Realistic treatments of sediment morphodynamics are critical to models of bedrock–alluvial river evolution. Incorporating conservation of sediment mass leads to process-based descriptions of sediment flux and sediment cover on bedrock channel beds. The two approaches to sediment mass conservation considered here are the Exner-type approach, in which temporal changes in sediment thickness are driven by changes in sediment transport capacity, and the erosion–deposition approach, in which erosion and deposition fluxes of sediment are calculated to determine changes in sediment flux and alluvial layer thickness. Either type of mass conservation statement may be used to model bedrock–alluvial rivers. used a modified 1-D Exner equation for sediment mass conservation incorporating both mobile bed load and an alluvial layer. and followed a similar approach, further modifying the sediment mass conservation framework to include below-capacity sediment transport. Exner sediment transport in bedrock–alluvial systems was expanded to 2-D by and , though the specifics of calculating sediment flux differ between the model of and the latter two. The Exner-based conservation framework allows calculation of bed cover by sediment as a function of the ratio of sediment flux to transport capacity (qs/qc) . predicted a range of cover functions depending on the ratio of grain roughness to bedrock roughness, ranging from an exponential decline in exposed bed fraction with increasing qs/qc to rapid alluviation above a qs/qc threshold. Other models for alluvial bed cover arise from simplifications to the Exner-style approach. made steady-state predictions for alluvial cover as a function of qs/qc on a non-erodible bedrock bed, and showed that bedrock roughness relative to sediment size is an important control on setting the influence of qs/qc on bed cover. assumed that sediment was transported at capacity at every point along a 2-D channel cross section in order to compute the thickness of alluvial cover in the cross section.

The erosion–deposition framework differs from the Exner approach in that it conserves mass on the bed and in the water column to treat simultaneous erosion and deposition of a single substrate . Erosion–deposition mass conservation is based on explicit calculations of sediment entrainment and deposition, rather than on the spatial divergence of sediment transport capacity. Erosion–deposition models may dynamically transition between transport-limited and detachment-limited behavior see discussion by. present such a model based on the relative influence of erosion from the bed into the water column (erosion flux) and deposition from the water column onto the bed (deposition flux). When the deposition flux is much smaller than the erosion flux, and the sediment transport length scale, which can be thought of as the average travel distance of a grain from entrainment to redeposition, is long, their model becomes equivalent to a basic detachment-limited model. When the deposition flux increases and the transport length scale is short, the model predicts transport-limited behavior in which sediment flux divergence controls the channel bed elevation. The erosion–deposition framework, validated for alluvial rivers by the laboratory experiments of , is capable of matching transient and steady-state longitudinal profile predictions made by both detachment-limited and transport-limited models, and smoothly transitions between the two types of model behavior . The model of improved on previous erosion–deposition models by using a sediment transport length scale that increases with water discharge. Erosion–deposition-type models that transition between detachment-limited and transport-limited behavior allow the exploration of both types of models, and intermediate cases, with simple parameter changes rather than changes to the model structure.

Erosion–deposition models, like Exner-based transport models, can yield predictions for the influence of sediment dynamics on erosion-inhibiting sediment bed cover. The models of , , and incorporate statements of sediment mass conservation to explore cover dynamics on a stationary bedrock bed. used a cellular automaton model of alluvial cover evolution that allowed probabilistic entrainment and deposition of sediment on a non-eroding bedrock bed. They found that either a linear or exponential decline in bedrock exposure with increasing qs/qc could occur if all grains had equal entrainment probability, with the exponential decline occurring at low entrainment probabilities and the linear decline occurring at high entrainment probabilities. Additionally, they found that when isolated grains have a higher entrainment probability than clustered grains, bedrock exposure declines sigmoidally with increasing qs/qc and runaway alluviation can occur. Their results support the experimental results of showing that both the linear decline in exposure with increasing qs/qc and the abrupt shift from fully exposed to fully alluviated bed are possible, with the former occurring at low slopes and high bed roughness, and the latter occurring at high slopes and low bed roughness. explicitly tracked erosion and deposition governing the transfer of particles between two mass reservoirs: the stationary particles on a non-erodible bedrock bed and the mobile particles in the water column. They showed that the relationship between bedrock exposure and qs/qc can take a wide range of forms depending on, among other factors, the probability of increasing bed cover for a given fraction of exposed bed.

Evolving the channel bed over time requires a description of the mechanics of bed erosion, which can be coupled with sediment mass conservation to model bedrock–alluvial channels. The most well-known approach to bed erosion is the stream power and shear stress family of models e.g.,, in which bed erosion is a function of water quantity (either depth or discharge) and channel bed slope. The major limitation of basic stream power and shear stress models is the lack of a description of how sediment influences channel bed erosion. A large number of sediment-flux-dependent incision models, in which the interaction of sediment flux with sediment transport capacity enhances or inhibits bedrock erosion, have been developed to fill this gap e.g.,. Sediment may act as “cover,” inhibiting bedrock erosion, or as “tools,” accelerating bedrock erosion see review by. In most sediment-flux-dependent models, erosion is influenced by a factor fqs,qc ranging between 0 and 1, where qs is sediment flux (either volume or mass flux per unit width) and qc is sediment transport capacity. The value of fqs,qc for any particular qs and qc depends on the choice of function f; proposed forms include a linear decline fqs,qc=1-qsqc;, parabola , near-parabola , and other similar shapes . showed that the fraction of exposed bed may decline exponentially with increasing qs/qc, leading to an exponential tail on fqs,qc. Any of the formulations for bedrock erosion detailed here are insufficient on their own for modeling bedrock–alluvial channels, as there is no mass conservation of sediment and therefore no development of an alluvial layer. Any bedrock erosion model may be coupled with a statement of mass conservation to capture the coupled evolution of bedrock–alluvial channels.

While several existing erosion–deposition models contain descriptions of both sediment mass conservation and bed erosion, some are limited to erosion through a single substrate, and are not capable of evolving a bedrock-alluvial channel with a distinct alluvial layer e.g.,. presented a model in which the erosion–deposition framework is applied to cases in which a layer of sediment overlies bedrock, but their model requires that one or the other material fully occupy the surface of a cell at a given time. Without substantial modification, erosion–deposition models are therefore unable to simultaneously treat sediment morphodynamics and bedrock erosion in mixed bedrock–alluvial systems. A small subset of existing channel evolution models combine sediment mass conservation with equations for bedrock incision to treat bedrock–alluvial channels. For example, used the Exner equation for sediment conservation to explicitly track a layer of alluvium of thickness Ts with median grain size D50 overlying a bedrock bed. He tested both the linear and the exponential forms of the decline in bed exposure with increasing qs/qc, both expressed as a function of the ratio of Ts to D50 (his Fig. 3). The model of partitions stresses between the channel bed and banks to allow dynamic channel width variations but does not incorporate the tools effect of sediment on bedrock erosion. , following in the footsteps of , combined the St. Venant equations with an Exner equation for sediment mass conservation. incorporated an equation for bed abrasion by bed load, assuming that abrasion rates should scale with the velocity of the bed load layer and should decline with increasing bed load layer thickness (yielding a form of the tools and cover effects). Several workers have adapted forms of the saltation–abrasion bedrock erosion model of in conjunction with sediment mass conservation to incorporate both the tools and cover effects into models of bedrock–alluvial river erosion . combined a stochastic erosion–deposition model with a saltation–abrasion-style incision rule and explored the sensitivity of bed cover and bedrock erosion rate to changes in sediment transport. showed an exponential decline in bed exposure with increasing numbers of sediment particles equivalent to the exponential decline in the deterministic model of . coupled the cross-section channel flow model of with a simple parameterization for the formation and destruction of an alluvial layer on the channel bed. The thickness and lateral extent of the alluvial layer vary in response to the ratio of sediment flux to sediment transport capacity. Unlike most models described here, explicitly resolved the shear stresses on the channel bed and banks, and applied the saltation–abrasion model of bedrock erosion to exposed portions of the channel margin. Their model allows dynamic channel width but is only solved at a single channel cross section with a prescribed bed slope. This at-a-station approach precludes the development of alluvial cover by sediment derived from bedrock erosion. presented a 1-D model of the co-evolution of an alluvial layer and a bedrock bed. They used a modified Exner equation for sediment mass conservation incorporating both mobile bed load and an alluvial layer, and a saltation–abrasion-style bedrock incision rule . expanded the sediment conservation and bedrock erosion formulations of to 2-D to allow spatial variability in alluvial thickness and bedrock erosion along both the channel length and width. advanced the model of by incorporating a simple parameterization for bedrock bank erosion to investigate meander bend migration. presented a channel longitudinal profile model coupling local sediment transport dynamics with the saltation–abrasion incision rule of . The model of can capture transience driven by changes in qs/qc, or downstream propagation of changes in sediment supply. Their model tracks the thickness of a sediment layer compared to the macro-roughness of the bedrock surface such that alluvial thickness less than the macro-roughness length scale results in exposed bedrock. Their geometric approach is similar to that of , but the model incorporates both tools and cover effects due to its derivation from the saltation–abrasion model . While note that their model could incorporate downstream variations in channel width, it does not do so in the form they presented.

The models reviewed above differ in their intended spatial and temporal scales of application. For example, the cross-section saltation–abrasion model of resolves flow hydraulics and channel width variations with high fidelity but cannot be feasibly applied at the landscape scale. Similarly, the bed cover evolution models of , , , , and focus specifically on cover dynamics at the reach scale. While principles from these models could be incorporated into larger-scale landscape evolution models, several of the models are formulated for a fixed bedrock bed, thus limiting their potential for application to problems of long-term landscape evolution. The 2-D mixed bedrock–alluvial models proposed by could potentially be incorporated into existing landscape evolution model frameworks, but the feasibility of such an integration is unclear as the models have been tested over hourly to daily timescales, not geologic timescales. The models most suited to landscape evolution applications are those of , , , and . The model realistically incorporates hydraulic roughness of both bedrock and alluvium but is designed to handle a constant sediment supply rate, a constraint which is unlikely to be satisfied in landscape evolution modeling applications. The model of assumes no supply limitation on sediment flux and is thus not suited to modeling mixed bedrock–alluvial cases. The two existing models that most closely match our goal of treating bedrock–alluvial channel dynamics over landscape evolution space scales and timescales are those of and . We advance on these models by using the erosion–deposition sediment conservation framework to craft explicit expressions for the entrainment and deposition of sediment as well as the erosion of bedrock.

We present a new model for simultaneous sediment and bedrock evolution that extends from the approaches of , , and , as well as from the foundation laid by the other models reviewed above. The SPACE model conserves sediment in two reservoirs, the bed and the water column, in the style of . It is therefore an erosion–deposition model in its treatment of sediment. SPACE also calculates bedrock erosion, incorporating progressive bedrock exposure with thinning of an alluvial layer in a similar way to and such that evolution of the alluvium and the bedrock may be simultaneously calculated. SPACE is unique in that it employs local analytical solutions for sediment flux changes in space and alluvium thickness changes in time. These local analytical solutions are derived by combining expressions for conservation of sediment in the water column e.g.,, conservation of sediment on the channel bed, and conservation of mass of bedrock. Our new model advances beyond sediment-flux-dependent bedrock incision models in which sediment transport and storage are not treated explicitly, as well as previous erosion–deposition models assumed to be eroding a single substrate. The SPACE model is designed to run over large spatial and temporal scales in two dimensions, and can be easily coupled to many other surface process models as part of the Landlab modeling toolkit.

To maintain ease of model coupling and a relatively low level of complexity, we neglect some processes treated in previous models. Because sediment cover and transport are averaged across a model cell, we do not explicitly model the spatial distribution of sediment cover within a cell, or the entrainment and deposition of individual sediment grains e.g.,. We do not employ dynamic channel width variations as in the models of , , , and , instead relying on empirical parameterizations of channel width as a function of drainage area or discharge. SPACE does not contain independent descriptions of the roughness of bedrock and sediment, and does not distinguish between the case in which bedrock is rougher than sediment and the one in which sediment is rougher than bedrock e.g.,. We also do not model the potential driving of bedrock erosion by bed load sediment (the tools effect), as used by , , , , , and other saltation–abrasion-type models. Both width dynamics and bed load abrasion could be incorporated without changing the underlying model structure but have been omitted in order to facilitate numerical model comparison to analytical solutions.

The SPACE model is intended to model river channel evolution over long timescales while honoring the reality that channels may transition between alluviated, bedrock, and mixed bedrock–alluvial states over geologic time. Below, we develop the SPACE model, verify our numerical solutions against analytical solutions, and show that SPACE can transition naturally between transport-limited and detachment-limited behavior. We briefly describe the Landlab modeling toolkit and show an example landscape evolution model in which SPACE is coupled to other surface process models.

showed that the rate of change in the volume of sediment in the water column per unit area of river bed ∂csh∂t may be written as ∂csh∂t=Es-Ds-∂Qs/w∂x, where cs is the concentration of sediment in the water column calculated by Qs/Q (where Q is volumetric water discharge), Es is the volumetric entrainment flux of sediment per unit bed area, Ds is the volumetric deposition flux of sediment per unit bed area, Qs is sediment flux in units of [L3/T], and w is channel width. Equation () is sufficient when considering a channel bed on which only a single material (i.e., sediment) is exposed. Yet many channels are partially alluviated, indicating that sediment and bedrock may be eroded and entrained into the water column simultaneously. Such a scenario requires an addition to Eq. () to account for entrainment of bedrock material into the water column: ∂csh∂t=Es+1-FfEr-Ds-∂Qs/w∂x. Er is the volumetric erosion flux of bedrock per unit bed area. Ff is a unitless fraction of fine sediment. Ff represents the volumetric fraction of bedrock that breaks into sediment small enough to be considered permanently in suspension, for which no further treatment of bed–water column interactions is needed. For bedrock that breaks only into sand and gravel fractions, Ff would be zero. Therefore, bed sediment thickness H and sediment flux Qs only include sediment coarse enough that it does not enter permanent suspension. Assuming that ∂csh∂t=0 (an important model assumption that potentially restricts applicability to hydrograph-scale modeling), the spatial gradient in sediment flux is a balance between sediment entrainment, rock erosion, and sediment deposition: ∂Qs/w∂x=Es+1-FfEr-Ds.

Change of channel bed elevation with time is the sum of changes in rock height and sediment thickness: ∂η∂t=∂R∂t+∂H∂t. Following Eq. () and , conservation of sediment on the channel bed with sediment thickness H may be written as ∂H∂t=Ds-Es1-ϕ, where ϕ is the porosity of the bed sediment. Because there is no deposition of bedrock, the rate of change of bedrock elevation R over time is ∂R∂t=U-Er, where U is the rock uplift rate relative to base level. For simplicity, we assume that the porosity of rock is zero, but rock porosity could easily be added to Eq. (). Steady sediment thickness requires equal erosion and deposition of sediment Ds=Es and steady bedrock elevation requires that erosion of rock is balanced by rock uplift relative to base level U=Er. Figure  shows a schematic of a model cell and defines relevant variables.

As discussed by , any number of accepted expressions for entrainment of bed sediment and erosion of bedrock could be used for Es and Er, respectively. We consider here the unit stream power formulation in which entrainment and erosion are functions of water flux and slope . We add to the unit stream power model a means by which the flow may simultaneously entrain sediment and erode bedrock, and by which the bed may smoothly transition from fully alluviated to bare bedrock. Consider that the volumetric entrainment rate per unit bed area of bed sediment depends on unit stream power but is also influenced by sediment thickness H relative to a reach-scale bedrock roughness length scale H∗. At low H/H∗, the lower points on the bedrock bed are mantled with sediment, but high points on the bedrock surface are still exposed. At high H/H∗, all areas of the bed are covered with sediment. Further, Es must be zero when H is zero and no sediment is available to be entrained, and Es must reach a maximum (for a given stream power) when there is enough sediment to fully cover the bed (Fig. ). This approach is conceptually similar to the models of , (his exponential model), and , and similarly eliminates the need to explicitly track bedrock exposure. Given the conceptual model described above, the entrainment rate of sediment Es may be written as Es=KsqSn-ωcs1-e-H/H∗, where Ks is the sediment erodibility parameter, S is channel-bed slope, and n is a scaling exponent. q is water discharge per unit channel width and may be calculated by any number of methods. The simplest is q=kqAm, where A is drainage area, m is a scaling exponent (generally ≈0.5) designed to reflect downstream width changes and discharge–area scaling , and kq is an empirical constant, with units dependent on the value of m, that is subsumed into Ks and the rock erodibility parameter Kr. Using the volumetric water flux per unit channel width q results in Ks and Kr having dimensions of L-1. ωcs is a threshold stream power required for sediment entrainment. In very simple cases, ωcs may be neglected, but the ability to include a threshold term is important in any river evolution model as threshold effects have been shown to significantly alter model outcomes . While we employ a simple closure for channel width in which width scales as the square root of water discharge e.g.,, it may be desirable for some applications to add dynamic channel width adjustments to the model, as previous work has suggested that width trades off with slope in transient channels e.g.,. One option for incorporating dynamic width is to calculate or approximate shear–stress distributions across channel cross sections e.g.,. A simpler dynamic width rule can be obtained by partitioning erosive power between the bed and banks under a trapezoidal channel assumption as detailed by . Different approaches have different numbers of parameters and computational costs, and further work will be necessary to elucidate which advances beyond the standard empirical width closure are tractable within the SPACE landscape evolution model framework.

If sediment entrainment declines with decreasing sediment thickness as a result of increased bedrock exposure, bedrock erosion should follow an inverse but conceptually similar pattern (Fig. ). Assuming that increasing mean sediment thickness leads to higher proportions of bedrock covered by sediment and that sediment cover inhibits erosion e.g.,, the volumetric erosion rate of bedrock per unit bed area may be written as Er=KrqSn-ωcre-H/H∗. Here, Kr is the bedrock erodibility parameter, which is generally expected to be substantially lower than Ks. ωcr is the threshold stream power for detachment of bedrock, which may vary significantly depending on the relative dominance of plucking or abrasion e.g., as well as the weathered state of the bedrock . , , , and employed a similar exponential decline in rock erosion rate with increasing sediment thickness. Equation () falls into the category of “cover” models, which treat erosion reduction by sediment shielding the bed without incorporating the potential erosive effects of mobile sediment e.g.,. The smooth transitions between bare-bedrock, bedrock–alluvial, and fully alluviated channels given by Eqs. () and () are both more stable and more realistic than models in which the existence of any alluvium fully covers the bedrock (implying a perfectly smooth, planar bedrock surface). Figure  shows the pattern of sediment entrainment and bedrock erosion over different values of H/H∗. fH/H∗ in Fig.  is the dimensionless exposure term that modifies stream power entrainment and erosion in Eqs. () and () (i.e., e-H/H∗ or (1-e-H/H∗)). Perhaps the most significant simplification in our model is that we do not include explicit treatment of the erosive effects of grains in transport e.g.,. Such an effect could enhance sediment entrainment if grains in saltation hit resting grains and enabled their entrainment, and could enhance bedrock erosion if sediment-rich water were flowing over well-exposed bedrock. The “tools effect” on bedrock erosion could be incorporated into the model by assuming that Er at a given H/H∗ increases with Qs until Qs reaches transport capacity (which is Qs when Es+Er=Ds). Because increases in H/H∗ already account for the “cover effect,” the Er dependence on Qs need only be positive and not decline with increasing Qs e.g.,. We assume for the purposes of model validation against analytical solutions that these effects are negligible relative to changes in unit stream power and bed cover.

The major advantages of the exponential entrainment and erosion approach outlined here are that (1) sediment and bedrock may be simultaneously entrained/eroded into the water column, (2) the presence of sediment does not completely inhibit bedrock erosion at low values of H/H∗, which is supported by modeling and observations of mixed bedrock–alluvial channels , and (3) model stability is improved because sediment thickness gradually approaches zero, preventing a sudden transition from sediment entrainment to bedrock erosion. Many different rules for sediment entrainment and bedrock erosion could be used in this model framework in place of stream-power-type equations. No matter how erosive power is calculated, the SPACE approach allows both erodibility and entrainment/detachment thresholds to be chosen independently for sediment and bedrock, unlike in strict detachment-limited/transport-limited models or the basic form of erosion–deposition models. This enables the treatment of systems with multiple erosion thresholds, such as a river for which bedrock erosion requires both mobilization of an alluvial cover (described by ωcs) and the plucking of bedrock blocks (described by ωcr).

The flux of sediment from the water column onto the bed is the product of sediment concentration averaged over the depth of the water column and effective sediment settling velocity V : Ds=csV=QsQV. V is not the still-water particle settling velocity but is the net effective settling velocity after accounting for the upward effects of turbulence. V also incorporates the vertical gradient in sediment concentration through the water column d∗ in. In an equivalent formulation, treated the sediment deposition rate as Ds=d∗csV, where d∗ is a dimensionless number that relates sediment concentration near the bed to mean sediment concentration in the water column. Equation () assumes that sediment and water move at the same speed such that all changes in QsQ are driven by erosion and deposition.

We develop steady-state analytical solutions for sediment flux Qs, channel slope S, and bed sediment thickness H, all of which are steady when ∂csh∂t=0. We assume for the purposes of this derivation that there are no entrainment or erosion thresholds, and that Ff and ϕ are both negligible – assumptions that are easily relaxed. We define steady state in this system as a state of time-invariant bedrock elevation and topographic elevation (which also implies time-invariant sediment thickness). This occurs when two conditions are satisfied. First, rock uplift must be balanced by bedrock erosion such that ∂R∂t=0=U-KrqSne-H/H∗. Second, sediment entrainment and deposition must balance each other such that sediment thickness H is unchanging in time: ∂H∂t=0=VQsQ-KsqSn1-e-H/H∗. At steady state, the volumetric sediment flux Qs at any point along the channel must balance the volume of newly uplifted rock in the area draining to that point: Qs=UA. To find steady-state channel slope, we begin by rearranging Eq. () and combining it with Eq. (): KsqSn1-e-H/H∗=VUAQ. Recognizing that Q=Ar where r is a runoff rate, KsqSn1-e-H/H∗=VUr. We rearrange Eq. () to isolate e-H/H∗, substitute it into Eq. (), and solve for S to yield S=UVKsqr+UKrq1/n, or if q=kqAm as in the simple stream power formulation, S=UVKsAmr+UKrAm1/n, where kq is subsumed into Ks and Kr. When n=1, the second term on the right-hand side is the slope predicted by a detachment-limited incision model in which slope increases with faster rock uplift, lower rock erodibility, or less water discharge (or drainage area). The first term on the right-hand side describes the additional component of slope required to transport sediment. That component of slope must increase with increasing settling velocity, lower sediment erodibility, and lower water discharge. derived a similar expression for slope–discharge scaling for their erosion–deposition model. The major difference between our result and theirs is that their expression is for slope of a single bed material with a single bed erodibility when erosion balances rock uplift, whereas Eq. () incorporates equilibrium in both sediment thickness and bedrock height. Equation () may be rearranged to show that SPACE predicts a standard stream power slope–area relationship modulated by Vr as well as sediment and bedrock erodibility: S=VKsr+1Kr1/nU1/nA-m/n. The ratio between the effective settling velocity V and the runoff rate r controls the relative importance of the bedrock and alluvial components of the steady-state channel slope. In the simplified case of Ks=Kr, a ratio of Vr=1 would indicate equal contributions from the two regimes. Quantifying Vr for natural systems could therefore give a valuable indication of process dynamics in natural channels.

Solving Eq. () for H gives steady-state bed sediment thickness as a function of channel slope: H=-H∗ln⁡1-VUrKsqSn. To obtain a slope-independent solution for H, we can combine Eqs. () and () and simplify them to H=-H∗ln⁡1-VKsrKr+V. The SPACE model therefore predicts constant sediment thickness along the channel at steady state as long as all parameters in Eq. () are constant in space. As settling velocity becomes larger, VKsrKr+V approaches 1 and H becomes large. As Ks increases and sediment is more easily entrained from the bed, VKsrKr+V and therefore H both approach 0. Increasing bedrock erodibility Kr causes an increase in steady-state H as more sediment is created from detached bedrock.

We present a non-dimensionalization of the model described above. For simplicity, we assume that sediment entrainment and bedrock erosion thresholds are negligible, though the model allows independent entrainment and erosion thresholds as shown above. The model contains three independent variables, Qs, H, and R, the latter two of which are summed to give land-surface elevation. Each of these variables requires a scale for non-dimensionalization. We begin by non-dimensionalizing sediment flux: Qs′=QsVKsq2Sw. Sediment thickness H and bedrock elevation R may both be scaled by the sediment layer length-scale H∗: H′=H/H∗ and R′=R/H∗, and downstream distance x by the length scale q/V (noted by and found to govern the transition between detachment-limited and transport-limited behavior): x′=xV/q. Finally, time t is non-dimensionalized by t′=tV/H∗. Replacing the dimensionless variables into the governing equations yields the following equations for dimensionless sediment flux, sediment thickness, and rock elevation, which are applicable for negligible erosion thresholds: ∂Qs′∂x′=S1-e-H′+1-FfKrKsSe-H′-Qs′∂H′∂t′=KsqVQs′-S1-e-H′∂R′∂t′=UV-KrKsKsqVSe-H′. Three dimensionless parameters appear in Eqs. ()–(): a normalized rock uplift rate UV, a ratio of erodibility KrKs, and a sediment entrainment ratio KsqV. The normalized rock uplift rate shows the relative importance of rock uplift and grain settling velocity, with increases in both driving increased channel slope. The erodibility ratio reflects the relative ease of eroding bedrock and entraining sediment, and is influenced in natural systems by bedrock and sediment lithology, grain size, and grain sorting on the channel bed. The sediment entrainment ratio encompasses competition between sediment entrainment, which is driven by high water discharge and high sediment erodibility, and sediment deposition, driven by high grain settling velocity. Notably, the sediment entrainment ratio contains the q/V length scale described by as the average travel distance of sediment grains from entrainment to redeposition. The model predicts detachment-limited behavior when q/V and therefore our entrainment ratio are large and transport-limited behavior when they are small. Specifically, following Davy and Lague (2009), we can define a dimensionless number Vr that governs the transition between detachment-limited and transport-limited dynamics. In the sediment-only case (when H≫H∗), or in the bedrock-only case (H=0), Vr>1 gives transport-limited behavior and Vr<1 results in detachment-limited behavior (Davy and Lague, 2009). In the cases where sediment and bedrock are eroded simultaneously, especially if there is a significant erodibility contrast between the two, the behavior is not so easily predicted and will generally contain contributions from both detachment and transport limitations.

As shown in Eq. (), the x-directed rate of change in sediment flux depends on sediment entrainment, bedrock erosion, and sediment deposition. The dependence of deposition rate on QsQ means that the deposition flux, and therefore the change in sediment flux, at a given node depends on the sediment flux entering that node from upstream (Eq. ). It is therefore critical to order nodes in upstream to downstream order and calculate sediment flux iteratively from upstream to downstream. This approach unfortunately precludes simultaneous calculation of sediment flux at all nodes. Landlab's flow-routing capabilities order all nodes into a “stack” following the methodology of . Because our sediment flux calculations must progress from upstream to downstream, we use their “inverted stack order” in which nodes are ordered from upstream to downstream, allowing the SPACE algorithm to efficiently sum water and sediment fluxes at tributary junctions. In addition, the downstream sediment flux calculation is written and compiled using the Cython library, giving it significant performance improvements over the same loop in pure Python. Numerical integrations of sediment entrainment and deposition are often significant sources of model inaccuracy and instability due to the spatial extrapolation of linear entrainment and deposition equations. Consider a river reach of length dx with clear water entering the reach at the upstream end. The initial sediment entrainment rate is Es=KsqSn-ωcs and the initial deposition rate is zero because Qs=0. However, two natural processes make the linear extrapolation of these initial entrainment and deposition rates over cell length dx inappropriate. First, sediment entrainment rate may decline over x as sediment thickness H declines, if H is not much greater than H∗. Second, as sediment is entrained over distance dx, Qs increases, which drives a progressive increase in deposition rate. Simply numerically integrating Eq. () to calculate sediment flux does not account for either the progressive decline in available sediment or the progressive saturation of the water column and increase in deposition flux. We have therefore developed a local analytical solution to account for such effects. This approach prevents severe overestimation of sediment entrainment into the water column, making SPACE more stable than models that do not account for within-cell changes in Qs. Our local analytical solution for sediment flux accounts for the fact that sediment flux from upstream Qsin and any net erosion (or deposition) in a model cell of area dx2 contribute to Qs, which drives deposition. Let Qsout represent the sum of sediment influx, erosion, and deposition such that Qsout is the net erosion rate in the cell multiplied by the cell area: Qsout=Qsin+1-ϕEsdx2+1-FfErdx2-VQsrAdx2. Because sediment deposition in a cell depends on both Qsin from upstream and sediment entrained from the cell itself, we can substitute Qsout for Qs in the deposition term. Equation () may then be solved to yield the local analytical solution for Qs within a model cell: Qsout=Qsin+1-ϕEsdx2+1-FfErdx21+Vdx2/rA. Equation () breaks down where rA=0, which is acceptable because Qs will always be zero where rA=0. Figure  shows Qs as a function of some of the relevant variables in Eq. ().

After calculation of Qs at every node, sediment thickness H(t) is calculated according to Eq. (). Similar to the solution of the sediment flux equation, solution of H(t) is subject to numerical inaccuracies and instabilities driven by the dependence of Es on H. Extrapolating ∂H∂t over a full time step using H0 (H at the beginning of the time step) causes overestimation of sediment entrainment, especially at larger time steps. We therefore develop a local analytical solution for Ht for a small time interval over which variations in H are important but Ds, Ksq, and S (as well as any entrainment threshold) may be considered steady. We find the analytical solution for Ht by integrating Eq. () with respect to time with the knowledge that H has some initial value H0 at the beginning of a time step (t=0): Ht=H∗ln⁡[1Ds/1-ϕ/E^s-1×(eDs/1-ϕ-E^st/H∗×Ds/1-ϕE^s-1eH0/H∗+1-1)], where E^s=KsqSn-ωcs. Inspection of Eq. () reveals that Ht may become undefined in two physically realistic situations. The first is where Ds/1-ϕ/E^s=1, and the second is where E^s=0. Equation () is therefore only applied at nodes with Ds/1-ϕ≠E^s and E^s>0. When Ds/1-ϕ=E^s, the change in alluvium thickness with time becomes ∂H∂t=e-H/H∗. Integrating with respect to time and applying H=H0 at t=0 gives the solution for Ht when Ds/1-ϕ=E^s: Ht=H∗ln⁡KsqSn-ωcsH∗t+eH0/H∗. When E^s≤0, no entrainment of sediment occurs and any changes in Ht are driven by deposition. In this case, changes in H are computed with a simple forward numerical solution: Ht=H0+Ds1-ϕdt. In all cases, the relevant equation for Ht is solved each time step using t=dt, where dt is the model time step length. Unlike for Qs, H may be simultaneously calculated at every node, allowing efficient solution of Eqs. ()–() over the entire model domain. Figure  shows Ht as a function of some of the relevant variables in Eq. ().

Bedrock erosion is calculated by combining Eqs. () and () and solving forward in time from a previous rock elevation R0: R=R0+U-KrqSn-ωcre-H/H∗dt. The simple forward numerical solution employed in Eq. () becomes inappropriate at very large time steps, as H may change significantly, influencing channel slopes and therefore bedrock erosion. However, because bedrock erosion is generally a much slower process than sediment entrainment in most cases, Eq. () is unlikely to introduce substantial instability.

Landlab is a flexible, open-source modeling framework written in Python that allows efficient model building and hypothesis testing across many subdisciplines in Earth system science . Landlab is a plug-and-play environment in which users can easily build two-dimensional numerical models consisting of any number of well-vetted components e.g., along with user-specific equations and functionality. The greatest advantages of using Landlab are (1) its built-in gridding engine, which creates model grids, efficiently stores spatially distributed variables, and handles boundary conditions, and (2) the ability to easily couple different components into a single model sharing a single grid. Landlab allows efficient coupling of components representing fluvial erosion, hillslope processes, basin hydrology , geodynamics, vegetation, and many other processes into novel surface dynamics models.

Landlab's gridding engine supports grids consisting of square and rectangular grids (“raster grids”), hexagonal grids, and Voronoi–Delaunay interlocked meshes . Every Landlab grid is made up of nodes, cells, and links. Nodes are points in (x,y) space. Cells are polygons surrounding all non-boundary (interior) nodes that may be rectangular, hexagonal, or defined by Voronoi polygons depending on the chosen grid type. Links connect adjacent pairs of nodes and are directional. Rectangular grids have four links per node, hexagonal grids have six, and Voronoi grids have a number of links per node equivalent to the number of faces on each Voronoi polygon. Links have default directionality, but this directionality does not determine the directions of fluxes in Landlab models, which are set by gradients along links. In this paper, we focus for simplicity on a square (Δx=Δy) raster grid, which is currently the only grid type supported by the SPACE model. A diagram of a generic raster grid is shown in Fig. . Nodes, cells, and links may all store model data in the form of NumPy arrays associated with one of the three grid elements. Each data field is defined by a keyword in a dictionary data structure attached to a certain grid element. The SPACE model, for example, tracks sediment depth at all grid nodes, a field that may be accessed by any component (this field is called “soil depth” in Landlab to keep terminology standard between hillslope and fluvial components). An array of sediment depths at all grid nodes could be found by typing grid.at_node[“soil__depth”]. The treatment of boundary conditions in Landlab grids is described thoroughly by and . In short, nodes may be set as “boundary” nodes and then defined as open, fixed-gradient, or closed boundaries. Non-boundary nodes are set as “core” nodes.

With no sediment (H=0 and cs=0) and Ff=1 (all bedrock eroded becomes wash load and is not included in model calculations), all changes in bed elevation are driven by changes in bedrock elevation: ∂η∂t=∂R∂t, where ∂R∂t=U-KrqSn-ωcr. When ωcr=0, Eq. () is the simple stream power model . At topographic steady state, when ∂R∂t=0 and U=KrqSn, the slope at every point in the channel is

S=UKrq1/n. Because we use a simple stream power formulation where q=kqAm for our test case, the slope–discharge relationship may be rewritten to yield a slope–area relationship: S=UKrAm1/n, where kq is subsumed into Kr. We test whether the SPACE component can replicate steady-state detachment-limited behavior by comparing slope–area relationships predicted by Eq. () with those calculated by the SPACE component. See Table  for the parameter values used. Figure  shows the results after the test model domain has achieved topographic steady state. Figure a shows the longitudinal profile of the longest drainage path in the model domain. As predicted by the theory described above, slope and drainage area trade off such that the outcome is a concave-up longitudinal profile with constant concavity. Figure b compares the slope–area relationship predicted by Eq. () (gray dashed line) to the slope–area relationship in the steady-state model landscape (black dots). All core nodes from the model domain are shown, and every node obeys the predicted detachment-limited slope–area scaling. The slope of the slope–area power-law scaling relationship (Fig. b) is the channel concavity, thus confirming that the channel concavity observed in the longitudinal profile is constant, and that the SPACE component agrees with theoretical predictions for detachment-limited rivers at steady state.

When sediment thickness H is large relative to H∗, changes in bed elevation are driven entirely by changes in sediment bed elevation, which is set by the balance between sediment erosion, deposition, and rock uplift: ∂η∂t=U+Ds-Es. At steady state, ∂η∂t=0 and Es-Ds=U. Substituting in the equations derived above for sediment erosion and deposition and assuming for simplicity that sediment porosity ϕ and the sediment erosion threshold ωcs are negligible, KsqmSn-VQsQ=U. Applying the steady-state mass conservation relationship (Qs=UA), recalling that Q=rA, and solving for S gives an expression for steady-state channel slope: S=UVKsqr+UKsq1/n. If q=kqAm as in our test case, the resulting slope–area relationship is then S=UVKsAmr+UKsAm1/n, where kq is subsumed into Ks. Equation () nicely distinguishes the contributions of sediment deposition (first term on the right side) and sediment entrainment (second term on the right side) to steady-state channel slope. If effective settling velocity is negligible, erosion is only limited by the efficiency of sediment entrainment and Eq. () gives the detachment-limited steady-state slope (though importantly the bed is still entirely composed of sediment). If entrainment and deposition of sediment are rapid enough that erosion is limited by transport capacity (i.e., the river has enough energy to erode more sediment but the water column is saturated), the system is transport limited and the left-hand term in Eq. () dominates in setting the steady-state slope. Note the subtle difference between Eqs. () and (); when H≫H∗ and all surface lowering is accomplished by sediment entrainment, both terms on the right-hand side of Eq. () reflect erosion of sediment (i.e., Ks is used in both). This occurs because when H≫H∗, change in bedrock elevation over time is not zero as in the true complete steady state but is equal to the uplift rate. Therefore, for topographic steady state to be achieved, both the transport and detachment terms of Eq. () must be accomplished through erosion of sediment. Equation () may be rewritten to show that it predicts a standard stream power slope–area relationship that is modified by the ratio of settling velocity to effective runoff: S=Vr+11/nUKs1/nA-m/n. We compare the slope–area relationships predicted by Eq. () with those extracted from the SPACE model. In order to achieve conditions in which the transport term in Eq. () dominates, we set initial soil depth to 100 m everywhere on our test grid so that H≫H∗. See Table  for all parameter values.

Figure  shows the results of the transport-limited model experiment. Figure a shows the longitudinal profile of the longest channel and shows that the SPACE component produces concave-up longitudinal profiles at steady state under transport-limited conditions. The appearance of constant concavity in the longitudinal profile is verified by the constant slope in log–log space of the slope–area data shown in Fig. b. Figure b compares the theoretical slope–area relationship (Eq. , gray dashed line) with data from the model run (black dots). All core nodes from the model domain are included, and all agree well with the theoretical prediction. In addition to matching the analytical prediction for channel slope, the model also matches the expected steady-state sediment flux relationship, Qs=UA (Fig. c). This indicates that the SPACE component is successfully matching expected transport-limited model behavior for both slope and sediment flux at steady state.

One major advantage of SPACE over many existing fluvial erosion models is its ability to simultaneously compute the evolution of an alluvial layer and a bedrock surface. True steady state in the mixed bedrock–alluvial case occurs when the thickness of the alluvial layer H and the bedrock height R are both unchanging in time (∂H∂t=0 and ∂R∂t=0). In such a scenario, Es=Ds and U=Er. As described in Sect. , steady-state analytical solutions exist for channel slope, sediment thickness, and sediment flux (here, again, we use q=kqAm with kq subsumed into Ks and Kr, and keep ϕ=0 and Ff=0): S=UVKsAmr+UKrAm1/n,H=-H∗ln⁡1-VKsrKr+V, and Qs=UA. Running the SPACE component to complete steady state in a case where both sediment entrainment and bedrock erosion contribute to setting channel slope should therefore result in a concave-up channel profile with a sediment layer of constant thickness, and sediment flux equal to the product of the rock uplift rate and drainage area. We use the same tilted plane initial model domain as described in Sect.  to test whether the model can replicate the expected behavior in the bedrock–alluvial case. Matching the steady-state analytical solutions requires both erosion of bedrock to generate the concave-up profile and accumulation of sediment to a constant thickness over the landscape. We begin the numerical experiment with zero sediment thickness at all nodes. Table  shows all parameter values used. The driver script used for this model experiment is included in the code guide for this paper.

Figure  shows the evolution of the longitudinal bedrock profile and alluvial cover layer in the longest channel over several model time slices. Beginning from a low-slope tilted plane, the channel incises bedrock and begins to build up a layer of alluvium on the channel bed. As model time progresses, the channel profile increases in concavity and the layer of alluvium thickens. Alluvial thickening progresses from downstream to upstream. By the final time slice, the alluvial layer has reached its equilibrium value, the channel profile has equilibrated to the imposed uplift rate, and the bedrock surface, alluvium thickness, and topographic surface are all at steady state. Figure a shows the final channel profile when the topographic surface, sediment thickness, and bedrock height are all at steady state. The topographic surface (top of the sediment layer) is parallel everywhere to the bedrock surface, and the bed sediment layer is 1.25 m thick at every point along the channel profile. Given V=5 myr-1, Ks=0.01, and Kr=0.005 m-1, as used in the model, the steady-state sediment thickness of H=1.25 m calculated by the model matches the analytical prediction of Eq. (). Figure b compares the theoretical prediction for slope–area scaling given by Eq. () (gray dashed line) with the model results (black dots). As in the detachment-limited and transport-limited cases, the model matches the analytical prediction. Figure c compares the theoretical steady-state relationship between drainage area and sediment flux (Qs=UA) with modeled sediment flux and shows that the model shows the predicted linear increase in Qs with drainage area. The ability of the SPACE component to treat both the detachment-limited and transport-limited end-members of fluvial systems as well as the mixed bedrock–alluvial case confirms that the model equations are being solved correctly, and importantly that our use of stabilizing, local analytical solutions does not compromise the ability of the model to replicate expected behavior. Below, we show how the SPACE component may be efficiently coupled with other surface process models in the Landlab modeling framework to provide insight into landscape evolution.

We use a Landlab raster model grid composed of 2500 nodes (50×50 grid). We use a node spacing (dx) of 100 m, resulting in a 25 km2 area grid. As with the model verification experiments described above, we close all model domain boundaries except for a single open outlet in the lower left (southwest) corner. The use of a single outlet means that the entire model domain will be a single watershed draining to the outlet, and the sediment flux leaving the outlet is the integrated sediment flux from the entire basin. Our landscape initial condition is a plane tilted slightly (initial regional slope of ≈1.4×10-5) towards the basin outlet. The tilted plane has initial, random sub-millimeter-scale surface roughness to initiate the formation of drainage pathways.

For simplicity, we use the simple stream power form of the SPACE model (m=0.5, n=1, q=kqAm) and keep ϕ=0 and Ff=0. We do not incorporate sediment entrainment and bedrock erosion thresholds. See Table  for all SPACE parameter values used in this example. We couple this parameterization of the SPACE component with Landlab's linear diffusion component, which computes the topographic-gradient-driven movement of mass at every node by the equation ∂η∂t=κ∂2η∂x2, where κ is a diffusivity in units of (L2T-1) (we use κ=0.005 m2yr-1). Programmatically, we incorporate Eq. () simply by running Landlab's linear diffusion component immediately after running the SPACE component in each model time step. One simplification made by simple linear diffusion models is that the entire landscape is made of the same material (i.e., no distinction between rock and soil). To realistically couple linear diffusion with our fluvial erosion model that explicitly separates the dynamics of sediment and bedrock, we assume that any material diffused from one model node onto another is sediment. Such an assumption is realistic given the purpose and limitations of the linear diffusion model, which has been shown to apply primarily to soil-mantled hillslopes.

We ran our coupled model with a 1-year time step for 300 000 years. Because our goal was to use SPACE and linear diffusion to explore topographic growth and decay, we used a rock uplift rate U relative to base level that was unsteady in time. We simulated a “pulse” of rapid rock uplift preceded and followed by periods of slower rock uplift. U was set to 0.0001 myr-1 for the first 100 kyr, increased to 0.0005 myr-1 for the second 100 kyr, and returned to 0.0001 myr-1 for the final 100 kyr. Two variables of broad interest that generally vary in response to changing rock uplift rates are topographic relief and total sediment flux out of the model catchment. We predict under such a scenario that topographic relief and sediment flux will increase as surface erosion responds to rock uplift for the first 200 kyr, and that relief and sediment flux will reach their maximum when the rock uplift rate is at its highest value. We then expect decays in relief and sediment flux as the high points in the landscape erode and topographic gradients shrink in response to a return to the lower rock uplift rate. We recorded the topographic elevation at all model nodes, along with relief and sediment flux from the domain, at 1 kyr intervals.

Figure shows several time slices of model topography showing the response to imposed rock uplift, and Fig.  shows sediment depth over the model domain at the same time slices. Over the first 100 kyr, bedrock incision into the initially non-alluviated domain dominates adjustment to rock uplift. However, because the rock uplift rate is low and topographic gradients are being lowered by fluvial erosion and hillslope diffusion, the landscape has <10 m of relief after 100 kyr. Sediment is first produced on the hillslopes (the products of diffusion are considered to be sediment) but not stored in the channels for the first 50 kyr as the channels incise into bedrock to accommodate the onset of rock uplift. By 100 kyr, the landscape is nearly equilibrated to the rock uplift rate, and alluviation has occurred in the channels as sediment thickness approaches its equilibrium value everywhere. Between 100 and 200 kyr, during which the rock uplift rate is 5 times its initial value, the network progressively develops higher and higher relief (up to ≈50 m) until the rock uplift rate is reduced again at 200 kyr. During this period, the channels initially strip their alluvial cover and incise bedrock to match the increased rock uplift rate. However, as in the first 100 kyr, as the landscape begins to equilibrate to the new rock uplift rate by 200 kyr, alluvium thickness again increases towards its equilibrium condition. Once the rock uplift rate is reduced, diffusion of material from the hillslopes into the channels results in sediment mantling of the channels and significant reduction of the rate of bedrock incision. By 300 kyr, the lowering of high points by diffusion, together with the inability of low-gradient, sediment-mantled rivers to effectively incise bedrock, has resulted in reductions in landscape relief such that the landscape at 300 kyr nearly mirrors the 100 kyr time slice (≈10 m of relief). At this point, the reduction in topographic gradients has reduced sediment delivery to the channels, which begin to strip their alluvial cover down to its equilibrium thickness.

The patterns observed in Figs.  and are quantified in Fig. . Relief and sediment flux out of the model domain increase initially as the landscape is adjusting, through bedrock incision, from its initial condition to become equilibrated with the imposed rock uplift rate. The rates of increase in both relief and sediment flux increase substantially as the rock uplift rate quintuples at 100 kyr, and while neither reaches its equilibrium value (i.e., steady in time), both relief and sediment flux begin to asymptote towards those equilibrium values. At 200 kyr, when the rock uplift rate is reduced to its initial value, diffusion rapidly reduces topographic gradients and alluviates channels, and channels become less erosive. Thus, both relief and sediment flux decline over the final 100 kyr of the experiment, again approaching, but not fully reaching, their steady-state values. The simple experiment performed here shows that the SPACE component may be easily coupled with other models of Earth surface processes in Landlab to explore any number of questions relating to landscape evolution. While 1-D models of river longitudinal profile evolution have proliferated widely over the past decades, models that explicitly incorporate sediment morphodynamics and bedrock erosion simultaneously e.g., are rare. Those that act over 2-D landscapes and are easily coupled with other models are even rarer. The SPACE algorithm and component fill an important gap in the quantitative geomorphologists's toolkit, and it is our hope that future users will apply it, potentially in conjunction with other Landlab components, to solve diverse geomorphological problems.

Models of river erosion are notoriously difficult to test against field data. Uncertainties in initial and boundary conditions, as well as spatial and temporal heterogeneity in rock type and sediment size, introduce substantial complexity into model verification exercises. Nevertheless, studies exploiting natural experiments, or landscapes where initial conditions, boundary conditions, and/or parameter values are particularly well constrained , have seen some success in validating models e.g.,. The SPACE model makes predictions for steady-state channel slope and sediment thickness that could be validated by field or experimental studies. In addition, the co-evolution of the bedrock surface and sediment layer thickness could be used to validate the SPACE model in transient cases.

Previous studies have addressed the validation of the detachment-limited and transport-limited end-members of fluvial erosion models e.g.,, so we focus here on the potential for evaluating the predictions of SPACE for the mixed bedrock-alluvial case. The SPACE model predicts that both sediment entrainment and bedrock erosion will contribute to setting the steady-state slope–area relationship (Eq. ), and that the relative importance of the two components will be set by the ratio Vr and the relative erodibility of sediment and bedrock. The steady-state sediment layer thickness (Eq. ) is also governed by Vr and the ratio of erodibility. As Vr or KrKs increases, the channel slope is dominated by sediment entrainment dynamics and the steady-state alluvial layer thickness increases. If the SPACE model is valid for natural settings, it should be possible to find (or create in the laboratory) steady-state channels with an alluvial layer with constant thickness along the channel. Further, across a gradient of rock and/or sediment erodibility (e.g., a sequence of changing metamorphic grade or a spatial gradient in grain sizes), both the steady-state channel slope and steady-state alluvial thickness should show predictable changes as a function of KrKs. Finally, an intriguing prediction of SPACE is that the steady-state alluvial thickness is independent of the rock uplift rate while the slope–area relationship is not. Field investigations across a range of rock uplift rates with consistent lithology, climate, and sediment properties should show constant steady-state sediment thickness imposed on rock-uplift-dependent slope–area scaling.

In the transient case, the relationship between the longitudinal profile of the bed sediment surface and that of the bedrock may be useful for validating SPACE model predictions. For example, Fig.  shows that for an uplifting landscape with zero initial sediment thickness, SPACE predicts bottom-to-top alluviation of the channel profile. In this case, the sediment surface does not reflect the steepened reach commonly associated with the propagation of transient signals up a river profile, while the bedrock beneath does. The prediction of SPACE is therefore that in a transient river profile with some amount of bed sediment, the concavity of the sediment surface is not expected to match that of the bedrock surface. The difference in concavity between the sediment surface and the bedrock surface should then decline as channels approach steady state, a prediction that is testable in a landscape where channels exist in different stages of transient adjustment. It is important to remember that the sediment thickness predicted by SPACE is a spatial average within a model cell. Further, using realistic (i.e., time-varying) flow distributions to force the model would result in temporal variability in sediment thickness , complicating the interpretation of sediment thickness values from a given field campaign. While testing the steady-state predictions of SPACE is likely feasible in well-constrained landscapes, the transient dynamics may be best explored in a laboratory setting.

The SPACE model is intended to provide a simple, extensible, easy-to-use tool to expand the set of questions that can be addressed with numerical models of river channel evolution. We have consolidated recent advances in the treatment of sediment erosion and deposition as well as simultaneous evolution of sediment and bedrock layers into a single model that has the additional advantages of being effective for modeling over large grids on landscape evolution timescales, easily accessible, and easily coupled to other surface process models in the Landlab modeling toolkit. As such, our model encompasses some of the same limitations as the previous work from which it is derived, including a lack of dynamic channel width and a lack of treatment of the tools effect.

To allow efficient solution of the SPACE model over large model grids and timescales, we do not incorporate the hydrodynamic calculations (i.e., explicit computation of stresses on the bed and banks) required to allow channel width to evolve freely e.g.,. Instead, we employ a common parameterization for channel width as a function of drainage area or discharge. In the examples in this paper, we used a width scaling of w∼Q0.5, which results in q∼Q0.5 or q∼A0.5 if drainage area is linearly related to volumetric water discharge. The width scaling exponent m, which we held equal to 0.5 in the examples discussed here, is a user-defined parameter. The use of an empirical width scaling means that (1) any given point along the channel occupies a single grid cell regardless of whether width is less than, equal to, or greater than grid cell size, and (2) SPACE does not capture temporal channel width variations in response to rock uplift, sediment flux, and discharge changes. In some cases, channel width may respond more significantly than channel slope to such forcings e.g.,. While our use of a downstream width relationship enables us to test the SPACE model against known analytical solutions and eliminates the need for hydrodynamic calculations, the future addition of dynamic width would allow exploration of the relative importance of slope and width adjustment.

The SPACE model as presented here follows previous models in treating reduction of bedrock erosion by sediment cover e.g., but does not include the enhancement of bedrock erosion by the presence of mobile bed load tools (e.g., , ; , ; , ; , ; , ). Such a simplification keeps expected model behavior conceptually simple and allows comparison of model results to known analytical solutions for the purposes of model validation. However, there is substantial field evidence indicating that mobilized bed load can be an important erosive agent, both by detaching bedrock particles and extending fractures through macroabrasion (, ; , ; , ). The major effect of excluding this effect from the SPACE model is that bedrock erosion is underpredicted at low to moderate sediment fluxes, where there is enough bed load to frequently abrade the bed but not enough to form a deep layer of alluvial cover. Adding the tools effect to the SPACE model could be accomplished by changing the form of the bedrock erosion function to increase with increasing Qs.

The SPACE model borrows and combines concepts from models for sediment-flux-dependent bedrock incision, bed cover evolution, sediment transport, and prior models of mixed bedrock–alluvial channels. Here, we briefly highlight the similarities and differences between our model and previous models. SPACE relaxes the traditional assumptions governing detachment-limited and transport-limited erosion models by incorporating both mass conservation of sediment and an incision rule for bedrock. Our model, like other recent models for bedrock–alluvial channels, explicitly incorporates sediment transport to move beyond the assumption that bed exposure depends on the ratio of sediment flux to transport capacity used in sediment-flux-dependent bedrock incision models (e.g., , ; , ; , ; , ). In doing this, our model follows a substantial number of previous contributions. Specifically, , , and used explicit treatments of sediment morphodynamics to explore spatial and temporal changes in bed cover on a non-erodible bedrock bed in response to different forcings. used a stochastic sediment erosion and deposition framework in conjunction with the saltation–abrasion model for bedrock erosion to explore both cover variation and its influence on bedrock erosion rates. Their models are formulated at the reach scale and focus on cover dynamics rather than landscape evolution. Unlike SPACE, however, their models allow different forms for the dependence of exposed bed fraction on sediment transport to arise dynamically, while we use a simple exponential decline in bed exposure with increasing sediment thickness. coupled sediment transport and saltation–abrasion rules in a model of channel cross-section evolution. Our model assumes a planar channel bed and parameterized channel width for application to landscape evolution problems and does not dynamically evolve channel cross-section shape as does the model by . Sediment transport in the SPACE model is computed with the erosion–deposition framework, using an approach almost exactly following . The major difference between SPACE and the models of and is that SPACE applies the erosion–deposition framework to the simultaneous erosion of sediment and bedrock. This allows the model to transition not only between transport-limited and detachment-limited behavior, as in those models, but also between fully alluviated, mixed bedrock–alluvial, and pure bedrock states. This is also the key difference between SPACE and the model, which used Exner-based sediment conservation and bed abrasion under an assumption of infinite sediment supply.

formulated three insightful models for the evolution of mixed bedrock–alluvial channels. There are several areas in which these models incorporate more physical realism in fluvial erosion processes than does SPACE. For example, incorporated the roughness of both the bedrock surface and the alluvial layer; their model is therefore able to treat both the case where the alluvial layer is rougher than the bedrock bed, and vice versa. SPACE does not currently have the ability to alter flow resistance based on the relative exposure of sediment and bedrock. While more advanced representations of the roughness of both bedrock and alluvium are possible, they are not currently incorporated in our model. However, the erosion–deposition treatment of sediment dynamics in SPACE means that it can move beyond the constant sediment supply assumption of . The models of contain substantially more reach-scale complexity than SPACE in that the flow and sediment transport equations are solved in 2-D, resulting in their model being able to resolve changes to alluvial thickness and bedrock elevation in the downstream and cross-stream directions. also added a simple parameterization for bank erosion. The SPACE model treats bedrock elevation and alluvial thickness as cell-averaged quantities, and channel width is empirically parameterized and therefore may at any point be smaller than, equal to, or greater than a cell width. As such, SPACE resolves downstream changes in sediment thickness and bedrock elevation but not cross-stream changes. On the other hand, the simplicity of treating the problem in 1-D allows SPACE to be applied to orogen-scale grids over landscape evolution timescales, whereas the models of have primarily been applied at the reach to kilometer scale.

SPACE is most similar to, and most based upon, the models of and , both of which model the full transition between bedrock, bedrock–alluvial, and fully alluvial channels over full channel profile scales. All three models contain similar treatments of the progressive exposure of bedrock with thinning of the alluvial layer. compared alluvial layer thickness to the median grain size, compared alluvial thickness to the macro-roughness of the bedrock surface, and SPACE similarly computes bedrock exposure based on the ratio of alluvial thickness to a bedrock roughness length scale. The three models differ primarily in that SPACE uses the erosion–deposition framework of to explicitly calculate sediment transport morphodynamics, while and use Exner-based sediment conservation approaches. The other differences among the three models are driven by which processes are included and which are neglected. The model of incorporates dynamic channel width by partitioning shear stresses between the bed and banks, while both the model of and SPACE do not incorporate dynamic channel width but use empirical width scaling parameterizations. use a saltation–abrasion rule for bedrock incision in their model such that they capture both the tools and cover effects. Both the model of and the SPACE model incorporate bed cover as a function of increasing alluvial layer thickness but do not include a dependence of bedrock erosion on sediment flux (the tools effect). Both dynamic width and the tools effect could be incorporated in future versions of the SPACE model.