25 Years of Self-Organized Criticality: Solar and Astrophysics

Shortly after the seminal paper {\sl "Self-Organized Criticality: An explanation of 1/f noise"} by Bak, Tang, and Wiesenfeld (1987), the idea has been applied to solar physics, in {\sl "Avalanches and the Distribution of Solar Flares"} by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.Comment: 139 pages, 28 figures, Review based on ISSI workshops "Self-Organized Criticality and Turbulence" (2012, 2013, Bern, Switzerland

The Influence of Sediment Transport on Stationary and Mobile Knickpoints in River Profiles

The analysis of stationary and mobile knickpoints has been one of the most powerful concepts in tectonic geomorphology for decades. While several models combining bedrock incision and sediment transport are available, the existing theory of knickpoints focuses on detachment‐limited erosion without consideration of sediment transport. On the other hand, fully transport‐limited systems do not maintain distinct knickpoints. This study addresses the existence and the properties of slope‐break knickpoints in fluvial systems between the two extremes with the help of the shared stream‐power model, which allows for a continuous shift between detachment‐limited and transport‐limited erosion. The most important result is that both stationary and mobile knickpoints remain sharp and are not smeared by sediment transport. The system responds to tectonic signals in two phases where the first phase is similar to the detachment‐limited model. Comparing rivers with the same equilibrium profile at a given uplift rate, the velocity of knickpoint migration increases toward transport‐limited conditions. The second phase of the response is dominated by changes in sediment flux and affects the entire river segment downstream of the mobile knickpoint including the fault where the tectonic signal was initiated. During this phase, the mobile knickpoint is flattened in the sense that the difference in steepness index across a slope‐break knickpoint decreases through time. It is also shown that large parts of the theory for the nonlinear model, in particular knickpoint sharpening and stretching, remain valid, but the effect of nonlinearity decreases with increasing contribution of sediment transport.Plain Language Summary: Knickpoints are either breaks in slope along a river or even almost vertical steps, or waterfalls. While knickpoints can be generated by a variety of phenomena, sudden changes in tectonics or climate typically generate knickpoints that are moving upstream over geological time scales. If preserved, such mobile knickpoints provide an archive of the tectonic or climatic history. The existing theory of knickpoints is based on a simple erosion model where the effect of sediment transport is neglected. This study extends the theory by the effect of sediment transport. As a main result, the occurrence of knickpoints is not affected by sediment transport, but their properties change. So we cannot conclude from the occurrence of mobile knickpoints that the effect of sediment transport on the evolution of the respective river segment is negligible. When analyzing a given river segment quantitatively, neglecting sediment transport can result in misinterpretations. In particular, the velocity of knickpoint migration will be underestimated. This also implies that a given river segment carries information over a shorter time span than usually assumed.Key Points: The simple stream power incision model extended by sediment transport does not predict knickpoint smearing in river profiles. Neglecting sediment transport in analyzing a given river profile underestimates the velocity of knickpoint migration. Stationary tectonic knickpoints are not immediately steady if sediment transport is taken into account

A Physical Statistical Approach to Erosion

Erosion is a complex process consisting of many components, like surface runoff, impact of raindrops, wind forces, soil, and rock mechanics, etc. Trying to integrate all these processes into a physical model seems to be hopeless. In order to understand the variety of natural shapes and patterns produced by erosion we present an integrated statistical approach. Our model is based on simple physical constraints for the separation of amalgamated particles (abrasion) and for the movement of loose particles (denudation) and on the laws of statistics. After some simplifications we obtain a nonlinear system of partial differential equations that is solved using finite volume techniques. The model is suitable for the formation of different types of rill systems and the episodic behaviour of erosion processes, a kind of self-- organized criticality. Besides effects of inhomogenities, e. g., the formation of terraces can be investigated. Introduction Erosion has been subject of theoretical and ..

Size–frequency distribution of shallow landslides in the Black Forest, Germany

Landslides are a major hazard in mountainous regions, represent a threat to human life, and cause substantial economic costs. While some landslide hazard assessments, including hazards maps, are available for Germany, their spatial coverage is not uniform. In the Black Forest, several landslides have recently attracted public attention, but the landslide hazard in this region has received limited consideration in literature to date. This study focuses on the spatial pattern and size distribution of soil‐borne landslides in the submountainous valley of Menzenschwand. A combination of remote sensing, geophysical surveying, and geotechnical testing was used to map and characterize the landslide inventory of two selected hillslopes. In the statistical analyses, we observe a larger proportion of small‐scale landslides with size below 100 m2 than usually reported in similar studies. This effect may be related to the low cohesion of the soil (glacial deposits). As a major result, a cutoff in the distribution at large landslide sizes compared with the expected power‐law tail of the distribution was found. While the maximum landslide size found in this study was about 1100 m2, the largest landslide should theoretically be at least 2500 m2 at 95% probability. The cutoff at large sizes is probably due to the limited soil thickness, where about 50% of the considered area has a depth to bedrock of 1 m or less. For the considered location, this result suggests that an increase in frequency and intensity of rainstorms should predominantly result in an increase of landslide frequency, but without increasing the size of the largest landslides. As a more general implication, the contribution of large landslides to the total hazard may be overestimated if soil thickness is not considered, in particular if the present‐day hazard is projected to future scenarios.The spatial pattern and size distribution of soil‐borne landslides in low mountain ranges investigated using a combination of remote sensing, geophysical surveying, and geotechnical testing reveal a cutoff in the distribution at large landslide sizes and possible overestimation of future scenarios

Morphometric Characterization of Longitudinal Striae on Martian Landslides and Impact Ejecta Blankets and Implications for the Formation Mechanism

Longitudinal striae are a shared characteristic of long run-out landslides and layered ejecta crater deposits. They appear to be a fundamental feature of disintegrated mass flows, but their formation and the required conditions are poorly understood. We evaluated their similarity using spectral analysis and assessed the possibility of a common formation mechanism. The topography of striae is scale-invariant in the form of a power law relationship of the power spectrum and the derived spectral exponent and amplitude factor, which are a measure for roughness, show similar correlations on both types of deposit. There is no correlation to geologic substrate units, latitude, or age. Parameter values are isotropic in horizontal direction for ejecta deposits and show a weak anisotropy for landslide deposits. Spectral parameter values of substrate topography match well with the values of the superposed deposit, which indicates that roughness is transferred from substrate to deposit surface during emplacement. Testing different geometric models, we find that a simple superposition of topography with a semideterministic, anisotropic pattern does not reproduce the patterns of our data. We find that phase locking of a surface with scale-invariant properties creates striae with fractal properties close to our natural data sets as well as recreating other morphological features that can form alongside striae. Although the transferal of substrate roughness cannot be fully conciliated with conventional flow models, we find that a model that combines advection with lateral diffusion accounts for the unidirectional preservation of phase information and is also consistent with the scale-invariance of striae

Tectonic geomorphology at small catchment sizes extensions of the stream-power approach and the method

Quantitative tectonic geomorphology hinges on the analysis of longitudinal river profiles. The model behind almost all approaches in this field originates from an empirical relationship between channel slope and catchment size, often substantiated in the form of the stream-power model for fluvial incision. Significant methodological progress was recently achieved by introducing the transform. It defines a nonlinear length coordinate in such a way that the inherent curvature of river profiles due to the increase of catchment sizes in the downstream direction is removed from the analysis. However, the limitation to large catchment sizes inherited from the stream-power approach for fluvial incision persists. As a consequence, only a small fraction of all nodes of a digital elevation model (DEM) can be used for the analysis. In this study we present and discuss some empirically derived extensions of the stream power law towards small catchment sizes in order to overcome this limitation. Beyond this, we introduce a simple method for estimating the adjustable parameters in the original method as well as in our extended approaches. As a main result, an approach originally suggested for debris flow channels seems to be the best approximation if both large and small catchment sizes are included in the same analysis.(VLID)214713

Earth Surface Dynamics / The destiny of orogen-parallel streams in the Eastern Alps : the SalzachEnns drainage system

The evolution of the drainage system in the Eastern Alps is inherently linked to different tectonic stages of the alpine orogeny. Crustal-scale faults imposed eastward-directed orogen-parallel flow on major rivers, whereas late orogenic surface uplift increased topographic gradients between the foreland and range and hence the vulnerability of such rivers to be captured. This leads to a situation in which major orogen-parallel alpine rivers such as the Salzach River and the Enns River are characterized by elongated eastwest-oriented catchments south of the proposed capture points, whereby almost the entire drainage area is located west of the capture point. To determine the current stability of drainage divides and to predict the potential direction of divide migration, we analysed their geometry at catchment, headwater and hillslope scale covering timescales from millions of years to the millennial scale. We employ mapping for different base levels, generalized swath profiles across drainage divides and Gilbert metrics a set of local topographic metrics quantifying the asymmetry of drainage divides at hillslope scale. Our results show that most drainage divides are asymmetric, with steeper channels west and flatter channels east of a common drainage divide. Interpreting these results, we propose that drainage divides migrate from west towards east so that the Inn catchment grows at the expense of the Salzach catchment and the Salzach catchment consumes the westernmost tributaries of the Mur and Enns catchments. Gilbert metrics across the SalzachEnns and SalzachMur divides are consistent with inferred divide mobility. We attribute the absence of divide asymmetry at the InnSalzach divide to glacial landforms such as cirques and U-shaped valleys, which suggest that Pleistocene climate modulations are able to locally obscure the large-scale signal of drainage network reorganization. We suggest that the eastward-directed divide migration progressively leads to symmetric catchment geometries, whereby tributaries west and east of the capture point eventually contribute equally to the drainage area. To test this assumption, we have reconstructed the proposed drainage network geometries for different time slices. mapping of these reconstructed drainage networks indicates a progressive stability of the network topology in the Eastern Alps towards the present-day situation.(VLID)473586

Quantification of the damming and sediment trapping capacity of landslides and their dammed lakes: the example of the Hintersee landslide dam

International audiencePerennial landslide dams interrupt the sediment connectivity of rivers. Although most landslide dams do not persist for more than a few days, those that do can exhibit significant sediment trapping capacity. While water can pass through or over the dam, the sediment load is trapped upstream of the dam until the dam breaks or gradually erodes, or is completely filled with deposits. The volume of sediment stored in this way can reach up to three times the volume of impounded water, as we find by back-analyzing the lake Hintersee in southeastern Germany. In this work, we reconstruct the pre-landslide topography using Petrel and then use the Gerris shallow-water flow solver with a Voellmy rheology to back-analyze this landslide-dammed lake in the Bavarian Alps. We test several landslide release scenarios and different landslide rheologies to obtain the best-fitting reconstruction of the dam topography. We then fill the landslide dam with water and sediment using simple slope algorithms and validate the results against the current topography. Finally, we compare the landslide deposit thicknesses, water depths, and trapped sediment thicknesses of our different scenarios in order to provide new insight into the damming and sediment trapping capacity of landslides