Spatial and temporal analysis of hillslope–channel coupling and implications for the longitudinal profile in a dryland basin

The long‐term evolution of channel longitudinal profiles within drainage basins is partly determined by the relative balance of hillslope sediment supply to channels and the evacuation of channel sediment. However, the lack of theoretical understanding of the physical processes of hillslope–channel coupling makes it challenging to determine whether hillslope sediment supply or channel sediment evacuation dominates over different timescales and how this balance affects bed elevation locally along the longitudinal profile. In this paper, we develop a framework for inferring the relative dominance of hillslope sediment supply to the channel versus channel sediment evacuation, over a range of temporal and spatial scales. The framework combines distinct local flow distributions on hillslopes and in the channel with surface grain‐size distributions. We use these to compute local hydraulic stresses at various hillslope‐channel coupling locations within the Walnut Gulch Experimental Watershed (WGEW) in southeast Arizona, USA. These stresses are then assessed as a local net balance of geomorphic work between hillslopes and channel for a range of flow conditions generalizing decadal historical records. Our analysis reveals that, although the magnitude of hydraulic stress in the channel is consistently higher than that on hillslopes, the product of stress magnitude and frequency results in a close balance between hillslope supply and channel evacuation for the most frequent flows. Only at less frequent, high‐magnitude flows do channel hydraulic stresses exceed those on hillslopes, and channel evacuation dominates the net balance. This result suggests that WGEW exists mostly (~50% of the time) in an equilibrium condition of sediment balance between hillslopes and channels, which helps to explain the observed straight longitudinal profile. We illustrate how this balance can be upset by climate changes that differentially affect relative flow regimes on slopes and in channels. Such changes can push the long profile into a convex or concave condition. © 2018 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd.

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understanding of the physical processes of hillslope-channel coupling makes it challenging to determine whether hillslope sediment supply or channel sediment evacuation dominates over different timescales and how this balance affects bed elevation locally along the longitudinal profile. In this paper, we develop a framework for inferring the relative dominance of hillslope sediment supply to the channel versus channel sediment evacuation, over a range of temporal and spatial scales.
The framework combines distinct local flow distributions on hillslopes and in the channel with surface grain-size distributions. We use these to compute local hydraulic stresses at various hillslope-channel coupling locations within the Walnut Gulch Experimental Watershed in SE Arizona, USA. These stresses are then assessed as a local net balance of geomorphic work between hillslopes and channel for a range of flow conditions generalising decadal historical records. Our analysis reveals that, although the magnitude of hydraulic stress in the channel is consistently higher than that on hillslopes, the product of stress magnitude and frequency results in a close balance between hillslope supply and channel evacuation for the most frequent flows. Only at less frequent, high-magnitude flows do channel hydraulic stresses exceed those on hillslopes, and channel evacuation dominates the net balance. This result suggests that WGEW exists mostly (~50% of the time) in an equilibrium condition of sediment balance between hillslopes and channels, which helps to explain the observed straight longitudinal profile. We illustrate how this balance can be upset by climate changes that differentially affect relative flow regimes on slopes and in channels. Such changes can push the long profile into a convex or concave condition.

Rationale
The interaction between hillslopes and river channels plays a fundamental role in fluvial system evolution and in the storage and export of water and sediment.
Hillslopes impose a sediment supply on river channels that is transported or stored, and which therefore impacts bed material grain size and local bed elevation (Attal and Lave, 2006, Korup, 2009, Sklar, et al., 2017. Channel behaviour in response to hillslope sediment supply depends on the mass and GSD of delivered sediment, its spatial and temporal characteristics (Benda andDunne, 1997, Gabet andDunne, 2003), as well as on the competence of the flow to transport the supplied sediment. Where hillslopes and channels are fully coupled (not buffered by a floodplain) (Bracken and Croke, 2007, Brunsden, 1993, Fryirs, et al., 2007, Harvey, 2001, sediment can be transported directly to the channel. If hillslope supply is greater than downstream channel transport, the result is net accumulation of sediment at that point, raising bed elevation. In contrast, if channel transport exceeds hillslope sediment supply, there will be net sediment evacuation and bed degradation.
Therefore, alluvial river bed elevation at a point along the longitudinal profile is determined by the net balance of sediment supply and channel sediment transport (Hack, 1957, Harvey, 2001, Leopold and Bull, 1979, Rice and Church, 1996, Simpson and Schlunegger, 2003, Singer, 2010, Slater and Singer, 2013. Sediment supply to any location in the channel is the sum of the contributions from upstream and from lateral sources. Over 10 1 -10 3 -yr timescales the divergence of sediment transport along the channel may be considered constant (Walling and Fang, 2003) and lateral sources of sediment thus become significant in determining the net channel sediment balance. However, lateral sediment supply to the channel (e.g.
from hillslopes) is poorly constrained in most river basins, which limits our understanding of its effect on this net balance, local bed elevation and by extension, its expression over the whole channel longitudinal profile (Tucker and Bras, 1998, Tucker and Slingerland, 1997, Willgoose, et al., 1991. Over individual storm cycles, the balance between hillslope supply and channel transport controls changes in local sediment storage and bed elevation. Over centuries to millennia it governs longitudinal profile evolution (Snow and Slingerland, 1987). The prevailing climatic regime determines whether this balance is dominated by hillslope sediment supply (net channel accumulation) or channel sediment transport (net evacuation) over a particular timescale. For example, in basins with perennial channel discharge and slow subsurface storm flow through vegetated slopes, hillslope sediment supply only typically results from catastrophic slope failure, and the net balance along the channel profile favours channel sediment evacuation. However, in basins characterised by Hortonian overland flow on hillslopes and ephemeral flow in channels (i.e. dryland basins), the sediment balance between hillslopes and channels becomes more equivocal.
The longitudinal profile is therefore shaped by the relative magnitude and frequency of erosion events on hillslopes and in the channel over time (Wolman andGerson, 1978, Wolman andMiller, 1960). A general question is whether more frequent sediment-moving events dominate the morphological expression in landscapes (Wolman and Miller, 1960), or whether topography is shaped by infrequent events that reorganize the landscape, followed by long period of 'recovery' (Baker, 1977, Wolman andGerson, 1978). When considering hillslope sediment supply versus channel sediment evacuation, it is currently unknown whether channel events dominate over hillslope events and how the balance of geomorphic work in these two landscape components over the spectrum of runoff-producing rainstorms affects the shape of the long profile. In drylands, long profiles are often straight , Powell, et al., 2012, Vogel, 1989, suggesting that the balance between hillslope and channel erosional events differs from humid environments that display the typical concave-up equilibrium profile. In dryland basins the stochasticity and spatio-temporal variability in rainfall  pose a challenge to anticipating the relative balance between hillslope and channel erosion.

Hillslope-channel coupling
Hillslope-channel coupling is particularly important for understanding the evolution of dryland basins for several reasons. 1) Overland flow during storms causes erosion of sparsely vegetated hillslopes that can deliver high and coarse sediment supply to the channel (Bull, 1997, Michaelides andMartin, 2012). 2) Spatial and temporal variability in rainfall means that hillslope sediment supply and channel evacuation may be out of phase such that one dominates the other over a particular timescale.
3) Channel sediment evacuation is accomplished by discrete flash floods travelling over dry streambeds with significant transmission losses (Hereford, 2002, Jaeger, et al., 2017.
These factors may result in net sediment accumulation in dryland channels as hillslope supply dominates over channel evacuation, except during rare, extreme events. Cycles of channel degradation or aggradation may persist in the landscape for decades to millennia (Bull, 1997, Slater and Singer, 2013, Slater, et al., 2015, Waters and Haynes, 2001, following changes in climate or base-level. However, due to the lack of theoretical understanding of the spatial and temporal expression of hillslope-channel coupling , progressive changes in landscape topography are challenging to anticipate. In dryland basins that are particularly sensitive to climatic changes affecting runoff, we need a better understanding of hillslope-channel coupling to predict landscape responses and evolution to exogenous perturbations such as climate or base level change.

Hydrological and erosional processes in dryland basins
Dryland valleys are shaped by a cascading set of interacting processes that are triggered during individual rainstorms. Rainfall is converted to runoff by infiltrationexcess overland flow on hillslopes, runoff erodes hillslope sediment, and this sediment is delivered to channels, some of which contributes to channel bed material. Runoff accumulates and generates flow in river channels, which in turn, transports bed material sediment. However, storm events in drylands are short-lived and spatially discontinuous, leading to sporadic water and sediment delivery from hillslopes to channels. In these desert environments, the interaction between rainfallrunoff, vegetation, and erosion affects grain size of material eroded from slopes (Michaelides, et al., 2009. In addition, channel flow undergoes significant transmission losses into the sedimentary bed such that flood discharge decreases with distance downstream and many floods do not reach the basin outlet (Renard and Keppel, 1966). These ephemeral channel flow processes in dryland basins leave a strong signal of inheritance from previous rainstorms, e.g., poorly sorted river beds lacking armouring (Laronne, et al., 1994), underdeveloped bar forms (Hassan, 2005), and generally simple topography . As channel transport rates are very sensitive to bed material GSDs, hillslope sediment supply may strongly influence subsequent channel sediment flux (Lekach and Schick, 1983) and thus, trends of sediment accumulation or evacuation in various parts of a dryland basin (Pelletier and DeLong, 2004).
The aim of this study is to investigate the net balance of hillslope sediment supply and channel sediment evacuation at distinct points along the channel, and to generalise this coupling within an entire river basin. Our analysis is based on the computation of a proxy for the net balance between sediment supply from hillslopes and channel sediment evacuation over a range of flows from the historical record.
The spatial and temporal manifestation of this net balance can be used to understand long-term evolution of the longitudinal profile under the impact of past or future climatic conditions.

This basin, situated in the transition zone between the Chihuahuan and Sonoran
Deserts, exists on a bajada sloping gently westwards from the Dragoon Mountains, reaching the San Pedro River at Fairbank, AZ. It is drained by Walnut Gulch, a sand and gravel-bedded ephemeral river. The climate of the region is semi-arid with low annual rainfallaverage 312 mm yr -1 for the period 1956-2005 (Goodrich, et al., 2008). Convective thunderstorms during the summer monsoon season (July-September) generate 60% of the annual precipitation and 90% of the runoff for WGEW and are the major driver of erosion and sediment redistribution (Nearing, et al., 2007, Nichols, et al., 2002, Osborn, 1983b, Osborn and Lane, 1969, Renard and Keppel, 1966. These storms are characterised by extreme spatial variability, limited areal extent, high intensity and short duration rainfall (Osborn, 1983a). It is not uncommon for storm events to exceed intensities of 100 mm hr -1 at the centre of the storm, lasting on the order of minutes (Nicholson, 2011, Renard andLaursen, 1975). During an event, channel flow decreases downstream due to transmission losses (Renard and Laursen, 1975). However, when considering the entire historical record of stream flow at various spatial scales within the basin, total annual discharge increases downstream ( Figure 2A).

Existing Data
WGEW has the longest global record of runoff in a semi-arid site  covering the period 1954-2015. Historical records of event discharge at WGEW exist for this period at 7 flumes along the main channel, and 7 on tributaries (Figures 1;   2A). Event based rainfall data exist for the same period at many of the 95 operational gauging stations across all of WGEW. (Goodrich, et al., 2008). These historical records of rainfall and discharge (http://www.tucson.ars.ag.gov/dap/) provide the opportunity to assess flow on hillslopes and in channels. A 1-m resolution Lidar DEM exists for WGEW obtained in 2007.

Approach
The accurate assessment of sediment transport at high spatial resolution over a basin is logistically difficult without a time series of topographic surveys (e.g. repeat Lidar), widespread measurements of sediment flux and/or erosion rates from geochronology. To better understand the spatial variability of hillslope-channel coupling, we compute hydraulic stress (i.e. the force applied to a substrate by flowing water) acting upon a template of measured surface grain size distributions as a proxy for potential sediment transport. We employ a rich historical record of rainstorm intensity and duration data and discharge measurements at various spatial scales in WGEW to extract characteristic values of flow in the channel and on the hillslopes for the stress calculations. We then multiply hillslope and channel stress metrics by the frequency of their occurrence in the historical record to generate a proxy for geomorphic work done by each flow. The net balance of these frequencynormalised stresses can be used as a comparison of relative sediment yield proxies, to infer local hillslope-channel coupling as the relative dominance of hillslope sediment supply or channel sediment evacuation. Finally, we generalise this analysis to assess the likely impact of hillslope-channel coupling over the last several decades on the longitudinal profile of Walnut Gulch.

Ingredients for analysis
Our subsequent analyses use the following data obtained from the historical records in WGEW and a field campaign: decadal records of event rainfall, decadal records of event discharge, hillslope and channel grain size measurements and topographic data. The rainfall data are used as inputs into a rainfall-runoff model to produce values of hillslope runoff. The 1-m Lidar DEM was used to calculate a flow accumulation raster in ArcGIS, from which upstream drainage areas for each transect were computed. Rainfall and channel discharge data were used in the calculation of hydraulic stress magnitudes and probabilities (frequencies).

Field Measurements of topography and grain size
We measured topography and grain sizes in the field to provide relevant information as input for calculating hydraulic stresses (Eq 1). We surveyed by real time kinematic GPS (accuracy: 1 cm vertically; 2 cm horizontally) channel centreline elevations at 72 locations spanning ~30 km of the drainage network. At a subset of 31 locations we measured channel width and adjacent hillslope profiles, of which 11 were fully coupled on both sides and 20 were partly coupled (hillslope-channel connection only on one side of the channel)giving a total of 42 hillslopes. Channel measurements were made at intervals of ~100 m in the headwaters and at ~500 m downstream ( Figure 1). The local channel slope, S, at each transect was computed as: in which z is centreline elevation, x is distance downstream, and j is the location identity. Channel slope in this basin is insensitive to sampling resolution, since the longitudinal profile is essentially straight. We have confirmed this by comparing slope obtained from 1-m, 10-m, and 30-m elevation data for WGEW.
We measured grain size of surface sediments at three locations on each of the 44 hillslopes and at 81 locations in the channel. A photographic method was used for grain size analysis (Buscombe, et al., 2010). Photographs of the surface were taken using a Nikon Coolpix S9700 16.0-megapixel (4608 x 3456 pixels) digital camera mounted to a survey pole at a height of approximately 25 cm and orthogonal to the ground under natural light. The camera was set to automatically reduce shake. A scale was placed in the field-of-view of all photographs near to the edge of image.
The image resolution varied between photos because modifications were needed to the apparatus to ensure that the photo was orthogonal to the ground and without shadows cast by the apparatus or nearby vegetation. The camera height therefore varied approximately ±0.15 m, resulting in image resolutions of approximately 0.1 mm/pixel in all photos. We employed an automated method of grain size distribution detection (Buscombe, 2013, Buscombe, et al., 2010).
This method was tested against a surface pebble count method for phi grain size classes between 2 mm and 512 mm using the Wolman method (Wolman, 1954). A selection of grain size distributions derived by both methods were compared and found to be statistically similar (Supplementary Table 1). Photographically derived grain size distributions were analysed using GRADISTAT software (Blott and Pye, 2001) to generate characteristic size percentiles (D 10 , D 50 and D 90 ).

Magnitude of hydraulic stress
We use stream power instead of shear stress as a metric of hydraulic stress, as it minimises data requirements and enables direct comparison of stress on hillslopes and in channels. Additionally, shear stress has been shown to be a poor predictor of sediment transport by overland flow on coarse-mantled desert hillslopes (Abrahams, et al., 1988). Stream power incorporates both runoff depth and velocity of the flow, which co-vary on hillslopes to affect sediment entrainment (Michaelides and Martin, 2012), so it is a more sensible metric of hydraulic stress in this context. While runoff depth and velocity measurements are not common, information on depth and velocity can be easily obtained from rainfall-runoff models in Hortonian overland flow environments , where event-based rainfall data are available.
Stream power (W/m) is defined as the product of discharge, slope, and weight of water: where ρ is the density of water (1000 kg/m 3 at 4°C), g is gravity (9.81 m/s 2 ), Q (m 3 /s) is discharge, and S is energy gradient (m/m), which is equivalent to the bed slope for uniform flow. Normalising by width (for channels), we obtain unit stream power, ω (W/m 2 ): where B is the width of flow (m). Discharge for a rectangular cross section of channel is defined as: Where U is mean stream velocity (m/s) and h is flow depth (m). Therefore, we can rewrite (Eq 3), replacing Q with its components as: Eq 5 can be applied to the channel by inverting discharge data with Eq 4, again assuming a rectangular cross section, which is a common feature of dryland channels (Leopold, et al., 1966, Singer and. It is applied to the hillslope using flow velocity, depth and discharge output from a rainfall-runoff model where, q = uh, and q is unit hillslope discharge (m 2 /s), h is overland flow depth (m), and u is downslope velocity (m/s) . Parker (1979) defined dimensionless depth (ℎ * = ℎ/ 50 ) and velocity ( * = / 50 ), where is the submerged specific gravity of the sediment, = − / , is sediment density and 50 is the median diameter of the surface sediment from field measurements. Eaton and Church (2011) combined h* and V* with a dimensionless slope term ( * = / ) to derive dimensionless stream power as * = ℎ * * * . After combining and simplifying, dimensionless stream power can be expressed as: * = 50 3/2 Using this metric (ω*), we can compare the relative magnitudes of hydraulic stress for the channel and adjacent hillslopes for any percentile of flow.

Channel
We retrieved from the online database information on discharge at each flume including runoff event start-time, duration (mins), total equivalent runoff depth (mm), and the peak runoff rate (mm/hr) for each discharge event measured at every flume in WGEW since 1953. We extracted discharge values for 25 th , 50 th and 75 th percentiles for 6 of the flumes to represent the low, medium and high discharges.
These were plotted against drainage area on a log-log plot and a linear regression line was drawn between the points (Figure 2A). Using these regression equations, we calculated discharge values for each flow percentile for each transect location in the channel (Figure 1) as a function of the upstream contributing area. Based on local discharge values generated by the relationship between discharge and drainage area we then computed ω by (Eq 5) and ω* by (Eq 6) for each transect location.

Hillslopes
Hillslope runoff is not measured directly in a systematic way, so we employed a rainfall-runoff model to convert measured rainfall events into runoff events utilising the 63-year historic record of rainfall in WGEW. We plotted the event rainfall intensity versus duration for every storm on record at all rain gauges in WGEW. We then thresholded this dataset at an intensity of 15 mm/hr ( Figure 3A), as a conservative estimate of the intensity above which runoff is generated. This threshold was based on a various values from previous work in this basin (Osborn andLane, 1969, Syed, et al., 2003). Figure 3B and 3C show the distributions of rainfall intensities and durations over all recorded events above 15mm/hr.
We then used the STOchastic Rainfall Model (STORM, ) to randomly sample rainfall events from this thresholded phase space of intensity-duration, such that a randomly selected value of total rainfall for each year is satisfied across the basin. Thus, these simulations are faithful to the hydroclimate of WGEW. We simulated three ensembles each of 30 years to broadly represent the range of rainstorms recorded at Walnut Gulch over the last several decades.
To convert these rainfall events into hillslope runoff, we employed the rainfall-runoff model COUP2D (Michaelides and Martin, 2012, Michaelides and Wainwright, 2008, Michaelides and Wilson, 2007, which simulates overland flow hydraulics on hillslopes in response to discrete rainfall event inputs.
Because runoff response to rainfall is significantly modulated by hillslope length (e.g. see Michaelides and Martin, 2012), we ran model simulations (using the same randomly selected rainfall events) on four hillslope lengths: 25, 50, 75 and 100 m to give us the signal of rainfall to runoff for different hillslope lengths (total simulations = 1832). Hillslope angle is important for determining the flow hydraulics (i.e. the depthvelocity split) but, for the same infiltration rate it does not affect the discharge, so we used a constant angle in our simulations (10°). We also used a constant value of Manning's n (0.056) in these ensemble model simulations. The distribution of all modelled runoff values is shown in Figure 3D.
We then used the modelled q values to calculate flow percentiles (25 th , 50 th and 75 th ) for each hillslope length. These values of flow percentile were plotted against hillslope length and a power law function was the best fit between the points ( Figure   3E). Using these equations, we calculated discharge for each flow percentile for each of the 44 hillslopes along the sampling transect (Figure 1 and Supplementary hillslope ω by (Eq 5) and hillslope ω* by (Eq 6).

COUP2D simulates infiltration-excess and saturation-excess overland flow as a result
of filling a fixed soil moisture store and infiltration is represented using the modified Green and Ampt (1911) infiltration model (Michaelides and Wilson, 2007). Runoff is routed on a 2D rectangular grid of a hillslope strip (hillslope length x 2 m width) using the kinematic wave approximation. This approximation is rated using the Manning's n friction factor, with flow routing from cell to cell defined by a steepest descent algorithm.
For simplicity we use one value of initial and final infiltration rates (2.2 mm and 0.25 mm/min respectively) for the model simulations based on reported measurements by Abrahams, et al. (1995) in WGEW. While we acknowledge that infiltration rates on hillslopes are highly variable, model sensitivity analysis has shown that rainfall rate is by far the most important determinant of runoff rates compared to infiltration rates (see Michaelides and Wainwright, 2002) and spatial variability in infiltration rates is only important where the runoff magnitude is low (i.e. rainfall and infiltration rates are similar). Even then, the sensitivity of runoff rates to spatial patterns in infiltration is relatively low (see Michaelides and Wilson, 2007). All hillslope variables are provided in Supplementary Table 2.

Probability of hydraulic stress occurrence
To assess the net balance of hydraulic stress over a multidecadal period, we normalise the magnitude of each value of the driving flow (q on hillslopes and Q in channels) by the probability (frequency) of its flow occurrence in the historical record to produce the computed value of ω*. We separately compute probabilities of occurrence for hillslope runoff and channel discharge.

Channel
In the channel, we calculate an exceedance probability for streamflow equalling or exceeding a particular value of channel discharge, Q (m 3 /s) as: where k is a flume identifier, f is the total number of flumes used (n=7), and subscript xx indicates the percentile of discharge (25 th , 50 th , 75 th ). In other words, we are computing the overall channel flow probability of occurrence as the average of all local (at each flume) channel probabilities of Q exceeding a particular value. In this case, we multiplied average of storm days per year by the number of years of record for each flume to obtain the total # of storm days in (Eq 7).

Hillslopes
On hillslopes, we calculate the probability of runoff occurrence equalling or exceeding a particular value of hillslope runoff, q as: where q indicates hillslope unit runoff (m 2 /s) and subscript xx indicates the percentile of runoff (25 th , 50 th , 75 th ). Based on a characterisation of the rainfall record, we computed an average of 37 rainstorms per year in WGEW , yielding 1110 storm events over 30 years (used in the denominator of Eq 8).

A proxy for geomorphic work
The magnitude of stress produced by a flow scaled by its likelihood describes its geomorphic effectiveness in shaping the landscape over longer timescales (Wolman and Miller, 1960). Thus, we compute a proxy for geomorphic work (Λ) done for each percentile of stress on either the hillslope or in the channel by multiplying (Eq 6) by either (Eq 7) or (Eq 8) as: and for the hillslopes and channel, respectively.

Quantifying geomorphic work balance at WGEW
For the hillslope and channel at each transect, we multiplied all ω* values calculated for each percentile magnitude by the probability of q or Q exceeding or equalling its respective magnitude, given by equation (7) and (8) to generate Λ_HS xx and Λ_CH xx (Eqs 9 and 10), respectively. We then calculated the net local balance (N Bal ) between hillslope and channel Λ at each hillslope-channel transect for paired values of Λ_HS xx and Λ_CH xx at each topographic cross-section along the channel as:

Morphological and sedimentary characteristics from field measurements
The field data reveal a straight longitudinal profile in the channel, where elevation monotonically declines downstream, with minimal impact of tributaries ( Figure 4A).
The straight long profile is consistent with previous work in drylands , but which has yet to be fully explained from a mass balance perspective. Channel width fluctuates and displays no downstream trend ( Figure   4B), which is again consistent with other dryland basins (Jaeger, et al., 2017, Michaelides and and may reflect a topographic expression of downstream transmission losses. Hillslope angles throughout WGEW are low (median = 5.7°; IQR = 2.9˚), and 90% of the measured slopes have angles <10° ( Figure 5A). Hillslope lengths vary greatly across our surveyed transects (median = 149.7 m; IQR = 131.5 m) ( Figure 5B).
Characteristic grain sizes in the channel and on hillslopes fluctuate with no downstream fining trend ( Figure 4C). Hillslope surface sediments are generally coarser than channel bed material sediment and there was no spatial correlation between the hillslope and channel GSD. However, we found that over all sites analysed the hillslope D 50 and channel D 90 are statistically similar (Kolmogorov-Smirnov statistic (KS) = 0.1844, p = 0.2842, n1/n2 = 71/44). This result is consistent with findings from another dryland environment, which suggested that sediment delivered from slopes to channels in drylands becomes the characteristic scale of hydraulic roughness . Figure 5C displays the aggregated channel and hillslope GSDs over nested drainage areas within the watershed. This analysis reveals that hillslope surface sediment GSD is scale invariant, despite variability in slope length and angles ( Figure 5A and 5B). In contrast, the channel GSDs display a coarsening trend with increasing contributing area. This finding contradicts most published channel sediment data which display downstream fining (Ferguson, et al., 1996, Menting, et al., 2015, Sternberg, 1875, but is consistent with some published work where sediment supply exceeds channel transport (Brummer and Montgomery, 2003) or where flow competence causes a winnowing of fines (Attal, et al., 2015, Singer, 2010. Figure 6 compares the distributions of ω*, p and Λ between hillslopes and the channel calculated from the entire dataset (all flow percentiles and all transects). Figure 6A shows that dimensionless stream power (ω*) in the channel is significantly higher than on the hillslopes (KS = 0.77, p = 9.5x10 -44 , n1/n2 = 207/135). In contrast, the probabilities of occurrence (p) associated with these stresses are significantly higher for the hillslope than for the channel (KS = 0.61, p = 4.8x10 -3 , n1/n2 = 18/12) ( Figure 6B). The product of the stress and associated probability, Λ, is significantly greater in the channel than on the hillslopes albeit they converge to being much closer in value (KS = 0.59, p = 3.1x10 -25 , n1/n2 = 207/135). This suggests that N Bal should be slightly negative overall. In other words, dimensionless stream power is found to be an order of magnitude greater in the channel than on hillslopes. Even when accounting for the higher probabilities of these stream powers occurring on the hillslopes than in the channel, the net effect in terms of potential geomorphic work is that the channel overall does more work than the hillslopes. Figure 7 presents comparisons of ω*, p and Λ between hillslopes and the channel organised by flow percentiles (Q xx and q xx ). Figure 7A shows that ω*_CH is systematically and significantly higher than ω*_HS for all percentiles of flow (Supplementary Table 3). However, the probabilities of hillslope p(q) and channel p(Q) hydraulic stress occurrence, show the reverse pattern and are systematically and significantly higher for hillslope flows than channel flows across all flow percentiles ( Figure 7B; Supplementary Table 3).

General analysis of ω* and Λ
The product of the hydraulic stresses and their respective probabilities yields a metric of geomorphic work (Λ) that indicates a tendency towards sediment transport.
At the lowest and highest flow percentiles (25 th and 75 th ) the channel has higher Λ values than the hillslopemeaning that channel sediment transport exceeds hillslope sediment supply to the channel under those flow conditions. However, at median flow conditions (50 th percentile) hillslope Λ exceeds that of the channel, suggesting that under the most commonly occurring flow conditions, hillslope sediment supply exceeds channel sediment evacuation. The differences between hillslope and channel Λ values are statistically significant across all flow percentiles (Supplementary Table 3).
The higher probability of all flows on hillslopes counterbalances the higher stream power in the channel, resulting in close balance between the potential geomorphic work in the two landscape components especially at the median flow conditions. At the high flow percentiles, which occur less frequently, the channel dominates over the hillslopes.

Discussion
This analysis revealed that the magnitude of ω*_CH is consistently higher than ω*_HS, regardless of flow percentile ( Figure 7A). However, once we multiplied these stress magnitudes by their respective frequency of occurrences in the historical hydrological record at WGEW, we find variations in the resulting geomorphic work metric (N Bal ) between the flow percentiles that flip between channel dominance to hillslope dominance. Particularly, at the low and high flow percentiles (25 th and 75 th ) channel geomorphic work tends to be higher than that of the hillslopes. However, at the 50 th flow percentile, hillslope geomorphic work exceeds that of the channel ( Figure 7C), a result that corroborates measurements in a first order sub-basin of WGEW showing hillslopes to be the dominant contributor to total sediment yield (Nichols, et al., 2013). This result suggests that WGEW exists mostly (~50% of the time) in this condition of hydraulic stress balance between hillslopes and channels.
Furthermore, the net local balance that is struck between these frequencynormalised stresses (N Bal ) on hillslopes and channels over the entire basin fluctuates around zero, over all spatial scales and over all recorded flows (Figures 8 and 9).
In this paper, we revealed longitudinal variations in N Bal , which depend on both the magnitude and frequency of driving flow events (Figure 8 and 9). Specifically, we interpret from these stress metrics and the flow probabilities that the common condition of this dryland landscape is one of infrequent flow in the channel and more frequent overland flow on slopes for the same rainfall events ( Figure 6B). However, when the channel does flow at higher than average levels (<25% of the time), channel hydraulic stress systematically exceeds that on adjacent hillslopes. Thus, it appears that the channel of WGEW operates under a regime of net sediment accumulation from hillslopes most of the time, followed by (less frequent) episodic transport of channel sediment.
Channel flows, however, are not generally long-lived enough to evacuate all the sediment supplied by hillslopes, especially considering that discharge declines in the downstream direction due to transmission losses (Renard and Keppel, 1966).
Instead, ephemeral channels incompletely sort the supplied hillslope sediment into diffuse coarse and fine patches that fluctuate down the channel ( Figure 4B and 4C), in a manner that is typically out of phase with hillslope-channel coupling loci and width fluctuations ).
Thus, the WGEW channel apparently inherits coarse patches from the bounding hillslopes and they accumulate such that the GSD coarsens with increasing drainage area ( Figure 5C). The coarse particles delivered from hillslopes become the hydraulic roughness of the channel , limiting river incision under moderate flow conditions.
Since the balance between hillslope sediment supply and channel sediment evacuation (N Bal ) exerts an important control on local channel bed elevation ( Figure   10A-C), we may infer that a net zero balance struck over a long enough time period (e.g. at least several decades) would produce a long profile that does not change appreciably in elevation (Leopold and Bull, 1979). While fluctuations in local bed elevations would be expected, there would be no long-term trend of aggradation or degradation, a condition supported by previous dryland research (Leopold, et al., 1966, Powell, et al., 2007. This idea is distinct from that of the graded river profile, where the river transports all the sediment supplied to it because of supply limitation Bull, 1979, Mackin, 1948). By contrast, a dryland system such as WGEW appears to have a very high supply of sediment that has likely persisted as long as the duration of the current hydrological regime. Ephemeral channels such as WGEW can thus be considered oversupplied with sediment, which are shaped by infrequent and discontinuous channel flow into a straight longitudinal profile and symmetrical channel cross sections (referred to as 'topographic simplicity', ). This interpretation of the equilibrium condition for ephemeral channels is consistent with observations in other dryland environments (Hassan, 2005, Leopold, et al., 1966, Powell, et al., 2012, Vogel, 1989 and with modelling of long profile development under different forcing conditions (Snow and Slingerland, 1987). This is a topic of ongoing research, so the first-order mechanisms driving this topographic condition have not yet been determined.
One might wonder how stable a straight long profile might be and how it might be perturbed into becoming concave or convex. Modelling of long profile evolution might help to address such questions. However, our spatially explicit analysis linking magnitude (ω*) and frequency (p) of hydraulic stresses suggests that climate change could have important consequences for the long profile. While the pdfs of the product of magnitude and frequency (Λ) for hillslopes and channels have limited overlap under the current hydrological regime at WGEW, these distributions could shift toward or away from each other, depending on how climate change is expressed in runoff regimes. Singer and Michaelides (2017) analysed historical hydrological trends at WGEW and found that rainfall intensity has declined significantly in recent decades and especially for high intensity rainfall (>15 mm/hr), yet total monsoonal rainfall is trending upward over this same time period. This has translated into a significant downward trend in runoff at the WGEW basin outlet . These findings suggest that there are more storms each monsoon delivering less intense rainfall, which would tend to increase the frequency of hillslope runoff and decrease the frequency of channel streamflow ( Figure 10D). If this climate change trend persists well into the future, it would tend to maintain a straight long profile, but could even yield a convex long profile by oversupplying the channel with sediment that is not evacuated ( Figure 10E). Indeed, there is some evidence for a trend of oversupply from repeat channel cross sections over multiple decades (Supplementary Figure A). However, it is worth noting that dryland environments often experience dry periods that are punctuated by catastrophic flooding, wherein the system can reset itself with hydraulic stresses in the channel that are large enough to cross a geomorphic threshold and ream out stored sediment (Baker, 1977, Baker, 1987, Wolman and Gerson, 1978.

Conclusions
We developed a framework for analysing the relative balance between hillslope sediment supply to the channel and channel sediment evacuation, over a range of temporal and spatial scales in a dryland basin, where erosional processes are driven by the flow of water. Our approach utilises historical records of rainfall and streamflows in combination with surface grain-size distributions, to compute local hydraulic stresses at 32 hillslope-channel transects. The magnitude of these stresses was multiplied by the frequency of their occurrence in the historical record to produce a proxy for geomorphic work. We then assessed the local net balance between hillslope and channel 'geomorphic work' at each transect over a range of flow conditions generalising decadal historical records. Our results reveal that overall there is a close balance between hillslope supply and channel evacuation for high frequency flows. Only at less frequent, high-magnitude flows does channel 'geomorphic work' exceed that of hillslopes, and channel evacuation dominates the net balance. While there are spatial patterns in the net balance, they tend to cancel out yielding an overall basin-scale balance that is close to zero. This result suggests that WGEW exists mostly (~50% of the time) in an equilibrium condition of balance between hillslopes and channels, which helps to explain the straight longitudinal profile. We also demonstrate that climate changes can affect this net balance and thus change the shape of the longitudinal profile.  . Q values were available for 14 sub-watersheds of varying areas within the basin (numbered in A and keyed to Figure 1). Histograms of discharge events for the three ovaled watersheds in A: a small watershed, Flume 103 (B), a medium-sized watershed, Flume 9 (C), a large watershed, Flume1 (D). Note: scales on x-axes differ between subplots.

Figure 3.
A) Phase space of rainfall intensity versus duration. These data were thresholded at 15 mm/hr, for all data measured at WGEW since 1953 (black dots). We sampled from this distribution and then used these data to drive COUP2D. B) Histogram of all rainfall durations for storm events > 15 mm/hr and the quartile values for the distribution. C) Histogram of rainfall intensities for rainfall events >15 mm/hr and the extracted intensities for each curve in (A). D) Histogram of all modelled hillslope runoff (n=1832), based on stochastic simulation of runoff on slopes of 4 different lengths. E) Relationships between hillslope length and the q percentiles of modelled runoff used later to calculate the q percentiles for the measured hillslopes in WGEW.