Watching rocks grow

Abstract

Nature abounds with beautiful and striking landscapes, but a comprehensive understanding of their forms requires examples where detailed comparisons can be made between theory and experiment. Geothermal hot springs 1 produce some of the most rapidly changing terrestrial landscapes, with reported travertine (calcium carbonate) growth rates as high as 5 mm per day 2-4 . Unlike most landscapes, the patterns of which are the result of erosion processes on timescales of millions of years, the hot-spring depositional landscapes exhibit a spectacular cascade of nested ponds and terraces 5 , for which the origins and quantitative characterization have remained elusive. Here, we take advantage of this millionfold difference in geological timescale to present a novel combination of data from time-lapse photography, computer simulation and mathematical modelling that explains the emergence of the large-scale pond and terrace patterns, predicts and verifies the dynamics of their growth and shows that these patterns are scale invariant.The dynamics of turbulent fluid flow coupled to the precipitation-driven growth of the travertine substrate was modelled on a discrete lattice of cells. Our cell dynamical system (CDS) is a set of rules, described in detail in the Methods section, that updates the lattice variables representing the heights of the landscape and fluid above each cell. The rules mimic fluid depositional dynamics and the influence of landscape features on the flow pattern 6,7 , enabling efficient computations of complex landscapes. Such a formalism complements analytic descriptions of carbonate precipitation patterns, such as our work on travertine domes 7 and recent studies of needle-like speleothem growth 8,9 . We were able to verify that our CDS model is quantitatively equivalent to the more conventional approach using differential equations, by using it to analyse the growth of single travertine domes 7 . The CDS enables long-time (28,000 steps), large-scale 600 × 500 cells) simulations of landscapes with arbitrary, complex structure. Statistical properties were computed by averaging over 130 independent simulations. The model was not 'tuned': as long as the parameters are not varied so much that the model ceases to make physical sense, it produces the same morphological and statistical results, showing that our findings are generic for this class of precipitation patterns. Figure 1 shows frames captured from a typical time-dependent simulation, initiated on a sloping plane with small initial roughness. Initial depositional instabilities grow to form dams, pond water pools behind the dams and complicated interactions arise as terraces grow and interact. The frames show a process of pond inundation, or 'drowning' . In this mechanism, which seems to be the dominant one through which larger ponds form, the lip of a downstream pond grows more rapidly than that of its upstream neighbour. Eventually the downstream lip becomes taller, inundating the upstream pond, and leaving a single large pond. Although we also obs...

Sculpting of Rocks by Reactive Fluids

“…The emergence of patterns in such systems is not trivial and intuition often fails us. Our group and the group of Nigel Goldenfeld at the University of Illinois at Urbana-Champaign have studied travertine terracing in detail using a fruitful combination of field observation, laboratory experiments and computer modelling (Jettestuen et al, 2006;Goldenfeld et al, 2006;Veysey and Goldenfeld 2008;Hammer et al, 2007;2008;2010). A particularly interesting aspect of this work was the way in which we attacked the problem at many different levels of abstraction, in order to answer several types of questions.…”

“…The model reproduces several peculiarities of travertine terrace growth observed in nature, such as downslope migration of steps, coarsening by merging and drowning of upstream steps and stretching of dams in the downslope direction. Goldenfeld et al (2006) and Veysey and Goldenfeld (2008) developed a somewhat similar model, with further simplification of the flow (using a discrete model for Stokes flow) but including several other processes such as surface tension and simplified chemistry and heat transport. These simulations also assumed a relationship between flow velocity and precipitation rate, producing very realistic terrace patterns.…”

“…4.2), the scaling behaviour was studied quantitatively by Veysey and Goldenfeld (2008). They found that the size of water filled pools can be represented by a power law where the number of dams with area, A, scales with A to the power of -2.2.…”

Travertine terracing: patterns and mechanisms

“…Veysey & Goldenfeld (2008) produced a movie based on a year-long data series from the Mammoth Hot Spring complex in Yellowstone National Park, showing progressive coarsening by pond inundation where the rim of a pool grows faster than the rim of the upstream pool, causing drowning of the upstream pool and the formation of a single large pool. Their movie also shows downstream migration of terraces.…”

“…This provides a positive feedback mechanism between hydrodynamics and precipitation on the rim, leading to localization of precipitation and the stabilization and growth of rims. Goldenfeld et al (2006) and Veysey & Goldenfeld (2008) developed a much more detailed cellular model, including simplified rules for water flow, surface tension, water chemistry and outgassing of CO 2 . Their precipitation rule included idealized terms incorporating oversaturation level, flux normal to the surface, and flow velocity.…”