Modelling the surface mass balance of the Greenland ice sheet (GrIS) in large-scale ice-sheet models using temperature parameterizations in relation with the positive degree-day (PDD) approach is highly sensitive to a parameter: the temperature standard deviation (Reference BraithwaiteBraithwaite, 1984; Reference ReehReeh, 1991). The PDD method is a statistical approach that relates the totals of positive near-surface air temperatures to the amount of snow or ice that melts. The standard deviation of the near-surface air temperature, σpdd, is important for PDD modelling because it indicates whether the temperature has been above freezing during a month even though the mean monthly near-surface air temperature value is below. Reference Fausto, Ahlstrøm, van As, Bøggild and JohnsenFausto and others (2009a) demonstrated that a uniform increase of σpdd from 2.5°C to 4.5°C results in a 33% increase in the modelled melt area over Greenland where melt is >1 mm. It is therefore important to constrain the σpdd value with observations. In large-scale ice-sheet and surface mass-balance models of Greenland, it is common that σpdd is assigned a single value which typically spans the interval 4.5–5.5°C (Reference GreveGreve, 2005; Reference Goelzer, Huybrechts, Loutre, Goosse, Fichefet and MouchetGoelzer and others, 2010; Reference Greve, Saito and Abe-OuchiGreve and others, 2011; Reference Sundal, Shepherd, Nienow, Hanna, Palmer and HuybrechtsSundal and others, 2011). The value of σpdd is often used as a tuning parameter, instead of using the temperature standard deviations observed at the automatic weather stations (AWSs) on the ice sheet. To add to the temperature parameterization presented by Reference Fausto, Ahlstrøm, van As, Bøggild and JohnsenFausto and others (2009a), it is proposed to construct a similar distributed parameterization for the temperature standard deviation using the same dataset.
Commonly, large-scale ice-sheet models over Greenland calculate the amount of melt using the PDD method by assuming an annual sinusoidal evolution of the near-surface air temperature (Reference ReehReeh, 1991). The number of PDDs from the normal probability distribution around the monthly mean temperatures during the years, following Reference ReehReeh (1991), is given as
where t is the time, T(°C) is the near-surface air temperature (2 m), Tanc (°C) is the annual near-surface temperature cycle and σpdd is the standard deviation of the near-surface air temperature. Tanc is assumed to vary sinusoidally over time,
where A is 1 year. Tj and Ta are the mean July and mean annual near-surface air temperatures. σpdd is also assumed to vary sinusoidally over time,
where u j and u a are the mean July and mean annual standard deviation of the near-surface air temperatures.
Based on Reference ReehReeh (1991) and following the study of Reference Fausto, Ahlstrøm, van As, Bøggild and JohnsenFausto and others (2009a), the annual mean (Ta) and July mean (Tj) near-surface air temperatures are parameterized as a linear function of altitude, latitude and longitude.
The standard deviation of the near-surface air temperature over the GrIS is parameterized using data from AWSs located on the ice sheet. The parameterization is expressed in terms of mean annual and mean July temperature standard deviations. Mean monthly values are calculated from hourly temperature observations for each month in a given year for the Greenland AWSs (for details see fig. 2 and tables 2 and 3 in Reference Fausto, Ahlstrøm, van As, Bøggild and JohnsenFausto and others, 2009a). The associated standard deviations around the monthly means were calculated and a least-square fit was applied to the observed σpdd values, assuming a linear dependence on altitude, z s, latitude, ϕ, and longitude, λ:
where σpdd is the standard deviation parameterization. The values of the coefficients found for Equation (4) are given in Table 1. The temperature- and temperature standard deviation parameterizations are based on mean monthly values which constrain Equations (2) and (3) for monthly time integration in Equation (1).
Table 1. Coefficients for Equation (4) and their root-mean-square difference (RMSD) relative to the observed standard deviation
Table 2 presents the modelled and observed values of the mean annual, mean July and mean summer near-surface air temperature standard deviation together with their differences for the 27 AWSs on the GrIS used in this study. The standard deviations show an annual cycle with the smallest values during summer and the largest during winter. Figure 1 shows an example of the annual cycle of the temperature standard deviation from three AWSs located at different elevations. The figure shows that the assumption of a sinusoidal function for the monthly temperature standard deviation is reasonable. Smaller values during summer can be explained by a limiting influence of the surface temperature over a melting snow and ice surface. When the surface temperature reaches the melting point, energy that could potentially raise the near-surface air temperature is used for melting. In the winter no melting occurs and the temperature variations are not limited by the surface temperature. July and August account for the smallest standard-deviation values of the ablation season (<2.0°C), and the highest values (3.0–6.0°C) are found in May, June and September. Furthermore, the standard deviation has a clear altitudinal dependence with minor influences by the latitude and longitude as indicated by the coefficients and their rootmean-square difference (RMSD) in Table 1. Both tables show that the smallest standard deviations are found at low elevation (σpdd <2°C). The highest standard deviations are found at high surface elevation (σpdd ∼7°C) (Table 2).
Fig. 1. Monthly temperature standard deviation for three AWSs located in the dry snow zone (Summit), near the equilibrium line (Swiss Camp) and in the ablation zone (JAR2). The lines are from Equation (3) using 
and 
from Table 2.
Table 2. A comparison between the modelled 
standard deviation distribution and observed data (σaj) from the stations. σa and σj are the mean annual and July standard deviation, respectively. The difference (Diff.) is calculated between the modelled and observed data. Acc. and Abl. denote stations located in the accumulation zone or in the ablation zone. The elevation (Elev.) of the stations is in m a.s.l.
To calculate reliable melt rates with the PDD method using a single mean annual value for the whole ice sheet would not be appropriate (Reference Fausto, Ahlstrøm, van As, Bøggild and JohnsenFausto and others, 2009a,Reference Fausto, Ahlstrøm, van As, Johnsen, Langen and Steffenb). The new spatially and temporally varying parameterization for σpdd addresses this problem from an empirical point of view. We suggest that the parameterization presented here should be validated using in situ observational records.
Acknowledgements
We thank M. van den Broeke for providing temperature data from the K-Transect. Many thanks to two anonymous reviewers for constructive criticism which significantly improved the manuscript. This is a PROMICE (Programme for Monitoring the Greenland Ice Sheet) publication and the paper is published with the permission of the Geological Survey of Denmark and Greenland (GEUS).
3 September 2011
