Ice-Wedges and Ice-wedge Polygons as thermal ratchet

Because they represent perhaps one of the purest forms of thermo-mechanical ratchet and because current explanations for their development are seemingly very close to actually describing this process (see for example: Leffingwell 1915; Black 1974; Mackay 1973b, 1974, 2000; Mackay and Burn 2002; Burn 2002), it might be helpful to start discussion of wider periglacial features with an outline of how the thermo-mechanical ratchet process appears to be at work in the formation of ice-wedges and ice-wedge polygons.       

Figure 2(a) shows a typical plan of a developed ice polygon network. It is interesting to note that these networks appear to divide up into either orthogonal arrays or those in which on average there is a tendency towards a hexagonal array. Were the problem of predicting the most likely form for these crack patterns to be approached in terms of defining the crack geometry that would release the maximum tensile stored energy over the least crack fracture length, then the nearer the system approaches an array of circles then the closer it will be to the optimal solution. On this basis the hexagonal network would appear to be the preferred natural selection. It is curious therefore as to why the orthogonal arrays so often provide the preferred natural choice. The depth, d_c, to which the cracks extend, seems to be closely related to the depth, d_t, of the seasonal thermal wave, as suggested in Figure 2(c). Depending upon the location and the weather conditions significant seasonal changes in maxima and minima of the temperature envelope appear to extend to depths of around, d_t =10 to 20m. These depths also have strong dependence upon factors such as the thickness and moisture content of the active layer which control the levels of latent heat needed to be extracted or given out before changes in surface insolation energy input can begin to be conducted into the underlying permafrost. The thickness and form of seasonal snow covers also appears to have a strong influence upon the depth reached by the seasonal thermal wave. Average spacing between adjacent, roughly parallel, cracks, are, amongst other factors, seemingly very dependent upon the depth of the thermal cracks. The crack depths are themselves governed by the depth of propagation of the thermal wave. This situation is suggested by the indicative section A shown in Figure 2(b). Theoretical models (Plug and Werner 2001,2002) of the topologies developed by naturally occurring ice-wedge polygons have been criticised for having invoked edge compression ramparts that are inconsistent with empirical reality (Burn 2004). But there are perhaps even more fundamental grounds for supposing that these theoretical models cannot possibly be capable of capturing the true nature of the ice-wedge polygon growth. They are based upon a two dimensional form of finite element representation of the stress and fracture patterns, employing a “two-dimensional periodic lattice of square cells representing the ground surface” (Plug and Werner 2001). Evidence suggests that crack spacing within polygonal networks is intimately linked to the depths reached by the thermal waves. This becomes apparent from observations of the related polygonal patterns, usually referred to as “alligator cracks”, that develop in asphalt pavements (Croll 2006a,b), or those being observed on the surfaces of some of the outer planets and their satellites. That being the case any theoretical model would as a necessary consequence need to model this three dimensional thermal stress-fracture system using a lattice of cubic, not square, cells. However, current interest centres not on the mechanics of initiation of the ice-wedge polygonal geometries but on the processes whereby they progressively grow once the basic geometric patterns have been set-down.

                                                    

Fig. 2    (a) Typical geometric forms of ice-wedge polygons, showing (b) sections through

opposite ice-wedges, and (c) envelopes of seasonal temperature variations through thickness

on permafrost

As surface temperatures vary over a typical annual seasonal cycle (see Figure 3(a) where t=0 is taken to represent the time at which maximum summer average surface daily temperatures occur) the following response might be expected. After a long period of subzero ground temperatures, surface temperatures will eventually exceed zero. It will be some time before this surface warming, at say a time (1), will have input enough solar energy to provide the latent heat energy required for thawing. Thawing of the active layer will gradually extend down to the top of the permafrost, z=d_tp, which will then effectively remain at a temperature T=0^oC. At the end of the summer period surface temperatures will again drop. They may need to drop to a level (2) before enough energy will have been extracted to allow release of the latent heat required for the refreezing of the active layer. The propagation of the cold wave into the permafrost will consequently lag behind the surface temperatures. But it is the temperature drops occurring after time t=t₂ that will start to induce tensile strain into the permafrost and active layers as a result of the restraint to the thermal contraction wanting to take place. Eventually, at time t=t₃ , point (3), these strains may have reached levels where the associated stresses are sufficient to cause first cracking at the locations of the ice-wedges and presumably elsewhere, as suggested in Figure 3(b). Any further thermal tensile stressing will be limited by a gradual opening up of these discrete and other smaller cracks. This will have the effect of limiting the tensile strain energy remaining in the permafrost when the temperature starts to again increase at point (4). It is likely that surface precipitation will have entered these thermal cracks, forming new ice veins within the existing ice-wedges, with the result that compressive strains will not be required to recover the previous tensile crack strains before compressive stress is again introduced into the permafrost. As warming continues after say point (4), any residual tensile stresses, indicated in Figure 3(b), will be soon overcome and compression stresses will accompany the compressive strains associated with the restraint to the thermal expansion wanting to take place. With ice and the frozen ground of permafrost being fairly resistant to compression, material failure will generally not occur until these compressive stresses reach relatively high values.

The nature of the compressive failure might be a form of local crushing in the vicinity of the ice-wedges, resulting in consolidation so that the permafrost in these zones eventually comes to have lower ice content. It might also take the form of an outward shoving of the active layers and possibly some upward deformation of areas of the active layers and permafrost adjacent to the ice wedges. The active layers themselves will be likely to experience shoving action associated with the restrained outward expansion during the warming period. This would result in upward shear and fold failures adjacent to and either side of the ice-wedges. Such behaviour, indicated in Figure 4, would account for the characteristic rims that so often develop either side of the propagating ice-wedge, sometimes referred to “ramparts”, around the edges of the polygons. Allowing for the loss of effective volume, accompanying any consolidation of permafrost near the polygon edges, then the volume of elevated material associated with the compressive failures must equal the volume of material replaced by ice during the gradual formation of the ice wedges. However, over time some of this raised material will undoubtedly have been eroded back into the low lying parts of the polygons, with the result that lowest central levels may themselves be higher than the ground surface before the ice-wedge process commenced. Sometimes this gradual rise in elevation of the average ground surface level can result in a gradual upward growth in the top level of the ice-wedge to form what have been termed “syngenetic ice-wedges” (Mackay 2000). Consequently, when long-term climate changes lead to melting of the ice-wedges the relic form might be the high centred polygons formed as a result of the material volumes of the slumped former ramparts not being sufficient to replace the melted ice around the edges of the polygons. 

In this model of the formation of ice-wedges and ice-wedge polygons, the thermal ratchet process relies upon the cyclic compressive and tensile energies being dissipated through two very different processes. During the warming, compression, phase the release is through an outward shoving of particularly active layer material, giving rise to upward shearing and folding. During the cooling, tensile, phase the dissipation of energy is the result of brittle tensile fracture. The outward and upward movement of permafrost and active layer material compensates the growth in volume of the peripheral ice wedges. As future posts will suggest this model may be closely related to that responsible for some of the features in the formation of patterned ground often referred to as sorted and unsorted circles, polygons, nets, steps and stripes, so eloquently described by Washburn (1979).