2.1. Aims and Limitations
 The construction of any model for a complicated natural system requires selection of relevant variables and processes, guided by intended use and time scale. This has relevance to our model, and its limitations, in several ways.
 First, because our aim is to address the fundamental behaviors of thaw lake expansion in general, we have included classes of processes which are common to all thaw lakes: heat transfer and thawing in permafrost, thaw-driven subsidence, and mass movement. Other geomorphic processes have been described to influence particular and/or an unknown number of lakes, including erosion by the expansion and rafting of lake ice (“ice push”) and lake shallowing by the growth of ice wedge polygons beneath already shallow lakes [Mackay, 1963]. Such processes are intentionally not included because they are not universal; hence, as with most models other than those that are highly “tuned” to match a particular natural instance of a system, our results are intended to be applicable to the average behavior of a large number of thaw lakes, rather than one in particular. Also, because our model is two-dimensional we do not attempt to include long-shore transport of sediment by water, which has been hypothesized to influence sediment budgets on some lake margins where there is a strongly dominant wind direction [Rex, 1961; Hinkel et al., 2005].
 Second, substrates around natural lakes can be highly heterogeneous, especially in terms of ice content. Unlike most spatially extended geomorphic models, we do explicitly include, and test the sensitivity to, spatially heterogenous material properties (sections 2.7.2 and 3.3). However, our treatment of heterogeneity is at a spatial scale of ≈0.5 m, the scale of discretization in the model (ie. variability is imposed by adding normally distributed random perturbations to ice content in individual cells). In nature, ice wedges and massive buried ice bodies [e.g., Mackay, 1971] may cause variability over 10s to 1000s of meters (the scale of lakes), which can cause neighboring lakes to exhibit variable dynamics and morphology.
 Third, the construction of a mass movement submodel requires the representation of processes over appropriate scales of space and time. On the one hand, a reductionist approach which uses mechanistic parameterization of processes might be attempted. For example, detailed calculations [e.g., Chandler, 1972; McRoberts and Morgenstern, 1974] and measurements [Harris and Lewkowicz, 2000] of pore fluid pressure and effective stress on thawing slopes have been used to predict when and where a particular planar slope is unstable to shear failure. A reductionist model of the evolution of subaerial and subaqueous thaw lake slopes over years to centuries is unwieldy, however, because it would require simultaneous treatment of processes of toppling, skin flows, mud flows, active layer detachment slides, slumps, and underlying processes including pore fluid migration, thaw consolidation and growth of binding roots. On the other hand, highly generalized methods are used by most landscape evolution models, such as linear diffusion, to simulate slope evolution by multiple processes over thousands of years or longer [e.g., Hanks et al., 1984; Hallet and Putkonen, 1994; Kooi and Beaumont, 1996; Howard, 1997; Dietrich et al., 2003]. These approaches may reproduce the essential character of a landscape over millennial time scales [Martin and Church, 1997], but the time scale of interest for thaw lakes is shorter. Because of coupling between mass movement and thawing, infrequent fast events may have a significant effect which is difficult to capture by a long-term-averaging approach. Hence, our approach is a middle ground between short-time-scale mechanistic models and landscape evolution models. Our model is not an attempt to reproduce a particular slope process (for example, fast retrogressive thaw slumps [Burn, 2000]) in detail, although geomorphic phenomena of this type do emerge in the model. We reserve discussion of the model's “geomorphic” behavior to section 3.1.
2.3. Mass Movement
 For natural thaw lakes, momentum-dominated processes including small debris flows and skin flows redistribute sediment from bluffs to the lake margin and into the basin [Hopkins, 1949; Hopkins and Kidd, 1988; McRoberts and Morgenstern, 1974]. Active layer detachment slides and retrogressive thaw slumps on low-angle slopes may cause long-range (1–100 m) transport of material to the lake shore [Rampton, 1982; Murton, 2001]. These have been described for other periglacial slopes, with headwalls migrating up to 10 m a−1 [Harris and Lewkowicz, 2000; Burn, 2000]. Intact blocks of peat and frozen soil may raft on the lake surface until thawed and waterlogged, then settle on the basin floor. The lake floor can be smoothed by wave- and gravity-driven disturbances [Hopkins, 1949; Hopkins and Kidd, 1988; Murton, 1996]. Turbidity currents, initiated by subaerial slope failures or below the water line, can cause long-range subaqueous transport [Hopkins and Kidd, 1988]. Superimposed on the episodic but fast advective long-range processes is creep by freeze-thaw cycles, shrink-swell, and faunal disturbances, processes treated as linear diffusion in a broad range of climates and substrates [e.g., Dietrich et al., 2003].
 To capture the net transport of sediment from the broad range of advective and diffusive processes, we seek a minimal model which (1) exhibits accelerated erosion as slopes approach a critical threshold gradient; (2) transports eroded material distances ranging from centimeters (creep) to meters (debris and mud flows) to hundreds of meters (turbidity currents and rafting) per year, weighted to the former but with increasing probability of long-range transport from higher and steeper lake bluffs; and (3) deposits sediment primarily in topographic lows nearest the erosion site but, because of rafting events and flows with significant momentum, also occasionally in more distant depressions.
 The time-averaged cumulative effect of diffusive slope transport (hillslope creep) and advective mass movement has been treated using a nonlinear sediment transport relation [Howard, 1998; Roering et al., 2001]
where z is elevation above an arbitrary datum (L), Sc is the critical slope angle of the ground material (which may be identified as the angle of internal friction), and K a rate constant (L2T−1). This produces near-pure creep behavior where ∣∇z∣ ≪ Sc (linear diffusion: = K∇z), and rapid increase in sediment flux with increasing ∣∇z∣, becoming infinite where ∣∇z∣ = Sc, to effectively capture small slope failures that do not travel long distances downslope [Roering et al., 1999, 2001]. One limitation of this approach, important for our purposes, is that sediment released from a hillslope must fill the nearest downslope depression before deposition in more distant depressions can occur. It does not treat advective movements such as toppling, rolling, and rafting in water which can transport material over one or more topographic depressions.
 In our discrete model, sediment flux from a position i on the ground surface has a similar nonlinear dependence on local slope Si, treated by setting the probability of slope failure PE,i to
For frozen material, the critical slope angle Sc is set to 90° because ice-cemented permafrost can maintain vertical faces for short periods, as shown by the near-vertical angles of retreating permafrost cliffs along the eroding coast of the NSP field site, and the headwalls of some slumps [Burn, 2000]. Sc is set to 45° for unfrozen sediment, because this value is intermediate between estimates of 31° to 50° from calibration of nonlinear transport models for unfrozen, unconsolidated soil against topographic measurements [Howard, 1997; Roering et al., 1999]. In surveys of the subaerial bluffs around active thaw lakes in the YCP and NSP, the mean slope of surveyed subaerial margins (measured from water line to bluff crest) was ≈33° and the maximum slope angle, over 1 m long distances, was 43° [Plug, 2003b] (section 3.2). Sc must be significantly greater than the mean gradient and ≥ the steepest gradient. In clay-rich muds, headwalls of recently stabilized thaw slumps may have slopes as low as 12.5° [Burn, 2000], suggesting that 45° is an overestimate of Sc for clay-rich soils during a short period following thaw (because their slow drainage causes positive pore water pressure and hence liquification). Fast retrogressive thaw slumping that can occur in thawing clay-rich soils [e.g., Burn, 2000; Harris and Lewkowicz, 2000] is probably insignificant or absent in most thaw lakes (section 3.1). A modification of the model to extend its application to this process is considered in section 3.6.
 As with K in (1), CE in (2) acts as a rate constant encapsulating substrate properties which determine shear strength (pore pressure, angle of internal friction ϕ, and density ρ), with weaker soil having higher CE. In general, barring pure clays where cohesion is important, the shear strength of a potential failure plane at depth h beneath a fully submerged soil column is
where θ is slope angle. The reduction in shear strength by pore pressure is proportional to the density contrast of dry soil solids, ρb, and water. In our model, the effect of pore pressure on slope stability is treated by making CE proportional to the density contrast of soil solids and water. For an unfrozen soil with porosity 30% and a solid component that is 80% mineral (2700 kg m−3) and 20% organic (400 kg m−3) (section 2.7.2), for example, CE is ∼40% greater below the water line than above. Subaqueous slopes relax more quickly than subaerial slopes, all other factors being equal. The rate of modeled bank retreat in our model generally increases with CE because more sediment detaches in a time step. However, interactions between erosion, thawing, and insulation by sediment accumulation at the base of lake bluffs preclude a straightforward dependence of bank retreat rate on CE.
 Once detached by slope failure, eroded material moves downslope from i until deposited at position k with a probability of %
where Δxi,k and Δyi,k are the horizontal and vertical distances between i and k respectively. The exponent n sets sediment transport distance, possibly expressing the rate of dissipation of flow energy as a function of transit distance.
 As with CE, n might, arguably, vary for transport through air versus water. The higher viscosity of water would increase dissipative losses and so might shorten travel distances. However, the greater density of water can maintain sediment in suspension, particularly silt-sized grains common in thaw lake terrain, and allow rafting of peat blocks into the basin. Both would counteract dissipative losses in other mass movement types. In the absence of a quantitative argument for varying n between subaqueous and subaerial settings, we use constant values. This assumption may be tested by comparing modeled to measured lake basin morphology: if subaqueous transport distances in natural lakes are significantly longer than subaerial transport, we expect that our use of constant values would produce spurious build-ups of sediment at the water line of modeled lake basins. The selection of n and CE are considered in section 2.7.3 and sensitivity of results to these parameters in section 3.3.
 For round natural lakes, sediment transported from a unit length along the margin bluff converges with proximity to a lake's center. To treat this three-dimensional effect in our cross-sectional model, the deposition rate at position k in the basin is
where Dkt is the rate of deposition at k in the cross-sectional mass movement model during the time step, rlt is the lake radius, and rk is the distance of k from the lake center.
2.4. Thawing and Subsidence
 Temperatures are updated as morphology evolves using a time-marching, Dufort-Frankel finite difference scheme to solve the two-dimensional heat equation [West and Plug, 2008; L. N. Trefethen, Finite difference and spectral methods for ordinary and partial differential equations, 1996, available at http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/pdetext.html]. The effect of latent heat on phase change is addressed by reducing thermal diffusivity across a -1 to 0°C envelope appropriate for silty soil [Comini et al., 1974; Ling and Zhang, 2003]. Ground cells have a thermal diffusivity α = k/C. k is thermal conductivity and C is volumetric heat capacity, which depends on the cell's phase (frozen or thawed) and ice or water content. In nature, thawing permafrost with a volumetric ice content in excess of its unfrozen porosity subsides to partly or fully recover its native porosity, owing to the phase transition and to drainage of excess water (the latter may be slow or partial). In the model, permafrost cells are removed when passing through the phase change envelope, with a probability that depends on ice volume. For example, a permafrost cell with 20% excess ice (in excess of native porosity) has a 21.8% chance of removal from the simulation (the 20% ice content, plus 0.09 of the 20% to account for the conversion of ice to water). Overlying sediment, if there is any, is shifted downward as cells beneath compact from thaw subsidence. Thaw subsidence occurs at the phase change interface of the talik and at the lake margin. Subsidence in the model is instantaneous on thawing, an appropriate treatment for the slow 1–2 cm a−1 rate of thaw front advance in the talik for lakes ≥100 years old (e.g., section 3.1) [Ling and Zhang, 2003]. Before this, and during annual thawing in the active layer, the more rapid advance of the thaw front may exceed the rate at which excess water can migrate away from the thaw front through low-permeability sediment. Under these conditions, thaw subsidence would occur over time after thawing, and may have greater spatial variability than is modeled owing to pore water escape processes of liquifaction and fluidization that rearrange sediment. Most water escape structures are submeter in scale [Lowe, 1975], so any morphology they generate is likely below the scale of thaw lake basins relevant here.
2.6. Initial Morphology and Ground Ice
 The simulation begins with a small half basin of radius ∼34 m, depth 4 m, and basin slope 40° in a 300 × 300 m lattice. The basin floor is overall flat but with ±0.3 m cells-to-cell irregularity. These dimensions were chosen because they are appropriate for an already developed but small natural thaw lake [e.g., West and Plug, 2008, Figure 7E].
 In the model, the initial ground ice content decreases with depth from a maximum value at the permafrost table. The volume of ground ice in i depends on its depth, di:
where υ is the unfrozen porosity of the soil and υE the maximum volumetric excess ice. Both υE and dE (the depth of the layer with excess ice) are simulation-specific parameters that depend on the geographic region modeled. ε is a random perturbation representing ground ice heterogeneity owing to ice lenses, veins, and wedges. Because our goal is to examine the sensitivity of lake dynamics to ground ice and climate, we use this generalized representation of ground ice. A measured depth profile of ice content might be used if available and if the goal were simulation of a specific lake, but would complicate comparisons between simulations and analysis of the intrinsic variability of lake expansion. We use the linear dependence in (6) because ground ice enrichment occurs over 1000s of years within the upper few meters of permafrost by formation of massive ice (e.g., ice wedges), lenses and pore ice by infiltration of unfrozen water under thermal gradients that cause a matric potential [Williams and Smith, 1991]: Hence, the slowly aggrading or static surfaces during deglaciation and the Holocene would accumulate greater ground ice within the few meters below their paleosurfaces, compared to faster aggrading deeper loess. Ice volume above the permafrost table depth, dp, is set to υ. This suppresses thaw subsidence in the active layer during seasonal thawing. Other than a slight effect on modeled active layer depths owing to latent heat effects, our results are not sensitive to this assumption.