scholarly journals Steady state, continuity, and the erosion of layered rocks

2016 ◽  
Author(s):  
Matija Perne ◽  
Matthew D. Covington ◽  
Evan A. Thaler ◽  
Joseph M. Myre
Keyword(s):  
Steady State ◽  
Inverse Function ◽  
General Description ◽  
Mathematical Tool ◽  
Uplift Rate ◽  
Stream Power ◽  
Direction Parallel ◽  
Base Level ◽  
Bedrock Channel ◽  
Special Case

Abstract. Considerations of landscape steady state have substantially informed our understanding of the relationships between landscapes, tectonics, climate, and lithology. Topographic steady state, where topography is fixed in time, is a particularly important tool in the interpretation of landscape features, such as bedrock channel profiles, within a context of uplift patterns and rock strength. However, topographic steady state cannot strictly be attained in a landscape with layered rocks with non-vertical contacts. Using a combination of analytical solutions, stream erosion simulations, and full landscape evolution simulations, we show that an assumption of channel continuity, where channel retreat rates in the direction parallel to a contact are equal above and below the contact, provides a more general description of steady state landscapes in layered rocks. Topographic steady state is a special case of the steady state derived from continuity. Contrary to prior work, continuity predicts that channels will be steeper in weaker rocks in the case of subhorizontal rock layers when the stream power erosion exponent n < 1. For subhorizontal layered rocks with different erodibilities, continuity also predicts larger slope contrasts than would be predicted by topographic steady state. Continuity steady state is a type of flux steady state, where uplift is balanced on average by erosion. If uplift rate is constant, continuity steady state is perturbed near base level. These perturbations decay over a length scale that is an inverse function of the contrast in erodibility, such that erodibility contrasts of more than approximately a factor of three lead to rapid decay. Though examples explored here utilze the stream power erosion law, continuity steady state provides a general mathematical tool that can be used to explore the development of landscapes in layered rocks using any erosion model.

scholarly journals Steady state, erosional continuity, and the topography of landscapes developed in layered rocks

2017 ◽  
Vol 5 (1) ◽  
pp. 85-100 ◽  
Author(s):  
Matija Perne ◽  
Matthew D. Covington ◽  
Evan A. Thaler ◽  
Joseph M. Myre
Keyword(s):  
Steady State ◽  
Vertical Direction ◽  
Implicit Assumption ◽  
Mathematical Tool ◽  
Stream Power ◽  
Direction Parallel ◽  
Erosion Model ◽  
Erosion Rates ◽  
Base Level ◽  
Erosion Processes

Abstract. The concept of topographic steady state has substantially informed our understanding of the relationships between landscapes, tectonics, climate, and lithology. In topographic steady state, erosion rates are equal everywhere, and steepness adjusts to enable equal erosion rates in rocks of different strengths. This conceptual model makes an implicit assumption of vertical contacts between different rock types. Here we hypothesize that landscapes in layered rocks will be driven toward a state of erosional continuity, where retreat rates on either side of a contact are equal in a direction parallel to the contact rather than in the vertical direction. For vertical contacts, erosional continuity is the same as topographic steady state, whereas for horizontal contacts it is equivalent to equal rates of horizontal retreat on either side of a rock contact. Using analytical solutions and numerical simulations, we show that erosional continuity predicts the form of flux steady-state landscapes that develop in simulations with horizontally layered rocks. For stream power erosion, the nature of continuity steady state depends on the exponent, n, in the erosion model. For n = 1, the landscape cannot maintain continuity. For cases where n ≠ 1, continuity is maintained, and steepness is a function of erodibility that is predicted by the theory. The landscape in continuity steady state can be quite different from that predicted by topographic steady state. For n < 1 continuity predicts that channels incising subhorizontal layers will be steeper in the weaker rock layers. For subhorizontal layered rocks with different erodibilities, continuity also predicts larger slope contrasts than in topographic steady state. Therefore, the relationship between steepness and erodibility within a sequence of layered rocks is a function of contact dip. For the subhorizontal limit, the history of layers exposed at base level also influences the steepness–erodibility relationship. If uplift rate is constant, continuity steady state is perturbed near base level, but these perturbations decay rapidly if there is a substantial contrast in erodibility. Though examples explored here utilize the stream power erosion model, continuity steady state provides a general mathematical tool that may also be useful to understand landscapes that develop by other erosion processes.

scholarly journals Landscape evolution models using the stream power incision model show unrealistic behavior when <i>m</i> ∕ <i>n</i> equals 0.5

2017 ◽  
Vol 5 (4) ◽  
pp. 807-820 ◽  
Author(s):  
Jeffrey S. Kwang ◽  
Gary Parker
Keyword(s):  
Steady State ◽  
Landscape Evolution ◽  
Drainage Area ◽  
Uplift Rate ◽  
Stream Power ◽  
River Incision ◽  
Vertical Incision ◽  
Small Scales ◽  
Insight Into

Abstract. Landscape evolution models often utilize the stream power incision model to simulate river incision: E = KAmSn, where E is the vertical incision rate, K is the erodibility constant, A is the upstream drainage area, S is the channel gradient, and m and n are exponents. This simple but useful law has been employed with an imposed rock uplift rate to gain insight into steady-state landscapes. The most common choice of exponents satisfies m ∕ n = 0.5. Yet all models have limitations. Here, we show that when hillslope diffusion (which operates only on small scales) is neglected, the choice m ∕ n = 0.5 yields a curiously unrealistic result: the predicted landscape is invariant to horizontal stretching. That is, the steady-state landscape for a 10 km2 horizontal domain can be stretched so that it is identical to the corresponding landscape for a 1000 km2 domain.

scholarly journals Coupling slope–area analysis, integral approach and statistic tests to steady-state bedrock river profile analysis

2017 ◽  
Vol 5 (1) ◽  
pp. 145-160 ◽  
Author(s):  
Yizhou Wang ◽  
Huiping Zhang ◽  
Dewen Zheng ◽  
Jingxing Yu ◽  
Jianzhang Pang ◽  
...  
Keyword(s):  
Steady State ◽  
Linear Regression ◽  
Profile Analysis ◽  
Integral Approach ◽  
Alluvial Channels ◽  
Stream Power ◽  
Bedrock Channel ◽  
Coupled Process ◽  
River Profiles ◽  
Channel Steepness

Abstract. Slope–area analysis and the integral approach have both been widely used in stream profile analysis. The former is better at identifying changes in concavity indices but produces stream power parameters with high uncertainties relative to the integral approach. The latter is much better for calculating channel steepness. Limited work has been done to couple the advantages of the two methods and to remedy such drawbacks. Here we show the merit of the log-transformed slope–area plot to determine changes in concavities and then to identify colluvial, bedrock and alluvial channels along river profiles. Via the integral approach, we obtain bedrock channel concavity and steepness with high precision. In addition, we run bivariant linear regression statistic tests for the two methods to examine and eliminate serially correlated residuals because they may bias both the estimated value and the precision of stream power parameters. We finally suggest that the coupled process, integrating the advantages of both slope–area analysis and the integral approach, can be a more robust and capable method for bedrock river profile analysis.

scholarly journals Landscape evolution models using the stream power incision model show unrealistic behavior when <i>m/n</i> equals 0.5

2017 ◽  
Author(s):  
Jeffrey S. Kwang ◽  
Gary Parker
Keyword(s):  
Steady State ◽  
Landscape Evolution ◽  
Drainage Area ◽  
Uplift Rate ◽  
Stream Power ◽  
Channel Network ◽  
River Incision ◽  
Vertical Incision ◽  
Small Scales ◽  
Insight Into

Abstract. Landscape evolution models often utilize the stream power incision model to simulate river incision: E = KAmSn, where E = vertical incision rate, K = erodibility constant, A =  upstream drainage area, S = channel gradient, and m and n are exponents. This simple but useful law has been employed with an imposed rock uplift rate to gain insight into steady-state landscapes. The most common choice of exponents satisfies m/n = 0.5; indeed, this ratio has been deemed to yield the “optimal channel network.” Yet all models have limitations. Here, we show that when hillslope diffusion (which operates only at small scales) is neglected, the choice m/n = 0.5 yields a curiously unrealistic result: the predicted landscape is invariant to horizontal stretching. That is, the steady-state landscape for a 1 m2 horizontal domain can be stretched so that it is identical to the corresponding landscape for a 100 km2 domain.

scholarly journals A probabilistic framework for the cover effect in bedrock erosion

2016 ◽  
Author(s):  
Jens M. Turowski ◽  
Rebecca Hodge
Keyword(s):  
Time Scales ◽  
Channel Morphology ◽  
Scale Model ◽  
Probabilistic Framework ◽  
Channel Dynamics ◽  
Base Level ◽  
Bedrock Channel ◽  
Bedrock Erosion ◽  
Long Time ◽  
Bedrock Channels

Abstract. The cover effect in fluvial bedrock erosion is a major control on bedrock channel morphology and long-term channel dynamics. Here, we suggest a probabilistic framework for the description of the cover effect that can be applied to field, laboratory and modelling data and thus allows the comparison of results from different sources. The framework describes the formation of sediment cover as a function of the probability of sediment being deposited on already alleviated areas of the bed. We define benchmark cases and suggest physical interpretations of deviations from these benchmarks. Furthermore, we develop a reach-scale model for sediment transfer in a bedrock channel and use it to clarify the relations between the sediment mass residing on the bed, the exposed bedrock fraction and the transport stage. We derive system time scales and investigate cover response to cyclic perturbations. The model predicts that bedrock channels achieve grade in steady state by adjusting bed cover. Thus, bedrock channels have at least two characteristic time scales of response. Over short time scales, the degree of bed cover is adjusted such that they can just transport the supplied sediment load, while over long time scales, channel morphology evolves such that the bedrock incision rate matches the tectonic uplift or base level lowering rate.

Transient-Phase and Steady-State Kinetics for Enzyme Activation

1973 ◽  
Vol 51 (6) ◽  
pp. 806-814 ◽  
Author(s):  
Nasrat H. Hijazi ◽  
Keith J. Laidler
Keyword(s):  
Steady State ◽  
Kinetic Data ◽  
Enzyme Activation ◽  
Transient Phase ◽  
Time Curve ◽  
Substrate Complex ◽  
Enzyme Substrate ◽  
Substrate Activation ◽  
Steady State Kinetics ◽  
Special Case

A non-steady-state analysis has been worked out for two mechanisms in which an activator Q can become attached to an enzyme–substrate complex EA, the species EAQ breaking down more rapidly than EA. It is shown that if EAQ breaks down into EQ + product there can be no steady state. If, however, EAQ breaks down into E + Q + product, the transient phase is followed by a steady state in which the product versus time curve is linear. A special case of this mechanism is when Q is the substrate (substrate activation). Some published kinetic data on carboxypeptidase are analyzed with reference to the equations derived.

scholarly journals Glacial limitation of tropical mountain height

2019 ◽  
Vol 7 (1) ◽  
pp. 147-169 ◽  
Author(s):  
Maxwell T. Cunningham ◽  
Colin P. Stark ◽  
Michael R. Kaplan ◽  
Joerg M. Schaefer
Keyword(s):  
Steady State ◽  
Mountain Range ◽  
Equilibrium Line Altitude ◽  
Glacial Erosion ◽  
Tropical Mountains ◽  
Base Level ◽  
Snow Line ◽  
Bedrock River Incision ◽  
High Tropical Mountains ◽  
The Tropics

Abstract. Absent glacial erosion, mountain range height is limited by the rate of bedrock river incision and is thought to asymptote to a steady-state elevation as erosion and rock uplift rates converge. For glaciated mountains, there is evidence that range height is limited by glacial erosion rates, which vary cyclically with glaciations. The strongest evidence for glacial limitation is at midlatitudes, where range-scale hypsometric maxima (modal elevations) lie within the bounds of Late Pleistocene snow line variation. In the tropics, where mountain glaciation is sparse, range elevation is generally considered to be fluvially limited and glacial limitation is discounted. Here we present topographic evidence to the contrary. By applying both old and new methods of hypsometric analysis to high mountains in the tropics, we show that (a) the majority are subject to glacial erosion linked to a perched base level set by the snow line or equilibrium line altitude (ELA) and (b) many truncate through glacial erosion towards the cold-phase ELA. Evaluation of the hypsometric analyses at two field sites where glacial limitation is seemingly marginal reveals how glaciofluvial processes act in tandem to accelerate erosion near the cold-phase ELA during warm phases and to reduce their preservation potential. We conclude that glacial erosion truncates high tropical mountains on a cyclic basis: zones of glacial erosion expand during cold periods and contract during warm periods as fluvially driven escarpments encroach and destroy evidence of glacial action. The inherent disequilibrium of this glaciofluvial limitation complicates the concept of time-averaged erosional steady state, making it meaningful only on long timescales far exceeding the interval between major glaciations.

The Gru¨neisen Parameter in Fluids

1984 ◽  
Vol 106 (2) ◽  
pp. 193-200 ◽  
Author(s):  
V. Arp ◽  
J. M. Persichetti ◽  
Guo-bang Chen
Keyword(s):  
Steady State ◽  
Single Phase ◽  
Equations Of State ◽  
Liquid Structure ◽  
Limited Range ◽  
Phase Flow ◽  
Single Phase Flow ◽  
Internal Pressures ◽  
Special Case ◽  
Fluid Hydrodynamics

The Gru¨neisen parameter has long been used in equations of state for solids to relate thermodynamic properties to lattice vibrational spectra [1]. A few papers have extended the concept to studies of liquid structure. Knopoff and Shapiro [2] have evaluated a Gru¨neisen parameter for water and for mercury, attempting to relate its temperature dependence in a limited range to atomic clustering within the liquid. Sharma [3], in a series of papers, has evaluated a pseudo-Gru¨neisen parameter in mercury and liquefied gases and related it to internal pressures, a solubility parameter, and clustering phenomena. In this paper we evaluate the Gru¨neisen parameter for a variety of fluids, and show how it occurs in many problems in compressible fluid hydrodynamics, without reference to concepts of liquid structure. The work extends that reported in an earlier paper for the special case of steady state, single phase flow [4].

Evaluating the effect of variable lithologies on rates of knickpoint migration in the Wutach catchment, southern Germany

2020 ◽  
Author(s):  
Andreas Ludwig ◽  
Wolfgang Schwanghart ◽  
Florian Kober ◽  
Angela Landgraf
Keyword(s):  
Transient Response ◽  
Stream Power ◽  
Main Trunk ◽  
Southern Germany ◽  
Level Change ◽  
Base Level ◽  
Bulk Parameter ◽  
River Profiles ◽  
Topographic Evolution ◽  
Headward Erosion

&lt;p&gt;The topographic evolution of landscapes strongly depends on the resistance of bedrock to erosion. Detachment-limited fluvial landscapes are commonly analyzed and modelled with the stream power incision model (SPIM) which parametrizes erosional efficiency by the bulk parameter K whose value is largely determined by bedrock erodibility. Inversion of the SPIM using longitudinal river profiles enables resolving values of K if histories of rock-uplift or base level change are known. Here, we present an approach to estimate K-values for the Wutach catchment, southern Germany. The catchment is a prominent example of river piracy that occurred ~18 ka ago as response to headward erosion of a tributary to the Rhine. Base level fall of up to 170 m triggered a wave of upstream migrating knickpoints that represent markers for the transient response of the landscape. Knickpoint migration along the main trunk stream and its tributaries passed different lithological settings, which allows us to estimate K for crystalline and sedimentary bedrock units of variable erodibility.&lt;/p&gt;

Sign in / Sign up

Export Citation Format

Share Document