ABSTRACT. The geomorphic evolution of many landscapes is fundamentally determined by the evolution of the river channels and their interactions with hillslopes. Consequently, models of landscape evolution ought to track the evolution of the channel geometry so as to quantify the rate of erosion of channel bottoms and to follow the changes in hillslope-channel coupling over time. Unfortunately, the spatial resolution required to describe channel morphology adequately is computationally impractical. It is also beyond the resolution of most digital elevation data. What is required is a parameterization scheme for approximating fine scale channel morphology at a coarse pixel scale. Such a parameterization is already implicitly employed in most models by assuming channel equilibrium, which ties the width and depth of a model channel to the square root of discharge through a pixel. Channel fluxes are thereby predictable, and a closed form of the governing equations is attained. In reality, mountain river channels do not take a simple equilibrium form and show great spatial variability and evident disequilibrium geometry. Since the time scales of changes in channel geometry, bedrock channel erosion, and hillslope response are all closely related, it is reasonable to infer that the spatio-temporal development of the landscape is determined by their interaction and that channel disequilibrium is a fundamental factor in the dynamics of landscape evolution. If this is the case, we need an alternative sub-grid scale parameterization that aggregates channel properties such as surface morphology, roughness, cross-sectional geometry, so that the time dependent behavior of these properties can be estimated at the coarse pixel scale. We introduce such a parameterization measure, which we term channelization, after extensive investigation of the pixel resolution dependence of topographic relief. We focus in particular on the effect of coarse graining on digital elevation data for derived measures such as channel slope and upstream area and demonstrate that we can approximately correct for this effect. We show that a very simple geomorphic model can be constructed around the channelization parameter and the resolution-invariant topographic measures. This model demonstrates that channel disequilibrium may play a significant role in mountain landscape dynamics. It also shows how geomorphic models in general could be modified to incorporate such sub-pixel scale complexities and to better model these dynamics.
Abstract. The rate of erosion of a landscape depends largely on local gradient and material fluxes. 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 self-constraint to convert a gradient-dependent 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 surface-normal slowness. By rewriting surface tilt in terms of normal slowness components and 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 to solve 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 surface-normal slowness at that point. In this context, gradient-dependent erosion involves two distinct directions: (i) the surface-normal 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. This dependence of erosional anisotropy on gradient scaling explains why, as previous studies have shown, model knickpoints behave in two distinct ways depending on the gradient exponent. Analysis of the Hamiltonian shows that the erosion rays carry boundary-condition information upstream, and that they are geodesics, meaning that surface evolution takes the path of least erosion time. Correspondingly, the time it takes for external changes to propagate into and change a landscape is set by the velocity of these rays. The Hamiltonian also reveals that gradient-dependent erosion surfaces have a critical tilt, given by a simple function of the gradient scaling exponent, at which ray-propagation behaviour changes. Channel profiles generated from the non-dimensionalized Hamiltonian have a shape entirely determined by the scaling exponents and by a dimensionless erosion rate expressed as the surface tilt at the downstream boundary.
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 self-constraint 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 surface-normal 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 surface-normal slowness at that point. In this context, erosion involves two distinct directions: (i) the surface-normal 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 boundary-condition 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 self-organization, neither energy nor energy dissipation is invoked in this variational principle, only geometry.
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