Introduction

In order to evaluate glacier ablation under a debris layer. Reference Nakawo and YoungNakawo and Young (1981) proposed a simple model which was successfully employed in analyzing experimental data. With this model, ablation under a debris layer can be estimated from meteorological variables when the thermal resistance of the layer is known. Since it is difficult to determine directly the thermal resistance of a layer of unknown material in the field, it was suggested that the surface temperature of the debris layer may be used for estimating the thermal resistance and consequently the ablation under the layer.

This paper presents the results of testing the validity of the proposed method by comparing estimated data with field measurements. The symbols used are denned in Table 1.

Table I.

Nomenclature

Model

The energy-balance equation at a debris surface, in which all the terms are taken to be positive downward, is given by

(1)

where

(2)

(3)

(4)

Assuming a steady temperature profile in the debris layer (i.e. a linear profile for a uniform layer), then

(5)

since the temperature at the ice-debris interface is 0°C and T _s is in degrees Celsius. Neglecting the variation of the stored heat in the layer, and assuming no conduction of heat into the ice beneath, then

(6)

When condensation takes place, it is assumed that e _s is equal to the saturation vapour pressure, which is a function of T _s. As long as the debris surface is wet, this assumption is also made for periods when evaporation occurs. For a dry surface, on the other hand, e _s is assumed to be equal to e_a .

By combining Equations (1) through (6) and eliminating T _s (and e _s with the above assumptions), one can estimate r for a given R when F (or G and A if α is known), u _a, T _a, p, and e _a are provided. This was demonstrated by Reference Nakawo and YoungNakawo and Young (1981). In most cases in the field, however, the value of R is unknown.

When T _s is given instead, F, H, and E can be estimated (Equations (2) through (4)), allowing R to be determined by combining Equations (1) and (5). Once R is determined, r can be estimated for other periods using the procedure mentioned above. This is the method to be tested.

Experimental data

Experiments were carried out at Peyto Glacier (lat. 51° 41′N., long. 116° 33′W.) in the Rocky Mountains, Alberta, Canada from 20 to 22 August 1979. Meteorological variables during the measurement period are summarized in Table II. The data were collected using the procedures reported by Reference Nakawo and YoungNakawo and Young (1981).

Table II.

Meteorological variables during the experimental period

The ablation rate under debris layers was observed at six plots prepared artificially with debris materials collected from the supraglacial debris of the glacier. Each plot was 0.3 m square with a layer thickness h given in Table III. The ablation at the plots during a given period was determined by measuring the increase in the relative distance between the debris surface and a taut string installed over the plots. The results are also compiled in Table III.

Table III.

Meteorological variables during the experimental period

Surface temperature was measured by a thermistor inserted within a few millimetres of the surface of the debris. This measurement was made only twice in the daytime, but this is considered to be satisfactory for a test of the method as the weather was very stable during the experiments (e.g. atmospheric pressure was almost constant at 803.1 ± 1 mbar). The observed values for T _s of the measurements are shown in Table IV.

Table IV.

Values for the meteorological variables when the surface temperature was measured

Test results and discussion

The albedo of the layers was not determined. The debris material had a relatively dark colour and its albedo, when dry, was considered to be about 0.1 to 0.2 (Reference PenndorfPenndorf, 1956; Reference GeigerGeiger, 1961). For a wet surface, the albedo decreases by about 20% (Reference GeigerGeiger, 1961), but it was still considered to be in the range of 0.1 to 0.2.

By substituting the values of T _s (Table III) and meteorological variables (Table IV) into Equations(2) through (5), and combining with Equation (1), R was estimated for each plot. The estimated values are shown in Table III. Uncertainty in R was caused both by the uncertainty in the albedo and by the difference in the two estimates for 21 and 22 August. It should be noted that E in Equations (1) and (4) was assumed to be zero in the calculation for plot F because its surface was dry. Thermal conductivity K_m estimated from these R values was in the range 1.4 to 2.6 W m ⁻¹ deg ⁻¹ m (Table III) which are comparable with the values obtained for various soils (e.g. Reference KerstenKersten, 1949; Reference Penner, Penner, Johnston and GoodrichPenner and others, 1975; Reference JumikisJumikis, 1977).

The ablation rate was calculated using the data on meteorological variables given in Table II, and assuming α = 0.1 and α = 0.2. The results are plotted against R in Figure 1 (a) for the first two periods and (b) for the latter two periods. The solid and dashed lines are for wet and dry surfaces respectively in the daytime. The short dashed lines correspond to the estimates for night time. The observed ablation rates in Table III are also plotted in Figure 1 using the R values given in Table III.

Fig.1.

Ablation rate versus thermal resistance during 20–21 August(a)and 21–22 August(b).Solid and open circles are observed data during night time and day time respectively. Short dashed lines show the estimation from meteoro logical variables for night–time. Solid and dashed lines represent the calculation for wet and dry surface respectively during daytime.

The agreement between calculation and observation is fairly good, although there are some discrepancies. It is considered that the disagreement could be attributed to uncertainty in the estimates of R because there were few measurements of T _s and there was the uncertainty as to whether the temperature was in a steady state at the time of the observations. The errors involved in the measurement of T _s could also cause an error in the determination of R, particularly when R is large. For plots E and F, for example, a 0.5 deg difference in T _s would result in 10 × 10 ³ and 5 × 10⁻³ deg W⁻¹ difference respectively in the value of R when the modified T_s is applied through Equations (1) to (5).

Another source of disagreement between the calculated and the observed data is the uncertainty in the value of /β. The value of 4.89 J m⁻³ deg⁻¹ is an average compiled by Reference NaruseNaruse and others (1970) from the data for β obtained at various surfaces of glaciers, snow fields, and artificial basins. The original data for β scattered in a range of ± 1.16 J m⁻³ deg⁻¹ around the mean value. Owing to the presence of the experimental plots, the surface roughness of these plots on Peyto Glacier was greater than that of a natural glacier surface. This would result in a larger value of β than for a natural surface. The value of β at the plot could therefore have been larger than 4.89 J m⁻³ deg ⁻¹. Using a larger value of β would result in a larger ablation rate for a given R; if R is large, however, an increase in the value of β has little effect. The value of β is also dependent on wind stratification. Log–linear profiles of wind speed and temperature were found applicable at the glacier (Reference DerikxDerikx, [1975]; Reference Munro and DaviesMunro and Davies, 1977, Reference Munro and Davies1978). In the present experiments advection could have played an important role in heat exchange at the surfaces of the plots, since the area of the plots was small. However, the determination of the value of β taking the advection term into consideration is a very complex problem.

Nonetheless, the general agreement between the calculated and observed values suggests that glacier ablation under a debris layer can be predicted from meteorological and surface temperature measurements. To obtain a good prediction, it is recommended, as pointed out by Reference KrausKraus ([1975 ]), that special attention be paid to surface roughness which is sometimes very large at stagnant areas near termini of glaciers (see e.g. Reference Iwata, Iwata, Watanabe and FushimiIwata and others, 1980). A continuous record of surface temperature as well as observations on temperature profile in the debris layer would also improve the prediction.