Introduction
Glacier net balance b n is the result of two processes, accumulation and ablation, which correspond approximately to routinely measured winter b w and summer b s balances. The altitude (z) variation of net balance b n at ten glaciers in Norway and two in northern Sweden is markedly linear, and the gradients b’n (denoting db n/dz) vary little from year to year (Reference RasmussenRasmussen, 2004). Reference Dyurgerov, Olshanskiy and ProkhorovaDyurgerov and others (1989) found in a study of balance curves for 13 glaciers, including five in Norway, that there is little variation of gradient from year to year for glaciers in maritime climates, whereas there is significant variation for those in continental climates.
Reference Meier and TangbornMeier and Tangborn (1965) concluded from seven b n(z) measured on South Cascade Glacier, Washington State, USA, that the profiles differed from year to year by an amount constant with altitude. Reference KuhnKuhn (1984) found the same for Hintereisferner, Austria, and Reference LliboutryLliboutry (1974) founded his linear balance model on that assumption. In contrast, several studies (Reference Oerlemans and HoogendoornOerlemans and Hoogendoorn, 1989; Reference Funk, Morelli and StahelFunk and others, 1997; Reference Dyurgerov and DwyerDyurgerov and Dwyer, 2000) found steeper b n(z) in warm years;that is, correlation r < 0 between b’n and glacier-total b n. For the 12 glaciers in Scandinavia (Reference RasmussenRasmussen, 2004), however, the correlation is weak and for six of them r > 0.
An issue related to balance gradients is to determine at how many sites on a glacier it is necessary to measure mass balance to obtain a reliable estimate of the glacier-total balance. Reference Fountain and VecchiaFountain and Vecchia (1999) analyzed the relation between uncertainty in the glacier-total b n and uncertainty in the measurement at a site, concluding that five to ten sites are needed for alpine glaciers <10 km2. Reference Trabant and MarchTrabant and March (1999) concluded from work on two glaciers in Alaska, USA, that three sites are needed: one in the ablation zone, one near the equilibrium-line altitude (ELA) and one in the accumulation zone. Reference Funk, Morelli and StahelFunk and others (1997) found that in the Alps one site per km2 is needed. By contrast, for the 12 glaciers in Scandinavia, correlation between the glacier-total b n and the value determined at only one site, near the middle of the altitude range of the glacier, was r ≥ 0.97 (Reference RasmussenRasmussen, 2004).
In this paper, we extend the analysis of Reference RasmussenRasmussen (2004) to profiles of seasonal balance components b w(z) and b s(z) for ten glaciers in Norway (Fig. 1). Our main objective is to examine the properties of the profiles as a first step towards understanding the separate roles of the two processes, accumulation and ablation, in determining b n(z). A second objective is to determine whether b n (z) is linear because b w(z) and b s(z) are both linear or because their curvatures cancel. Another is to determine whether b w or b s at one site on the glacier correlates well with the glacier total. In investigating these questions, we use measured profiles for the glaciers (Table 1).
Fig. 1. Glacier locations. See Table 1 for names.
Table 1. Years with b w(z) and b s(z) profiles used in this analysis. Altitude interval is 50 m, except for glaciers 1, 2 and 6, where it is 100 m
Norwegian Mass-Balance Program
A comprehensive review of the variation of glaciers in Norway since 1900 is given by Reference Andreassen, Elverøy, Kjøllmoen, Engeset and HaakensenAndreassen and others (in press). Altitude profiles of b w, b s and b n for most years are published in Reference KjøllmoenKjøllmoen (2004) and in previous reports by the Norwegian Water Resources and Energy Directorate (NVE). Profiles for a few other years are available from the NVE archives. Values are given at 50 m altitude intervals for seven glaciers and at 100 m intervals for the other three (Table 1). Methods used to measure mass balance have changed little over the years, except that the fieldwork has become more efficient as more experience has been gained. Mass balance is calculated by the stratigraphic method (Reference Østrem and BrugmanØstrem and Brugman, 1991), i.e. between successive ‘summer surfaces’. Consequently, the measurements describe the state of the glacier after the end of melting and before fresh snow starts to fall.
Winter balance, which is the accumulated snow during the winter season, is measured each spring by probing to the previous year’s summer surface at typically 50–150 sites on each glacier. Snow density is measured utilizing coring and pits dug at one location, or at two locations at different elevations, to determine the water content of the snow. Manual coring is used to verify the location of the summer surface.
Summer balance is obtained from stake measurements. The number of stakes, typically 5–15, varies from glacier to glacier. In general, stake density is highest on the smallest glaciers and declines with increasing glacier size (Reference Andreassen, Elverøy, Kjøllmoen, Engeset and HaakensenAndreassen and others, in press).
Net balance is defined as the algebraic sum b n = b w + b s of the seasonal balances, in which bs is a negative quantity. Melting after the ablation measurements may occur in warm periods late in autumn. On glaciers with large altitude ranges, such as Nigardsbreen and Engabreen, melting after the ablation measurements is commonly observed on the lower parts. This melting will not greatly affect the total specific balance, due to the small area involved, but will influence the lower parts of the profile. Melting after the last ablation measurements and new snow deposited at these measurement sites are considered part of the next year’s winter balance.
Profiles are made by plotting point measurements of winter and summer balance vs altitude, and their mean values for each 50 or 100m elevation interval are determined (Fig. 2). Snow probing and stake locations are chosen to sample the glacier area overall, and their readings are extrapolated to the lower and upper parts. Another approach, however, was used until the 1980s when hand- contoured maps of accumulation and ablation were made from the observations. The areas within each height interval (50 or 100 m) were measured using a planimeter, and the total amount of accumulation and ablation was calculated for each height interval. Profiles bw(z), bs(z) and bn(z) were thus created. There was no appreciable change in the linearity of the profiles between the earlier and later periods.
Fig. 2. Storbreen 2003 probing sites (small circles), stakes (large circles) and snow pit (square). Coordinates are Universal Transverse Mercator zone 32 (meters).
Mean Glacier Balance
Mean and variance of the seasonal components b w and b s are given in Table 2 along with their correlation with net balance r nw and r ns. For the maritime glaciers (numbered 1–7) interannual variation of b w is larger than for b s, so r nw > r ns, and for continental glaciers (8–10) it is smaller, so rns > r nw. The ratio r nw/r ns differs slightly from that of the standard deviations σ w/σ s because of the small correlation r ws between the two seasonal components (Reference Rasmussen and ConwayRasmussen and Conway, 2001). The r ws are consistent with those of Reference Dyurgerov and MeierDyurgerov and Meier (1999), although their r are negative because they defined b s to be positive, and with those of Reference Braithwaite and ZhangBraithwaite and Zhang (1999). The r ws are all positive, but only one is significant at 99%.
Table 2. Means and standard deviations of mass-balance components, and r correlations between components (boldface significant at 99%)
Balance Profiles
Because b n = b w + b s, their profiles are also related by b n(z) = b w(z) + b s(z). From any of the profiles, the glacier- total balance b is obtained as a linear combination of the values at several discrete altitudes. If the profiles from year to year differ only by an amount ∆b that is constant with altitude, regardless of the shape of those profiles, the glacier- total balance also will differ from year to year by ∆b.
Various curves of b w(z) and b s(z) have been made over the years, and curves from prior years have often been used as a guide, especially for years with few or questionable observations. The 19 years of profiles at Storbreen are shown in Figure 3. The b w(z) are quite linear, with little variation of gradient from year to year, but there is slight (d2 b s/dz 2 < 0) curvature in the b s(z).
Fig. 3. Storbreen balance profiles for 19 individual years.
Mean values over the period of record of the three profiles are shown in Figure 4 for each of the glaciers. For each glacier, the sets of b w(z) for all the years were fit with two different functions: (1) a separate linear function each year, and (2) a family of parallel linear functions. The same was done with b s(z) and b n(z). With only a few exceptions, which will be noted below, the profiles are fit very well by the linear functions (Table 3). Calculation of the goodness of fit r2 is detailed in Reference RasmussenRasmussen (2004).
Fig. 4. Mean altitude profiles b s (open circles), b w (solid circles) and b n (connected solid circles). Vertical line is b = 0, and horizontal line is ELA. See Table 1 for glacier names and period of record.
Table 3. Results of fitting b
n(z), b
w(z) and b
s(z) profiles for n individual years. The fit of individual linear functions with generally different gradients each year is 
and the fit of a family of linear functions with the same gradient each year is 
All r
2 are significant at 99%
Balance gradients, as approximated by slopes of the individual linear functions fit to the b(z) profiles, vary little from year to year (Table 4). Here gradients are denoted b’w for db
w/dz, etc. Gradients of net balance are greatest (6.4–8.7mw.e.km–1) at maritime glaciers, as noted by Reference Oerlemans and BalkemaOerlemans (2001), and lowest (2.0–6.1) at more continental glaciers. The ratio 
does not vary systematically from west to east in southern Norway for b
n, b
w or b
s. Gräsubreen is unusual, in that distributions of both accumulation and ablation are strongly dependent on glacier geometry, with the result that the mean b’w < 0 (Fig. 4).
Table 4. Means 
and standard deviations σ of vertical gradients of balance b’n, b’w and b’s in m w.e. per km of altitude. Correlation of b’n with b’n is r
nn, of b’w with b’w is r
ww, of b
s with b
s is r
ss, of b’w with b
n is r
wn and of b’s with b
n is r
sn (boldface significant at 99%)
Correlation r nn between b’n and b n is weak and of varying sign from glacier to glacier. Correlation r ww between b’w and b w is generally positive, and r ss between b’s and b s is generally negative. Because b’n = b’w + b’s, these two relations generally tend to counteract each other to weaken correlation between b n and b’n. Reference Beniston, Keller and GoyetteBeniston and others (2003) found that increase in snowfall amount with altitude in the Swiss Alps is greater in warm than in cold winters.
If b n < 0 is caused by b s being more negative than normal, it will make b s more positive and thus strengthen b n. If b n < 0 is caused by b w being less positive than normal, it will make b w less positive and thus weaken b n. The opposite occurs when b n > 0 is caused by either b w being more positive than normal, strengthening b n, or b s being less negative than normal, weakening b n.
Conversely, if b w is more positive than normal and b s is more negative than normal in the same balance year, they will both tend to strengthen b n. If b w is less positive and b s less negative, they will weaken b n. The first case is high turnover b t = b w – b s, and the second is low turnover, so b’n will correlate better, positively, with b t than with b n (Reference RasmussenRasmussen, 2004).
The foregoing cancellation effect is illustrated by Figure 5, showing two years with b n > 0 (1989, 1990) and two with b n < 0 (2002, 2003) at Hellstugubreen, which has strong r ww and r ss compared with other glaciers (Table 4). In both b n > 0 years, b w was more positive than normal and b s less negative, whereas in both b n < 0 years b w was less positive than normal and b s more negative. The b s gradient increases from ~4 m w.e. km–1 in the b n > 0 years to ~5 m w.e. km–1 in the b n < 0 years, and the bw gradient decreases from ~2 m w.e. km–1 to ~1 m w.e. km–1, but the b n gradient remains at ~6 m w.e. km–1.
Fig. 5. Hellstugubreen b(z) profiles for four extreme years. In each panel, the more positive curves are 1990 (thick) and 1989 (thin); the more negative curves are 2003 (thick) and 2002 (thin).
Glacier-to-glacier correlations of balance gradients are weak, whereas correlations of balance itself are generally strong and are positive for b n, b w and b s (Fig. 6). For b n, 35 of the 45 correlations are significant at 95%;for b n, only 5 of the 45 correlations are significant at 95%, and 16 are negative. For b w, 33 are significant at 95%;for b w, only 8 are significant at 95%, and 12 are negative. For b s, 39 are significant at 95%;for b s, only 2 are significant at 95%, and 14 are negative. In all 15 cases in which the b correlations are significant at 95%, they are positive.
Fig. 6. Glacier-to-glacier correlation of balance b (solid circles) and of its gradient b’ (open circles) vs distance D between the glaciers.
Glacier-to-glacier correlation of b is positive for b w, b n and b s and is a generally decreasing function of the distance D between the glaciers (Fig. 6). The correlations of b w with D and of b s with D are both r = –0.76. Correlations of gradients b are of both signs and are poorly correlated with D.
Glacier-Total Balance from Balance at One Altitude
For all glaciers, the altitude where the published b w correlates best with the glacier-total b w is near the ELA, rather than in the middle of the accumulation zone (Table 5). The altitude where the published b s correlates best with the glacier-total b s is also near the ELA, rather than in the middle of the ablation zone. Using b s from the optimum altitude for using b n has nearly as good a correlation as using that from the optimum altitude for b s, and for b w it is nearly as good as using that from the optimum altitude for b w except for Langfjordjøkelen.
Table 5. Altitude z
n where reported balance b
n(z
n) has best correlation 
with glacier-total balance b
n, and similarly for 
and 
Correlation of b
w(z
n) with b
w is 
and of b
s(z
n) with b
s is 
All r
2 are significant at 99%
These results are a direct consequence of the properties of the balance gradients. Because linear functions all with the same gradient fit the observed profiles so well (
in Table 3), change in balance from one year to the next is much the same at all altitudes. Change in the glacier-total balance is then approximately the same.
A related issue arises from the combination of correlation of b between glaciers (Fig. 6) and correlation between b at one altitude and glacier-total b (Table 5). Published balance at 1650m on Nigardsbreen is used here to illustrate the relations for all three components (Table 6). Comparable results would occur if balance at z n (Table 5) for any other glacier in southern Norway were used to estimate glacier-total balance at the other glaciers.
Table 6. The r correlation between (1) balance at 1650 m on Nigardsbreen and (2) glacier-total balance at each glacier over n years of common record. The rms error is from the regression of (2) on (1). All r are significant at 99% except Langfjordjøkelen
Discussion
Balance profiles were created by different workers during the NVE mass-balance program, which may have increased their variation over the years. Conversely, objective measurements were supplemented by glaciological judgment, including using profiles from previous years as a guide, especially in years with few or uncertain measurements, which may have decreased their variation over the years. In view of the very high spatial density of measurements, however, it is unlikely that methodological inconsistencies have introduced significant spurious variation in the published profiles.
Variation of the gradient b’n of net balance is weak, with an apparently large random component. For most of the glaciers (Table 4), positive correlation of b’w with b w counteracts negative correlation of b’s with b s so that correlation of b’n with b n is weak. Year-to-year variations of all three gradients are poorly correlated from glacier to glacier, whereas the variations of the balance itself are strongly positively correlated for b w, b s and b n.
Absence of curvature in b s(z) might result from solar radiation not accounting for as much ablation in this cloudy, maritime climate as in other regions. Its effect is non-linear with altitude largely because of the altitude distribution of mean seasonal albedo. Solar radiation is more important at the eastern glaciers (numbered 8–10), but not as important as in strongly continental regions such as in the Alps or central Asia.
Turbulent fluxes might then dominate even in the low- albedo ablation zone of the glacier. Because temperature is a generally linear function of altitude and because wind speed might be relatively constant with altitude, as found by Reference Greuell and SmeetsGreuell and Smeets (2001) on Pasterzenkees, Austria, the effect of the turbulent fluxes would also be roughly linear with altitude. The fact that the ten glaciers have little southern exposure would also contribute to the dominance of turbulent fluxes.
Linearity of b w(z) might be the result of two factors. One is that mountains in Norway are not so high that there is pronounced decrease of precipitation at higher altitude, which would impart d 2 b w/dz 2 < 0 curvature. Another is that winters are so cold that precipitation generally falls as snow over the entire altitude range of a glacier. Altitudinal variation of accumulation would then follow that of precipitation, which Reference OerlemansOerlemans (1992) found to be linear at glaciers 3, 6 and 9.
If strongly negative b s is produced by summers that are uncommonly sunny, the amount of solar radiation received by the glacier will be increased. Because surface albedo is lower in the ablation zone, the increase in melt there is expected to be greater than the increase higher on the glacier. This resulting steepening of the b s(z) profile contributes to the negative correlation between b s and b’s. No mechanism is apparent, however, for why b’w correlates positively with b w.
Conclusions
At the ten glaciers in Norway with long records, vertical profiles of seasonal mass balance are remarkably linear. The gradients of both b w and b s have little variation from year to year. A direct consequence of this property is that balance near the ELA correlates strongly with the glacier-total balance. This property, in combination with the strong correlation from glacier to glacier in the glacier-total balance, results in the strong correlation of seasonal balance near the ELA on one glacier with the glacier-total seasonal balance of other nearby glaciers.
It does not follow, however, that the regularity of the profiles is a basis for making a major reduction in the number of stakes used in the monitoring program. Stakes often disappear during winter due to heavy precipitation, as well as being prone to sinking and bending, and they sometimes melt out in summer due to the large mass turnover. Although Norway is better covered by glacier measurements than other glacierized regions, long-term mass-balance measurements exist for only 10 out of about 1600 glaciers. Continuing to monitor profiles of seasonal mass balance is also important for detecting deviations that might occur in extreme years or in a changed climate.
Acknowledgements
This work was funded from US National Science Foundation grant OPP-0240861. Discussion with H. Conway improved interpretation of the results. We thank M. Dyurgerov, R. Naruse and an anonymous reviewer for helpful comments.
