It would seem that I am not the only one who has concerns about the adequacy of current hypotheses for explaining the genesis and growth of pingos. Whether the pressures recorded in the underlying unfrozen ground are sufficient to account for pingo growth has been seriously questioned by Muller (1963). But my own concerns as to the completeness of current hypotheses go somewhat further as I will try to explain – again, without the help of explanatory sketches which would undoubtedly aid clarity.
Overall Heave versus Local Pingo Deformation: The first somewhat problematic area with the current models for the formation of pingos is the tendency for their growth to be in the form of localised geometric distortions rather than a more general uplift, or heave, of the lake bed. The many photographs and summarised field data, see for example Mackay (1998), suggest that the extents of the recently drained lake-beds or estuarine areas from which pingos typically grow can be measured in terms of kilometres. In contrast, the ponds and the eventual regular dome deformations characterising the pingo may be at most a few 10’s to 100’s of metre in horizontal extent. Even allowing for local thinning of the permafrost beneath a residual pond relative to that over the rest of the aggrading permafrost of the old lake or estuary bed, it seems probable that the pressure would find relief through an overall heave deformation of the whole lake bed. An overall heave deformation (what in an engineering context might be referred to as an overall plate deformation) of the thicker but relatively newly formed permafrost layer, will for a given excess pore water pressure, be rather more likely than a deformation into the typically more local dome shape of the pingo. The local plate mode, or dome shape of the pingo, would unless there was a very considerable difference in average permafrost thickness across the deformed shapes require very much higher pressures to develop. I have calculated elsewhere that the pressure required to produce a given level of elastic upward pingo bulge deformation for pingo 14, summarised by Mackay (1998) in his Figures 34 to 40, would be nearly 30 times higher than for this same upward deformation to be produced through an overall heave of the lake bed. For these comparative calculations the pingo thickness, at the start of its growth is taken as 15 m with horizontal radius 70 m, and rather conservatively the thickness of the bed of the pre-existing lake is taken to be 40 m, over an average horizontal radius of 600 m. The pressures for the heave deformation are likely to be even lower in the sense that at the start of the pingo growth the depth of the permafrost was probably rather less than that recorded in 1973. Furthermore, the average of the considerably varying permafrost thickness is, as clearly shown in Figure 35, Mackay (1998), a lot less than the 1973 maximum of 40 m. Even if the calculations were to be carried out using visco-plastic collapse models, and the development of dilation cracks were to be more properly taken into account, the relative pressures would still be of the same orders of magnitude. A similar imbalance between the pressures required and the observed development of open system pingos has been highlighted by Muller (1963).
If the preferred mode were to be assessed more realistically in terms of which mode, the local (pingo) or the overall (bed heave), would for a given excess pore pressure require the release of the lower energy, then the situation would be even worse for the local pingo mode. With the upward deformations required to achieve a particular change in volume being in the ratio of the inverse squares of their respective radii, the local deformation for pingo 14 discussed above, would require a pressure of almost 75 times that required for the overall, heave, mode. If, as suggested above the pressure required to produce a unit displacement in the pingo mode is around 28 times higher than for the same deformation to be produced in the heave mode, it seems reasonably clear that without the intervention of some other fairly powerful mechanical cause there would be a tendency for the system to relieve the pressure energy through overall heave deformation rather than the local pingo distortion. It would require much less energy to develop the overall bed heave mode than it would the local pingo dome mode of deformation. The result would be layers of stratified ice beneath the entire bed of the drained lake, as occurs at smaller scale within the active layer during seasonally induced frost heave. But this does not appear to happen, or if it does the reasons why the local bulge can form simultaneously will be suggested in a later posting.
Shape of Local Pingo Mode: A second troubling aspect for ground water pressure being the primary cause of pingo growth is the dominance of a deformation geometry that exhibits an unusual high propensity towards an axisymmetric form, particularly for closed system pingos. Given the usually irregular shapes of the lake beds, or the relatively thinned residual pond areas, from which they emerge it would seem more probable that the geometry into which the excess of underlying pore water pressure would distort the permafrost layers would reflect this base irregularity. In many instances shown in photographs (Mackay, 1998), it is apparent that “most pingos tend to be more or less circular” (Washburn, p180, 1979). There seems to be a tendency for a single or sometimes a cluster of relatively small diameter, often circular plane-form, regular dome-like pingo geometries to form (see for example Figures 14, 22, Mackay, 1998). If these are associated with localised areas where the permafrost layer is thinner it is difficult to account for the generally regular nature of their geometry.
Localised thinned areas of permafrost would be most unlikely to exhibit the very regular planar shapes necessary for a pressure induced upward distortion to account for the regular dome-like configuration typical of so many reported pingos. Much more likely if this explanation were to be the cause, the distortions induced by a relatively uniform underlying excess pore pressure would reflect the inherent irregularity of the thinner than average regions of permafrost. This does not appear to be the case. Even where they are reported to have developed in an irregularly shaped residual pond, the pingos that develop do not appear to reflect this irregularity. Instead they have a fairly robust tendency towards the characteristic regular dome shape. In the most recent growth recorded for pingo 14, the centre of growth is no longer at the point of maximum elevation of the pingo. Recent growth is as shown in Figure 39, Mackay (1998), maximum in the vicinity of bench-marks 48 and 49. These, as shown in Figure 40, Mackay (1998), are a considerable distance to the North of the top of the pingo at bench-mark 50. Even though this recent growth is from an irregular base, indeed it appears to be on the side of the earlier growth, it exhibits as shown in Figure 39, op cite, and indeed by the subsidence experienced when pressure was relieved shown in Figure 38, op cite, a remarkable degree of symmetry in its recent growth.
Pore Water Pressure Just Enough for Overall Heave: But there is a third aspect of the models suggested to account for pingo growth that for me is even more troubling. At least for open system pingos, Muller (1959, 1963) was similarly vexed by this same problem. As reported by Washburn (1979) there is “an objection to the purely artesian explanation of (open system) pingos” in that the “calculated pressures required to dome a pingo are extreme compared to most measured artesian pressures” concluding that “therefore, additional pressure effects are probably involved”. Muller’s (1959,1963) ingenious attempts to invoke a form of mechanism analogous to the operation of a hydraulic press received little support and as far as I can tell seems to be based upon empirically unproven pressure levels being developed. Even for closed system pingos the empirical evidence suggests that the build-up of underlying pore water pressures would not be adequate to induce the material distortions associated with the growth of the pingos.
There is of course clear evidence that build-up of pore water pressure in the talik does occur. However, from all the evidence I have been able to gather, this pressure build-up is generally not even sufficient to exceed the weight of the saturated sand and gravel that forms the permafrost overburden. Even if it did the preferred deformation of the permafrost would, as discussed above, presumably be an overall heave rather than the localised form of pingo deformation mode.
That the pore water pressure often does not exceed the overburden weight is for example made clear from the evidence of the water heads exhibited in bore hole tests and pressure transducers reported by Mackay (1998). In many bore holes the water head was recorded to have been just enough to produce a gentle surface runoff of ground water. Even in the so called “gusher” emanating from the 7.5 cm diameter bore hole driven to a depth of 22 m beneath the growing pingo 14, the height of the water spout only reached 2.6 m above ground surface level, see Figure 5, Mackay (1998). Based upon even conservative estimates of the specific weight of the permafrost overburden (taken to be 15kN/m3), and allowing for pipe friction losses, from a depth of 22 m the gusher would have had to reach a height in excess of 10 m for the underling pore water pressure to be approaching that required to even carry the overburden weight. The upward pore water pressure acting on the underside of the permafrost would be just a little over two third that required to support the weight of the permafrost overburden. Equilibrium would demand that the missing upward force be supplied by the “effective pressure” exerted between the particles of the granular talik material at the lower permafrost boundary.
Seemingly, there would appear to be insufficient pore water pressure to initiate a lifting of the permafrost layer in either a local pingo dimple or the generalised bed heave. The pressure transducer data summarised for this same pingo in Figure 37 of Mackay (1998) is similarly conclusive in this regard. Over the period 1977 to 1991 the pressures recorded at a depth of 22 m were consistently below 35 m of water head, falling to as low as 30 m. Even allowing for a reduced average density to take account of the sub-pingo water lens, this pressure would at its maximum still seem to have been just sufficient to support the weight of the material above the pressure transducers. Admittedly the pressures seem to have been recorded during summer field investigations, so that peak pressures presumably reached at the ends of any downward aggrading of permafrost might have been somewhat higher.
And of course before the pressures can start to produce relative distortions of the shape characterising pingo geometries, the pressures would need to be well in excess of those required to just support the weight of the overburden material.
Insufficient Excess Pore Water Pressure for Dilation: Which brings us to yet another area of concern regarding the completeness of the model relying upon pore water pressure alone to provide the energy needed to distort permafrost into the shapes characteristic of pingos. To induce an upward incremental deformation in the form of a local bulge the pore water pressure would have to support not only the overburden weight but also the elastic-plastic energy required to be produced for this incremental deformation. At small deformations, and assuming a bending rather than membrane resistance, the pressure p required to elastically deform a circular disk of radius a by an amount w will be given by
(if any one can tell me how to adequately reproduce mathematical eqns I would be grateful)
p = 64 D w / a4 = 5.33 E t3 w / (1-nu2) a4 (1)
where D=Et3/12(1- nu2) is the elastic bending stiffness of the disk having thickness t, modulus of elasticity E and Poisson’s ratio nu. For Pingo 14 taken to have a thickness 22m, radius 70 m, an assumed Poisson’s ratio of 0.4 and elastic modulus taken conservatively to be 5 GPa, a 0.03 m deformation increment, reported to have been the average annual increment over the period 1973 to 1976, Mackay (1998), would require an over-pressure p=423 kN/m2. With this being equivalent to an hydraulic head of 42 m, the total sub-pingo pore water pressure required to both equilibrate the overburden weight and to overcome the elastic resistance would therefore need to be around (22 x 1.5 + 42 =) 75 m of hydraulic head. Even making allowance for the somewhat reduced energy needed when visco-plastic effects are taken into account this required pressure head is well beyond anything recorded in bore hole or pressure transducer tests. Under these circumstances it would appear to be somewhat less than “clear (that) the water pressure beneath Pingo 14 was great enough to uplift and deform more than 25 m of superincumbent material (ie the frozen pingo overburden and subjacent ice core)”, as suggested by Mackay (1998). Had a permafrost plus ice thickness greater than 22 m been used then of course the pressures required to induce pingo growth would be even higher than the 423 kN/m2. But had a higher radius of say 120 m been used instead of 70 m then the pressure needed for this incremental growth would be 49 kN/m2 and not 423 kN/m2. The cross section through pingo 14 (Figure 35, Mackay, 1998) would suggest that the areas most distorted, and this is reinforced by the incremental deformations shown in Figure 39 or the subsidence of Figure 38, Mackay (1998), corresponds with the deformation growth being more closely related to the 70 rather than the 120 m. It appears therefore that excess pore water pressure alone cannot be accounting for the growth of pingos. Accordingly, it is difficult to conceive that the development of the domed characteristic in the growth of pingos can be entirely attributed to the action of the excess pore water pressure acting alone. Just as was concluded by Muller (1963) there seems to be a need for some other mechanism capable of providing the missing energy that is driving the growth of both open and closed system pingos.
References referred to:
Muller, F. (1959) Beobachtungen uber pingos. Detailuntersuchungen in Ostgronland und in der Canadischen Arktis: Medd. Om Gronland, 153(3), 127pp.
Muller, F. (1963) Observations on pingos (Beobachtungen uber pingos), Canada Natl. Research Council, Tech Translation, 1073, 117pp.
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