Stefan Hergarten

25 Years of Self-Organized Criticality: Solar and Astrophysics

Shortly after the seminal paper “Self-Organized Criticality: An explanation of 1/fnoise” by Bak et al. (1987), the idea has been applied to solar physics, in “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.

Extension during continental convergence in the Eastern Alps: The influence of orogen-scale strike-slip faults

In the Miocene, the European Eastern Alps extruded laterally along orogen-scale strike-slip faults due to both extensional gravitational collapse and compressional tectonic forcing. Horizontal extension is most prominently evidenced by detachments east and west of the Tauern Window; it is commonly explained by a retreating slab beneath the Carpathian arc hundreds of kilometers east of the orogen. Horizontal compression is shown by north-south shortening in the Tauern Window and the entire Eastern Alps in response to the convergence of the Adriatic plate with Europe. It is interesting that analogue and numerical models for the Eastern Alps designed to describe the east-directed lateral extrusion have failed to explain the extensional regime in the region of the Tauern Window. Using a numerical model for plan-view deformation that considers internal faults, we show here that orogen-scale strike-slip faults are mechanically required to cause extension during plate convergence in the Miocene Eastern Alps. We test the idea by coupling this model with a landscape evolution model and by comparing modeled and observed drainage system geometries.

THERMAL HISTORY: A new software to interpret diffusive zoning profiles in garnet

Mineral grains can record the cooling history of metamorphic terrains by preserving characteristic chemical zoning profiles caused by diffusion. A range of analytical and numerical models have been used to describe the relationship between the cooling rate and the shape of chemical zoning profiles. Most of these models are characterized by a deficit of usability to external users. This problem is overcome by the code THERMAL HISTORY presented here. The code is platform independent and runs without compilation or a hard disk install. Model results are stored in a basic database and displayed graphically. The code is controlled by an intuitive graphical user interface and uses a very fast diffusion algorithm. THERMAL HISTORY can be used to model zoning profiles as a function of a series of cooling histories, and is written so that it is particularly applicable for the Fe-Mg exchange between garnet and biotite. The code takes into account mass balance so that the volumetric ratio of garnet and biotite can be considered explicitly and it provides a facility to calculate sections through the grains. As some of these facilities have never been published before, the impact of cooling histories, mass balance and section position is explained in some detail. As an application example, THERMAL HISTORY is used to demonstrate that highly non-linear cooling histories, small biotite-garnet ratios and the section effect may result in zoning profiles that are misleading if interpreted in terms of the cooling history.

Analytical approximations of discharge recessions for steeply sloping aquifers in alpine catchments

The validity and applicability of various methods to infer hydraulic properties of sloping aquifers in alpine settings using the power law relationship between the discharge recession and its first time derivative is explored. For this purpose, a synthetic spring catchment implemented in the numerical groundwater flow model MODFLOW as well as the example of a relict rock glacier in an alpine setting is examined. The various approaches are found to differ particularly in the late time domain, whereas most of them agree fairly well in the early time domain and at the transition point between the two time domains. As the early recession may be affected by uncertainties from inappropriate initial conditions, it is proposed to use the transition point for estimating aquifer thickness and transmissivity. Using only prolonged winter recessions in the analysis of the field data from the relict rock glacier yields estimates of aquifer thickness and hydraulic conductivity consistent with results from a geophysical survey and tracer tests, respectively. In the other seasons, the recession is frequently interrupted by minor recharge events, and using the lower envelope of the entire data is found to yield estimates that are too high in the given case. It is thus recommended to focus on the winter recession in the analysis of hydrograph data from alpine settings.

The topography of a continental indenter: The interplay between crustal deformation, erosion, and base level changes in the eastern Southern Alps

Abstract The topography of the eastern Southern Alps (ESA) reflects indenter tectonics causing crustal shortening, surface uplift, and erosional response. Fluvial drainages were perturbed by Pleistocene glaciations that locally excavated alpine valleys. The Late Miocene desiccation of the Mediterranean Sea and the uplift of the northern Molasse Basin led to significant base level changes in the far field of the ESA and the Eastern Alps (EA), respectively. Among this multitude of mechanisms, the processes that dominate the current topographic evolution of the ESA and the ESA‐EA drainage divide have not been identified. We demonstrate the expected topographic effects of each mechanism in a one‐dimensional model and compare them with observed channel metrics. We find that the normalized steepness index increases with uplift rate and declines from the indenter tip in the northwest to the foreland basin in the southeast. The number and amplitude of knickpoints and the distortion in longitudinal channel profiles similarly decrease toward the east. Changes in slope of χ‐transformed channel profiles coincide spatially with the Valsugana‐Fella fault linking crustal stacking and uplift induced by indenter tectonics with topographic evolution. Gradients in χ across the ESA‐EA drainage divide imply an ongoing, north directed shift of the Danube‐ESA watershed that is most likely driven by a base level rise in the northern Molasse basin. We conclude that the regional uplift pattern controls the geometry of ESA‐EA channels, while base level changes in the far field control the overall architecture of the orogen by drainage divide migration.

Ejecta thickness and structural rim uplift measurements of Martian impact craters: Implications for the rim formation of complex impact craters

The elevated rim in simple craters results from the structural uplift of preimpact target rocks and the deposition of a coherent proximal ejecta blanket at the outer edge of the transient cavity. Given the considerable, widening of the transient cavity during crater modification and ejecta thickness distributions, the cause of elevated crater rims in complex craters is less obvious. The thick, proximal ejecta in complex impact craters is deposited well inside the final crater rim and target thickening should rapidly diminish with increasing distance from the transient cavity rim. Our study of 10 complex Martian impact craters ranging from 8.2 to 53.0 km in diameter demonstrates that the mean structural rim uplift at the final crater rim makes 81% of the total rim elevation, while the mean ejecta thickness contributes 19%. Thus, the structural rim uplift seems to be the dominant factor to build up the total amount of the raised crater rim of complex craters. To measure the widening of the transient cavity during modification and the distance between the rim of the final crater and that of the transient cavity, we constructed balanced cross section restorations to estimate the transient cavity of nine complex Martian impact craters. The final crater radii are ~1.38–1.87 times the transient cavity radii. We propose that target uplift at the position of the final crater rim was established during the excavation stage.

Nonlinear Flow Process: A New Package to Compute Nonlinear Flow in MODFLOW

A new MODFLOW package (Nonlinear Flow Process; NLFP) simulating nonlinear flow following the Forchheimer equation was developed and implemented in MODLFOW-2005. The method is based on an iterative modification of the conductance calculated and used by MODFLOW to obtain an effective Forchheimer conductance. The package is compatible with the different layer types, boundary conditions, and solvers as well as the wetting capability of MODFLOW. The correct implementation is demonstrated using four different benchmark scenarios for which analytical solutions are available. A scenario considering transient flow in a more realistic setting and a larger model domain with a higher number of cells demonstrates that NLFP performs well under more complex conditions, although it converges moderately slower than the standard MODFLOW depending on the nonlinearity of flow. Thus, this new tool opens a field of opportunities to groundwater flow simulation with MODFLOW, especially for core sample simulation or vuggy karstified aquifers as well as for nonlinear flow in the vicinity of pumping wells.

A New Equation Solver for Modeling Turbulent Flow in Coupled Matrix-Conduit Flow Models.

Karst aquifers represent dual flow systems consisting of a highly conductive conduit system embedded in a less permeable rock matrix. Hybrid models iteratively coupling both flow systems generally consume much time, especially because of the nonlinearity of turbulent conduit flow. To reduce calculation times compared to those of existing approaches, a new iterative equation solver for the conduit system is developed based on an approximated Newton-Raphson expression and a Gauß-Seidel or successive over-relaxation scheme with a single iteration step at the innermost level. It is implemented and tested in the research code CAVE but should be easily adaptable to similar models such as the Conduit Flow Process for MODFLOW-2005. It substantially reduces the computational effort as demonstrated by steady-state benchmark scenarios as well as by transient karst genesis simulations. Water balance errors are found to be acceptable in most of the test cases. However, the performance and accuracy may deteriorate under unfavorable conditions such as sudden, strong changes of the flow field at some stages of the karst genesis simulations.