Patterns of subsurface wedges of ice that form along cooling-induced tension fractures, expressed at the ground surface by ridges or troughs spaced 10–30 m apart, are ubiquitous in polar lowlands1. Fossilized ice wedges, which are widespread at lower latitudes, have been used to infer the duration2,3,4 and mean temperature5,6 of cold periods within Proterozoic2 and Quaternary climates3,4,5,6,7,8,9,10,11,12,13, and recent climate trends have been inferred from fracture frequency in active ice wedges14. Here we present simulations from a numerical model for the evolution of ice-wedge networks over a range of climate scenarios, based on the interactions between thermal tensile stress, fracture and ice wedges. We find that short-lived periods of severe cooling permanently alter the spacing between ice wedges as well as their fracture frequency. This affects the rate at which the widths of ice wedges increase as well as the network's response to subsequent climate change. We conclude that wedge spacing and width in ice-wedge networks mainly reflect infrequent episodes of rapidly falling ground temperatures rather than mean conditions.
Ice wedges originate with fractures opened in perennially frozen ground by tensile stress induced by falling winter temperatures15,16,17. Fractures partially fill with ice from blowing or thawing snow, and then close when the ground warms16,17. Because ice in fracture cavities is generally weaker than frozen ground, fractures usually follow the same path from year to year3,16,17,18. Over thousands of years, V-shaped wedges of ice 3–5 m deep and up to several metres wide develop along recurring fracture paths3. Upward deformation at the ground surface forms ridges with heights ranging from decimetres to 1 m, revealing the position of growing ice wedges below3.
Although these basic processes relating to a single ice wedge have been the focus of previous investigations15,16,17,18, the development of network patterns over longer timescales and their relationship to varying climates remains uncertain. The nonlinearity of fracture initiation and propagation, the influence of open fractures on subsequent fracturing during a single winter, and the long-term memory of fracture patterns from past winters stored in ice wedges all constitute building blocks of a system that could exhibit complex dynamical behaviour diverging from that of a single ice wedge.
Reflecting the essentially two-dimensional nature of ice-wedge networks19, we model fractures and ice wedges using a two-dimensional array of cells representing the top approximately five metres of frozen ground. Each cell contains either frozen ground or ice, and can contain a segment of an open fracture. All cells initially are assigned unfractured frozen ground, and thereafter evolve through repeated application of an algorithm that corresponds to one year in the model (Methods). To represent the thermal tensile stress that gives rise to fractures in frozen ground, a pre-fracture tensile stress (the thermal stress without reduction by fractures) that is uniform across the model lattice is incremented from 0 MPa to a specified maximum value. Fracture segments locally reduce this tensile stress by an amount that is proportional to the inverse square of distance and is maximized along a line perpendicular to the segment15,19. As thermal stress increases, fractures initiate in cells where local stress exceeds strength, and propagate cell-by-cell in two opposing directions along directions determined by stress imposed by open fracture segments, parameterized heterogeneity, and the energetic cost of propagation (higher in frozen ground than in ice). At the end of a fracture episode, the width of ice wedges is increased in cells with open fractures, all open fractures are closed, and values of tensile strength and propagation threshold are set to those for ice in all cells that contain fractures. The effect of a growing ice wedge on the ground surface is modelled by deforming the surface over the wedge by a volume equal to the volume of ice added. The relaxation of morphology by soil creep is modelled with linear diffusion. The values of parameters were chosen to be consistent with existing measurements (Methods), many with broad uncertainties. Model results are not highly sensitive to parameter values19.
Our goal is to model the development of ice-wedge networks over thousands of years. Details pertaining to the shorter-timescale dynamics of fracture propagation, tensile stress field and frozen ground deformation are simplified or omitted, on the basis of the hypothesis that dynamics of these processes dissipate on the longer timescales of the pattern of interest20,21, the ice-wedge network. This assumption is supported by comparisons of modelled to natural ice-wedge networks (Figs 1 and 2). In both modelled and natural networks, ice wedges range from long and sinuous to short and straight. Many ice wedges terminate at orthogonal intersections with other wedges, but some paths cross, forming four-way junctions. The spacing (15–20 m) and relative orientation (Fig. 3) of surface ridges and the mean width of ice wedges after 2,000 yr (1–3 m) are consistent with measured values from ice-wedge networks in northern Alaska and Canada3,19. The similarity between natural networks and networks modelled using reasonable parameter values (Methods), and the relative insensitivity of model results to parameter values19, suggest that dynamics of modelled networks can be used to investigate natural networks, despite some parameters being poorly constrained by measurements (Methods). Qualitative model-based inferences are most robust.
The response of modelled networks to changes and variability in climate was investigated using four numerical experiments in which only the sequence of maximum annual stress was varied. In the first experiment, the severity of each winter's cooling episode was fixed by setting maximum tensile stress to the same value, 2 MPa, each year. The resultant ice-wedge network diverges from the initial fracture pattern (Fig. 2a), as illustrated by a decrease in spacing between ice wedges from 21.7 m in the first year to 19.2 m after 800 yr (Fig. 3a). Although most fractures are constrained to ice wedges, the divergence in spacing arises as different permutations in the order of ice-wedge fracturing permit the introduction of new fracture paths and hence new wedges. These new fractures open at locations where tensile stress exceeds the strength of frozen ground before fractures that reduce the regional tensile stress open in nearby ice wedges. Switching the sequence of fracturing in two ice wedges can convert an orthogonal three-way (T-type) intersection into a crossing four-way intersection; in this case, the wedge that terminates at the T can be extended if it fractures before the perpendicular wedge. After approximately 800 yr, the modelled ice-wedge network attains a steady state, characterized by consistent spacing between fractures, ice wedges and surface ridges. 88% of ice wedges fracture each year, a frequency sufficient to form ridges more than 0.3 m high at the surface over all ice wedges. Long ice wedges tend to fracture more frequently because they contain more potential sites for fracture initiation, and because a fracture, once initiated in a wedge, generally propagates along its entire length. Open fractures in these long wedges reduce tensile stress in nearby short wedges, lowering their fracture frequency. The width of ice wedges reaches 2.8 ± 0.7 m after 2,000 yr, the high standard deviation reflecting the variation in frequency with which ice wedges fracture and the addition of new wedges during the first 800 yr. The width and frequency of fracture of ice wedges decrease as the ratio of ice strength to ground strength increases in the model; we expect that the values provided here are upper limits.
To investigate the sensitivity of ice-wedge width and spacing to climate variability, two experiments with maximum annual stress normally distributed around a mean of 2 MPa were performed, with standard deviation (s.d.) = 1 MPa (narrow-range) and s.d. = 1.75 MPa (broad-range). The pattern of ice wedges in the narrow-range network reaches an approximate steady state after about 1,200 yr, with a spacing between ice wedges of 13.1 m (Fig. 3b), whereas the broad-range network only attains approximate steady state after 1,800 yr, with ice wedges separated by 12.0 m (Fig. 3c). The fraction of ice wedges that fracture in a given year is reduced from, and more variable than, the fraction for the uniform-stress network: 57 ± 15% for the narrow-range network, and 48 ± 23% for the broad-range network. Spacing between surface ridges after 2,000 yr is 15.7 m for the narrow-range network and 16.8 for the broad-range network (Fig. 3b, c). These spacing values are intermediate between mean annual fracture spacing and ice-wedge spacing, because not all wedges added during high-stress winters fracture with sufficient frequency during subsequent winters to form ridges. Counter to intuition, ridge spacing in the broad-range network is greater than ridge spacing in the narrow-range network, despite reduced ice-wedge spacing. This reflects greater over-saturation by ice wedges in the broad-range network; therefore, fewer ice wedges fracture with sufficient frequency to form ridges. The patterns of surface ridges of both broad- and narrow-range networks achieve steady state more than 600 yr after the corresponding ice-wedge patterns; this is because ice wedges added during high-stress years are expressed at the surface only after many episodes of refracture.
The sensitivity of ice-wedge networks to conditions at their time of initiation is illustrated in an experiment in which maximum tensile stress was set to 4 MPa for the first 4 yr of network development and to 0.8 MPa (below the fracture initiation threshold in frozen ground) thereafter. This scenario corresponds to network development following draining of a thaw lake22, in which minimal insulating snow accumulates in the wind-swept basin during the first winters after desiccation, but accumulation increases to approximately 0.5 m after 5 yr as snow-trapping plants colonize the basin16. Spacing of modelled wedges reaches 16.3 m after 4 yr (Figs 2c, 3d), and remains constant thereafter because the maximum tensile stress is below frozen ground strength. Despite developing principally under a maximum stress insufficient to fracture frozen ground, ice wedges in the lake-floor network grow at a rate comparable to that in other numerical experiments. More so than in the other experiments, ice wedges that are long, and therefore contain more potential sites for fracture initiation, tend to fracture with greater frequency than short wedges. Short ice wedges that rarely fracture do not appear (or barely appear) in the visual pattern of surface ridges because of relaxation of ground surface morphology (Fig. 2c). The spacing between surface ridges, 19.4 m, is consequently 20% greater than ice-wedge spacing (Fig. 3d).
We interpret these results as having the following implications. First, metre-wide fossil wedges indicate the existence (for at least thousands of years) of permafrost and conditions sufficient to fracture wedge ice; but they do not necessarily indicate sustained conditions sufficiently harsh to fracture frozen ground, because a brief exposure to high-tensile-stress conditions can lead to the formation of an ice-wedge network that then continues to develop under conditions insufficient for initial formation. Because networks are sensitive to initial and extreme conditions, the geographical distribution of networks might not correlate with mean climate, as has been suggested for modern10 and fossil2,3,4,5,6,7,8,9,10 networks. Second, wedge spacing7 mainly reflects infrequent severe winter conditions (periods of rapidly falling air temperature at low absolute temperature) during the formation of the network, because episodes of high maximum tensile stress add new wedges to a network. Third, measurements of ice-wedge width2,3,4 might provide minimum limits to the duration of cold climates, but the maximum duration is uncertain because growth rates are reduced by climatic variability and are widely variable between ice wedges. Long wedges are the most reliable measure of network duration because they form early in network development. Narrow fossil wedges are consistent with a cold, inherently variable climate, or with a moderate climate marginally suitable for ice-wedge development5,6. Fourth, low rates of fracture in networks4,18 are consistent with a stationary but inherently variable climate, as well as with other climate scenarios, and so alone cannot be used to infer climate change. However, changes in fracture frequency across a population of ice wedges14, sustained over many winters, are consistent with climate change.
The characteristics of ice-wedge networks reflect an interplay between fracturing, ice-wedge growth and network development, all nonlinear processes with differing intrinsic timescales. Our model for the evolution of ice-wedge networks shows complex behaviour under a range of climate scenarios. This suggests the need for caution in inferring the response of ice-wedge networks (and that of other, more complicated, geomorphic systems) to climate change using simple models or conceptions of their behaviour.
Thermal tensile stress in frozen ground is a complicated function of material-dependent rheology, absolute temperature and rate of cooling, and is highly uncertain in field settings. The maximum annual stress used for the uniform stress model, 2 MPa, is based on a viscoelastic rheological model for frozen ground before fracture15, a cooling rate of 10 °C d-1 and a quasi-viscous parameter, n̄, of 8 × 1025 N3 m-6 s, where n̄ relates the rate of permanent quasi-viscous strain Ė under a constant sustained stress P in P3 = 2n̄ Ė (ref. 15). An increase of maximum annual stress to 3.75 MPa, used as the +1 s.d. limit in the broad-range network, might be caused by a decrease in the absolute ground temperature from -10 °C to -30 °C, with a corresponding increase in n̄ to 15 × 1025 N3 m-6 s (ref. 15). Changes in absolute ground temperature are one route to changes in maximum annual tensile stress, and changes in the rate of temperature change are another15.
Local tensile stress, σkl(θ), the stress perpendicular to angle θ in cell kl, is taken to be pre-fracture thermal tensile stress, σT, minus the sum of reductions from nearby open fracture segments in cells ij, according to
where dijkl is the distance (in m) from the centre of cell ij to the centre of cell kl, βijkl is the difference between the angle of the fracture in cell ij and angle θ, and αijkl is the angle between the line segment connecting cell kl and cell ij and the normal to the fracture segment in cell ij (ref. 19). Parameters are set to n = 2 and C = 3 m2 so that far-field modelled stress approximates the solution to two-dimensional elastic equations for stress around an infinitely long, straight fracture 4 m deep15,19.
Fractures are initiated where local tensile stress exceeds strength (set to 1 MPa for frozen ground and 0.3 MPa for ice wedges, values chosen from the small range of laboratory measurements of small samples of ice and frozen silt19,23,24, because in situ measurements of wedge ice and heterogeneous soil are lacking). Once initiated, fractures continue to propagate if the elastic strain energy released by opening of the fracture, G, exceeds the energy required to create new fracture surfaces, Go, set to 1.5 J m-2 in ice-wedge cells25 and to 8 J m-2 in frozen ground cells19. Both values have large uncertainties; their ratio determines the degree to which fractures tend to follow wedges. G is calculated from the pre-fracture tensile stress along the 5 m of fracture path preceding the position of the fracture tip, linearly weighted toward stresses near the tip, and from modelled shear stress across the fracture plane at the fracture tip19. The angle of fracture propagation is re-evaluated as a fracture tip enters a new cell, with P(θ,Δθ), the probability that propagation angle changes from θ to θ + Δθ, given by a Boltzmann factor exp[ΔG(θ + Δθ)/Grandom]. Here ΔG(θ + Δθ) is the change in net energy (G - Go) released as a function of change in propagation angle and Grandom parameterizes the magnitude of heterogeneity in frozen ground and in tensile stress. Grandom is set to 0.5 J m-2 so that isolated fractures in the model follow irregular paths similar to paths of first fractures in frozen ground16,19.
Ice-wedge and surface dynamics
The width of ice wedges is increased in cells with open fractures by a positive amount drawn from a normal distribution with both mean and standard deviation 0.0015 m (ref. 3). The effect of a growing ice wedge on the ground surface is modelled by raising the surface over the wedge by a volume equal to the volume of ice added, with the depth of fractures assumed to be 4 m. This elevation increment has a gaussian shape centred on the axis of the ice wedge, with standard deviation equal to the wedge width3. This approach assumes the common natural case of a ridge over an ice wedge. Other modes of deformation can result in different morphologic responses1; however, the cause of differences between them is poorly understood and is not the subject of our investigation. Measurements of morphologic relaxation in permafrost are lacking; we model relaxation as diffusion26 with diffusion constant 5 × 10-3 m2 yr-1.
We thank R. Anderson and B. Hallet for reviews, and the late D.M. Hopkins for comments on an earlier version. This work was supported by the NSF, Arctic Natural Sciences Program, the Andrew W. Mellon Foundation, and the National Park Service, Bering Land Bridge National Preserve.
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Bulletin of Engineering Geology and the Environment (2019)