Gregory E. Tucker

Dynamics of the stream-power river incision model: Implications for height limits of mountain ranges, landscape response timescales, and research needs

The longitudinal profiles of bedrock channels are a major component of the relief structure of mountainous drainage basins and therefore limit the elevation of peaks and ridges. Further, bedrock channels communicate tectonic and climatic signals across the landscape, thus dictating, to first order, the dynamic response of mountainous landscapes to external forcings. We review and explore the stream-power erosion model in an effort to (1) elucidate its consequences in terms of large-scale topographic (fluvial) relief and its sensitivity to tectonic and climatic forcing, (2) derive a relationship for system response time to tectonic perturbations, (3) determine the sensitivity of model behavior to various model parameters, and (4) integrate the above to suggest useful guidelines for further study of bedrock channel systems and for future refinement of the streampower erosion law. Dimensional analysis reveals that the dynamic behavior of the stream-power erosion model is governed by a single nondimensional group that we term the uplift-erosion number, greatly reducing the number of variables that need to be considered in the sensitivity analysis. The degree of nonlinearity in the relationship between stream incision rate and channel gradient (slope exponent n) emerges as a fundamental unknown. The physics of the active erosion processes directly influence this nonlinearity, which is shown to dictate the relationship between the uplift-erosion number, the equilibrium stream channel gradient, and the total fluvial relief of mountain ranges. Similarly, the predicted response time to changes in rock uplift rate is shown to depend on climate, rock strength, and the magnitude of tectonic perturbation, with the slope exponent n controlling the degree of dependence on these various factors. For typical drainage basin geometries the response time is relatively insensitive to the size of the system. Work on the physics of bedrock erosion processes, their sensitivity to extreme floods, their transient responses to sudden changes in climate or uplift rate, and the scaling of local rock erosion studies to reach-scale modeling studies are most sorely needed.

Landscape response to tectonic forcing: Digital elevation model analysis of stream profiles in the Mendocino triple junction region, northern California

The topographic evolution of orogens is fundamentally dictated by rates and patterns of bedrock-channel incision. Quantitative field assessments of process-based laws are needed to accurately describe landscape uplift and denudation in response to tectonics and climate. We evaluate and calibrate the shear stress (or similar unit stream-power) bedrock-incision model by studying stream profiles in a tectonically active mountain range. Previous work on emergent marine terraces in the Mendocino triple junction region of northern California provides spatial and temporal control on rock-uplift rates. Digital elevation models and field data are used to quantify differences in landscape morphology associated with along-strike northwest to southeast changes in tectonic and climatic conditions. Analysis of longitudinal profiles supports the hypothesis that the study-area channels are in equilibrium with current uplift and climatic conditions, consistent with theoretical calculations of system response time based on the shear-stress model. Within uncertainty, the profile concavity (𝛉) of the trunk streams is constant throughout the study area (𝛉 ≈ 0.43), as predicted by the model. Channel steepness correlates with uplift rate. These data help constrain the two key unknown model parameters, the coefficient of erosion ( K ) and the exponent associated with channel gradient ( n ). This analysis shows that K cannot be treated as a constant throughout the study area, despite generally homogeneous substrate properties. For a reasonable range of slope-exponent values ( n ), best-fit values of K are positively correlated with uplift rate. This correlation has important implications for landscape-evolution models and likely reflects dynamic adjustment of K to tectonic changes, due to variations in orographic precipitation, and perhaps channel width, sediment load, and frequency of debris flows. The apparent variation in K makes a unique value of n impossible to constrain with present data.

Hillslope processes, drainage density, and landscape morphology

Catchment morphology and drainage density are strongly influenced by hillslope processes. The consequences of several different hillslope process laws are explored in a series of experiments with a numerical model of drainage basin evolution. Five different models are considered, including a simple diffusive-advective process transition, a runoff generation threshold, an erosion threshold, and two types of threshold-activated landsliding. These different hillslope processes alter both the visual appearance of the landscape and the predicted relationship between slope and contributing area. On the basis of the different threshold theories, we derive expressions for the relationships between drainage density and environmental factors such as rainfall, relief, and mean erosion rate. These relationships vary depending on the dominant hillslope threshold. In particular, the sign of the predicted relationship between drainage density and relief is positive in semiarid, low-relief landscapes and negative in humid landscapes dominated by a saturation threshold and/or in high-relief landscapes dominated by simple threshold landsliding.

Drainage basin responses to climate change

Recent investigations have shown that the extent of the channel network in some drainage basins is controlled by a threshold for overland flow erosion. The sensitivity of such basins to climate change is analyzed using a physically based model of drainage basin evolution. The GOLEM model simulates basin evolution under the action of weathering processes, hillslope transport, and fluvial bedrock erosion and sediment transport. Results from perturbation analyses reveal that the nature and timescale of basin response depends on the direction of change. An increase in runoff intensity (or a decrease in vegetation cover) will lead to a rapid expansion of the channel network, with the resulting increase in sediment supply initially generating aggradation along the main network, followed by downcutting as the sediment supply tapers off. By contrast, a decrease in runoff intensity (or an increase in the erosion threshold) will lead to a retraction of the active channel network and a much more gradual geomorphic response. Cyclic changes in runoff intensity are shown to produce aggradational-degradational cycles that resemble those observed in the field. Cyclic variations in runoff also lead to highly punctuated denudation rates, with denudation concentrated during periods of increasing runoff intensity and/or decreasing vegetation cover. The sediment yield from threshold-dominated basins may therefore exhibit significant variability in response to relatively subtle environmental changes, a finding which underscores the need for caution in interpreting modern sediment-yield data.

Erosional dynamics, flexural isostasy, and long‐lived escarpments: A numerical modeling study

Erosional escarpments are common features of high-elevation rifted continents. Fission track data suggest that these escarpments form by base level lowering and/or marginal uplift during rifting, followed by lateral retreat of an erosion front across tens to hundreds of kilometers. Previous modeling studies have shown that this characteristic pattern of denudation can have a profound impact upon marginal isostatic uplift and the evolution of offshore sedimentary basins. Yet at present there is only a rudimentary understanding of the geomorphic mechanisms capable of driving such prolonged escarpment retreat. In this study we present a nonlinear, two-dimensional landscape evolution model that is used to assess the necessary and sufficient conditions for long-term retreat of a rift-generated escarpment. The model represents topography as a grid of cells, with drainage networks evolving as water flows across the grid in the direction of steepest descent. The model accounts for sediment production by weathering, fluvial sediment transport, bedrock channel erosion, and hillslope sediment transport by diffusive mechanisms and by mass failure. Numerical experiments presented explore the effects of different combinations of erosion processes and of dynamic coupling between denudation and flexural isostatic uplift. Model results suggest that the necessary and sufficient conditions for long-term escarpment retreat are (1) incising bedrock channels in which the erosion rate increases with increasing drainage area, so that the channels steepen and propagate headward; (2) a low rate of sediment production relative to sediment transport efficiency, which promotes relief-generating processes over diffusive ones; (3) high continental elevation, which allows greater freedom for fluvial dissection; and (4) any process, including flexural isostatic uplift, that helps to maintain a drainage divide near an escarpment crest. Flexural isostatic uplift also facilitates escarpment retreat by elevating topography in the vicinity of an eroding escarpment, thereby increasing channel gradients and accelerating erosion which in turn generates additional isostatic uplift. Of all the above conditions, high continental elevation is common to most rift margin escarpments and may ultimately be the most important factor.

A stochastic approach to modeling the role of rainfall variability in drainage basin evolution

We develop a simple stochastic theory for erosion and sediment transport, based on the Poisson pulse rainfall model, in order to analyze how variability in rainfall and runoff influences drainage basin evolution. Two cases are considered: sediment transport by runoff in rills and channels and particle detachment from bedrock or cohesive soils. Analytical and numerical results show that under some circumstances, rainfall variability can have an impact equal to or greater than that of mean rainfall amount. The predicted sensitivity to rainfall variability is greatest when (1) thresholds for runoff generation and/or particle detachment are significant and/or (2) erosion and transport are strong nonlinear functions of discharge. In general, sediment transport capacity is predicted to increase with increasing rainfall variability. Depending on the degree of nonlinearity, particle detachment capacity may either increase or decrease with increasing rainfall variability. These findings underscore the critical importance of hydrogeomorphic thresholds and other sources of nonlinearity in process dynamics. The morphologic consequences of rainfall variability are illustrated by incorporating the pulse rainfall theory into a landscape simulation model. The simulation results imply that, all else being equal, catchments experiencing a shift toward greater climate variability will tend to have (1) higher erosion rates, (2) higher drainage density (because of increased runoff erosion efficiency), and ultimately (3) reduced relief. The stochastic approach provides a useful method for incorporating physically meaningful climate data within the current generation of landscape evolution models.

An object-oriented framework for distributed hydrologic and geomorphic modeling using triangulated irregular networks

We describe a newset of data structures and algorithms for dynamic terrain modeling using a triangulated irregular network (TINs). The framework provides an efficient method for storing, accessing, and updating a Delaunay triangulation and its associated Voronoi diagram. The basic data structure consists of three interconnected data objects: triangles, nodes, and directed edges. Encapsulating each of these geometric elements within a data object makes it possible to essentially decouple the TIN representation from the modeling applications that make use of it. Both the triangulation and its corresponding Voronoi diagram can be rapidly retrieved or updated, making these methods well suited to adaptive remeshing schemes. We develop a set of algorithms for defining drainage networks and identifying closed depressions (e.g., lakes) for hydrologic and geomorphic modeling applications. We also outline simple numerical algorithms for solving network routing and 2D transport equations within the TIN framework. The methods are illustrated with two example applications, a landscape evolution model and a distributed rainfall-runoff model. # 2001 Elsevier Science Ltd. All rights reserved.

The Channel-Hillslope Integrated Landscape Development Model (CHILD)

Numerical models of complex Earth systems serve two important purposes. First, they embody quantitative hypotheses about those systems and thus help researchers develop insight and generate testable predictions. Second, in a more pragmatic context, numerical models are often called upon as quantitative decision-support tools. In geomorphology, mathematical and numerical models provide a crucial link between small-scale, measurable processes and their long-term geomorphic implications. In recent years, several models have been developed that simulate the structure and evolution of three-dimensional fluvial terrain as a consequence of different process “laws” (e.g., Willgoose et al., 1991a; Beaumont et al., 1992; Chase, 1992; Anderson, 1994; Howard, 1994; Tucker and Slingerland, 1994; Moglen and Bras, 1995). By providing the much-needed connection between measurable processes and the dynamics of long-term landscape evolution that these processes drive, mathematical landscape models have posed challenging new hypotheses and have provided the guiding impetus behind new quantitative field studies and Digital Elevation Model (DEM) -based analyses of terrain (e.g., Snyder et al., 2000). The current generation of models, however, shares a number of important limitations. Most models rely on a highly simplified representation of drainage basin hydrology, treating climate through a simple “perpetual runoff” formulation.

Bedrock channel adjustment to tectonic forcing: Implications for predicting river incision rates

We present detailed data of channel morphology for a river undergoing a transient response to active normal faulting where excellent constraints exist on spatial and temporal variations in fault slip rates. We show that traditional hydraulic scaling laws break down in this situation, and that channel widths become decoupled from drainage area upstream of the fault. Unit stream powers are ∼4 times higher than those predicted by current scaling paradigms and imply that incision rates for rivers responding to active tectonics may be significantly higher than those heretofore modeled. The loss of hydraulic scaling cannot be explained by increasing channel roughness and is an intrinsic response to tectonic forcing. We show that channel aspect ratio is a strongly nonlinear function of local slope and demonstrate that fault-induced adjustment of channel geometries has reset hillslope gradients. The results give new insight into how rivers maintain their course in the face of tectonic uplift and illustrate the first-order control the fluvial system exerts on the locus and magnitude of sediment supply to basins.

Channel response to tectonic forcing: field analysis of stream morphology and hydrology in the Mendocino triple junction region, northern California

Abstract An empirical calibration of the shear stress model for bedrock incision is presented, using field and hydrologic data from a series of small, coastal drainage basins near the Mendocino triple junction in northern California. Previous work comparing basins from the high uplift zone (HUZ, uplift rates around 4 mm/year) to ones in the low uplift zone (LUZ, ∼0.5 mm/year) indicates that the HUZ channels are about twice as steep for a given drainage area. This observation suggests that incision processes are more effective in the HUZ. It motivates a detailed field study of channel morphology in the differing tectonic settings to test whether various factors that are hypothesized to influence incision rates (discharge, channel width, lithology, sediment load) change in response to uplift or otherwise differ between the HUZ and LUZ. Analysis of regional stream gaging data for mean annual discharge and individual floods yields a linear relationship between discharge and drainage area. Increased orographic precipitation in the HUZ accounts for about a twofold increase in discharge in this area, corresponding to an assumed increase in the erosional efficiency of the streams. Field measurements of channel width indicate a power-law relationship between width and drainage area with an exponent of ∼0.4 and no significant change in width between the uplift rate zones, although interpretation is hampered by a difference in land use between the zones. The HUZ channel width dataset reveals a scaling break interpreted to be the transition between colluvial- and fluvial-dominated incision processes. Assessments of lithologic resistance using a Schmidt hammer and joint surveys show that the rocks of the study area should be fairly similar in their susceptibility to erosion. The HUZ channels generally have more exposed bedrock than those in the LUZ, which is consistent with protection by sediment cover inhibiting incision in the LUZ. However, this difference is likely the result of a recent pulse of sediment due to land use in the LUZ. Therefore, the role of sediment flux in setting incision rates cannot be constrained with any certainty. To summarize, of the four response mechanisms analyzed, the only factor that demonstrably varies between uplift rate zones is discharge, although this change is likely insufficient to explain the relationship between channel slope and uplift rate. The calibrated model allows us to make a prediction of channel concavity that is consistent with a previous estimate from slope–drainage area data. We show that the inclusion of nonzero values of critical shear stress in the model has important implications for the theoretical relationship between steady-state slope and uplift rate and might provide an explanation for the observations. This analysis underscores the importance of further work to constrain quantitatively threshold shear stress for bedrock incision.