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Edited by: Michael Lehning, École Polytechnique Fédérale de Lausanne, Switzerland

Reviewed by: Sebastian Schlögl, WSL Institute for Snow and Avalanche Research SLF, Switzerland; David Loibl, Humboldt-Universität zu Berlin, Germany

This article was submitted to Cryospheric Sciences, a section of the journal Frontiers in Earth Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Surface energy balance models are common tools to estimate melt rates of debris-covered glaciers. In the Himalayas, radiative fluxes are occasionally measured, but very limited observations of turbulent fluxes on debris-covered tongues exist to date. We present measurements collected between 26 September and 12 October 2016 from an eddy correlation system installed on the debris-covered Lirung Glacier in Nepal during the transition between monsoon and post-monsoon. Our observations suggest that surface energy losses through turbulent fluxes reduce the positive net radiative fluxes during daylight hours between 10 and 100%, and even lead to a net negative surface energy balance after noon. During clear days, turbulent flux losses increase to over 250 W m^{−2} mainly due to high sensible heat fluxes. During overcast days the latent heat flux dominates the turbulent losses and together they reach just above 100 W m^{−2}. Subsequently, we validate the performance of three bulk approaches in reproducing the observations from the eddy correlation system. Large differences exist between the approaches, and accurate estimates of surface temperature, wind speed, and surface roughness are necessary for their performance to be reasonable. Moreover, the tested bulk approaches generally overestimate turbulent latent heat fluxes by a factor 3 on clear days, because the debris-covered surface dries out rapidly, while the bulk equations assume surface saturation. Improvements to bulk surface energy models should therefore include the drying process of the surface. A sensitivity analysis suggests that, in order to be useful in distributed melt models, an accurate extrapolation of wind speed, surface temperature and surface roughness in space is a prerequisite. By applying the best performing bulk model over a complete melt period, we show that turbulent fluxes reduce the available energy for melt at the debris surface by 17% even at very low wind speeds. Overall, we conclude that turbulent fluxes play an essential role in the surface energy balance of debris-covered glaciers and that it is essential to include them in melt models.

Debris-covered glaciers constitute a significant minority of the total glacierized area in High Mountain Asia. Although about 11% of the glaciers is debris-covered, about 18% of the total ice mass is stored under a debris mantle (Nuimura et al.,

To estimate melt at the debris-ice interface, debris energy balance models are employed, although simpler degree day models (Mihalcea et al.,

Fluxes at the debris surface are generally derived using low-frequency measurements of an Automatic Weather Station (AWS). Turbulent heat fluxes are evaluated using the assumptions of similarity theory (Monin and Obukhov,

Direct measurements of turbulent fluxes require the use of eddy correlation systems, and to our knowledge this has only been attempted twice previously over debris-covered glaciers. Yao et al. (

We hypothesize that turbulent fluxes play an important role on a debris-covered glacier surface that has a distinct drying and wetting cycle due to this surface. We however do not know how well suited the commonly applied bulk parametrizations are for this case. We present measurements of surface energy fluxes during the transition period from the wet monsoon to the dry post-monsoon on a debris-covered glacier in the Himalayas. We compare direct turbulent fluxes measured with an eddy correlation system to different bulk approaches. This enables us to (a) provide an assessment of the suitability of commonly used bulk approaches to estimate these fluxes when an eddy correlation system is not available and subsequently (b) assess the relative contribution of turbulent fluxes to the energy balance of a debris-covered glacier. We conclude the study by assessing the importance of the turbulent fluxes in the surface energy balance by running the best performing model for a full melt season.

Measurements were conducted on Lirung Glacier in the Nepalese Himalaya (location of the AWS at 28.23967N, 85.55711E, 4250 m a.s.l.; Figure ^{2}, approximately 30% of which is glacierized. Debris cover accounts for approximately 25% of the total glacierized area (Ragettli et al., ^{2}. The debris-covered tongue has a length of 3.5 km and is on average 500 m wide (Immerzeel et al.,

The local climate is dominated by monsoon circulation, with 68–89% of the annual precipitation falling between June and September (Immerzeel et al.,

The location of the AWS was chosen on a location of the glacier that had several hundreds of meters relatively unobstructed fetch from the main wind direction.

The eddy correlation system was mounted onto the AWS tower. The tower is based on a modular design using three vertical anchors drilled into the ice and horizontal cross-pieces for stability (Jarosch et al.,

At the AWS we measured incoming and outgoing shortwave and longwave radiation (1.6 m), relative humidity, air temperature measured at 10 min intervals (3.12 m) and wind speed components using an IRGASON sonic anemometer at 10 Hz (3.42 m, see Table

Sensor specifications for relevant measurements at the AWS and eddy correlation system (EC) and in the debris.

SW ↑↓ | 1.6 | CNR1 | Kipp&Zonen | 305−2,800 nm | ±10% |

LW ↑↓ | 1.6 | CNR1 | Kipp&Zonen | 5−50 nm | ±10% |

RH_{air} |
3.12 | HC2S3 | Campbell Scientific | 0−100 % | ±0.8% |

T_{air} |
3.12 | HC2S3 | Campbell Scientific | −50−100°C | ±0.1°C |

u_{x},u_{y},u_{z} |
3.42 | IRGASON | Campbell Scientific | - | <±0.08m s^{−1} (u_{x},u_{y}) |

<±0.04m s^{−1} (u_{z}) |
|||||

T_{air} HF |
3.42 | IRGASON | Campbell Scientific | −50−60°C | ±0.025°C |

RH_{air} HF |
3.42 | IRGASON | Campbell Scientific | −50 −60°C | ±2% |

SM | −0.35 | CS650 | Campbell Scientific | 0−100 % | <0.05% |

T_{deb} |
−0.15/−0.35/−0.55/−0.75 | PB−5006−1M5 | Gemini | −40 −125°C | ±0.2°C |

RH_{deb}/T_{deb} |
Surface | HOBO U23 Pro | Onset | −40 −70°C | ±0.21°C |

0 – 100% | ±2.5% |

_{air} is relative humidity of the air and T_{air} air temperature. u_{x}, u_{y} and u_{z} are the wind components and T_{air} HF and RH_{air} HF are the sonic air temperature and humidity respectively. SM, T_{deb} and RH_{deb} are soil moisture, temperature and relative humidity of the debris

In the subsequent year during the same season relative humidity and temperature measurements were carried out on the surface at the same location as the AWS, using a HOBO U23-Pro sensor.

Data from an AWS operational previously on the same glacier (Figure

Turbulent fluxes were estimated with the bulk aerodynamic method (Munro,

where ^{−2}], ρ_{a} is air density at the measurement site [kg m^{−3}] calculated as _{0} is density [kg m^{−3}] at standard sea level pressure _{0} [1013.25 kPa] and p is pressure [Pa] measured at the site (Cuffey et al., ^{−1}], T_{z}, T_{s} and q_{z}, q_{s} are the temperature [K] and specific humidity [-] at measurement height z [m] and the surface respectively. T_{s} is determined from the outgoing longwave radiation by inverting the Stephan Boltzman law, assuming an emissivity of 1. This estimate of T_{s} covers a larger footprint than a focused IR sensor. L_{e} is the latent heat of vaporization for T_{s} >273.15 K (2.476 × 106 kg J^{−1}) and sublimation for T_{s} <273.15 K (2.830 × 106 kg J^{−1}) otherwise.

C_{p} is the specific heat capacity [J kg^{−1} K^{−1}] of humid air calculated as

where c_{ad} is the specific humidity of dry air (1006 J kg^{−1} K^{−1}).

C_{bt} denotes the bulk transfer coefficient. We test the three most commonly applied parametrizations of this bulk transfer coefficient (Fitzpatrick et al.,

Some studies assume neutral stability over the debris cover (Nakawo and Young, _{bt} excluding a stability function

where k is the von Kármán constant (0.41), z_{u} and z_{t} are the measurement heights [m] for wind speed as well as temperature and relative humidity respectively. z_{0m} [m] is the surface roughness length of momentum, and z_{0t} and z_{0q} are the surface roughness lengths [m] for temperature and water vapor respectively, further discussed in section 4.2. Considering the large variations of surface temperature induced by the strong heterogeneous heating of the surface during the day (Brock et al., _{ib}), where C_{bt} becomes

The stability functions are given by Brutsaert (

where ϕ_{m/h/v} are the stability functions for momentum, heat and water vapor respectively and the Richardson number _{ib} (Moore,

g is acceleration due to gravity [m s^{−2}] and _{m} [K] is the mean temperature between the measurement height z and the surface. Only values close to neutral atmospheric conditions (|R_{ib}|<0.2) are considered.

The last approach is based on the MO stability parameter (z/L) resulting in

where

where _{*} [m s^{−1}] is the friction velocity. As L is a function of

Accurate modeling of turbulent fluxes depends on representative values for the roughness lengths of momentum (z_{0m}), temperature (z_{0t}) and water vapor (z_{0q}) (Smeets et al., _{0m} (Brock et al., _{0t} and z_{0q} were either assumed equal to z_{0m} or an order of magnitude smaller. Due to the highly variable grain size of the debris cover, surface roughness varies considerably in space Miles et al. (

where z_{0m} is the effective surface roughness at the tower location, z_{l} the local topographic roughness derived for the respective pixel and Ω denotes the integration domain, in this case the complete footprint. We estimated the boundary layer height to be at a maximum of 300 m, which corresponds to estimates based on Nieuwstadt (_{0m} derived from wind tower measurements (Miles et al., _{0m} with those using a constant value of 0.03. We derived z_{0t} and z_{0q} applying a scaling relationship on z_{0m} described in Smeets and van den Broeke (

Turbulent latent heat flux calculations are a function of specific surface humidity (_{s}) which is difficult, if not impossible, to measure for the entire footprint of the flux measurements. No specific humidity measurements were available for the study period at the location of the AWS. Moreover, _{s} is highly variable due to variations in texture, from silt particles to large boulders, and the heterogeneous drying of the debris. Earlier studies have assumed the debris surface to be always fully saturated (Reid and Brock,

where _{s} - _{a} is the water vapor gradient through the sublayer [kg m^{−3}], _{w} is the molar mass of an evaporating liquid [kg mol^{−1}], R is the universal gas constant [8.314 J mol^{−1} K^{−1}] and RS is the air relative humidity at the top of the viscous sublayer, which corresponds to the effective relative humidity of the surface used to calculate the latent heat flux using the bulk approach. P_{sat}(T) is the saturated vapor pressure estimated by the Antoine equation, a semi-empirical equation derived from the Clausius-Claperon relation.

where T is in [°C].

We solve equation 11 for RS and introduce parameters _{sat}(T_{a}) and _{sat}(T_{s}) leading to

Turbulent fluxes were measured for 13 consecutive days during the transition period between monsoon and post-monsoon (Figure

Fluxes and variables measured at the AWS, SW_{net} denoting net shortwave radiation, LW_{net} net longwave radiation, H sensible and LE latent heat flux

As incoming shortwave radiation increases, the debris surface quickly heats up to 30°C. Valley wind circulations lead to increased wind speeds and enhanced turbulent fluxes during the daytime. Relative humidity during the day quickly drops and rises again during clear days but remains saturated during overcast days often also during the day, resulting in a median full saturation. Winds on Lirung glacier are predominately anabatic, not in the direction of the glacier tongue but the valley itself, hence driven by larger convective patterns rather than local temperature gradients (Figure

Wind velocities

On clear sky days the sensible heat flux is more negative than latent heat due to the strong heating and quick drying of the surface, which is typical for debris (Steiner and Pellicciotti, ^{−2} around noon to −32 ± 18 W m^{−2} in the late afternoon (Figure ^{−2}. Latent heat fluxes reach values of about −102 ± 29 W m^{−2} at noon and decrease to −33 ± 9 and −3 ± 2 W m^{−2} at night (Figure ^{−2} and the total net radiative balance is reduced between 17 and 125%.

Measured sensible

Median fluxes and meteorological data on

On an overcast day incoming shortwave radiation decreases from a median 846 ± 118 W m^{−2} to 221 ± 123 W m^{−2} (Figure ^{−2} at noon up to positive values of 1.5±2 W m^{−2} at dawn. Latent heat plays a relatively larger role on overcast days, as daytime values reach beyond −50 W m^{−2} at noon (Figure

The resulting observed evaporation rates, calculated by dividing the latent heat flux by water density and the latent heat of vaporization (2.476 MJ kg^{−1}) over the debris surface are on average 2.8 and 1.8 mm w.e. day^{−1} on clear and overcast days, respectively. These values are higher than evaporation rates measured in the Tien Shan where Yao et al. (^{−1} and considerably higher than what has been observed on a debris-covered glacier in the European Alps, where Collier et al. (^{−1}. Observations on a clean ice glacier show values of one order of magnitude lower (Kaser,

Sensible heat fluxes are comparable to the Tien Shan (Yao et al.,

The total energy available for melt on the debris surface on clear (1.8 mm w.e. day^{−1}) and overcast days (0.7 mm w.e. day^{−1}) is considerably lower than if turbulent fluxes would be ignored (3 and 1.3 mm w.e. day^{−1} respectively).

The large negative flux between 1 pm and 7 pm could have implications for refreezing of liquid water present in the debris, a process so far rarely accounted for in debris energy balance models (Lejeune et al.,

Turbulent fluxes are closely correlated to the temperature difference between air and the surface both on clear (

Half-hourly values for temperature difference between air and surface measurements

Debris temperatures in the upper 15 cm respond quickly to surface changes (Figure

As the surface heats quickly, unstable conditions develop, resulting in mixing and higher wind speeds and increased turbulent fluxes (Figures

The combination of slow cooling of the debris and sustained relatively high wind speeds into the afternoon (Figures

Using the model proposed by Kljun et al. (^{*} [m s^{−1}] and the Obukhov length L [m] and the standard deviation of lateral velocity fluctuations [m s^{−1}] derived from the eddy correlation measurements and the boundary layer height assumed to be 300 m. The footprint provides a raster over the area around the AWS with each raster cell having a weight between 0 and 1 and all cells summing to unity (Equation 10). The representative roughness value of the whole footprint is then calculated by multiplying the weights with the respective topographic roughness from the gridded product. The resulting time series is relatively close to the initial constant value chosen and that provides confidence in the use of a constant value only (Figure _{0t}) and vapor (z_{0q}) to be either equal to Schlögl et al. (_{0m}). However, this factor is expected to be considerably lower for rough flow and high Reynolds numbers (Andreas,

_{0m} is shown. _{0m} obtained from the respective footprints (black) for each time step as well as the constant value used in this study (0.03, red) and in Brock et al. (

Bulk model results with the assumption of a fully saturated surface are shown in Figures _{ib}|>0.2. While the model results correlate well with measured fluxes for sensible heat (^{2}>0.8), the root mean square error (RMSE) is in the order of magnitude of the fluxes measured during the day. The models based on the Richardson number and without a stability correction also produce extreme spikes for latent heat fluxes, which have been reported previously (Brock et al., ^{−2}, which occurred 16 times for the latent heat flux in the these models, specifically on clear days, but only once for the model iterating over the MO length.

Model results for all bulk models over the whole study period for ^{−2}.

Diurnal cycles for all model results on all days ^{−2}.

Statistics of model performance for all days for both the latent and sensible heat flux for the three different models.

^{2} |
||||
---|---|---|---|---|

H nSC | 0.92 / 0.81 | 0.6 / 0.7 | 24 / 8 | −14 / −3 |

LE nSC | 0.67 / 0.59 | −1.7 / 0.2 | 115 / 28 | 50 / 2 |

H rSC | 0.88 / 0.82 | 0.4 / −0.1 | 59 / 21 | 19 / 8 |

LE rSC | 0.34 / 0.65 | −0.8 / −1.3 | 96 / 69 | 36 / 26 |

H iSC | 0.92 / 0.89 | 0.7 / 0.8 | 26 / 7 | 3 / 0 |

LE iSC | 0.53 / 0.64 | −0.9 / −0.4 | 95/ 42 | 36 / 13 |

^{2} is the coefficient of determination, KGE the Kling Gupta efficiency, RMSE the root mean square error and MBE the mean bias error. Values are given for clear sky and overcast days respectively. nSC denotes the model with no stability correction, rSC based on the Richardson number and iSC iterating over the Monin-Obukhov length

Additionally to the coefficient of determination (R^{2}), the root mean square error (RMSE) and the mean bias error (MBE) we use the Kling-Gupta efficiency as an indicator for model performance, with 0 describing a situation where any random guess is as useful as the model and 1 a perfect prediction (Gupta et al., ^{2}>0.8 and a RMSE<30 W m^{−2} for clear and <10 W m^{−2} for overcast days and a KGE>0.6 for all cases. They also perform well for the latent heat flux on overcast days, but overestimate the flux on clear sky days as the debris dries out, resulting in a RMSE between 95 and 115 W m^{−2} for the iterative approach and model without a stability correction respectively which is in the range of the highest measured values on these days. The MBE is 36 and 50 W m^{−2} respectively.

Comparing model performance to the temperature difference over the debris-covered tongue (Figure _{ib} becomes < 0, the offset increases. The empirical parameters in the function when derived for such large differences likely coincided also with higher wind speeds, typical for clear sky days with strong incoming solar radiation. The hummocky terrain of a debris-covered glacier prevents such high velocities to occur and hence makes the application of the approach using the Richardson number problematic. Similarly, the fact that the MO length measured at the eddy correlation system does not agree with the bulk model for which the Monin-Obukhov theory was developed (assuming constant flux layer and stationarity), introduced model uncertainties, which are difficult to quantify.

Differences in modeled and observed sensible

A similar relation between the temperature difference and the offset in the latent heat flux also is observed for all models. In this case, as the relation is especially strong on clear sky days, it can be explained with the indirect relation to the drying of the surface, as high surface temperatures caused by the incoming solar radiation cause the difference to rise and the debris to dry. In these cases the models, as they assume full surface saturation, overestimate the flux. Finding a relation between surface temperature and the observed specific humidity of the surface could therefore likely provide a possible improvement.

Given the difficulty in measuring accurate surface temperatures, the strongly heterogeneous surface roughness as well as the variable wind speed due to the hummocky terrain, we conduct a sensitivity analysis on these variables. We run the models with ±10% of the original value of temperature and wind speed. To account for the logarithmic effect we changed the log(z_{0m}) ±50%, which corresponds to a range of z_{0m} between 0.005 and 0.17, in line with observations of the observed spatial variability in the field Miles et al. (

Sensitivity of both turbulent fluxes to changes in input variables surface temperature (T_{s}), wind speed (ws) and surface roughness (z_{0m}).

T_{s} ±10% |
±14 % | ±22 % | −26 / 14% | −42 / 23% |

ws ±10% | ±10 % | ±10 % | ±8% | ±8% |

log(z_{0m}) ∓50% |
±106 / 41% | ±106 / 41% | ±100 / 40% | ±101 / 40% |

Results show that the models are very sensitive to surface temperature, as it is a strong direct driver of modeled fluxes (Figure

Surface temperatures measured by the UAV over the whole footprint at the corresponding time (boxplot), the corresponding weighted footprint temperature and the measurement from the AWS.

The fluxes are less sensitive to wind speed but the results are non-negligible as changes of ±10% can equally be caused by sensor inaccuracies. The hummocky terrain causes wind speeds to vary over the surface, as wind speeds are possibly larger on top of the debris mounds and lower in the depressions. Dadic et al. (^{−1}, which is largely beyond the wind speeds measured in our case. While further detailed analysis of the stability corrections is beyond the scope of this study, considering the large variety in debris surfaces when it comes to the local hummocky terrain and grain size, further sensitivities on idealized surfaces could provide insights for different climatic settings. Considering the variability of z_{0m} observed on the footprint (Figure _{0t} to be possibly larger Calanca (_{0m} Smeets et al. (

The sensitivities reported in Table _{0m} can be estimated with a topographic method (Miles et al.,

It is obvious from the results above that the models fail to reproduce latent heat fluxes during clear sky days when the surface dries out and the assumption of a saturated surface becomes invalid. Some of this offset can very likely be explained by inadequate stability corrections. One way to improve this would be to decrease z_{0q}. However even decreasing it from z_{0q} = 0.0015 to 10^{−5} only improves results marginally. We hence believe that the larger part of this discrepancy can be explained by our insufficient knowledge about surface drying. Since measuring surface specific humidity on a debris-covered glacier is difficult, we use surface temperatures as a proxy.

We derive the relative humidity of the surface required to model latent heat accurately with the bulk approach from the model without a stability correction by solving Equations 2 and 4 for the relative humidity, replacing the bulk flux of latent heat with the measured values. Placing these in equation 13, allows us to derive the parameters

Relative humidity at the surface, necessary to fit the bulk model to eddy correlation measurements (RH_{bulk}), measured embedded in the debris surface during a subsequent measurement campaign at the same location (RH_{meas}) and obtained using a regression based on (Haghighi and Or, _{mod}).

Using a relative humidity value based on this equation, we are able to reduce the modeled latent heat fluxes considerably (Figure ^{−2} and the MBE decreases from 50 to 10 W m^{−2} on clear days. On overcast days the RMSE improved only marginally from 28 to 27 W m^{−2} and the MBE decreased from 2.1 to 1.4 W m^{−2}. While still not accurately reproducing the fluxes on the hourly scale as the KGE only improved to 0.3 and R^{2} even dropped from 0.67 to 0.49 on clear days, it gives us more confidence of estimating the contribution of latent heat fluxes over a longer period of time.

Diurnal cycles for the latent heat flux on clear days assuming full saturation

To assess the importance of turbulent fluxes over a complete melt season, we use the model without a stability correction to compute sensible and latent heat fluxes based on data from an AWS from May until October 2014 on the same glacier (Figure

Energy balance on the glacier, with measured radiative fluxes and modeled turbulent fluxes based on AWS from 2014 during pre-monsoon

The location of the AWS in 2014, although only several hundred meters away, results in two distinct differences affecting the energy balance. First, net shortwave radiation is significantly higher, reaching 650 W m^{−2} in the post-monsoon season on average, while at the eddy correlation system such values were only reached on completely clear days. As the location is further up-glacier, it is shaded from the steep head walls most notably in the afternoon. Second, the AWS in 2014 was located in a depression resulting in wind speeds nearly half as low as at the location in 2016. As a result turbulent fluxes are considerably lower than at the measurement site from 2016. Nevertheless turbulent fluxes reduce the radiative fluxes between 7 a.m. and 5 p.m. by 5–66% in pre-monsoon, 0–12% in monsoon and 0–41% in post-monsoon. Over the entire melting season the turbulent fluxes reduce the total energy for melt available at the debris surface by 17%. During monsoon turbulent fluxes reach only 40 W m^{−2}, largely because of a decreasing temperature gradient as the debris heats up less and an often saturated surface and very high air humidity, while wind speeds stay similar to the drier seasons. During both pre- and post-monsoon turbulent fluxes reach around 100 W m^{−2} between 10 a.m. and 4 p.m., with the latent heat flux being more important in the drying period in post-monsoon.

The aim of this study was to quantify turbulent fluxes over a debris-covered glacier by direct measurements with an eddy correlation system, test the accuracy of bulk models against these measurements and evaluate the contribution of turbulent fluxes to the energy balance at the debris surface. Direct measurements over a period of 16 days in the post-monsoon season in 2016 were investigated and compared against other climatic variables. Three separate bulk models, employing different parametrizations of the bulk transfer coefficient, were compared to eddy correlation measurements under different weather conditions. The best performing model was then used to evaluate turbulent fluxes over a complete melt season.

Our direct measurements of turbulent fluxes at the point scale show that turbulent fluxes are an essential part of the energy balance. Turbulent fluxes reduce the average net radiative flux input by 80% on clear days and 72% on overcast days, causing the total net surface balance to become negative very early in the afternoon, leaving only 5–7 h per day where the energy flux toward the debris is positive. While sensible heat fluxes are clearly the dominant turbulent flux on clear sky days, with values reaching up to −180 W m^{−2} during the day, the latent heat flux is dominant during overcast days and takes up more than 35% of the radiation input, resulting in evaporation rates between 1.8 mm day^{−1} on overcast and 2.8 mm day^{−1} on clear days.

We show that a frequently applied bulk approach using stability corrections based on the Richardson number performs poorly for large surface to air temperature differences, which are common on clear sky days. Since the original parametrization was likely developed for large differences in combination with high wind speeds, it fails for a debris-covered glacier where wind speeds remain relatively low even during the day. As a consequence the model overestimates turbulent fluxes by at least 100% during the day. The best performing models based on the same climate input data result in strong correlations for the sensible (>0.9 on clear sky and >0.8 on overcast days) and less strong for the latent heat flux (>0.5). The mismatch between model and observation is much larger for the latent heat flux because the specific humidity at the surface is below saturation during clear sky days, while the models assume full saturation. We however propose a simple parametrization based on the surface temperature that provides reasonable estimates for surface humidity and considerably improves the results on clear sky days. On overcast days the models overestimate the latent heat flux much less, with the best results provided by the model without a stability correction. This is encouraging, since most days during the melting season in High Mountain Asia are overcast, hence minimizing the error accrued over seasonal model runs. We conclude therefore that the bulk approach without a stability correction is the most suitable option for these hummocky surfaces with strong temperature differences and relatively low wind velocities.

We have used this model to calculate turbulent fluxes over a complete melt season at another location on the glacier. Because of its location in a depression at the lower tongue wind speeds were lower but incoming solar radiation higher. As a result absolute and relative turbulent fluxes were much lower at this location. However turbulent fluxes still reduced the radiative fluxes by up to 50% during the day and reduced the total energy for melt by 17% over the complete season.

It is therefore concluded that it is essential to include turbulent fluxes in melt calculations for debris-covered glaciers because they play a major role. Energy balance models for debris-covered surfaces as well as for cliffs and ponds that make use of the bulk approach should consider using the more suitable approach with no stability correction or a correction based on the MO length, rather than the Richardson parametrization, which could lead to significantly underestimated melt rates.

We also conclude that these models are very sensitive to both surface temperatures as well as wind speed and surface roughness, variables that strongly vary spatially over the glacier surface. Since the footprint for these turbulent fluxes covers a considerable surface even at these low wind speeds, an understanding of these variables in space is essential. While surface roughness can be estimated from the topography and we can show that point measurements of surface temperatures are good representations of temperatures in the footprint of the eddy correlation measurements, more efforts have to be invested to understand the surface boundary layer dynamics in order to produce reliable estimates of temperature and wind fields over such heterogeneous surfaces potentially using high resolution large-eddy simulations.

JFS wrote the initial version of the manuscript. ML, ES, JS, MB, and WI commented on the initial manuscript and helped improving this version. JFS developed the methodology with inputs from WI, MB, ML, and ES. JFS performed the analysis with support from WI, ML, and ES. All authors participated in fieldwork.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We are grateful to the trekking agency Glacier Safari Treks and their staff without whom this work would not have been possible. We are grateful to GEMINI for providing the TinyTag sensors for this fieldwork. We are also grateful to our research partners, specifically Francesca Pellicciotti and Pascal Buri, for maintaining the AWS setup in 2014.