Introduction

The Geological Survey of Greenland (GGU) has made nearly daily readings of ablation on two outlet glaciers from the Greenland ice sheet, as well as collecting simple climate data at nearby field stations (Reference Olesen, Braithwaite and OerlemansOlesen and Braithwaite, 1989). We have previously used these data to develop statistical relations between ice ablation and temperature (Reference Braithwaite and OlesenBraithwaite and Olesen, 1989), and we now analyse these ablation-temperature relations with a simple energy-balance model. This is useful because ablation-temperature equations will continue to be used for many practical purposes despite efforts to develop a more physically based alternative.

Data

The data used in the present study comprise (1) observed daily ablation data from two glaciers, (2) simple daily climate data from field stations close to the glaciers, and (3) energy-balance components calculated from the simple climate data by an energy-balance model (Reference Braithwaite and OlesenBraithwaite and Olesen, 1990).

The ablation data come from stake 53 (at 880 m a.s.l.) on Nordbogletscher and stake 751 (790 m a.s.l.) on Qamanàrssûp sermia (Fig. 1). The data cover 415 d in June-August at Nordbogletscher (1979–83) and 512 d at Qamanàrssûp sermia (1980–86). The data mainly refer to ice ablation, as both sites have little or no winter snow although traces of new snow occur occasionally during cold periods in the summer and are taken into account in calculating the energy balance. The daily turbulent fluxes of sensible and latent heat are calculated from air temperature, wind speed, and vapour pressure using equations given by Arnbach (1986). The calculation takes account of the difference in surface roughness between ice and snow surfaces, e.g. turbulent-heat fluxes to a snow surface are 30% less than to an ice surface under the same climate conditions. The calculations assume a melting glacier surface but there are days when this is not the case, and the calculated ablation is then set to zero.

Fig. 1. Locations of two glacier-climate stations: stake 53 is on Nordbogletscher at 880 m a.s.l. and lat. 61 28'N.. and stake 751 is on Qamanàrssûp sermia at 790 m a.s.l. and lat. 64°28'N

The absorbed short-wave radiation is calculated from measured incoming radiation assuming albedos of 0.3 and 0.7, respectively, for ice and snow, i.e. a snow surface absorbs 57% less short-wave radiation than an ice surface. The long-wave radiation is calculated from air temperature and cloud amount using an equation fromReference Ohmura Ohmura (1981).

For convenience, all energy-balance components are expressed in ablation units, i.e. as mm water d⁻¹ or as kgm⁻²d⁻¹.

The errors in calculating ablation from the simple energy-balance model are summarized in Table I for different months. Although there are substantial errors on a day-to-day basis, the model is surprisingly accurate considering its simplicity.

Table I

The Problem

The observed daily ablation at any place is assumed proportional to the daily mean air temperature at the same place, and at about 1–2 m above the glacier surface, as long as air temperature is at or above melting:

(1)

and daily ablation rate is zero for temperatures below melting:

(2)

where a_t and T, are respectively the daily ablation rate and temperature on day t, and α ₀ and β ₀ are parameters. The above relations give correlation coefficients from 0.71 to 0.83 for seven summers 1980–86 at Qamanàrssûp sermia (Reference Braithwaite and OlesenBraithwaite and Olesen, 1989).

The α and β parameters in the ablation-temperature relation vary because the surface-energy balance is implicit in them. This can be a serious drawback for studying ablation under past or future climates, or in unstudied regions, as an ablation–temperature equation for one situation need not apply to another situation. It is therefore important to understand ablation–temperature models in terms of the more fundamental energy balance.

Method

The method is based on Reference AmbachBraithwaite (1981). By the least-squares algorithm, the temperature response of ablation, or slope of Equation (1), is given by:

(3)

where N is the number of days with ablation and temperature data, a and T are the means of the respective N day samples, and S _T is the standard deviation of temperature. It is understood that the samples are arranged to include only days with temperatures at or above the melting point. The intercept in the regression equation is given by:

(4)

Ablation can be simulated by the energy-balance equation involving M energy sources:

(5)

where á _t is the simulated ablation, not necessarily identical to observed ablation q_t , and q_it is the ith energy source on the ith day. By hypothesis, each energy source is also linked to air temperature by a regression equation:

(6)

where α _i and β _i are the intercept and temperature response for ith energy source. Equation (6) is quite general if β _i = 0 is not excluded a priori. Substitution of Equation (6) into Equation (5) gives:

(7)

which shows that the simulated ablation is given by the sum of regression equations linking individual energy sources to temperature. Alternatively, the α _i and β _i parameters express the contribution to the overall ablation-temperature relation by the ith energy source.

Results

For the Nordbogletscher data set (386 d with T ≥ 0 deg), the regression equation linking observed daily ablation a_t to daily temperature T_t is:

(8)

The regression equation for simulated ablation á _t is:

(9)

where r is the correlation coefficient. The corresponding equations for Qamanàrssûp sermia (480 d, T ≥ 0 deg) are:

(10)

and

(11)

The two sets of equations are broadly similar but the

Qamanàrssûp sermia equations indicate higher ablation rates than for the same temperature at Nordbogletscher. Simulated ablation à_t is the sum of the energy sources, and differences between the a_t and à_t , equations reflect the effects of errors in the energy-balance calculation. These differences are small, supporting the accuracy claimed for the energy-balance model in Table I.

The possibility that parameters in the ablation-temperature equation vary according to season, e.g. with global radiation as suggested byReference Gottlieb Gottlieb (1980) and Reference Lundquist Lundquist (1982), was examined by repeating the above calculation for each month separately. Although the results (Table II) for different months are not the same, there is no clear pattern of seasonal variation, thus agreeing with Reference Braithwaite and OlesenBraithwaite and Olesen (1989, fig. 3), and seasonal variations are neglected in the following discussion.

Table II

Energy sources and temperature

Regression equations were calculated for each individual energy source and temperature, and are plotted in Figures 2 and 3. The large contribution to ablation energy by short-wave radiation at 0 deg is almost entirely offset by negative values in the other sources, so that the intercept in the ablation–temperature equation is close to zero. The numerical values of the corresponding temperature responses, i.e. slopes of the regression equations, are given in Table III. The most sensitive energy source at both sites is sensible-heat flux whose temperature response accounts for about one-half of the temperature response of ablation. Net radiation (short-wave + long-wave radiation) accounts for about one-quarter of the temperature response of ablation at both sites, while latent-heat flux and error account for the remainder.

Fig. 2. Temperature variations of simulated energy balance at Nordboglelscher.

Fig. 3. Temperature variations of simulated energy balance at Qamanàrssûp sermia.

Table III.

Effect of wind

The temperature response of sensible-heat flux is larger at Qamanàrssûp sermia than at Nordbogletscher, i.e. 4.22 compared to 3.37 mm water d⁻¹ deg⁻¹. This partly reflects higher wind speeds at Qamanàrssûp sermia, i.e. averages of 4.8 and 3.3 ms⁻¹, respectively, and contributes to higher ablation rates there.

There is an association of high wind speeds with high temperatures under Föhn conditions which increases the sensible-heat flux at higher temperatures. This introduces an apparent non-linearity into the relation between sensible-heat flux and temperature which may explain the negative intercepts in the (assumed linear) regression equations for sensible-heat flux. The effect of this temperature-wind correlation was studied by re-running the energy-balance model with the wind speed artificially held constant (equal to the mean wind speed). This is equivalent to suppressing the temperature-wind correlation. The resulting regression lines are plotted in Figures 4 and 5 where “variable wind” models are the same as the “simulated ablation” models in Figures 2 and 3. The temperature responses are respectively 21 and 19% less for the “constant wind” cases than for “variable wind”. The effect of wind variations, and especially the temperature-wind correlation, is therefore to increase ablation at higher temperatures.

The regression lines for the “variable wind” simulation and observed ablation are not in perfect agreement, suggesting that the “variable wind” simulation still does not take full account of wind variations. This may be because the model uses daily means of temperature and wind speed for the simulations and their product might not fully reflect the actual temperature-wind associations in Föhn events. This could be avoided in future by calculating sensible-heat flux with sub-daily data for wind and temperature.

Fig. 4. Ablation-temperature regression lines for different wind conditions at Nordbogletscher.

Fig. 5. Ablation-temperature regression lines for different wind conditions at Qamanàrssûp sermia.

The present results demonstrate the quantitative importance of the temperature—wind correlation for ablation but research on boundary-layer processes is required to understand the mechanisms of Föhn events.

Effect of snow cover

The observed ablation data mainly refer to ice ablation, so it is difficult to assess directly the effects of snow cover. However, the effect was simulated by re-running the energy-balance model with artificial assumptions about the glacier surface, i.e. ablation was calculated for notional ice and snow surfaces. The resulting ablation-temperature lines are plotted in Figures 6 and 7 where the “trace of snow” lines are the same as the “simulated ablation” lines in Figures 2 and 3.

Fig. 6. Ablation-temperature regression lines for different surface conditions at Nordbogletscher.

Fig. 7. Ablation-temperature regression lines for different surface conditions at Qamanàrssûp sermia.

Snow is fairly rare at the two sites, i.e. with average frequencies of only 4 and 3 d/month, respectively, so that “ice” and “trace of snow” simulations are similar. The fact that the “trace of snow” lines do not coincide with the “observed ablation” lines suggests that the simulations still do not take full account of the effects of snow. By comparison, assumption of a complete snow cover reduces ablation dramatically. For example, at an air temperature of 5 deg, snow ablation is nearly 60% less than ice ablation, while the difference drops to less than 50% at 10 deg. This partly reflects the high albedo and low surface roughness assumed for snow surfaces but also the greater difficulty in maintaining snow surfaces at the melting temperature compared to ice surfaces. The model still assumes an identity between melting and ablation that may not be true for a thick snow cover, e.g. refreezing of melt water as ice lenses or superimposed ice could further reduce snow ablation compared with ice ablation.

Conclusions

Ice ablation at Nordbogletscher and Qamanàrssûp sermia is related to air temperature by a two-parameter equation which explains about one-half of daily ablation variance.

The ablation–temperature relation can be explained in terms of energy balance whereby the biggest contribution to the temperature response of ablation is provided by sensible-heat flux with smaller contributions from radiation and latent-heat flux.

Under Föhn conditions, high wind speeds are associated with high temperatures which increases the sensible-heat flux at high temperatures.

A complete snow cover reduces ablation dramatically compared with an ice surface at the same temperature, partly reflecting high albedo and low surface roughness of snow surfaces.

Acknowledgements

This paper is published by permission of the Geological Survey of Greenland. The field station at Nordbogletscher 1978–83 was partly funded by the European Economic Community (EEC) and by the Danish Energy Ministry, while the Qamanàrssûp sermia station was wholly funded by the Geological Survey of Greenland. The work at Nordbogletscher was led by P. Clement in the period 1980–83.