Ahlmann (1948) first developed mass-balance concepts in a series of pioneering measurements in Nordic countries. The longest continuous measurements were started in 1946 on Storglaciaren in northern Sweden (Schytt, 1962) and similar measurements started in the 1940s and 1950s on other glaciers in Norway, the Alps, western North America, and on numerous glaciers in the former USSR. The number of glaciers studied and their geographical coverage expanded rapidly in the 1960s under the impetus of the International Hydrological Decade (IHD) 1965-1974. Although the IHD involved an emphasis on the hydrological role of glaciers (Collins, 1984), interest has more recently shifted to the possible increased melting of glaciers as a cause of rises in global sea level, and recent analyses of mass-balance data exclusively stress this aspect of mass-balance study.

Regular tabulations of mass-balance data have been published in hard copy since 1967 as part of The Fluctuations of Glaciers series (Kasser, 1967, 1973; Müller, 1977; Haeberli, 1985; Haeberli & Müller, 1988; Haeberli & Hoelzle, 1993; Haeberli et al., 1998b). The World Glacier Monitoring Service (WGMS), sponsored by UNESCO, now publishes this series and also distributes bi-annual summaries of mass balance data over the Internet, e.g. WGMS (2001). However, the latter data are very basic summaries and many prefer the more extensive information (to say nothing of the beautiful free maps!) provided by Fluctuations of Glaciers.

The format for the dissemination of data by the World Glacier Monitoring Service (WGMS), and its predecessors, reinforces the impression that 'true' mass-balance data are measured with stakes and snowpits, i.e. using the 'glaciological' method of Ahlmann (1948), and that hydrological and geodetic methods (Hoinkes, 1970) measure something else. The amount of published massbalance data based on stakes and snowpits has steadily increased over the years, both with respect to number of glaciers studied and length of series for individual glaciers. Collins (1984) could only find annual mass-balance data for 95 glaciers, Dyurgerov & Meier (1997) identify 257 glaciers, Cogley & Adams (1998) identify 231 glaciers and Braithwaite (2002) analyses data for 246 glaciers. Dyurgerov (2002) compiles some previously overlooked data, especially from American and Russian glaciers, and corrects many published mass-balance series to give data for 280 glaciers. There are numerous minor discrepancies between the different data compilations but we can say that annual mass balance has been measured for at least one year on about 300 glaciers. This latter figure is reached by combining data from Braithwaite (2002) and Dyurgerov (2002).

In principle, it should be possible to calculate the present contribution to sea-level rise from glaciers and ice caps by averaging available measurements, but we must recognize the relatively large variability of mass balance in both space and time; see figs 8 & 9 in Braithwaite (2002). Available data for 300 glaciers are strongly biased to western Europe, North America and the former USSR and are further biased to wetter conditions; see figs 6 & 7 in Braithwaite (2002). It therefore seems implausible to me that averages of available data can adequately describe mass-balance variations over the whole globe. For example, the chance inclusion/exclusion of a particular glacier for a particular year will bias the average mass balance for that year.

Despite the above scepticism, results from Dyurgerov & Meier (2000, 2004) and Meier et al. (2003) deserve mention. Dyurgerov & Meier (2000) compare average mass balances for 1961-1976 and 1977-1997 with global average temperatures and infer a temperature sensitivity of -0.37myr-1 K-1 for global glacier mass-balance. This estimate of mass-balance sensitivity to global temperature is remarkably close to the global mass-balance sensitivity estimated by Oerlemans (1993c); see below. This cosy agreement seems to be contradicted by table 16.2 in Dyurgerov & Meier (2004) but we might agree that the average of available massbalance measurements (Dyurgerov & Meier, 2000, 2004; Meier et al., 2003) is somehow picking up real variations in global mass balance.

For the future, we should establish new measurements in key areas where feasible but also supplement the 'glaciological' method with modern implementations of the 'geodetic' method, for example, using laser altimetry (Sapiano et al., 1998; Arendt et al., 2002). This will not greatly improve the global coverage of mass balance data, however, and we will always deal with 'a data set that is sparse geographically' in the words of Meier (1993). If existing measurements are not adequate for inferences on a global scale they must be supplemented by modelling.

The simplest model for mass balance and climate is the regression model, where an annual series of mean specific balance is correlated with parallel summer temperature and annual, or winter, precipitation series at some nearby climate station. Liestol (1967), Martin (1975), Braithwaite (1977), Tangborn (1980), Günther & Widlewski (1986), Holmlund (1987), Letréguilly (1988), Laumann & Tvede (1989), Chen & Funk (1990), Trupin et al. (1992) and Müller-Lemans et al. (1994) have made such studies. The regression coefficient can be interpreted as the sensitivity of mass balance to changes in the corresponding climate variables. The regression model is simple to use and provides a correlation coefficient to give an idea of'goodness of fit'. The main practical disadvantages are that a few years of observed data (5-10yr?) are needed to give meaningful results and that a nearby climatic station is needed. The need to have a few years of massbalance record drastically reduces the number of glaciers available for modelling. Wood (1988) could find only 53 glaciers with at least 7yr of record and even up to 1999 there are only 143 glaciers with at least 7yr of record (188 glaciers with at least 5yr of record).

The fundamental objection to the regression model is that its coefficient reflects conditions prevailing during the period in which the data are measured and will not be valid under a greatly different climate with a changing surface-area distribution. Ultimately there can be no unique relation between glacier mass balance and temperature because, under a constant temperature, glaciers will always tend towards a steady state. Nevertheless, regressions of mass balance on summer temperature yield sensitivities that are comparable, although slightly lower, to those obtained by more advanced models. For examples, see table 6 in Braithwaite & Zhang (2000) and table 1 in Braithwaite et al. (2003) for details. High-latitude glaciers have a low regression coefficient for summer temperature, whereas Alpine glaciers have a higher coefficient (Braithwaite, 1977).

If you want a mass-balance model to drive an ice-dynamics model, you have to calculate mass balance at different altitudes, i.e. to use a distributed mass-balance model rather than a lumped model such as the regression model (above). The most obvious models here are degree-day and energy-balance models, where the terms 'degree day' and 'energy balance' refer to the ways in which melting is calculated, although the models must also have routines to calculate the accumulation of snow.

Energy balance models for single glaciers are very demanding in terms of data and are best used for glaciers that are well instrumented. Examples of the approach are by Escher-Vetter (1985),

Degree-day factors

| Gridded climatology!

Glacier

Temporary model i

| Gridded climatology!

Glacier

Braithwaite & Olesen (1990), van de Wal & Russell (1994), Arnold et al. (1996), Hock & Holmgren (1996), Hock (1999), van den Broeke (1996) and Braithwaite et al. (1998). There is little doubt that the future of improved mass-balance models for single glaciers, e.g. for runoff forecasting, belongs to detailed energy-balance modelling although there are still problems to be solved, especially with respect to turbulent fluxes and albedo variations. By contrast, the energy-balance model of Oerlemans (1993b) is highly parameterized and hardly uses more than temperature and precipitation as input data. The model has been applied to only a few glaciers (Oerlemans & Fortuin, 1992; Fleming et al., 1997), which might suggest that its application is laborious.

For a large-scale assessment of the sensitivity of glacier mass balance to climate change, we can also use the degree-day model. Different examples of this approach are by Braithwaite (1985), Braithwaite & Olesen (1989), Reeh (1991), Huybrechts et al. (1991), Laumann & Reeh (1993), Johannesson et al. (1995), Johannesson (1997), Braithwaite & Zhang (1999, 2000) and Braithwaite et al. (2003). Although de Woul & Hock (in press) use a degree-day model, their approach is regression based. The following outline is based on the version used by Braithwaite et al. (2003).

The accumulation for each month is calculated from monthly precipitation multiplied by the probability of below-freezing temperatures, which is in its turn calculated from monthly mean temperature assuming that temperatures are normally distributed within the month. Monthly melt is assumed proportional to the positive degree-day sum that is calculated from monthly mean temperature (Braithwaite, 1985). The proportionality factors (positive degree-day factors) linking melt to positive degree-day sum depend upon whether the melting refers to ice or snow. There is a range of values in the literature (Braithwaite & Zhang, 2000; Braithwaite, et al. 2003; Hock, 2003). Refreezing is estimated for subpolar glaciers by assuming that all annual melt is refrozen within the snow cover up to a definite fraction of the annual accumulation. This factor is taken to be 0.58 if snow density has to change from 375 to 890 kgm-3 before runoff can occur (Braithwaite et al. (1994), but other estimates of initial snow density and ice density for runoff give melt/accumulation ratios of 0.4 to 0.7. The monthly temperature and precipitation are extrapolated to the altitude in question from a nearby weather station, or from a gridded climatology (New et al., 1999). Consistent values of model parameters (degree-day factors for snow and ice and temperature lapse rate) and precipitation are chosen by 'tuning' the model, i.e. varying conditions in the model until the modelled distribution of mass balance with altitude agrees with the field data. Coarse tuning involves establishing suitable degree-day factors and temperature lapse rate for the whole glacier, and fine tuning involves varying the precipitation with altitude to obtain a nearly perfect fit between modelled and observed mass balance as a function of altitude (Fig. 83.1). There is an element of subjectivity in this tuning as several different combinations of model parameters will produce a similar fit of model to data (Braithwaite & Zhang, 2000). The precipitation obtained by fine tuning the model is the effective precipitation over the glacier, e.g. resulting from snow drift and topographic channelling, which is often higher than the precipitation value available in a gridded climatology, or read off a weather map. Oerlemans & Fortuin (1992) tune their energy balance model by a similar variation of model precipitation.

13 2

Figure 83.2 Mass-balance sensitivities for 61 glaciers in different glacierized regions. Results from Braithwaite et al. (2003).

After you have tuned the degree-day or energy-balance model to agree (more-or-less) with the field data, you can simulate the effects of climate change by changing the temperature and/or precipitation in the model. This necessarily involves the assumption that the model parameters do not change as climate changes. This may not be the case but it is difficult to know how the parameters should be changed.

In the following discussion we consider only the sensitivity of mass balance to a 1K temperature change that is applied equally to every month in the year. This is a good 'standard' change for comparing results from different glaciers and different regions although real climate changes will involve different changes for different times of the year.

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