Volume–area scaling parameterisation of Norwegian ice caps: A comparison of different approaches

Abstract

Over the recent decades, glaciers have in general continued to lose mass, causing surface lowering, volume reduction and frontal retreat, thus contributing to global sea-level rise. When making assessments of present and future sea-level change and management of water resources in glaciated catchments, precise estimates of glacier volume are important. The glacier volume cannot be measured on every single glacier. Therefore, the global glacier volume must be estimated from models or scaling approaches. Volume–area scaling is mostly applied for estimating volumes of glaciers and ice caps on a regional and global scale by using a statistical–theoretical relationship between glacier volume (V) and area (A) (V = cA^γ) (for explanation of the parameters c and γ, see Eq. 1). In this paper, a two-dimensional (2D) glacier model has been applied on four Norwegian ice caps (Hardangerjøkulen, Nordre Folgefonna, Spørteggbreen and Vestre Svartisen) in order to obtain values for the volume–area relationship on ice caps. The curve obtained for valley glaciers gives the best fit to the smallest plateau glaciers when c = 0.027 km^3−2γ and γ = 1.375, and a slightly poorer fit when the glacier increases in size. For ice caps, c = 0.056 km^3−2γ and γ = 1.25 fit reasonably well for the largest, but yield less fit to the smaller.

Keywords

area glacier ice caps Norway scaling volume

Introduction

Global sea-level rise is a major challenge for coastal, lowland areas worldwide. According to the IPCC report published in 2013, glaciers continued to retreat and lose mass. Most of the ice loss between 2003 and 2009 was from glaciers in Alaska, Canadian Arctic, the periphery of the Greenland ice sheet, southern Andes and in Asia (Vaughan et al., 2013). The total global mass of all glaciers (excluding the Antarctic and Greenland ice sheets) is likely between 114,000 and 192,000 Gt, corresponding to a glacio-eustatic sea-level equivalent of 31.4–52.9 cm (Vaughan et al., 2013). With such large uncertainties, more precise estimates of glacier volume are important when making assessments for future sea-level change, the climate impact on glaciers, management of water recourses and projections for river runoff in glaciated catchments. The estimates of future sea-level rise because of melting glaciers, except Greenland and Antarctica, lie between 0.35 and 0.60 cm (Grinsted, 2013; Radic and Hock, 2010; Raper and Braithwaite, 2005; Van de Wal and Wild, 2001). The uncertainty that these numbers reflect is because of several factors: incomplete data of the world’s glaciated area and volume, uncertainty in the volume–area relationship, different divisions between plateau glaciers and outlet glaciers and/or plateau glaciers, which means uncertainty in the volume–area relationship to be used (ice cap or valley glacier).

At present, digital inventories yield reasonably good estimates of the area of the glaciers around the globe (e.g. Kargel et al., 2014; Pfeffer et al., 2014). There are few glaciers where ice volume is calculated on the basis of field data that provide ice thicknesses directly, such as radar measurements and/or seismic data. The global glacier volume must therefore to a large extent be estimated by means of models or scaling methods.

Physics-based glacier models have been developed in recent years (e.g. Clarke et al., 2009; Farinotti et al., 2009; Huss and Farinotti, 2012; Linsbauer et al., 2012). These use digital terrain models, glacier extent and/or different flow lines to estimate glacier thickness distribution and thus their volume. Andreassen et al. (2015) recently estimated the total volume of Norwegian glaciers using the new glacier inventory for Norway (Andreassen and Winsvold, 2012) and a distributed physics-based model (Huss and Farinotti, 2012). Their calculations yielded a total glacier area of 2692 ± 81 km² and a total glacier volume of 257–300 km³, corresponding to 236–275 Gt = 0.66–0.76 mm sea-level equivalents. Martín-Español et al. (2015) estimated, by means of regionally based scaling relationships, the total volume of Svalbard glaciers to 6700 ± 835 km³, which corresponds to a potential sea-level rise of 17 ± 2 mm.

These calculations, however, are labour-intensive, time-consuming and require a significant amount of work on each glacier. Therefore, the much simpler, but more uncertain, volume–area scaling method is commonly used to estimate the glacier volume. This is an analytical technique that has become an increasingly important and applied approach for calculating volume of past and present glaciers and ice caps, and for estimating the volume changes of glaciers and ice caps to future environmental and mass balance changes. In particular, volume–area scaling is mostly used for the purpose of estimating regional and total global glacier and ice cap volumes and their contribution to sea-level change and water resource management (e.g. Bahr et al., 2014, 2015 and references therein).

In order to make glacier volume estimates, the area of the glacier is needed, and the volume is calculated using a statistical–theoretical relationship between the volume (V) measured in cubic kilometres and area (A) in square kilometres:

V = c A^{γ}

(1)

where c = 0.03 and γ = 1.36 originally were determined as empirical constants derived from data for 144 valley glaciers (Bahr et al., 1997) (Figure 1). In theoretical analyses by Bahr (1997) and Bahr et al. (2015), γ is considered a constant, and for a valley glacier, the value is 1.375. The parameter c differs and is, for example, dependent on the basal topography, sliding parameters and ice-flow parameters. The mean value for valley glaciers is c = 0.034 km^3−2γ. Several researchers have, however, suggested different values for both γ and c found from regression of data. An overview of the various values was presented by Grinsted (2013).

Figure 1.

Glacier volume–area distribution for 144 glaciers (adapted from Bahr et al. (2015) based on a compilation of Bahr et al. (1997)). A power law relationship is evident with the linear trend (log–log plot). For further details, see figure caption in Bahr et al. (2015: 97).

For ice caps, γ = 1.25 by assumption of plastic deformation on a horizontal circular surface that is thickest in the middle (Paterson, 1994). The constant c (Eq. 1) depends on the value of the basal shear stress (τ). For example, τ = 50 kPa gives c = 0.0426 km^3−2γ and τ = 100 kPa yields c = 0.0602 km^3−2γ (see Appendix 1). A theoretical approach by Bahr et al. (2015) yields same values for γ without inferring plastic ice deformation. The parameter c is considered a scaling parameter and is different from the c on valley glaciers.

Although the physics-based models provide promising results, the volume–area relationships provide affordable good results if it is properly calibrated locally (Bahr et al., 2015). In this article, we have tested values of area–volume scaling parameters using a two-dimensional (2D) dynamic glacier model on four Norwegian plateau ice caps with mapped subglacial topography and estimated ice volumes in order to find the most appropriate values used to calculate the volume–area scaling relationships.

The glaciers in this study

Hardangerjøkulen (Figure 2, Table 1) is the sixth largest glacier in Norway with an area of 71.28 km² (Andreassen and Winsvold, 2012) and the glacier has an altitudinal range from 1020 to 1863 m a.s.l. The glacier is an ice cap with 14 outlet glaciers with Rembesdalskåka and Midtdalsbreen being the two largest. The mass balance measurements have been conducted since 1963 by Norwegian Polar Institute (NPI) and Norwegian Water Resources and Energy Directorate (NVE) (Kjøllmoen, 2011). The thickness measurements have been carried out by NVE using glacier radar and thickness maps and a subglacial bed topography map has been created (Andreassen et al., 2015; Giesen, 2009).

Figure 2.

Location map Norwegian glaciers. Map: norgeskart.no.

Table 1.

The glaciers in this study (for location, see Figure 2).

Glacier	Latitude/longitude	Area (km²)	Max. alt. (m)	Min. alt. (m)
Hardangerjøkulen	60°32′N 7°25′E	71.28	1863	1020
Nordre Folgefonna	60°12′N 6°27′E	26.43	1644	1000
Spørteggbreen	61°36′N 7°29′E	23.03	1760	1455
Vestre Svartisen	66°39′N 1355′E	218.73	1590	10

Nordre Folgefonna (Figure 2, Table 1) is the second largest of the three Folgefonna plateau glaciers and the glacier has an area of 26.43 km² (Andreassen and Winsvold, 2012) with an altitudinal distribution from 1000 to 1644 m a.s.l. No mass balance measurements have been carried out on this glacier, but measurements on several parts of the other two parts of Folgefonna in different periods have been carried out (nve.no/bre) and a comparison between the different parts of the glacier was performed by NVE (Elvehøy, 1998). The ice thickness measurements have been performed by NVE in collaboration with the University of Bergen, and ice thickness maps have been constructed (Førre, 2012). A basal topography map of Nordre Folgefonna has also been produced (Bakke et al., unpublished data) together with a discussion of glacier extent in a future warmer climate.

Spørteggbreen is a small plateau glacier east of Jostedalsbreen (Figure 2, Table 1) with an area of 23.03 km² (Andreassen and Winsvold, 2012) and an altitudinal distribution from 1455 to 1760 m a.s.l. The mass balance measurements were conducted by NVE during the period 1988–1991 (Kjøllmoen, 2011). The subglacial topography was measured by Kennett (1989), and the estimates of glacier geometry in the past and in the future were presented by Laumann and Nesje (2014).

Vestre Svartisen (Figure 2, Table 1) has an area of 218.73 km² (Andreassen and Winsvold, 2012) and is the second largest glacier in Norway. The glacier spans an altitudinal range from 10 to 1590 m a.s.l. The ice thickness measurements have been performed by NVE and mass balance measurements have been carried out on parts of the glacier from 1970 (Kjøllmoen, 2011).

Methods

The 2D glacier model applied here is a combination of a model described by Le Meur and Vincent (2003) for Glacier de Saint-Sorlin in France and a one-dimensional model described by Oerlemans (2001). In Norway, the model has been applied on Spørteggbreen (Laumann and Nesje, 2014) and Nordre Folgefonna (Bakke et al., unpublished data). The model has previously been described in detail and will therefore only be given a brief description here. The basic equation in the model is the continuation equation for mass conservation:

\frac{\partial h}{\partial t} = - \nabla (v h) + b

(2)

v = v_x + v_y, where v is the mean velocity vector when ignoring the vertical component, v_x and v_y are the velocity in x- and y-directions, respectively (right-hand system with z vertical up), h is the ice/glacier thickness, b is the specific annual mass balance and t is the time. The velocities are a combination of internal deformation and subglacial sliding. Both require empirically found constants, and by introducing these in Eq. 1, this can be transformed into:

\frac{\partial h}{\partial t} = \frac{\partial}{\partial x} (D \frac{\partial s}{\partial x}) + \frac{\partial}{\partial y} (D \frac{\partial s}{\partial y}) + b

(3)

where D = (ρgh)³(∇s)²[f₁h² + f₂] with (∇s)² = (∂s/∂x)² + (∂s/∂y)².

ρ is the density of ice, g is the acceleration of gravity, ∂s/∂x is the glacier surface slope in the x-direction, ∂s/∂y is the glacier surface slope in the y-direction, f₁ is a constant that refers to internal glacier deformation (Oerlemans, 2001) and f₂ is a sliding parameter based on a ‘sliding law’ (Budd et al., 1979; Oerlemans, 2001). The equation is solved in every grid point, the time step is 0.01 year and the grid space is 100 m.

In model calculations, a dynamic calibration is needed to find these dynamic parameters, trying to find the best values of f₁ and f₂, such that the geometry of the glacier fits with the measured values. Giesen (2009) calculated the values for Hardangerjøkulen: f₁ = 2.4 × 10⁻¹⁷ Pa⁻³ y⁻¹ and f₂ = 0.29 × 10⁻¹² Pa⁻³ m² y⁻¹. Bakke et al. (in preparation) used f₁ = 2.4 × 10⁻¹⁷ Pa⁻³ y⁻¹ and f₂ = 0.20 × 10⁻¹² Pa⁻³ m² y⁻¹ that gave reasonable results on Nordre Folgefonna. To obtain comparable values, we have in this study used the figures that were found by Giesen (2009) on Hardangerjøkulen.

In order to obtain a glacier volume–area relationship, several points in a volume–area plot are needed. These are found using the 2D glacier model that has been run under different climate scenarios. For all glaciers in this study, the mean specific mass balance versus elevation is known (measurements performed by NVE (nve.no/bre)). Reanalyses of long-term series of glaciological and geodetic mass balance for 10 Norwegian glaciers were recently published by Andreassen et al. (2016). These curves are parallel displaced so that the total annual balance of each glacier is equal to 0 with the spatial distribution for 2007 for Nordre Folgefonna, 1995 for Hardangerjøkulen, 2011 for Spørteggbreen and 2012 for Svartisen. These profiles (Figure 3) are here called ‘zero balance profiles’ and yield start values for our calculations. A warmer climate with more negative annual (net) mass balance is simulated by successively subtracting 0.2 m water equivalents from the ‘zero balance profile’. For each ‘climate’ state, the model is run for 100 years and the volume and surface area are calculated until the mass balance is so negative that the glacier disappears. To obtain the corresponding values for larger areas, the model is driven by successively adding 0.2 m water equivalents to the zero balance profile. Areas are found by summing all pixels (a = 100 × 100 m), where glacier thickness (h) is greater than 0 and the respective quantities are found by adding all the corresponding ‘pixel volumes’ (a × h).

Figure 3.

Zero mass balance profiles for the glaciers in this study. See text for explanation.

The exponent γ in the scaling relationship is here treated as a constant (either 1.375 or 1.25) and the parameter c is treated as a proportionality factor determined by different geometric parameters for each glacier, such as sliding, internal deformation, shape factors, slope angle and basal topography (Bahr et al., 2015). For example, a glacier with significant internal deformation and/or major sliding along the subglacial bed is commonly long and gentle, whereas glaciers without basal sliding will grow thicker in the upper areas and become steeper. It is therefore important to find ‘real’ dynamic parameters for the glaciers to be studied for volume–area relationships.

Results

Glacier volume–area relationships

Figure 4 shows approximately 60 points in a glacier volume–area plot distributed on the four plateau glaciers in this study. To the points, two curves are displayed: one adjusted curve by the theoretical exponent value γ = 1.375 for valley glaciers and an adjusted curve with the theoretical exponent value γ = 1.25 for ice caps. ‘The valley glacier curve’ yields the best fit to the smallest plateau glaciers when c = 0.027 km^3−2γ and a slightly poorer fit when the glacier increases in size. For ice caps, c = 0.056 km^3−2γ fits reasonably well for the largest glaciers, but gives rather bad fit to the smaller ice caps.

Figure 4.

Volume–area plot for the ice caps Nordre Folgefonna, Hardangerjøkulen, Spørteggbreen and Vestre Svartisen. Volume–area relationships for ice caps and valley glaciers are also displayed.

The parameter c

As pointed out by Bahr et al. (2015), the exponent in the volume–area relationship for valley glaciers is a constant and the parameter c is therefore regarded as a variable dependent on different glaciological conditions. The dynamical 2D model used in this study has two dynamical parameters (f₁ and f₂) applied to calibrate the model. By changing these parameters, the contribution from, for example, sliding and internal deformation on the parameter c is displayed. This has been done for Hardangerjøkulen and Nordre Folgefonna by first choosing f₁ = 2.4 × 10⁻¹⁷ Pa⁻³ y⁻¹ (as above) and f₂ = 0, that is, no sliding along the base. Thereafter, f₂ = 0.20 × 10⁻¹² Pa⁻³ m² y⁻¹ was selected (as above) and f₁ = 8.6 × 10⁻¹⁷ Pa⁻³ y⁻¹, which represents temperate ice. In both cases, the area and volume are calculated at different times. The results of these calculations are displayed in Figure 5, with y = 1.375 and the best-adjusted values for c. As expected, the value is higher for a cold-based glacier (c = 0.03 km^3−2γ) than for a warm-based glacier (c = 0.023 km^3−2γ). In relation to the best-adjusted values for f₁ and f₂, this yields a difference on c of approximately ±10%.

Figure 5.

Suggested volume–area relationships for cold-based and warm-based glaciers.

Outlet glaciers from ice caps

Most scaling parameters in the literature are found from studies on valley glaciers (Grinsted, 2013 and Table 2). It is unlikely that these can be applied to individual outlet glaciers from the ice caps, because the thickness in the accumulation zone is larger at the plateau glaciers than valley glaciers and the boundary between adjacent outlet glaciers consists of ice and not bedrock faces. To investigate this effect, we used the same method as described above on seven individual outlet glaciers. These are Juklabreen on Nordre Folgefonna, Rembesdalskåka, Blåisen, Midtdalsbreen and Vestre Leirebottsskåka on Hardangerjøkulen and Engabreen and Storglombreen at Vestre Svartisen. The results are displayed in Figure 6 together with the values Andreassen et al. (2015) found from a corresponding parameter set on the basis of 79 Norwegian glaciers, mostly outlet glaciers. For these glacier types, ice thickness in the accumulation zone may increase or decrease, even though the area is constant. This is relatively more important for outlet glaciers than for an entire plateau glacier. Assuming a constant exponent in the scaling law, the scaling factor c is therefore greater for outlet glaciers than for entire ice caps (0.060 and 0.027 km^3−2γ, respectively). Andreassen et al. (2015) chose to let both exponent and scaling parameter being variables and found the best fit for their data sets (c = 0.0511 km^3−2γ and γ = 1.4492 km^3−2γ). The results of these two different methods show still reasonably good coherence, particularly for small outlet glaciers.

Table 2.

Selection of volume–area scaling laws (units: area (km²) and volume (km³)).

Study/reference	Glacier	Ice cap
Macheret and Zhuravlev (1982)		0.0597A^1.12
Chizhov and Kotlyakov (1983)		0.04A^1.25
Macheret et al. (1988)	0.0298A^1.379
Chen and Ohmura (1990)	0.0285A^1.375
Bahr et al. (1997)	0.0276A^1.36
Van de Wal and Wild (2001)	0.0213A^1.375
Radic and Hock (2010)	0.0365A^1.375	0.0538A^1.25
Bahr and Radic (2012)	0.033A^1.36
Grinsted (2013)	0.0433A^1.29	0.0538A^1.23
Andreassen et al. (2015)	0.0511A^1.4492

Figure 6.

Suggested volume–area relationships for Norwegian outlet glaciers used in the study by Andreassen et al. (2015) and in this study.

To test our volume–area scaling for small (<~120 km²) ice caps with volume–area scaling for outlet glaciers, we calculated the total volume of Hardangerjøkulen using formulas for outlet glaciers and small ice caps. The area for each outlet glacier (Figure 7) is from the latest glacier inventory in Norway (Andreassen and Winsvold, 2012) and the total volume of Hardangerjøkulen is calculated as the sum of all outlets. The results are shown in Table 3 and provide reasonable coherent values for total ice volume.

Figure 7.

Hardangerjøkulen with outlet glaciers.

Table 3.

Calculated total volume of Hardangerjøkulen by means of different formulas.

Study	Andreassen et al. (2015) for outlets V = 0.0511A^1.442	This study for outlets V = 0.06A^1.375	This study (small ice caps) V = 0.027A^1.375
Volume (km³)	9.4	9.4	9.7

The glacier area–volume relationship of Jostedalsbreen

Jostedalsbreen is the largest glacier on mainland Europe (473.75 km²) (Andreassen and Winsvold, 2012) and is a complex of glaciers that together form an elongated plateau glacier with around 20 valley outlet glaciers (e.g. Tunsbergdalsbreen and Nigardsbreen being the longest). It is therefore unlikely that our volume–area scaling for ice caps fits for the complex Jostedalsbreen ice mass.

Two approached were used to estimate the glacier volume of Jostedalsbreen. First, the volume of each outlet glacier was estimated by means of volume–area scaling for outlets, and thereafter the volumes for all of them were added. In the second approach, Jostedalsbreen was subdivided into several small ice caps and our formula for small ice caps was used on each of them, as done by Meister (2010). The area for each outlet glacier was found from the values given in the last glacier inventory (Andreassen and Winsvold, 2012) (glaciers <1 km² were not included). The division of the entire glacier into smaller ice caps requires, however, extended knowledge of the glacier geometry and behaviour. Figure 8 shows the subdivision of Jostedalsbreen used in our calculations and Table 4 shows the results of the calculations, ranging from 67.1 to 75.1 km³ for the entire glacier. Apart from zones 3 and 4, the values are fairly identical for the different methods. Zones 3 and 4 contain long outlet glaciers and therefore it is not unreasonable that the ‘ice cap method’ does not work well here. Using different approaches, Meister (2010: 107) calculated total volumes of Jostedalsbreen to 65.94–73.05 km³, with a distinct outlier of 46.17 km³.

Figure 8.

Sectors of Jostedalsbreen used for volume calculations. Satellite image: NASA World Wind.

Table 4.

Area and volume of outlet glaciers and ice caps of five sectors on Jostedalsbreen.

Sector (see Figure 8)	1	2	3	4	5	Total
Area (km²)	82.5	64.8	108.5	124.7	71.3
Volume (km³) Andreassen et al. (2015), outlets V = 0.0511A^1.55	10.1	9.1	22.5	24.5	8.9	75.1
Volume (km³); this study, outlets V = 0.06A^1.375	10.2	9.0	20.7	22.9	8.9	71.7
Volume (km³); this study, V = 0.027A ^1.375	11.6	8.4	17.0	20.6	9.5	67.1

Discussion

Andreassen et al. (2015) estimated the total ice volume in Norway to be 280 ± 34 km³. According to their calculations, the volume of the 15 largest glaciers and glacier complexes in Norway adds to 212 km³. These calculations were based on the distributed model by Huss and Farinotti (2012). By utilising our volume–area scaling relationships on the same 15 glaciers, and area data from Andreassen and Winsvold (2012), we obtained approximately 230 km³, and for the 40 largest glaciers in Norway (from the glacier atlas by Andreassen and Winsvold (2012)) (glaciers >7 km²), we obtained a volume of 254 km³. By using different scaling laws from the literature, Andreassen et al. (2015) obtained values from 137 to 432 km³ for the total glacier volume in Norway. This clearly shows that by using scaling laws, local knowledge about glaciers is highly beneficial. Furthermore, it is important to select the appropriate subdivision in glacier categories and apply appropriate constants in volume–area scaling laws. As pointed out by Andreassen et al. (2015), the physics-based models will probably provide the most correct result for volume estimates. However, they are time- and labour-intensive and appropriate use of area–volume scaling laws will, as shown above, provide reasonably good glacier volume estimates.

The volume–area relationships obtained by modelling four different ice caps using different climate scenarios have produced a total of approximately 60 different glacier/ice cap geometries. Calculating the surface area and volume for each of these configurations allowed estimating the value of the scale factor c in the volume–area relationship. The goal was to determine whether to use the value of γ determined for valley glaciers (1.375) or that determined for ice caps (1.25). Our calculations gave a value of c = 0.027 km^3−2γ to be used with γ = 1.375 for the smallest ice caps, and a value of c = 0.056 km^3−2γ to be used with γ = 1.25 for the largest ice caps. However, even with approximately 60 glacier geometries to work with, the glaciers used to constrain c and decide which value of γ to use span a small range of magnitudes. As the relationship between V (scale magnitude for glacier volume) and S (scale magnitude for glacier surface area) is exponential, several orders of magnitude of data are needed in order to find the most appropriate values to obtain the volume–area scaling relationships (e.g. Bahr et al., 2015).

The theoretical value of γ for ice caps is obtained with the assumption of an ice cap laying on a horizontal surface. Few Norwegian plateau glaciers, however, fit the assumption. They commonly cover upland plateaus that are intersected by deep valleys filled with glacier ice. With increasing size, however, the subglacial relief commonly becomes relatively progressively smaller, and volume–area laws for ice caps will likely be dominating.

It is a bit strange that the volume–area values for valley glaciers fit reasonably well on the Norwegian ice caps in this study. However, the basal topography for most of the plateau glaciers/ice caps in Norway shows that they consist of several outlet glaciers filling deep tributary depressions. Commonly, the ice thickness along the ice divide is restricted and does not fit to typical ice caps that are normally thickest in the central part. When such a plateau glacier melts down, the valley sides will to some extent determine the area distribution, as for a valley glacier (Figure 9), whereas the increasing size of an ice cap makes the basal topography relatively progressively smaller, and volume–area laws for ice caps will likely be dominating. Within our area segments, the difference between the volumes is not particularly large by various choices of scaling laws, but for larger areas the differences can be substantial (e.g. Grinsted, 2013).

Figure 9.

The geometry of Hardangerjøkulen with increased melting in a warmer climate.

Conclusion

The global glacier volume can be estimated from direct measurements, models and scaling approaches. On a regional and global scale, volume–area scaling is mostly used for volume calculations of glaciers and ice caps by applying statistical–theoretical relationships between glacier area and volume.

In this study, a 2D glacier model was used on four Norwegian ice caps (Hardangerjøkulen, Nordre Folgefonna, Spørteggbreen and Vestre Svartisen) to obtain proper glacier volume–area relationships. A total of approximately 60 different valley glacier/ice cap geometries have been produced in order to estimate the value of the scale factor c and the exponent γ to use in the volume–area scale relationship. The volume–area curve obtained for valley glaciers in the literature (γ = 1.375) yields best fit to the small plateau glaciers when c = 0.027 km^3−2γ and a slightly poorer fit for larger plateau glaciers/ice caps. Using γ = 1.25 for ice caps, c = 0.056 km^3−2γ fits reasonably well for the largest ice caps, but yields rather bad fit to the smaller ones. Even by using approximately 60 points in our volume–area plot, they span too small orders of magnitude. We still think that the results are reasonable taking into account the values referred to in the literature.

Applying our volume–area scaling relationships on the same 15 glaciers as Andreassen et al. (2015) (they calculated a volume of 212 km³), we obtained a glacier volume of approximately 230 km³. We calculated the total volume of the 40 largest glaciers in Norway (glaciers >7 km², data from Andreassen and Winsvold, 2012), to 254 km³.

Footnotes

Appendix 1

The profile of a perfect plastic ice cap half-width R on a horizontal surface and the origin at the base at the centre is theoretically (Paterson, 1994):

h (x) = {(\frac{2 τ}{ρ g})}^{1 / 2} {(R - x)}^{1 / 2}

where ρ is the ice density, g is the gravity acceleration and τ is the basal shear stress. By integrating this as a body of revolution around the vertical axis, the volume of the glacier on a circular plan can be obtained as follows:

V = \frac{8 π}{15} = {(\frac{2 τ}{ρ g})}^{1 / 2} R^{5 / 2}

The substitution R = (A/π)^1/2, where A is the area of the circular plane and inserted values of the constants, except for the value of the basal shear stress (τ) yields:

V = 6 \times 10^{- 3} {(τ)}^{1 / 2} A^{1.25}

Choosing square kilometres and cubic kilometres as units for area and volume, respectively, yields:

V = 0.0426 A^{1.25} for τ = 50 kPa

V = 0.0602 A^{1.25} for τ = 100 kPa

V = 0.0737 A^{1.25} for τ = 150 kPa

Acknowledgements

The authors are grateful to the reviewers who provided useful and constructive comments on the manuscript.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

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