developed a model describing the long-profile evolution of transport-limited, gravel-bed rivers. Their approach brings together established theory relating water flow, sediment transport and channel hydraulic geometry that has been extensively tested in laboratory and field settings. The result is a model grounded in first principles and consisting only of parameters that are physically defined (i.e., parameters that can, in principle, be measured). We envisage a self formed channel meandering through a gravel valley with width B and length L (Fig. ). Over time, the channel sweeps from side to side, moving downstream sediment from the entire valley cross section. Then, following , change in elevation, z, through time, t, is controlled by along-stream variations in bedload sediment discharge, Qs: 1∂z∂t=-1B(1-λp)∂Qs∂x+U, where x is distance down valley, λp is sediment porosity and U is a source/sink term that can account, for example, for uplift and subsidence, along-stream sources of sediment, or bedload loss due to downstream fining (all notation is summarised in Appendix ). derived an expression for Qs in terms of bankfull water discharge, Qw, and down-valley slope: 2Qs=-kQsIQwS7/6∂z∂x∂z∂x1/6, where I is the intermittency of bankfull discharge, S is sinuosity, and kQs≈0.041 is a coefficient combining terms relating to sediment transport and equilibrium hydraulic geometry. This expression assumes that sediment discharge depends on bed shear stress according to the relationship of , which was later updated by ; that gravel-bed channels adjust their widths such that the bed shear stress is maintained at a fixed ratio of the threshold for bedload motion ; and that bed roughness follows a grain-size dependent Manning–Strickler formulation .

Combining Eqs. () and () gives 3∂z∂t=kQsIS7/6B(1-λp)∂∂xQw∂z∂x∂z∂x1/6+U, a non-linear diffusion equation, in which we have assumed that kQs, I, and S do not vary along stream. Throughout this work, we solve Eq. () using the boundary conditions: 4∂z∂x|x=0=-SQs,0kQsIQw,06/7 and 5z(x=L,t)=0. Additionally, for single-segment models, we apply the initial condition: 6z(x,t=0)=z‾=-SQs,0kQsIQw,06/7(L-x), where L is the x (i.e., down-valley) position of the valley outlet. Equation () states that the slope at the valley inlet is set by the ratio of the sediment and water supplies (denoted Qs,0 and Qw,0, respectively). Equation () states that the elevation at the valley outlet is fixed to zero. Equation () states that the long profile begins in equilibrium with the supplied sediment to water ratio.

explored the behaviour of Eq. () for the simple, single-segment case in which all water and sediment is supplied at the valley inlet (Fig. a). They showed that, if the system is subjected to small variations in water and sediment supply, such that 7Qw(x,t)=Qw‾+δQw(t) and 8Qs,0(t)=Qs,0‾+δQs,0(t), where 9δQw(t)≪Qw‾ and 10δQs,0(t)≪Qs,0‾, then variation in elevation can be approximated by 11∂δz∂t≈κ∂2δz∂x2, where 12κ=76kQsIQw‾B(1-λp)∂z‾∂x1/6, where κ is a sediment-transport diffusivity. In Eqs. ()–(), overlines indicate mean values (with respect to time), while “δ”s indicate small variations about those means. Note that, while variations in sediment supply are imposed only at the valley inlet, changes in water supply are assumed to affect the entire valley instantaneously (Eqs. –).

further showed that if sinusoidal fluctuations in water and sediment supply are imposed, so that 13δQw(t)=AQwQw‾sin⁡2πtP and 14δQs,0(t)=AQs,0Qs,0‾sin⁡2πtP, then resulting fluctuations in elevation and sediment discharge are approximated by 15δz(x,t)≈z‾(AQs,0-AQw)Gzsin⁡2πPt-φz and 16δQs(x,t)≈Qs,0‾(AQs,0-AQw)GQssin⁡2πPt-φQs. AQs,0 and AQw are dimensionless amplitudes, normalised by their mean values, of sediment supply and water discharge, respectively, while P is the period of the imposed signal. The valley's response to sinusoidal variations in water and sediment supply is itself approximately sinusoidal, modulated by two parameters: “gain”, Gz and GQs, and phase shift, φz and φQs obtained a similar result for a generic linear system. Gain describes the response amplitude relative to the amplitude of the imposed signal. A value of zero indicates that, despite imposed variation in water or sediment supply, there is no variation in elevation or sediment discharge. A value between zero and one indicates that amplitudes of variation in elevation or sediment discharge are lower than those imposed (i.e., the signal is damped, or buffered). A value of one indicates that the response and imposed signal have the same amplitude, while a value greater than one indicates the that imposed signal is amplified. The phase shift, or lag, describes the offset in time between the imposed signal and the valley response.

provided analytical expressions for Gz, GQs, φz and φQs, and showed that they are principally controlled by two key parameters: the relative distance along stream, x/L, and the forcing period relative to the valley's equilibration time, P/Teq, where Teq=L2/κ e.g.,. This result implies that for a given external forcing, the likelihood of terrace formation, related to Gz, and its timing, related to φz, depend on the timescale of that forcing, the size of the system, and the position along stream. Similarly, the likelihood of a detectable signal reaching downstream sedimentary basins, related to GQs at the valley outlet, and its timing, related to φQs at the outlet, also depend on the forcing timescale and the system size.

We supply water along stream so that water discharge increases continuously according to a power law: 17Qw=kx,Qw(x+x0)px,Qw, where kx,Qw is the power-law coefficient linking distance downstream to bankfull water discharge, px,Qw is the power-law exponent, and x0 is a distance from the drainage divide at the valley inlet. This approach is similar to that of several previous studies e.g.,. Equation () is related to Hack's Law, which connects downstream distance to upstream drainage area though commonly expressed in the opposite sense, with downstream distance as a function of drainage area;. However, the relationship between drainage area and bankfull water discharge is not linear, being influenced, for example, by catchment hydrology and the catchment's size relative to that of the footprint of major rainfall events . These effects result in bankfull water discharge increasing more slowly downstream than drainage area . We adapt the earlier definitions of equilibration time, Teq, e.g., to account for along-stream variation in water discharge as follows: 18Teq=L2〈κ〉, where the angled brackets indicate a spatial average.

In an extension to previous studies, we also supply sediment along stream. We set the source term, U, in Eq. (), to the sediment supply per unit distance along stream. We choose this along-stream sediment supply so that sediment discharge at steady state also increases downstream according to a power law with the same form as Eq. (). As such, 19U=1B(1-λp)∂Qs∂x=kx,Qspx,Qs(x+x0)px,Qs-1B(1-λp)=1B(1-λp)Qs,0Qw,0∂Qw∂x.

We simulate five single-segment valleys in which sediment and water discharge increase at different rates, with 0.8≤(px,Qw=px,Qs)≤2.4 (Fig. a). This range corresponds approximately to an inverse Hack's law exponent in the range 1.6–2.4 typical values from global compilations, e.g., combined with an exponent linking drainage area to bankfull water discharge in the range 0.5–1 . For each valley, we choose kx,Qw and kx,Qs so that the ratio of sediment to water discharge remains constant along stream at steady state and the average water and sediment discharge is equal across all five simulated valleys (specifically, we set the spatial and temporal mean water discharge, 〈Qw‾〉, to 26 m3s-1, and supply sediment from upstream and along stream such that the spatial and temporal mean sediment discharge, 〈Qs‾〉, is 2.6×10-3 m3s-1). As such, each valley has the same steady-state slope, which is constant along stream (Fig. b). We set valley width, B, to a uniform value of 256 m, so that each valley has the same equilibration time of 100 kyr (as defined in Eq. ).

For each valley, we performed a series of numerical simulations in which we varied sediment or water supply sinusoidally (both at the inlet and along stream), with a range of periods between P/Teq=10-2–102. We define gain numerically as 20Gz(x)=max⁡(z(x,t))-min⁡(z(x,t))2z(x)‾(AQs-AQw) and 21GQs(x)=max⁡(Qs(x,t))-min⁡(Qs(x,t))2Qs(x)‾(AQs-AQw), which we compute directly from the simulated time series of elevation and sediment discharge. To avoid the influence of transient effects at the onset of periodic forcing, we measure gain only after two complete cycles (i.e., with t>2P). We estimate the phase shift by extracting peaks and troughs in the simulated time series of elevation and sediment discharge and measuring the difference in time between them and peaks and troughs in the imposed signal. We then explore how gain and lag vary along stream, as functions of forcing period and discharge exponent, and compare with equivalent results for the single-segment, upstream supply case derived from numerical simulations and analytical solutions previously presented by.

We first show simulated aggradation, incision and sediment discharge of an example valley with px,Qw=px,Qs=1.6 at three forcing periods (Fig. ). For comparison, we also include results from an equivalent simulation with sediment and water supplied only upstream i.e., with px,Qw=px,Qs=0;. To elucidate further how the valley response varies spatially, with forcing timescale, and with the power-law exponent, we then show gain and lag for each valley as functions of downstream distance (Fig. ) and forcing period (Fig. ), again with comparison to the upstream supply case . We found that, in all cases, patterns of long-profile evolution are similar regardless of whether sediment supply or water supply is varied. We therefore only show variation in elevation driven by variation in sediment supply here (equivalent results for variation in water supply are shown in Figs. S1–S3 in the Supplement). To further simplify the presentation of results, while we consider how the long profile varies along stream, we focus on variation in sediment discharge only at the valley outlet, which is most relevant for downstream sedimentary basins.

Broad patterns of aggradation and incision are similar for both the upstream and along-stream supply cases Figs. g–l, and a and d; see also. In both cases, when the forcing period is short relative to the valley's equilibration time (P≪Teq), amplitudes of aggradation and incision are low (Gz≈0) and lag significantly behind the forcing (φz>0). At intermediate forcing periods (P≈Teq), amplitudes are increased and lag times reduced. When forcing periods significantly exceed the equilibration time (P≫Teq), aggradation and incision occurs with similar amplitudes to, and close to in phase with, the imposed variation in sediment or water supply (Gz≈6/7≈0.86, φz≈0; the value of 6/7 is related to the 7/6 power on valley slope in the sediment-discharge equation, Eq. ).

Nevertheless, some important differences in patterns of aggradation and incision do arise between the upstream and along-stream supply cases, most evident in the distributions of gain and lag as functions of downstream distance and forcing period (Figs.  and ). First, in the along-stream supply case, Gz is larger and φz is generally lower than in the upstream supply case, for a given forcing period. Second, in the upstream supply case, Gz and φz decrease and increase continuously downstream, respectively (Fig. ). In contrast, in the along-stream supply case, while Gz initially decreases and φz increases away from the inlet, both reach turning points so that Gz increases and φz decreases towards the outlet. Third, while in the upstream supply case φz decreases continuously as the forcing period increases, in the along-stream supply case, it briefly increases to a maximum for periods close to the valley's equilibration time (P≈Teq; Fig. d). These differences generally become more pronounced as the water- and sediment-discharge power-law exponents increase (Figs.  and ).

When sediment supply is varied, variation in sediment output follows similar patterns to aggradation and incision (Figs. a–c and b and e); however, when water supply is varied, different behaviour emerges Figs. d–f and c and f; cf.. When the period of variation in water supply is short relative to the valley's equilibration time (P≪Teq), variation in sediment output has the same amplitude and is in phase with the forcing (GQs=1, φQs=0). In the upstream supply case, when the forcing period is increased, the amplitude of variation in sediment output briefly exceeds that of the imposed variation in water supply (GQs>1 while P≈Teq), before decreasing to zero for long forcing periods. In the along-stream supply case, variation in sediment output is not amplified at periods close to the valley's equilibration time, instead decreasing continuously as the forcing period increases. In both cases, negative lag times are introduced, so that peaks in sediment output precede corresponding peaks in water supply.

Two concepts help explain physically the variations in valley response that arise due to changing sediment and water supply (Figs. –). First, the extent to which the valley can adjust to external perturbations depends on the relationship between the perturbation timescale and the timescale over which the valley can aggrade and incise e.g.,. At the valley scale, the approximate timescale over which the valley can adjust is given by the equilibration time, Teq. At the local scale, however, the adjustment efficiency is also influenced by the local diffusivity (Eq. ). In the upstream case, diffusivity is uniform along the valley, while in the along-stream supply case, it increases downstream with water discharge, so that the capacity for the valley to adjust also increases downstream. Second, variation in sediment and water supply can be thought of as a signal introduced to the valley that can propagate along stream. In the upstream supply case, this signal takes the form of variations in slope at the inlet, that must then propagate downstream. In the along-stream supply case, as well as signal originating from the valley inlet, signal is introduced continuously along the valley, and propagates both up- and down-stream.

We first apply these concepts to patterns of aggradation and incision (Figs. g–l, and a and d). At the valley scale, when sediment or water supply vary with periods much smaller than Teq, aggradation and incision cannot keep pace with the forcing and amplitudes are low. As the forcing period increases to values similar to or greater than Teq, aggradation and incision can increasingly keep pace and amplitudes increase. In the upstream supply case, signal is introduced only at the inlet, so that lag increases continuously downstream as the signal propagates. For short forcing periods relative to Teq, incomplete adjustment causes the signal to diffuse as it propagates, so that gain decreases downstream. In contrast, in the along-stream supply case, signal is introduced continuously along stream, and diffusivity increases downstream as a result of increasing water discharge. As such, downstream parts of the valley do not need to “wait” for signal to propagate from the inlet, and have a greater capacity to adjust to the forcing. Away from the inlet, local signal sources therefore increasingly dominate, so that gain increases and lag decreases towards the outlet (i.e., the signal appears to propagate upstream).

Peaks in lag at forcing periods close to Teq for the along-stream case reflect the interaction between local and upstream signal sources (Fig. d). At shorter forcing periods, signal introduced at the inlet quickly diffuses away, so that aggradation and incision further downstream is controlled only by local signal sources. At periods around Teq, signal introduced at the inlet maintains some amplitude downstream, but takes time to propagate; the increasing influence of this upstream signal causes lag times to increase briefly with increasing period. As the forcing period increases further, the signal propagates more efficiently downstream, so that lag times again steadily decrease.

When sediment supply is varied, variation in sediment discharge is driven only by variation in valley slope (Eq. ). Thus, in both the upstream and along-stream supply cases, variation in sediment output closely follows that of aggradation and incision at the valley outlet: GQs and φQs behave similarly to Gz and φz (Figs. a–c and b and e).

In contrast, when water supply is varied, it influences the valley's capacity to transport sediment directly as well as triggering variations in slope (Eq. ). Interactions can then arise between the imposed variation in water supply and the resulting variations in slope that can be damped and lag behind the forcing (Figs. a–c and b and e). At forcing periods much shorter than Teq, slopes do not adjust along much of the valley, so that variation in water supply is translated directly into to variation in sediment output. Meanwhile, at forcing periods much longer than Teq, slopes can adjust to maintain an equilibrium with sediment and water supply, so that sediment output varies little see also. At intermediate forcing periods, slopes partially adjust, but lag behind the imposed variation in water supply. In the upstream supply case, variation in water supply and slope combine to amplify the imposed signal, while in the along-stream supply case, amplitudes simply decrease from short to long forcing periods.

In both the upstream and along-stream supply cases, the delayed adjustment of valley slope leads to a negative lag in sediment output, such that peaks in sediment output arise prior to imposed peaks in water supply (Figs. d–f and f). Consider the onset of sinusoidal variation in water supply. Initially, water supply increases without any adjustment of valley slope, so that sediment output also increases. Valley slope then begins to decrease, so that sediment output increases more slowly and then begins to decrease. This transition occurs prior to water supply reaching its maximum, resulting in the negative lag or appearance that variation in sediment output leads that of water supply.

We model alluvial valley networks as a series of interconnected network segments, each of which follow Eq. (). We supply sediment and water to inlet segments (i.e., segments without any segments upstream), and impose base level at the outlet segment (i.e., a segment without any segments downstream). In some cases, we also supply water and sediment along stream, analogously to the single segment, along-stream supply case presented above (Fig. c). Within the network, a segment's sediment supply is set by the sum of sediment discharges from its upstream segments, while elevation is required to be continuous across segment junctions see also. To solve Eq. (), we use the semi-implicit finite-difference scheme presented by adapted for a network .

To define the network topology (i.e., the position of each segment in the network relative to its adjacent segments), we use an approach introduced by , following the algorithm outlined by . This algorithm randomly generates binary trees with a given number of valley inlet (or “exterior”) segments; an example is shown in Fig. a–d. The algorithm treats all possible binary trees as equally likely. It has been argued that populations of binary trees generated with this approach are not realistic representations of real river-network populations, because, unlike river networks, the trees produced are not necessarily space filling e.g.,. discusses observations from real networks that deviate from the predictions of Shreve's model, often associated with space-filling constraints, but also occurring in regions with high relief. We note that several other frameworks for randomly generating synthetic networks have been developed, including, for example: undirected or directed random walks on lattices ; models of headward growth and branching ; “Random Self-similar Networks”, that can be constructed with specified statistical properties ; and “Optimal Channel Networks”, that seek to minimise energy expenditure involved in the transport of water across the landscape . Several of these approaches avoid the space-filling problem by constructing networks explicitly on two-dimensional grids, but no consensus has emerged as to which most successfully recreates realistic network populations. Furthermore, our model of long-profile evolution and sediment transport does not require any spatial information beyond distances along stream and the topological relationships between segments. We therefore persist with Shreve's approach, since it is relatively simple and samples all possible binary trees, with the caveat that the statistics of network populations we obtain may deviate from those of natural networks.

Once the network topology has been defined, lengths and properties such as water discharge and valley width need to be assigned to each segment. In natural networks, segment lengths and widths vary, and water discharge is derived from upstream catchment areas as well as locally from groundwater and surface runoff. Such complexities may influence how alluvial river networks respond to external variation in sediment and water supply, but they may also obfuscate more general aspects of network behaviour. We therefore start with more simplified synthetic representations of alluvial river networks, before progressively building complexity towards more naturalistic representations.

Specifically, we consider four network scenarios. In the simplest scenario, we set segment lengths to a uniform value of 5 km and supply water and sediment only at the valley inlets (Figs. c and a and e). In the second scenario, we draw random segment lengths from a gamma distribution with a shape parameter of two and a mean of 5 km, following and , but still supply water and sediment only at the valley inlets (Fig. b and f). Note that, unlike and , we do not use different length distributions for inlet and interior segments, for simplicity and due to a lack of evidence for systematic differences between them . also showed that a segment's local contributing area scales approximately linearly with its length, with a factor of around 300 m2m-1. In the third scenario, we therefore return to uniform segment lengths, but add water and sediment along each segment as well as at the valley inlets (Figs. c and c and g). In the fourth scenario, we use non-uniform segment lengths and also add water and sediment along each segment (Fig. d and h). In all scenarios, we set values of upstream and along-stream water supply so that the mean across each network, 〈Qw‾〉, is 26 m3s-1. We also fix the ratio of sediment to water supply to a constant value of 10-4, and neglect any downstream fining, so that the network long profiles are linear at steady state Fig. i–l; cf.. We do not expect this choice to influence significantly our results, since the diffusivity is only weakly dependent on valley slope (Eq. ). In all scenarios, we fix the valley width to a uniform value of 254 m; we return to implications of this choice in the Discussion (Sect. ).

To explore how a network's geometry influences its response to external forcing, we first need to be able to describe its geometry in a quantitative way. This topic has been the focus of considerable effort for the last few decades, so that many network metrics have been proposed and applied. In the absence of a priori ideas regarding controls on network responses to changing sediment and water supply, we test a range of existing metrics that quantify different aspects of network structure.

developed an influential set of statistics describing network geometries that are now known as “Horton's laws”. These laws rely on the concept of “stream order”, which was later modified by . Exterior or inlet segments are defined as first order streams. Where two first order streams combine, a second order stream is formed. Where a second order stream meets a first order stream, the second order stream is lengthened; only when two second order streams meet is a third order stream initiated, and so on. Thus streams of order two or greater can consist of multiple network segments. showed that, within a given network, the number of streams with order ω, Nω, decreases with stream order such that 22NωNω+1≈RB,  or  Nω≈RBΩ-ω, where Ω is the maximum stream order within the network. RB is termed the “bifurcation ratio”, and can be visualised as the gradient of a linear function relating ω to Nω in linear-logarithmic space (Fig. m–p). also showed that stream lengths increase with stream order, such that 23LωLω-1≈RL,  or  Lω≈RLω-1. showed that a similar rule applies to stream drainage areas: 24AωAω+1≈RA,  or  Aω≈RAω-1. Lω and Aω are mean lengths and drainage areas, respectively, of streams with order ω; and RL and RA are termed the length and area ratios, respectively. Since the synthetic networks used here have no inherent upstream drainage area, we define a related statistic using water discharge: 25Qw,ωQw,ω+1≈RQw,  or  Qw,ω≈RQwω-1, where Qw,ω is mean water discharge of streams with order ω and RQw is the discharge ratio. Typical values of Horton's ratio in natural networks are: 3<RB<5, 1.5<RL<3 and 3<RA<6, and showed that populations of random networks generated using his approach (which we adopt here) return similar values. However, showed that these ranges are in fact characteristic of all possible binary trees, while showed that Horton's laws retain relatively little information about network topology. Both and therefore suggest that Horton's Laws are not particularly useful measures of network structure. do show, nevertheless, that they can in some cases be used to discriminate between network populations with different characteristics. We include them here for completeness and consistency with the existing literature.

Another limitation of Horton's laws is that they do not distinguish between streams of a given order that flow into streams of different higher orders (e.g., a first-order stream that flows into a second-order stream is treated the same as one that flows into a third order stream). As such, they only hold exactly for networks in which streams of each order exclusively flow into streams exactly one order higher . devised an alternative scheme which does account for all possible relationships between streams of different orders. Tokunaga defines the parameter ϵi as the average number of streams of order ω flowing into streams of order ω+i. They then define the parameter K as average decrease in ϵi as i is incrementally increased, i.e., the average of ϵ1/ϵ2, ϵ2/ϵ3, …, ϵΩ-1/ϵΩ. K is therefore a more sophisticated version of Horton's RB.

Another prominent series of network metrics are built around the concept of “topological length” e.g.,. This framework is particularly attractive for the present problem since timescales of diffusive processes are known to scale with a system's length e.g.,. A segment's topological length, which we denote l, is defined as the number of segments from it to the outlet (including it and the outlet segment). Commonly used metrics making use of the topological length are: the maximum topological length of all segments in the network, lmax (also known as the network “diameter”); the average topological length of all segments in the network, 〈l〉; and the average topological length of all inlet segments, 〈lI〉 (also known as the “mean source height”). Since some of the networks we analyse have variable segment lengths, we also define an equivalent set of metrics that use absolute rather than topological lengths, i.e.: the maximum length to the outlet, Lmax; the mean length of all segments in the network to the outlet, 〈L〉; and the mean length from all inlet points to the outlet, 〈LI〉 cf. the catchment “centre of gravity”;.

Similar to the topological length is the concept of “topological width”, defined as the number of segments positioned at a given topological length from the outlet, w e.g.,. Typically, the maximum topological width in a network, wmax, is taken as a measure of a its structure; we also include the mean topological width, 〈w〉.

Lastly, we make use of the widely documented power-law relationship between upstream drainage area and distance downstream from the drainage divide known as Hack's law . The exponent of this power law provides a measure of the rate at which drainage area accumulates downstream. As in the case of RA above, since the networks we analyse have no inherent upstream drainage area, we fit a power-law function relating a point's distance from the furthest inlet to its water discharge (insets in Fig. e–h). The exponent of this power law, p, is then equivalent to a combination of an inverse Hack's exponent and an exponent linking drainage area to bankfull water discharge.

For each of the four network scenarios described in Sect. , we performed two sets of simulations. First, to assess the range of behaviour of networks with a fixed number of segments, we generated 200 networks each with 40 valley inlet segments (corresponding to 69 segments in total). Second, to assess how network behaviour varies with the number of segments, we generated a set of networks with numbers of inlet segments between 2 and 150. We generated four distinct network topologies for each number of inlet segments, for a total of 596 networks. For each network, we varied sediment and water supply sinusoidally with seven periods logarithmically spaced between one hundredth and one hundred times the equilibration time, where as an initial estimate, we define network equilibration time with the maximum stream length, Lmax (denoted Teq,max). We ran each simulation for four full cycles with timesteps of one thousandth of the forcing period. After each simulation, we measured elevation gain, Gz, and lag, φz, throughout the network, and sediment-discharge gain and lag at the network outlet (GQs,L and φQs,L, respectively).

In preparatory simulations, we found that networks responded to cyclical variations in sediment and water supply in broadly similar ways to the simpler, single-segment cases described earlier see also. Nevertheless, we also found significant variability between networks with the same number of segments, but different segment configurations, implying that a network's configuration influences its response time. To quantify this variability, we compared GQs,L as a function of forcing period for each network to the predictions for the simple single-segment case in which all sediment and water are supplied at the inlet. This single-segment, upstream supply case has a well defined relationship between GQs,L and the forcing period normalised by the valley equilibration time, P/Teq, which has been determined analytically Fig. b;. We therefore used an iterative optimization scheme to find, for each network, an empirical network equilibration time, Teq^, that minimises the difference between its GQs,L, obtained numerically, and the analytical solution for the single-semgment, upstream supply case. From this equilibration time we define an “effective length”, L^, for the network, following the definition of equilibration time in Eq. (), given by 26L^=Teq^〈κ〉. We can then compare empirical network equilibration times and effective lengths obtained in this way with network properties to assess which features control the timescales of the network responses.

We first assess timescales of network responses to variation in sediment and water supply and their controls (Figs. –). We then present broad patterns of aggradation, incision and sediment output as functions of forcing period, providing an overview of the possible range of behaviour across catchments with different properties (Fig. ). Lastly, we explore in detail spatial patterns of aggradation and incision, which influence the distribution and timing of terrace formation within networks (Figs. –).

As in the single-segment case, network responses to variation in sediment and water supply can be understood in terms of the relative timescales of the forcing and network aggradation and incision e.g.,, and of signal propagation through the network (Figs. –). In both network cases, signal is now introduced at multiple valley inlets, while, in the along-stream supply case, signal is also introduced internally along each network segment. On individual segments, variation in sediment and water delivery from upstream segments transmits signal downstream, while aggradation and incision of downstream segments drives local base-level variation and transmits signal upstream. As in the single-segment, along-stream supply case, accumulation of water downstream means that diffusivity, and the capacity for segments to adjust to external forcing, is greater in downstream, higher order parts of the network.

It is well understood that, in the single-segment case, the timescale over which a diffusive system evolves is set by the square of its length e.g.,. In the network case, signal is introduced at range of distances from, and therefore must propagate a range of distances to, the network outlet. Our results suggest that the timescale over which a network can aggrade and incise is controlled by the average of these distances (in our terminology, the mean inlet length, 〈LI〉; Figs. –). This result may also be related to that of , who showed that metrics related to a network's mean length retain more information about network structure than those focused on branching statistics (i.e., Horton's and Tokunaga's laws) or the accumulation of drainage area and water discharge (i.e., Hack's law). For example, two networks can have the same maximum length but very different distributions of segments; the mean length, in contrast, takes into account the entire network structure.

This framework also helps make sense of spatial variation in the network response (Figs. –). On lower order streams, the response is dominated by signal propagating from upstream inlets: Gz therefore generally decreases and φz generally increases downstream. Higher order streams, in contrast, are influenced by signal propagating from a range of distances up- and down-stream, as well as, in the along-stream supply case, local signal sources. Signals introduced far upstream take time to propagate to high order segments and lose amplitude as they do so. However, signals introduced on short tributaries can propagate efficiently to higher order streams, behaving similarly to the local sources of the single-segment along-stream supply case. Combined with increased diffusivity on higher order segments, these effects result in a general increase in Gz and decrease in φz along higher order segments, similar to single-segment, along-stream supply case this general reduction of φz on high order segments, such that the network response is more spatially uniform than the single-segment, upstream supply case, was also noted by. Higher amplitudes and lower lags on higher order segments also feed back to short adjacent tributaries by varying their base level. Thus, Gz is maximised and φz minimised on short tributaries close to the valley outlet.

The networks with upstream and along-stream supply of sediment and water behave similarly in many respects. In particular, we obtained very similar estimates of equilibration times and effective lengths for the two cases, with those of the along-stream supply networks being only slightly shorter (Figs.  and ). These patterns likely reflect the fact that, even when along-stream supply is included, an increasing fraction of a segment's sediment supply is still derived from upstream as the segment order increases. Thus, for higher order segments such as the outlet, whose behaviour we used to estimate Teq^, the signal propagating from upstream dominates in both cases. An exception arises in patterns of φz,L, for both variation and sediment and water supply, and φQs,L, for variation in sediment supply, at low forcing periods (i.e., P≲Teq^; Fig. e–f, m–n). Here, signals propagating from upstream are strongly damped, so that, in the along-stream supply case, local sources begin to dominate, and φz,L and φQs,L even decrease with decreasing period (similar to the single-segment, along-stream supply case; Fig. d and e). In contrast, in the upstream supply case, lag times at the network outlet strongly diverge as P/Teq^, decreases, so that a wide range of φz/P and φQs/P occurs even for simulations with similar P/Teq^.

This last result suggests that, for the upstream supply networks, the mean inlet length is not a good predictor of lag time at the outlet for short forcing periods. If progressive damping of signals as they propagate downstream results in inlets positioned far upstream having increasingly limited influence on aggradation and incision at the outlet, a corollary is that inlets close to the outlet should exert greater influence. Indeed, distinguishing networks by the distance upstream from the outlet to their closest inlet segment reveals a clear pattern (Fig. ). For the networks with uniform segment lengths, three populations exist, in which the first inlet occurs two, three or four segments upstream from the outlet; lag times increase as the distance to the first inlet increases (Fig. a). For the networks with non-uniform segment lengths, there is more scatter, but the same general pattern is present (Fig. b). This scatter may arise due to differences in the distribution of water discharge, and hence diffusivity, along the paths from the first inlet downstream to the outlet. In the networks with uniform segment lengths, water discharge increases downstream at consistent intervals, so that networks with equal distances to the first inlet also have similar distributions of discharge along the path. Conversely, in the networks with non-uniform segment lengths, discharge and distance are decoupled, so that two networks with similar distances from the outlet to the first inlet can have very different distributions of discharge along the path, and hence very different rates of signal propagation. These results confirm that, at short forcing periods, aggradation and incision, and, consequently, sediment export, at the network outlet are mainly influenced by network structure close to the outlet. Similarly, we expect the response to short period external forcing at different points within the network to be mainly controlled by local network structures.

We have shown that simplified, single-segment models with only upstream supply of sediment and water can accurately predict amplitudes of variation in sediment export from networks with varying complexity, provided an appropriate lengthscale is chosen to characterize the network (Figs. –). We also showed that an appropriate lengthscale for a given network is predictable from its mean inlet length. As such, we propose a modified definition of alluvial river equilibration time appropriate for networks: 27Teq=L^2〈κ〉≈kL2〈LI〉2〈κ〉, where kL is a coefficient scaling mean inlet length, 〈LI〉, to effective length, L^. The value of kL depends on the number of network segments and on how discharge varies throughout the network. In our simulations, it increased from close to one for networks with few segments to approximately 1.4–1.45 for networks with many segments (Fig. ). It may continue to rise slowly for larger networks than those tested here.

This result implies that, if our main concern is the amplitude of variation in sediment discharge at the network outlet (e.g., for studies of downstream stratigraphic records), we can use simplified single-segment models, and avoid the complexity and computational cost of network implementations. We note that, for variation in sediment supply, single-segment models with only upstream supply of sediment and water do not predict network lag times as effectively as their amplitudes (Fig. e–f, m–n). When sediment supply is varied at relatively short forcing periods (P<Teq), lag times for networks with only upstream supply are lower than predicted by the single-segment case and considerable scatter is introduced. We found that this pattern reflects a decoupling of aggradation and incision at the outlet from distant parts of the network, and a greater influence of network structure close to the outlet, such as the distance from the outlet to its closest inlet (Fig. ). Such nuances are not captured by single-segment models. For networks with along-stream supply of sediment and water, lag times follow the single-segment, along-stream supply case more closely than the single-segment, upstream supply case across the range of forcing periods we tested. We also stress that, in all our simulations, we varied sediment and water supply uniformly across the network. In reality, forcing could occur more locally, for example if a large landslide delivers excess sediment to a specific tributary, or if orographic effects influence precipitation patterns across a catchment. Such localised forcing may affect sediment discharge at the outlet in different ways from the simulations shown here, depending, for example, on the distance from the outlet at which the forcing occurs.

, analysing a single example network configuration, also noted a broad similarity between its behaviour and that of a single-segment model. Comparing a single-segment model with a network of equal maximum length and equal water discharge at the outlet, he estimated the response time to be approximately six times longer in the network, based on the time needed to approach equilibrium following a step change in the hydraulic regime. This result may appear to conflict with our analysis, in which network equilibration times are typically similar to or shorter than single-segment models with the same maximum length (Figs.  and a–d). The difference, however, arises due our different treatments of water discharge. Rather than comparing networks and single-segment models with equal maximum discharge, we used mean discharge in our definitions of network effective length and equilibration time (Eqs.  and ). In our set of networks with forty inlet segments, maximum discharge was on average six times larger than mean discharge, decreasing our estimated equilibration times by an average factor of around six relative to . Meanwhile, 's network has 39 inlet segments, a maximum topological length of 30, and a mean inlet topological length of around 18; in this case, the network maximum length provides a reasonable approximation of its effective length (too long by a factor of ≈1.15). Thus, our estimates of network equilibration time are broadly consistent with that of .

What controls a network's mean inlet length? In our simulations, we constructed networks with specified numbers of segments and specified (average) segment lengths, approximately corresponding to a consistent drainage area. In such network populations, the shortest mean lengths occur in the most compact networks, while the longest mean lengths occur in the most elongate (Fig. ). However, in nature, spatial constraints such as coastlines or fault-bounded mountain ranges may be more relevant, and could instead fix a network's main stream or maximum length. Comparing networks with equal main stream lengths, the relationship is reversed: the shortest mean lengths occur in the most elongate networks, which will also have smaller drainage areas, while the longest mean lengths occur in the broadest networks. argued that, while tributaries tend to be arranged in a regular fashion in tectonically quiescent areas, this pattern can be disrupted by faulting in tectonically active areas. This result implies that network structures in tectonically active regions can be particularly irregular, which could lead to a wide range of mean lengths, equilibration times, and responses to external forcing in adjacent catchments of similar sizes. also showed that basins tend to be more elongate in more arid regions. Depending on whether networks of similar drainage areas or similar main stream lengths are compared, this effect could introduce a tendency towards longer network mean lengths and response times, or shorter mean lengths and response times, respectively. These patterns may therefore exacerbate or counteract to some extent the effect of reduced water discharge on the response time of alluvial networks in arid regions Eq. ; e.g..

Although networks' integrated responses to external forcing, as expressed in variations in sediment discharge at their outlets, appear to be predictable using simplified, single-segment models, patterns of aggradation and incision within them show considerable complexity (Figs. –). Specifically, amplitudes of aggradation and incision (Gz) and the extent to which they lag behind the imposed forcing (φz) vary along stream and between streams of different orders. This complexity arises despite relatively simple, spatially uniform, sinusoidal variations in sediment and water supply, as a consequence of the dynamics of signal propagation in networks (in contrast to the relatively simple, unidirectional signal propagation associated with single-segment models). Complex distributions of terraces and their ages in natural networks therefore need not imply complex and local variations in sediment and water supply. In detail, patterns of Gz and φz throughout a specific network depend on its specific structure. We therefore argue that, if a detailed understanding of dynamics of aggradation and incision in a specific network are desirable – for example, to aid interpretation of terrace ages or geometries – then that specific network should be modelled using approaches like the one we employ here. Nevertheless, there are some general features in patterns of Gz and φz that appear to be common to most networks and have implications for the distribution and timing of terrace formation.

Gz describes variations in the amplitude of aggradation and incision, relative to imposed variations in sediment and water supply, which in turn influence the likelihood of terrace formation and preservation; higher amplitudes of aggradation and incision are more likely to result in long lived terrace sequences (Figs. b, d–f and a–d, i–p, q–t). Note that, to obtain absolute amplitudes of aggradation and incision, Gz is multiplied by steady state elevation (Eq. ), so that it modulates a general tendency towards higher amplitudes at greater distances from the outlet . Gz is generally higher closer to the network outlet, canceling out to some extent the natural tendency for amplitudes of aggradation and incision to decrease towards the outlet, and thus promoting terrace formation further downstream. On lower order segments that join higher order segments near the outlet, Gz decreases rapidly downstream, so that amplitudes of aggradation and incision decrease particularly quickly (Figs. e and f and a–d). On such tributaries, terrace formation is therefore more likely than on adjacent higher order segments, but this likelihood decreases quickly away from valley inlets towards higher order segments. Appreciable amplitudes of aggradation and incision (Gz≈0.2) persist on such segments even for short forcing periods, suggesting that terrace formation is possible in specific parts of a network even when the timescales of variation in sediment or water supply are much shorter than its equilibration time (P≈Teq/10; Fig. a–d).

Network φz is typically lower and more uniform along stream, particularly along higher order segments, compared to the simple unidirectional signal propagation predicted for the single-segment, upstream supply case as noted by. Nevertheless, φz can vary spatially by factors of 2–3 in simulations with significant amplitudes of aggradation and incision (e.g., with P≈Teq; Figs. m–p and i–l), implying that terrace formation can occur diachronously throughout a network, and result in a single terrace surface whose age varies spatially despite a simple, spatially uniform forcing (Figs. c, d–f and e–h, m–p, u–x). On higher order segments, φz generally increases upstream, so that terrace formation will occur progressively later away from the network outlet. This pattern may superficially resemble upstream propagation of a signal introduced at the valley outlet (i.e., base-level variations), despite being driven by variation in sediment and water supply evenly distributed across the network and its inlets. It arises because signal can arrive at higher order segments relatively efficiently along short tributaries; high order segments that can then, due to their larger discharges and diffusivities, adjust most efficiently. On lower order segments, φz generally increases away from valley inlets, so that terrace formation will appear to propagate downstream. This effect is particularly pronounced on lower order segments that join higher order segments close to the network outlet, where φz can increase rapidly downstream and could generate a wide range of terrace ages over a relatively short distance (Figs. e and f and e–h, m–p). This pattern arises because these low-order segments experience both direct forcing at their upstream ends, as well as significant variation in local base level associated with the efficient aggradation and incision of higher order segments at their downstream ends.

Throughout our analysis thus far, we have treated valley width, B, as a constant. In natural alluvial valley networks, it can vary both in time and space. That valley widths evolve through time is clear from the preservation of paired terrace sequences, which necessarily record valley narrowing over time e.g.,. Models describing the lateral evolution of alluvial valleys have been developed, and could in future be coupled with our network implementation of the long profile model of to explore the effects of such behaviour e.g.,.

Several studies have also explored how valley width varies along stream, and in particular as a function of upstream drainage area e.g.,. Such measurements can contain significant noise, with considerable short wavelength variation, but typically take a power-law form on average. combined a compilation of observed power-law parameters with theoretical arguments to conclude that plausible power-law exponents are in the range 0.03–0.9, with a likeliest value around 0.4–0.5. This broad range suggests that valley width can remain essentially constant downstream, or increase almost as quickly as upstream drainage area (and, by extension, water discharge). Our preceding analysis therefore represents one end member of along-stream valley-width behaviour. To explore the other end member, we ran a second set of simulations in which valley width increased downstream at the same rate as water discharge. This formulation is mathematically convenient because the increasing valley width cancels out the effects of increasing water discharge on the efficiency of sediment transport, so that the diffusivity becomes constant across the network (Eq. ).

The majority of results for this non-uniform valley width case are very similar to those for the uniform valley width case (see Figs. S10–S15 in the Supplement for single-segment results and Figs. S16–S28 in the Supplement for network results). Again, amplitudes of variation in sediment discharge at the network outlet (GQs) are well approximated by analytical predictions for the single-segment, upstream supply case (Fig. S16). Effective lengths obtained for the non-uniform valley width case also correlate closely with network mean length (Fig. S18–S20). Ratios of effective length to mean length, kL, are lower than in the uniform valley width case (up to 1.16 for upstream supply cases and 1.13 for along-stream supply cases; Fig. S21).

Some differences between the uniform and non-uniform valley width cases do arise, because, in the former, diffusivity increases downstream as water accumulates, whereas in the latter, diffusivity is spatially constant. In the uniform valley width case, signal propagation is relatively inefficient upstream or on lower order segments where discharge is low, whereas it is more efficient downstream or on higher order segments with larger discharge. This pattern contributes to a general increase in Gz and decrease in φz towards the network outlet. In contrast, in the non-uniform valley width case, signal propagation is equally efficient throughout the network, so that signal propagating from upstream valley inlets has more influence farther downstream, and distributions of Gz and φz are more symmetrical along stream. In the single-segment, along-stream supply case, minima in gain and maxima in lag occur further downstream than in equivalent models with uniform valley width (Figs. S12 and S13). In the network case, Gz and φz are more uniform along higher order segments, and differences between lower order segments near the outlet and those further upstream are less pronounced (Figs. S26–S27).

This analysis accounts for the effects of smoothly varying valley widths, likely to be representative of alluvial valley networks on average. However, significant local variability can arise, due to, for example, lithological variations, distributions of tectonic structures such as active faults, or geologically recent drainage integration, so that natural networks can contain anomalously narrow or wide reaches e.g.,. Very narrow sections will efficiently transmit signals up- and down-stream, increasing response amplitudes and decreasing response lag times in surrounding parts of the network. Meanwhile, very wide sections will adjust slowly and, depending on the timescale, could act as local points of fixed elevation, decoupling upstream sections from the rest of the network. How such dynamics play out in nature needs to be assessed on a network-by-network basis.

We used a relatively simple approach for randomly generating networks . Although this approach theoretically samples all possible binary trees, the number of possible networks far exceeds the number we can practically analyse, such that the network populations we obtained are necessarily incomplete. End-member configurations such as perfect binary trees (in which every internal segment has two internal segments upstream, until a final level of inlet segments) are very unlikely to be selected. Statistics of network populations generated in this way also likely diverge, at least to some extent, from those of natural network populations, since, for example, the networks are not required to be space filling . The high fidelity of the relationship we obtained between network equilibration times and their mean inlet lengths strongly suggests that it will apply to branching networks more generally (Fig. ). Nevertheless, the statistical results we obtained, such as precise values of the coefficient kL scaling network mean length to effective length (Eq. ; Fig. ), and their variation with network size (Fig. ), likely depend at least to some extent on our choice of network generator. We also note that, since we did not sample the full range of possible network configurations, we may not have observed the full range of possible behaviour in, for example, spatial patterns of aggradation and incision, or lag at the network outlet (Figs. –). Repeating our analysis using more involved approaches to network generation that appear to produce more naturalistic network populations, such as random walks on lattices or Optimal Channel Networks , or indeed using network populations extracted from real landscapes, may provide results more representative of natural alluvial river systems. Approaches that allow random network populations to be generated with prescribed statistical properties may also facilitate more detailed and systematic exploration of structural controls on network behaviour . In the meantime, these issues highlight the importance of modelling the specific structures of specific networks of interest, particularly when an understanding of spatial patterns of aggradation and incision is required.

Lastly, we note a superficial similarity between the reduced amplitude of sediment-supply signals we observe at short forcing periods (i.e., Gz, GQs,L<6/7 for P≲Teq) and the concept of signal “shredding”, as proposed by . This concept builds from the observation that sediment transport in many environments is a threshold process subject to large, autogenic, stochastic fluctuations. demonstrated that, if an external signal is imposed with a timescale similar to or shorter than the timescales of these internal fluctuations, it will be smeared across a range of timescales and, consequently, its amplitude will be severely reduced. The original signal is therefore unreconisable in time series of sediment discharge (i.e., it will be “shredded”). This process is distinct from the diffusive damping or buffering we observe, in which signal amplitudes are reduced but their frequency content is preserved see also. The shredding concept has subsequently been applied extensively, particularly in studies focused on conditions for stratigraphic preservation of environmental signals in net depositional settings such as fans and deltas e.g.,.

In their model of long-profile evolution and sediment transport by alluvial rivers, which we apply here, derived a deterministic relationship between valley slope, water discharge, Qw, and sediment discharge, Qs (Eq. ). That such a deterministic relationship exists is consistent with the preservation of terrace sequences recording Milankovitch periodicities, which implies that alluvial rivers can respond systematically to external change e.g.,. The model accounts for the efficient adjustment of alluvial river-channel width in response to water-discharge variability , which leads to the approximately linear form of Eq. (), and tempers the influence of extreme events on river long-profile evolution. The model also assumes that the channel migrates laterally, mobilising sediment across the full valley width, B; the associated lateral migration timescale likely sets a lower temporal bound on the model's applicability. The model uses a characteristic bankfull discharge, which combined with the assumption of a hydraulic geometry that is in equilibrium with this discharge, embeds the assumption that stochastic or local events – such as floods, landslides, or avulsions – do not affect the hydraulic geometry or sediment-transport efficiency significantly in the long term, such that Qs remains a near-linear function of long-term average slope and Qw.

Nevertheless, it is clear that short-term fluctuations in sediment-transport rates do occur, and that these fluctuations may limit transmission of external signals through alluvial valleys. proposed that, below a characteristic maximum time scale for internal variability, autogenic processes dominate morphologic and/or sediment-output signals. Using an experimental hillslope (rice pile) with a sediment feed at the upslope boundary, demonstrated that signals with timescales shorter than the duration of the largest stochastic events are severely degraded, whereas those with timescales longer than the duration of the largest stochastic events but shorter than the largest events' repeat time are degraded but, in some cases, still recoverable. For fluvial aggradation and incision under variation in either sediment or water supply, the diffusive and stochastic frameworks make similar predictions: short period signals are damped or degraded, while long period signals are faithfully transferred into aggradation or incision that may be preserved in terrace sequences. As such, if autogenic processes act on timescales shorter than our diffusive Teq, they will add noise to our predicted aggradation and incision, but otherwise have a limited effect. If, however, autogenic timescales exceed our Teq, these processes will impose a stricter limit on the translation of external signals into aggradation and incision. For variations in sediment discharge, which could contribute to the development of downstream stratigraphy, similar behaviour will arise under varying sediment supply. However, when water supply is varied, the two frameworks make opposite predictions about patterns of sediment discharge: we predict that short period variation in water supply will translate directly into short period variation in sediment discharge and that long period signals will be damped Figs.  and ; see also; meanwhile, autogenic processes will still act to degrade short period signals. As such, if autogenic timescales are shorter than Teq, water-supply signals will be preserved only at periods between the autogenic timescales and Teq, whereas if the autogenic timescales exceed Teq, water-supply signals may be suppressed across all input periods. Ultimately, placing tighter constraints on both autogenic and diffusive timescales in real alluvial valley systems is critical for understanding the relative importance of each process, and we hope that our physically based, network approach can facilitate this goal.