The underlying approach for our integration of grain size fining into a landscape evolution model is based on grain size solution (D) by as a function of dimensionless distance downstream (x*) within a depositional area for gravel, 1D(x*)=D0‾+ϕ0C2C1e-C1y*-1, and sand, 2D(x*)=D0‾e-C3y*, where x* is the distance along the river profile, x, normalized by its total length (x*=x/L). D0‾ and ϕ0 are the initial mean grain size and standard deviation at the source, respectively. C1,C2, and C3 are constants that represent the change in mean grain size and standard deviation downstream. Cv is defined as the ratio of the downstream change in the standard deviation relative to mean grain size, i.e. Cv=C1/C2, and referred to as the coefficient of variation. C1 ranges between 0.5 and 0.9, C3 between 0.1 and 0.45, and Cv between 0.7 and 0.9 . note that a Cv value of 0.8 is commonly observed. Note that, to avoid repetition, this work will focus on the application of the gravel equation (Eq. ), but all methods described further below could be modified and applied to solving the sand equation (Eq. ).

As defined in , y* is the dimensionless distance transformation given by 3y*(x*)=∫0x*R*(x*)dx*, where R* is the dimensionless downstream distribution of deposition defined as 4R*=(1-γ)RL, where γ is the porosity of the sediment deposit, and R is the ratio of deposition rate, r, to sediment flux, qs, i.e. R=r/qs. At each x* point along the river profile, R* relates the sediment deposition into the substrate to the sediment flux in the river, taking into account a given length and porosity. Finally, deposition rate, r, is the sum of the rate of change of elevation of the landscape and the subsidence rate, σ, i.e. r=dh/dt+σ.

In non-mathematical terms, the approach of postulates that grain size fining over long timescales is controlled by (1) the source sediment supply and grain size distribution; (2) the deposition rate throughout the system length; (3) the hydraulic mobility of different grain size types (gravel vs. sand) assuming a constant shield stress at the bed; (4) self-similarity between the subsurface and surface of the distribution of gravel grain size clasts; and (5) a mass balance of the transport, substrate, and an active layer . We refer the reader to for a more detailed explanation of the assumptions behind this model and how it has been derived from these assumptions. However, the self-similarity of gravel grain sizes has now been repeatedly demonstrated from field data , and this model has been successfully applied to stratigraphic examples .

We use FastScape as a basic Landscape Evolution Model (LEM) that solves the Stream Power Law (SPL) in two dimensions following the efficient algorithm described in . The SPL states that the rate of bedrock elevation change is the sum of uplift (or subsidence) rate U and erosion rate assumed proportional to upstream drainage area, A, used as a proxy for discharge, and local slope S : 5dhdt=U-KAmSn, where K is a rate coefficient or erosivity that depends mostly upon lithology and precipitation rate. m and n are exponents that generally maintain a ratio of m/n≈0.5. Sediment transport and deposition are incorporated into FastScape using the implementation of of the ξ-q's algorithm by , which states that the rate of sediment deposition is proportional to sediment flux and inversely proportional to upstream drainage area. This leads to the following evolution equation that integrates the processes of uplift/subsidence, erosion, transport, and deposition into a simple framework: 6dhdt=U-Kp̃mAmdhdsn+Gp̃A∫AU-dhdtdA. In this formulation, G is a dimensionless deposition coefficient, and p̃ represents spatial or temporal variations in precipitation rate with respect to a mean value that is contained in both K and G. In this work, we will vary topography through G and discuss its impact on model results. G controls the transport- or detachment-limited nature of the depositional system . have demonstrated that many natural river systems tend to be on the transport limited side. Note that, for numerical reasons, G=1 is the maximum transported-limited value computed efficiently and reliably by FastScape , although larger values can be used but may require multiple iterations and reduce efficiency. Reducing G reduces the transport limited nature of the system and moves into more of a detachment limited setup.

In FastScape, in order to compute the transport of water and sediment, nodes are ordered along flow paths through the landscape. This is performed by creating a stack order using single- or multiple-receiver algorithm. Both algorithms allow for the formation of multiple channels, but the single-receiver algorithm is limited to flow convergence, while the multiple-receiver algorithm permits flow divergence. Both are therefore different from the single-channel approach that is common to all previous applications of the self-similar grain size fining model. In all the model results presented here, we have used the multiple-receiver algorithm, in which the proportion of water and sediment that is given from one node to its downslopes' neighbouring nodes is proportional to slope.

In FastScape, uplift and subsidence rate are imposed as functions of space and time, but they can also be obtained by solving the flexure equation representing flexural isostasy of the lithosphere/crust system.

We have generalized self-similar approach of to two dimensions, i.e. following flow path, s, or the stack node ordering computed in FastScape. We call our algorithm GravelScape. For this, we replace any spatial integration or differentiation along the normalized river distance by its equivalent value along the normalized steepest descent flow path, s*=s/Ls, where Ls is the total length of the flow path. For example, the deposition rate is the spatial derivative of the flux along that flow path: 7∂qs∂s=1Ls∂qs∂s*.

In practice, we first compute the flow path and erosion/deposition rate along the flow path using the multiple-receiver routing in FastScape. Next, sediment flux is computed by summing the erosion/deposition rate, r, down the flow path. Note that, where deposition takes place, r is negative and consequently decreases the sediment flux qs in the summation along the flow path to the limit of qs=0. Once qs is computed, the downstream distribution of deposition R can be calculated according to 8R=(1-γ)rqsifr,qs>00otherwise. Next, the dimensionless distance transformation y* is computed from a weighted summation of R through the nodal stack. Where flow converges, a weighted mean is used to compute the receiver y* values, where the weights are proportional to the upstream drainage area size of the donor nodes. In this way, when multiple flow paths converge, the largest drainage area flow path will dominate the downstream grain size distribution. Such an approach matches observations that grain size distributions of larger catchment rivers tend to dominate downstream grain size distributions .

Finally, using y*, we compute the mean grain size using Eqs. () or (). In these equations, the value for D0 is needed at every grid point within the model. For this, we compute a field D0 that is obtained by propagating the value of the grain size last deposited on the bed (which we call Dt) down the flow path of a given time step from where the channel originates. Using D0 defined in this way at each grid point, we use Eq. () or () to calculate a grain size value, Dx, at all points of the grid where deposition occurs, which we also use to update Dt for the following time step. At the start of the simulation, D0 and Dt are set equal to the mean grain size as produced by bedrock erosion in the mountain catchment. Note that deposition does not take place at every grid point at every time step, so Dt is only updated to the newly computed grain size Dx where net deposition takes place. Finally, local erosion may take place in the basin, causing an increase in sediment flux and a reduction in Dx fining rates.

In summary, we carry three grain size values at every grid point: (1) the grain size of material deposited at the current time step (Dx); (2) the grain size at the surface of the model, i.e. of the material last deposited at that point (Dt); and (3) the grain size in the “source” area, i.e. where the flow path traversing the grid point originated (D0). We show in Fig. S1 in the Supplement. computed values of D0, Dt, and Dx for a given time step under the assumption of single and multiple-receiver routing. In the work presented here, we will only show computed grain size values using the multiple-receiver algorithm.

To illustrate the behaviour of the new model and, most importantly, to compare it to previous studies which computed grain size fining along a single, main channel, we chose a setup that is similar to that used by . As shown in Fig. , this setup is composed of two regions, one subjected to uplift (the source or mountain region of length LM) and the other to subsidence (the basin region of length LB). Although our model is two-dimensional, the uplift and subsidence are assumed to be functions of one of the two horizontal components only, i.e. the x coordinate. Following the setup of , the subsidence, σ(x), in the basin area is given by 9σ(x)=σ0e-αx/LM, where σ0 is the subsidence at the orogenic front (i.e. at the limit between the source and basin areas, where x=0), and α is the rate of decay of the subsidence away from the front. Large values of α result in most of the subsidence near the source, while small values of α result in a more distributed subsidence across the basin. This exponential function mimics subsidence curves resulting from flexure of the underlying lithosphere/crust under the weight of the topography created by uplift in the source area relatively well, without any lateral heterogeneity.

do not explicitly model the source area but assume a constant sediment flux, qs,i, originating from it. To reproduce this setup, we will assume that the source area uplifts at a rate U chosen such that qs,i=(1-γ)ULM. In our setup, we will need to run the model for a sufficiently long period of time for it to reach steady state so that the flux is constant, and our results become comparable to those of . Note that the values of the various LEM parameters, i.e. K, m, n, and G, do not influence the value of the steady-state flux coming out of the source area, only the time it will take to reach the steady state.

We can then follow and introduce a quantity, F (see Fig. ), that is the ratio of the flux coming out of the source, qs,i, to the flux of sediment that is trapped in the basin. As demonstrated by , F and α are the major controls on the relative rate of grain size fining across the basin. F can be written as 10F=qs,i/∫0LBσ(x)dx, where F indicates the under-filled, over-filled, or filled state of the basin. When F is larger than 1 there is more incoming flux, qs,i, than the basin subsidence can accommodate, producing an outflow of sediment from the basin (qs,f>0). In contrast, when F is smaller than 1, the basin is under-filled, and no sediment leaves the basin (qs,f=0). Here, systems that are characterized by F values lower than or equal to 1 will be referred to as under-filled or limited-bypass systems. Systems where F is greater than 1 but less than or equal to 10 will be referred to as filled or low-bypass systems (Fig. a). Systems where F is greater than 10 will be referred to as over-filled or high-bypass systems (Fig. b). In Fig. c we show how varying σ0 while keeping both α and qsi constant leads to different values for F.

From the definition of the flux at steady state and by performing the integral of the subsidence given by Eq. (), we obtain the following relationship between F, U, and σ0: 11F=(1-γ)ULMασ0LB(1-e-α). In practice, we will impose values for σ0, α, and U and deduce the corresponding value of F from Eq. ().

have shown that for a given value of the incoming sediment flux from the source, qs,i, and a given subsidence pattern, i.e. given values of F and α, the self-similar grain size fining model of  predicts a unique distribution of grain size in the basin. If we note that the sediment flux at any point, x, along the surface of the basin is the upstream integral of the deposition/subsidence rate, we can substitute Eq. () into Eq. (), to obtain 12y*(x*)=-ln⁡1-1-e-αx*F(1-e-α), which, in turn, can be used to obtain an analytical expression for the grain size fining curve. Examples of such solutions are given in Fig. d. We will now compare results obtained with GravelScape with this “theoretical” solution. For this we will perform two types of model runs, one using the full two-dimensional nature of GraveScape, i.e. with a large number of nodes in the y direction, and one with only two nodes in the y direction to mimic the behaviour of a single channel. To simplify the description of the results we will use the following naming convention: the theoretical solutions (assuming deposition as equal to subsidence in a single channel) will be named DULLER; the full planform, multi-channel GravelScape solutions will be named GravelScapeMCH; and the single-channel (2 cells wide) GravelScape solution will be named GravelScape1CH. Model inputs used for standard GravelScape1CH and GravelScapeMCH setups used to validate our model against DULLER are given in Table .

In Fig.  we show an example of the result of a GravelScape model run. We used model parameter values as given in Table ) with an imposed subsidence producing a very high bypass basin (F=10000) and moderately transport-limited conditions (G=1). More specifically, in Fig. , we show computed topography, erosion rate, drainage, grain size distribution over time (Dt), and grain size at an instantaneous time step (Dx) for a reference model experiment at a series of arbitrary consecutive time steps selected once the system has reached flux steady state; i.e. the computed values of qs,i and qs,f do not change significantly with time. Note that the red colours in the erosion rate map correspond to areas where net erosion takes place, whereas the blue areas correspond to areas where net deposition takes place. By comparing panels b and e, we see that GravelScape predicts instantaneous grain size (Dx) only in regions of net deposition. We see that, even though the model has reached a quasi steady state, important fluctuations in topography, discharge and grain size take place between time steps in response to what appears to be large avulsions. We note that, on average, the grain size tends to decrease from the mountain front to the base level. We also note that the predicted grain size is coarser in channels (regions of high discharge) and is highly variable spatially and temporally. Since the largest channel is often coarser than the smaller channels, the fining is greater when averaged across all channels (in the y direction) than when considering the largest channel only.

This strong variability implies (1) that the predicted grain size in the channels under high bypass should be used to compare our model's predictions to the predictions of the model of where predictions are limited to a single, main channel and (2) that the grain size predictions must be averaged over many time steps (e.g. >10 kyr) for comparison with the model of , which assumes that deposition rate is equal to subsidence rate and is thus a very smooth function of x and t.

We also note that there is a small, yet non-negligible, grain size fining in the mountain area, such that the mean grain size of the material leaving the mountain front is not exactly at a value of unity, corresponding to the assumed grain size at the source. This also implies that to compare our grain size predictions to those of , we must scale our predicted values in the basin area by the relative grain size predicted at the exit of the source area.

In Fig. , we compare grain size predictions made by DULLER, GravelScape1CH, and GravelScapeMCH. In the top panels (a and b), we show the predicted surface and bedrock topographies obtained in GravelScapeMCH. In the middle panels (c, d, e, and f), we show the predicted grain sizes, and in the bottom panels (g and h), we show the predicted standard deposition rate for GravelScapeMCH compared to the imposed subsidence rate. In the three left panels (a, c, e, and g), we show the results of different model runs performed by varying the subsidence rate, σ0, leading to different values of the parameter F ranging from 2.5 to 100, i.e. from low to high bypass. In the three right panels (b, d, f and h), results are shown for three values of the transport parameter G, i.e. 0.5, 1 and 1.5.

We see that the GravelScape1CH predictions are, within numerical precision, identical to DULLER for all values of F and G (Fig. c and d), implying that when reduced to a single channel, GravelScape reproduces the predictions of the self-similar model of  exactly when deposition rate is equal to subsidence rate, as done by . Conversely, the GravelScapeMCH predictions deviate markedly from DULLER, and the difference between the two model predictions increases with F (Fig. e) and G (Fig. f). With a constant G=1, we tested additional high bypass values of F=10–10 000 and observed the similar fining trends as the F=100 scenario in Fig. e. For values of roughly F>10, GravelScapeMCH predictions converge to a total grain size fining across the basin of approximately 15 % (i.e. from 1 to 0.85 in relative grain size) regardless of the value of F, whereas DULLER predicts less and less fining as F increases to the point where for F=100, there is almost no fining across the basin. Similarly, varying G (with a constant F value) in GravelScapeMCH leads to grain size fining that is 2 to 3 times larger compared to DULLER (or GravelScape1CH).

We also see that varying F has little to no effect on the surface topography at steady state (Fig. a), whereas changing G strongly affects it (Fig. b). This is because, for values of F⪆1, the surface topography is set by surface processes almost regardless of the basement subsidence as shown in . G, on the other hand, affects the distance over which deposition takes place and therefore strongly influences the height and slope of the depositional system (Eq. 20 in ). We also see that varying G causes variations not only in predicted surface topography but also in the standard deviation in deposition rate within the multi-channel model (Fig. h), implying that steeper sedimentary systems are affected by higher-amplitude spatial variations in deposition rate. Recall that deposition rate is a direct parameter within the equations such that any changes in deposition rate will be reflected in grain size fining.

We can therefore conclude that the deviations in grain size fining from DULLER observed in GravelScapeMCH are related to the degree of bypass of the system, when varying F, or to the shape of the surface topography changing the variation in deposition rate, when varying G. Further analysis on the internal dynamics of the system is conducted in .

Using GravelScape with values of F<1.25 leads to predicted surface geometries that are so flat (less than a few tens of metres over a 200 km distance) that local depressions appear in the topography that lead to the formation of lakes where the modified SPL equation cannot be used to predict deposition or erosion. In the depressions, the SPL is replaced by a simple depression filling algorithm, and the resulting deposition rate cannot be used to compute grain size fining according to Eq. (). We checked that these local depressions or minima do not influence our findings by computing the frequency of occurrence of such minima. This frequency is computed by counting the number of local minima in the main channel at a given position x over N time steps and dividing it by N−1. We found that the number of local minima is negligible for all model runs where F>1.25. In Fig.  we show the computed frequency of the local minima as a function of the x coordinate for the model runs shown in Fig. . We see that this frequency varies with the amplitude of the topography but also that it remains relatively low and does not vary significantly between the different model runs shown in Fig. .

We performed another set of model runs to verify the sensitivity of GravelScapeMCH to the assumed spatial and temporal resolution as well as the timescale over which the results are averaged to compute the grain size that we compare to DULLER. The results are shown in Fig. . In panel a, we see that the grain size fining computed by GravelScapeMCH depends on the spatial resolution but that the results converge with increasing resolution. This mesh dependence is related to the mobility of channels that strongly affects the standard deviation in deposition rate. We will discuss this point in greater detail in the discussion. We also see that there is little dependence of the solution on the temporal resolution (value of the time step, Δt – panel b) or on the timescale over which the grain size is averaged in GravelScapeMCH to be compared to DULLER, whether we perform this averaging over 5 Myr but with a varying number of time steps (panel c) or over 25 time steps of varying length (panel d), as long as this timescale remains larger than a characteristic timescale for channel avulsion (≈250 kyr).