Erosion Rule Research Articles

Numerical modeling of glacial erosion and headwall processes in alpine valleys

How rock relief is developed by glacial erosion in alpine landscapes is a key question in understanding the controls on mountain range topography. Any generation of relief by glaciers at the highest elevations in mountain ranges can affect the distribution of precipitation in alpine landscapes, creating feedbacks between glacier mass balance and glacial erosion. Relief production at high-elevation mountain crests is addressed using a numerical model that incorporates glacier dynamics and subglacial erosion, as well as relevant hillslope processes. Blowing snow from a valley-bounding plateau, and avalanching onto the accumulation zone of the glacier is included in the model. Furthermore, temperature-dependent frost cracking is used as a proxy for headwall backwearing. The parameterization of subglacial erosion produces about 500 m of erosion over the 400 ka of each numerical simulation. In steady and varying climate scenarios, valley floors flatten as headwalls increase in height and retreat headward over time. Headwall slopes increase in simulations in which quarrying is explicitly incorporated; rates of quarrying are high near the base of the headwall, increasing headwall length and slope as a result of valley floor lowering. High rates of subglacial erosion and production of headwall relief also result when elastic flexural uplift occurs in response to mass unloading. Small overdeepenings form at the base of the headwall in steady-climate simulations, when the glacier is small at the end of the model run. While the final profiles are relatively insensitive to the erosion rule used, quarrying is most effective near the head of the glacier, whereas rates of abrasion reflect the instantaneous pattern of integrated ice discharge, or ice flux. In simulations in which cirques are formed, the equilibrium line altitude is hundreds of meters above the cirque floor; however, the time-averaged horizontal position of the equilibrium line corresponds well with the downvalley position of the cirques.

Modeling river delta formation

A model to simulate the time evolution of river delta formation process is presented. It is based on the continuity equation for water and sediment flow and a phenomenological sedimentation/erosion law. Different delta types are reproduced by using different parameters and erosion rules. The structures of the calculated patterns are analyzed in space and time and compared with real data patterns. Furthermore, our model is capable of simulating the rich dynamics related to the switching of the mouth of the river delta. The simulation results are then compared with geological records for the Mississippi River.

Conditional Simulation of Multi-Type Non Stationary Markov Object Models Respecting Specified Proportions

Although Boolean model simulation has been widely used during the last two decades to simulate sedimentary bodies (especially in fluvio-deltaic environments), a key issue has subsisted. One of the most important parameter for object model simulation, namely the (non stationary) intensity of the underlying object process, is not a parameter provided by the end user but must instead be computed from other input parameters, such as local proportions of lithofacies, erosion rule between objects of different types and interaction between objects. This paper revisits a birth and death algorithm for simulating conditional, non stationary, multi-type objects models with interaction. It provides workable approximations for computing the local intensity of the underlying point process to respect proportion maps. Simulated examples show that this algorithm is able to reproduce the desired proportions. Important issues for implementing this algorithm are discussed.

Morphologies of knickpoints in submarine canyons

The question of how turbidity currents erode their beds is important for understanding how submarine canyons develop, how they maintain continuity in tectonically active margins to ensure sediment bypass, and for knowing how knickpoints (reaches of anomalously steep gradient) record tectonic information. The problem is potentially more complex than fluvial erosion, because flow vigor is also affected by the flow entraining ambient water and incorporating or depositing suspended load, which can significantly affect its excess density. However, in canyon sections where the total sedimentary mass passing through the canyon is much larger than the locally excavated mass, the solid loads of eroding currents change little during passage down-canyon. Canyon morphology can then potentially reveal how gradient and other factors affect erosion rate. Simple bed erosion models are presented herein, which are analogous to the detachment- and transport-limited erosion models of fluvial geomorphology, which predict that the channel topography should advect or diffuse (smooth out), respectively. Data sets from continental slopes off Alaska, New Jersey, Oregon, Chile, the Barbados accretionary prism, and published maps from other areas show these tendencies. Although knickpoints may arise from spatially varied resistance to erosion, some of those described here lie upstream of faults or anticlines and within uniform turbidites, implying that they can advect upstream. A forward numerical model is developed for knickpoints in the southern Barbados accretionary prism, which appear to have been created in a simple manner by the frontmost thrusts. If the erosion rules are applied continuously, the channel profiles are well represented with both advective and diffusive components. If a boundary condition of nondeposition/erosion is imposed on the base of the knickpoint slope (representing scour associated with a hydraulic jump, for example), the upstream profiles can be reproduced solely by diffusion. In these channels, the threshold stress for transport or erosion is probably small relative to stress imposed by the currents, because modeling shows that a threshold sharpens the knickpoint lip rather than rounds it. For the other, mostly smaller, knickpoints studied, however, the lip varies from sharp to rounded. This varied morphology could arise from a number of influences: effects of flow acceleration, differing threshold stress, differing sediment flux affecting flow power, or depth-varying substrate resistance to erosion. Despite the diversity of forms, upstream migrations imply that erosion can be enhanced where flow is more vigorous on steep gradients, implying that the body rather than the head of turbidity currents is responsible for erosion in those cases. Also discussed is how bed failure, quarrying, and abrasive scour lead to knickpoint evolution in submarine channels that is analogous to that in fluvial channels, but also likely differences are noted.

Testing a glacial erosion rule using hang heights of hanging valleys, Jasper National Park, Alberta, Canada

In most models of glacial erosion, glacier sliding velocity is hypothesized to control rates of bedrock erosion. If this hypothesis is correct, then the elevation difference between hanging and trunk valley floors, the hang height, should be dictated by the relative sliding velocities of the glaciers that occupied these valleys. By considering sliding velocity to be proportional to balance velocity and using mass continuity, hang height is expressed in terms of glacier catchment areas, slopes, and widths, which can be estimated for past glaciers from the morphology of glacial valleys. These parameters were estimated for 46 hanging valleys and their trunk valleys in three adjacent regions of Jasper National Park. The variability in valley morphology can account for 55–85% of the hang height variability if erosion rate scales with balance velocity raised to a power of 1/3. This correspondence is in spite of spatial variations in glaciation duration, snow accumulation rates, and other variables that likely affected hang heights but cannot be readily estimated and so are not included in our formulation. Thus it appears that balance velocity, and by extension, sliding velocity if the two are proportional, may be a reasonable control variable for assessing erosion rate.

Quaternary relief generation by polythermal glacier ice

AbstractThe juxtaposition of wet‐based erosive ice in valleys and cold‐based, non‐erosive ice atop felsenmeer‐covered interfluve plateaus has generated relief in the Torngat Mountains of northeastern Canada. Measurements of in situ terrestrial cosmogenic nuclide (TCN) concentrations from 31 bedrock sites, coupled with soils and geomorphology, indicate that erosion of the valleys has been >2 m during a single glacial–interglacial cycle. However, on summit plateaus the long‐term (over several glacial–interglacial cycles) erosion rate is <1·4 m Ma−1. TCN ratios reveal that the exposure plus ice‐cover history retained on some summit surfaces probably spans more than 800 ka despite complete ice cover as recently as 11 ka. A thermodynamic ice sheet model with a basal water calculation is used to calculate the sliding distance normalized by the duration of ice cover for the region. We formulate a general glacial erosion rule for the Torngat Mountains, which correlates TCN‐derived erosion rates for terrain once partially covered by cold‐based ice with modelled average ice basal sliding velocities. Erosion rates vary linearly with average sliding velocity by a glacial erosion coefficient of 5 × 10−7. Due to the significant distribution of cold‐based ice cover in this high latitude region, our estimates of net regional glacial erosion and glacial erosion coefficient are orders of magnitude lower than a previously published value. Copyright © 2005 John Wiley & Sons, Ltd.

Alpine landscape evolution dominated by cirque retreat

Despite the abundance in alpine terrain of glacially dissected landscapes, the magnitude and geometry of glacial erosion can rarely be defined. In the eastern Kyrgyz Range of central Asia, a widespread unconformity exhumed as a geomorphic surface provides a regional datum with which to calibrate erosion. As tectonically driven surface uplift has progressively pushed this surface into the zone of ice accumulation, glacial erosion has overprinted the landscape. With as little as 500 m of incision into rocks underlying the unconformity, distinctive glacial valleys display their deepest incision adjacent to cirque headwalls. The expansion of north-facing glacial cirques at the expense of south-facing valleys has driven the drainage divide southward at rates of as much as two to three times the rate of valley incision. Glacial erosion rules based on ice flux incompletely explain this dominance of cirque retreat over valley incision. Local processes that either directly sap cirque headwalls or inhibit erosion down-glacier appear to control, at least initially, alpine landscape evolution.

Controls on the strength of coupling among climate, erosion, and deformation in two‐sided, frictional orogenic wedges at steady state

Many important insights regarding the coupling among climate, erosion, and tectonics have come from numerical simulations using coupled tectonic and surface process models. However, analyses to date have left the strength of the coupling between climate and tectonics uncertain and many questions unanswered. We present an approximate analytical solution for two‐sided orogenic wedges obeying a frictional rheology, and in a condition of flux steady state, that makes explicit the nature and sensitivity of the coupling between climate and deformation. We make the simplifying assumption that the wedge grows in a self‐similar fashion consistent with Airy isostasy such that topographic taper is invariant with orogen width, tectonic influx rate, and climate. We illustrate first how and why the form of the erosion rule matters to orogen evolution and then derive a physically based orogen‐scale erosion rule. We show that steady state orogen width, crest elevation, and crustal thickness are controlled by the ratio of accretionary flux to erosional efficiency to a power dictated by the erosion process. Remarkably, we show that for most combinations of parameters in the erosion law, rock uplift rate is more strongly controlled by erosional efficiency than it is by the accretionary flux. Further, assuming frontal accretion with no underplating, the spatial distribution of erosional efficiency dictates the relative rock uplift rates on the pro‐wedge and retro‐wedge and the time‐averaged trajectories of rocks through the orogen. The restriction to invariant frictional properties is conservative in these respects; systems subject to positive feedback between erosion and rheology will exhibit even stronger coupling among climate, erosion, and deformation than shown here.

Rights Eroding by Past Breach

Author(s): Ben-Shahar, Omri | Abstract: Legal rights may erode as a result of past, uncontested, breach. In light of ongoing violations, the rightholder's enforcement may result in the loss of the entitlement. The doctrines, of course, of performance in contract law and adverse possession in property law are prominent examples of this widespread erosion phenomenon. In analyzing the effects of such laws, the Article confronts two conflicting intuitions. On the one hand, the license to continue breach prospectively encourages opportunism. On the other hand, the risk of erosion may reinforce the rightholder's motivation to take anti-erosion measures, bolstering the credibility of the threat to enforce, thus better preserving the entitlement. The article proves that these two effects of erosion rules always balance out. The same amount of value will be extracted from the right holder irrespective of the law's erosion doctrine. The article also demonstrates the limits of this irrelevance claim and the factors that may lead to its collapse. It applies the analysis to offer new perspectives on various prominent legal rules.

Self-organized critical drainage networks.

We introduce time-dependent boundary conditions in a model of drainage network evolution based on local erosion rules. The changing boundary conditions prevent the model from becoming stationary; it approaches a state where fluctuations of all sizes occur. The fluctuations in the sizes of the drainage areas show power law behavior with an exponent that differs significantly from that of the static distribution of the drainage areas. Thus, the model exhibits self-organized criticality and proposes a novel concept for predicting fractal properties of drainage networks.

Eroding Entitlements as Litigation Commitment

This Comment on Omri Ben-Shahar's excellent article makes two points: 1. Entitlements might be more valuable under an erosion rule than under a no-erosion rule. 2. In contractual settings, any observed irrelevance is likely to be the product of forces that are independent of Omri BenShahar's 'itigation effect. But before discussing these two issues, let me briefly reformulate Ben-Shahar's contribution.' First, erosion rules can give rise to an antierosion effect because erosion rules are a litigation commitment device that helps entitlement holders commit to prosecuting a broader class of opportunistic breaches. In BenShahar's model, erosion is caused by promisors who intentionally chisel on their duties, knowing, because of the costs of litigation, that the promisees cannot credibly threaten to sue. Second, Ben-Shahar's irrelevance claim is both stronger and weaker than he acknowledges. The irrelevance claim is stronger, because his model shows not just that erosion costs will be the same under different erosion rules, but that we know that these costs will always be exactly equal to litigation costs saved. The promisors will always chisel to the point where the promisee is almost indifferent between paying litigation costs and swallowing the erosion costs. The irrelevance claim is weaker than Ben-Shahar suggests because, while total erosion costs stay the same, different erosion rules produce different profiles (or cash flows) of erosion. The model predicts that no-erosion rules will produce constant amounts of erosion in each period, but that erosion rules will produce increasing amounts of erosion in each period (less erosion than under a no-erosion rule in early periods and more erosion than under a no-erosion rule in later periods). These different erosion profiles predicted by the model provide an important

The erosion of rights by past breach

Legal rights may erode as a result of past, uncontested, breach. In light of ongoing violations, the rightholder's lackluster enforcement may result in the loss of the entitlement. The doctrines of course of performance in contract law and adverse possession in property law are prominent examples of this widespread erosion phenomenon. In analyzing the effects of such laws, the article confronts two conflicting intuitions. On the one hand, the 'licence' to continue breach prospectively encourages opportunism. On the other hand, the risk of erosion may reinforce the rightholder's motivation to take antierosion measures, bolstering the credibility of the threat to enforce, thus better preserving the entitlement. The article proves that these two effects of erosion rules always balance out. The same amount of value will be extracted from the rightholder, irrespective of the law's erosion doctrine. The article also demonstrates the limits of this 'irrelevance' claim and the factors that may lead to its collapse. It applies the analysis to offer new perspectives on various prominent legal rules.

Mechanism for global optimization of river networks from local erosion rules.

We show that landscapes and their drainage networks evolve under erosion rules to obey a variational principle. Starting from hydrodynamics we first find by separation of variables a general relationship between erosion and a modified slope-area law. This law encompasses previous work and observations, and we confirm it by simulation. Secondly we show that this corresponds to a solution of a variational minimization model, generalized from Rodriguez-Iturbe et al. Thus a mechanism by which the local process of erosion acts to globally optimize a river network is produced.

Catalytic interface erosion.

We study interface erosion processes: catalytic erosions. We present two cases. (1) The erosion of a completely occupied lattice by one single moving particle starting from somewhere inside the lattice, considering deterministic as well as probabilistic erosion rules. In the latter case, the eroded regions appear to have interfaces with continuously tunable fractal dimensions. (2) The kinetic roughening of an initially flat surface, where ballistic or diffusion-limited particles, which remain intact themselves, erode the surface coming from the outside, using the same erosion rules as in (1). Many features resembling realistic interfaces, for example, islands and inlets, are generated. The dependence of the surface width on the system size is due to both the erosion mechanism and the way particles move before reaching the surface.