1. Introduction

Glacier termini are commonly characterized by the presence of large quantities of debris, existing both within and on top of the glacier ice. The surface debris cover largely controls the rates of ice melt which, in turn, affects the mode of formation of glacially deposited landforms. The purpose of this paper is to investigate the effect of surface debris on rates of ice melt.

Quantitative assessment of ablation rate under a debris layer can also be significant for studies of glacier mass balance (Reference InoueInoue 1977), glacier dynamics (Reference GlazyrinGlazyrin 1975, Reference NakawoNakawo 1979), and glacial history (Reference BondarevBondarev 1961, Reference KiteKite and Reid 1977, Reference WhalleyWhalley 1979). It is an essential factor in any understanding of the formation and decay of dirt cones on glacier surfaces (Reference SharpSharp 1949, Reference BoultonBoulton 1967, Reference DrewryDrewry 1972). It is possible, furthermore, that it could lead to useful application of artificial ablation control and be adopted for engineering and hydrological purposes (Reference Grove, Grove, Aidun and KingeryGrove and others 1963, Reference Williams and GoldWilliams and Gold 1963, Reference HiguchiHiguchi 1973)

The ablation rate under a debris layer is a function of external variables including radiation and air temperature, as well as physical characteristics of the layer such as thickness, albedo, and thermal conductivity. The effect of the debris layer on ablation of glacier ice should be studied in association with these variables

Observations have been made to evaluate the effect of a debris layer on ablation of glacier ice, snow, and lake ice (Reference østremØstrem 1959, Reference WijngaardenWijngaarden 1961, Reference BakerLoomis 1970, Reference Moribayashi and HiguchiMoribayashi and Higuchi 1972, Reference Small and ClarkSmall and Clark 1974, Reference FujiiFujii 1977). These studies suggest that a very thin debris cover may accelerate melting, while an increasing thickness of debris inhibits melting. Few studies, however, have considered the effects of both the external variables and the physical properties of the debris material. Reference KrausKraus (1971) proposed a comprehensive theory for the ablation under a debris layer, taking various external variables into consideration. To the present authors' knowledge, however, the theory has not been tested against observed data

In the past the major difficulty in examining theoretical prediction has been lack of reliable data. It is difficult to determine, in particular, the physical properties of debris material in the field. Artificial debris layers were therefore prepared using a known material in order to examine the effect of these layers on ablation of glacier ice. Ottawa sand (American Society for Testing and Materials C-109) was chosen as the material since many of its physical properties are known.

2. Experiments

The experiments were carried. out at Peyto Glacier (51° 41'N, 116° 33'W) in the Rocky Mountains, Alberta, Canada, from 20 to 22 August 1979. Six plots were prepared with dried Ottawa sand on relatively flat ice near the snout of the glacier. They were aligned perpendicular to the prevailing wind direction and spaced about 0.1 m apart. Each plot was 0.3 m square and of thicknesses h, given in (Table I). The site was levelled so that, initially, the surface of each plot was horizontal.

Table I Artificial Debris Layer At Experimental Site Site

Precautions were taken so that the debris surface merged smoothly with the surrounding ice surfaces, but this was not successful because of local irregularities and general inclination (7° leeward) of the original glacier surface. As the experiment progressed, moreover, differential ablation accentuated the irregularities between the debris surface and the surrounding ice surface. At plots E and F the debris surface, which was originally prepared to be horizontal, including gradually towards the south as ablation proceeded. The inclination was 17° and 9° at E and ⌈, respectively, at the end of the experiments; no slant developed at the other plots.

The dry density of the Ottawa sand γ_d was determined to be 1 560 ± 120 kg m^-3 when the sites were prepared (evening of 20 August). On the morning of 21 August, however, the debris surface appeared to have become saturated with water at plots A, B, C, and D. This continued throughout the observation period and the water content (ω = mass of water W _w/mass of dry sand W _d) for each plot was determined at the end of the experiments (Table I). The water seemed to be distributed uniformly in the debris at each plot. The debris surface was apparently dry at plots E and F; however, a wet layer was observed near the bottom of the debris material at both plots. A distinct boundary was found between the dry layer near the surface and the wet layer near the bottom. The thickness of the dry layer h _d as well as the bulk water content ω were measured at both plots (Table I).

Ablation during a given period was determined by measuring the increase in the distance between the debris surface and a string installed horizontally over the site at the beginning of the experiments. For each measurement five readings were taken at marked locations within each plot. The ablation of the bare ice beside the plots was measured by the same procedure.

Atmospheric pressure p, air temperature T _a, and humidity were continuously recorded at the base camp established on a lateral moraine about 600 m from the site. The temperature and humidity readings were correlated with the data, measured with an Assmann ventilated psychrometer at the site to take into account the differences between conditions at the base camp and those at the site. Incoming short-wave radiation was measured with a Sol-A-Meter (Matrix Inc.), which provides a read-out of total integrated insolation. Readings were taken every 1 to 3 h during the day. In addition, wind speed u _a, cloud amount a _c, and cloud type were recorded.

3. Ottawa sand

Ottawa sand (ASTM C-109) is a typical silica sand of medium grain size. The average grain diameter is 0.4 mm and more than 90% by weight falls into a range of 0.2 to 0.6 mm in diameter (Reference BakerBaker 1976). Specific gravity and specific heat of dry Ottawa sand are 2 650 kg m^-3 and 0.84 J g^-ldeg^-1 respectively(Reference JumikisJumikis 1977).The specific heat capacity of the debris layer C _w is shown in Table I.

The thermal conductivity of Ottawa sand K _m was measured intensively by Reference KerstenKersten (1949) for a variety of water contents ω, dry densitiesγ_d, and temperatures. His results for γ_d = 1 560 kg m^-3 at 4°C are plotted against water content ω in (Fig. 1). Reference Slusarchuk and FoulgerSlusarchuk and Foulger (1973) also measured conductivity at 4°C for extreme cases of water content, but their dry density value was larger than that obtained at the plots on the glacier. Their results were therefore corrected for γ _d = 1 560 kg m^-3 by a relation between K _m and γ_d given by Reference KerstenKersten (1949) and are presented in (Fig. 1).

Fig. 1. Thermal conductivity of Ottawa sand for γ d = 1 560 kg m^-3 at 4°C versus water content.

The variation of K _m with increase in ω is very small when ω <0.01. For ω> 0.01, on the other hand, K _m increases rapidly; the trend may be represented by

(1)

where K _mis given by W_m ^-1deg^-1. This form of relation between K _m and ω for relatively large water contents was established by Reference KerstenKersten (1949) for various kinds of soils.

Thennal conductivity K _m at plots A. B, C, and D was estimated by means of Equation (1) using the bulk water content ω given in Table I. The water content in the dry layer at plots E and F was assumed to be 0.01, since the variation of K _m with ω is very small for ω>0.01. This value was used to calculate the water content of the wet layer at the two locations. The value of K _m was obtained for each layer using Equation (1). The bulk conductivity for the sites was then estimated using a series model of two layers. All the estimated K _m values for each plot are given in Table I, in which the thermal resistance R (= h/K _m) for each location is also shown

The albedo of Ottawa sand was measured later for wet and dry surface conditions in relation to solar altitude. It decreased for dry (wet) conditions uniformly (almost linearly) from 0.51 (0.31) to 0.32 (0.28) as solar altitude increased from 15 to 50 deg, this range in solar altitude being encountered during the observation period. The results compared favourably with those reported by Reference Büttner and SutterBüttner and Sutter (1935) and Reference FriedrichFriedrich (1965).

4. Model

The energy balance equation at a debris surface is given by

(2)

where C, F, H, and E are conductive heat flux into the debris, radiation flux, sensible heat flux, and latent heat flux, respectively. All the terms are taken to be positive downward. The energy balance equation at the debris/ice interface is

(3)

where M is the heat used for ice ablation during a unit period, and C' and C" are conductive heat fluxes from the debris and towards the ice, respectively, again taken to be positive downward. The difference between C and C' gives the change in heat stored in the debris layer. M is given by

(4)

where L _f is the latent heat of fusion (334 J g^-1), ρi is the density of ice (assumed to be 900 kg m^-3), and r is the ablation rate.

Three assumptions are made, that: C"= 0, which is valid for a temperate glacier and is certainly valid for the site during August; C = C', stored heat in the debris layer is constant; and the temperature in the debris layer is in a stationary state: a linear profile for a uniform debris layer. Using these assumptions

(5)

and

(6)

where h is the debris thickness, K _m is the thermal conductivity, R is the thermal resistance, and T _s is the surface temperature of the debris in °C.

The following expressions were assumed for F, H, and E (Reference SuttonSutton 1953, Reference GeigerGeiger 1965, Reference MunnMunn 1966):

(7)

where α is the surface albedo of the debris, G is the global radiation, A is the atmospheric radiation, and σ is the Stephan-Boltzmann constant (5.67 x 10^-8 W _m ^-2 K⁴);

(8)

where β is the coefficient of heat transfer (4.89 ± 1.16 J m^-3 deg^-1 compiled by Reference Naruse, Dura and KojimaNaruse and others (1970) from the data for β obtained at various surfaces of glaciers, snow fields and artificial basins), u _a is the wind speed, and T _a is the air temperature at the site; and

(9)

where L _e is the latent heat of evaporation (2 494Jg^-1), p is the atmospheric pressure, c _p is the specific heat capacity of air (1.0 J g^-ldeg^-1), e _a is the vapour pressure in air, and e _s is the vapour pressure at the surface.

With Equations (4) to (9) one can estimate r for a given R when the external variables G, A, u _a, T _a, p, e _a, and e _s are given. This model is rather more practical than the one proposed by Reference KrausKraus (1975), although the basic assumptions and the expressions are similar.

5. Analysis

Values of the external variables for the observation period are compiled in Table II. Reflected radiation is a function of albedo, which is dependent on solar altitude and, accordingly, on time. Total reflected radiation for each period was estimated by summing each reading of global radiation, which had been multiplied by albedo at the time of the reading. Average reflected radiation in Table II was obtained by dividing the total reflected amount by the length of each period. Nominal average albedo for each day depends on cloud cover and other factors because albedo is time-dependent. The values calculated for 21 August, however, were similar to those calculated for 22 August (0.288 and 0.287 for wet surface on the 21st and 22nd, respectively, and 0.374 and 0.372 for dry surface on the 21st and 22nd, respectively).

Atmospheric radiation A was estimated by a Brunt-type empirical equation proposed by Reference KondoKondo (1967)

(10)

where T _a is in degrees Celsius, e _a is in 10³P_a and C is a parameter determined by e_a, cloud amount a_c, and cloud type. The estimated results for the periods of daylight are also tabulated in Table II. As cloud amount and type were not observed during night time, A was assumed to be as shown in parentheses in Table II.

5.1 Wet surface

It was assumed that the air at the debris surface was saturated at T _s for a wet surface, that ² was equal to 4.89 J m^-3deg^-1 and that external variables had the values given in Table II. The dependence of the ablation rate r on thermal resistance R was thus determined, using Equations (4) to (9), by eliminating T _s. The results are shown by solid lines in (Fig. 2)(Fig. 3). They indicate a general asymptotic decrease of r as R increases.

Fig. 2. Ablation rate versus thermal resistance for 20–21 August. Calculations for wet surface with 8 = 4.89 J m^-3deg^-1 shown by solid fines, 8 = 4.89 ± 1 .16 J m^-3deg^-1 shown by dotted Tines; broken Tine shows calculation for dry surface.

Fig. 3. Ablation rate versus thermal resistance for 21–22 August. Calculations for wet surface with B = 4.89 J m^-3deg^-1 shown by solid lines, B = 4.89 ± 1.16 J m^-3deg^-1 shown by dotted lines; broken line shows calculation for dry surface

For a given R, one can also estimate each flux component. F is essentially a constant because |T _s| ≪ 273 for a realistic range of R. H and E, on the other hand, decrease from positive to negative with increase in R. For a small R (very thin debris layer), T _s being close to the ice temperature (0, °C), heat is transported from the ambient air to the debris surface (positive H and E (condensation)), leading to a large ablation rate. For a large R, M (and hence r) being close to zero owing to the insulation effect of the layer, the energy is almost balanced at the debris surface by the three flux components F, H, and E. Consequently, H and E become negative (see Equation (5)). H = 0 when T _a=T _s and E = 0 when e _a = saturation vapour pressure for T _s.

Dotted lines in (Fig. 2)( 3) show the estimated ablation rates with upper and lower B values of 4.89 + 1.16 J m^-3deg^-1 and 4.89 - 1.16 J m^-3deg^-1, respectively. The larger β value predicts a larger ablation rate. It may be seen that the effect of β value (deviation of the dotted lines from the solid line) decreases as R increases for night time. The same tendency may be observed during the day, but the deviation becomes zero at an R value where H + E = 0. As R increases further, the deviation becomes larger, but in the opposite sense; the larger β value is associated with a smaller ablation rate.

Table II External Variables During Observation Period

5.2 Dry surface

The assumption of saturated vapour pressure at the surface, made for the above calculation, would be valid for wet-surface conditions; it would be valid for apparently dry surfaces, as well, when condensation occurs. This was the case during the night time. It was considered, therefore, that the ablation rate for a dry surface during night time was the same as for a wet surface. During the day time, however, it would be reasonable to assume for a dry surface that e _s = e _a, i.e., that E = 0.

The calculated ablation rate for a dry surface (with this assumption) is shown by dashed lines in (Fig. 2)(Fig. 3). The value of β was taken to be 4.89 J m^-3deg^–1 in the calculations. When values of β = 4.89 ± 1.16 J m^-3deg^-1 were used, the variation in r for given R was almost the same as for the calculations for wet surfaces.

The value of F for dry surfaces is smaller than that for wet surfaces because of the larger albedo of a dry surface. The predicted ablation rate is therefore smaller for dry surfaces than for wet surfaces, when R is small, where condensation occurs for both conditions. At a large R value, however, the ablation rate for dry surfaces is predicted to be larger than for wet surfaces. For large R, the difference in evaporation loss between wet and dry surfaces is too large to be compensated by the difference in F for the two conditions.

6. Discussion

Observed ablation rates at the site are plotted in (Fig. 2)(Fig. 3) using solid circles for the night-time observations and open circles for day. Ablation rates of bare ice beside the plots are also shown with solid (night time) and open (day time) arrows as reference. The bare ice surface was dirty and its albedo observed to be 0.32 ± 0.02 near noon when solar altitude was about 50°.

The predicted ablation rate compared reasonably well with the observed results. (Note that results atplots E and F during day time are to be compared with the broken lines, not with the solid lines.)

For small R values a larger ablation rate than predicted was observed at plots A, B , and C during day time, suggesting that the actual value of β could be slightly larger than the value used for the calculations. This is reasonable since surface roughness is greater at the site than at the natural surface owing to the presence of the plots. The predicted values for plots E and F during day time on 22 August are also smaller than the observed ablation rate. This discrepancy is probably caused by an increase in inclination of the debris surface towards the south, as mentioned in section 2.

The general agreement between the predicted and observed results suggests that one can predict ablation if the external variables and thermal resistance of the debris layer are known. Determination of thermal resistance, however, is a most difficult problem for an unknown debris material in the field.

The model described in section 4 could also be used to predict surface temperature T _s by eliminating M from Equations (5) and (6). Estimation was made using the data for external variables obtained between 11.15 and 13.45 h local time on 21 August. The predicted surface temperature is shown in (Fig. 4a) with solid (wet surface) and broken (dry surface) lines. The ablation rate was also estimated for this rather short period by means of the same procedure (Fig. 4b). Observed ablation rates during this period and surface temperature measured at 12.40 h local time are also plotted for comparison. The estimated and observed values agree favourably for both surface temperature and ablation rate. It should be possible, therefore to estimate thermal resistance from surface temperature measurements using Equations (5) to (9).

Fig. 4. Surface temperature (a) and ablation rate (b) versus thermal resistance from 11.15 to 13.45 h local time, 21 August; surface temperature measured at 12.40 h local time. Calculations for wet surface with β = 4.89 J m^-3deg^-1 shown with solid lines; calculations for dry surface with S = 4.89 J m^-3 deg^-1 shown with broken lines; calculations for both wet and dry surfaces with 8 = 4.89 ± 1.16 J m^–3deg^–1 shown by dotted 1ines

The estimates were made using observations for the noon period on 21 August. Estimated thermal resistance is compared in (Fig. 5) (solid circles) with the resistance given in Table I. Thermal resistance was obtained also from the data on surface temperature and ablation rate using Equations (4) and (6), but this method is not of practical interest because ablation rate is not usually known and is the term to be determined. Values obtained by this method are also compared in (Fig. 2) (open circles) with the values given in Table I. Thermal resistances obtained by the three different methods compare reasonably well with each other.

Fig. 5. Comparison of thermal resistances at sites estimated from surface temperature with values obtained from the physical constants of Ottawa sand. Estimations made from ablation rate (open circles) and external parameters (solid circles)

It is fairly easy to determine the surface temperature of a debris layer over a wide area by means of remote sensing. One can estimate, therefore, the thermal resistance and consequently the ablation rate under the debris layer from a knowledge of external variables only.

The model now proposed neglects the change of stored heat in a debris layer, although it could be significant for a thick layer with a large heat capacity. Neither is the effect of rainfall included. Significant heat could be transported by water percolation into a porous, thick debris layer (Reference YoshidaYoshida, unpublished). For a very thin, or scattered, debris layer on a glacier surface, albedo must be considered as a variable. Such a situation is outside the scope of the present paper, but it is an important consideration since accelerated ablation takes place when this condition prevails (Reference Megahan, Meiman and GoodellMegahan and others 1970, Reference Higuchi and NagoshiHiguchi and Nagoshi 1975. Further study will be necessary on this subject.

Acknowledgements

The authors are grateful for the assistance given in the field by N. Maeno and H. Narita of the Institute of Low Temperature Science, Hokkaido, Japan, and are indebted to L.W. Gold of the Division of Building Research, National Research Council of Canada, for reviewing the manuscript.

This paper is a contribution from the Division of Building Research, National Research Council of Canada, and is published with the approval of the Director of the Division.