Episodic sediment supply to alluvial fans: implications for fan incision and morphometry

. Sediment supply is widely believed to be a key control on alluvial fan morphology and channel dynamics. Although the sediment supply to natural fans is rather episodic, experimental studies of alluvial fans often use constant sediment supply rates, making it difficult to relate fan dynamics to the magnitude and frequency of sediment supply in the field. This paper presents a series of experiments designed to test the impact of episodic sediment supply on fan evolution and dynamics. We compare four experiments, each with the same mean sediment supply but different durations of high-and low-supply periods. 5 The experiments show that fan morphology and channel dynamics respond systematically to the temporal elongation of sediment supply oscillations: longer supply cycles generate flatter fans with more trenched channels. These results highlight how different basin conditions might generate different fan morphologies: supply limited basins with intermittent sediment supply might generate fans that are flatter than expected. Our results raise the question of whether a constant sediment supply in experimental models can adequately characterise the dynamics of natural fans in the field. We therefore suggest that experimental 10 modellers should include variability when investigating fan responses to sediment supply.

. Experimental design (not to scale). Water and sediment mix in the funnel and drop into the experiment at the head of the feeder channel, where sediment aggrades and degrades freely. The hillshaded topography and flow map example are from Run CON repeat 2, 20 hours into the experiment.

Experimental Approach
Our experimental alluvial fan is a "similarity-of-process" or "analogue" model (c.f. Hooke, 1968a;Paola et al., 2009), as are 90 most physical models of fans (e.g. Bryant et al., 1995;Clarke et al., 2010;Davies and Korup, 2007;Van Dijk et al., 2009;De Haas et al., 2016Hamilton et al., 2013;Hooke, 1967Hooke, , 1968bHooke and Rohrer, 1979;Miller et al., 2019;Piliouras et al., 2017;Reitz and Jerolmack, 2012;Schumm et al., 1987). The key processes in the model (fluvial sediment entrainment, transport and deposition) are similar to those on fans in the field. In alluvial fan experiments, it is challenging to maintain Froude scaling between the model and any field prototype, due to the large scaling ratio that is necessary to build a conveniently small 95 laboratory model. This lack of Froude scaling means that it is inappropriate to extrapolate rates and volumes measured in the experiment to field settings. Nevertheless, comparisons between the different experiments demonstrate how natural fans might respond to different frequencies and durations of sediment delivery. Such comparisons also highlight the distortions introduced through temporal averaging in the experimental inputs.
The slope of the fan itself, and the dimensions of the channels upon it, were self-formed. Therefore, it was not possible 100 to control the Froude or Reynolds numbers during the experiments. To give readers an idea of the flow dynamics, we have Each experiment was~20 hours long, which was approximately the duration over which the fan prograded to the far walls of the experimental table. For Run CON, we then conducted two additional repeats (i.e. three repeats in total); the data for Run CON therefore appear more dense in Figures 5-9 and we have increased the transparency for Run CON to account for this.
Using a length scale of 1:128, we approximated the experimental grain size distribution (GSD) from a surface gravel sample 130 collected in the channel at Three Sisters Creek fan, Canada, which is a typical gravel-cobble fan (located at 51.055108, -115.333515; see also Figure A1). The experimental mixture ranged from 0.25-8 mm (Figure 3). This sandy GSD encouraged subsurface flow, which sometimes generated seepage channels on the lower fan during the experiments. Such processes are common on fans in the field; for instance, both down-fan channel narrowing and spring formation have been attributed to infiltration on fans (Davidson et al., 2013;Kesel and Lowe, 1987;Woods et al., 2006). All four experiments had a constant flow of 150 ml s -1 . This flow is approximately equal to the 20-year flood in the stream where grain size data were collected. While the model is generic and does not represent a specific field prototype, the above relation provides context for the size of this flow relative to the size of the sediment used in the experiments.
The sediment concentration was 3.6% by volume in the high-supply periods.The experimental grain size mixture was truncated at 0.25 mm, omitting the finest~40% of material in the field sample, so the true bedload sediment concentration could 140 be expected to be around 6% by volume. In mountain streams, bedload makes up between 10% and 99% of the total sediment load, with a mean of 44.5% and standard deviation of 31.1% (see Table A1 for values reported in the literature). Consequently, the maximum experimental truncated-bedload sediment concentration of 3.6% is roughly equal to a total volumetric sediment concentration of 13.5% (7.9-45.0%) in the field.
The 150 ml s -1 flow rate was relatively high, as was the sediment concentration. As a result, these experiments can be thought sediment concentration. This paper therefore explores the effect of abrupt, large-scale variations in the sediment concentration, and the timing thereof.

Data analysis
Our photogrammetry system generated a topographic point cloud and co-registered 1 mm orthomosaic for each minute of each 150 experiment. We interpolated the point cloud (using a nearest neighbour approach) to generate a 1 mm resolution digital elevation model (DEM). DEM accuracy was assessed in Leenman (2021); elevation values for an individual cell in a topographically inactive area varied by less than −0.7, +0.8 mm (90% confidence) over any 30 minute period. Following Leenman and Eaton (2021) and Leenman et al. (2022), only data from 12 hours and onward were used, as the fan slope and wet fraction were scale-dependent earlier in the experiment.

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Fan slope was measured from 88 equally-spaced down-fan profiles, extracted from the DEM (see Figure A2 for their locations). For each profile, slope was taken from a linear regression of elevation against distance down-fan (profiles were quasi-linear). For each time-step, the representative fan slope was taken as the median of these 88 measurements.
Fan-head entrenchment was measured from arcuate cross-fan profiles extracted from each DEM at 0.25 m down-fan (see Figure A2 for profile location). Fan-head entrenchment was measured as the difference between maximum and minimum 160 elevations along the profile, for each time-step. We located the profile at 0.25 m because the method was not applicable farther down-fan: the convex nature of cross-fan profiles (e.g. Blair and McPherson, 1994a) meant that the minimum elevation was not always in the active channel for transects that were farther down-fan.
The orthomosaics were processed to generate binary (wet-or-dry) maps of the fan and channels at each time-step. These flow maps quantified the proportion of the fan area covered by flow (referred to as the "wet fraction"). The flow maps were 165 also used to count the number of channels at an arcuate cross-section 1 m down-fan (discounting seepage channels that were not connected to the fan-head by surface flow; see Figure A2 for cross-section location). We chose the 1 m cross-section as this was far enough down-fan that changes in flow pattern there represent changes on the fan as a whole. In addition to the cross-section analyses, change detection between successive flow maps revealed the area newly inundated in each minute, which was normalised by fan area to give F n (equation 1), the percentage of the fan newly inundated in a minute: Change detection between the DEMs produced DEMs of Difference (DoDs) that quantified the erosion and deposition in each minute. The DEMs were first smoothed with a 7 × 7 mm moving average filter (approximately the size of the largest grains). In the resulting DoDs, patches of erosion or deposition with a planform area of less than 2 cm 2 were discounted; elevation change of < 2 mm was also discounted.

Results
A general understanding of the fan responses to constant or oscillating sediment supply can be gained from the time-lapse videos (these show the fan from~12 hrs onward): Run CON, Run OSC10, Run OSC20 and Run OSC40. Figure 4 also has links to these videos. Note that frames were collected at one-minute intervals for Run CON, and 10 s intervals for Runs OSC10-OSC40, so the video speed differs. These videos demonstrate how the fan responded to constant sediment supply (Run CON) 180 or sediment supply oscillations (Runs OSC10-OSC40). Flow became more diverging when the sediment supply was turned on (also see upper row, Figure 4), and more channelised when the supply was cut off (middle row, Figure 4). The videos show that rapid lateral migration and channel readjustment followed each change in the sediment supply.

Fan morphology
Down-fan gradient is one of the simplest descriptors of fan morphology. As fans are self-formed, it provides a useful metric 185 for their self-organised adjustment to the input conditions or changes thereof. Figure 5 shows how the median down-fan slope differed across the four experiments, and how it adjusted during high-and zero-supply conditions. Figure 5 shows that, compared to Run CON (constant sediment supply), the fan was steeper with short-duration sediment supply oscillations (Run OSC10); it became flatter as the duration of the oscillations increased (Runs OSC20-OSC40). The fan steepened during high-supply periods and regraded to a lower slope during zero-supply periods. This partly explains the 190 Figure 4. Example fan morphology after a period of high sediment supply (upper row) and zero supply (middle row). Note that Run CON had no supply oscillations but has been sampled at the same timestamps as Run OSC10 for comparison. Lower row: QR codes that link to each experimental time-lapse video. The videos are also available at https://youtu.be/ML2LV28MQEM (Run CON), https: //youtu.be/jXjWIkLU-7A (Run OSC10), https://youtu.be/T4JbZC9YkXQ (Run OSC20) and https://youtu.be/EcCWYGIbsqA (Run OSC40).
trend in slope across the four experiments: during high-supply periods, sediment was deposited on the fan-head, steepening the fan, but during zero-supply periods the channel incised this material, lowering the fan slope. There was a lag of 2-3 minutes between the onset of a new sediment supply rate and the geomorphic response. In the 10-minute cycle in Run OSC10, the 5-minutes of zero-supply was insufficient to completely incise the new material at the fan-head, leading to a fan that was steeper than it would be with constant sediment supply (Run CON). Conversely, in the 40-minute cycle in Run OSC40, the 195 fan-head was deeply incised during the 20-minutes of zero-supply, reducing overall fan gradient. The spatial pattern of erosion and deposition, linked to these slope adjustments, is considered further in section 3.4.   These values indicate that channel gradients (as opposed to fan gradients) at the end of the zero-supply periods were likely even lower than the median fan gradients shown in Figure 5.

Channel patterns
Channel patterns (and channel pattern change) provide a metric for how flow on the fan self-organises to transport the available sediment supply. Here, channel pattern is characterised using two variables: the number of connected channel threads at 1 m 205 down-fan ( Figure 6), and the portion of the fan occupied by flow (Figure 7).  Figure 6 shows that flow became more branching in high-supply periods, with the number of channel threads increasing.
Conversely, flow became more channelised in zero-supply periods, with the number of channel threads decreasing.
The number of channel threads was slow to respond to a change in sediment supply rate. Consequently, the variability in the mean number of channels is similar between Run CON (constant sediment supply) and Run OSC10 (10-minute supply 210 oscillation cycle). Moreover, there is insufficient time for flow to channelise in the 5-minute zero-supply periods, so that flow maintains a divergent pattern in Run OSC10 with a fairly high number of channel threads (5-6 channel threads on average, at 1 m down-fan). In comparison, the 20-minute zero-supply duration in Run OSC40 means that fan-head trenching extends down-fan to the 1-m cross-section, bringing the mean number of channels down to around 2 at the end of the 40-minute cycle.
Comparing Figure 5 and Figure 6 indicates that fan gradient adjusted more readily than the channel pattern: fan gradient has fluctuate more than in Run CON (with constant sediment supply). This is likely because channels were counted at 1 m downfan, but the channel response to a change in sediment supply typically started at the fan-head and propagated down-fan to the 1 m cross-section (e.g. see time-lapse video for Run OSC40). Extracting the number of channels at the 0.5 m cross-section confirms that the fan-head channel pattern responded more readily to changes in the sediment supply rate (see Figure A3).

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The second metric for channel pattern was the "wet fraction": the proportion of the total fan area that was inundated with flow in a given minute. Variations in the wet fraction ( Figure 7) were more subtle than those in the number of channels, and provide a nuanced measure of how the flow pattern adjusted to changes in the sediment supply rate.  Figure 7 shows that, as with the number of channel threads, there was a lag of a few minutes before the wet fraction began to adjust to a change in the sediment supply. Consequently, the rapid supply oscillations in Run OSC10 (10 minute cycle) 225 had little impact on the wet fraction; its variation through time was similar to that with constant sediment supply (Run CON).
During Runs OSC20 and OSC40, the longer durations of high-and zero-supply show more clearly how the wet fraction adjusted to the sediment supply rate. At the start of high-supply periods, the wet-fraction increased after an initial lag, as flow widened and shallowed and/or slowed. A close examination of the experimental videos reveals that this increase was not through sheetflow or the complete abandonment of channelised flow, but through flow divergence into numerous channel the wet fraction decreased, suggesting that flow ultimately organised into fewer, deeper channels (although still more channels than during zero-supply periods). Close examination of Figure 6 supports this notion.
At the start of zero-supply periods, the wet fraction increased briefly, showing that a larger portion of the fan was inundated as flow re-adjusted to the reduced sediment concentration. However, the number of channel threads decreased during this period, 235 as did the sector of the fan occupied by the flow (Figure 6 and Figure A4). Their decrease implies that flow was collecting into fewer channels, but that those channels were initially shallow (or slow) and wide as the total area of flow was high. Toward the end of the zero-supply period, the wet fraction decreased, suggesting that channels were becoming deeper (or faster) and narrower as the flow area decreased. These changes can also be seen in the experimental videos.

Lateral mobility 240
In addition to characterising channel patterns, we also monitored channel change. Here, we characterise lateral channel mobility by measuring F n , the percentage of the fan area that is newly inundated in a given minute. High F n values imply either avulsion, a rapid channel sweep across the fan, or a rapid change in channel pattern (e.g. divergence from single-to multi-threaded flow).
Conversely, a low F n reflects a relatively stable channel.  sediment supply, lateral mobility peaked, with the channel adjusting rapidly to the altered sediment concentration. The experimental videos show that, when the sediment supply was turned on, rapid divergence into a multi-threaded flow pattern caused this peak in lateral mobility. Conversely, when the sediment supply was cut, the videos show rapid fan-head trenching, cutting off the flow to the diverging channel threads and rapidly redirecting flow into a single (and often new) channel, again raising mobility rates. This peak decayed gradually during both high-and zero-supply periods. Lowest F n values, implying the most 250 stable channel pattern, were attained in Runs OSC20 and OSC40 at the end of the zero-supply periods. This stability reflects an incised channel, which must be filled with sediment in the next high-supply period before rapid migration can re-commence.
Such channel filling thus delayed the onset of peak mobility in the following high-supply period, with the high-supply mobility peak occurring later in Runs OSC20 and OSC40, as those runs had longer zero-supply periods in which the channel became more entrenched at the fan-head. These results mirror a set of experiments by Vincent et al. (2022), in which longer duration 255 low-flow periods between debris floods increased the time required for a debris flood to cause avulsion.

Morphologic reworking
In addition to the lateral mobility computed by comparing flow maps, we calculated rates of vertical change (i.e. erosion and deposition) by performing change detection between successive DEMs. Figure 9 shows the volumes of sediment deposited or eroded in each minute of the sediment supply oscillation cycle.
260 Figure 9 indicates that the addition of short sediment supply oscillations in Run OSC10 increased the erosion and deposition rates on the fan, compared to Run CON with constant sediment supply. The temporal variation in erosion and deposition was similar in Run OSC10 and Run CON, although the timing of peaks in Run OSC10 followed the same pattern as Runs OSC20 and OSC40, described below.
In Runs OSC20-OSC40, deposition (top row) increased during high-supply periods. The increase was gradual, with peak 265 deposition coming several minutes after the onset of high-supply. This reflects the fact that deposition began in the feeder channel upstream of the fan-head, which buffered the fan from the immediate effect of changes in environmental conditions in the same way that a confined upstream reach would in a natural system. When the sediment supply was turned off, deposition rates remained high for a few minutes as sediment in the feeder channel was mined. Deposition rates then decreased toward the end of the zero-supply period.

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Erosion (bottom row) also increased at the start of the high-supply period, even though deposition rates were high. This reflects a period of channel adjustment in response to the sudden increase in sediment supply. After the initial peak, erosion decreased to a minimum in the second half of the high-supply period. When the zero-supply period commenced, erosion accelerated, reflecting the rapid lateral migration (Figure 8) at this time. After the initial increase, erosion rates stabilised during the zero-supply period. This was particularly evident in Runs OSC20 and OSC40, where erosion and deposition rates were 275 approximately equal toward the end of the zero-supply. Their similarity implies that, through fan-head trenching during the zero-supply period, the fan reached a form of equilibrium with the imposed flow rate and the supply of sediment from incision.
Although erosion and deposition occurred, there were no major peaks, suggesting that there was little channel reorganisation. Coloured text indicates each experiment's time-averaged mean and standard deviation. Figure A9 also overlays these data for comparison.
Instead, a condition of relative stability persisted, with the channel gradually incising the fan-head and depositing sediment on the lower fan.

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The average spatial patterns of erosion and deposition are revealed in Figure 10, which demonstrates the effects of lengthening the sediment supply oscillations. During high-supply periods (top row), deposition was concentrated at the fan-head, and particularly in the fan-head trench (most visible in Runs OSC20 and OSC40). Downstream of this deposition was a focused erosion zone. As the duration of the sediment supply oscillation cycle increased, this deposition-erosion couplet extended farther downstream. The deposition zone became more elongate, emphasising the filling of the fan-head trench, while the erosion 285 zone shifted farther down-fan. Figure 10. Spatial patterns of erosion (red) and deposition (blue). All one-minute DoDs for high-supply (above) and zero-supply periods (below) were averaged to show the typical pattern. Data from Run CON are split into the same groups as Run OSC10, although the former had no sediment supply oscillations; the Run CON data show how deposition and erosion varied randomly in comparison to the spatially organised erosion and deposition in Runs OSC10-OSC40. The semicircle at 2 m down-fan shows how the central fan became more elongate as the sediment supply oscillations lengthened. To maintain the same data density (and therefore, signal-to-noise ratio), the data for Run CON displayed here are only from the second repeat of this experiment, which was closest to the mean slope and area across the three repeats of Run CON.
During zero-supply periods, trenching at the fan-head was again clear, with a zone of focused erosion at the fan-head. The erosion zone and fan-head trench extended down-fan as the duration of the zero-supply period increased. A zone of focused deposition radiated from the downstream end of the fan-head trench, and is particularly visible in Runs OSC20 and OSC40.
This pattern indicates that, during the longer zero-supply periods in Runs OSC20 and OSC40, the fan-head trench incised and 290 acted as a conduit for sediment eroded from the feeder channel. The morphology of the fan-head trench is also visible in the cross-fan topographic profiles in Figure A5-A7.
The cumulative effect of this coupled fan-head trenching and lower-fan deposition can be viewed by comparing the fan shapes in Figure 10. Each DoD compilation in this figure was masked to the fan planform at 18 hours of experimental run time; the same total volume of water and sediment had been delivered to each fan. Nevertheless, as the durations of sediment 295 supply oscillations increased, the fan shape became more elongate: the mid-fan (axial) radius increased from Run OSC10 to OSC40, while the side radius remained comparable across all runs. This reflects fan-head trenching and sediment transfer from the upper to lower fan during the zero-supply periods, which lengthened the central fan.

Discussion
By alternating between a high sediment supply rate and no sediment supply, and by varying the duration of oscillations, These experiments have demonstrated how fan-channels respond to an abrupt increase in the sediment supply rate (and 305 therefore, the sediment concentration). The primary responses were steepening, an increased number of channels, and increased inundated area. This was accompanied by a rapid increase in lateral mobility that tailed off as the flow pattern stabilised, and by a peak in erosion rate during this channel adjustment phase. Deposition was generally high during high-supply periods.
When sediment supply was abruptly cut off, the fan responded through slope reduction and channelisation. The inundated area was high at first, while flow adjusted toward a single channel, but then decreased as that channel incised. Similarly, there 310 was a steep but short-lived peak in lateral mobility as the flow pattern adjusted. This pulse of lateral mobility was accompanied by an increase in erosion, that stayed high during the zero-supply period due to fan-head incision. Meanwhile, the deposition rate gradually decreased, eventually equilibrating with the erosion rate in the runs with longer zero-supply periods.
Extending the length of the sediment supply oscillations altered fan morphology, even though the mean sediment supply rate was unchanged. Short oscillations led to sediment accumulation at the fan-head, steepening the fan in comparison to the 315 constant supply experiment, and generating diverging, laterally active channels. Conversely, longer oscillations allowed the channel to incise the fan-head, shifting the zone of geomorphic activity farther down-fan and giving rise to a flatter gradient and more elongated fan shape.

Sediment concentration variability on fans in the field
These "similarity-of-process" experiments are a simple but useful model of fan dynamics (Hooke, 1968a;Paola et al., 2009).

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The different scenarios explore the limits of fan behaviour: while there are few situations in which fans abruptly alternate between high-supply and clearwater floods, wide temporal variations in sediment concentration do occur on fans and in steep, gravel-bed streams (e.g. Garcia et al., 2000;Habersack et al., 2001;Hayward and Sutherland, 1974;Kuhnle, 1992;Mao et al., 2014;Meyer and Wells, 1997;Reid et al., 1985;Wasson, 1974;Wells and Harvey, 1987). Our experiments show how fan channels respond to some of the widest possible variations in sediment concentration; one could expect lower-amplitude 325 variability to generate dampened but qualitatively similar responses.
On fans in the field, wide variations in sediment concentration can occur during a single flood event, or between successive events. For instance, sediment exhaustion over the course of a storm can rapidly decrease sediment concentration during a flood event (Meyer and Wells, 1997;Wasson, 1974;Wells and Harvey, 1987). Similarly, debris flow control nets are designed to trap coarse sediment but transmit water and fine sediment (Wendeler et al., 2007;Wendeler and Volkwein, 2015); therefore, their 330 emplacement in source basins can lower the bedload sediment concentration in flows reaching fans. Lower amplitude sediment concentration variations have also been observed during floods in gravel bed rivers (e.g. Garcia et al., 2000;Habersack et al., 2001;Hayward and Sutherland, 1974;Kuhnle, 1992;Mao et al., 2014;Reid et al., 1985).
The experiments presented in this paper provide a controlled analogue to the scenarios described above; the experimental results indicate how fans in the field may respond to different scales of variability. For instance, the changes in channel pattern 335 and lateral migration in Run OSC10 (see Figures 6 and 8) were minor, suggesting that short-term fluctuations in the sediment concentration do not strongly influence channel pattern. Although Run OSC10 had the steepest slope ( Figure 5), this likely reflects the high amplitude of the sediment supply rate variations, from 0 to 10 g s -1 . A preliminary experiment with lower amplitude variations but the same mean sediment supply rate and oscillation duration generated a less-steep fan ( Figure A8).
Consequently, it seems that small-scale, short variations in sediment concentration during a flood will have little influence on While these experiments highlight the impact of rapid changes in sediment concentration, they also underscore the importance of clearwater flow for shaping fan and channel morphology, and particularly the fan-head trench. Such trenches are ubiquitous in natural fans of varying ages in varying climates (e.g. Bowman, 1978;Bluck, 1964;Davies and Korup, 2007;350 Dorn et al., 1987;Harvey, 1987;Mather and Hartley, 2005, among others). Previous experimental studies have suggested that an incised fan-head trench can develop in the absence of external perturbations; that is, when water and sediment are supplied at constant rates (Clarke et al., 2010;Van Dijk et al., 2012;Schumm et al., 1987;Whipple et al., 1998;Zarn and Davies, 1994).
Nevertheless, the constant flow and sediment supply experiment presented here (Run CON) had a relatively high sediment supply and wide grain size distribution relative to previous experiments, so that the channel planform was highly dynamic 355 throughout the experiment (see Leenman and Eaton, 2021). This dynamism meant that a single channel never persisted for long enough to allow significant fan-head trenching.
In contrast, a distinct fan-head trench did develop in Runs OSC20-OSC40, with longer sediment supply oscillations. It is interesting to note that there was no notable fan-head trench in a related set of experiments with short-duration flow variability (published in Leenman et al., 2022) or in the experiment with short-duration sediment supply variability (Run OSC10). These 360 results suggest that longer-term sediment concentration variations (and particularly periods of lower-than-average sediment supply) encourage the formation of an incised fan-head trench. Figure 6 suggests one explanation for this phenomenon: the number of channels decreased during periods of low sediment concentration, as flow collected into a few main channel threads.
This flow contraction then concentrated the erosive activity of the channel into a narrow sector of the fan (see also Figure A4), which likely enhanced the rate of down-cutting and trench formation. This down-cutting further concentrated flow at the fan-365 head in a positive feedback, as lateral migration was most restricted at the end of the zero-supply periods (especially in Run   OSC40). The zero-supply periods in Run OSC10 were too short for this down-cutting mechanism to be effective and so a distinct trench did not form all the way down to the 1 m transect we analysed. This result highlights how fan responses to disturbance are governed by the fan's adjustment timescale. Disturbances may not alter fan morphology if the duration of or time between disturbances is shorter than the adjustment timescale. Figure 6 therefore suggest that, at least for the formation 370 of a fan-head trench in response to sediment supply exhaustion, our experiments' adjustment timescale must be between five and ten minutes.
In our experiments and in some active fans in the field, the fan-head trench fills when sediment supply is high; in nature, this may occur when landslides temporarily raise the sediment supply (e.g. Davies and Korup, 2007). A fan-head trench does not necessarily require a clearwater flood event in order to develop: experiments by Vincent et al. (2022) showed that even low flow 375 periods with no sediment supply were capable of incising a fan-head trench, given a sufficiently long period. An additional implication of the experiments in this paper, supported by the work of Davies and Korup (2007) and Vincent et al. (2022), is that the impact of floods with high sediment concentration is mediated by the depth and length of the fan-head trench, as this depression must be filled before avulsion can occur at the fan-head.
In the field, depositional events on fans have formed deposits of a comparable size to that generated by the high-supply 380 periods in these experiments. In Runs OSC10-OSC40, fan-head trenches were on average 9-14 mm deep respectively; over all areas of topographic change (including erosion) the mean depth of topographic change in high-supply periods was 2-4 mm respectively. The ratio of deposition depth to trench depth therefore ranged from 0.2-0.3. On fans in southwest New Zealand, Davies and Korup (2007)  Parallels can be drawn between our experimental fans and natural fans formed by basins that are supply-or transport-limited.
For instance, a transport-limited basin is less likely to have frequent or prolonged clearwater floods. Such a basin is most similar to Run CON or OSC10, and one might therefore expect it to generate a steeper fan with diverging, laterally active channels, all other things being equal. Conversely, a supply-or "weathering-limited" basin (Bovis and Jakob, 1999) might have more 395 frequent or prolonged clearwater floods, or floods with low sediment concentration. It is therefore more similar to Runs OSC20 or OSC40, and one might expect it to generate a fan with a lower gradient and more incised fan-head trench that is relatively stable, punctuated by periods of high lateral activity and more divergent flow when hillslope erosion generates peaks in the sediment concentration (such as that observed on the Poerua fan by Davies and Korup (2007)).
Climate change is increasing the frequency and severity of extreme weather events that lead to floods (IPCC, 2022). In 400 transport-limited basins, such changes could increase the frequency of flood events with high sediment concentration, making fan behaviour more similar to Run CON or OSC10. Conversely, in supply-limited basins, this hydroclimatic change could increase the frequency of floods capable of reworking the fan, and may have less impact on the sediment supply (depending on the nature of hillslope erosion). Such a change might make fans more similar to Runs OSC20 or OSC40.

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Studies of alluvial fans have sought for many decades to relate fan area and slope to source basin morphometry (e.g. Al-Farraj and Harvey, 2005;Beaumont, 1972;Bull, 1964;Crosta and Frattini, 2004;Denny, 1965;De Scally and Owens, 2004;Harvey, 1984;Hooke, 1968b;Kostaschuk et al., 1986;Lecce, 1991;Melton, 1965;Milana and Ruzycki, 1999;Oguchi and Ohmori, 1994;Saito and Oguchi, 2005;Silva et al., 1992;Stokes and Mather, 2015;Stokes and Gomes, 2020;Tomczyk, 2021). A key goal of those studies was to untangle the sources of scatter in such relationships, thereby elucidating the controls on fan slope While studies of fan and catchment morphometry have contributed a great deal to our understanding of alluvial fans, linking catchment and fan variables rests upon the assumption that fans have reached (or will reach) an equilibrium with their source 415 catchment. This is not always the case; the sediment supply to fans oscillates at periods ranging from single events (e.g. Cabre et al., 2020) to orbital cycles (e.g. Blechschmidt et al., 2009). In fact, this temporal variability gives rise to the concept of alluvial fans as environmental "indicators" that record changes in sediment supply (Harvey, 2012). Our experiments show how, over many successive cycles, such oscillations in the "upstream" conditions of an alluvial fan can alter fan morphology, even for the same average sediment supply rate. Consequently, our results reveal another potential source of scatter in relationships 420 between basin and fan morphometry: the periodicity of sediment inputs.
The flattening and lengthening of our experimental fans with increasing oscillation duration (see Figure 10 in particular) also suggests that, in the field, we can infer something of the sediment-supply histories of fans based on their elongation.
For instance, fans that have a shorter, more "stacked" morphology might form from high-frequency oscillations. Conversely, fans with more "telescoping" morphologies (like that of Run OSC40) might reflect a more intermittent sediment supply, with 425 longer-term periods of high and then low sediment supply. In particular, longer periods of low sediment supply likely allow for sediment redistribution from the upper to lower fan, forming a fan-head trench and "telescoping" lower fan. Figure 10 suggests that the fan-head trench becomes more elongate (and the intersection point farther down-fan) as the duration of a low sediment supply period lengthens. One example of a set of natural fans with elongate, "telescoping" morphologies is the mountain front fans along the west of the Musandam Mountains (Al-Farraj and Harvey, 2005). Prolonged periods of higher monsoonal 430 rainfall throughout the Late Quaternary have driven long-term changes in the sediment supply in this region (Blechschmidt et al., 2009), which may account for these fans' telescoping morphology.

A representative sediment supply rate?
These experiments show how sediment supply fluctuations can affect fan morphology and channel patterns. However, they also highlight a problem in the common approach to physical models of alluvial fans: that of constant sediment supply as 435 an approximation for a range of variable sediment supply rates in the field. While the temporal variations imposed in Runs OSC10-OSC40 were almost as simple as possible, they nevertheless generated fans with morphology, channel patterns and behaviour that were different from Run CON, with constant sediment supply. Moreover, fan morphology varied systematically with the duration of high-and zero-supply periods. These findings indicate that constant sediment supply rates poorly represent fans in the field that are governed by episodic sediment supply.

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It might be possible to find a "representative" sediment supply rate which generates the same fan slope as would a series of episodic sediment delivery events. For instance, median fan slope was similar in Run CON (constant sediment supply) and Run OSC20 (20-minute oscillation cycle). One could infer that the 20-minute cycle in Run OSC20 may represent some kind of "characteristic event periodicity" for the given sediment supply and flow rates, with the 10-minute cycle being too short (and therefore producing steeper fans) and the 40-minute cycle being too long. However, while Runs CON and OSC20 445 produced similar fan slope, geomorphic activity was much more variable in Run OSC20, with higher extreme values of lateral and topographic change. Consequently, when seeking to model natural hazards on fans, it might be appropriate to choose a higher "representative" sediment supply, or to include oscillations about the mean. Although doing so may misrepresent fan slopes, it is more likely to capture the extreme values of erosion or lateral migration which are of interest for understanding and managing natural hazards.

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Other experimental studies have explored how sediment supply alters fan gradients and channel dynamics (Ashworth et al., 2004;Bryant et al., 1995;Delorme et al., 2018;Whipple et al., 1998). However, those experiments featured constant sediment supply rates. The experiments presented here show that both fan gradient and channel dynamics vary systematically with the duration of sediment supply oscillations. These results raise the possibility of equifinality: natural fans with steeper gradients and laterally active channels may result from high sediment supply, from rapid fluctuations between high and low sediment 455 concentrations, or from some combination of both these influences. Moreover, it is possible that long-term average sediment supply and the intermittency of sediment inputs could covary in natural fans, making these two controls difficult to separate.
Series of shorter and longer sediment supply periods likely occur as well, which may generate legacy effects depending on their sequencing. Further experiments varying both the mean sediment supply and the duration of supply oscillations could aid in addressing these questions. The experiments showed that, when sediment concentration increased abruptly, fans steepened and flow diverged into more channel threads, inundating a larger fraction of the fan. Lateral mobility and erosion rates were high at first, before the channel pattern stabilised. Deposition rates remained high while sediment concentration was high. Conversely, when sediment concentration decreased abruptly, fans adjusted through slope reduction and channelisation. The inundated area, lateral mobility and erosion rate were high at first, until flow adjusted toward a single-channel state. The erosion rate remained elevated while 470 sediment concentration was low, due to fan-head trenching.
The duration of high-and zero-supply periods systematically affected fan morphology and channel dynamics. Short-term oscillations promoted fan-head deposition, steepening the fan and generating diverging, laterally active channels. Long-term oscillations promoted fan-head incision, shifting the zone of geomorphic "activity" down-fan and generating a flatter, more elongate fan. Finally, the duration of sediment supply oscillations produced systematic variation in fan slope and area, even though all experiments had the same mean sediment supply rate. This raises the question of how closely experimental fans built with constant sediment supply can be said to represent fans in the field. While such experimental studies have revealed the importance of autogenic fan dynamics, our work builds on this foundation by modelling more "real-world" scenarios that are relevant for hazard management in changing hydroclimates. Future studies of alluvial fans could consider including supply variability Code and data availability. Basic data processing steps were conducted using the code at https://github.com/a-leenman/phd_code/releases/ tag/v1.0. Code to conduct further analysis and produce the figures can be found at https://github.com/a-leenman/Leenman_Eaton_2022/ releases/tag/v1.2. Data underlying the figures, as well as back-up copies of the timelapse videos and code, can be downloaded at https: //doi.org/10.5281/zenodo.7100814. The raw data from these experiments is still under analysis for subsequent publications; please email the 495 corresponding author if you would like a copy.

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B. C. Eaton conceptualised the project, edited the manuscript, and provided supervision throughout the project.
Competing interests. The authors have no competing interests to declare.