2.1. Measurements

To investigate glacier–climate interactions on Kilimanjaro, we have been operating two automatic weather stations (AWSs) on the massif’s central peak Kibo since February 2005, and a third AWS has been operated by the University of Massachusetts since February 2000 (Fig. 1). AWS3, which is located in the upper part of the steep Kersten Glacier at 5873 m a.s.l., provides all the necessary data to run physically based mass-balance models (Reference Mölg, Cullen, Hardy, Kaser and KlokMölg and others, 2008). Two-year records without any gaps are now available from this station for half-hourly global radiation (incoming shortwave radiation with respect to a horizontal reference surface), albedo, incoming and outgoing longwave radiation, air temperature and humidity, wind speed and direction, surface height change, and barometric air pressure. As for the entire Kilimanjaro region, the climate on Kibo is characterized by hygric seasonality, with wet seasons from March to May (MAM) and October to December (OND). A pronounced dry season occurs from June to September, and a moderately dry season in January and February. Reference Mölg, Cullen, Hardy, Kaser and KlokMölg and others (2008) present the weather conditions recorded at AWS3, as well as instrument details. Mean air temperature is −7°C and hardly varies on a monthly scale, while the time series of air humidity and precipitation (snowfall) reflect the hygric seasonality. A Kipp & Zonen CNR1 net radiometer installed horizontally at (initially) 1.64 m above the glacier surface (Fig. 1) measures separately the downward fluxes of (1) solar radiation in the spectral (shortwave) range 0.3–2.8 μm, and (2) longwave radiation in the range 5–50 μm. The nominal accuracy of the sensors is ±10% on daily totals. Experience in cold environments has shown that the actual accuracy is higher and remains within ±5% (Reference Van den Broeke, van As, Reijmer and van de WalVan den Broeke and others, 2004; Reference Van As, van den Broeke, Reijmer and van de WalVan As and others, 2005), but occasional errors above 10% have recently been reported for a mid-latitude site (Reference Michel, Philipona, Ruckstuhl, Vogt and VuilleumierMichel and others, 2008). Figure 2 depicts the time series of global radiation between February 2005 and January 2007, the dataset on which this study proceeds. Measurements clearly reflect the variations in the top-of-atmosphere (TOA) extraterrestrial solar radiation. The period around December 2005–January 2006 features a few daily means that are almost as high as TOA values. A possible explanation of this is the anomalously dry conditions that coincided with this period (Reference Mölg, Cullen, Hardy, Kaser and KlokMölg and others, 2008), but the definite reason is not clear. However, this anomaly in global radiation was also recorded at AWS1 (Fig. 1), so is unlikely to be related to an instrument error.

Fig. 1. Glacier extent on Kilimanjaro’s central part Kibo in 2003 (Reference Cullen, Mölg, Kaser, Hussein, Steffen and HardyCullen and others, 2006) and the location of AWSs and vertical ice walls (UTM zone 37S projection; contours at 200 m spacing). Africa’s highest point, Uhuru Peak (5895 m a.s.l.), the eruption cone (Reusch Crater) and Kersten Glacier (KG) are also indicated. The photo shows AWS3 (5873 m a.s.l.) in February 2005, with the CNR1 net radiometer circled (photo: N.J. Cullen).

Fig. 2. Daily means of global radiation at AWS3 and TOA radiation (100% and 85%) over Kilimanjaro between 9 February 2005 and 9 January 2007.

A set-up similar to AWS3 on Kibo has been maintained at an AWS on the tongue of Glaciar Artesonraju (4850 m a.s.l.), Peruvian Andes, since March 2004 by the Institut de Recherche pour le Développement (IRD), France, in cooperation with the Tropical Glaciology Group at Innsbruck. While Kibo stands very close to the equator (Fig. 1), Glaciar Artesonraju is located further away from it at ∼9° S (for maps see Reference JuenJuen, 2006). At this site, there is only one core wet season (January–March) and one core dry season (June–August). The remaining months are regarded as transitional months. Mean air temperature is close to freezing point and shows little month-to-month variability. Reference JuenJuen (2006) documents in detail the instrument specifications of, and weather conditions at, the AWS on Glaciar Artesonraju. Measurements from this station are available from 25 March 2004 to 31 March 2005 for the validation part of this study. Importantly, radiative fluxes are measured with the same sensors as at AWS3 on Kibo (see above). The characteristics of the local climate are summarized in Table 1. Briefly, the warmer and more humid environment at the Artesonraju AWS (ablation area), compared to the Kilimanjaro summit region (accumulation area), leads to substantially lower global radiation, G, but higher incoming longwave radiation, L ↓, and to a less variable moisture climate (discussed further in section 3.3).

TOA outgoing longwave radiation (TOA-OLR) from the US National Oceanic and Atmospheric Administration (NOAA) satellite (Reference Liebmann and SmithLiebmann and Smith, 1996), available at 2.5° horizontal resolution, is also used for the gridcell containing Kilimanjaro (centered at 2.5° S, 37.5° E). In the tropics, TOA-OLR is a good indicator of (and negatively correlated with) convective cloudiness and precipitation (e.g. Reference Garreaud, Vuille and ClementGarreaud and others, 2003), and therefore helps assess the cloud variable deduced from the solar radiation model (see section 3.2).

2.2. The solar radiation model

Previous solar radiation modelling for the three glacierized mountains of East Africa (Mount Kenya (Reference HastenrathHastenrath, 1984), Rwenzori (Reference Mölg, Georges and KaserMölg and others, 2003a) and Kilimanjaro (Reference Hastenrath and GreischarHastenrath and Greischar, 1997; Reference Mölg, Hardy and KaserMölg and others, 2003b)) explored the spatial pattern of incident solar radiation qualitatively, but did (or could) not validate simulated global radiation quantitatively. The basic equation to calculate G in this paper reads as follows (Reference HastenrathHastenrath, 1984):

where S _cs is the clear-sky direct solar radiation, D _cs clear-sky diffuse solar radiation, k a constant, and n _eff the ‘effective’ cloud-cover fraction (0–1). The impacts of typical cloud properties (e.g. height, type, optical thickness) on solar radiation are parameterized by k, which therefore varies with geographical latitude (Reference HastenrathHastenrath, 1984). Still, since k in the real world is not constant temporally, n _eff also parameterizes such impacts partly and the condition ‘effective’ expresses the difference to true cloud-cover fraction.

Clear-sky direct solar radiation is computed by:

where S ₀ is the solar constant (1367 W m⁻²), E ₀ the eccentricity correction factor, h the solar elevation above the plane of the horizon, and τ _cs the clear-sky transmissivity of the atmosphere for the (broadband) direct sun beam. If S _cs is not computed for a horizontal reference surface (e.g. for a gridcell in a digital terrain model), sin h is replaced by cos ζ _p, where ζ _p is the zenith angle of the sun with respect to an arbitrarily oriented and inclined plane (Reference Mölg, Georges and KaserMölg and others, 2003a). We refer to Reference IqbalIqbal (1983) for calculation of E ₀, ζ _p and h, a summary of which is provided by Reference Mölg, Georges and KaserMölg and others (2003a). τ _cs is the product of four transmission coefficients (e.g. Reference Klok and OerlemansKlok and Oerlemans, 2002) that describe extinction of solar radiation due to Rayleigh scattering (τ _r), absorption by gases (τ _g), absorption by water vapour (τ _w) and attenuation (absorption and scattering) by aerosols (τ _a). The terms τ _r and τ _g are functions of optical air mass and air pressure, while τ _w can be estimated from optical air mass and precipitable water (a function of screen-level temperature and humidity (Reference PrataPrata, 1996)). Expressions of the transmissivities are taken from Reference IqbalIqbal (1983) and Reference Meyers and DaleMeyers and Dale (1983), with attenuation by aerosols given by:

where m is the optical air mass, and x is an altitude-dependent constant which decreases downslope as the aerosol concentration increases. Here the relation proposed by Reference Klok and OerlemansKlok and Oerlemans (2002) for high mountains is employed. Their linear fit predicts x = 0.88 at sea level and x = 1 above 4925 m a.s.l. (∼565 hPa), so τ _a = 1 at the Kibo AWS3. This is consistent with the general vertical pattern of aerosol optical depth in the tropical Andes, which approximates zero in the 500–600 hPa region (Reference HastenrathHastenrath, 1997). Also, Reference HastenrathHastenrath (1984) concluded from solar radiation measurements on Mount Kenya (∼590 hPa) that turbidity by aerosols is negligible.

Diffuse solar radiation in clear sky is parameterized in one term, derived from Reference IqbalIqbal (1983) who resolves D _cs in Rayleigh-scattered, aerosol-scattered and sky-reflected diffuse radiation:

Besides the variables that are already involved in the computation of S _cs above, τ _aa is the transmissivity due to absorption of solar radiation by aerosols (a function of m and τ _a (Reference IqbalIqbal, 1983)), p is local air pressure (hPa) and p ₀ is 1013.25 hPa. K _dif is a constant which incorporates the forward scatterance of Rayleigh and aerosol scattering, and ranges between 0.4 (for Rayleigh scattering) and 0.66 (for aerosol scattering) in Reference IqbalIqbal’s (1983) expressions.

We first model clear-sky global radiation (S _cs + D _cs) and optimize K _dif for a selection of clear-sky days from the AWS3 data using the following three criteria: (1) Mean daily G must be at least 85% of TOA radiation, which captures the peak values in daily G (Fig. 2). (2) Daily net longwave radiation must be smaller than its 10% percentile value. This criterion follows an approach used by Reference Van den Broeke, Reijmer, van As and BootVan den Broeke and others (2006), who showed that clear skies coincide with highly negative values of net longwave radiation at different elevations in Dronning Maud Land, East Antarctica. Energy balances in Antarctica (during summer) are, in turn, very similar to those on high-altitude tropical glaciers (Reference Wagnon, Sicart, Berthier and ChazarinWagnon and others, 2003). (3) Daily relative humidity (RH) must be <35%, which appears to be a useful threshold as identified in a scatter plot of daily G versus daily RH (not shown). Of 700 days, 59 pass this test for the optimization of K _dif.

All-sky G is modelled by optimizing the product kn _eff for the entire global radiation record from AWS3 (Fig. 2), i.e. for 700 days. Hence, different n _eff series are produced from Equation (1) for k values between 0.40 and 0.80 (by inserting measured G on the left-hand side and resetting resultant n _eff <0 or n _eff >1 to n _eff = 0 and n _eff = 1, respectively), and the optimal pair (k and its n _eff series) is determined. Reference HastenrathHastenrath (1984) suggests k = 0.65 for the equator (i.e. G under a completely overcast sky is 35% of clear-sky G), which he adopted from the literature. A big advantage of having AWS3 for the optimization procedure is its location very close to Kibo’s peak (Fig. 1), which almost completely eliminates shading of the CNR1 sensors by surrounding terrain (discussed further in section 3.1).

2.3. The longwave radiation model

Based on the Stefan–Boltzmann law, it is well known that incoming longwave radiation from the atmosphere (L ↓) depends on atmospheric emissivity (∊ _A), the Stefan–Boltzmann constant (σ) and absolute temperature of the air (T). For L ↓ in clear sky (expressed below by the subscript ‘clear’ ) the relation may be written as

The presence of clouds increases the atmospheric emissivity by a cloud factor F _cl ≥ 1, which is thus positively correlated with n _eff (Reference Niemelä, Raisanen and SavijärviNiemelä and others, 2001a). Hence, for all-sky conditions ε _A = ε _A−clear Fcl and L ↓= L ↓_clear F _cl, and therefore F _cl = L↓ =L↓_clear (e.g. Reference Sicart, Pomeroy, Essery and BewleySicart and others, 2006). For the clear-sky emissivity, Reference BrutsaertBrutsaert (1975) gives an expression (with parameters P ₁ = 1.24 and P ₂ = 7) based on water-vapour pressure, e (hPa), and T (K), which both refer to screen level:

To formulate an optimal longwave radiation model for L ↓ data from AWS3 on Kibo, the relation between n _eff (obtained from the optimized solar radiation model) and F _cl is examined first. F _cl is obtained by dividing measured all-sky L↓ by L↓_clear calculated from Equations (5) and (6) (using measured e and T). This is followed by an optimization of the clear-sky model (parameters P ₁ and P ₂) over the 59 clear-sky days defined in section 2.2, and the determination of an optimal function F _cl(n _eff) over the entire 700 day period (section 3.3).

2.4. The mass-balance model

A detailed description of the mass-balance model is found in Reference Mölg, Cullen, Hardy, Kaser and KlokMölg and others (2008) who employed it to simulate mass-balance response at AWS3 to climate fluctuations. The model computes the specific mass balance as the sum of snowfall, surface deposition, internal accumulation (refreezing of meltwater in snow), meltwater runoff and surface sublimation. This approach is based on the surface energy balance (SEB) of a glacier in the following form:

where S ↓ is incoming shortwave radiation (equivalent to G for a horizontal glacier surface), α is surface albedo, L ↓ and L ↑ are incoming and outgoing longwave radiation, QS and QL are the turbulent fluxes of sensible and latent heat, respectively, QC is the conductive heat flux in the subsurface, and QPS is the energy flux from shortwave radiation penetrating to the subsurface. The sum of these fluxes yields a resulting net flux F which represents the latent heat flux of melting if glacier surface temperature (T _sfc) reaches 273.15 K. The model’s SEB module interacts through T _sfc with the subsurface module, which solves for the englacial temperature on a numerical grid (see Reference Mölg, Cullen, Hardy, Kaser and KlokMölg and others, 2008, fig. 4). The sign convention used is that a flux is positive (negative) when it induces a heat gain (sink) at the surface.

As a minimum input, the model requires air temperature, air humidity, air pressure, wind speed and precipitation rate. S ↓, α, L↓ and L ↑ (a function of T _sfc) can be measured input or, optionally, are parameterized by the model. Here, the model is run in hourly time-steps for the 1 year dataset from Glaciar Artesonraju. S ↓ and L ↓ are supplied by measurements for the reference run, but are generated from the optimized radiation scheme in sensitivity runs. This adds to the validation part of this study (section 3.4), to explore whether the proposed radiation scheme has skill for mass-balance modelling of tropical glaciers other than those found on Kilimanjaro (section 3.5).

The set-up of the mass-balance model is as in Reference Mölg, Cullen, Hardy, Kaser and KlokMölg and others (2008), except for the turbulent heat-flux computation (see Reference Möolg and HardyMölg and Hardy, 2004) where roughness lengths are set to 0.3 × 10⁻³ m (Reference JuenJuen, 2006), and for the constant bottom ice temperature (at 3 m depth) which is set to melting point because the ice at the AWS site on Glaciar Artesonraju is temperate. In addition, the precipitation rate is derived from the station Llupa (3800 m a.s.l.) and corrected for the AWS altitude (4850 m a.s.l.) with a vertical gradient of 0.035 mm m⁻¹ month⁻¹ (Reference JuenJuen, 2006). Local slope at the AWS is negligible, so S ↓ = G.

3.1. Solar radiation model: optimization

For the 59 clear-sky days, the optimization of K _dif yields 0.66. Equation (4) does not explicitly account for diffuse radiation reflected from the sky, so it seems reasonable that the value of the optimal K _dif is at the upper threshold and not in the centre of the commonly used K _dif range (see section 2.2). The outcome of modelling clear-sky G is shown in Figure 3. The largest daily deviation (20 W m⁻²) only represents 5.4% of measured G, which is within the measurement uncertainty. The initial lower cluster of clear-sky G (dots and crosses within the dashed rectangle in Fig. 3a) exhibits a positive model bias. Closer examination showed that this is due to a high overestimation of G by the model (factor >4) in the hour 0700–0800 local time (LT) on 9 of the 59 days (Fig. 3b). These nine hours showed almost identical characteristics of sun elevation (11–12°), illumination direction (68–74° from north) and time of year (day 200–226), which suggests strongly that the sensor is shaded in this very narrow range of conditions. This does not have a large impact over the entire period (700 days), but the model performance for clear-sky conditions may be better assessed if modelled G is set equal to modelled D _cs in these nine hours. Assuming shading for the nine hours in question removes the systematic model error, and the lower cluster spreads around the 1 : 1 line (dots and open circles within the dashed rectangle in Fig. 3a).

Fig. 3. Measured and modelled clear-sky global radiation for (a) daily means (N = 59 days) and (b) hourly means (N = 1416 hours) at AWS3 on Kilimanjaro. Different symbols for the lower cluster in the daily plot (dashed rectangle) are explained further in the text.

For all-sky conditions, the optimal performance in fact occurs for an n _eff that is solved from k = 0.65 (daily r = 0.99; RMSD = 7.0 W m⁻²), so the original value adopted from the literature (Reference HastenrathHastenrath, 1984) can be maintained. The derived cloud pattern is depicted in Figure 4, with a mean n _eff of only 0.2. The major peaks nicely coincide with the wet seasons MAM 2005, MAM 2006 and OND 2006. Reference Mölg, Cullen, Hardy, Kaser and KlokMölg and others (2008) note that the OND wet season in 2005 was anomalously dry, which is corroborated by the pattern of n _eff. Figure 4 also shows the monthly effective transmissivity (τ _eff) for shortwave radiation (measured G/TOA radiation). Given the relatively short path of the solar beam through the tropical atmosphere and low n _eff over Kibo, τ _eff is generally high. On a typical Alpine glacier, τ _eff is considerably lower and lies in the range 0.35–0.59 (Reference Oerlemans and KnapOerlemans and Knap, 1998). Values on Kibo (0.67–0.83) resemble much more those in the atmosphere over East Antarctica, where Reference Van den Broeke, Reijmer, van As and BootVan den Broeke and others (2006) report a range of 0.64–0.79. The minima of τ _eff on Kibo correspond to the wet seasons when n _eff is relatively high (Fig. 4), with the exception again of OND 2005.

Fig. 4. Daily estimates of the daytime effective cloud-cover fraction n _eff (grey bars, with the bold line plot showing the 30 day running average) at AWS3 on Kilimanjaro between 9 February 2005 and 9 January 2007. The point-symbol plot gives the monthly effective transmissivity of the atmosphere for shortwave radiation, τ _eff.

The optimized K _dif and k also allow an estimation of diffuse solar radiation to be made, which reaches a maximum of 35% of potential G during overcast conditions (Equation (1)). ‘Potential’ refers to a clear sky and a non-obstructed direct sunbeam. In the clear-sky run (Fig. 3), D _cs only amounts to 4.6% of potential G. Since the optimization for clear-sky conditions only concerns D _cs (through K _dif) our results suggest that parameters from the literature (for τ _cs) appear to work adequately (Fig. 3), which is not unexpected for the highly transparent atmosphere over Kibo in clear sky. The small contribution of D _cs to potential G agrees with high-altitude measurements by Reference HastenrathHastenrath (1978, Reference Hastenrath1984) on Quelccaya ice cap, Peruvian Andes, and Lewis Glacier, Mount Kenya. Hastenrath found that D _cs accounts for only 2–10% of potential G. Specifically, for nearby Mount Kenya (∼590 hPa) he reports a range of 5–10%; thus, the smaller value on the more elevated Kibo (∼500 hPa) seems consistent. For intermediate n _eff, a linear relation may be used to approximate the amount of diffuse solar radiation (Reference HastenrathHastenrath, 1984), here 4.6% of potential G at n _eff = 0 and 35% at n _eff = 1. Mean measured G on Kibo over 2 years (333 W m⁻²) would, on average, therefore consist of 286 W m⁻² direct and 47 W m⁻² diffuse solar radiation.

3.2. Assessing the relation between n _eff and longwave radiation

Based on theory and measurements around the world (Reference Niemelä, Raisanen and SavijärviNiemelä and others, 2001a), clouds increase L↓ as described by the cloud factor F _cl (see section 2.3). Figure 5 shows the scatter plot of F _cl versus n _eff. Only the hours 1100–1500 h LT are considered, since small errors in modelled clear-sky G (or measured G) at relatively low solar elevations may lead to a large error in n _eff as defined by Equation (1). This is apparent on some dry days when n _eff is close to zero the whole day but >0.7 in the early morning and in the hour before sunset. The chosen time window helps reduce the scatter by restricting observations to relatively high solar elevations (>49°). Figure 5 demonstrates the expected positive correlation between F _cl and n _eff. The scatter plot, however, indicates a polynomial shape for the relation, which implies the sensitivity of L ↓ to n _eff decreases at greater n _eff. Plotting the relation for daily means (not shown) suggests a somewhat stronger linearity than the hourly values in Figure 5, but a polynomial fit leads to a higher r ² than a linear fit for the daily data as well. A similar non-linear relation has been shown by Reference Sicart, Pomeroy, Essery and BewleySicart and others (2006) in their study from Yukon, Canada, at ∼60° N, where an overcast sky also amplifies longwave emissivity by roughly 50% compared to a clear sky (Reference Sicart, Pomeroy, Essery and BewleySicart and others, 2006, fig. 4).

Fig. 5. Cloud factor F _cl (all-sky divided by clear-sky incoming longwave radiation) versus the effective cloud-cover fraction n _eff derived from the solar radiation model (Equation (1)) between 1100 and 1500 h LT (N = 2800 hours) at AWS3 on Kilimanjaro. Clear-sky incoming longwave radiation is calculated after the model of Reference BrutsaertBrutsaert (1975), while all-sky incoming longwave radiation is a measurement. The black curve shows the polynomial fit.

On Kibo, the decreasing slope in the F _cl–n _eff relation at n _eff >0.5 (Fig. 5) is probably linked to the observed negative correlation between T and L ↓ for relatively high water-vapour pressures (Fig. 6) and, thus, counteracts the factor T ⁴ in the general formulation of L ↓ (section 2.3). High e presumably coincides with cloud coverage most of the time, and the atmospheric emissivity therefore becomes increasingly a function of the number of hydrometeors per volume air, a rise in which is favoured by low T. The example of e from 3.5 to 3.6 hPa (Fig. 6b) supports this interpretation by the high concentration of data points at T ≈ 267.5 K and L↓ ≈ 290 W m⁻², which indicates radiative equilibrium between the local atmosphere and surface boundary layer (σ × 267.5 K = 290 W m⁻²), an unlikely situation if conditions are not near-overcast. The analysis (Fig. 6) is also in line with the somewhat stronger linearity in the daily F _cl–n _eff relation (see above), since the hours 1100–1500 h LT (Fig. 5) feature a greater fraction of high water-vapour pressures than the entire day.

Fig. 6. (a) Correlation coefficient between screen-level air temperature T and incoming longwave radiation L ↓ for water-vapour pressure (e) bins between 0.1 and 4 hPa (bin size is 0.1 hPa; p values based on t test) at AWS3 on Kilimanjaro. Bins of N < 100 hours are not shown. (b) Two bins are chosen for the scatter plot L ↓ versus T, one characteristic of low e and one of relatively high e.

For low-latitude glaciers, the only attempt to relate longwave radiation and cloudiness (to our knowledge) was made for Glaciar Antizana, Ecuador, by Reference Francou, Vuille, Favier and CáceresFrancou and others (2004). They define a cloud index as 1.3–1.4τ _eff, and illustrate a clear correlation to the TOA-OLR on a monthly scale (r = −0.68). As shown above, τ _eff and n _eff strongly correlate (Fig. 4), so are equal measures of the cloud impact on clear-sky solar radiation. Our n _eff correlates with TOA-OLR over Kilimanjaro at r = −0.72 on a monthly basis, which is very similar to Antizana conditions. These results, along with the similarity of the F _cl–n _eff relation (Fig. 5) to that in other cold regions (Reference Sicart, Pomeroy, Essery and BewleySicart and others, 2006), corroborate that n _eff derived from Equation (1) is a useful term to characterize cloudiness on tropical mountains.

3.3. Longwave radiation model: optimization

Brutsaert’s formula, based on the 59 selected clear-sky days, performs reasonably using the original parameters P ₁ = 1.24 and P ₂ = 7 (RMSD is 19.9 W m⁻² between daily measured and modelled L↓_clear). An optimal performance (RMSD = 6.6 W m⁻²) is reached if P ₁ = 1.24 and P ₂ = 6, which we therefore propose for conditions over tropical glaciers. The new P ₂ value conforms to Reference BrutsaertBrutsaert’s (1975) suggestion for climates of low water-vapor amounts, which is met in these conditions. This leads to the following relation between incoming longwave radiation, F _cl and n _eff on Kibo (with parameters fitted to the 2800 hours shown in Fig. 5):

To make this relation applicable to lower-lying tropical glaciers as well, we propose an altitude correction to the above equation, because the relative range of moisture variability in the tropical atmosphere (compared to the mean state) seems to increase with altitude (and therefore affects the range of the ratio L ↓/L ↓_clear as discussed below):

Table 1 complements these facts by showing that measured L↓ at 5873 m a.s.l. on Kilimanjaro (502 hPa) shows an even greater absolute range than at the 1000 m lower-lying measurement site on Glaciar Artesonraju (570 hPa). Since both datasets cover at least one complete annual cycle, it is very likely that the 5% percentile reflects clear-sky days, and the 95% percentile overcast days. This implies that the range of F _cl from n _eff = 0 to n _eff = 1 decreases with decreasing altitude. An altitude-dependent form of Equation (8) then reads

where p is the local air pressure at the measurement site (in hPa), and 502 hPa is the (reference) mean air pressure at AWS3 on Kibo. Since Equation (9) can be solved iteratively for n _eff, all-sky G can now be parameterized through n _eff derived from L ↓ measurements or, in the opposite direction, L↓ can be parameterized (for daytime) by n _eff obtained from measured G (Equation (1)), provided that screen-level measurements of T and e are available.

3.4. Validation for a different site: Glaciar Artesonraju

Here we maintain the model parameters optimized for Kilimanjaro data (K _dif = 0.66; k = 0.65; P ₂ = 6; Equation (9) for F _cl) and formulations used from the literature (Reference BrutsaertBrutsaert (1975) for P ₁; Reference IqbalIqbal (1983), Reference Meyers and DaleMeyers and Dale (1983) and Reference Klok and OerlemansKlok and Oerlemans (2002) for τ _cs and D _cs) but apply the radiation scheme to the 1 year dataset from the AWS on Glaciar Artesonraju (section 2.1). Shading of this AWS by the surrounding relief only occurs, if at all, in the early-morning hours (Reference JuenJuen, 2006), so has a negligible impact on modelling G.

Table 2 presents the statistical validation for both G and L↓ parameterizations, and as a comparison the same statistics are shown for the optimization dataset from Kilimanjaro. For L ↓, only the daytime validation makes sense (when n _eff can be derived from measured G), and how to treat n _eff during night in that case should be an individual decision (e.g. linear interpolation between evening and morning (Reference Klok and OerlemansKlok and Oerlemans, 2002) or by assuming mean n _eff of the preceding afternoon (Reference Lhomme, Vacher and RocheteauLhomme and others, 2007)). RMSD in the order of 100 W m⁻² for G appears large at first glance, perhaps since similar validations are usually given for daily means in the literature (e.g. Reference Klok and OerlemansKlok and Oerlemans, 2002), which greatly reduces the RMSD (e.g. Fig. 3). For hourly values, however, such a RMSD is common in all-sky solar radiation schemes (cf. Reference Niemelä, Raisanen and SavijärviNiemelä and others, 2001b, table 2). Further, a high hourly RMSD is favoured for our tropical sites where G in the early afternoon regularly exceeds 1000 W m⁻². For L↓, the performance strongly resembles that of other broadband schemes in all-sky conditions, where the explained variance drops from ∼75% to 55% between optimization and validation data that refer to different sites but a similar climate zone (cf. Reference Lhomme, Vacher and RocheteauLhomme and others, 2007, table 3). Hence, there is no indication that the radiation scheme developed in this paper underperforms those for other environments.

Figure 7 shows the scatter plots associated with Table 2 for the validation site. The only sign of a systematic discrepancy appears for L ↓ > 320 W m⁻², where model values are lower than those obtained from measurements. The mean difference of 25 W m⁻² is still smaller than the nominal accuracy of the sensor for this data range (33 W m⁻²). From the model’s perspective, the mean n _eff for these cases (0.75) appears rather low for such high L ↓. On the other hand, CNR1 measurements may overestimate the actual L ↓ if riming occurs (Reference Van den Broeke, van As, Reijmer and van de WalVan den Broeke and others, 2004). On tropical glaciers, with little variation in daily air temperatures and strong solar irradiance year-round (Table 1), this is most likely on moist days with high L↓. The broad scatter of measurements for modelled G around 150 W m⁻² (Fig. 7a) also indicates that the mostly lower model values could originate from a too high n _eff due to a too high L↓ measurement. Nine of the ten days with modelled G <160 W m⁻² but measured G >200 W m⁻² indeed show measured daytime L ↓ >320 W m⁻² as well. In summary, though, Table 2 and Figure 7 demonstrate that the model parameters optimized for Kilimanjaro are transferable to other tropical mountain sites.

Fig. 7. Measured and modelled mean daily global radiation G (a) and daytime incoming longwave radiation L ↓ (b) at the Artesonraju AWS over 25 March 2004 to 31 March 2005 (validation dataset). Daily means are calculated from the hourly values evaluated in Table 2.

3.5. Implications for mass-balance modelling

Here we consider the radiation scheme’s suitability for incorporation into mass-balance modelling at the validation site. This deserves exploration, in order to highlight the importance of correctly simulating the diurnal cycle of incoming radiation. Periods of strong ablation on tropical glaciers depend mainly on short early-afternoon periods – scattered over the whole year – when melting may contribute to mass loss in addition to the more-or-less continuous sublimation (Reference Mölg, Cullen, Hardy, Kaser and KlokMölg and others, 2008). This contrasts with extratropical mountain glaciers, where strong ablation is confined to the summer seasons when the surface is almost constantly at melting point (and, thus, a correct simulation of daily radiation may be sufficient).

Figure 8a presents the application of the AWS data to the mass-balance model, where all required input data (section 2.4) are measurements (reference run). There is very good agreement between the measurements and model in terms of the net specific mass balance (b _n). The most important mass fluxes over the 372 days at this site are surface melt (6.75 m w.e.), solid precipitation (1.15 m w.e.) and surface sublimation (0.15 m w.e.). Details about the microclimate in the ablation area, where the AWS is situated, can be found in Reference JuenJuen (2006).

Fig. 8. Measured and modelled specific mass balance at the Artesonraju AWS between 25 March 2004 and 31 March 2005, using (a) measured global radiation G and measured incoming longwave radiation L ↓ as model input; (b) parameterized G (case 1 in Table 2) and measured L ↓; and (c) measured G and parameterized L ↓ (case 2 in Table 2). The grey envelopes illustrate the range of the reference run when (center) measured G or (right) measured L ↓ is offset by ±5%.

Figure 8b and c present two further runs, which use parameterized G or L ↓ as input obtained from the radiation scheme. For the L ↓ parameterization, night-time n _eff (not deducible from G measurements) is interpolated linearly between n _eff in the sunset and sunrise hours, as in Reference Klok and OerlemansKlok and Oerlemans (2002). There are minor positive but no significant deviations from the reference run, evaluated with respect to the effects that a 5% measurement error in G or L ↓ would have (i.e. the black curve favours the upper part of the shaded error margin in Figure 8). Interestingly, using parameterized G with a positive bias (Table 2) also leads to this positive deviation, while one could anticipate a negative deviation because of the higher energy supply to the glacier. The positive bias in G (Table 2), however, results mainly from an overestimation of G in the morning and late afternoon, while G is underestimated in the early afternoon when melting takes place – which connects to the opening discussion of this section. This suggests increased sensitivity of modelled b _n to a shortening of sub-daily melt periods, since diurnal variations in the scheme’s performance (under-versus overestimation of measurements) have less time to compensate each other. Again, the differences in modelled b _n introduced by the radiation parameterizations (Fig. 8) are small and, in a physically based mass-balance model which treats feedback processes, are most probably concealed by errors from other sensitive parameterizations like albedo, turbulent heat transfer and surface temperature schemes (e.g. Reference Mölg, Cullen, Hardy, Kaser and KlokMölg and others, 2008; Reference Reijmer and HockReijmer and Hock, 2008). Still, the length and timing of melt periods of a particular tropical glacier (site) should be considered as critical if the radiation scheme is to be adopted.