Predicting landscape-scale erosion rates using digital elevation models

Functional relationships between erosion rates and topography are central to understanding controls on global sediment flux and interactions among tectonics, climate, and erosion in shaping topography. Based on such relations digital elevation models (DEMs) allow predicting landscape-scale erosion rates to the degree that process models can be calibrated and to the extent that such processes reflect elevation, drainage area, and aspect, or their derivatives such as slope and curvature. Digital elevation models allow investigating the influence of erosional processes on landscape form and evolution through generalized quantitative expressions often referred to as ‘erosion laws’. The analytical forms of such expressions are derived from physical principles, but only limited data are available to guide calibration to particular landscapes. In addition, few studies have addressed how different transport laws interact to set landscape-scale erosion rates in different environments. Conventionally, landscape-scale sediment flux is considered to be linearly related to slope or relief, but recent analyses point toward non-linear relations for steep terrain in which changes in the frequency of landsliding accommodate increased rates of rock uplift. In such situations, landscape-scale erosion rates are more closely tied to erosion potential predicted by models of bedrock river incision. Consequently, I propose that using DEMs to predict absolute or relative erosion rates at the landscape-scale counter-intuitively involves the rate of fluvial processes as governing the sediment flux from steep landscapes, and rates of hillslope processes as governing sediment flux from low-gradient landscapes. To cite this article: D.R. Montgomery, C. R. Geoscience 335 (2003).     

Influence of a glacial buzzsaw on the height and morphology of the Cascade Range in central Washington State, USA

Analysis of climatic and topographic evidence from the Cascade Range of Washington State indicates that glacial erosion limits the height and controls the morphology of this range. Glacial erosion linked to long-term spatial gradients in the ELA created a tilted, planar zone of 373 cirques across the central part of the range; peaks and ridges now rise ≤600 m above this zone. Hypsometric analysis of the region shows that the proportion of land area above the cirques drops sharply, and mean slopes >30° indicate that the areas above the cirques may be at or near threshold steepness. The mean plus 1σ relief of individual cirque basins (570 m) corresponds to the ∼600-m envelope above which peaks rarely rise. The summit altitudes are set by a combination of higher rates of glacial and paraglacial erosion above the ELA and enhanced hillslope processes due to the creation of steep topography. On the high-precipitation western flank of the Cascades, the dominance of glacial and hillslope erosion at altitudes at and above the ELA may explain the lack of a correspondence between stream-power erosion models and measured exhumation rates from apatite (U-Th/He) thermochronometry.     

Relative size of fluvial and glaciated valleys in central Idaho

Quantitative comparisons of the morphometry of glaciated and fluvial valleys in central Idaho were used to investigate the differences in valley relief and width in otherwise similar geologic and geomorphic settings. The local relief, width, and cross-sectional area of valleys were measured using GIS software to extract information from USGS digital elevation models. Hillslope gradients were also measured using GIS software. Power-law relationships for local valley relief, width, and cross-sectional area as a function of drainage area were developed. Local valley relief in glaciated valleys relates to drainage area with a power-law exponent similar to fluvial valleys, but glaciated valleys are deeper for a given drainage area. Local valley width in glaciated valleys is greater than in fluvial valleys, but the exponent of the power-law relationship to drainage area is similar in both valley types. Local valley cross-sectional area in glaciated valleys increases with drainage area with a power-law exponent similar to fluvial valleys, however, glacial valleys have roughly 80% greater cross-sectional area. Steep valley walls in glaciated basins increase the potential for bedrock landsliding relative to fluvial basins. Both the Olympic Mountains of Washington and valleys in central Idaho show relationships in which glaciated valleys are up to 30% deeper than fluvial valleys despite differences in lithology, tectonic setting, and climate.     

Influence of vertical channel change associated with wood accumulations on delineating channel migration zones, Washington, USA

We combine hydraulic modeling and field investigations of logjams to evaluate linkages between wood-mediated fluctuations in channel-bed-and water-surface elevations and the potential for lateral channel migration in forest rivers of Washington state. In the eleven unconfined rivers we investigated, logjams were associated with reduced channel gradient and bank height. Detailed river gauging and hydraulic modeling document significant increases in the water-surface elevation upstream of channel-spanning wood accumulations. Logjams initiated lateral channel migration by increasing bed-or water-surface elevations above adjacent banks. Because the potential for a channel to avulse and migrate across its floodplain increases with the size and volume of instream wood, the area of the valley bottom potentially occupied by a channel over a specified timeframe — the channel migration zone (CMZ) — is dependent on the state of riparian forests. The return of riparian forests afforded by current land management practices will increase the volume and caliber of wood entering Washington rivers to a degree unprecedented since widespread clearing of wood from forests and rivers nearly 150 years ago. A greater supply of wood from maturing riparian forests will increase the frequency and spatial extent of channel migration relative to observations from wood-poor channels in the period of post-European settlement. We propose conceptual guidelines for the delineation of the CMZs that include allowances for vertical fluctuations in channel elevation caused by accumulations of large woody debris.     

Local relief and the height of Mount Olympus

A three‐dimensional assessment of the net volume of rock differentially eroded from below mountain tops to form valleys yields a range‐wide constraint on feedback between valley development and the height of mountain peaks. The ‘superelevation’ of mountain peaks potentially attributable to differential removal of material from below peaks in the Olympic Mountains, Washington, was constrained by fitting a smoothed surface to the highest elevation points on a 30 m grid digital elevation model of the range. High elevation areas separate into two primary areas: one centred on Mount Olympus in the core of the range and the other at the eastern end of the range. The largest valleys, and hence areas with the greatest volume of differentially eroded material, surround Mount Olympus. In contrast, the highest mean elevations concentrate in the eastern end of the range. Calculation of the isostatic rebound at Mount Olympus attributable to valley development ranges from 500 to 750 m (21 to 32 per cent of its height) for a 5 to 10 km effective elastic thickness of the crust. Comparison of cross‐range trends in mean and maximum elevation reveals that this calculated rebound for Mount Olympus corresponds well with its ‘superelevation’ above the general cross‐range trend in mean elevation. It therefore appears that the location of the highest peak in the Olympics is controlled by the deep valleys excavated in the centre of the range. Copyright © 2000 John Wiley & Sons, Ltd.     

Vortex evolution due to straining: a mechanism for dominance of strong,interior anticyclones

In this article we address two questions: Why do freely evolving vortices weaken on average, even when the viscosity is very small? Why, in the fluid's interior, away from vertical boundaries and under the influence of Earth's rotation and stable density stratification, do anticyclonic vortices become dominant over cyclonic ones when the Rossby number and deformation radius are finite? The context for answering these questions is a rotating, conservative, Shallow-water model with Asymmetric and Gradient-wind Balance approximations. The controlling mechanisms are vortex weakening under straining deformation (with a weakening that is substantially greater for strong cyclones than strong anticyclones) followed by a partially compensating vortex strengthening during a relaxation phase dominated by Vortex Rossby Waves (VRWs) and their eddy–mean interaction with the vortex. The outcome is a net, strain-induced vortex weakening that is greater for cyclones than anticyclones when the deformation radius is not large compared to the vortex radius and the Rossby number is not small. Furthermore, when the exterior strain flow is sustained, the vortex changes also are sustained: for small Rossby number (i.e., the quasigeostrophic limit, QG), vortices continue to weaken at a relatively modest rate, but for larger Rossby number, cyclones weaken strongly and anticyclones actually strengthen systematically when the deformation radius is comparable to the vortex radius. The sustained vortex changes are associated with strain-induced VRWs on the periphery of the mean vortex. It therefore seems likely that, in a complex flow with many vortices, anticyclonic dominance develops over a sequence of transient mutual straining events due to the greater robustness of anticyclones (and occasionally their net strengthening).     

A-star envelopes: a test of local and non-local models of convection

We present results of a fully non-local, compressible model of convection for A-star envelopes. This model quite naturally reproduces a variety of results from observations and numerical simulations which local models based on a mixing length do not. Our principal results, which are for models with T _eff between 7200 and 8500 K, are the following. First, the photospheric velocities and filling factors are in qualitative agreement with those derived from observations of line profiles of A-type stars. Secondly, the He  ii and H  i convection zones are separated in terms of convective flux and thermal interaction, but joined in terms of the convective velocity field, in agreement with numerical simulations. In addition, we attempt to quantify the amount of overshooting in our models at the base of the He  ii convection zone.     

Free time minimizers for the three-body problem

Free time minimizers of the action (called “semi-static” solutions by Mañe in International congress on dynamical systems in Montevideo (a tribute to Ricardo Mañé), vol 362, pp 120–131, 1996) play a central role in the theory of weak KAM solutions to the Hamilton–Jacobi equation (Fathi in Weak KAM Theorem in Lagrangian Dynamics Preliminary Version Number 10, 2017). We prove that any solution to Newton’s three-body problem which is asymptotic to Lagrange’s parabolic homothetic solution is eventually a free time minimizer. Conversely, we prove that every free time minimizer tends to Lagrange’s solution, provided the mass ratios lie in a certain large open set of mass ratios. We were inspired by the work of Da Luz and Maderna (Math Proc Camb Philos Soc 156:209–227, 1980) which showed that every free time minimizer for the N-body problem is parabolic and therefore must be asymptotic to the set of central configurations. We exclude being asymptotic to Euler’s central configurations by a second variation argument. Central configurations correspond to rest points for the McGehee blown-up dynamics. The large open set of mass ratios are those for which the linearized dynamics at each Euler rest point has a complex eigenvalue.