some mechanical aspects of pingo growth and failure, western arctic coast, canada

Some mechanical aspects of pingo growth and failure, western Arctic coast, Canada

J. Ross MACKAY Department of Geography, The University of British Columbia, Vancouver, B. C . , Canada V6T 1 W5

Received November 12, 1986

Accepted December 9, 1986

Many closed-system pingos are underlain by sub-pingo water lenses, and the same is probably true of numerous open- system pingos. In the early growth stage the bending of the frozen overburden of a pingo by a sub-pingo water lens can be compared to the bending of a thin elastic plate. Although the assumptions of elastic plate theory do not apply fully to a growing pingo, because time-dependent plastic and creep deformation are involved, the application of elastic plate theory nevertheless helps to explain the peripheral normal faulting and spring flow of pingos, summit failure, the ease with which elongated pingos appear to collapse, and the changing roles played by the radius and overburden thickness of pingos from early growth to the cessation of growth.

Plusieurs systbmes rapprochCs de pingo sont sus-jacents B des lentilles d'eau de type sub-pingo, et il en est probablement de mCme pour de nombreux systkmes ouverts de pingo. En dCbut de croissance, la flexion du sol gel6 d'un pingo sous la pression des lentilles d'eau du sub-pingo peut &tre comparte B la flexion de minces plaques tlastiques. M&me si les fondements de la thCorie des plaques Clastiques ne s'appliquent pas entibrement au phtnomkne de croissance des pingo, B cause du fait que la dCformation plastique et le glissement varient avec le temps, il n'en demeure pas moins que l'application de la thtorie tlastique des plaques contribue B expliquer la prtsence des failles normales et l'tmergenge de sources B la pCriphCrie des pingo, l'appari- tion des fissures au sommet, la facilitt avec laquelle s'effondrent les pingo allongCs et le changement des r6les jouts par le rayon des pingo et leur Cpaisseur de sol entre le dCbut et la cessation de la croissance.

[Traduit par la revue]

Can. J. Earth Sci. 24. 1108 - 11 19 (1987)

Introduction Pingos are ice-cored hills (Fig. 1) that can only grow and

persist in a permafrost environment. There are about 1450 pingos along the western Arctic coast of Canada (Mackay 1962; Stager 1956); most are concentrated in the Richards Island and Tuktoyaktuk Peninsula area, Northwest Territories (Fig. 2). Seven hundred pingos have also been identified along the Arctic coastal plain of Alaska (Hamilton and Obi 1982). Two main types of pingos can be distinguished: closed-system (hydrostatic) pingos and open-system (hydraulic) pingos (Mackay 1979; Miiller 1959). Pingos are domed either by the growth of segregated ice, by the intrusion and progressive freezing of a sub-pingo water lens, or by both processes. A pingo that is domed by the freezing of a sub-pingo water lens resembles a laccolith domed by the intrusion of magma. For this reason, pingos have often been referred to as hydrolaccoliths (e.g., Brown and Kupsch 1974; Gary et al. 1972; Holmes et al. 1968; Shumskii 1964; Solov'ev 1952). Field studies show that the failure of many pingos is mechanically induced, similar to that of laccoliths (Gilbert 1877; Johnson 1970; Johnson and Pollard 1973; Pollard 1968; Pollard and Johnson 1973). The collapse and failure of other pingos are primarily from a thermal disturbance to ice-rich permafrost. This paper discusses some mechanical aspects of pingo growth and failure for the closed-system pingos of the western Arctic coast.

Closed-system pingos Most pingos of the western Arctic coast are of the closed-

system type with much intrusive ice (Mackay 1979, 1985~) . The great majority have developed or grown up in shallow residual ponds left by rapid lake drainage in areas underlain by sands. A few have developed in the alluvium of the modem Mackenzie Delta (Mackay 1963; Mackay and Stager 1966). Because most western Arctic coast pingos are unnamed, pingo numbers are used, the same as in previous publications (e.g., Mackay 1979).

Permafrost growth

The mean annual air temperature along the western Arctic coast is about - 10°C or lower, the mean annual ground surface temperature of undisturbed sites is about -6°C or lower, and the thickness of undisturbed permafrost is about 400 m or more (Mackay 1979). Prior to lake drainage, the thermal mass of a lake maintains an unfrozen basin (talik) beneath the lake bottom. When drainage exposes a lake bottom to subfreezing air temperatures, permafrost commences to aggrade downward. In addition, permafrost aggrades upward from the bottom of the talik, the rate being greatest in nearshore areas, where the depth to permafrost is least (Mackay 1984, 19856).

Rate of permafrost growth The rate of permafrost growth below a drained lake bottom

suddenly exposed to subfreezing air temperatures can be estimated from Stefan's formula (Johnston 1981; Lunardini 1981). Stefan's formula contains simplifying assumptions that make it convenient to use under field conditions, where detailed information on soil and temperature conditions is rarely available. The formula overestimates the freezing depth, particularly for long (e.g., 500 years) time periods. Stefan's formula is

The nomenclature for [l] and subsequent equations is given in Table 1. From [I] the growth rate is

The ratio between the freezing depth for a saturated sand to that of ice for the same period is

Printed In Canada I Imprimt au Canada

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MACKAY 1109

FIG. 1 . Ibyuk Pingo, 49 m high, located just to the southwest of Tuktoyaktuk, Northwest Temtories. See Fig. 2, pingo 18.

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11 10 CAN. J. EARTH SCI. VOL. 24, 1987

TABLE 1. Definitions of symbols used 2:l ratio calculated by Brown (1964). Upon lake drainage, permafrost aggradation will commence below the exposed lake

Symbol Definition bottom. On the other hand, pingo growth is usually delayed a

b Stefan's b few years because the growth site tends to be a shallow bi Stefan's "b" for ice residual pond, the thermal mass of which retards permafrost

b, Stefan's "b" for frozen sand growth in comparison with the surrounding exposed lake C Circumference bottom. D Flexural rigidity [D = E P / 1 2 ( 1 - vZ)] E Young's modulus of elasticity; creep parameter Subpermafrost pore-water pressures H Height above ground level to which subpermafrost water The total stress (Oz) at the the frozen fringe

could rise in a vertical standpipe (i.e., the zone where most of the pore water freezes) in h Uplift height of a point on the lake bottom during pingo downward aggrading permafrost (Fig. 3) is

growth i Ice [4I 0, = YSZ k Thermal conductivity of frozen ground k, Thermal conductivity of ice k, Thermal conductivity of frozen sand L Length L, Final length Lo Original length n Exponent Q Volumetric latent heat of fusion q Effective upward driving pressure of subpermafrost pore

water R Pingo radius s Frozen sand T Thickness I; Thickness of pingo ice T, Thickness of overburden above pingo ice To Thickness of overburden (T,) and pingo ice (I;) combined

( g = + T,) t Time t, Time required to freeze to depth z u Pore-water pressure u, Pore-water pressure beneath overlying material including

overburden and any pingo ice u, Pore-water pressure at depth z w Maximum deflection of a circular plate; height of sub-pingo

water lens x Semi-major axis of ellipse y Semi-minor axis of ellipse z Depth; thickness of permafrost zi Thickness of ice z, Thickness of frozen sand a Angle p Parameter y Bulk unit weight y,, Mean bulk unit weight of pingo ice and the overburden y, Bulk unit weight of frozen sand 6 Parameter

and from [I] and [4] for time t and depth z

151 0, = ysbJ t

The effective stress (a:) is

If all pore water freezes in situ as permafrost aggrades, then the ground surface will heave about 9 % of the volume of pore water frozen. In situ freezing must therefore separate many soil particles that were formerly in contact. Thus the local effective stress ( [6]) must approach zero (Miller 1980, p. 290). That is, in situ pore-water freezing creates a pore-water pressure approaching that of the total normal stress. Balduzzi (1959), for example, measured a pore-water pressure of 95% of the normal stress in closed-system freezing. If, however, pore water is free to move in a closed system in response to a pressure gradient, pore water can be expelled from a site with a high pressure and move towards a site with a lower pressure. Pore-water expulsion in sands with drainage has been well documented for many decades (Beskow 1935; Janson 1964; McRoberts and Morgenstern 1975; Tsytovich 1975).

Subpermafrost pore-water pressures, as measured with strain gauge pressure transducers installed beneath ice-bonded permafrost in three drained lakes, one with a growing pingo, one with two growing pingos, and one with three growing pingos, are plotted in Fig. 4. If the bulk unit weight (y,) of the frozen lake bottom sands is assumed to be 20 kN/m3 and the thickness of permafrost is z , then the subpermafrost pore-water pressure (u,), estimated from Fig. 4, is about

[7] u, - 0 . 7 5 ~ ~ ~ or 0.750,

and from [5] and [7] t Linear strain [8] uz~0.75ySbz&, 6 Absolute value of negative ground-surface temperature - - v Poisson's ratio a Normal stress a, Tensile yield strength a, Normal stress at depth z a' Effective stress a Effective stress at depth z

Because the bulk unit weight of the frozen lake bottom sands is about twice that of water, the height (H) to which subpermafrost water could rise above ground level in a vertical standpipe would be about

Figure 5 shows permafrost thicknesses and piezometric heads for Stefan's b of 3 and 4 m/year1/2. These values encom-

Stefan's b has been determined for the growth of permafrost pass the range for the most rapid permafrost growth so far beneath five recently drained lakes in the Richards Island and measured along the western Arctic coast. The height of a pingo Tuktoyaktuk Peninsula area and also for the growth of some for a b of 1.5 m/year1'2 is included for comparison. Results pingos (Mackay 1979, 1983, 1985a, 1985b). The range is show that the piezometric head can exceed the top of a pingo. from about 4.0 m/year'IZ for a sand without excess ice to about Indeed, water loss from drill-hole flow (Fig. 6) from a sub- 1.5 m/year112 for ice-rich permafrost. These field values agree pingo water lens causes pingo subsidence (Mackay 1983). This with those derived from [3] and are similar to the approximate demonstrates that the water pressure is sufficient to both lift

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MACKAY 1111

I METRES FIG. 3. Cross section of pingo 9 as it was in 1977 (Mackay 1978). The change from 1977 to 1983 was reported in Mackay (1985~). The

unbonded permafrost, below the ice-bonded permafrost, is at a temperature of about -O.l°C.

and bend the pingo ice core and the overburden above the ice 0 core (Fig. 3).

Pingos as hydrolaccoliths

Johnson and Pollard (Johnson 1970; Johnson and Pollard 1973; Pollard 1968; Pollard and Johnson 1973) discussed in detail the bending of the overburden of laccoliths (Fig. 7) by adapting plate theory for the bending of thin elastic plates (Timoshenko and Woinowsky-Krieger 1959; Ugural 1981). The deformation of frozen ground, like that of glacier ice, is not solely elastic but also involves plastic and creep deforrna- tions (Johnston 198 1 ; Ladanyi 1981 ; Morgenstern 1985; Soo et al. 1986; Tsytovich 1975). Therefore, although the deformation of frozen ground does not meet the assumptions of elastic plate theory, the application of elastic plate theory nevertheless provides a useful framework within which to examine pingo growth.

Plate theory Many pingos are approximately circular in planimetric

shape, so the original pingo overburden can be represented by a thin, circular plate. The boundary condition for the edge of the plate can be viewed as either simply supported or clamped. For the initial period of pingo growth, a simply supported edge may be appropriate. Because the thickness of permafrost surrounding a growing pingo will usually be two to three times the height of the pingo (Fig. 3), a clamped edge is more appropriate very shortly after pingo growth has commenced. For simplicity, a uniformly distributed load (i.e., upward direc- tion) is assumed to apply beneath a sub-pingo water lens.

In plate theory, the body force of a thin plate is omitted in computing the effective driving pressure. However, in the growth of laccoliths (Pollard and Johnson 1973) and also in the growth of pingos, the weight of the overburden must be included because it greatly reduces the effect of the upward driving pressure of the fluid. The effective driving pressure (q) is given by

\ I I

\ \ \ \ \ Pingos 8, 9 - l \

- \

\ e Pingo 15 \ \

\@pingos 13, 14, 17 - \ - \ \ \ c

\%

- - \ \ \ \ \ - \ -

\ \ \ \ \

I I \

PORE-WATER PRESSURE (u,) BENEATH PERMAFROST (MPa)

FIG. 4. The pore-water pressure beneath aggrading permafrost was measured below ice-bonded permafrost by means of pressure transducers of the strain gauge type.

The maximum deflection (w) of a circular plate with a simply supported edge (Fig. 7 ) in response to a uniform driving pressure is at the centre and is (Ugural 1981, p. 34)

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11 12 CAN. 1. EARTH SCI. VOL. 24, 1987

YEARS

FIG. 5 . Growth of permafrost and pingos using Stefan's formula, which, for periods of hundreds of years, will overestimate permafrost depths. See text.

60

4 0

O n

The flexural rigidity (D) in [ l l ] is

- - piezomctric Head for h = 4

p i e r ; C 0 = 3 pingo eight for h = 1 5 \li -

Ground ~ p ~ ~ /

FIG. 6. Drill-hole flow in 1977 from a sub-pingo water lens 22 m below the side of pingo 14. In 1976 and 1977, as a result of flow from the sub-pingo water lens, the pingo subsided, with the greatest subsidence being 60 cm at the top.

FIG. 7. Diagrammatic cross section of a laccolith (Johnson 1970) or a pingo (Mackay 1979) where the overburden is bent by fluid under pressure. See text.

If the edge of the plate is clamped, the maximum deflection is also at the centre and is (Ugural 1981, p. 33)

Poisson's ratio (v) for frozen ground is usually in the range of 0.3 -0.4, so the maximum deflection of a simply supported plate is about four times that of a plate with a clamped edge. Moreover, the terms involving Poisson's ratio are so close to one that they can be omitted without serious error in [ l 11, [12], and [13]. Equation 13 can then be rewritten as

The deformation of frozen ground depends upon many factors such as soil type, strain rate, and time (Grechishchev 1970, 1978; Ladanyi 1981 ; Morgenstern 1985; Parameswaran 1980; Tsytovich 1975). Young's modulus (E) in the above equation is temperature dependent (Grechishchev 1970, 1978; ~ s ~ t o v i c h 1975) and can be estimated from

For sands down to a temperature slightly below - 10°C, the exponent n can be put equal to unity (Tsytovich 1975). According to Grechishchev (1970, 1978), [15] can be used for both an instantaneous elastic deformation and a long-term creep deformation, although the values of the constants will differ. The implication of [15] is that if the overburden is thin and warms up close to O°C in summer, Young's modulus will be very much smaller in late summer than in late winter. Therefore, when both the overburden and the pingo ice warm up towards 0°C in late summer, an increase in the height (w) might be expected in comparison to late winter.

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TABLE 2. Pingo data required to test the plate theory

Thickness Driving (overburden Young's Tensile yield

Height, pressure, Radius, plus ice core), modulus, strength, w 4 R TO E "Y

(m, approx.) (kPa, approx.) (m, approx.) (m, approx.) (MPa, approx.1 (MPa, approx.)

Pingo 9 2.2 40 50 12 12 - Pingo 14 3.0 70 100 24 32 - Pingo 15 1 .O 50 50 16 14 0.33 Pingo 17 3.0 70 130 24 9 1.4

The state of stress in the overburden of a laccolith (Fig. 8) has been discussed by Pollard (1968) and Pollard and Johnson (1973). In laccoliths, failure tends to occur around the periphery where the overburden is in tension, either by diking or normal faulting. The observed peripheral failure patterns of pingos 4, 15, and 17 are similar, as illustrated in Fig. 9.

The maximum principal stresses in a clamped circular plate occur at the edge. If the plate is made of an isotropic material of tensile yield strength ay, the value of the effective driving pressure governing the onset of yielding is (Ugural 1981, P 35)

If the effective driving pressure at the time of peripheral failure is known, then the tensile yield strength (a,,) may be estimated from [16].

Testing the plate theory Ideally, the plate theory should be tested just as the over-

burden of a young pingo becomes domed, but complete data are not available for that condition. However, some data are available for a number of older growing pingos, as given in Table 2. The heights (w) of the water lenses of pingos 9 and 14 have been measured by drilling, and reasonable estimates can be given for pingos 15 and 17. The effective driving pressure (q) has been estimated from data in Fig. 4 and [7]. The criterion used to define the radius (R) is the distance, measured from the top of the pingo, where drill-hole flow caused a subsidence exceeding 0.5 cm for pingos 14 and 15. For pingos 9 and 17, the radius was measured on a map. The thickness given in Table 2 is the total thickness (Ti,), that is, the overburden thickness (To) plus pingo ice thickness (T), as shown in Fig. 3 near the top of the pingo.

Young's modulus (E), or what might better be termed a creep parameter, calculated from [14] is given in Table 2. If strain (E) in pingo growth is defined as the stretch of the overburden from an original length Lo to a final length & then

Insofar as is known, the most rapid growth period of a pingo is in the first few years. If the height of the sub-pingo water lens is w and the pingo is circular, then the longitudinal strain (E) of the overburden from stretching, derived by Pollard and Johnson (1973) for a laccolith from [13], is

The great majority of laboratory tests of Young's modulus for frozen soils and ice involve strain rates of more than lo-'

Tension Tension

FIG. 8. Diagrammatic cross section of the overburden of a laccolith (Pollard 1968; Pollard and Johnson 1973) showing areas that are in compression and tension. Peripheral failure, expressed by normal faulting and diking, occurs in association with some laccoliths. Similar peripheral failure occurs with both closed-system and open- system pingos. See Fig. 9.

FIG. 9. Peripheral failure as observed in pingos. (a) Spring flow at the periphery of a pingo has been observed in pingos 4, 15, and 17. (b) Normal faulting has been observed in pingos 4 and 17. At pingo 17, spring flow has issued along the fault line.

s-' (e.g., Parameswaran 1980; Traetteberg et al. 1975; Tsytovich 1975). These rates are much higher than those expe- rienced in pingo growth. For example, pingo 9 has a radius of about 50 m, and the maximum growth is unlikely to have exceeded 0.5 m year1. From [18], the maximum strain rate

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CAN. J. EARTH SCI. VOL. 24, 1987

I 1 I

Canadian Pingos A Alaskan Pingos 0 /

Siberian Pingos 0 0 To = 0.13R - 1.26 - - r =0.97

5' 0

/

/ 0'

0' 0

/

v0 - - m 0 /'

0 / @

m. a,/ :0 ,8 /*';

/A - 2

PING0 RADIUS, R (metres)

FIG. 10. Pingo overburden thickness plotted against pingo radius.

would then be about 2 x lo-'' s-' or orders of magnitude smaller than the smallest strain rates customarily used in laboratory tests. Thus, the value of Young's modulus or a long-term creep parameter for pingo growth should be much smaller than the short-term values determined in laboratory tests. Grechishchev (1970) for example, gave a long-term value for frozen sand in tension at - 3 OC of 40 MPa, a value comparable to the computed values in Table 2.

The long-term tensile yield strength (uy) has been calculated from [16] for pingos 15 and 17 for periods when both pingos underwent peripheral failure. The long-term tensile strength of a frozen sand (Grechishchev 1970, p. 15; Johnston 1981, p. 95) is about 0.2 MPa, so the field values calculated for the pingos are in good agreement with those obtained in the laboratory.

Mechanically induced failure When the frozen bottom of a residual pond domes to become

a pingo, overburden stretching usually results in both summit failure, as seen in a vertical cross section through the pingo summit, and circumferential failure, as seen in a horizontal cross section.

Overburden thickness The general relationship between the total pingo overburden

thickness and radius for those pingos where data are available is plotted in Fig. 10. The data for Canada are from Mackay (1973, 1977, 1978, 1979), Mackay and Stager (1966), Miiller (1959), and Pihlainen et al. (1956). Siberian data are from Solov'ev (1952), and Alaskan data are from Kovacs and Morey (1985), D. A. Walker (personal communication, 1985), and Walker et al. (1985). A least-squares fit with measure- ments in metres gives

Dilation Cracktnn

FIG. 11. Diagrammatic sketch of the growth pattern of a pingo. Location "a" on the flattish bottom of a residual pond of a drained lake is uplifted to location "b" by pingo growth. See text and Mackay (1979).

bottom to top. This rupture produces the so-called pingo crater (Fig. 1). The mode of summit failure is shown schematically in Fig. 11, where a point on the bottom of the residual pond at location "a" is uplifted approximately along an arc of a circle to location "b" (Mackay 1979, 1985~) . Therefore, if point "a" was a distance "L" from the centre, after uplift to a height "h," the outward planimetric displacement (AL) is

r201 AL = h(cosec a - cotan a)

When h is the pingo height, then 2AL gives the approximate overburden failure from stretching, the distance corresponding closely to the diameter of the summit crater. In addition to overburden (i.e., mineral soil) failure, the pingo ice core also undergoes dilation cracking at the summit where the ice core is in tension (Fig. 8). The vertical dilation cracks so produced, when infilled with surface water, form vertically banded dilation crack ice with a total width approximating the width of the pingo crater (Mackay 1985~) .

Circumferential failure Doming of the flattish bottom of a residual ~ o n d into the "

Summit failure three-dimensional surface of a pingo results not only in summit Radial stretching of the pingo overburden is relieved pri- failure but also in circumferential failure. From Fig. 11, the

manly by rupture at the pingo top rather than uniformly from circumference of the circle described by all points represented

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---, - 7, - - , 'I" ,=

k&L--=- *. +, . ,.$," *I-- - - , - Spring Flow

1 -. - *- ..>* , -,

;, , . , '

10 20 30 40 50

METRES

FIG. 13. The cross section is to the left of the frost blister on which the man in Fig. 12 is standing. When active faulting takes place in wmnter, willow branches, frozen in the snow, are broken and vertically offset.

FIG. 12. Spring flow issuing along the line of a normal fault on the side of pingo 17 in May 1978. Spring flow has saturated the snow. Field evidence shows that episodic spring flow had been issuing along the same fault line for at least several decades. Note man, standing on an ice-cored frost blister, for scale.

by "a" on the pond bottom at a radial distance L from the centre of the pingo prior to uplift is 2xL. After point "a" is uplifted to point "b", the radial distance from the planimetric centre has increased by AL. Therefore, the increase in the circumference (AC) is

[21] AC = 2~h(cosec a - cotan a )

The circumferential stretching tends to be relieved by radial dilation cracks that trend downslope from the top to the bottom of the pingo and often far onto the adjoining lake flats (Mackay 1978, 1979, 1 9 8 5 ~ ) . The radial dilation cracks, like the summit dilation cracks, can become infilled with surface water to form dilation crack ice.

Peripheral failure Peripheral faulting has been observed along the periphery of

pingo 17 (Fig. 12) where there is a 100 m long concave scarp, presumably a fault scarp. In the winter of 1974- 1975, willow branches, solidly frozen into the snowpack, were broken and differentially uplifted 10-30 cm along a 13 m long section that also yielded spring flow (Fig. 13). Excellent high-angle peripheral normal faults have also been observed in sectioned pingos (French et al. 1982; Mackay and Stager 1966; Rampton and Mackay 1971).

Rupture and collapse The only known example of nearly "instantaneous" pingo

collapse is that of pingo 12 (Mackay 1979). In 1935 the site of pingo 12 was occupied by a lake (airphoto A5025-49C). By 1943, the lake had drained and pingo 12 had grown large enough to be visible on an airphoto (3025-300R-49). By 1947 (airphoto A10988-111) the pingo had collapsed. Thus the maximum period between the existence of the lake and the collapse of the pingo was only 12 years. If mid-dates between successive photographs are used for estimates of the times of lake drainage and pingo collapse, then lake drainage took place in about 1939 and pingo collapse in about 1945, for a 6 year

interval between lake drainage and pingo collapse. These dates will be used for subsequent calculations.

In plan view, pingo 12 forms an asymmetric 200 m long crescent with a 50 m mean central width (Fig. 14). Remnant ridges, which rise to 1.3 m above the lake flats, enclose two small ponds with maximum depths of 1 m below the lake flats. The simplest counterpart in plate theory to the shape of pingo 12 is a clamped elliptical plate with a semi-major axis (x) of 100 m and a semi-minor axis (y) of 25 m. The maximum deflection of a clamped elliptical plate (Ugural 1981, p. 101) is

If pingo 12 commenced growth in 1939 with a Stefan's b value of 3 m/year1'2, by 1943 when the pingo was visible on an airphoto the depth of permafrost around the pingo would have reached about 6 m. The surface of the 1.3 m high ridges of pingo 12 initially formed part of the bottom oVf a shvallow residual pond. An overburden thickness of about 2 m has been estimated from 1986 drilling on the ridges. From [22], the maximum height of the water lens of pingo 12 would be about 2.5 times that of a circular pingo with a radius of 25 m, which is that of the semi-minor axis of the pingo. The suggestion made here is that a thin 2 m overburden combined with a large height (w) of the sub-pingo water lens caused rupture and collapse.

In July 1986, the remains of a frost blister with intrusive ice (Fig. 15) grown the previous winter was found at the site of the large central pond of pingo 12. Drilling, which resulted in drill-hole flow, showed that permafrost by the pond was only 7 m thick. Water-quality analyses of the frost-blister ice and drill-hole flow show a common origin. Evidently, the thin permafrost beneath the pond is still under sufficient pressure to rupture occasionally. Nevertheless, it should be stressed that many elongated pingos have grown to "full" heights (French 1976; French and Dutkiewicz 1976; Mackay 1963; Pissart and French 1976), so an elongated shape is not the only factor in pingo failure.

Collapse and regrowth Pingo 4 (Fig. 16) in plan view is kidney shaped with a length

of 200 m and a mean width of 50 m, the overall dimensions being similar to those of pingo 12. In profile, pingo 4 consists of a steep central hill 9 m high and 35 m in basal diameter, a moat that partially surrounds the central hill, and two flanking 80 m long ridges that are several metres in height (Fig. 16). The field and airphoto evidence suggests the following

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FIG. 14. Photograph of the collapsed remnant of pingo 12. The collapsed remnant is kidney shaped, about 200 m long and 50 m wide.

FIG. 15. Photograph of a frost blister with intrusive ice formed in the winter of 1985 - 1986 at the site of the large central pond of pingo 12 shown in Fig. 14. Much of the water in the frost blister came from the rupture of thin permafrost and spring flow at the site of the pond. Note the shovel for scale.

sequence of events. Pingo growth started not long before 1890, when the oldest dated willows on the two low flanking ridges commenced growing. The central portion then collapsed. Regrowth in the collapsed centre started shortly before 1935, because the pingo appears as a low mound, flanked by the present ridges, on a 1935 airphoto (A5020-39L). The ages of the willows on the pingo hill also show that growth started there not long prior to 1940. The collapse, as with pingo 12, is attributed to the elongated shape and resulting large height of

the sub-pingo water lens. The pingo has been under observa- tion since 1967 and was under precise survey from 1969 to 1981. During the 1979- 1981 period there was peripheral failure with spring flow and pingo subsidence. The peripheral failure is attributed to tensile failure, as shown in Figs. 8 and 9.

Discussion

Although the deformation of a pingo's overburden does not meet the basic assumptions of elastic plate theory, the theory

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Low Rio

FIG. 16. Photograph of pingo 4. The pingo hill in the centre is 9 m high and it is flanked by two low ridges, remnants of a former 200 m long kidney-shaped pingo that grew, collapsed, and then regrew in the centre. See text.

does provide a useful framework within which the growth and mechanical failure of pingos can be examined, for the following reasons.

First, peripheral failure (i.e., normal faulting) is explained by plate theory. According to the stress distribution shown in Fig. 8, if a crack opened at the periphery of a sub-pingo water lens, where the overburden is in a state of tension, the crack would propagate upward into a zone of compression, where cracking might then cease. However, as ~achenbmch (1962) showed in the comparable situation of thermal contraction (ice wedge) cracks, a crack can propagate from a zone of tension into a zone of compression because of the dissipation of strain energy by plastic deformation near the leading edge of the crack. It also seems clear that peripheral spring flow, originating in a sub-pingo water lens, must exit along a failure surface, because the water, which has a negative temperature, could not thaw a conduit for itself through the overlying ice- bonded permafrost. Spring flow has been most in evidence during the winter. This suggests that in winter peripheral failure is favored both by mechanical tension at the base of the overburden (Fig. 8) and by thermally induced tension at the top of the overburden.

Second, plate theory helps explain why elongate pingos may collapse soon after growth starts. According to plate theory, the maximum height of a sub-pingo water lens should increase as a pingo shape changes from circular to elliptical. Thus the early collapse of pingos 4 and 12 can be attributed to their elongated elliptical shapes.

Third, both the field-derived values of Young's modulus (E) using plate theory and the long-term tensile strength (a,) of the pingo overburden are of the right order of magnitude relative to the results of laboratory studies.

In view of the preceding, plate theory as expressed by [13] may help to explain other mechanisms of pingo growth and failure. Clearly, if the pore-water pressure below the overburden is less than the overburden pressure, there can be no sub-pingo water lens. Therefore, uplift is from segregated, not

intrusive, ice. The change in pore-water pressure with time is difficult to predict. In the early stage of pingo growth, when downward permafrost aggradation is most rapid, the subpermafrost pore-water pressure appears to be governed primarily by the thickness of the overlying permafrost. With the passage of time, the effect of upward permafrost aggradation from the base of the lake-bottom talik becomes increasingly important (Mackay 1984, 1985a; Nixon 1983). Therefore, the subpermafrost pore-water pressure probably increases more rapidly with time than the rate of downward permafrost growth ([2]) would suggest.

Pingo radius plays a most important role in pingo growth because the maximum deflection prior to yielding increases as the fourth power of the radius. Thus a doubling of the radius would increase the maximum deflection 16 times.

Young's modulus (or the creep parameter) increases with a decrease in temperature ([15]). Therefore, if the overburden is relatively thin (e.g., 3 m), so that ground temperatures can approach 0°C in the late summer, an increase in the height of the sub-pingo water lens could be a seasonal phenomenon. The total overburden thickness, which helps to resist uplift, enters as the third power. However, in the early stage of pingo growth, the overburden above the sub-pingo water lens comprises only lake-bottom sediments. As an ice core grows, the overburden above the sub-pingo water lens comprises both the lake-bottom sediments and that of the ice core. Therefore, the resistance term in the denominator increases with time. By way of contrast, the flexure from the pingo radius decreases with time, because pingo growth ceases centripetally from the periphery to the centre (Mackay 1979). Field data ([19]) show that on average, the pingo radius is about eight times the overburden thickness. The implication is that if the pingo radius is much larger than eight times the overburden thickness, the flexural stresses would cause the overburden to rupture. Con- versely, if the pingo radius were much smaller than eight times the overburden thickness, there would be insufficient flexure in the overburden for a sub-pingo water lens to form. Thus,

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1118 CAN. J. EARTH SCI. VOL. 24. 1987

plate theory provides an acceptable framework for assessing the various factors associated with the growth and failure of closed-system pingos.

Plate theory, discussed above for closed-system pingos, seems equally applicable to open-system pingos. Peripheral failure can be seen in photographs of open-system pingos (e.g., Muller 1959, P1. 11), and summit failure with spring flow has been observed in many pingos (e.g., Holmes et al. 1968; Muller 1959; O'Brien 1971). In fact, peripheral and summit failure occur more frequently in open-system than in closed-system pingos. This suggests that the sub-pingo water pressure of open-system pingos can be higher than for closed- system pingos. This may be related to the origin of the water pressure, which, in the open system, is external to the pingo but in the closed system is linked to pingo dynamics.

Conclusion Plate theory, describing the bending of thin, elastic, clamped

plates, provides a useful framework within which the growth and mechanical failure of closed-system pingos with intrusive ice can be explained, even though the basic assumptions of plate theory are not fully satisfied in the case of pingo growth. Plate theory explains the peripheral normal faulting, the spring flow associated with peripheral faulting, and the collapse of some elongated pingos. Calculations based upon plate theory give reasonable estimates of Young's modulus (or a creep parameter) and the long-term tensile strength of the pingo overburden. According to plate theory, the flexure exerted by a sub-pingo water lens varies directly with the fourth power of the pingo radius, whereas the resistance to bending increases as the third power of the overburden thickness. These tentative relationships suggest that pingos with relatively large radii and thin overburdens are most subject to early rupture, whereas pingos with relatively small radii and thick overburdens may grow into only a low lake-bottom bulge. Because the centri- petal encroachment of permafrost reduces the radius of the sub-pingo water lens and increases the ice thickness above the water lens, the application of plate theory helps to explain the growth pattern of pingos where growth is initially nearly linear from bottom to top, then growth eventually ceases from bottom to top.

Improvements in the application of engineering mechanics to the study of pingos might be sought by applying a creep con- stitutive relationship for the frozen overburden, rather than the time-dependent elastic approach implied here. This would allow a better balance to be obtained between the time- dependent freezing and water expulsion, and the relaxation of stresses due to creep.

Acknowledgments The fieldwork has been supported by the Geological Survey

of Canada, the Polar Continental Shelf Project, Canada, the Natural Sciences and Engineering Research Council of Canada, and the Inuvik Scientific Resource Centre, Inuvik, N.W.T. Professor J. Solecki has been of great assistance in the translation of material from the Russian. The paper has been improved by the constructive comments of C. R. Bum, H. M. French, J. A. Heginbottom, J. F. Nixon, V. R. Pararneswaran, and R. 0. van Everdingen.

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some mechanical aspects of pingo growth and failure, western arctic coast, canada

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