Fundamentals of erosion

Banner image is a photo of Dettifoss in NW Iceland (courtesy Gaia Stucky de Quay).

Understanding how landscapes are generated from the field-scale to continents

At small scales the erosional process is often extremely complicated. There exists an apparent disagreement between an assertion that long wavelength uplift dictates most of the shapes of fluvial landscapes and many geomorphological studies that highlight the importance of erosional complexity at shorter wavelengths. For example, our work measuring the evolution of Europe's 'most powerful' waterfall, Dettifoss in Iceland, shows that erosion, at scales up to kilometres and tens of thousands of years, can be controlled by the strength of substrate and maximum annual discharge (Stucky de Quay et al., 2019). At large scales simplicity appears to emerge and simple erosional models can be used to fit large inventories of drainage data (e.g. Roberts & White, 2010; Rudge et al., 2015; Roberts et al., 2019; Roberts, 2021). Stochastic theory appears to provide the means to bridge the 'scale-gap' and incorporate physics into the evolution of landscape at large scales (Roberts & Wani, 2024).

Figure. (a)–(c) Photographs of the three largest waterfalls in the Jokulsargljufur canyon, NE Iceland. Locations are shown in panel (d). (d) Drone model of the canyon draped by orthomosaic imagery (2x vertical exaggeration). Coloured surfaces = mapped terraces. Model is 5 km long in field of view. Blue arrow indicates flow direction. (e) Elevation of canyon with key geological/geomorphological observations and sample locations: S = Selfoss, D = Dettifoss, H = Hafragilsfoss, FI = fissure, SC = scoria cone. White circles = samples collected in this study used to calculate waterfall retreat rates (see Stucky de Quay et al., 2019).

Wavelet Power Spectral Analysis of Landscapes

In an attempt to unify the different views of landscape evolution we have developed wavelet spectral techniques to map the distribution of signal power from drainage patterns across the scales of interest. This work shows that nearly all of the power (> 90%) of large rivers (e.g. Niger, Orange, Columbia, Colorado) resides at wavelengths > 100 km, where their longitudinal profiles have self-similar scaling best characterised as red noise. At shorter wavelengths there is a transition to spectra best described as pink and perhaps blue noise. These observations indicate that river shapes and their evolution can be thought of as systems that have simple large signals that emerge through local complexity. Whilst some landscape evolution models (e.g. those employing a stream power law and erosional 'diffusivity') appear to be able to generate the red noise parts of river profiles they are less good at generating pink noise, which instead is attributable to local changes in, e.g., substrate (lithology; Wapenhans et al., 2021). This work indicates that river profile evolution is scale dependent.

Figure: Wavelet spectral analyses of the longitudinal profile of the Niger river (grey curve in panel a). Labelled arrows = man-made dams. Red solid/dashed curves = inverse wavelet transforms for low pass filters in which wavelengths < 100 or 1000 km have been removed. (b) Wavelet power spectrum of Niger river. Solid/dashed black lines show 100 and 1000 km wavelengths (see Roberts et al., 2019; Roberts 2019; Wapenhans et al., 2021; for details).

A stochastic theory of fluvial erosion

A general challenge is development of a theory of landscape evolution that embraces scale-dependent insights. We do so by incorporating randomness and probability into a theory of fluvial erosion. We have explored the use of stochastic differential equations of the Langevin type, and the Fokker-Planck equation, for predicting migration of erosional fronts. We are also developing analytical approaches, incorporating distributions of driving forces, critical thresholds and associated proxies. Finally, we have introduced a linear programming approach that, at its core, treats evolution of longitudinal profiles as a Markovian stochastic problem. The theory was developed essentially from first principles and incorporates physics governing fluvial erosion. Predictions of this theory, including the natural growth of discontinuities and scale-dependent evolution, including local complexity and emergent simplicity have been explored in Roberts & Wani (2024).

Figure. Demonstration of local erosional complexity and emergent simplicity for an evolving theoretical river profile subject to stochastic forcing. (a-b) Examples of inserted forces (grey), associated expected values and variance (solid and dashed), critical threshold (red), and resultant analytical probabilities of erosion (thick black). (c) Example of profile evolution in one simulation. Grey and black curves = profiles every 50 and 200 time steps, respectively. (d) Zoom into panel c. Black outlined bars = river profile at time step 600. (e) Zoom into a separate simulation that has different random driving forces (but same distribution). Black outlined bars = time step 600. Note differences between evolution and resultant profiles in panels d and e. (f) ‘Fuzzy’ black curves = profiles at 200, 400, 600 and 800 time steps (e.g. black curves in panel c) from 10 simulations with different random driving forces (but same distribution). Straight line = starting condition. Circles = expected displacements of erosional front originating at the mouth of the ‘river’ at (1000, 0); colors = time steps (see panel c for scale); calculated variance is smaller than symbol size. Small grey square centred at (190, 250) = position of zoomed-in region shown in panels d, e and g. (g) Calculated positions of longitudinal profiles at time step 600 for the 10 simulations shown in panel f. Colors and line widths simply indicate profiles from different simulations for clarity. Note their variability.

Erosional Thresholds and the Emergence of Simplicity in Eroding Landscapes

I have been exploring how simplicity could emerge despite local complexity using simple threshold-based models of erosion (see Roberts, 2021, for more details).

I think this work is a step towards unifying process-based models of landscape evolution that emphasis local complexity and phenomenological approaches (e.g. inverse modelling of drainage networks) necessary for understanding landscape evolution at larger scales. In short, it appears that landscape evolution can be extremely dynamic and complicated ('shocky'; non-linear) at small scales, whilst being relatively simple and broadly deterministic at large scales.

Top: Fluvial erosion at large scales from stream power model (a-c), and scale dependent river profile evolution from simple physics-based models (d-i). Boxes in panels h & i indicate position of panels g & h, respectively. Note emergent simplicity at large scales.

Left: Comparison of predicted river profile shapes from stream power (large-scale simplicity) and block toppling (locally complex) models.

Rules of thumb:

Landscape response times

We have developed various rules of thumb for estimating landscape response to changes in uplift (or base-level). This example for Austalia from Czarnota et al. (2014) shows a look-up chart for estimating timing of uplift events from knickzone distances along river profiles. Amplitude of event can be estimated from knickzone height.

Figure. (a) River length plotted as a function of knickzone distance from mouth. Numbered lines = uplift event isochrons for a given river length-knickzone distance pair. (b) Squares = knickzone estimate for Snowy River is shown. Circles = knickzone estimates of uplift timing for rivers east of Great Divide; color = latitude at head of river. Notice presence of youthful knickzones between 50 and 15 Ma.

Another example of estimating landscape response times at continental scales is shown in this map of Madagascar from Roberts et al. (2012). It is generated by integrating velocities (1/v) of knickzone retreat rates with respect to distance upstream. As far as I am aware, this was the first example of estimating landscape response at large scales. Other examples from us exist for the Americas, Africa, Australia and Eurasia. Various other groups now do something similar via 'chi' analyses. A caution is that uplift inserted upstream changes the distribution of landscape response times. My view is that inverting for uplift as a function of space and time is crucial for understanding landscape evolution.

Figure. (a) Landscape response time of Madagascar for knickzone created at coastline. Contours every 2 Ma. Gray shading = slope between 10–25; black shading = slope > 25 . Highest slopes occur within the central and northern highlands. (b) Enlarged section of east coast. High slopes between isochrons of 15–10 Ma (orange to yellow contours) record effects of headward migration of base-level change.

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