Introduction

The Tuomuer mountain district is the largest glacierized area in the Tien Shan, China, with a total of 629 glaciers covering an area of 3850 km² (Reference Su and GaoSu and Gao, 1985; Reference HanHan, 2007). A striking glaciological feature of this region is the development of many huge dendritic valley glaciers (called Tuomuer-type valley glaciers by Chinese glaciologists for their unique characteristics). Another impressive characteristic of these glaciers is a heavy supraglacial debris cover in the ablation zone. The data compiled in the Tien Shan volume of the Chinese Glacier Inventory (Reference Xie, Ding, Liu, Jiao, Wang and WangXie and others, 1987) show that >60% of the ablation area of Tuomuer-type valley glaciers is covered by supraglacial debris. Since, due to thermal isolation, a debris layer with thickness greater than ∼20 cm may inhibit melting of ice underneath (Reference ØstremØstrem, 1959; Reference LoomisLoomis, 1970; Reference FujiiFujii, 1977; Reference Mattson, Gardner and YoungMattson and others, 1993), the extensive debris entrainments on Tuomuer-type valley glaciers (thickness typically 20–100 cm) greatly suppress melting and prevent rapid glacier wastage. Reference Wang, Liu, Ding, Xie, Ding, Liu, Jiao, Wang and WangWang and others (1987) reported that, at locations around 4000 m a.s.l. on West Qiongtailan Glacier, ice ablation was 20–30% less under a 10 cm thick debris cover, and about 50% less under a 20 cm thick debris cover, than under a clean ice layer. This implies that, compared with glaciers covered by clean ice and snow, glaciers with extensive supraglacial debris cover may be more stable against the warming climate, in terms both of dimensions (length and area) and of meltwater production.

The hydrology of this debris-covered area is, however, complicated by the existence of many ice cliffs and exposed supraglacial ponds or lakes. Backwasting of ice cliffs can be significant during the ablation period due to more intense melting over the bare-ice slopes than under the debris mantle. Melting of ice cliffs may therefore be an important meltwater source in the debris-covered area. Reference Sakai, Nakawo and FujitaSakai and others (1998) reported that ice cliffs, with only 1.8% of the areal coverage, contributed about 69% of the total ablation within the debris-covered area on Lirung glacier, Himalaya. A similarly significant mass loss due to rapid backwasting can be found in numerous debris-covered glaciers worldwide (e.g. Tasman Glacier, New Zealand (Reference Purdie and FitzharrisPurdie and Fitzharris, 1999); Donjek Glacier, Yukon, Canada (Reference JohnsonJohnson, 1992); and Holmströmbreen, Svalbard (Reference Schomacker and KjærSchomacker and Kjær, 2008)). In the Tuomuer mountain district, however, quantitative knowledge of ice-cliff backwasting over debris-covered glaciers is limited.

This paper describes quantitative assessments of ice-cliff backwasting in the debris-covered area on Koxkar glacier on the basis of energy-balance modeling. Field measurements of the backwasting rate of ice cliffs were carried out for model calibration and validation on 38 ice cliffs in the debris-covered area of Koxkar glacier between 5 August and 5 September 2008. Our main aims here are to assess the predictability of backwasting rate using the energy-balance model and to quantify the annual backwasting rate on the basis of model calculation.

Study Area

Koxkar glacier (41°42′ -41°53′ N, 79°59′ -80°10′ E; Fig. 1) is a typical Tuomuer-type glacier (Reference Su and GaoSu and Gao, 1985) originating from Koxkar mountain (6342 m a.s.l.), and flows southeast to the terminus at 3060 m a.s.l. The glacier is 25.1 km long and covers an area of 83.56 km². The equilibrium line occurs at 4300 m a.s.l. in the icefall, from the foot of which a 15.5 km long, debris-mantled glacier tongue appears. The supraglacial debris covers ∼15.6 km², accounting for 83% of the total ablation area. Debris thickness ranges from <0.01 m on the upper reach of the ablation area and on ice-cliff slopes to >3.0 m near the glacier snout (Reference Han, Ding and LiuHan and others, 2006).

Fig. 1. Location of Koxkar glacier, Tuomuer mountain, China.

Figure 2 is a map of the ice cliffs and supraglacial ponds/lakes, based on a panchromatic Système Probatoire pour l’Observation de la Terre (SPOT) image (5 m resolution) taken in late April 2006. The ice cliffs are mainly exposed on the debris-covered area 1.5–10 km up-glacier from the terminus (3060 ma.s.l.) and tend to be developed in the medial part of the glacier. This may be due to higher ice speed in the centre of the glacier than at the margins (Reference Benn and EvansBenn and Evans, 1998). Calculation based on Figure 2 shows that the projected area of the ice cliffs over the horizontal plane was ∼221 330 m² by the end of April 2006, accounting for about 1.13% of the total debris-covered area.

Fig. 2. Distribution of ice cliffs and supraglacial ponds/lakes in the debris-covered area on Koxkar glacier, mapped from a panchromatic SPOT image of 27 April 2006.

The mean annual air temperature observed near the terminus of the glacier is 0.77°C, while the mean summer temperature is as high as 7.74°C (Reference HanHan and others, 2008). The melt season over Koxkar glacier starts in May and continues for 6 months until the end of October. Mean annual precipitation near the terminus is ∼608 mm, of which >80% occurs during the ablation period (Reference HanHan and others, 2008). The melt runoff measured near the snout is ∼102.86 × 10⁶ m³ a⁻¹, of which 93.6% occurs during the melt season (Han and others, in press).

Field Observations

Backwasting, defined as the lateral retreat of steep ice-walls or ice-cored slopes (Reference PickardPickard, 1984; Reference KrügerKrüger, 1994; Reference Krüger and KjærKrüger and Kjær, 2000), is the term commonly used to quantify the wastage of a sloped ice mass with debris mantle. On Koxkar glacier, ice cliffs for backwasting measurements were selected so that the main ice-cliff characteristics (e.g. altitudinal distribution, scale, orientation and inclination angle) were represented. The elevations of observed ice cliffs ranged from 3180 to 3578 m. Generally a fixed outstanding boulder behind the top edge of the ice cliff was painted as a benchmark so that a tape could be used to measure the distance from the ice cliff to this benchmark (Fig. 3). Alternatively, when such a fixed benchmark was not available, a graduated ablation stake was inserted perpendicularly into the bare-ice slope so that the melt rate of the ice cliff could be measured. The backwasting rate was then calculated from the perpendicular melt and slope angle of the cliff. The slope angles for all selected ice cliffs were measured using a geological compass. Since the bare-ice slope of an ice cliff is often presented as a curved surface, slope angle measurements were conducted over different parts of the ice cliffs to derive mean values. It was observed that slope angles of measured ice cliffs were 30–64°, with a mean of 46.4°. Azimuth angles of all 38 ice cliffs were measured near the center of the top edge using a geological compass. The observed ice cliffs were found to face in a variety of directions, the number near north being larger than near south. This concurs with a previous study (Reference Han, Ding and LiuHan and others, 2006). In addition, the face area, mean height and mean width of each ice cliff were recorded using a tape measure. Backwasting measurements of all selected ice cliffs began between 5 and 11 August. Repeat measurements were carried out between 31 August and 5 September, allowing mean backwasting rates for each ice cliff to be measured over at least 25 days. To facilitate model calculation, the vertical ice melt (Fig. 3) is also employed because the analysis of energy-balance elements is generally computed over a unit horizontal area. The observed vertical melt rate can be inferred from the observed backwasting rate through a trigonometric transformation.

Fig. 3. Schematic diagram of the measurements of ice-cliff backwasting.

The meteorological variables were captured by three automatic weather stations (AWSs) at 3212 m a.s.l. (site A₁), 3433 m a.s.l. (site A₂) and 3730 m a.s.l. (site A₃) (Fig. 1). Sites A₁ and A₂ are covered by continuous supraglacial debris, and the AWS at site A₃ was installed on narrow clean ice with debris cover distributed on the western and eastern sides. The meteorological sensors on each AWS are listed in Table 1, and mean weather conditions during the observation period are given in Table 2.

Table 1. AWS meteorological sensors

Table 2. Statistics of meteorological variables measured by AWS on Koxkar glacier from 5 August to 5 September 2008

Energy-Balance Model

The energy-balance equation for a unit horizontal area over an ice-cliff slope can be written as

(1)

where Q _m is heat for vertical ice melt (W m⁻²), I _n and L _n are net short- and longwave radiation fluxes (W m⁻²), and H and LE are turbulent sensible and latent heat fluxes (W m⁻²). Heat conduction into the glacier ice and the specific heat of rainfall are neglected since they are small compared with the above terms. All fluxes are taken positive when they are toward the ice-cliff face.

The net shortwave radiation flux on an ice-cliff slope in a unit horizontal area is given by

(2)

where I _s is the direct solar irradiance from the sky (W m⁻²), D _s is the diffuse sky irradiance (W m⁻²), D _t is the diffuse irradiance from the surrounding terrain (W m⁻²), α is the surface albedo of the ice-cliff slope, taken to be 0.37 according to our observation at site A₃, and β is the angle of ice-cliff slope. Considering the shading effect of surrounding topography, the direct shortwave radiation reaching a slope can be computed by (Reference Garnier and OhmuraGarnier and Ohmura, 1968)

(3)

(4)

where I _b is the direct normal irradiance which is the part of the extraterrestrial radiation penetrating the atmosphere (W m⁻²), φ is the latitude of the slope (°), ω is hour angle, solar noon being zero and the morning being negative (°), A is the azimuth of the slope (°), δ is the sun’s declination, positive when the sun is north of the Equator (°), θ is the solar zenith angle, and H _s is the horizon angle (°) in the direction of solar beam. The horizon angle, defined as the angle subtended by the lines extending from a point to the zenith and to the summit of the surrounding terrain in a given direction (Reference Dozier and FrewDozier and Frew, 1990), was computed from the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) global digital elevation model (GDEM; 30 m resolution) in this study. The full solution of direct normal irradiance, I _b, involves determination of a number of transmittances of solar radiation due to absorption and/or scattering by ozone, gas, water and aerosols (e.g. Reference IqbalIqbal, 1983; Reference Chou and SuarezChou and Suarez, 1999; Reference Wong and ChowWong and Chow, 2001) and thus is derived indirectly in this study using global solar radiation observed at site A₁ or A₃ depending on the distance to the calculated ice cliff. It gives

(5)

where I _o is observed global solar radiation (W m⁻²), D _h is diffuse irradiance to a horizontal surface (W m⁻²) and h is the solar elevation (°). Following Reference Reindl, Beckman and DuffieReindl and others (1990), D _h is obtained using the diffuse fraction k _d and clearness index k_t (both dimensionless) by

(6)

(7)

(8)

(9)

(10)

where the extraterrestrial radiation I _E (W m⁻²) is given by

(11)

I _sc is the solar constant, taken as 1367 W m⁻², and E _o (dimensionless) is the eccentricity correction-factor of the Earth’s orbit. This is given by (Reference Wong and ChowWong and Chow, 2001)

(12)

where Γ is the day angle (rad) which is calculated as

(13)

where N is the day number of the year. The diffuse sky irradiance incoming to the ice-cliff slope, D _s in Equation (2), can be obtained by introducing the isotropic sky-view factor, V_d which accounts for the portion of the overlying hemisphere visible to a sloping surface (Reference Dozier and FrewDozier and Frew, 1990)

(14)

(15)

where H_φ is the horizon angle (rad) for direction φ. The sky-view factor for a given ice cliff was calculated such that the shading effects of local inclination of ice cliff and high-relief terrain at the glacier margin were better presented, while the effect of supraglacial topography around the ice cliff was loosely included due to the low resolution of the ASTER GDEM used in the calculation. Shortwave contribution from the surrounding terrain to the ice-cliff surface is

(16)

where α _t is the average surface albedo of the surrounding terrain, taken to be 0.24 according to Reference Han, Ding and LiuHan and others (2005).

Net longwave radiation can be split into incoming atmospheric longwave irradiance from the unobscured portion of sky, L _s, longwave irradiance from surrounding terrain, L _t, and outgoing longwave radiation, L _o, which gives

(17)

where all irradiances are expressed in W m⁻². L _s is determined based on the work of Reference Sridhar and ElliottSridhar and Elliott (2002):

(18)

where σ is the Stefan–Boltzmann constant (5.67 × 10⁻⁸ W m⁻² K⁻⁴), and e _a and T _a are the vapor pressure (kPa) and air temperature (°) at screen height (2 m) respectively. Since air temperatures at the observed ice cliffs were not measured, temperatures for each ice cliff were obtained indirectly, using temperature lapse rates calculated from the air temperatures recorded at the three AWSs. This treatment is validated by the fact that during the study period the hourly air temperatures at the AWS are highly correlated (coefficient of determination R ² = 0.99 between sites A₁ and A₂ and R ₂ = 0.96 between sites A₂ and A₃). The mean lapse rates during the observation period were found to be −1.29°C (100 m)⁻¹ between sites A₁ and A₂ and −0.52°C (100 m)⁻¹ between sites A₂ and A₃. The air temperatures at observed ice cliffs can therefore be interpolated from temperature records measured at the nearest AWS. Since ground surface temperatures were also not available, the longwave irradiance from surrounding topography can be adequately computed assuming the ground radiates as a black-body radiator at screen air temperature (Reference ColeCole, 1979), which gives

(19)

The outgoing longwave radiation from an ice-cliff surface is calculated by

(20)

where ε is the effective emissivity of the glacier ice, assumed to be 0.97 in this study, and T _s is the surface temperature of the ice cliff (°C), assumed constant at 0°C.

The turbulent heat fluxes were calculated using formulations developed by Reference AmbachAmbach (1986). The sensible heat flux was computed from temperature and wind speed:

(21)

where P is atmospheric pressure (kPa), and u is the wind speed derived from the nearby AWS (m s⁻¹). K _s is the exchange coefficient that, for a neutral boundary layer with logarithmic profiles for wind speed and temperature, is given by

(22)

where c_p is the specific heat of air at constant pressure (1004 J kg⁻¹ K⁻¹), k is von Kármán’s constant (0.41), ρ ₀ is the standard density of air (1.29 kg m⁻³), P ₀ is the standard atmospheric pressure (101.3 kPa), z is the height above the ice surface at which the mean meteorological variables for the ice cliffs were retrieved from the adjacent AWS (2 m in the present case) and z ₀ and z _0T are the roughness lengths for momentum and temperature profiles (m), respectively. The latent heat flux is given by:

(23)

where e _s is the vapor pressure at the ice-cliff face, assumed to be the saturated vapor pressure at 0°C (0.611 kPa), and K_L is the exchange coefficient for latent heat under the same assumptions as K _s. K_L is given by

(24)

where L is the latent heat of evaporation (2.514 × 10⁶ J kg⁻¹) and z _0e is the roughness length for a logarithmic profile of vapor pressure (m). The roughness length was calculated using Lettau’s (1969) model modified for high-relief terrain, by

(25)

where h _c is the mean height of the ice cliff (m), and λ _f is the frontal area density which is presented in this study as the degree of a slope exposed to the wind (dimensionless) and is calculated by

(26)

where W _D is the wind direction (°). z _0T and z _0e are assumed to be equal and are derived using Reference AndreasAndreas’ (1987) model,

(27)

where Re _* is the roughness Reynolds number. This is given by

(28)

where v is the viscosity of air (1.35 × 10⁻⁵ m² s⁻¹)and u _* is the friction velocity (m s⁻¹), estimated by

(29)

The heat for vertical ice ablation, Q _m, is calculated from the energy-balance model. The daily ice-cliff backwasting, r _b (cm d⁻¹), is obtained by

(30)

where L_f is the latent heat of phase change of ice (0.334 MJ kg⁻¹) and ρ _i is the density of ice (900 kg m⁻³).

Results and Discussions

Model validation

Figure 4 shows the relationship between estimated and observed backwasting rates, r _b, for all 38 ice cliffs during the observation period. In general, the estimate is satisfied with errors of ±1.96 cm d⁻¹ for r _b, with root-mean-square errors (RMSEs) of 0.99 cm d⁻¹, but the results are by no means perfect. A number of factors can lower the performance of the proposed energy-balance model (e.g. errors in surface albedo, turbulent fluxes, roughness length and field observations).

Fig. 4. Scatter diagram of estimated versus observed backwasting rates for all 38 ice cliffs during the observation period.

The surface albedo, α, is an important parameter that characterizes the reflectivity of the ground surface to the incident solar radiation. Since solar radiation generally contributes the majority of energy available for ice melt, determination of the surface albedo of glacier ice is of great importance in the model parameterization. In general, the albedo depends on the incident angle of the incoming radiation and surface properties. However, in areas with complex terrain and significant topographic shading, the numerical determination of ice surface albedo on the basis of incidence angle may be difficult. As demonstrated in Figure 5, no obvious relationship is found between the observed ice surface albedo and the solar elevation, and most observed albedo is distributed in the range 0.27–0.5, with a mean of 0.37. We supposed that the low correlation between ice surface albedo and solar elevation in the studied area may be due to the significant diffusivity of the surrounding terrain to the solar radiation. In our calculation the surface albedo was taken uniformly for all 38 ice cliffs. Note that the constant setup of the surface albedo in the model may overestimate the net shortwave radiation, and thus the backwasting rate, over the ice-cliff slope, especially for those slopes with northerly azimuth and large inclination angle, where the angle of incidence is acute. In the light of the model performance, however, this overestimation is acceptable.

Fig. 5. Relationship between solar elevation, h, and albedo of ice surface observed at site A₃ from 5 August to 5 September 2008. Days with snow-covered surface were excluded.

Turbulent fluxes are also important components in the energy-balance equation over the ice-cliff slope. By applying the standard procedure to estimate turbulent fluxes over complex terrain, the large uncertainty is the exchange coefficient which may change significantly with slope orientation. It was pointed out that when the wind is perpendicular to a smooth surface the exchange coefficient, or more exactly the drag coefficient, is one order greater than when it is parallel (e.g. Reference Thom and MonteithThom, 1975). In our model the effect of exchange coefficient with slope orientation was handled by introducing a topographic-dependent roughness length which varies with wind direction and slope angle. Even though Reference LettauLettau’s (1969) model cannot reproduce the effect of mutual sheltering for higher topography (Reference BottemaBottema, 1996), it is very simple and the relevant variables can be readily measured in the field or from satellite images, which may be important when applying the model in an area with limited data availability. The wind-tunnel experiments and practical trials showed that Reference LettauLettau’s (1969) model performs well (Reference PetersenPetersen, 1997; Reference DuijmDuijm, 1999), and it also runs reasonably in the proposed model according to our calculation.

Solar radiation and energy balance

On the basis of the model calculation for all 38 ice cliffs during the observation period, Figure 6 shows the dependence of daily direct shortwave irradiance for a unit sloped area on the slope orientation of the ice cliff, considering the effect of cloud and scatter in the atmosphere. It shows that, for any given slope angle, the maximum direct solar radiation over the ice-cliff surface occurs at the southwest orientation. When the azimuth increases or decreases from this direction, the direct shortwave radiation decreases, reaching a minimum at around north as expected. Generally, the highest insolation on a clear day is found on the slope facing due south. The shift in orientation receiving the maximum insolation in the present case may be due to the asymmetric distribution of the hourly global solar radiation during the study period. As shown in Figure 7, the observed global solar radiation was significantly larger in the afternoon than at the corresponding times in the morning; this increases the accumulated shortwave radiation reaching a slope with a westerly azimuth compared with a slope facing east, so that the western shift in orientation receives the maximum insolation. The larger proportion of direct solar radiation found at ice cliffs facing west rather than east (Fig. 6) is similarly explained. The steep contours in Figure 6 indicate that, to some extent, the variation of shortwave radiation may be more sensitive to azimuth angle than to ice-cliff inclination angle, and this is more true for ice cliffs facing west than for those facing east.

Fig. 6. Dependence of daily shortwave radiation for a unit sloped area on the slope orientation of ice cliffs, considering the effect of cloud and scatter in the atmosphere, mapped using calculated daily shortwave radiation for all 38 ice cliffs in the debris-covered area on Koxkar glacier from 5 August to 5 September 2008.

Fig. 7. Variations of mean global solar radiation (solid curve) during the study period compared with a normal distribution curve (dotted curve, σ = 2.1), showing the asymmetry of global solar radiation with time.

As well as direct solar radiation, diffusive shortwave irradiance from the atmosphere and the surrounding topography contribute significantly to ice-cliff melt. As shown in Figure 8, the proportion of diffusive sky irradiance to total shortwave radiation over the ice-cliff surface ranged from 43.5% to 62.2%, and diffusive ground irradiance from 4.0% to 19.6%. Due to the high frequency of cloudy hours in the debris-covered area, the quantity of direct solar beam arriving at the ice-cliff slopes varied significantly with coefficient of variation, C _v = 0.73. In comparison, diffusive shortwave irradiance shows less variation, with averages of 7.1 and 1.5 MJ m⁻² d⁻¹ for diffusive sky irradiance (C _v = 0.25) and diffusive irradiance from surrounding terrain (C _v = 0.49).

Fig. 8. Proportions of shortwave components reaching the ice-cliff slope for different azimuth slope angles, plotted based on calculation for all 38 ice cliffs from 5 August to 5 September 2008.

For clarification, the daily averaged heat-balance elements for ice cliff F5 (Fig. 1; elevation 3191 m, slope angle 55°, azimuth angle 292°) from 5 August to 5 September are shown in Figure 9, as an example of the proportion and variation of heat components for an ice slope. It is seen that the net shortwave radiation is the most important heat source for ice-cliff ablation, contributing about 76% of the total heat available for ice melt. Due to the frequent influence of cloud, solar radiation varies significantly from 4.2 to 14.3 MJ m⁻² d⁻¹ in the case of F5, and for all the studied ice cliffs it varies in the range 2.0–15.5 MJ m⁻² d⁻¹ in our calculation. Owing to the large differences between air temperature and ice-cliff surface temperature during the melt season, the sensible heat flux is the secondary contributor to ice-cliff ablation, being responsible for nearly 24% of total heat for ice-cliff wastage. The latent heat flux was, however, small during the study period, compared to the sensible heat flux. Likewise, net longwave radiation is not significant for ice-cliff melting. Daily mean net longwave radiation is found to be −2.6 MJ m⁻² d⁻¹ for ice cliff F5. Note that when air temperatures are low, as observed from 28 August to 5 September, heat loss due to net longwave radiation may be considerable. This was attributed to the significant drops both in diffusive sky and in diffusive ground irradiances reaching the ice-cliff surface.

Fig. 9. Heat-balance elements at ice cliff F5 from 5 August to 5 September 2008. Dates are month/day.

Backwasting rate for the melt season

On the basis of the above proposed energy-balance model, backwasting rates for all 38 ice cliffs during the 2008 melt season, i.e. 1 May–31 October, were estimated using observed meteorological records at the three AWSs. The procedures for model data preparation for the ablation period duplicated those for the experimental period 5 August–5 September. The assumption of a consistent 0°C temperature at the ice-cliff face was maintained to facilitate the calculation, though the surface temperature of the ice cliff may be lower than 0°C in early May and late October.

As an example, the daily variation of backwasting rates for ice cliff F5 is shown in Figure 10, together with interpolated daily mean air temperature at F5. It is observed that the fastest backwasting of F5 during the 2008 melt season occurs in early August, with a maximum rate of 9.6 cm d⁻¹. The backwasting is also retarded significantly in October due to a rapid drop in temperature. The relationship between backwasting rates and daily mean temperatures (Figs 10 and 11) suggests there is a fairly strong correlation between backwasting rate and temperature. This may allow backwasting rate to be estimated using statistical backwasting–temperature models (e.g. a positive degree-day model) for areas where meteorological observations are limited.

Fig. 10. Variations of estimated backwasting rate and daily mean air temperature at ice cliff F5 during the 2008 melt season. The black curve shows the variations in backwasting rate. The grey curve is the corresponding daily mean temperature at the site.

Fig. 11. Estimated backwasting rate, r _b, versus daily mean air temperature, , at ice cliff F5 during the 2008 melt season.

Overall, the backwasting of all 38 ice cliffs ranges from 4.43 to 11.42 m during the 2008 melt season, and gives a mean value of 7.64 m, or, alternatively, 7.64 m a⁻¹ given that winter ablation is neglected. The annual backwasting rate of ice cliffs is comparable with that on Kötlujökull, Iceland (7.1 m a⁻¹ (Reference Krüger and KjærKrüger and Kjær, 2000)), and Larsbreen, Svalbard (7 m a⁻¹ (Reference Lukas, Nicholson, Ross and HumlumLukas and others, 2005)), but is lower than that on glaciers in the Himalaya (∼15 m a⁻¹ (Reference Sakai, Nakawo and FujitaSakai and others, 1998, Reference Sakai, Nakawo and Fujita2002; Reference Benn, Wiseman and HandsBenn and others, 2001)) where meteorological conditions are comparable but the melt season lasts much longer than in the study area.

Based on the snapshot of ice-cliff distribution in 2006 (Fig. 2) and given a fixed inclination angle of 46.4°, the mean backwasting over all ice cliffs in the debris-covered area on Koxkar glacier will produce about 1.60 × 10⁶ m³ of meltwater. According to mass-balance measurements, the total meltwater from the debris-covered area is of order 22 × 10⁶ m³. Therefore, ice-cliff backwasting accounts for about 7.3% of the total from the debris-covered area. This is a much lower ratio than for Lirung glacier, Himalaya (69% (Reference Sakai, Nakawo and FujitaSakai and others, 1998)), due to more intense melting of the buried ice on Koxkar glacier (∼0.88 cm d⁻¹ on average) than on Lirung glacier (0.19 cm d⁻¹ (Reference Rana, Nakawo, Fukushima and AgetaRana and others, 1997)). The large discrepancy in daily melt rate between the two glaciers may be attributed to the differences in average debris thickness (0.53 m for Koxkar glacier and 0.5–1.0 m for Lirung glacier (Reference Rana, Fukushima, Ageta and NakawoRana and others, 1996)), and in altitude of the ablation area (3060–4250 m a.s.l. for Koxkar glacier and 4000–4400 ma.s.l. for Lirung glacier (Reference Gades, Conway, Nereson, Naito and KadotaGades and others, 2000)) which denotes the different air temperatures between two alpine environments.

Summary

Revision of the Chinese Glacier Inventory is in progress, the main goal being to improve knowledge of the current status of glaciers in China and assess recent glacier changes. One of the challenges in this work is to assess ongoing changes in debris-covered glaciers in areas such as the Tuomuer mountain district, the Pamir plateau and the Himalaya where glaciers with extensive supraglacial debris are widespread. Thus, there is a need to improve the current understanding of the meltout of debris-covered glaciers through long-term field observations as well as quantitative assessments.

We have described a physically based energy-balance model to estimate the backwasting rate of ice cliffs in a debris-cover area with improved parameterization of solar radiation for a sloped ice surface. The model was tested against observations carried out on 38 ice cliffs within the debris-covered area on Koxkar glacier from 5 August to 5 September 2008.

In general, the energy-balance model gives a good estimate of the backwasting rates, with errors in the range ±1.96 cm d⁻¹ and RMSEs of 0.99 cm d⁻¹. Errors arising from using fixed surface albedo and turbulent flux parameterization were limited. We calculated that the maximum incoming shortwave radiation to the ice-cliff surface occurs from the southwest, and as the azimuth increases or decreases from this direction, the incoming shortwave radiation decreases, reaching a minimum at around north. The incoming shortwave radiation decreases with increased slope angle at any given azimuth angle, and, to some extent, the variation of shortwave radiation over a sloped ice surface may be more sensitive to azimuth angle than to ice-cliff inclination angle.

Shortwave radiation is the most important heat source for ice-cliff ablation, contributing about 76% of the total energy available for ice melt, while the sensible heat flux supplies nearly 24% of total energy for ice-cliff wastage. According to the model calculation, latent heat flux and net longwave radiation are comparatively small.

The mean backwasting rate of ice cliffs within the debris cover on Koxkar glacier is estimated to be 7.64 ma⁻¹ in 2008 when winter ablation is neglected. With this annual back-wasting rate and given a mean slope angle of 46.4°, the backwasting of ice cliffs will produce about 1.60 × 10⁶ m³ of meltwater in 2006, and accounts for about 7.3% of total melt runoff from the debris-covered area.

The results are based on short-term field observations and are thus preliminary. The errors in the model calculation suggest that there is still much scope for model improvement. However, this model can provide detailed information on how energy is distributed on an inclined ice cliff and how ice melt is influenced by meteorological variables and glacier topography, which is very helpful for improving our understanding of the physical processes of ice-cliff backwasting. Though a physically based model may not always yield better results than a statistical model (e.g. degree-day model), the latter is usually based on the statistical relationship between ice melt and meteorological forces and thus is ambiguous in explaining how melt is affected.

Continued long-term monitoring of ice-cliff backwasting is in progress in order to further validate the modeling results and improve the model for running on remotely sensed datasets. In the future, increased focus will be needed on improving the parameterization of turbulent heat fluxes, which may require observations of wind and temperature profiles perpendicular to the ice cliff, and comparison with horizontal measurements.