In a recent paper by Dr. J. F. Nye of the Cavendish Laboratory, Cambridge, a method is presented by which the thickness of ice sheets can be calculated.Reference Nye
1
The tendency of the ice to spread laterally and to flow downhill is balanced by the inward shear force exerted by the rock floor. By Nye’s method it is possible to make an approximate calculation of the variation of the shear stress on the bed of an ice cap if the bed and ice surface profiles are known.
For a sheet of ice resting on a bed the slope of which changes both in magnitude and direction, the shear stress on the bed is calculated by Nye to be approximately τ = ρgh sin α, provided that the local values of the thickness of ice (h) and surface slope (α) are used, and that these values do not change much in distances of order h. ρ is the density of the ice, and g is the gravitational acceleration.
Calculations on alpine valley glaciers show that the shear stresses on their beds are between 0.5 and 1.5 bars.
2
(1 bar= 106 gm./cm. sec.2=106 dynes/cm.2). Nye assumes for a first calculation that τ is constant ~1 bar over the floor of a moving ice sheet, “— as shear stresses much smaller than 1 bar produce extremely small rates of strain, while shear stresses much greater than 1 bar produce very much larger rates of deformation than those existing in glaciers and ice-sheets.” In the case of an almost stationary mass of ice, a considerably smaller shear stress can exist on the bed.
With a constant value of τ (shear stress), and if the height of the bed is known, the absolute height of a (former) ice surface can be calculated. Another type of problem that can be solved by Nye’s method is the calculation of the thickness of ice of a glacier, when the heights and slopes of the ice surface are known. One needs to assume a value for τ in this case also.
Nye found that τ = 0.88 bar gave the best fit when he compared his calculated heights of the Greenland Ice Cap with the observed surface as reported by the French Greenland Expedition of 1948–51. He therefore assumed τ to be 0.88 bar everywhere in Greenland, and calculated the height of the bed of the Greenland Ice Cap by using surface contours from a survey map of scale 1:5,000,000. The values for τ and α were uncertain, but Nye found some significant features. Even with the value of τ reduced to 0.5 bar, the floor of the ice cap appears to be as much as two m. below sea. level, at its lowest point.
It is now possible to apply Nye’s formula in another area. The southern lobe of the Barnes Ice Cap in Baffin Island is well known as a result of the observations in 1950 by members of the Arctic Institute of North America expeditionReference Baird
3
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A gravimetrist from the Dominion Observatory in Ottawa took part in the expedition, and the results of his work on the ice cap have now been publishedReference Littlewood
4
Littlewood’s work aimed at making a survey of the southern lobe of the Barnes Ice Cap; this lobe is roughly circular with a radius of approximately 10 miles (16 km.) (see Fig. 1, p. 244). Three long and a number of shorter traverses were made, establishing 155 gravity stations. The objects of the survey were to attempt to determine the thickness of the ice and to outline the topographical features of the bed. Little-wood gives for all the stations: latitude, longitude, distance from a reference station, ice-surface elevation, rock elevation, ice thickness, and observed gravity. In his calculations he uses an ice density of 0.91 gm./cm3. This is the value found by the glaciologists at Camp A, near the centre of the lobe,Reference Baird, Ward and Orvig
5
and has been used in the present calculations. The observed gravity on the ice cap varied between 982.4742 cm./sec2 (near the edge on the north side) and 982.3674 cm./sec2 (at Camp A). The difference is too small to influence the calculated value of τ in any more than the third decimal. The value for g as observed at Camp A is therefore used at all stations in calculating τ from Nye’s formula.
Fig. 1. The south-eastern lobe of the Barnes Ice Cap showing the lines of the gravity traverses
(Reproduced by courtesy of the Editor of Arctic)
The slope has been calculated from the distance and the difference in elevation between neighbouring stations. Littlewood notes in his paper that the absolute elevation for each station may be in error by as much as 10 per cent, since all elevations are relative to the elevation of Camp A, which was determined by a number of aircraft altimeter readings. However, this error in no way influences the calculations, as only differences in elevation are used in the two cases. Assuming that the combined errors in the computed ice thickness is within ±35 feet (10.7 m.) (Littlewood, p. 121), the error in the calculated value of τ will only be present in the third decimal.
Fig. 1 shows the southern lobe of the Barnes Ice Cap with the lines of the gravity traverses. Fig. 2 (p. 245) shows the cross-sections on these traverses. The whole ice cap is shown in Fig. 2, p. 3, Journal of Glaciology, Vol. 2, No. 11, March 1952.
(Reproduced by courtesy of the Editor of Arctic)
Fig. 2. Cross-sections of the Barnes Ice Cap along the lines of the gravity survey
The tables on page 246, numbered I to IV, give the calculated values for τ (the shear stress) for a number of stations, using Nye’s formula: τ = ρgh sin α. The traverses A–B and A–C slope from Camp A down to the edge of the ice cap. Camp A was not located on the highest point of the southern lobe, however, and the traverse A–D is therefore treated in two separate parts, the first is the slope from station A-142 (the highest point) to Camp A, and the second is the north slope from A-142 to D.
Table III Traverse A–D Part 1: From Station A-142 To Camp A
The traverse L–K showed that the ice surface between these two stations forms a wave. Only seven stations sloping down eastwards (towards the edge) have been used.
These values for the shear stress on the bed have been calculated along the traverses which do not necessarily correspond to the lines of greatest slope or lines of flow.
The values of τ along the traverses do not differ greatly, except along A-142 to D where the values are quite high, and where they are caused by a combination of steep slope and relatively thick ice. The average slopes along the traverses are: Traverse A–B, 1° 23′; Traverse A–C, 1° 07′; Traverse A–D (part 1), 0° 43′; Traverse A–D (part 2), 2° 27′; Traverse L–K, 0° 44′.
It is reasonable to conclude, therefore, that there is a considerably greater movement of ice in the direction A-142 to D, i.e. towards the north-east side of the southern lobe, than in the other directions, especially as Glen has shown that the rate of strain in ice is proportional to a high power of the stress.Reference Glen
6
This conclusion is in agreement with the observations of Goldthwait, who found that there was a general recession of the ice on the south-west and advance on the northeast. The net effect is a very slow shift north-eastwards of this end of the ice cap.Reference Goldthwait
7
The mean value of τ along the other traverses is around 0.40 bar, which is below the lower limit cited by Nye from alpine valley glaciers (0.5–1.5 bars), and should produce small rates of strain in the ice, particularly because the temperature of the ice is thought to be below the pressure melting point throughout8 The investigations on the Barnes Ice Cap showed that it is nearly stationary, and all signs indicated only small rates of deformation where slow retreat was in progress.Reference Baird, Ward and Orvig
8
It is probable that 0.40 bar is a reasonable magnitude of the shear force exerted by the rock floor under an ice cap in a topographical setting similar to that of the Barnes Ice Cap, with higher values to be expected in directions where the activity of the ice is greater, as is the case in a northerly direction from the neighbourhood of station A-142.
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In a recent paper by Dr. J. F. Nye of the Cavendish Laboratory, Cambridge, a method is presented by which the thickness of ice sheets can be calculated.Reference Nye 1 The tendency of the ice to spread laterally and to flow downhill is balanced by the inward shear force exerted by the rock floor. By Nye’s method it is possible to make an approximate calculation of the variation of the shear stress on the bed of an ice cap if the bed and ice surface profiles are known.
For a sheet of ice resting on a bed the slope of which changes both in magnitude and direction, the shear stress on the bed is calculated by Nye to be approximately τ = ρgh sin α, provided that the local values of the thickness of ice (h) and surface slope (α) are used, and that these values do not change much in distances of order h. ρ is the density of the ice, and g is the gravitational acceleration.
Calculations on alpine valley glaciers show that the shear stresses on their beds are between 0.5 and 1.5 bars. 2 (1 bar= 106 gm./cm. sec.2=106 dynes/cm.2). Nye assumes for a first calculation that τ is constant ~1 bar over the floor of a moving ice sheet, “— as shear stresses much smaller than 1 bar produce extremely small rates of strain, while shear stresses much greater than 1 bar produce very much larger rates of deformation than those existing in glaciers and ice-sheets.” In the case of an almost stationary mass of ice, a considerably smaller shear stress can exist on the bed.
With a constant value of τ (shear stress), and if the height of the bed is known, the absolute height of a (former) ice surface can be calculated. Another type of problem that can be solved by Nye’s method is the calculation of the thickness of ice of a glacier, when the heights and slopes of the ice surface are known. One needs to assume a value for τ in this case also.
Nye found that τ = 0.88 bar gave the best fit when he compared his calculated heights of the Greenland Ice Cap with the observed surface as reported by the French Greenland Expedition of 1948–51. He therefore assumed τ to be 0.88 bar everywhere in Greenland, and calculated the height of the bed of the Greenland Ice Cap by using surface contours from a survey map of scale 1:5,000,000. The values for τ and α were uncertain, but Nye found some significant features. Even with the value of τ reduced to 0.5 bar, the floor of the ice cap appears to be as much as two m. below sea. level, at its lowest point.
It is now possible to apply Nye’s formula in another area. The southern lobe of the Barnes Ice Cap in Baffin Island is well known as a result of the observations in 1950 by members of the Arctic Institute of North America expeditionReference Baird 3 .
A gravimetrist from the Dominion Observatory in Ottawa took part in the expedition, and the results of his work on the ice cap have now been publishedReference Littlewood 4 Littlewood’s work aimed at making a survey of the southern lobe of the Barnes Ice Cap; this lobe is roughly circular with a radius of approximately 10 miles (16 km.) (see Fig. 1, p. 244). Three long and a number of shorter traverses were made, establishing 155 gravity stations. The objects of the survey were to attempt to determine the thickness of the ice and to outline the topographical features of the bed. Little-wood gives for all the stations: latitude, longitude, distance from a reference station, ice-surface elevation, rock elevation, ice thickness, and observed gravity. In his calculations he uses an ice density of 0.91 gm./cm3. This is the value found by the glaciologists at Camp A, near the centre of the lobe,Reference Baird, Ward and Orvig 5 and has been used in the present calculations. The observed gravity on the ice cap varied between 982.4742 cm./sec2 (near the edge on the north side) and 982.3674 cm./sec2 (at Camp A). The difference is too small to influence the calculated value of τ in any more than the third decimal. The value for g as observed at Camp A is therefore used at all stations in calculating τ from Nye’s formula.
Fig. 1. The south-eastern lobe of the Barnes Ice Cap showing the lines of the gravity traverses
(Reproduced by courtesy of the Editor of Arctic)
The slope has been calculated from the distance and the difference in elevation between neighbouring stations. Littlewood notes in his paper that the absolute elevation for each station may be in error by as much as 10 per cent, since all elevations are relative to the elevation of Camp A, which was determined by a number of aircraft altimeter readings. However, this error in no way influences the calculations, as only differences in elevation are used in the two cases. Assuming that the combined errors in the computed ice thickness is within ±35 feet (10.7 m.) (Littlewood, p. 121), the error in the calculated value of τ will only be present in the third decimal.
Fig. 1 shows the southern lobe of the Barnes Ice Cap with the lines of the gravity traverses. Fig. 2 (p. 245) shows the cross-sections on these traverses. The whole ice cap is shown in Fig. 2, p. 3, Journal of Glaciology, Vol. 2, No. 11, March 1952.
Fig. 2. Cross-sections of the Barnes Ice Cap along the lines of the gravity survey
The tables on page 246, numbered I to IV, give the calculated values for τ (the shear stress) for a number of stations, using Nye’s formula: τ = ρgh sin α. The traverses A–B and A–C slope from Camp A down to the edge of the ice cap. Camp A was not located on the highest point of the southern lobe, however, and the traverse A–D is therefore treated in two separate parts, the first is the slope from station A-142 (the highest point) to Camp A, and the second is the north slope from A-142 to D.
Table I. Traverse A–B
Table II Traverse A–C
Table III Traverse A–D Part 1: From Station A-142 To Camp A
Table IV Traverse L–K
The traverse L–K showed that the ice surface between these two stations forms a wave. Only seven stations sloping down eastwards (towards the edge) have been used.
These values for the shear stress on the bed have been calculated along the traverses which do not necessarily correspond to the lines of greatest slope or lines of flow.
The values of τ along the traverses do not differ greatly, except along A-142 to D where the values are quite high, and where they are caused by a combination of steep slope and relatively thick ice. The average slopes along the traverses are: Traverse A–B, 1° 23′; Traverse A–C, 1° 07′; Traverse A–D (part 1), 0° 43′; Traverse A–D (part 2), 2° 27′; Traverse L–K, 0° 44′.
It is reasonable to conclude, therefore, that there is a considerably greater movement of ice in the direction A-142 to D, i.e. towards the north-east side of the southern lobe, than in the other directions, especially as Glen has shown that the rate of strain in ice is proportional to a high power of the stress.Reference Glen 6 This conclusion is in agreement with the observations of Goldthwait, who found that there was a general recession of the ice on the south-west and advance on the northeast. The net effect is a very slow shift north-eastwards of this end of the ice cap.Reference Goldthwait 7
The mean value of τ along the other traverses is around 0.40 bar, which is below the lower limit cited by Nye from alpine valley glaciers (0.5–1.5 bars), and should produce small rates of strain in the ice, particularly because the temperature of the ice is thought to be below the pressure melting point throughout8 The investigations on the Barnes Ice Cap showed that it is nearly stationary, and all signs indicated only small rates of deformation where slow retreat was in progress.Reference Baird, Ward and Orvig 8
It is probable that 0.40 bar is a reasonable magnitude of the shear force exerted by the rock floor under an ice cap in a topographical setting similar to that of the Barnes Ice Cap, with higher values to be expected in directions where the activity of the ice is greater, as is the case in a northerly direction from the neighbourhood of station A-142.