pingos as a form of thermally induced upheaval buckling

Whether it be the bulges observed on lake ice or those so commonly experienced on asphalt pavements the thermal cycles described in previous blogs as being responsible for their development occur over time scales measured in terms of hours, days or possibly a few days. Such short term fluctuations in temperature are clearly not going to provide the driving force for the development of pingos. Pingos emerge from permafrost that can, even at the start of the upward growth, be many metres thick. Short term surface temperature changes will propagate just a few centimetres into the permafrost so that any expansion and contraction forces associated with these near surface temperature changes will be small, and insufficient to induce the levels of force required to cause an upheaval buckle to form. However, it is possible that the temperature variations experienced over typical annual seasonal cycles could be enough to induce levels of force sufficient to induce uplift of the permafrost.

Reports indicate that average seasonal changes of surface temperature could be as high as 15oC, and in the areas covered by Mackay et al (2002) and Burn (2004) the measured annual changes in ground temperature show that at ground surface, fluctuations in the region of 10oC can occur which with attenuation still show small changes at depths just above the aggrading lower permafrost boundary. MacCarthy (1952) reports seasonal maximum and minimum average daily temperatures varying at the ground surface by almost 20oC, with some changes still being experienced to a depth of 20 m. Ground water temperatures in the saturated talik remain fairly constant at just below zero. That being so it is more than conceivable that during the spring to late summer warm-up periods the average temperature through the permafrost thickness will rise sufficiently to induce in-plane compressive forces great enough to either initiate a thermal uplift buckle or, where one has already begun, allow it to further propagate. This would be especially likely if the pore water pressure increases that might have taken place over the autumn to winter period had approached the levels required to reduce the effective specific weight of the permafrost overburden. Let us consider in a little more detail how this process might occur and why any incremental uplift experienced by the permafrost during warming is not simply reversed when it is subsequently cooled.

Figure 1(a) depicts an area of locally thinned permafrost. This could be the result of an old lake being drained. The lake would have acted as a damper on the propagation of permafrost so that beneath the lake bed the permafrost remains relatively thin. Suppose that the permafrost layer has over the period of possibly a few winters extended some metres into the old bed of the lake, but the thickness beneath the old lake bed remains less than the surrounding more ancient permafrost. It is possible that frost heave or frost mounds could develop over this initial period. Naturally the bed of the lake will not be perfectly flat, and it is likely to contain areas where residual ponds occur. These could continue to act as a an insulator to the downward propagation of the permafrost so that an even thinner area exists. Frost heave will have resulted in mild upward convexity over both the old lake bed as well as the residual pond area. This will be especially so as a result of the differential frost heave that will have occurred as a consequence of the differential rates of aggradation of the lower permafrost boundary.This situation is depicted in Figure 1(b) within an area of the recently formed permafrost layer.

It has been observed in many cases that pingos initiate in areas where the drained lake has left shallow ponding. With the relatively thinner permafrost beneath this pond any lake bed convexity within the thinned area would be a prime target for the development of a thermal buckle. That the resulting upward bulges of the pingo are so often of a generally regular axisymmetric form, even within irregular ponds, is again highly suggestive that an important element in their origins might be from thermal buckling effects rather than as currently proposed just being pressure driven. As the permafrost layer warms during the spring to summer period, compressive forces will be generated over the entire area as a result of the restraint provided by the older and deeper permafrost surrounding bed of the lake, as suggested in Figure 1(c). Figure 1(c) represents an expanded horizontal scale of the central portion of the sketch of Figure 1(b). As seems to often be the case the pingo bulge does not necessarily occupy the full area of the thinner residual pond area. This too is suggestive of a mechanical cause other than, or at least additional to, underlying excessive ground water pressure.

To appreciate the levels of compressive force that may be generated in the permafrost layer consider the effect of an average temperature increase of 10oC, in the sense of representing the average of the change over the depth of the permafrost, from the midwinter minimum to the summer maximum temperatures. This might be considered reasonable in regions where, say, the surface temperatures exhibit seasonal surface temperature variations from minimum to maximum of say 30oC. With a coefficient of thermal expansion taken to be 50x10-6 / oC (coefficients of expansion for permafrost are not terribly well recorded but some reports have quoted values as 90x10-6 / oC or even higher, but a more representative figure of 50x10-6 / oC has, for example, been given by Washburn, p39, 1978) a fully restrained sheet will develop a stress producing, average, compressive strain of 500x10-6 which for a permafrost layer having an average modulus of elasticity E=5 GPa (again, the permafrost literature is a little bashful in terms of giving values of elastic modulus. One of the few publications making use of E values was that of Mackay (1986) which used a value of 19 MPa which seems extraordinarily low, so here I have adopted a value of 5 GPa that seems to be somewhat more representative - perhaps more on this in a later contribution) will generate an in-plane isostatic stress of 2.5 MPa. This would be more than enough incidentally to cause cracking when similar levels of temperature decrease occur over the next six month cycle. For a permafrost layer of thickness 5 m an average compressive stress of 2.5 MPa will result in compressive forces of 12.5 kN (one and a quarter tonne) for every 1 mm width of permafrost, or 12500 kN/m (1250 tonne for every 1 m strip width of permafrost). These extremely high levels of compressive force could conceivably produce an uplift buckle of the form shown as a detail to Figure 1(c).

Continuing with the above example, and assuming that the hypothetical pingo had over the previous years reached a state in which a total uplift of 5 m had occurred for the pingo of base radius a0 = 50 m. Under the extreme assumption that there were to be no underlying pore water pressure, so that q=75 kN/m2 (it being assumed that the specific weight of the permafrost is 15 kN/m3), then the temperature required to initiate first uplift from the talik would be around 2.25oC. An underlying ground water pressure having a head equal to the thickness of the permafrost layer would reduce this to around 0.7oC. For the case of pingo 14, having an assumed radius of the deforming portion of the pingo of 70 m and an uplift amplitude of 10 m above the surrounding ground surface, a local thickness t=22 m, and taken to also have coefficient of thermal expansion of 50x10-6 / oC, modulus of elasticity E=5 GPa, and Poisson’s ratio nu = 0.4, the temperature increase required to initiate uplift is effectively unchanged at T=2.2oC for no pore pressure and 0.7oC when the pore pressure reaches a head equal to the thickness of the permafrost sheet, ie the pore water would be enough to cause a gentle surface run-off from a borehole drilled into the underlying talik.

It appears that the amount of thermal energy associated with typical seasonal increases in temperature through the full thickness of the agrading permafrost layers, would be more than enough to induce the incremental seasonal uplifts of pingos. For more extensive discussion of the temperatures required to induce the typically observed levels of seasonal incremental uplift I would refer you to Croll (2004, 2005, 2007 pingos1). But a subsidary question must be, why do the pingos not just subside when during the late autumn to winter cooling period the average temperatures drop?

References:

Burn, C. R. (2004) A field perspective on modelling of “single-ridge” ice wedge polygons, Permafrost and Periglacial Processes, 15, 59-65.

Croll, J. G. A. (2004) An Alternative Model for “Pingo” Formation in Permafrost Regions, Paper presented at 21st Int Congress of Theoretical and Applied Mechanics, ICTAM-04, Warsaw, 15-21 Aug., 2004.

Croll, J. G. A. (2005) Aspects of the mechanics of pingo formation in permafrost regions, Internal UCL Research Report, 2004, submitted to Proc Royal Society for possible publication.

Croll, J. G. A. (2007) Mechanics of thermal ratchet uplift buckling in periglacial morphologies, Proceedings of the SEMC Conference, At Cape Town, September, 2007 (pingos1)

Mackay, J. R. (1998). Pingo growth and collapse, Tuktoyaktuk Peninsula area, Western Arctic Coast, Canada: a long-term field study, Geographie physique et Quarternaire, 52, 271-323.

Washburn, A. L. (1979) Geocryology: A survey of periglacial processes and environments, Edward Arnold, 406pp.

thermal ratchetting of restrained ice-sheets

My first posting on this blog discussed some observations of curious ice-rings that had appeared on lake ice in New York's Central park reservoir. My attention had been drawn to these features as a result of my earlier attempts to explain the initiation and growth of pingos as a form of thermal uplift buckling, similar to that described in my most recent posting. Some of these ice rings seemed to have evolved from the collapse of uplift buckles caused by the restraint of the expansion that would have otherwise occurred when the ice was subjected to a diurnal increased temperature. This put me in mind of some work I had read while trying to find evidence in support of the hypothesis that pingos might also be a form of thermal upheaval of the frozen ground caused by seasonally induced temperature changes through the thickness of the pernennially frozen ground - referred to as permafrost.  

In-plane Forces Developed by Restrained Thermal Expansion of Ice-sheets: When a floating ice sheet undergoes an increase in temperature it will either expand freely if unrestrained around the shore line or it will develop massive in-plane forces when restrained. The extent of these deformations and/or the levels of force that can be developed have been described by Frellson (1963). Based upon his extensive experience as a Director of the Division of Waters at the Minnesota Department of Conservation, he describes how in one case at Cotton Lake near Detroit daily temperature fluctuations of 25 to 30oF were enough to cause shoreline expansion as much as 18 feet. Where they are restrained these expansions developed compressive forces sufficient to buckle the ice into ridges up to 3.5 feet in height. Where they were in the path of the expanding ice, lakeside cottages were severely damaged and in some instance their foundations were “so badly crushed that it was evident the entire ground on which they stood had been removed”. Other instances cited include cases where trees were overturned, massive masonry bridge piers moved out-of-line, retaining walls tipped over, and huge boulders moved considerable distances. Further evidence of the extreme levels of force generated by restrained thermal expansion of ice is that even very thick ice sheets can experience buckling and crushing at the edges. With coefficients of linear expansion some eight times higher than steel, and estimated, Frellson (1963) at 90x10-6 / oC, the behaviours of ice sheets are clearly extremely sensitive to changes in temperature.

Frellson (1963) also describes the “ratchet” or “jack” actions that can be caused by cyclic variations in temperature. Where the ice sheet has been attached around the shoreline a decrease in temperature by night can result in cracking before any appreciable tension force is generated, on account of ice being relatively weak in tension. Water penetrating these cracks soon freezes so that the next time there is an increase in temperature the now integral ice sheet will undergo a further expansion related deformation. Repetition of this cycle can result in the edges being gradually jacked out further and further. Similar behaviour occurs in the development of ice wedges and ice polygons within permafrost. Indeed, it is well recognised that decreased temperatures during the late summer to midwinter cooling period are sufficient to cause major tension cracking in permafrost. In this regard the long term field studies undertaken by Mackay and others, see for example Makay et al (2002), Burn (2002), are particularly illuminating. It is also well documented that these cracks attract water which subsequently turns to ice, see for example Washburn (1979), French (1996), Mackay (2002) and Burn (2002), forming well defined ice wedges that over time and in a cyclic annual pattern can grow up to significant widths and depths. It is also seems fairly self evident that if tension stresses are sufficient to cause cracking during the cooling, contraction, phase of thermal loading, that major levels of compression will similarly be developed during the warming, expansion phase of the thermal cycles. In the case of ice polygon formations these high compressions would appear to be the cause of the material shoving that is responsible for the gradual growth of the ramparts either side of the ice wedges around the periphery of the ice polygons, Burn (2004). There would even appear to be evidence that in high centred polygons, Washburn (1979), a form of mini-pingo is sometimes developed possibly as a result of the compression stress causing a form of general uplift, doming, rather than the local, edge rampart, form of buckling failure. There seem to be good reasons for supposing that the ratchet process responsible for the development of ice polygons, and particularly some examples of high centred ice polygons, might be a close analogy for the proposed model for the development of pingos.

thermal upheaval buckling of heavy sheets

If a thin heavy sheet of very large in-plane dimensions lying on a flat surface, is subject to an increase in temperature, it will if restrained at its outer edges experience the development of an in-plane compressive force. Indeed, if the friction between the sheet and the surface upon which it lies is great enough, these compressive forces could develop even if the outer edges are not restrained. Under certain circumstances these compressive forces can induce an upward buckle against the weight of the sheet tending to prevent the buckle. This phenomenon of thermally induced uplift buckling of thin heavy heets has been considered by Hobbs (1989, 1990) and more recently by Croll (2004a, b). It is closely related to the more extensively studied thermal buckling of one dimensional components such as railway tracks, subsea oil and gas pipelines, concrete road pavements (see for example Martinet (1934), Kerr (1974,1984), Hobbs (1984), Taylor and Gan (1986), Croll (1997, 1998)). A full description of the mechanics is beyond the scope of this short posting, but a summary of the behaviour required to appreciate its applicability to the development of pingos is as follows.

Before considering the buckling of a heavy sheet it might be worth briefly discussing the buckling of a circular disk for which the effects of its own weight are neglected.

Perfect Behaviour: If a circular disk of finite radius, a, is restrained at its outer edges against both out-of-plane deformation and rotation and is subject to an in-plane stress that is equal in all directions (sometimes referred to as a hydrostatic or isostatic stress state), it will, at a certain critical level of this stress, develop an out-of-plane buckling mode taking the shape depicted in Figure 1(a). When the plate is initially perfectly flat the behaviour, exhibited in the form of a stress resultant n against out-of-plane displacement, w, is as shown in Figure 1(b). The stress resultant, n, is simply the average stress through the thickness of the plate times the plate thickness – it can be thought of as the amount of force per unit length. The plot of Figure 1(b) shows that there are two possible equilibrium paths. A fundamental path for which there is no deformation for ever increasing in-plane stress resultant, n, and a post-bucking path that shows increasing deformation, w, for little change in stress resultant, n. The two paths intersect at a stress level that is usually referred to as the critical stress (or sometimes the bifurcation stress although strictly it is an intesection rather than a bifurcation). In reality, for such a restrained plate there will be an increase in stress resultant for increasing out-of-plane displacement on the post-bucking path. For present purposes however, this complication will be ignored. The situation summarised in Figure 1 is often referred to idealised or “perfect” behaviour; it is not realistic in the very practical sense that it cannot actually be observed.

Imperfect Behaviour: Even the smallest out of flatness, sometimes described as an imperfection, will cause the above bifurcation behaviour of Figure 1 to be modified as shown in Figure 2. For example, the weight of the plate will cause a downward deformation (represented here as a negative deformation) of say magnitude, w0, even before any in-plane load is applied. As the in-plane load, n, is increased the non-linear equilibrium path will take the form shown as a-b in Figure 2. That is, the in-plane load will amplify the imperfection and tend to push the plate down at a rate which increases as the stress approaches that of the critical or bifurcation stress – or strictly as the imperfect path a-b asymptotically approaches the perfect post-buckling path. However, if the plate is sitting on a rigid foundation this downward displacement is not possible. Under these circumstances the only option open to the plate is for it to buckle upwards. These deformations will be controlled by the upper path d-e, which is sometimes referred to as the complementary equilibrium path for this same initial imperfection. While for most buckling problems these complementary paths are of little consequence, for the case of uplift buckling from a rigid or semi-rigid base support they are of controlling significance.

Perfect Uplift Buckling: If the plate of very large in-plane dimensions is sitting on the rigid foundation then the buckling behaviour will be close to that described above except for one important complicating factor. As the amplitude of the uplift buckle, w, increases the radius, a, of the uplifted buckle will also increase. The result is a propagation path that takes the form shown by path a-b in Figure 3. As demonstrated by Croll (1997, 2004a, 2004b) the approximate form for this propagation curve may be expressed in terms of the projections, shown by the heavy dashed lines in Figure 3, for which the equations take the forms shown. What these equations show is that the in-plane stress resultant required to propagate the uplift buckle, np, is inversely proportional to the inverse square root of the amplitude, w, of the uplift buckle. The radius, a, of the uplifted disk increases in direct proportion to the quarter power of the deflection, w - or it is proportional to the square root of the square root of w. In these equations q is the net downward weight of the plate per unit area and D is the bending stiffness of the plate defined in Figure 1. It is apparent from Figure 3  that if the plate starts out as perfectly flat, it would take an infinite load np before the system could get onto the falling propagation path a-b. How in reality the system will develop a propagating uplift buckle is all to do with initial imperfections.

Imperfect Uplift Behaviour: Probably of most direct relevance to our subsequent consideration of how the thermal uplift buckling model could help to explain the mechanics of pingo development, is that of an unstressed continuous geometric imperfection. This has been outlined by Croll (2004a) but considered in greater detail and possibly greater clarity in Croll (2004b). Imagine that the foundation were not perfectly flat but contained an out-of-flatness of the form shown in Figure 4(a). Studies have shown that for a given amplitude of this out-of-flatness, w0, the worst possible choice of wavelength would be that which has an affinity to the equivalent propagation shape given by eqn (3). This means the wavelength, a0, giving the lowest stress, and therefore temperature since we seem to have forgottren that the stress is being produced by a temperature increase, required to induce an uplift buckle would be related to the amplitude of imperfection, w0, given by eqn(5) shown in Figure 4(a).

With this amplitude and shape of continuous geometric out-of-flatness the stress required to cause the plate to initiate a lift-off condition, nL, would be exactly one quarter the equivalent propagation load that would apply for a displacement equal to the initial imperfection amplitude. The load required to reach the maximum or buckling load, nb, would be exactly twice this value and would occur at a total displacement twice the amplitude of the initial out-of-flatness, as depicted in Figure 4(b).

Lift-off From a Geometrically Imperfect Base: Railway track and subsea pipelines are in certain situations prone to a form of possibly sudden upheaval or restrained lateral buckling when the temperature becomes high enough that the associated in-plane stress reaches the magnitude nb. In these circumstances there will be a sudden jump in deformation with the railtrack or pipeline becoming highly distorted when it finally stops buckling. Just as for the asphalt lift-off discussed in an earlier posting, the evidence for pingos seems to point to their undergoing incremental increases over each annual thermal cycle. In these circumstances the temperature would reach a value in excess of that needed to induce a stress resultant, nL, leading to an incremental lift-off, but probably not great enough to reach nb and thereby cause a dynamic jump in deformation. Here the stress n is related to the temperature T through the relationship n = alfa.T.E/(1 - nu) where alfa is the coefficient of linear thermal expansion, E is the elastic modulus for the sheet, and nu is the Poisson's ratio for the material of the sheet.


I will return to some of the these considerations in the next posting where I hope to discuss how such a thermal upheaval of a heavy sheet might help to explain the emergence of pingos.

This form of 2-dimensional thermal buckling is briefly covered at pavement-blisters

is there another explanation for pingo growth?

I believe there is a very credible alternative model which in addition to the build-up of excess pore water pressure relies upon the development of high levels of in-plane tensile and compressive stress in the permafrost layer. These stresses are due to the restraint of the expansions and contractions that in the absence of lateral restraint would occur when the permafrost layer is subject to seasonal increases in temperature. A thermally induced upheaval buckling under these conditions will be argued in a future blog to be consistent with the observed local distortions that characterise the typical geomorphic features of the pingo. But more inportant it will be demonstrated that the average annual temperature changes through the thickness of the pingo required to induce an increment of pingo growth are well within the ranges experienced in the areas where pingos typically occur.

However, before describing this alternative explanation of how pingos might form, it might be helpful to establish the background to this new model in relation to the thermally induced, uplift, buckling of heavy sheets. For this reason my next posting will represent a temporary deviation from the pingo theme of my current postings.

might there be problems with current hypotheses for pingo growth?

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.

what are the current hypotheses for the development of pingos?

I am currently preparing this posting while on holiday and do not have access to scanning facilities which would allow me to incorporate sketches. On the basis that a sketch can often convey more meaning than many thousands of words, this makes descriptions much more difficult, but let me try. Later I will try to add some sketches to illustrate the following more clearly.

It is widely accepted that the formation of a “pingo” largely results from the development of an excess of pore water pressure in the unfrozen ground, usually referred to as the “talik”, underlying the permafrost layer. This explanation for their formation was seemingly first suggested by Porsild (1938), who is also credited with the name “pingo” derived from the Inuit word for hill. The mechanics believed to account for the initiation and subsequent growth of pingos has been greatly elaborated in the many works of Mackay - see for example his extensive summary of his own work, covering a period of nearly 50 years of continuous research into pingos, as well as a comprehensive summary of the contributions of others, which is contained in Mackay (1998). There are also very useful summaries of pingos in their wider periglacial context in the works of for example Washburn (1979) and French (1996).

What most of the explanations have in common is that pore water pressure build-up, beneath the usually recently frozen and normally aggrading lower boundary of the permafrost, pushes up a locally weakened area into the characteristic dome or, less commonly, ridge formation. The local weakening is often attributed to the presence of a locally thinned area of permafrost, which is thought to sometimes arise as a result of any remaining shallow pond water. The build-up of pore water pressure in the underlying unfrozen “talik” is said to result from one of two different causes.

In the hydraulic model, see French (1996), Mackay (1998), Washburn (1979), sometimes referred to as an “open system”, a ground water flow and artesian pressure beneath the permafrost layer is thought to be induced by the hydraulic head associated with the elevated water tables in the surrounding higher landforms. Mackay (1998) however, observes that “despite the decades of field studies by many individuals on hydraulic (open) system pingos, particularly in Greenland and Spitzberen, there are no surveyed growth data, anywhere, on even one pingo”.

More typical of the ranges of pingo occurring in the areas extensively studied by Mackay and his associates are those that result from an hydrostatic model, otherwise known as the “closed system” – see for example Mackay (1998). In these cases the permafrost will have enclosed the underlying unfrozen talik, so that any downward growth of the underside of the permafrost ice layer will result in a roughly 10% volume expansion when the enclosed ground water turns to ice. As a consequence of the confinement this expansion will result in an increase in the hydrostatic pressure within the underlying ground water - in much the same way that a bottle of wine left unintentionally in a freezer will as the wine freezes develop very high liquid pressure, often enough to push the cork out or in more extreme situations burst the bottle.

A common precondition for the development of a closed system pingo is the existence of a shallow lake or estuary overlying a saturated, fluvioglacial sand and gravel, in what is otherwise a well developed permafrost region. This shallow water will act as a thermal buffer to the downward penetration of permafrost, preventing the permafrost from developing under the lake or estuary bed. This will result in a bowl of unfrozen ground beneath the lake or estuary, possibly overlying a much deeper and well formed layer of permafrost. The genesis of the formation of pingos is commonly the sudden draining of the lake or a change in sea level relative to the estuarine sediments. This exposes the relatively flat lake or estuarine bed to the development of permafrost that over a period of years gradually propagates downward to form a thickening layer, that could enclose the unfrozen talik. Any residual ponds will cause the growth of this upper layer of permafrost to be retarded, providing it is argued a relatively flexible thin layer at which the underlying pore water pressure can be relieved by inducing an upward bulge. Water will be extruded into the space created by the upward deformation of the locally bulged permafrost layer as a result of the pore water pressure in the underlying unfrozen talik. Much of this water will subsequently freeze to form the ice lens, which is an additional feature of the developing pingo.

This process will continue over a period of many years, perhaps centuries, until some sort of stasis is reached between the downward growth of the lower permafrost boundary, the fluctuations of the underlying hydrostatic pressure within the unfrozen talik, and the upward deformation of the pingo bulge. While the causal mechanics has been less well developed, surges in the growth of open system pingos are also believed to result from surges in ground water pressures.

Some references mentioned above and in other blogs that might be helpful in understanding more about the nature of pingos and the currently accepted hypotheses for their development are:

French, H. M. (1996) The periglacial environment, 2nd Edition, Longman, 341pp.
 
Mackay, J. R. (1998). Pingo growth and collapse, Tuktoyaktuk Peninsula area, Western Arctic Coast, Canada: a long-term field study, Geographie physique et Quarternaire, 52, 271-323.
 
Porsild, A. E. (1938). Earth mounds in unglaciated Arctic nothwest America. Geographic Review, 28, 46-58.


Washburn, A. L. (1979) Geocryology: A survey of periglacial processes and environments, Edward Arnold, 406pp.

what are pingos?

Pingos are a characteristic geomorphologic feature of certain high latitude and also some high altitude regions. They take the form of dome-shaped mounds, well often quite large hills, that emerge from areas characterised by having permanently frozen ground within what are otherwise flat landscapes. I have for example seen examples of pingos on the Seward Peninsular in Alaska, not too far from the Arctic Circle but at sea level, and on the Qinghai Plateau in SW China, relatively close to the equator but at around 5000m elevation. Although not always the case they have a tendency to be circular in plan with base diameters that depend upon their height but can be up to 300m across. They often form in areas of recent exposure to the effects of permafrost surface penetration, such as the beds of lakes or the alluvial delta of rivers, which have for various reasons been drained of the overlying water that previously insulated the bed layers from the penetration of permafrost experienced by surrounding terrain. The upward and outward growth of the pingo is gradual occurring over periods measured in 100s to 1000s of years with annual growth spurts generally taking place at certain restricted periods of the year. Annual upward growth spurts depend upon the age of the pingo with maximum rates seemingly occurring over the first few decades of growth. They are generally underlain by layered ice, whose annual increase in layer thickness appears to relate to the annual upward growth of the pingo – much like tree rings reflect the annual growth of trees.

Pingos have a fairly clear life cycle. Gradually the rates of annual upward growth slow down. As the pingo matures, fractures appear over the crown of the dome and the altered drainage characteristics lead to a gradual erosion of the upper areas of permafrost, especially near the crown of the pingo dome. Eventually, the erosion of the permafrost becomes so severe that it might expose the ice core. This together with climatic warming eventually leads to a thawing of the permafrost and a collapse of the ice core leaving a relic pingo taking the form of a central depression sometimes surrounded by a doughnut shaped ring. These relic pingos are important indicators of climate change and provide evidence of the latitudes to which frozen ground (permafrost) extended during one or other of the past glacial periods. For example, relic pingos have been observed in areas such as East Anglia and Wales within the British Isles and New Jersey in the USA.