28 Jul 2021
28 Jul 2021
The Direction of Erosion
 ^{1}BlueMarbleSoft
 ^{2}University of Cambridge
 ^{1}BlueMarbleSoft
 ^{2}University of Cambridge
Abstract. The rate of erosion of a geomorphic surface depends on its local gradient and on the material fluxes over it. Since both quantities are functions of the shape of the catchment surface, this dependence constitutes a mathematical straitjacket, in the sense that – subject to simplifying assumptions about the erosion process, and absent variations in external forcing and erodibility – the rate of change of surface geometry is solely a function of surface geometry. Here we demonstrate how to use this geometric selfconstraint to convert an erosion model into its equivalent Hamiltonian, and explore the implications of having a Hamiltonian description of the erosion process. To achieve this conversion, we recognize that the rate of erosion defines the velocity of surface motion in its orthogonal direction, and we express this rate in its reciprocal form as the surfacenormal slowness. By rewriting surface tilt in terms of normal slowness components, and by deploying a substitution developed in geometric mechanics, we extract what is known as the fundamental metric function of the model phase space; its square is the Hamiltonian. Such a Hamiltonian provides several new ways of solving for the evolution of an erosion surface: here we use it to derive Hamilton's ray tracing equations, which describe both the velocity of a surface point and the rate of change of the surfacenormal slowness at that point. In this context, erosion involves two distinct directions: (i) the surfacenormal direction, which points subvertically downwards, and (ii) the erosion ray direction, which points upstream at a generally small angle to horizontal with a sign controlled by the scaling of erosion with slope. If the model erosion rate scales faster than linearly with gradient, the rays point obliquely upwards; but if erosion scales sublinearly with gradient, the rays point obliquely downwards. Analysis of the Hamiltonian shows that these rays carry boundarycondition information upstream, and that they are geodesics, meaning that erosion takes the path of least erosion time. This constitutes a definition of the variational principle governing landscape evolution. In contrast with previous studies of network selforganization, neither energy nor energy dissipation is invoked in this variational principle, only geometry.
Colin Peter Stark and Gavin John Stark
Status: final response (author comments only)

RC1: 'Comment on esurf202159', Anonymous Referee #1, 24 Sep 2021
Overview
The manuscript considers the erosion of surfaces using a threedimensional generalization of the Stream Power Model (Royden and Perron (2013)), which expresses the temporal change in surface elevation in terms of the drainage area and the stream gradient. The generalization treats the landscape as an implicit surface and describes the erosion in the normal direction as a function of the surface tilt (Eq. (24)). This extension has practical advantages, it allows for the handling of steep or concave geometry, which is not uncommon in geological settings.
The underlying assumptions (constant external forcing and erodibility) lead to a pure geometric PDE, in which the evolution of the surface is determined solely by the surface geometry. As the geometric PDE is of the HamiltonJacobi type, the corresponding Hamiltonian is presented and as it is common for the eikonal equation, it is found that the variational principle associated with the evolution is least erosion time. At the moment, the model is constrained to the 2D setting, in which the trick in (26) can be carried out. Hence, the numerical examples are solved only in 2D by using the ray tracing equations. The manuscript concludes that the existence of the Hamiltonian has some insights regarding erosion, and claims that the Hamiltonian setting clarifies the phrase ‘direction of erosion’.
The manuscript builds on wellestablished concepts and techniques ranging from classical mechanics to erosion models, (aparat from an elegant trick) the novelty of the presented material is uncertain.
About the erosion model
The application of the Hamiltonian machinery is elegant, however, it strongly relies on the erosion model introduced in Eq. (24), which is identical with the model in Royden and Perron (2013), but it is expressed in the normal instead of the vertical direction. Later, Eq. (76) defines the erosion model by specifying the flow component of Eq. (24). The physical background and interpretation of this formula are missing. Back to Eq. (24): why not assume the surface normal speed in the form (x)f(sin), where f:[1,1], a suitably regular function?
Figure (5) shows an example of the ray tracing approach, but also introduces a physically different problem: fault slipping, which was not mentioned before in the text. It is unclear why it is chosen from the possible physical problems listed in the Introduction.
About the findings of the paper
The abstract suggests that one of the conclusions of the work is that "erosion takes the path of least erosion time" but later in Sect. 2.14, the concept is introduced as a proposal. It is unclear whether it was an assumption or an outcome of the model. The manuscript lacks any physical interpretation of the least erosion time, as a variational principle.
The significance of the rays is also unclear as the proposed equations could be solved with other numerical methods without difficulty. The manuscript points out that the slope exponent \eta and the direction of the rays are correlated, but the physical interpretation of this finding and the parameter \eta itself are missing. Figure (5) suggests that the ray tracing approach would not be appropriate if there were overhangs on the surface.
One of the findings of the manuscript is to show the anisotropy of landscape erosion, however, anisotropy is not defined in the manuscript. This emergence of anisotropy stems from the applied normal speed in Eq. (24). As the authors state in the introduction, surface erosion cannot be tracked by tracers lying on the surfaces, and therefore no true point pairs exist on the beforeafter states. It means, any wellposed pairing (i.e., a bijection) between the points of two, consecutive shapes in the evolution establishes an erosion model. Nonetheless, most models in the literature use the local normal to establish that bijection, and the present manuscript is not an exception. Still, the sentence in the conclusion ‘the rate of surface erosion is a velocity normal to the surface: differential geometry tells us that no other direction has any meaning’ is false and misleading.
The introduction suggests that no previous examples of modeling erosion as the evolution of implicit surfaces exist, but there are some papers on the evolution of implicit surfaces under erosion equations (Kraft et.al. (2011), Bencheikh et.al. (2020), The main result stating that the geometry of a surface determines its erosion is also a concept that constitutes the base of many works in the literature (Bloore (1977), Wilson (2009), Sipos et al. (2011), Domokos et al. (2014a), Domokos et al. (2014b)).
Although the paper presents a simple, two dimensional equation, it misses to compare the computational outcome to available experimental results. For example, the simple flume experiments on cuboid marble blocks in Wilson (2009) might be reproduced.
Other comments
Section 2 mixes topics that are tightly and loosely connected to the main ideas of the paper. It suggests that it gives us an overview of the background of the work, but it also contains statements about the model and the implementation (e.g., it refers to the Hamiltonian that is introduced in section 3.8). This makes it hard to differentiate between the known results, new ideas, related works, and suggestions for future investigations. Some of the sentences of this section would fit better in the conclusion section. It is hard to follow the line of thought of the text due to the interruption by remotely related topics. Although the beginning of Section 2 mentions the 2+1D streampower law models, there is no explanation or mention of them.
There are many explanations and statements that are not used and it would be enough to reference them or move them to the appendix. Some textbook concepts (such as covectors in Section 3.1.) are superfluous and should be omitted entirely. Valid parameter ranges are not discussed in detail and regarded only with the jargon "on a shell."
Overall impression
Despite the strengths of the manuscript, the reviewer does not recommend publication in the present form. Although the methods and results are discussed and described in detail, the motivation and the significance of the work remain unclear. The erosional model is not new and the results seem to be consequences of the chosen solution technique without any physical significance.
References
Bencheikh, I., M. Nouari, and F. Bilteryst. "Multistep simulation of multicoated tool wear using the coupled approach XFEM/multiLevelset." Tribology International 146 (2020): 106034.
J. Bloore, The shape of pebbles. J. Int. Assoc. Math. Geol. 9, 113–122, 1977
Domokos, Gábor, Gary W. Gibbons, and András A. Sipos. "Circular, stationary profiles emerging in unidirectional abrasion." Mathematical Geosciences 46.4 (2014a): 483491.
Domokos, Gabor, et al. "How river rocks round: resolving the shapesize paradox." PloS one 9.2 (2014b): e88657.
Kraft, Susanne, Yongqi Wang, and Martin Oberlack. "Large eddy simulation of sediment deformation in a turbulent flow by means of levelset method." Journal of Hydraulic Engineering 137.11 (2011): 13941405.
Royden, L., and Taylor Perron, J. (2013), Solutions of the stream power equation and application to the evolution of river longitudinal profiles, J. Geophys. Res. Earth Surf., 118, 497– 518, doi:10.1002/jgrf.20031.
Sipos, András A., et al. "A discrete random model describing bedrock profile abrasion." Mathematical Geosciences 43.5 (2011): 583591.
Wilson A: Fluvial bedrock abrasion by bedload: process and form. Dissertation, University of Cambridge, UK, 2009

RC2: 'Comment on esurf202159', David J. Furbish, 01 Nov 2021
The comment was uploaded in the form of a supplement: https://esurf.copernicus.org/preprints/esurf202159/esurf202159RC2supplement.pdf
Colin Peter Stark and Gavin John Stark
Model code and software
Geomorphysics Python library (GMPLib) Colin P. Stark https://cstarkjp.github.io/GMPLib
Geometric Mechanics of Erosion software package (GME) Colin P. Stark https://cstarkjp.github.io/GME
Zenodo archive of Geomorphysics Python library (GMPLib) Colin P. Stark https://doi.org/10.5281/zenodo.5105574
Zenodo archive of Geometric Mechanics of Erosion software package (GME) Colin P. Stark https://doi.org/10.5281/zenodo.5105597
Colin Peter Stark and Gavin John Stark
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