5.1. Observations of Thermal, Hydrological, and Meteorological Conditions
 We observed large temporal and spatial variability in stream temperatures in the Damma forefield, similar to results reported in studies of other proglacial field sites [Uehlinger et al., 2003; Brown and Hannah, 2008]. The observed stream temperatures were well above the freezing point at the location closest to the dead ice body (site S1), indicating that much of the water reaching this location was warmed as it crossed the rock face below the glacier, before passing underneath the dead ice body (see Figure 1). The even higher measured temperatures in the northern tributary stream (site S9) showed that the melt water from the glacier, which had already traveled between 1 and 2 km across an open rock face and a small proglacial area before reaching this location (see Figure 1), responded strongly to atmospheric heat fluxes during daytime. Thus, the cold glacier melt water can warm quickly (up to roughly 10°C) during daytime over short distances (less than roughly 2 km). The observed stream temperatures at the site below the confluence (site S8) were roughly 1.5°C higher than the volume-weighted average of the waters passing the two upstream measurement locations (S1 and S9). The differences demonstrate that the streams on the forefield are not in equilibrium with the surroundings, and that inferences regarding the stream's energy balance can be gathered by studying longitudinal temperature variations in the streams.
 We find different diurnal patterns in the temperature increases measured over two different stream reaches on the forefield (Figure 4). For the upstream reach (site S2 to S4), the stream temperatures reacted quickly to changes in meteorological conditions, and did not demonstrate any long-term memory of past variations in either solar radiation or air temperature. This is not paradoxical because cold melt water continually flushes the system quickly, resulting in residence times shorter than one solar heating cycle. The strong correlation between the temperature increase and shortwave radiation along the upstream reach suggests that direct solar heating is an important process warming the stream. The temperature increase along the short downstream reach (site S4 to S5) was relatively large during nighttime, particularly compared to the longer upstream reach. The large nighttime temperature increase indicates that heating through the streambed is an important factor warming the water. We think that the most likely explanation for the nighttime temperature increase along the downstream reach is groundwater inflow and hyporheic exchange. First, dye tracers injected into the groundwater monitoring wells at site S5 were detected in small springs on the nearby stream bank after a short time (<20 min), indicating subsurface water flow toward the stream. Second, we do not expect that heat conduction through the streambed could create differences in stream warming between the two reaches as large as those we observe during nighttime.
 Groundwater inflows often influence longitudinal stream temperature patterns [Westhoff et al., 2007]. However, the sum of discharges in the two tributaries (at sites S1 and S9) matched the discharge observed below their confluence (at site S7) within the accuracy of the measurements (Figure 3). Thus, if stream-groundwater interactions play an important role across the forefield, any gains and losses of stream water approximately balance between the upstream (sites S2 and S9) and downstream (site S7) discharge measurement sites. On proglacial fields, surface water can often infiltrate into the riparian zone along the streams, and return back to the channels further downstream [Malard et al., 2002], potentially having absorbed additional heat from the riparian subsurface.
5.2. Processes Influencing Stream Warming: Identify Dominant Processes
 The stream energy balance equation shows that surface heat fluxes per unit discharge should influence the longitudinal temperature increase along a reach (see equation (1)). To illustrate the relationship between stream water warming, surface heat fluxes and discharge in more detail, we analyzed the results obtained in the beginning of September when snow fell on the glacier resulting in a decrease in discharge (Figure 5). The surface heat fluxes were similar before and after this snowfall but the stream warming suddenly increased because the flowrate q(t) decreased more than the stream surface area, which is given by the (variable) stream width w(t) times the (fixed) reach length L. Thus, not only variations in radiative forcing influence the stream warming, but variations in the hydrological conditions, and in this case discharge variations, play an important role in the stream's temperature dynamics.
 With the regression analysis based on the energy balance equation, we can quantify how strongly the surface heat flux per unit discharge influences the stream warming (Figure 6, Table 1). The regression results indicate that heat fluxes across the stream surface are the major cause of the observed temperature increases over three different reaches (S1 to S2; S2 to S3; S3 to S4) including the whole reach stretching from site S2 to S4. We also found that the slopes of the regression lines always became steeper with increasing discharge because the stream width increased with discharge. The differences between the intercepts of the regression lines and the calculated frictional warming were small, and may indicate uncertainties within the calculated surface heat fluxes, measurement errors, or the influence of heat fluxes included in the residual temperature change (see equation (1)). For the downstream reach (S4 to S5), the weak correlation between ΔT and Q/q indicate that surface heat fluxes only weakly influence stream warming, and that one or more of the energy balance terms included in the residual temperature change instead warms the stream water.
 To explore the possible role of stream-groundwater interactions in warming the stream between site S4 and S5, we measured groundwater levels, temperatures, and gradients using two monitoring wells, located along a transect perpendicular to the stream (2 and 4 m from the channel) at site S5. Stage measurements 4 m away from the stream were used because the head level differences between the groundwater and stream were more clearly defined in this location than in the near-stream well. Conversely, groundwater temperatures in the near-stream well were used because they better represent the temperature of the subsurface water entering the stream. In the following, we only analyzed observed temperature changes between 21:00 and 06:00 to minimize the influence of radiative fluxes on stream temperature dynamics. We also subtracted the frictional warming constant from the observed stream warming.
 The average nighttime temperature increase along the reach stretching from site S4 to S5 correlated well with the head difference between the groundwater and stream level (Figure 12). The weak correlation between variations in air temperature and stream warming indicates that surface heat fluxes do not largely influence the stream temperature dynamics during nighttime. Similarly, the longitudinal increase in stream temperature did not show any obvious relationship with the remaining explanatory variables presented in Figure 12. However, the positive relationship between the temperature increase and the head difference indicates that subsurface inflows influence the stream temperature change. These subsurface inflows could include groundwater flowing from the surrounding catchment toward the stream, and/or hyporheic return flow of stream water that has infiltrated into the riparian aquifer upstream from the well transect and has been warmed in the subsurface before returning to the channel. We cannot distinguish between these two mechanisms with the measurements that we have.
Figure 12. Nighttime average temperature increase (21:00–06:00) from S4 to S5, corrected for frictional warming, compared to several variables that could influence stream warming: air, stream, and groundwater temperatures (Tair, Tstream, and Tgroundwater, respectively), stream discharge (q), head difference between the stream and a groundwater well 4 m from the channel (Δh), and temperature difference between the stream and a groundwater well 2 m from the channel (ΔTG). The observed temperature increase shows a clear relationship to the stream-groundwater head difference, but no strong correlation to the other variables.
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 If subsurface water inflow causes the longitudinal temperature increase, the ratio between observed nighttime stream warming (ΔT) and the temperature difference between the groundwater and stream water (ΔTG) also gives the ratio between subsurface inflow and stream discharge (provided that groundwater temperatures measured 2 m away from the stream are representative for the inflowing water). We found that a seasonal declining trend in the temperature ratio ΔT/ΔTG (and thus in the ratio of subsurface inflow to streamflow) corresponded to a similar trend in both groundwater level and the head difference between the groundwater and the stream (Figure 13). The seasonally decreasing head difference is a result of a decline in the riparian zone groundwater level. The results indicate that the ratio between subsurface water inflows into the channel and stream discharge is low (roughly 3 to 12%). If other heat fluxes also contribute to the nighttime temperature increase, the calculation above will overestimate the contribution of subsurface inflows to the stream. Note that the heterogeneous sediments and complex topography of the field site lead to highly complex stream and groundwater flow patterns (J. Magnusson et al., submitted, 2012). Therefore, any inferences drawn from our local measurements of head and temperature gradients should be treated with caution. More work is needed to explore the spatial heterogeneity of groundwater levels and stream-groundwater interactions in complex mountainous sites like the Damma glacier forefield.
Figure 13. (top) The solid black line shows the ratio between the stream warming (with frictional heating removed) and the stream-groundwater temperature difference (ΔT/ΔTG), and the dashed red line shows the head difference between the groundwater level and stream stage. (bottom) The solid line shows the same temperature ratio, but the dashed line shows the groundwater level in the monitoring well (in meters above sea level). The figure shows nighttime averaged values (21:00 to 06:00) for the reach from site S4 to S5. Groundwater levels were measured 4 m from the stream at site S5. The temperature ratio (ΔT/ΔTG) declines during the season, and follows the downward trend in groundwater levels in the nearby riparian zone. If subsurface inflows to the channel cause the longitudinal temperature increase, ΔT/ΔTG represents the ratio between subsurface water inflow and stream discharge.
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5.3. Inferring Hydraulic Geometry Relationships From Stream Temperature Measurements
 The exponents obtained for the three hydraulic geometry relationships are comparable with observations made elsewhere. The fitted width exponent is toward the upper end of the range usually observed in natural channels [Park, 1977]. However, high width exponents are typical for shallow gravel bed streams, as noted by Smith and Pavelsky , who found a similar exponent (b = 0.48 ± 10%) in a remote sensing study of a braided channel in Siberia. In another study, Chikita et al.  reported a slightly lower value than ours (b = 0.42) for a glacial forefield stream in Alaska. To our knowledge, the highest obtained width exponent (b = 1.05) was reported by Ashmore and Sauks  for a proglacial braided stream in British Columbia. Our velocity exponent (m = 0.29) falls well within the range of values summarized by Park . Our depth exponent (f = 0.22) is slightly lower than the majority of observations reported by Park , but is higher than the depth exponent reported by Chikita et al.  for their proglacial stream (f = 0.17). These studies, based on direct measurements, suggest that the exponents derived here may be realistic. Our results indicate that the studied stream reach accommodates discharge mostly through widening rather than large changes in depth and flow velocity.
 Note that in the comparison between the reach-averaged depth-discharge relationship and observed stream stage (see Figure 10), the numerical values differ (note the different scales on the left and right axes), which should be expected for at least three reasons: (1) the average depth over an entire reach will generally differ from the average depth at any cross-section, (2) the average depth at any cross-section will be less than the total depth, and will increase more slowly as a function of discharge (because the channel is not rectangular), and (3) there will be a constant offset between total depth and stream stage (because stream stage is measured relative to an arbitrary datum rather than the thalweg elevation). Nevertheless, the stage variations observed at the two sites are consistent with the inferred hydraulic geometry relationship.
 Our analysis shows that stream channel geometry has a direct and quantifiable effect on thermal relations in proglacial streams. Stream width has a strong enough effect on stream temperature dynamics (particularly when radiative heat fluxes are large) that the method outlined here can be used to derive reasonable estimates of the relationship between stream width and discharge. Of course, direct measurements of stream width would be preferable to indirect estimates obtained from stream temperature dynamics. However, measuring stream widths can be very difficult in shallow, irregular, highly braided channels like ours, which lack well-defined banks. Furthermore, our method gives estimates of average effective widths over long stream reaches; such estimates are difficult to obtain by most other methods.
5.4. Processes Influencing Stream Warming: Analyzing the Stream Energy Balance
 With ongoing glacier retreat, a good understanding of physical processes influencing the stream temperatures in glacial forefields are important, because stream temperatures in these environments are sensitive to changes in climate and runoff regimes, along with changing distances between the streams and the glaciers themselves [Webb et al., 2008; Chikita et al., 2010]. Our results show that variations in stream temperature are inversely related to discharge (Figure 11), which is consistent with previous results from alpine streams [Constantz, 1998]. In the case of proglacial streams, however, one might not expect an inverse relationship between discharge and longitudinal warming, because rates of glacial ablation (and therefore discharge) are highest precisely when air temperatures and solar fluxes (and therefore rates of heat transfer at the airstream interface) are also highest. Our results show that along with surface heat fluxes, stream morphology and discharge variations strongly influence the heating of proglacial streams on the Damma glacier forefield.
 We find that solar radiation contributed the greatest to the heat budget of the stream reach between sites S2 and S4 (Table 2). This energy balance component depends only on the albedo of water, which is fairly constant, and the amount of sunlight reaching the stream surface. In steep mountain streams, frictional warming can be large because this heating component is independent of time and discharge and only depends on the down-valley gradient [Meier et al., 2003]. The two energy fluxes mentioned above may not change substantially in the future because they are independent of the water temperature in the stream, which depends largely on the distance to glacier (compare, for example, the temperatures at sites S1 and S9 in Figure 2). On the other hand, the sensible and latent heat fluxes are driven by temperature and vapor pressure gradients above the stream surface. Our data show that the average sensible and latent heat fluxes are positive over the monitoring period, during both daytime and nighttime (Table 2). The calculated latent heat fluxes imply, perhaps surprisingly, that more water vapor condenses on the stream surface (warming the stream) than evaporates from the stream (cooling the stream). This condensation occurs because the air is warm and relatively moist, and the surface of the stream is very cold. With further glacier retreat, the stream temperatures on the field site will likely increase and, at the same time, we might expect drier air with higher temperatures in the future. As a result, we can expect that the latent heat flux will become negative causing water to evaporate from the stream (thus cooling the stream water) instead of condensing. For the particular stream reach from S2 to S4, we cannot identify the processes that underlie the residual temperature increase. This residual heating term is an important component in the stream energy balance, and more work is needed to quantify its possible sources.
 In the following, we discuss different uncertainties that may influence our analysis using the energy-balance equation. We are confident in our estimates of net shortwave radiation flux to the water surface; solar flux is measured nearby at the weather station on the forefield, and the albedo of water varies only slightly between different studies [Chikita et al., 2010; Leach and Moore, 2010]. We are also confident in our estimates of frictional warming, because they depend only on the elevation difference along the reach (which can be measured precisely), and on the assumption that the available gravitational potential energy is completely dissipated to heat (which is accurate because the change in kinetic energy of the flow is trivial by comparison). The calculated incoming longwave radiation was consistent with measurements made at a weather station located 2.6 km northeast of the forefield and 600 m above it (mean bias of −8 W m−2 and root mean square error of 28 W m−2). The calculated turbulent heat fluxes are more uncertain, mainly because the parameters in the equations were determined under conditions different from those prevailing at our study site [see Webb and Zhang, 1997; Leach and More, 2010]. However, at our alpine study site solar radiation dominates the surface heat fluxes, with relatively small contributions from turbulent heat fluxes. Finally, the measured discharge record at site S2 displayed a root mean square error of 0.10 m3 s−1 compared to the manual discharge observations, where discharges were measured with dilution gauging up to 2.4 m3 s−1.
 The residual temperature change primarily includes heat transport due to groundwater inflow, hyporheic exchange and streambed conduction (see equation (1)). Conductive heat flux through the streambed is negligible in gravel-bedded streams [Brown, 1969], and was also assumed negligible in a previous energy-balance study of proglacial streams [Chikita et al., 2010]. This heat flux can be calculated from streambed temperature measurements [Story et al., 2003], but temperature sensors are difficult to install in the stony and heterogeneous sediments that characterize proglacial streambeds. In any case, convection typically transports much more heat than conduction through the hyporheic zone [Malard et al., 2001]. Thus, we think that energy transfer at the streambed interface is probably dominated by heat inputs from groundwater discharge and hyporheic exchange. Some studies suggest that hyporheic flow may store and release heat through the streambed interface and influence the stream temperature dynamics [Evans and Petts, 1997; Alexander and Caissie, 2003; Story et al., 2003]. Nielson et al.  and Westhoff et al.  outlined methods for inferring hyporheic exchange from in-stream temperature observations, and validated their simulation results against independent streambed temperature measurements. Thus, such approaches may be particularly useful for proglacial streams such as ours, but require testing at field sites where validation measurements are possible.