Secular glacier mass balances derived from cumulative glacier length changes

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1 Global and Planetary Change 36 (2003) Secular glacier mass balances derived from cumulative glacier length changes M. Hoelzle a,b, *, W. Haeberli a, M. Dischl a, W. Peschke b a Department of Geography, Glaciology and Geomorphodynamics Group, University of Zurich, Winterthurerstr. 190, CH-8057 Zurich, Switzerland b Laboratory of Hydraulics, Hydrology and Glaciology, Federal Institute of Technology, Gloriastr. 37/39, CH-8092 Zurich, Switzerland Received 5 April 2002; accepted 17 September 2002 Abstract Glacier mass changes are considered to represent natural key variables with respect to strategies for early detection of enhanced greenhouse effects on climate. The main problem, however, with interpreting worldwide glacier mass balance evolution concerns the question of representativity. One important key to deal with such uncertainties and to assess the spatiotemporal representativity of the few available measurements is the long-term change in cumulative glacier length. The mean specific mass balance determined from glacier length change data since 1900 shows considerable regional variability but centers around a mean value of about 0.25 m year 1 water equivalent. D 2003 Elsevier Science B.V. All rights reserved. Keywords: Glacier fluctuations; Glacier length changes; Glacier mass changes; Climate change 1. Introduction Observation of worldwide glacier changes as compiled by the World Glacier Monitoring Service (WGMS) are presently being built into Global Climate Observing Systems (GCOS, WMO, 1997; Haeberli et al., 2000). Especially glacier mass changes are considered to represent natural key variables with respect to strategies for early detection of enhanced greenhouse effects on climate (Kuhn, 1980; Haeberli et al., 1999). * Corresponding author. Department of Geography, Glaciology and Geomorphodynamics Group, University of Zurich, Winterthurerstr. 190, CH-8057 Zurich, Switzerland. Tel.: ; fax: address: hoelzle@geo.unizh.ch (M. Hoelzle). The latent heat required to cause the measured glacier wastage can be compared with the estimated excess radiation income and with changes in sensible heat as calculated by numerical climate models. Several attempts have recently been undertaken to regionally or globally summarize the available data using various approaches such as area-weighting with glacier inventory data, spatial interpolation based on global ice extent and correlations between mass balance time series, comparison with integrated geometric changes as determined by laser altimetry flights and GPS surveys on selected flowlines, or cumulative length changes as combined with glacier inventory data (Cogley and Adams, 1998; Dyurgerov and Meier, 1997a,b, 2000; Dyurgerov, 2002; Echelmeyer et al., 1996; Gregory and Oerlemans, 1998; IAHS, 1999; Kuhn, /03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi: /s (02)

2 296 M. Hoelzle et al. / Global and Planetary Change 36 (2003) , 1995; Haeberli and Hoelzle, 1995, Letréguilly and Reynaud, 1989, 1990; Oerlemans, 1994; Rabus and Echelmeyer, 1998; Zuo and Oerlemans, 1997a). The results all confirm the order of magnitude (a few decimeters per year) characterizing worldwide annual ice thickness loss during recent decades. Presently observed rates of change in glacier mass and corresponding acceleration trends could well contain maninduced effects on greenhouse forcing. The anthropogenic influences on the atmosphere could now and for the first time represent a major contributing factor to ongoing glacier shrinkage at a global scale (Haeberli et al., 1999). The main problem with interpreting worldwide glacier mass balance evolution concerns the question of representativity, i.e. the possibilities of comparing the small sample of values measured during a few decades with the evolution in unmeasured areas and during previous time periods. One important key to deal with such uncertainties and to assess the spatiotemporal representativity of the few available measurements is long-term changes in cumulative glacier length. Corresponding possibilities had long remained unexploited because of two main reasons: (1) the time necessary for glacier adjustment to changed climatic conditions had been overestimated by earlier theoretical approaches (Nye, 1960) and (2) the straightforward averaging of annual length changes as percentages of advancing/retreating glaciers or as mean annual length changes (Paterson, 1981) had left undetected the important information contained in the observed data. Theoretical developments by Jóhannesson et al., (1989a,b) and the potential of numerically modelling time-dependent glacier evolution (Oerlemans, 1988, 1997, 1998; Oerlemans et al., 1998; Zuo and Oerlemans, 1997b) helped to overcome such problems. Today, the following three approaches can be applied for the interpretation of data concerning cumulative glacier length change: (a) intercomparison between curves from geometrically similar glaciers; (b) application of continuity concepts for assumed step changes between steady-state conditions reached after the dynamic response time; and (c) dynamic fitting of time-dependent flow models to present-day geometries and observed long-term length change. The advantage of the first two approaches is the simplicity of the procedure and its applicability to glaciers with strongly limited data (for instance, glacier inventory information). The third approach (cf. especially Oerlemans et al., 1998 for coordinated model experiments) requires a thorough parameterization involving a number of uncertainties but allows for better time resolution, gives information on variations in equilibrium line altitude, and helps testing the simpler first two approaches. An ideal concept, therefore, consists in combining all three approaches and comparing the results as far as possible with measured data and observed evidence. The present article deals with the first two approaches and attempts to establish a baseline of global/long-term information as available from the database of the World Glacier Monitoring Service. 2. Scientific background The curves of cumulative glacier advance and retreat are converted into time series of temporally averaged mass balance by applying a continuity model originally proposed by Nye (1960). This approach considers step changes after full dynamic response and new equilibrium of the glacier. Thereby, mass balance disturbance (yb) leading to a corresponding glacier length change (yl) depends on the original length (L o ) and the annual mass balance (ablation) at the glacier terminus (b t ): yb ¼ b t yl=l o : The dynamic response time (s r )ish max /b t (Jóhannesson et al. 1989a,b), where h is a characteristic ice thickness, usually taken at the equilibrium line where ice depths are near maximum. Assuming a linear adjustment of the mass balance b to zero during the dynamic response, the average mass balance < b> is taken as yb/2. The so-obtained value < b> is given in annual ice thickness change (meters of water equivalent per year) averaged over the entire glacier surface, and can be directly compared with values measured in the field. The method is simple and the results compare very well with long-term observations (Herren et al., 1999). The main limitation is the resolution in time: with a characteristic value for b t

3 M. Hoelzle et al. / Global and Planetary Change 36 (2003) at the snout of Grosser Aletschgletscher of 12 m year 1 and a maximum thickness of about 900 m, the response time is somewhere in between 50 and 100 years. The calculated mass balance values are therefore half-secular to secular averages. These mass balance values are therefore calculated for the glaciers according to their individual characteristic response time or multiples thereof. 3. Data compilation and processing Data compilation was performed in two steps. Swiss data on glacier length change was compiled by Peschke (1998) and corresponding worldwide data by Dischl (1999). Both data sets were combined to enable intercomparison of cumulative length changes (see Intercomparison of regional glacier length evolution) and estimates of regional mass balances for secular time periods (see Derived secular mass balances). The map in Fig. 1 shows the mountain regions where length change data were compiled. International glacier data collection has been coordinated since At that time, the Swiss limnologist F.A. Forel started periodical publishing of the Rapports sur les variations périodiques des glaciers on behalf of the then established Commission Internationale des Glaciers (Forel, 1895). Up to 1961, the data compilations constituting the main source of length change data worldwide were published in French, Italian, German, and English. Since 1967, the publications are all written in English. The first reports contain mainly qualitative observations with the exception of the glaciers in the Alps and Scandinavia, which are quite well documented by quantitative measurements from the very beginning (Brückner and Muret, 1908, 1909, 1910, 1911; Hamberg and Mercanton, 1914; Finsterwalder and Muret, 1901, 1902, 1903; Forel and Du Pasquier, 1896; 1897; Rabot and Muret, 1911, 1912, 1913; Rabot and Mercanton, 1913; Richter, 1898, 1899, 1900; Reid and Muret, 1904, 1905, 1906). After the First World War, Mercanton (1930, 1934, 1936, 1948, 1952, 1954, 1958, 1961) edited the more rarely appearing publications since 1933 on behalf of the International Commission on Snow and Ice (ICSI) of the International Association of Hydrological Sciences (IAHS). Starting with 1967, the data are published in five yearly intervals under the title Fluctuations of Glaciers, first by the Permanent Service on the Fluctuations of Glaciers (PSFG, Kasser, Fig. 1. Mountain regions with long-term glacier length change data. Selected glaciers in the regions with names are presented in Fig. 3.

4 298 M. Hoelzle et al. / Global and Planetary Change 36 (2003) ) and after merger of PSFG with the Temporary Technical Secretariat for the World Glacier Inventory (TTS/WGI) in 1986 by the World Glacier Monitoring Service (WGMS). The corresponding publications are IAHS (ICSI)/UNESCO (1967, 1973, 1977, 1985) and IAHS (ICSI)/UNEP/UNESCO (1988, 1993a, 1998).In addition to this data collection, information in other sources such as the Journal and the Annals of Glaciology or other scientific publications was collected and integrated in the database (cf. Baird and Field, 1952; Bouverot, 1958; Casassa et al., 1998; Desio, 1967; Ding and Haeberli, 1998; Hofmann, 1958; Jóhannesson and Sigurdsson, 1998; Johnson, 1954; Kaser, 1996, 1999; Matthes, 1934; Ommanney et al., 1998; Sigurdsson, 1998; Tsvetkov et al., 1998; Vanni, 1954). The field method of data collection for the frontal glacier-tongue variations in most cases consists of simple tape readings and sometimes of geodetic/photogrammetric surveys using reference points marked in the glacier forefield. The accuracy of annual measurements is in a range of about F 1 2 m. Such an accuracy is by far good enough for the order-ofmagnitude estimates presented here. Possibilities of intercomparison between the documented times series, however, are sometimes reduced due to intermittent interruptions and methodological heterogeneities within the recorded time series. Complete time series are available in the European Mountain ranges, but large gaps exist in most other mountain regions. Additional problems with data compilation and interpretation relate to information sometimes presented in text form rather than tables, to different languages used, or to glaciers having changed their names (for instance, Belengi Bezengi in CIS) or political state (for instance, Fürkele-Austria to Forcolo-Italy). Especially old records also contain numerous typing errors (for instance, 1930/31 instead of 1931/32 in a table given by Mercanton, 1936). As far as possible, such uncertainties were eliminated by careful examination of the situation. In addition to data on annual front variations for each glacier, the following variables were collected: the glacier code of the WGMS database, the political unit (country abbreviation), the general and specific location, latitude and longitude, highest, median or mean and lowest elevation, and length of the glacier around 1960 to 1975 as based on inventory data and aspect (Hoelzle and Trindler, 1998). A parameterization scheme earlier developed for analyzing glacier inventory data (cf. Haeberli, 1991; Haeberli and Hoelzle, 1995; Hoelzle and Haeberli, 1995 for more detailed discussion) was used. This scheme builds on measured data about total length (L 0 ) as well as maximum, mean or median and minimum altitude (H max, H mean or median, H min ) of the investigated glaciers. From these basic parameters, mean altitude was calculated from H max and H min where not available. Vertical extent (DH = H max H min ) and average surface slope (a = arctan {DH/L 0 }) were then derived as a first step. Average ice depth along the central flowline (h f ) was estimated from a and a mean basal shear stress along the central flowline (s f = fqgh f sina, with q = density and g = acceleration due to gravity), whereby s f depends in a nonlinear way on DH as a function of mass turnover (cf. Driedger and Kennard, 1986; Haeberli, 1985; Haeberli and Hoelzle, 1995). The shape factor f was chosen as 0.8 for simplicity in all cases. Glacier long profiles along the central flowlines are generalized as two simple wedges pointing up-slope in the accumulation area, down-slope in the ablation area and having in common the side representing the maximum thickness (h max ) at the equilibrium line. The value for h max is very roughly determined at 2.5h f instead of 2h f as estimated from known ice thickness measurements on various Alpine glaciers (Müller et al., 1976 and unpublished radio-echo soundings/hot water drillings by VAW/ETH Zurich) in order to account for some longitudinal variations in a. Mean altitude is taken as an approximation for equilibrium-line altitude ELA (cf. Braithwaite and Müller, 1980), and the mass balance (annual ablation) at the glacier tongue is computed as b t =db/dh (H mean H min ) where the mass balance gradient db/ dh receives values of 0.3 to 1.2 m water equivalent per 100 m and year for the ablation area. The determination of the mass balance gradient may represent the most delicate point in the parameterization. Realistic values for the gradients were sought by taking direct measurements from various mountain areas as a guide where available. The gradients used are based on IAHS (ICSI)/UNEP/UNESCO (1991, 1993b, 1994, 1996, 1999, 2001), Oerlemans and Fortuin (1992) and Oerlemans and Hoogendoorn (1989). Disturbances in mass balance (yb) were calculated from cumulative glacier length changes (see Scientific

5 M. Hoelzle et al. / Global and Planetary Change 36 (2003) background, cf. Paterson, 1994; Haeberli, 1991) in the sense of step functions between assumed steady-state conditions with respect to time periods corresponding to the characteristic dynamic response time s r = h max /b t, (cf. Jóhannesson et al., 1989b) of the involved glaciers. Average mass balance over the considered time interval is then taken as half the disturbance, assuming linear adjustment to new equilibrium conditions. 4. Intercomparison of regional glacier length evolution Length change measurements of more than 1000 glaciers worldwide were compiled. The here-presented intercomparison is based on 68 glaciers from the Swiss glacier network and 90 selected glaciers worldwide. The Swiss glaciers were treated separately because of the large sample size with highly variable glacier characteristics and exceptionally complete long-term records. The dynamic response to climatic forcing of glaciers with variable geometry involves striking differences in the recorded curves (Haeberli, 1994). Such differences reflect strong effects of size-dependent filtering, smoothing, and enhancing of the delayed tongue response with respect to the undelayed input (mass balance) signal. As a consequence, the sometimes still popular straight averaging of annual length change data (annual percentage of advancing/retreating glaciers, average annual length change) destroys essential aspects of the observed signal and must be avoided. The sample of Swiss glaciers shows that length and slope of a glacier constitute the predominant factor controlling glacier tongue reaction (see Fig. 2). Fig. 2. Total cumulative length in the Swiss Alps classified after the total length of the glaciers.

6 300 M. Hoelzle et al. / Global and Planetary Change 36 (2003) For intercomparison purposes, therefore, values of cumulative length change are presented with respect to size categories chosen in a way to optimally reflect common characteristics of the tongue-reaction signal. Glaciers with heavy debris cover, periodical surge activity, or calving instability in deep water were excluded from the analysis because of the strong non-climatic effects influencing them. Small, somewhat static, low-shear stress glaciers (cirque glaciers) have altitudinal extents comparable with the interannual variability of equilibrium-line altitude and hence reflect yearly changes in mass balance practically Fig. 3. Cumulative length change in different mountain regions of the earth.

7 M. Hoelzle et al. / Global and Planetary Change 36 (2003) without any delay (Fig. 2a). Larger, dynamic, highstress glaciers (mountain glaciers) react with enhanced amplitudes but a delay of several years to decadal fluctuations in climatic and mass-balance forcing (Fig. 2b and c). Large valley glaciers in the Alps give with a delay of several decades strong and most efficiently smoothed signals of tongue reactions to secular trends (see Fig. 2d). In fact, long glaciers such as Grosser Aletschgletscher never had an advancing period since the 19th century in contrast to smaller mountain glaciers such as Trient or Oberer Grindelwald which show two marked advancing periods in the 1920s and in the s. The smallest glaciers like Pizol directly respond to annual mass balance and snow line variability through deposition/ melting of snow/firn at the glacier margin. Considering all different types of glacier response obviously gives the best information on secular, decadal, and annual developments. Fig. 3 clearly shows the well-known fact that glacier retreat in the 20th century is a worldwide phenomenon. Large glaciers have suffered from the largest absolute length change measured since Long glaciers (>10 km) retreated continuously or remained stationary except in western Iceland. Glaciers in the size category of 2 to 10 km show clear decadal reactions. Advance periods in the s could not only be observed in the European Alps, but also in the Pamir-Alai, Tien- Shan, Olympic, and Coast Mountains. Advance tendencies continued into the 1990s for glaciers near the Norwegian West Coast and in Iceland. This development in the North Atlantic appears to parallel a similar development in the New Zealand Alps and forms a strong contrast to the European Alps, Rocky Mountains, Coast Mountains, and Cordillera Central where general retreat in the s is pronounced. Consideration of the cumulative length change curves in more detail reveals distinct differences between evolutions in various mountain ranges at decadal time. The worldwide glacier signal of climate change seems to be more or less homogenous at multi-decadal to secular time scales only. 5. Derived secular mass balances Reconstructed mass-balance values can be compared much easier than length change because the complex size effects on flow dynamics are removed to a certain degree: the direct response to climate forcing can be considered in a standard format, the mean annual mass change expressed as an average thickness change in meters of water equivalent. The following presents average mass-balance values reconstructed from multi-decadal to secular length change data of 68 Swiss glaciers and, correspondingly, calculated secular mass balances of 50 selected glaciers in different countries of the world Swiss glaciers Sixty-eight glaciers with their overall length and mean slope were subdivided into five classes as follows: Class 1 (long and flat valley glaciers, sample: 4 glaciers): glaciers longer than 10 km with a mean slope of < 15j; glaciers in this class reveal constant retreat since the beginning of the measurements. Class 2 (intermediate valley and mountain glaciers, sample: 11 glaciers): glaciers with a length between 5 and 10 km and a mean slope between 10j and 25j; such glaciers show strong fluctuations with large amplitudes and up to three advance and retreat periods since Class 3 (steep mountain glaciers, sample: 19 glaciers): glaciers with a length between 1 and 5 km and a mean slope ranging from 15j to 25j; these glaciers show moderate fluctuations and amplitudes but exhibit quite large variability and strongly individual reaction. Class 4 (flat mountain glaciers, sample: 14 glaciers): glaciers with a length between 1 and 10 km and a mean slope < 15j; glaciers of this type underwent weak fluctuations with small amplitudes but a clear overall retreat. Class 5: (extremely small and extremely steep glaciers, sample: 20 glaciers): glaciers shorter than 1 km with a mean slope larger than 15j or with a length between 1 and 5 km and a mean slope larger than 25j; glaciers at the extremes of size and slope show a pronounced high-frequency variability with moderate to large amplitude. For all glaciers, individual response times were calculated and mean specific mass balances for two

8 302 M. Hoelzle et al. / Global and Planetary Change 36 (2003) Table 1 Comparison of direct measured and from the length change calculated mean mass balances < b> m year 1 for time intervals z the response time of the glaciers Glaciers Time period Mean specific mass balance (m year 1 ) Reference Rhone Chen and Funk (1990) 0.28 calculated from yl Gries direct measurement 0.22 calculated from yl Silvretta direct measurement 0.02 calculated from yl Grosser direct measurement Aletsch 0.22 calculated from yl different time periods ( and around ) were determined according to the abovedescribed parameterization scheme. The whole procedure was verified by comparing directly measured mean specific mass balances at the four glaciers Grosser Aletsch, Rhone, Silvretta, and Gries with those calculated on the basis of the length change measurements. The results displayed in Table 1 confirm the method and prove that at least reliable orderof-magnitude estimates can be performed in this way. Information on the first time period is based on data from the 1850 glacier inventory as reconstructed and compiled by Maisch et al. (1999). The second time period (around 1890) is covered by the direct observations of the Swiss Glaciological Commission. Table 2 shows that average mass losses of long and flat glaciers have exceeded those of smaller glaciers: typical values center around 0.25 m year 1 for larger glaciers and around 0.11 m year 1 for the Table 3 Mean mass balance < b> m year 1 sorted after four length classes (since ca. 1900) Variables Class 1 Class 2 Class 3 Class 4 Total length (km) V V V 8.0 >8.0 Mean specific mass balance (m year 1 ) smaller ones. The main reason for large/flat glaciers to have higher mass losses may probably be that the larger thickness limits long-term ice losses to a lesser degree than in small glaciers where the bed is reached relatively soon. This result confirms that other factors being equal length and slope exert a predominant influence not only on flow dynamics but also on overall mass losses of glaciers an interesting feedback between mass balance and flow dynamics over decadal to secular time scales Glaciers worldwide On the worldwide database, similar calculations as for the Swiss glaciers were carried out. Determination of realistic mass balance gradients in each mountain region constitutes the most uncertain step in the procedure. The gradients applied for the 50 glaciers selected worldwide were estimated by using directly measured data relating to glaciers in the vicinity or in the same mountain range. Where such information Table 2 Different mean specific mass balances < b> m year 1 for the five classes and for the periods (a) 1850 to 1996 and (b) ca to ca Fig. 4. Mean specific mass balance < b> m year 1 in different mountain regions (since ca. 1900) calculated from length change data.

9 M. Hoelzle et al. / Global and Planetary Change 36 (2003) was unavailable, climatic data was used to estimate characteristic values for dry, continental-type (e.g. Altai) climatic conditions with gradients between 0.3 and 0.5 m year 1 per 100 m altitude, transitional climates (e.g. Caucasus) with gradients between 0.6 and 0.8 m year 1 per 100 m altitude, and humid, maritime-type conditions (e.g. Western Norway) with gradients between 0.9 and 1.2 m year 1 per 100 m altitude (cf. IAHS (ICSI)/UNEP/UNESCO, 1991, 1993a,b, 1994, 1996, 1999, 2001; Oerlemans and Fortuin, 1992; Oerlemans and Hoogendoorn, 1989). The results of the parameterization confirm the trend observed in the sample from the Swiss Alps for smaller glaciers to have lost mass at a slower rate than larger ones (Table 3). On average of the worldwide sample, larger glaciers have lost around 0.25 m year 1, a value which is identical to the value calculated for the larger Swiss glaciers. The reconstructed rates of secular mass losses strongly differ between humid-maritime-type glaciers such as those of western Scandinavia and dry-continental type glaciers in the Altai area, for instance (Figs. 4 and 5). This primarily results from the choice of the mass balance gradient in the calculation. The climatic dependence of the chosen gradients, however, is a well-established fact (Oerlemans and Hoogendoorn, 1989; Oerlemans, 2001) and must certainly be considered to be realistic, even though absolute values are somewhat uncertain. The sensitivity with respect to secular trends in global warming of maritime-type glaciers is much higher than the one of continentaltype glaciers. Fig. 5. Mean specific mass balance < b> m year 1 classified after different climate types (since ca. 1900). 6. Discussion The study presented here clearly shows that for direct intercomparisons of cumulative glacier length changes, shorter time scales and high temporal measurements are necessary. Especially, to derive the annual (very small glaciers) or decadal (mediumsized glaciers) fluctuations, such measurements have to be done. The high temporal measurements in the Alps and in Scandinavia are good examples. New technologies like satellites offer new possibilities to derive in the future long-term length changes, especially for deriving secular trends in mean mass balances. The concept of Jóhannesson et al. (1989a,b) presents the possibility to roughly estimate secular mass balance changes by using length change measurements. This means that length change measurements are, for the future, one of the most important key variables in global glacier monitoring strategies. The secular mass loss is a worldwide phenomenon in the period since Future changes will affect firstly the maritime ones and then, with a certain delay, the continental ones, which are mostly of polythermal or cold stage. 7. Conclusions and recommendations In addition to mass balance, this study shows that length observations of a representative subset of the world glaciers are and will be, in the future, a very valuable key factor, among others, for assessing climate change effects at regional or worldwide scale (Haeberli, 1998). In the strategy of the Global Terrestrial Network for Glaciers (GTNet-G) within the Global Climate Observing System (GCOS)/Global Terrestrial Observing System (GTOS), long-term observations of glacier length change data at a minimum of about 10 sites within each mountain range are attributed highest priority. These glaciers should be selected according to size and dynamic response from the existing set of sites where glacier length is monitored. At this level, spatial representativeness is very important. Today, approximately 800 glaciers where only length is measured are compatible with Tier 4 of the GTNet-G-strategy. Because access is infrequent, they can be located wherever necessary to ensure representativeness.

10 304 M. Hoelzle et al. / Global and Planetary Change 36 (2003) Acknowledgements We would like to thank M. Maisch for providing us with the length data of 1850 and G.H. Gudmundsson for many interesting discussions about this topic. Very much appreciated and helpful were the very constructive comments of L. Braun and J.O. Hagen. References Baird, P.D., Field, W.O., Report on the Northern American Glaciers. IAHS 32, Bouverot, M., Notice sur les variations des glaciers du Mont- Blanc. IAHS 46, Braithwaite, R., Müller, F., On the parameterization of glacier equilibrium line altitude. IAHS 126, Brückner, E., Muret, E., Les variations périodiques des glaciers. XIIme Rapport, Zeitschrift für Gletscherkunde und Glazialgeologie II, Brückner, E., Muret, E., Les variations périodiques des glaciers. XIIIme Rapport, Zeitschrift für Gletscherkunde und Glazialgeologie III, Brückner, E., Muret, E., Les variations périodiques des glaciers. 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