The study of supraglacial streams and rivers atop the Greenland Ice Sheet (GrIS) is an emerging subfield with implications for the physical understanding of ice sheet subglacial hydrologic systems, ice motion, and sea level rise (Irvine-Fynn et al., 2011; Rennermalm et al., 2013; Chu, 2014; Flowers, 2018; Pitcher and Smith, 2019). When the GrIS surface melts, meltwater that is not evaporated, stored, or refrozen moves through what is now understood to be a complex perennial hydrologic system distinct from terrestrial hydrology (Yang et al., 2016; Pitcher and Smith, 2019). Recent advances in mapping (Lampkin and VanDerberg, 2014; Rippin et al., 2015; Smith et al., 2015, 2017; Yang and Smith, 2016), modeling (Banwell et al., 2012, 2016; Clason et al., 2015; Karlstrom and Yang, 2016; Yang et al., 2018) and measuring (McGrath et al., 2011; Legleiter et al., 2014; Gleason et al., 2016; Smith et al., 2017) supraglacial channel networks have revealed numerous similarities to terrestrial watersheds, but their scale and remoteness have limited the number of field studies.

This new appreciation for supraglacial hydrologic processes has emerged at a time of increasing accuracy and sophistication of surface mass balance (SMB) modeling of the GrIS. SMB models use regional atmospheric forcing to simulate GrIS surface mass balance components, including the amounts of meltwater production and of liquid water in excess of evaporation and retention and refreezing (termed “runoff”) available for hydrologic functions (Fettweis et al., 2020; Vernon et al., 2013). SMB models here refer to any global and/or regional circulation model (G/RCM) or reanalysis that explicitly simulates ice sheet surface runoff. These models are grid-based and operate at pan-GrIS scales, producing a single runoff value for a given model grid and time step. Note that the terrestrial hydrology community commonly uses the term “water excess” to represent the volume of water available for routing after hydrologic processes, while the glaciology community uses the term “runoff” to represent this same quantity specific to ice sheets. Most existing SMB models do not route this runoff and instead assume that all runoff not refrozen in snow or firn leaves the ice sheet as soon as it is produced (Fettweis et al., 2020). In reality, observations of the GrIS surface indicate that lake impoundment (e.g., Arnold et al., 2014), flow through weathering crust (e.g., Cooper et al., 2018), and transport through supraglacial stream and river networks modify the timing and magnitude of excess water reaching moulins or the ice sheet edge (Smith et al., 2017). Modeling these processes is precisely analogous to the use of land surface models in terrestrial hydrology, whereby a land surface model (SMB model here) produces gridded water excess (runoff here) and then routes this water with a coupled routing model. Coupling surface water processes to SMB models, loosely or tightly, is thus needed for a fuller representation of GrIS supraglacial hydrology to align this field with practices in terrestrial hydrology (e.g. Bates et al., 1997; Beighley et al., 2009; Wood et al., 2011; Lin et al., 2019).

Previous studies have begun to stich these two research avenues together. For example, Banwell et al. (2012) used Darcy's law to describe meltwater flow routing through snow and Manning's equation to describe lateral runoff transport across bare ice and then later used this meltwater to fill supraglacial lakes or supply surface meltwater to moulins (Banwell et al., 2013, 2016). Leeson et al. (2012) similarly used Manning's equation to transport water in a 2D grid-based routing scheme, assigning all grids a uniform Manning's n while not explicitly defining flow differences between flow in channels and flow over bare ice. Liston and Mernild (2012) also applied mass conservation at the grid cell level to route runoff between grid cells and did not account for the presence of channels that convey this runoff with distinct hydraulics. Smith et al. (2017) attempted to address this channel routing via the classic empirical Snyder synthetic unit hydrograph (SUH) model (Snyder, 1938) to calculate discharge hydrographs for the terminal moulins of 799 internally drained surface catchments in the southwest GrIS. Yang et al. (2018) used a similar classic empirical model, the rescaled width function (RWF; Rinaldo et al., 1995), to partition the ice surface into slow-flowing interfluvial (i.e., hillslope) and fast-flowing (open-channel) zones and calculated moulin discharge while improving the physical realism of the supraglacial routing process. Importantly, Yang et al. (2020) demonstrated the likelihood of subsurface unsaturated zone flow even through bare glacial ice, a phenomenon confirmed by field (Cooper et al., 2018; Irvine-Fynn et al., 2011; Munro, 2011) and theoretical (Karlstrom and Yang, 2016) studies. Yang et al. (2020) recently compared several of these empirical models and found they introduce significant variability in diurnal moulin discharges and corresponding subglacial effective pressures.

These previous efforts demonstrated successful meltwater transport modeling on the GrIS ablation zone and its necessity, but their relative simplicity allows space for the application of sophisticated routing models from terrestrial hydrology to be applied to ice sheet surfaces more generally. For instance, Lin et al. (2019) used gridded estimates of water excess (analogous to runoff) to simulate daily flows in nearly three million river reaches between 1979 and 2013 with fully conserved mass and momentum in realistic river networks globally. This undertaking was the first demonstration of this capability at global scale following years of well-established theoretical work and advances in hydrologic representation for big data. This routing approach is suitable for representing GrIS surface water transport processes as gridded runoff on ice sheets must be routed through supraglacial rivers, lakes, and hillslopes (which include firn atop the GrIS), as on land. Building and calibrating models to route water through landscapes and channel networks while obeying fundamental principles of mass and momentum conservation is an established practice in terrestrial hydrology that may readily be applied to ice sheet surfaces as well.

There are several barriers to applying such routing for the GrIS at the catchment scale. First, routing models require a well-defined channel network with explicit and continuous topology. There have been demonstrations of network mapping (Yang et al., 2016) and topology generation (King et al., 2016), but to our knowledge no automated, large-network-scale (i.e., catchments with thousands of channels or more) coupled extraction and topological connection work exists for the GrIS. Existing terrestrial routing models like the hillslope river routing model (HRR; Beighley et al., 2009) stand ready to route runoff “off the shelf”, yet these cannot be applied until a generalizable automated extraction and topological connection process is available. Applying a model such as HRR could also further understanding of GrIS river networks, which is currently underdeveloped (Pitcher and Smith, 2019). For instance, the relative importance of hillslope flows and channel density on runoff transport have not been explored on a first-principles basis at network scales, and model parameters controlling hillslope friction, channel friction, and runoff reduction and augmentation could reveal how these physical processes interact to produce channel discharges.

In this paper we use HRR to advance the physical understanding of GrIS supraglacial meltwater transport processes as follows. (1) We automatically generate spatially explicit topological networks of varying drainage densities for a supraglacial catchment for which a brief (72 h) in situ record of outlet channel discharge is available. (2) We route water runoff generated by four different SMB models through these networks at an hourly timescale. (3) We constrain and calibrate the routing via hourly in situ discharge measurements and previously published field measurements of supraglacial channel frictions and velocities. Our initial routing results immediately revealed a mismatch between modeled and routed runoff and measured channel flows, so our philosophy for this study is to assume that measured discharge at the outlet is correct and calibrate SMB runoff volumes and channel properties to match discharge observations as mediated through the physics of the routing model. (4) To advance understanding of hillslope processes and channel density on meltwater transport, we design an experiment to test how the representation of hillslope processes and network density (as derived by our automated network generation process) affects the routing model. We ultimately route meltwater through thousands of supraglacial channels every hour, and we solve (via conservation of mass and momentum inherent to routing) for the roles of channel friction, hillslope delay, and network density in controlling the magnitude and timing of water fluxes through supraglacial channels and ultimately moulin injection in our test watershed. These procedures and results form a blueprint for the general coupling of runoff modeling, water transport, and channel processes atop the GrIS.

We develop our routing model for Rio Behar, a previously studied, internally drained supraglacial river catchment in southwest Greenland. First introduced by Smith et al. (2017), the Rio Behar catchment is approximately 63 km2 and centered at 67.04∘ N and 48.55∘ W with a highly developed perennial and well-drained supraglacial stream and river network during peak flow periods of late summer. Smith et al. (2017) report that the basin elevation spanned approximately 1200–1400 m in 2015, with air temperatures in the summer measurement period ranging from -3 to 2 ∘C and net radiation ranging from approximately -100 to 300 W/m2. Previous work in the basin includes (i) a comparison of SMB runoff and field-measured discharge using a simpler routing method (Smith et al., 2017), (ii) a study of subsurface water storage in bare-ice weathering crust (Cooper et al., 2018), (iii) albedo mapping (Ryan et al., 2017), and (iv) satellite and uncrewed aerial vehicle (UAV) remote sensing work to map the catchment's supraglacial channel network (Ryan et al., 2017; Yang et al., 2018). Readers are referred to these published works for more information on the physical setting of the basin. Here we use the Rio Behar specifically because it is the only known large GrIS supraglacial river catchment with an hourly in situ record of channel discharge (see Sect. 2.2). Other discharge records exist, as, for instance, McGrath et al. (2011) who provide hourly discharge records for a small (1.1 km2) catchment, while Chandler et al. (2013) give hourly moulin (fed by a channel) discharge for another small catchment, but the size of Rio Behar and the wealth of previous work therein makes it an ideal setting for this study. Using high-resolution remote sensing, the watershed is delineated to an in situ streamflow measurement point (Sect. 2.2) that defines the outlet located less than 1 km upstream of the catchment's terminal moulin. Because all meltwater runoff passing out of our watershed penetrates the ice sheet via a moulin, accurate modeling of this water flux is important for studies of GrIS subglacial hydrology and ice dynamics (Chu, 2014; de Fleurian et al., 2016; Banwell et al., 2016; Flowers, 2018; Davison et al., 2019)

We have successfully calibrated a hillslope river routing model capable of simulating hourly flows through thousands of supraglacial channels atop the GrIS while conserving runoff mass and momentum. The most accurate models to emerge from our experiments were those that employed a fine-channel network and/or inclusion of hillslope flow routing. We assert that our results support the inclusion of realistically fine river and/or stream networks and hillslope-enabled routing models for supraglacial runoff modeling applications that require the realistic representation of runoff timing and magnitude. While we cannot validate in-channel flows upstream of the outlet, this level of hydrological simulation could, for instance, be coupled with SMB models to calculate hourly moulin discharge rates, lake fill-and-spill volumes, channel incision rates (e.g., following Karlstrom and Yang, 2016, or Koziol et al., 2017), and supraglacial contributions to subglacial water pressures (e.g., following Banwell et al., 2016 or Yang et al., 2020). These processes have important implications for GrIS surface hydrology, surface mass balance, and subglacial hydrological systems. We believe this work represents a promising step toward coupled SMB-routing modeling that can be used to generate more realistic predictions of these processes and their sensitivity to changing surface meltwater forcings or surface topography.

The goal of this study was not to interrogate individual SMB models or suggest one is better than another. A recent synthesis (Fettweis et al., 2020) showed that SMB models vary considerably given the same forcing, and readers are referred to this and other literature for further information on why these models might disagree. Smith et al. (2017) and Mankoff et al. (2020) have both explored what these differences mean for water exiting the GrIS, but our purpose is to demonstrate the importance of coupling SMB model output with a surface flow routing model to understand runoff transport before it enters the englacial system. This enables rigorous estimation of supraglacial flow accumulation and routing delays to moulins atop the GrIS that route meltwater into a dynamically varying subglacial hydraulic system that influences ice sheet acceleration in response to the timing and magnitude of input discharges, which is imperative to accurately estimate diurnally varying moulin discharges using climate models. Second, this work advances the physical understanding of ice sheet surface hydraulic properties, for example, our finding hillslope friction values (Table 2) well outside typical terrestrial values of 0.01 to 1 (Hergarten and Neugebauer, 1997; McCuen, 2004; Kalyanapu et al., 2010). Yang et al. (2018) similarly estimated slow transport of meltwater on ice interfluves (similar to the hillslopes studied here) some 2–3 orders of magnitude slower than open-channel flow (∼10-1 m/s). Observations of ice density and saturation in shallow ice cores within the Rio Behar catchment indicate that substantial subsurface meltwater is stored within the upper decimeters of bare-ice weathering crust and was anecdotally observed to percolate through the crust (Cooper et al., 2017, 2018). If so, this unsaturated flow would move orders of magnitude slower than bare-ice overland flow. These convergent findings are consistent with conceptual models of unsaturated subsurface porous media flow and support the very slow lateral transport we observe here (on the order of 10-5 to 10-1 m/s) to the channel from the ice surface, but we cannot make any further conclusions on physical processes or mechanisms given our experiment design and model setup. That is, since we lump all flow over and through the ice, firn, snow, and crust before it reaches channels into a single “hillslope” flow with a single friction, we can be confident in the speed of this transport but not its flowpaths or mechanism. This result highlights the need for further basic research on the supraglacial hydrological process to further understand the importance of these velocities.

The importance of including hillslope process is also clearly manifested through calibrated channel frictions generated in model experiments that exclude it. There are discernible changes in channel friction when hillslopes are or are not modeled, and the results are intuitive: channels lacking hillslopes have much higher friction, especially in large channels (Fig. 7). Further, for the largest channels (i.e., upstream areas greater than 1.260 km2), models without hillslopes take channel friction values almost uniformly at the maximum of the realistic constraints we set (Fig. 7) while at the same time having a poor match to observed flows (Table 2, Fig. 3). HRR is not a glaciological model, and therefore it is agnostic about sources of friction and can trade off channel and hillslope friction to produce correct outflows if unconstrained. We have constrained the channel friction to match literature field observations closely and allowed hillslope frictions to vary over a much wider range of values given the longer history of study and larger databases of Manning's n values for ice channels relative to transport through the crust and/or bare ice. Therefore, non-hillslope models would likely improve only by including physically unrealistic channel frictional values given the results in Fig. 7. This is in line with mass conservation, as without hillslopes to slow water upstream, HRR needs to slow water using extreme friction near the outlet in order to match the hydrograph. This pattern is observed across both coarse and fine networks.

Ideally, we would have enough data to calibrate and validate the model over separate time periods and at more locations than the outlet. HRR produces an individual hourly discharge at each of our thousands of channels, but we can only verify these at the outlet. However, we believe that model calibration statistics at the outlet indicate the physical realism of the process we are attempting to model: since we modeled an accurate outlet hydrograph, the fully mass- and momentum-conserved physics of HRR mean that upstream flows must be realistically represented or we could not have produced a quality outlet hydrograph. Our results show that HRR is capable of matching outlet flows extremely well (calibration Kling–Gupta efficiency, KGE, as high as 0.96 and NSE as high as 0.92), and thus we believe this assumption well-founded. Recall also that the ADCP data were collected from July 20 to 23, but we model hourly flows for the entire month. We focus our evaluation only on this 72 h calibration period to discuss our experimental results without discussing the rest of the month's unverified results. Results for these other times are of course an ultimate end goal of future GrIS water routing as we look toward future coupled SMB-routing models that can be used to study interactions between surface hydrologic routing processes and subglacial processes. While we have here only reported flows during a verifiable 72 h period, in theory our model parameters should be able to accurately route water in similar areas of the GrIS with similar network drainage patterns in similar seasons.

Our results also support earlier assertions of the mismatched timing and magnitude of SMB runoff and observed discharges entering the Rio Behar terminal moulin (Smith et al., 2017). The routing model is unable to assign glaciologic process to mass gaps, so we can only suggest plausible mechanisms for closing that mass balance gap. Mass gaps could perhaps result from subsurface retention and/or refreezing in bare-ice weathering crust (Cooper et al., 2018), a process not currently well-represented in SMB models, or the mass imbalance could come from transport processes: filling lakes, drainage through fractures (there are no crevasses in the study area), or the breach of topographic divides are all plausible transport process gaps. Topographic breach is unlikely given that we use an observed (via image) channel network, and thus if breaches did occur, they are accounted for. Further, total depression storage (including true lakes and DEM artifacts) was 6.92×106 m3, which is 2 orders of magnitude less than the observed ADCP flux during this time (integrated into a bulk volume, 241×106 m3) and 1 order of magnitude less than the maximum runoff deficit (obtained by subtracting the ADCP from the largest SMB input, 17.5×106 m3). Therefore, if all depressions were dry at the start of routing and were completely filled by runoff before beginning to flow in the channel network, this would still only account for roughly one-third of extra runoff production mass. Given that we know lakes are full during this time period, we assert that this lake filling effect is not the cause of mass imbalance. Further, errors in our outlet hydrographs are dominated by the underestimation of nighttime low-flow periods as peak flows are modeled well across nearly all 16 trials. These nighttime low flows are particularly important for mass balance in the Rio Behar watershed as a large driver of mismatches in total mass balance (Fig. 4) comes from these low-flow periods. Error could come from the ADCP itself, and this instrument is generally less certain at lower flows. However, the ADCP record here is taken from Smith et al. (2017) and represents a well-documented procedure carried out by expert field personnel, and thus we are confident that ADCP errors are too small to explain Rcoef. We affirm that all SMB models examined here produce too much excess water relative to ADCP observations (at least at peak times, Fig. 4 shows MERRA2 total runoff is less than the ADCP total discharge but still requires Rcoef<1 to reduce the peak daytime volume of water) and do not model nighttime flows without routing, consistent with Smith et al. (2017). Our results suggest that hydrologic process modeling (i.e., routing) can correctly reproduce these nighttime low flows.

The workflow presented here is repeatable for any supraglacial stream and river network on the GrIS, but the in situ discharge datasets needed for calibration are not readily available. Future studies attempting to repeat this model setup elsewhere need an in situ discharge record (ideally longer than our 3 d record and ideally collected at multiple locations across stream orders), a high-quality DEM, and a fine-scale remotely sensed image. Modeling is efficient with these data in hand, yet the collection of in situ discharge in particular presents a major hurdle for widespread application to the GrIS. It is possible to use assumed discharges for calibration, but as our results clearly support a difference between predicted and measured fluxes, we believe measured calibration data are best. We suggest that the collection and publication of a repository of supraglacial channel discharges and hydraulic properties atop the GrIS would be an invaluable resource and that future studies should explore the transferability of key parameters (e.g., channel and hillslope frictions) to other locations on the ice sheet.

We confirm earlier assertions of the importance of terrestrial hydrological processes, specifically hillslope water transport and open-channel flow, on GrIS surface meltwater routing. Unlike previous studies routing meltwater, our results are generated using the hillslope river routing model (HRR) which uses an explicit kinematic wave to conserve water mass and momentum in hillslopes and channels and represents hourly flow in nearly 10 000 individual channels in a fully topological network. This first-principles investigation shows that observed supraglacial river discharges (and thus moulin hydrographs) cannot be accurately simulated without both reducing the volume of surface runoff generated by SMB models and accounting for hydrologic transport processes. We investigated two process-level controls on this modeling – modeling coarse- vs. fine-scale channel networks and inclusion/exclusion of hillslope process – and found that incorporating fine-scale channel networks and hillslopes yields superior results. Calibrated model parameters are intuitive and align with field observations and theory. The automated methods developed here could readily be deployed elsewhere atop the GrIS bare-ice ablation zone but require in situ supraglacial discharge data for calibration. More of these data should be collected if GrIS surface hydrology processes are to be fully understood.