Numerical modelling offers a unique approach to understand how tectonics, climate and surface processes govern landscape dynamics. However, the efficiency and accuracy of current landscape evolution models remain a certain limitation. Here, I develop a new modelling strategy that relies on the use of 1D analytical solutions to the linear stream power equation to compute the dynamics of landscapes in 2D. This strategy uses the 1D ordering, by a directed acyclic graph, of model nodes based on their location along the water flow path to propagate topographic changes in 2D. This analytical model can be used to compute in a single time step, with an iterative procedure, the steady-state topography of landscapes subjected to river, colluvial and hillslope erosion. This model can also be adapted to compute the dynamic evolution of landscapes under either heterogeneous or time-variable uplift rate. This new model leads to slope–area relationships exactly consistent with predictions and to the exact preservation of knickpoint shape throughout their migration. Moreover, the absence of numerical diffusion or of an upper bound for the time step offers significant advantages compared to numerical models. The main drawback of this novel approach is that it does not guarantee the time continuity of the topography through successive time steps, despite practically having little impact on model behaviour.

While the elevated but incised landscapes of mountain belts testify to the cumulated actions of tectonics, erosion and climate, unravelling how these processes act and interact to shape the Earth's surface remains one of the most challenging issues in Earth sciences (e.g. Molnar and England, 1990; Willett, 1999; Whipple, 2009; Steer et al., 2014; Croissant et al., 2019). Numerical models have been pivotal to understanding how topography and erosion respond to spatial and temporal changes in climate and tectonics (e.g. Howard et al., 1994; Whipple and Tucker, 1999; Tucker and Whipple, 2002; Carretier and Lucazeau, 2005; Thieulot et al., 2014; Croissant et al., 2017). At the mountain scale, numerical models generally account for geomorphological processes using effective and reduced-complexity erosion laws such as the stream power incision model (SPIM) for rivers (e.g. Howard et al., 1994) and diffusion for hillslopes (e.g. Roering et al., 1999). In particular, the SPIM is popular in landscape evolution models (LEMs) as its physical expression resolves to a non-linear kinematic wave equation, which offers simple finite-difference or finite-volume solutions in 1 and 2D (e.g. Pelletier, 2008; Braun and Willett, 2013; Campforts and Govers, 2015; Campforts et al., 2017). Despite these benefits, these numerical solutions have several drawbacks: (1) their stability or consistency requires the use of a small time step that must respect the Courant condition, i.e. that an erosional wave cannot travel over a distance greater than one or a few node spacings during one time step, and (2) they are prone to numerical diffusion and therefore only offer approximate solutions. Numerical schemes in 2D have recently been developed to reduce the time-step dependency on grid spacing (Braun and Willett, 2013) or numerical diffusion (Campforts and Govers, 2015). In 1D, evolution of river profiles can be derived using analytical solutions determined by the method of the characteristics (Luke, 1972, 1974, 1976; Weissel and Seidl, 1998; Whipple and Tucker, 1999; Lavé, 2005; Pritchard et al., 2009; Royden and Taylor Perron, 2013). These solutions have been successfully used in formal inversion of river profiles (Goren et al., 2014a; Fox et al., 2014; Goren, 2016), but they have been largely ignored in forward landscape evolution models, despite their inherent exact accuracy. This likely results from the apparent absence of an analytical solution in 2D.

In this study, I extend the applicability of these 1D analytical solutions to 2D problems by developing a new type of landscape evolution model based on analytical solutions. I first demonstrate how this model, that I refer to as Salève, can be used to compute – in a single time step – a steady-state topography in 2D. I then develop a dynamic version of Salève to solve for transient landscape changes under heterogeneous or time-variable uplift. Last, I demonstrate the ability of Salève to accurately model the propagation of knickpoints in LEMs and to account for river, colluvial and hillslope erosion.

Most LEMs require the computation of river water discharge as the main
driver of river erosion and sediment transport. While flow
algorithms based on physical considerations offer more accurate solutions (e.g. Davy et al., 2017), water
routing in 2D LEMs is generally achieved using simple flow algorithms, like
the steepest slope (O'Callaghan and Mark, 1984) or the multi-flow direction
(Quinn et al., 1991; Freeman, 1991). The Fastscape algorithm, and other
graph-based approaches, offers a very efficient means to order nodes along
the steepest water flow path and to compute river discharge and drainage
area (Braun and Willett, 2013; Schwanghart and Scherler, 2014). A single receiver and potentially several donors are attributed to each node of the topographic grid to recursively build a node stack (or graph) from the outlet node to the crest nodes of each catchment. Each node is therefore associated with its outlet node through a single flow path. These flow paths represent 2D trajectories in the (

In 1D, a classical detachment-limited approach to describe the rate of
change in river elevation change

Overview of the algorithms used for the

This solution (Eq. 4) can be extended to spatially variable uplift rate

Modelled steady-state topographies obtained after

The obtained solution looks very roughly like a classical steady-state topography, and yet it is not strictly at steady state (Fig. 2). Indeed, during this first iteration, the scheme used (Fig. 1a) imposes the constraint that rivers develop over the flow network defined by the initial topography and, in turn, does not ensure that the nodes located on the same crest of two juxtaposing catchments share the same response time or the same elevation. This leads to an excessive elevation as some rivers have planar length greater than predicted. This is the main limit of this 1D algorithm that cannot ensure the optimality of the 2D organization of the river network at steady state after only one iteration (Fig. 2).

However, repeating this operation by computing the topography and then
updating the flow network (i.e. by computing the steepest slope, node order,
and drainage area or discharge) after each iteration leads to a steady-state
topography after a few tens of iterations

Changing

Influence of model parameters and geometry on the convergence
towards a steady-state landscape.

The new algorithm developed in Salève presents significant advantages
compared to finite-difference schemes, which are fundamentally limited by
the time step

I now explore the use of this analytical model in dynamic simulations with
Salève (Fig. 1b). I first consider the case of potentially heterogeneous
but constant uplift rate

I here run a simulation, using the same parameters as in the steady-state
simulation, over a duration of 500 kyr (Fig. 4). The time steps

Dynamic behaviour of the Salève model.

The final topographies, i.e. at steady state, obtained with Salève or
with the implicit solution share roughly the same statistical properties in
terms of vertical and horizontal organization. The time evolution of the
mean elevation (

Moreover, erosion rates

Erosion rates in Salève, calculated by differencing elevation between
successive time steps and subtracting the contribution of uplift, are
significantly more variable, in particular for the model with the shorter
time step

In terms of horizontal organization, all the Salève and implicit models
lead to the same Hack's law (Hack, 1957), which relates, through a power-law
relationship, the downstream maximum river length

Dynamic evolution of the topography and knickpoint migration over
500 kyr. The initial uplift rate

I now investigate the case of time-variable but homogeneous uplift rate

Discrete temporal changes in uplift rates or in base-level elevation can
lead to sharp ruptures in the slope of river profiles, generally referred to
as knickpoints (e.g. Rosenbloom and Anderson, 1994; Whipple and Tucker,
1999; Steer et al., 2019). Finite-difference solutions to the stream power
equation inherently lead to a progressive numerical diffusion of knickpoints
during their migration, even with

In previous sections, I have considered the steady-state and dynamic
solutions of landscapes subjected only to river erosion following the SPIM.
However, these analytical solutions can be extended to simulate the dynamics
and morphology of colluvial valleys and hillslopes. Indeed, a power-law
scaling for the slope–area relationship is observed in colluvial valleys,
which suggest they could obey a similar erosion law as Eq. (1), but
with different

Practically, considering three different erosion laws, for river, colluvial
valleys and hillslopes, simply requires changing the value of

Steady-state topographies obtained with Salève considering
only

Based on previous analytical developments (e.g. Royden and Taylor Perron, 2013), I have designed a new method to solve for the steady-state topography or the dynamic evolution of a landscape in 2D, following the SPIM, with analytical precision. The model can solve in a single time step, using an iterative scheme, the steady-state topography of a landscape under homogeneous or heterogenous conditions (i.e. uplift rate, erodibility and runoff). Iterations are required to optimize the planar organization of the river network and crest positions, starting from a random network. The number of iterations required for the convergence of the scheme only depends on the
number of nodes discretizing the surface topography and only scales with

The developed scheme, that uses 1D analytical solutions, is limited
to flow networks that can be topologically classified as 1D node stacks or
graphs (Braun and Willett, 2013), as resulting from a steepest slope flow
routing algorithm. This excludes, for instance, recent models accounting for
flow algorithms based on physical considerations (Davy et al., 2017). The main limitation of
this new approach is that reorganizations of the river network, such as
catchment piracy, will not lead to transient phases of erosion, as the river
elevation is directly updated to its optimal elevation for each node where

The Salève model is also not designed for horizontal tectonic
displacement (e.g. Braun and Sambridge, 1997; Steer et al., 2011; Miller et
al., 2007) that displaces nodes relative to the location of the base-level
condition. Moreover, Salève is a purely detachment-limited model which
does not consider the role of sediment transport and deposition in landscape
dynamics. Only the linear SPIM with

Extending the Salève algorithm to non-linear SPIMs represents a
challenging and non-trivial perspective that requires accounting for more
complex analytical solutions with overlapping or stretching river profiles
for

A MATLAB version of the model can be accessed through a Zenodo repository:

The author declares that there is no conflict of interest.

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The two reviewers, Liran Goren and Sébastien Carretier, as well as the editor, Joshua West, and associate editor, Greg Hancock, are acknowledged for their constructive comments that helped with improving this article. I am also grateful to Sean Willett for his insightful comments on an earlier version of this article. I thank Dimitri Lague, Philippe Davy, Jean Braun, Boris Gailleton, Joris Heyman, Alain Crave, Thomas Croissant and Edwin Baynes for their helpful comments and for discussions about this work.

This research has been supported by the H2020 European Research Council (grant no. 803721).

This paper was edited by Greg Hancock and reviewed by Sebastien Carretier and Liran Goren.