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.
All very interesting T-J but surely it was shown back in 1987 by Professor Mackay that ground water pressure is sufficient to explain the deformation of permafrost to form pingos? Why do we need the additional mechanical effects of in-plane compression to induce the uplift?
ReplyDeleteThanks for raising this sottovoce1 since I did not feel it appropriate to cover this aspect in detail in the posting. I presume the work you are referring to is Ross Mackay’s paper dealing with “Some mechanical aspects of pingo growth and failure, western Arctic coast, Canada”, published in the Canadian J of Earth Sciences in 1987. In this paper Mackay attempts to account for pingo growth by treating the permafrost as a thin elastic plate deforming in flexure under the action of the overpressure in the underlying talik.
ReplyDeleteWhen this paper had been first drawn to my attention I looked forward to reading it since I had been having great difficulty finding directly relevant information on the elastic modulus of permafrost. I had hoped that the paper would provide me with some good data gathered from the field on this vital material property. It was surprising to find that rather than input the values of Young’s modulus in order to calculate the deformations arising from a prescribed level of pressure, Prof Mackay treats E as an output to be obtained from a backtrack of known or assumed deformations. The values that he obtains for E are of the order of 9 – 32 MPa. Although he suggests that these values are in broad agreement with those given for the long term deformation of frozen sand by Grechishchev, they would appear to be extremely low. For example, rubber and very soft saturated clays have E values at around these levels. Ice and presumably permafrost would surely be a lot stiffer than a soft clay or rubber? Other sources that I have seen give values of E for ice at around 5 to 10 GPa; that is between 2 and 3 orders of magnitude higher than those inferred by Mackay. In most heterogeneous matrix materials the composite E value lies somewhere between that of the two materials making up the composite; in this case this would be somewhere between the sand or gravel particles and the ice. With typical rocks having E values well in excess of 10 GPa one might therefore reasonably expect the permafrost E value to be in excess of the 5 – 10 GPa of ice. This is of course a vital issue in terms of the assessment of whether any excess ground water pressure is capable of lifting the permafrost into a pingo type deformation. That such low values of E are needed to account for a ground water pressure origin of pingos must surely be a source of concern?
Will continue these comments later.
This continues from the previous comment:
ReplyDeleteAnother issue that interested me was the level of pore water pressure underlying the pingo. Mackay (1987) is very explicit on this aspect which is very helpful. At eqn (7) and in Figure 4 the pore water pressure head at a depth of z is given as 0.75 times the bulk unit weight of the frozen lake bottom sands times the depth z. That is, the pore water pressure beneath the lake bottom is around 75% of the weight of the overburden of frozen sand and ice. As is shown in Fig 35, Mackay (1998) – written M98 for convenience – the underside of the lake bottom is at a depth of around 40m. Fig 4 (M87) shows the pore water pressure at this depth was measured as 600 kPa, which is 75% of the overburden weight when as Mackay suggests the average overburden bulk unit weight is 20 kN/m3. At the top of the water lens beneath the pingo, which from Fig 35 (M98) is at roughly 23m above the underside of the lake bottom permafrost, this pressure would produce a pressure of at maximum (600-23x10=) 370 kPa if losses due to permeability of the underlying saturated talik are neglected. When pressure losses due to the permeability within the talik are taken into account, these water pressure levels are starting to be consistent with the water pressures measured in the sub pingo water lens over the period 1977 to 1991, which as shown in Fig 37 (M98) have varied from 300 to 370 kPa. As suggested at Fig 37 (M98) these pressures were associated with a “hydraulic head … above the top of the pingo” explaining the gusher of 2.6m shown in Fig 5 (M98) and again at Fig 6 (M87). But at a depth of 22m the pressures of between 300 and 370 kPa are less than the overburden weight of (22x20=) 440 kPa that would be extant at the underside of the pingo. Even if there were no friction losses between the underside of the lake bottom and the top of the water lens the maximum water pressure consistent with the 600 kPa of Fig 4 (M87), the sub-pingo water pressure would be insufficient to overcome the weight of the permafrost overburden of the pingo. It is therefore not clear to me where the 70 kPa surplus pressure taken to be deforming pingo 14 in Table 2 (M87) comes from? This would require a sub-pingo water pressure of (440+70=) 510 kPa. Such a pressure level is consistent with neither the pressures beneath the lakebed permafrost, Fig 4 (M87), nor the pressure transducer data of water pressures beneath the pingo, Fig 37 (M98). Indeed, on the basis of the published evidence there would appear to be a shortfall in sub-pingo water pressure of at least (440-370=) 70 kPa, so that the active pressure within the sub-pingo sand would be around 70 kPa. Similar shortfalls in ground water pressure exist for the other pingos treated in Mackay (1987). On this basis I repeat what I said in an earlier blog, there appears to be “insufficient excess pore water pressure for dilation” to occur (see also Croll, 2004).
And this also continues the previous comments:
ReplyDeleteA further issue that worries me a lot is the use of the total, or historically accumulated deformations, rather than incremental, or annual increments, deformations. All the growth data seems to point to small annual increases in the deformation of the permafrost layers within the pingo. Whatever is producing these annual increments is doing so in some regular pattern every year – unless of course interrupted by some other factor. Ground water pressures fluctuate but seemingly around a fairly steady average ground water pressure, as shown in Fig 37 (M98). So that if the annual deformation were to be the product of the underlying ground water pressure the plate deformation resulting from this pressure should be the annual growth. For pingo 14 the annual growth between 1971 and 1977 was at the rate of 3 cm per year (M87). This is 100 times less than the 3 m assumed in Table 2 (M87) to represent the deformation being produced by the 70 kPa surplus(?) underlying ground water pressure. Even the 3m taken in Table 2 (M87) to represent the accumulated deformation is something of a mystery to me. Fig 35 and 40 (M98) shows pingo 14 having a total elevation above the datum lake bed level that is in excess of 10m at the maximum height near BM 50 and over 5m in the vicinity of the location of the bore hole. It would be interesting to know what was the basis of the 3 m? But it is my view that this historically accumulated deformation is less relevant than the annual incremental deformations.
One or two other aspects I feel are also worth clarifying. If as I believe it is more appropriate to use a measure of the incremental deformation rather than total deformation, then for pingo 14 the radius characterising the horizontal extent of the incremental deformations would appear to be nearer 70m rather than the 100m taken in Table 2 (M87). This is shown not only by the growth data but also the incremental subsidence data shown in Figs 38 and 39 (M98). I have recently re-plotted this data using a more sensitive form of nonlinear interpolation of the data. With the pressures required to deform a circular disc of permafrost being dependant upon the inverse of the forth power of the radius a reduction from 100 to 70m radius would involve a four-fold pressure increase to produce a given level of deformation. The effect of taking a deformation radius lower than the 100m adopted in Table 2 (M1987) would be to reduce even further the inferred E value, which is already seemingly very low.
Re: “insufficient excess pore water pressure for dilation”
ReplyDeleteRespectfully, please read Chapter 7: Thermodynamic behaviour of frozen soils, in The Frozen Earth: Fundamentals of Geocryology. It is a primer on this topic. The key is essentially the well known Clausius-Clapeyron equation and it is the pressure of the ice - not the water - that is the source of the expansive forces in a frost heaving soil, including heaving up of pingos. Pingos form in saturated soils, typically with fine grained "frost susceptible" material over coarse grained material. A pressure of MegaPascals is required to stop frost heave in fine grained soils.
You could always write Professor Mackay or Professor Burn a note if you want the system explained.
Peter (I hope you don't mind) as you might have gathered I have not been very active on this blog for the past couple of years - see posting earlier today, 21.2.12, entitled "Where was I?". This will make it clear that I have been in extensive correspondence not just with Ross Mackay but a number of other who have long worked on pingos, permafrost etc. I do not agree with your comments as to the relevance of frost heave to the growth of pingos but it being very late I will defer until tomorrow my reasons for not being able to agree with what you assert.
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