Walter et al., 2008]; also observed in Lake Claudi. Assuming all methane diffusing out of bubbles is oxidized before spring ice melt, and that the lake is ice covered for 2/3 of the year, then the annual simulated production would be a 20% overestimate of the annual methane emission.

[51] The primary goal of this model is to simulate the mobilization of Pleistocene carbon beneath thermokarst lakes; however, in order to make comparisons with observed methane emissions, recent carbon sources must also be considered. Field observations of Lake Claudi indicate negligible primary productivity with low water column chlorophyll, sparse submerged macrophytes and no methane-emitting emergent vegetation; therefore, we do not include a process-based model of methane production by the lake biomass. Instead, we assume the methane emission per unit area due to the modern biomass in thermokarst lakes is approximated by the total methane emission per unit area in nonthermokarst northern lakes, accordingly, we increase the methane emission rate in lake cells by a constant value (0.268 g m−2 yr−1) characteristic of nonthermokarst northern lakes [Schneider et al., 2009]. Vegetated lake margins also emit methane (3.4838 g C m−2 yr−1) [Schneider et al., 2009]. Although not observed in first generation lakes, like Claudi and Pear, later generation lakes frequently have methane-emitting emergent vegetation on 10–20% of the lake margin within ∼5 m of the shore. The model does not simulate the spatial variation in vegetation growth on the margin, but instead assumes a uniform lower rate, which reflects that only 15% of the margin and only half the 10 m cell width is vegetated. A constant production rate of 0.26 g C m−2 yr−1 is added to lake margin cells of second and later generation lakes. In contrast to the calculated methane production from sequestered carbon, these lake biota methane rates are for observed emission, thus account for oxidation during transport.

[52] Peatlands in drained lake basins are another hot spot of methane emission in thermokarst terrains. We assume peatland cells produce methane at a rate that decreases linearly over time since lake drainage due to plant community changes and topography evolution [Zona et al., 2010; Jones et al., submitted manuscript, 2011]. We use an initial rate characteristic of very wet sedge- and moss-dominated tundra (6.6 g m−2 yr−1 [Schneider et al., 2009]) that declines over 2000 years to a rate characteristic of polygonal ground containing both wet and dry regions (1.5 g m−2 yr−1 [Schneider et al., 2009]). The timescale for this topographic and plant community transition is not well constrained; however 2000 years since drainage is considered an old basin in reported lake classifications [Zona et al., 2010; Jones et al., submitted manuscript, 2011]. After 2000 years peatlands in the model continue to produce methane at this low rate. The emission rates used here were observed on the Lena Delta in Siberia [Wagner et al., 2003; Schneider et al., 2009], which has a lower mean annual air temperature (−14.7°C) than the Seward Peninsula, but a similar mean summer temperature of 7°C [Wagner et al., 2003]. It should be noted that complex interactions between topography, plant communities, water and nutrient availability, and temperature result in large spatial and temporal diversity of methane flux observations in peatlands, ranging from 0 g C m−2 yr−1 to well over 100 g C m−2 yr−1 [Moore and Roulet, 1995; Lai, 2009; Limpens et al., 2008]. In an effort to include a reasonable and conservative methane contribution from peatlands in this model we have oversimplified them, and acknowledge that their potential methane contribution warrants a more complete surface vegetation model.

3. Simulation Setup

[53] Simulations were initialized with the surface topography, ice content, and soil properties reflecting the current conditions in the vicinity of Lake Claudi and Pear Basin. Since only a few point observations of ice content and soil properties have been made, we assume an initially flat upland plateau with uniform subsurface ice and soil properties, and that the topography and subsurface ice content reflect the removal of ice by taliks beneath thermokarst lakes.

3.1. Initial Topography

[54] In order to simulate the evolution of Pear Basin and Lake Claudi we created a synthetic landscape that approximates the pre-Pear topography. Prior to the formation of Pear Basin and Lake Claudi this area was an upland plateau, evidenced by remnant uplands, bounded to the north and southwest by extinct thermokarst lake basins and to the east by the Kitluk River valley. To recreate the pre-Pear upland plateau, we removed Pear Basin and Claudi Lake from a 60 m NED (National Elevation Dataset) DEM [Gesch et al., 2002; Gesch, 2007]. Given the substantial modification of the DEM needed, we replaced the entire upland with a tilted plane that starts at 42 m on west side, representing the uplands west of Lake Claudi, and slopes downward toward the Kitluk River at 2 m/km reflecting the observed gentle west-east slope. Where the current Pear Basin encounters the eastern edge of the plateau, we smoothed the high plateau edge to reproduce an average topography of the plateau edge south and north of the current Pear basin as it slopes down into the Kitluk Valley. Outside the upland plateau the simulation retains the NED DEM topography, which includes the extinct thermokarst lake basins on the western and northern edges of the plateau, and the Kitluk river on the eastern edge (Figure 1).

[55] Since the topography results from compaction after excess ice melting, the elevation range of Pear basin (∼20 m measured from a DGPS transect; Figure 6) approximates the total thickness of excess ice, with a small correction for lake sediments. During the Pear simulation experiment, we found a total excess ice thickness of 22 m produces a bed that fits the profile observed with DGPS surveys of Pear Basin. We assumed a prethermokarst ice rich soil layer thickness of 44 m and a uniform 50% excess ice by volume [Zimov et al., 2006a]. We removed ice from the surface downward to create the existing thermokarst basins to the west and north of the upland plateau. Both soil and ice were removed to create the Kitluk river valley assuming hillslope erosion instead of thermokarsting was the primary process. The surface layers of the western basin were left with no excess ice because the outline and topography of Pear Basin indicate that it encountered the western edge of the plateau and drained westward while still expanding eastward toward the Kitluk River. We interpret this slow retreat of a low subaerial bank as an indication of low excess ice content. In contrast, the surface layers of the northern basin were assumed to have regrown excess ice in the near surface layers (10 m) because Lake Claudi is rapidly expanding northward [Jones et al., 2009].

Figure 6.

Measured DGPS transect through Pear Basin (black), shifted vertically to match the plateau elevation, and three transects through basins simulated with different diffusive mass movement constants of 0.75 m2 yr−1 (green), 1.0 m2yr−1 (red), and 1.25 m2 yr−1 (blue).

3.2. Soil Properties

[56] In addition to excess ice content, the model requires three soil properties in each subsurface cell: soil porosity (40%), labile carbon fraction (1/3), and mineral/organic fraction (97.4/2.6% percent of dry mass), which is an estimated mean value for yedoma loess derived from observations of soils in Siberia and Alaska [Zimov et al., 2006a, 2006b] and yields an initial carbon stock of 42 kg m−3 for dry soil or 21 kg m−3 including excess ice, consistent with other field observations [Kholodov et al., 2003; Dutta et al., 2006; Walter et al., 2007b; Schirrmeister et al., 2011]. We used a labile fraction of 1/3 as it corresponds to the observed ∼30% lower carbon content within soils beneath former thermokarst lakes relative to permafrost that remained frozen [Kholodov et al., 2003; Zimov et al., 2006a; Walter et al., 2007b]. As the observed methane emission rates in lake Claudi are typical of average thermokarst lake emission observations [Walter et al., 2006; Walter Anthony et al., 2010], we assume the soil has typical values of organic carbon content and labile fraction; however, it should be noted that these estimates linearly scale the predicted emission rates.

[57] During the creation of the initial topography, we reduced the carbon content in proportion to the excess ice reduction used to produce the topography; since rapid depletion of labile carbon occurs during talik formation (decades [Zimov et al., 2006a]), formerly thawed layers have little or no labile carbon.

[58] We calculate the methane age as a sum of the soil age in each layer weighted by the methane production in that layer. The active layer is assumed to have young labile carbon (age = 0 years) consistent with radiocarbon measurements on CO2 respired from the base of the active layer (<100 years) [Nowinski et al., 2010], reflecting the prior depletion of older organic matter not the bulk carbon age. Below the active layer, the simulated soil age increases linearly with depth from 10 kya to 56 kya, respectively reflecting the end of yedoma production at the beginning of the Holocene and the beginning of yedoma production, roughly approximated using the oldest organic matter dates obtained from Siberian yedoma at the base of the Duvanni Yar site along the Kolyma River [Walter et al., 2008]. This later date is near the upper limitations of AMS radiocarbon dating; deep yedoma may be much older. Since the modeled age of methane released at the surface scales linearly with the age profile, this poorly constrained age profile introduces substantial uncertainty in the methane age predictions, which increases with methane production depth due to the greater uncertainty of the deeper age boundary. In peatland cells, the carbon age in the sediment immediately below the active layer decreases as carbon is sequestered; however, these former talik cells have very little labile carbon after lake drainage. Soil age is tracked with soil transfer between surface cells due to slope diffusion and aggrading.

3.3. Climate

[59] In the model, climate is defined by four properties: mean annual air temperature (−5.8°C) and seasonal amplitude (16.3°C), precipitation rate (0.4 m yr−1) and evaporation rate. We used a summer land evaporation rate of 1.5 mm d−1 and a lake evaporation rate of 3.0 mm d−1 consistent with observations from the Alaskan arctic coastal plane [Mendez et al., 1998]. Temperature values approximate the current conditions on the Seward Peninsula as recorded at the nearby Kotzebue weather station (Western Regional Climate Center, Historical climate for western region of the United States, 2010, http://www.wrcc.dri.edu). Summer temperature, Tsumm, and summer length, Δtsumm, are calculated assuming a sinusoidal annual variation in temperature with the mean annual air temperature and seasonal amplitude given above. These parameters then define the thermal climate in the model and factor into the thermal bank degradation and the subsurface thermal algorithms by controlling the boundary temperature and total summer melt time for banks and shallow water. These parameters are kept constant throughout the simulations shown here, but could be varied in future simulations to explore the impact of climate change.

[60] In addition to these direct climate parameters, the model requires three indirect climate parameters: active layer depth, maximum lake ice thickness and mean annual deep water temperature; values were chosen to approximate modern Seward Peninsula conditions. The simulated active layer depth and maximum lake ice thickness are 0.6 m [Hopkins, 1949; Hopkins et al., 1955; Plug and West, 2009] and 1.5 m [Zhou and Huang, 2004; Plug and West, 2009]. The simulated mean annual deep water temperature is 3°C, which is within the range of observed thermokarst lake bottom temperatures (1.5°C to 5.5°C) [Burn, 2002, 2005], is the suggested average for deep water [Burn, 2005], and is within the range of values used in previous simulations [Ling and Zhang, 2003, 2004; Zhou and Huang, 2004; West and Plug, 2008; Plug and West, 2009; Taylor et al., 2008]. These later parameters also can vary systematically in the model with a change in mean annual temperature or amplitude from an assigned reference value using the accumulated degree days of thaw/freeze; however, no climate change was imposed in the simulations shown here.

4. Simulation Results and Discussion

[61] We report on three simulation experiments. First, a simulation of the formation of Pear Basin from a small protolake is used to calibrate the model. Second, the formation of Lake Claudi from a small protolake is simulated. Third, the long-term evolution of Pear Basin and Lake Claudi is simulated.

4.1. Simulation of Pear Basin

[62] As an initial test of the model, we simulated the evolution of Pear Basin (Figure 7a) from a 30 × 30 m, 1 m deep lake near the center of the current basin. A series of simulations were run, adjusting the lake initiation point, the subaqueous mass wasting diffusion constant, and the total thickness of excess ice to find a final lake that approximated the outline of Pear Basin and a DGPS transect of the basin floor. Pear Basin makes an excellent test case both because of its simple plan view geometry, which is mostly circular with a slight westward elongation (Figure 1), and because of its interesting floor profile, which contains a central bulge and lateral ramps (Figure 6). Another diagnostic feature of an accurate reconstruction is that Pearlake flowed westward at some time during its development, as can be seen in the IKONOS image (Figure 1), then the final drainage flowed eastward into the Kitluk River.

Figure 7.

(a) Plan view of Pear Basin after drainage at year 1390 with surface elevation in false color (scale in meters). Black contours indicate lake size at years 100, 400, 700, 1000, and 1390. Blue contour is the observed outline of Pear Basin. (b) Mean bank retreat rate over time (blue) and 50 year running average (black).

4.1.1. Calibration

[63] The prominent lateral ramps in Pear Basin result from Pleistocene excess ice buried deeper than the maximum depth of talik penetration, which decreases radially, and indicates radial lake growth and talik deepening of comparable timescales. The width and slope of these ramps allow us to roughly calibrate the subaqueous diffusion constant, which is a primary unknown constant controlling bank erosion rate. We compared a differential GPS transect across Pear Basin with simulated transects. Figure 6 shows the measured DGPS transect and three modeled transects from lakes that approximate the outline of Pear Basin when they drain but differ in the subaqueous diffusion constant used. The best fit results are from a subaqueous diffusion constant of 1 m2 yr−1, or 100 times the subaerial diffusion constant, indicating high erosion in the near shore.

[64] This calibration is only as good as the talik deepening approximation, and the accuracy of the assumed depth and distribution of excess ice; however, we checked this fitted subaqueous diffusion constant with a series of simulations of a circular lake expanding on a flat plane with different imposed subaerial bank heights. Bank heights of 0.2 m, 0.5 m, 1 m, 2 m and 5 m resulted in bank retreat rates of 0.67 m yr−1, 0.44 m yr−1, 0.31 m yr−1, 0.23 m yr−1 and 0.14 m yr−1, respectively. These values are within the range of measured long-term bank retreat rates seen on the Seward Peninsula [Jones et al., 2011] and the central coastal plain of northern Alaska [Jorgenson and Shur, 2007].

4.1.2. Morphology

[65] The initial expansion of Pear Lake in the model is rapid, and the lake is roughly circular (Figure 7). Bank retreat is ∼0.5 m yr−1, but as the eastern end of the lake expands downslope the sill elevation and water surface elevation decrease, effectively raising the bank height around the remainder of the lake and decreasing the mean bank retreat rate. Lower bank retreat on the northern, western and southern margins as the eastern margin moved downslope caused the elongated pear shape of Pear Basin. Furthermore, the lake narrows downslope because the increase in subaerial bank height away from the sill causes the sill to pull away from the remainder of the lake. This basic bank height dependence indicates that the morphology of the initial plateau could be fine tuned with further field data to enhance the similarity between the simulated and natural thaw lake basins.

[66] In this model, the amount of lake elongation is impacted by a number of variables, including the subaqueous diffusion constant, the surface slope, and the evaporation rate. High near shore erosion causes more nearly circular lakes because elongation relies on shielding of shorelines with high subaerial banks. Higher evaporation produces more circular lakes because it raises the bank heights and lowers the ratio of high to low bank heights around the lake. It should be noted that this model assumes a single near shore erosion constant, thus simulated lake shape is determined largely by bank height as suggested by Pelletier [2005]; however, other processes that make the near shore erosion rate nonuniform, such as wind driven waves [Carson and Hussey, 1962] or variability in snow accumulation, which has a strong influence on lake ice thickness [Duguay et al., 2003; Brown and Duguay, 2010] are not included, but may be significant factors in determining the plan view shape for some natural lakes.

[67] A topographic bulge, as noted by West and Plug [2008] and Mackay [1997], sometimes exists in the center of drained thermokarst lake basins. In the model, a similar bulge results from the redistribution of soil from the banks toward the center as the lake expands. While the talik expands downward through ice rich sediments, the older central region of the lake is lower than the edges due to greater thaw depth and compaction, thus the soil moves downslope toward the center. Since the total buried excess ice is uniform, complete thawing of the ice rich layer beneath the lake results in uniform surface lowering, which in the absence of lateral soil transport or soil deposition would result in a flat lake bed; however, prior removal of soil from the banks and soil deposition in the center results in a higher final surface elevation in the center. It should also be noted that continual soil deposition from an external sediment source would also produce a bulge in the oldest regions of the lake due to the greater time over which they experience deposition.

4.1.3. Methane

[68] The first generation of Pear Lake is an ideal thermokarst lake in which to explore methane production, showing only a couple simple patterns associated with radial lake growth and the differential lake expansion rates that cause elongation.

[69] First, simulated methane production is highly concentrated on the outer edge of the talik (Figure 8) where new labile carbon is being mobilized. The depth-integrated methane production (Figure 8d) shows a similar pattern with a zone of high methane production at the lake edge. After a rapid drop from the lake edge, methane production decreases slowly toward the center reflecting the slowly decreasing talik deepening rate. Depending on the lake age, depth-integrated methane production within one cell (10 m) of the lake edge is 2 to 3 times the mean interior production rate, in agreement with observations of much higher methane flux near lake edges in thermokarst lakes in Siberian yedoma where 15% of the lake area at the margin accounted for 79–90% of whole-lake methane emissions [Walter et al., 2006]. However, in simulations, the exact fraction of the methane released at the margin depends on lake age and whether the talik has exceeded the maximum depth of buried labile carbon, which takes about 1000 years (Figure 5). At year 700 (Figure 8b), the lake average methane production is 190 g CH4 m−2 yr−1, and the marginal 15% area produces 25% of the methane; at year 1390, the lake average methane production is 114 g CH4 m−2 yr−1, and the marginal 15% area produces 43% of the methane. If the talik had penetrated the carbon rich layer across the entire interior, the outer margin would produce between 80% and 90% of the methane depending on the width of the high-production zone. Depending on lake age, the simulated lake average methane production rate is up to 10 times the methane emission rates observed byWalter et al. [2006] and Walter Anthony et al. [2010] (25 g CH4 m−2 yr−1), which were high compared to previous estimates [Zimov et al., 1997] due to the novel inclusion of point-source and hot spot ebullition seeps. This discrepancy will be discussed in the presentation of the Lake Claudi simulation.

Figure 8.

(a–c) Methane production and (d) depth-integrated methane production at years 200 (Figure 8a, red in Figure 8d), 700 (Figure 8b, green in Figure 8d), and 1390 (Figure 8c, blue in Figure 8d) in an east-west cross section through the talik down the axis of Pear Basin with false color in Figures 8a–8c indicating the methane production rate (scale in g CH4 m−2 yr−1) in each cell on a log scale.

[70] The second pattern exhibited is that the thickness of the high–methane production zone decreases with talik depth (Figure 8), since talik deepening decelerates due to the lower thermal gradient (depth increases with the square root of time [West and Plug, 2008]). Methane production is also more concentrated on the slower moving western margin then on the faster moving eastern margin, again because the rate of mobilization is lower on a slower moving margin.

4.1.4. Flow

[71] Water flow in these simulations is a small factor in lake dynamics because the lake sizes are not determined by water availability, with full basins and sill overtopping being the most common condition; this is in contrast to some thermokarst lakes in which the annual precipitation largely determines lake size [Plug et al., 2008]. However, lake outflow direction is diagnostic of model performance and channelized flow from lake drainages plays a critical role in lake dynamics. This simulation replicates the east-west flip-flopping of the surface water flow in Pear Lake; however, it should be noted that this behavior is very sensitive to the calibrated lake initiation position. Drainage is initially east toward the Kitluk River due to the eastward slope of the upland plateau. At year 980, the western edge of Pear Lake expanding into the plateau edge drops below the eastern drainage outlet and the drainage switches to the west into an extinct thermokarst basin. The steep western edge of the upland plateau causes a rapid lowering of the sill and water surface elevation, increasing the mean bank height and decreasing the mean bank retreat rate (Figure 7b). However, the bank retreat into the western edge is slowed by the low excess ice fraction that we assumed resulted from depletion by the extinct thermokarst lake.

[72] At year 1390, the eastern edge of Pear lake drops below the western drainage outlet triggering a catastrophic lake drainage event (Figure 7a). Prior to the drainage event the lake area was 7.6e5 m2 and the lake volume was 9.5e6 m3, well in line with the observed lake area, 8e5 m2, and lake volume, 1.1e7 m3. The event produced an eroded channel volume of 9e4 m3, half the observed eroded channel volume, 1.8e5 m3 (Plug and Gardner, unpublished manuscript, 2008), a reasonable fit to observation given this crude drainage model.

[73] The timing of this event is sensitive to the scarp profile, which in this case is steep enough to trigger the event as soon as the drainage transitions to the east. This drainage event was a complete drainage of the lake; however, incomplete drainage events are also common in this model depending on the size and depth of the lake, as well as the drainage channel profile.

4.2. Simulation of Lake Claudi

[74] In contrast to Pear Basin, which is a mostly circular, fairly uncomplicated thermokarst lake basin, Lake Claudi presents a much greater challenge to the model. Lake Claudi is elongated north-south, with low banks on the northern and southern edges expanding rapidly into previous thermokarst basins of different ages and high banks on the eastern and western edges expanding slowly into virgin yedoma. This complex scenario results in lake dynamics that are interesting, but are also sensitive to the topographic details and initial conditions.

[75] We simulated the formation of Lake Claudi (Figure 9) using the same initial conditions, upland plateau topography and climate parameters as used for Lake Pear formation. Lake shape and the proximity of its southern margin to the edge of Pear Basin constrain the initiation point of Lake Claudi. Since Claudi currently drains to the north, and had a much lower water level than Pear Lake given its current incision into the plateau, we assume Claudi and Pear were distinct lakes prior to the drainage of Pear, otherwise Pear would likely have drained to the north through the Claudi sill. The edge of Lake Claudi has migrated ∼80 m into the edge of Pear basin in the ∼250 years since Pear drained [West and Plug, 2008]. Since the southern margin of Claudi would have been a slowly retreating high bank margin before it reached the edge of Pear, it must have been within meters of Pears margin when Pear drained.

Figure 9.

(a) Plan view of simulated Lake Claudi topography at year 2000 with elevation in false color (scale in meters). Black outlines indicate the lake margins at years 100, 300, 1200, and 2000, and red outlines indicate the present Lake Claudi margin. (b) Simulated mean bank retreat rate over time.

[76] Through a series of simulations, we determined a likely initiation time and position for Lake Claudi. We found the initiation time of Lake Claudi was roughly coincident with the initiation of Pear Lake. Lake Claudi has a slight north-south elongation, with a larger southern section (Figure 2). This asymmetrical elongation can be formed by two initial lakes that merge after an early expansion phase. The northern lake encounters the plateau edge to its north shorty after initiation. Lowering of the sill and water level as the lake expands into the plateau edge, raises subaerial banks on the other margins and slows lake expansion. The southern lake expands rapidly (∼0.5 m yr−1; Figure 9) across the virgin upland until it intersects with the northern lake. Once these two lakes join, higher erosion on the jutting shoreline smooths it. Further evidence for this two lake initiation hypothesis can be seen in bathymetric transects which exhibit slight northern and southern bulges (Figure 2).

[77] In Lake Claudi, the observed bank retreat rate is low on the high bank eastern and western margins (∼0.1 m y−1 and ∼0.2 m yr−1, respectively), and high (0.5 to 0.7 m yr−1) on the low bank southern and northern margins [Jones et al., 2009, 2011]. Simulated Lake Claudi at year 2000, when the lake margin most closely matches the current lake margin, exhibits a subdued form of this pattern with bank retreat rates of ∼0.05 m yr−1 on the eastern and western banks, and ∼0.1 m yr−1 on the southern margin; however, the low bank northern margin has retreat rates <0.02 m yr−1, in contrast to observations that the highest bank retreat rates are observed on the northern margin of Lake Claudi. The low retreat rate coincides with a water depth on the northern margin of <1 m; however, the low retreat rate results from prior bank migration northward into the scarp lowering the water level 14 m and leaving behind a thick ice depleted layer. Lack of buried excess ice inhibits the lowering of the lake bed through thaw compaction, while a very low slope shore results in little erosion of bank sediment by subaqueous diffusion.

[78] Model behavior is sensitive to the initial topography and buried excess ice, which may not accurately reflect the true initial condition. Relatively high bank retreat rates are simulated in conjunction with the low northern bank when the lake first encounters the scarp. Prior to year 1200, sustained retreat rates of 0.2 m yr−1 over several hundred years draws out the northern margin (Figure 9a).

[79] The model predicts that the observed high bank retreat rates on two margins of Lake Claudi are not sustainable because retreat rates decrease monotonically with bank height; greater retreat and lowering of the initially lower margin quickly increases the differential rate. Either observations are capturing a short-duration configuration, or the model is missing a negative feedback that limits retreat rate at very low bank heights. This missing negative feedback could also reduce the sensitivity of the Pear Lake simulation to the initiation location, working to conform lake margins to contours where lakes encounter steep slopes (e.g., the western margin of Pear Basin) without requiring decreasing excess ice with elevation as in the current model; this is an area for future model improvement. Additionally, the large observed difference in bank retreat rates between the high eastern and western banks and the low northern and southern banks must be a fairly recent development as the current shape of Lake Claudi is not reflective of those rates over hundreds of years, which would cause more substantial elongation.

[80] Further observations need to be made to disentangle whether the model is improperly simulating the excess ice content beneath the edge of lakes or if the relation between excess ice content and retreat rate is incorrect. Attempts to simulate 0.5 m yr−1 bank retreats rates on the low banks of Lake Claudi required a subaqueous diffusion constant that was 5 to 10 times the calibrated diffusion constant, resulting in Pear Lake bank retreat rates of ∼1.5 m y−1. This bank retreat rate results in a Pear Lake formation time that is too short (∼400 years) for the talik to deplete the buried excess ice, creating a basin that is only ∼7 m deep and does not exhibit a substantial central bulge. We therefore suggest that the current expansion rates for Claudi are not be representative of the climate during Pear's (or most of Claudi's) formation.

[81] In a separate simulation, we spliced Lake Claudi bathymetry into the NED DEM topography [Gesch et al., 2002; Gesch, 2007], and assumed uniform excess ice content. Simulated expansion rates more closely match observed expansion rates with maximum expansion rates of ∼0.2 m yr−1 and ∼0.1 m yr−1 on the low southern and northern banks and ∼0.03 to ∼0.05 m yr−1 on the high eastern and western banks (Figure 2); however, they are still low by a factor of two to four from the observed rates [Jones et al., 2009, 2011]. Also, this simulation exhibits more rapid expansion to the south than to the north indicating probable inaccuracy in the initial topography; higher-resolution DEMs are needed to develop and test a more accurate model of bank retreat.

4.2.1. Methane

[82] Methane production in this simulation of Lake Claudi has the same basic pattern as Pear Lake, with high production near the lake margin diminishing toward the center (Figures 10a and 10b). This robust pattern of methane production is also seen in the observed methane emissions in Lake Claudi (Figure 10b), which resembles the simulated methane production around year 1600. The simulated concentration of methane production near the shore is even greater in Claudi than Pear Lake. In the Lake Claudi simulation, the zone of high production extends only ∼30 m from the shore due to the low lake expansion rates, which allow for a depletion of the near surface carbon (∼7 m deep) as the bank moves across the 10 m cell. Methane emission was not measured at the shore due to obstructing ice conditions during field measurements; however, observations of hot spot ebullition seeps close to thermokarst shorelines in late October 2008 indicate substantially higher methane release there, consistent with observations on other yedoma thermokarst lakes in Siberia [Walter et al., 2006] and Alaska [Walter et al., 2008].

Figure 10.

(a) Plan view of lake Claudi at 1600 years after initiation with the depth-integrated methane production in false color on a log scale (scale in g CH4 m−2 yr−1). (b) Depth-integrated methane production through A–A′ at years 1100 (red), 1200 (green), 1300 (blue), and 1600 (black) and the observed methane emission in Lake Claudi (circles) calculated as seep ebullition plus a combined background ebullition and diffusive flux of 11 g CH4 m−2 yr−1.

[83] In addition to the basic decrease away from the shore, a central region of very low methane production was observed as a paucity of ebullition seeps in the center of Lake Claudi as well as in the simulated lake due to the depletion of the labile carbon throughout the carbon rich layer (Figure 10b), which happens about 200 years after the talik extends beyond the depth of labile carbon. The residual methane production in the lake center after Pleistocene carbon depletion is from surface sediments derived from the upper soil layers at the bank, as can be seen in the young simulated methane age in the lake center (Figure 11b). Lake biota and Holocene carbon accumulation maintain a low rate of methane production in surface layers over long timescales (Figure 5).

Figure 11.

(a) Plan view of lake Claudi at 1600 years after initiation with the depth-integrated methane age in false color (scale in 1000 years) and (b) depth-integrated methane age through A–A′ and methane age observations (circles) from Lake Claudi.

[84] Simulated methane age increases from the shore until about 80 m offshore due to the increasing depth of methane production. Beyond 80 m the methane age declines rapidly reflecting the depletion of Pleistocene carbon and greater role of Holocene carbon in methane production (Figure 11b). 14C ages observed in CH4 collected from non–hot spot ebullition seeps are consistent with simulated ages; however, the pattern of decreasing age with distance from the shore begins at the lake margin, from 40,000 years at 17 m from the margin to 22,400 years at 92 m from the margin (Figure 11b). This pattern of decreasing 14C age with distance from shore was also observed in a yedoma-type thermokarst lake in interior Alaska [Brosius et al., 2012] and in thermokarst lakes formed in Siberian yedoma [Walter et al., 2006].

4.2.2. Comparison of Simulated CH4 Production and Observed CH4 Emission

[85] Although the simulated pattern of methane production shows marked similarity with the observed methane seep ebullition release in Lake Claudi, the simulated rates are roughly twice as high as the field-based estimate of methane emissions. At year 1600, simulated lake average CH4 production was 67 g CH4 m−2 yr−1, with a maximum of 197 g CH4 m−2 yr−1near the lake margin, while the field-based estimate of methane emissions as the sum of seep ebullition, nonseep ebullition, diffusion and storage release is 39 g CH4 m−2 yr−1 [Walter Anthony et al., 2010; Walter Anthony and Anthony, submitted manuscript, 2011]. Several possible factors could contribute to this difference, including: estimation of hot spot ebullition, initial carbon content, labile fraction, ice content of yedoma, talik deepening rate, production rate constant and temperature dependence, the CH4 to CO2 production ratio, and microbial CH4 oxidation.

[86] The primary factors influencing methane production in the model are: initial carbon content (2.6%, 21 kg m−3), labile fraction (1/3), and CH4 to CO2production ratio (1:1). Since methane production scales linearly with each of these parameters, errors in these values could easily explain a 2× discrepancy between prediction and observation. 2.6% is a Siberian-Alaskan mean organic carbon fraction in yedoma [Zimov et al., 2006a, 2006b], which is consistent with a data set of pan-Siberian yedoma almost an order of magnitude larger [Schirrmeister et al., 2011], and with our own observations of organic carbon content of yedoma in the vicinity of Lake Claudi (1.6% to 3.3%). Using the lower value (1.6%) would decrease methane production by ∼40%. The labile fraction is a poorly constrained soil property, with different measurement techniques reporting widely diverging values [McLauchlan and Hobbie, 2004]; however, the fraction used here is based upon a small but consistent set of observations of ∼1/3 carbon depletion in taberal sediments beneath drained thermokarst lakes within yedoma permafrost in both eastern and western Siberia [Kholodov et al., 2003; Walter et al., 2007b]. The 1:1 production ratio is based upon steady state methanogenesis with a cellulose substrate [Conrad et al., 2002].

[87] The current procedure for estimating thermokarst lake methane emissions, which is a combination of observation and estimation from average values, contains considerable uncertainty. Based on high temporal resolution, year-round flux measurements of methane diffusion and different types of ebullition in Siberian thermokarst lakes,Walter et al. [2006]found that ebullition from discrete point-source and hot spot seeps constituted 70% of whole-lake emissions. Diffusive flux and background (nonseep) ebullition accounted for 6% and 24%, respectively. Assuming the same proportions of seep ebullition, background ebullition and diffusive flux from Claudi, field measurement derived whole lake emission is 39 g CH4 m−2 yr−1. Poor winter lake ice conditions near the lake margin during field surveys may have resulted in underrepresentation of hot spots, which were only 2% of total emissions in Lake Claudi compared to 4% and 18% on similar-sized Siberian yedoma thermokarst lakes [Walter et al., 2006]. The ground-based estimate of emission could be as high as 79 g CH4 m−2 yr−1, if unmeasured hot spots account for 49% of whole lake emissions as they did at Goldstream Lake, a small, intensively studied, thermokarst lake in interior Alaska (Walter Anthony and Anthony, submitted manuscript, 2011).

[88] A simple calculation can be done to check that the model is simulating reasonable methane emission rates for the soil parameters used. A 22 m thick column of soil with a bulk density of 1540 kg m−3 and an organic carbon fraction of 1.6% to 3.3% by dry mass, contains between 540 and 1100 kg C m−2, the lower end of which is close to the yedoma carbon stock reported by Zimov et al. [2006a] (528 kg C m−2). If 1/3 of this is highly labile [Kholodov et al., 2003; Zimov et al., 2006a; Walter et al., 2007b], and 1/2 of the labile C is converted to CH4 [Conrad et al., 2002; Walter et al., 2007a, 2007b] over the simulated talik thawing time of ∼1200 years, then 76–155 g CH4 m−2 yr−1 would be produced on average, confirming the simulated methane production is in line with the parameters being used. The lower end of these values is similar to the upper end of the methane emission rates estimated from observations assuming 49% unmeasured hot spots. This simple calculation points to a possible discrepancy between soil parameters available in the literature and reported methane emission rates, and the need for further investigation into carbon content, the labile fraction, and the CH4 to CO2 production ratio (1:1) in yedoma soils.

[89] In addition to the initial carbon content, labile fraction and CH4 to CO2 production ratio, which scale the methane production linearly, other variables affecting the methane production rate in the model are: the talik deepening rate, and the production rate constant RC and temperature dependence (equation (7)).

[90] High simulated methane production cannot be explained solely by inaccuracy in the talik deepening rate because this is constrained by the limited range of water temperatures and the depth of carbon rich soils. The talik deepening rate varies linearly with deep water temperature and talik depth, thus the methane production rate will be sensitive to these parameters; however, the water temperature is constrained by measurements from other Alaskan lakes and the 3°C water temperature used in the simulation is a conservative value. Additionally, the talik depth would need to be a factor of ∼2 deeper to account for a factor of ∼2 lower observed methane emission due to the slower incorporation of sequestered carbon; however, the talik would likely be beyond the depth of carbon rich sediments.

[91] We investigated the sensitivity of the simulated methane release to the methane production rate constant RC (equation (7) and Figure 12). Away from the lake edges, the depth-integrated methane production is fairly insensitive to a reduction inRC from 0.02 to 0.01 because methanogenesis depletes the labile carbon faster than thaw deepening incorporates new labile carbon into the talik, thus production is controlled by downward thaw progression. For lower RC values, production is influenced by RC and the total active carbon stock. Additionally, changes in the methanogenesis rate constant changes the depletion time and thickness of carbon rich soil producing methane, thus reducing the impact of RC changes when methane production is integrated over the soil column. Halving the methane production rate constant from 0.02 to 0.01 only reduces the methane production away from the lake edge by ∼10%. In contrast, at the edge of the lake where the talik is deepening rapidly, methane production is controlled by the methanogenesis rate constant; halving the rate constant halves the production rate. An interesting result of this parameter variation is that the mean methane production rate across the lake is highly insensitive to RC, only dropping by 7% with a factor of ten reduction in RC at year 1600. This robustness reflects the compensating effect of a larger active carbon pool when RC is smaller, due to the longer time it takes carbon to be mobilized.

Figure 12.

Depth-integrated methane production across simulated Lake Claudi for four different methane production constants, 0.02 (black), 0.01 (cyan), 0.005 (green), and 0.002 (red), and the observed methane emission in Lake Claudi (circles) calculated as seep ebullition plus a combined background ebullition and diffusive flux of 11 g CH4 m−2 yr−1.

[92] Another possible explanation for this missing methane is that microbial oxidation eliminates a substantial fraction of the methane produced at depth. Dissolved CH4 diffusing from anaerobic lake sediments into the water column is subject to aerobic oxidation [Rudd et al., 1976; Kankaala et al., 2006], and potentially to anaerobic oxidation, though the later has rarely been demonstrated in lake ecosystems [Zehnder and Brock, 1980; Schubert et al., 2011]. Due to the relatively rapid ascent of bubbles from sediments through the oxygenated portion of the water column, oxidation is thought to have a negligible affect on reducing ebullition emission in lakes, which comprised 94% of the methane emission in two intensively studied thermokarst lakes [Walter et al., 2006]. In winter, which comprises roughly two thirds of the year, methane within bubbles trapped beneath ice diffuses into the water and is then subject to oxidation. Observations of methane depletion in trapped gas by Walter et al. [2008] and at the Lake Claudi field site indicate ∼30% winter methane loss. Even assuming all of the methane that dissolved into the water oxidized, the mean annual oxidized methane would be ∼20% of the methane produced, small compared to the discrepancy between predicted production and observed emission rates.

4.3. Millennial Timescale Evolution

[93] We simulated one evolution path of this upland plateau for ten thousand years using the same conditions and parameters of the Pear and Claudi formation simulations. Although this simplified topography is not expected to accurately predict the detailed evolution of these two lakes, general lake interactions with the landscape and methane production over millennial timescales and multiple lake generations can be explored.

4.4. Morphology

[94] Over 10,000 years, the simulated Claudi and Pear lakes expand, merge, and drain several times (Figures 13a and 14). These simulations exhibit a number of properties commonly seen in natural thaw lakes, and illuminate several complex but fundamental dynamical behaviors of lakes that, without the model, would be difficult to explore.

Figure 13.

(a) Time series of simulated lake area in Pear (black) and Claudi (blue). (b) Time series of methane production in Pear (black), Claudi (blue), peatlands (red), and ten times the peatlands production rate (green). (c) Time series of total (solid) and lake biota (dashed) methane production in Pear (black) and Claudi (blue) and ten times the peatlands methane production rate (green).

Figure 14.

Six snapshots of lake water depth in pairs spanning catastrophic (Lake Pear between 1390 and 1400) and partial drainage events of Lake Pear and Lake Claudi (scale in meters).

[95] 1. Topography has a strong influence on lake expansion and hence carbon mobilization, and topographic controls can make lakes more or less sensitive to climate changes. Under identical climate conditions, the simulated Pear Lake expands to a large lake and catastrophically drains in just under 1400 years, while Lake Claudi radially expands slowly over most of the simulation (Figure 13a) and has several partial drainages. Pear's expansion, with its low banks, is limited by the thermal degradation of the bank, whereas expansion of Claudi is topographically controlled by high banks requiring removal of collapsed bank sediment. Because of this topographic control, Claudi would be much less sensitive to climate changes. Sustained rapid expansion is only possible where there is low relief, as expansion downslope lowers the sill elevation and raises high subaerial banks. It should be noted that the absence of substantial later generation lake coverage in Pear basin (>4000 years) is partly due to the expansion of lake Claudi into Pear basin, thus classifying that region as part of Lake Claudi.

[96] 2. Following from behavior 1, in the absence of climate controlled changes in lake water level, the sill elevation and water level generally drop monotonically as a lake expands. This causes continually increasing subaerial bank heights, which results in decreasing expansion rates. Increased mean expansion rates are indicative of a change in bank material properties or climate.

[97] 3. Distribution of excess ice, which can be influenced by previous lake dynamics including expansion and drainage, can strongly control current bank retreat rates and plan view shape. In simulations, the abundance (or lack thereof) of excess ice limits Pear's expansion rate to the southwest and Claudi's to the north and south. Under conditions of uniform excess ice, lakes expand more rapidly downslope because lower subaerial banks require less thermal/mechanical erosion. However, the depletion of near surface excess ice by earlier thermokarst lakes inhibits bank retreat by eliminating thaw compaction as a process in bank height reduction, leaving only mechanical erosion to drive bank retreat. By year 9580 (Figure 14) the remnant Lake Claudi follows the slope contour on the northern border due to limited excess ice impeding northward expansion.

[98] 4. Lake area controls probability and frequency of draining. Whereas Pear catastrophically drained, Claudi experienced several failed drainage attempts despite neighboring slopes which were similar for the two lakes. Claudi's failed drainage occurred because the lake area was much smaller than Pear lake prior to drainage (∼0.11 km2 vs ∼0.76 km2). Since erosion depends on drainage discharge, the same lowering in lake surface elevation produces 1/7 of the discharge and erosion. At this lower erosion rate, the water surface elevation lowering rapidly exceeds the sill elevation lowering, and drainage ceases. The lake water level in Claudi is currently ∼15 m below the plateau elevation, indicating that the surface elevation in Claudi has lowered slowly over time without catastrophic drainage, consistent with model predictions.

[99] 5. Both complete (Figure 14, year 1390 to 1400) and partial lake drainage (Figure 14, year 3225 to 3250) can occur during the lifespan of a lake, the sequence and frequency are sensitive to topography, and drainage history poses a significant control on expansion rates and C mobilization. Since channel erosion lowers the sill elevation the new water surface will be lower, remnant lakes tend to expand slowly because their banks have already been depleted of excess ice near the surface. Although slowly, they can continue to expand by mechanical erosion of the bank and ice depletion at depth in deep thaw bulbs. Lake expansion shortly after catastrophic drainage also occurs by sediment filling the eroded channel raising the sill elevation and water level, in contrast to the general monotonic fall of the sill elevation, as seen at year 1400 with the small crescent lake in Pear Basin (Figure 14, year 1400), which formed rapidly after lake drainage. As the lake expands additional lake drainage events can occur, reeroding the drainage channel and lowering the sill elevation. Over longer timescales, lake formation within drained lake basins can occur after excess ice formation elevates the ground surface, enabling the primary thermokarst lake feedback with topography and water pooling to reinitiate.

[100] 6. Lake drainage events can be linked, with drainage of one lake triggering drainage in another downslope. If Pear Basin contains a later generation lake and Lake Claudi drains into Pear Basin (Figure 14, year 3225 to year 3250), then the water entering Pear can trigger a drainage of Pear Lake (Figure 13a). In the suite of Pear/Claudi simulations that we ran, this behavior occurred but was uncommon because it requires a large later generation Pear Lake that reaches nearly to the edge of the former basin, and this configuration only persists for a short time before Claudi and Pear merge into a single lake.

[101] 7. Lake expansion transfers soil from the banks into the lake, which creates relief once the buried excess ice is depleted, resulting in significantly increased complexity of lake shape and bathymetry with time. As discussed above, a central bulge forms on the basin floor for a rapidly expanding, low bank lake like Pear. The slowly expanding, high banked Lake Claudi transfers a larger volume of soil over time, developing a topographic trough around the margin (Figure 14). Lowering of the sill elevation over time gives Claudi an annular shape (Figure 14, year 3225). Drainage of lakes with marginal troughs can also result in annular lakes and horseshoe lakes (Figure 14, year 3250), a lake shape common to the Seward Peninsula (e.g., unnamed lake at 66°14′N, 164°58′W). The nonlake portion of the annulus that forms the horseshoe results from lake expansion down a scarp; much less soil is removed from the low bank margin near the sill as the lake expands, thus leaving behind a higher bed elevation after complete thawing of the buried excess ice in that region. It should be noted that the floor of Lake Claudi does not currently exhibit a substantial central bulge or marginal trough, which develops in the simulation around year 1400, indicating that either the model overestimates the transfer of soil, the thaw depth near the lake margin is less then predicted, or Claudi is younger than predicted.

[102] 8. Lake depth shows regular trends both within a single lake and over multiple generations (Figure 14). In young, first generation lakes, lake depth change is dominated by thaw deepening and compaction. As the lake encounters greater topography, falling sill elevation and increasing bed aggradation act to shallow the lake. Once the total buried excess ice has been depleted in the center of the lake these processes dominate. Drainage of the lake erodes the sill dropping the lake level. Water depth in later generation lakes is dominated by changes in sill elevation and climate. Due to the prior depletion of buried excess ice and cutting down of the sill elevation, later generation lakes are often shallower [Brosius et al., 2012].

4.5. Methane Production

[103] Simulated methane production over 10,000 years (Figures 13b and 13c), including multiple lake generations, is primarily indicative of the rate of talik expansion into virgin yedoma, which overwhelms both the peatland and lake biota methane production. This overarching control explains both the steep increase in methane production with lake area during the early rapid expansion of Pear Lake and the fairly constant methane production during the long slow expansion of Lake Claudi (Figure 13). In Pear lake, rapid incorporation of virgin yedoma into the talik occurred through both lake expansion and talik deepening resulting in a steep rise in methane production. After year 1000, methane production begins to decline due to the depletion of the carbon rich layer and the slowing expansion. After drainage, methane production plummets and remains low. In contrast, methane production in Lake Claudi remains relatively constant throughout most of the 10,000 year simulation, decreasing by a factor of two between year 1000 and year 7000. Methane production is stable because Lake Claudi experiences partial drainage with remnant annular lakes. Since the lake edge is where Claudi is eating to virgin yedoma the high methane production zone remains intact after drainage. For example, partial drainage at year 5605 reduces the lake area by 78% but only reduces the methane production by 32%; one of the largest partial drainages, occurring at year 6115, leaves a remnant lake with only 13% of the former lake area, but 54% of the methane production (Figure 13).

[104] With the parameters used in this model methane production from peatlands is a small fraction of the total landscape methane (about one tenth); however, methane emission observations vary over several orders of magnitude [Lai, 2009; Limpens et al., 2008]. The green line in Figures 13b and 13cindicates a peatland methane emission rate that is ten times the simulated rate, which is still within the range of observed methane emission. With this higher rate, long-timescale peatland emission is comparable to the lake methane emission rate. Early in the simulation, spikes in peatland methane emission rates in recently drained lakes, associated with the establishment of fen vegetation, can even exceed the lake emission rates.

[105] While second generation lakes (i.e., lakes forming in drained lake basins) produce methane from Holocene carbon throughout the lake, sourced from the near surface layers of the bank and lake biota, the release of Pleistocene carbon only occurs near the lake periphery and is highly dependent upon the topography. Again, later generations of Pear and Claudi differ significantly. Lakes form in Pear basin due to infilling of the drainage channel and lake expansion up into the extinct basin, resulting in very little incorporation of virgin yedoma into the later generation taliks, except on the outer flanks of the lateral ramps, thus there is minimal release of sequestered carbon. In contrast, later generations of Claudi Lake form remnant edge ponds after drainage, which continue to expand outward into high virgin yedoma banks, thus they continue to mobilize substantial Pleistocene carbon.

[106] Methane production in the later generation lakes simulated here is limited by carbon depletion by deep first generation taliks, thus methane production in the later generation lakes is largely a function of lake expansion beyond the first generation basin limits. Over time, production from sequestered carbon decreases while total emission from lake biota increases (Figure 13c), becoming 3% and 10% of the whole lake methane emission in Claudi and Pear, respectively.

[107] First generation depletion of sequestered labile carbon is not a universal condition of thermokarst lakes. In contrast to the deep lakes simulated here, shallow first generation lakes with no permanent talik mobilize only a small fraction of buried labile carbon during lake expansion. Climate induced transition from a shallow to a deep thermokarst lake would result in intense methane production across the entire lake area at the high rates generally only seen at the edges of the lakes simulated here (Figures 5, 8, and 10).

5. Conclusions

[108] This first application of a 3-D numerical model of thermokarst terrains has demonstrated that this model can simulate to first order the morphology and methane emission from thermokarst lakes. This model can also act as a framework for experimenting with soil and environmental parameters that control the morphodynamics and carbon mobilization within these landscapes, thus guiding future research. Three major insights have been brought to light:

[109] First, the primary soil parameters controlling methane production in taliks need further investigation. The factor of two discrepancy between predicted and observed methane emission may indicate that current observations of methane emission underestimate the methane production beneath thermokarst lakes in thawed yedoma soils. However, since the modeled methane production is based upon rough estimates of carbon content, labile fraction and CH4:CO2 production ratios, a factor of two is within the uncertainty due to these parameters.

[110] Second, thermokarst lake expansion is tightly tied to topography, and thus is variable from lake to lake even within the same landscape and climate. This strong interaction between thermokarst lakes and topography necessitates higher-resolution digital elevation maps than are currently available for thermokarst terrains. This model produces plausible results when run with existing observational data sets and estimates of unknown properties; however, higher-resolution ground surface elevation and subsurface soil property/ice content data sets are necessary to fully test this model.

[111] Third, methane production in thermokarst terrains is largely a function of the rate of expansion of thermokarst lakes into virgin yedoma, and is thus dependent on topography, lake generation and the details of lake expansion and drainage events during lake history, which act to slow long-term expansion and protect virgin yedoma. This strong dependence bears on the question of whether the remaining carbon stocks on thermokarst landscapes in Asia and North America are significantly accessible through thermokarst lake expansion. In regions like the Seward Peninsula in which virgin yedoma outcrops are sparse, the model predicts these outcrops will remain largely untouched, or slowly touched, by future thermokarst lakes as in the case of simulated Lake Claudi. However, enhanced thermokarst lake development in regions with large continuous stretches of virgin yedoma, potentially Siberia, could see substantial increase in methane production as these carbon stocks are mobilized, as was seen in the simulation of Lake Pear.

Appendix A:: Equation (2) Derivation

[112] Equation (2) (equation (A1)) is derived by substituting the mean thaw depth (Hthaw + ΔHthaw/2) over a time step (Δt) for Hthaw in equation (1) (see text) resulting in equation (A2):

display math

in which ktalik is the mean thermal conductivity of the talik, Tbed is the lake bed temperature, Tpcb is the temperature at the phase change boundary (PCB) (= 0°C), Hthaw is the PCB depth or talik thickness, L is the latent heat of fusion of water, and ρice is the density of ice.

[113] ΔHice can also be expressed as the sum of the change in pore ice (first term in (A3)) and the change in excess ice (second term in (A3)), which can then be solved for ΔHthaw (A5) in terms of ΔHice, soil porosity (Ps) and volumetric excess ice fraction (Fxsice):

display math
display math
display math

in which 1 − Fxsice is the volumetric fraction of soil including pore space, and Rice is the ratio of total ice volume to soil volume. Substituting this expression for ΔHthaw (A5) into equation (A2) and collecting constants into C, yields a quadratic equation that can be solved using the quadratic formula:

display math
display math
display math
display math

Expanding C, simplifying and choosing the positive solution yields equation (2) (equation (A1)):

display math

Acknowledgments

[114] We acknowledge support from the NSF OPP grant 0732735 and NASA Carbon Cycle Sciences grant NNX08AJ37G. Field support was provided by CH2M Hill Polar Services. We thank Guido Grosse, Ben Jones, Peter Anthony, Mike LaDoceur, Louise Farquharson, two anonymous reviewers, and Melissa Smith for field assistance and Jeffrey Chanton for isotope analysis and CH4combustions for radiocarbon dating. We thank the National Park Service Fairbanks Office for providing high-resolution satellite imagery of parklands and permits to do fieldwork in the Bering Land Bridge National Preserve.

Ancillary