Slow closure of Earth's carbon cycle.

Carbon near the Earth's surface cycles between the production and consumption of organic carbon; the former sequesters carbon dioxide while the latter releases it. Microbes attempt to close the loop, but the longer organic matter survives, the slower microbial degradation becomes. This aging effect leaves observable quantitative signatures: Organic matter decays at rates that are inversely proportional to its age, while microbial populations and concentrations of organic carbon in ocean sediments decrease at distinct powers of age. Yet mechanisms that predict this collective organization remain unknown. Here, I show that these and other observations follow from the assumption that the decay of organic matter is limited by progressively rare extreme fluctuations in the energy available to microbes for decomposition. The theory successfully predicts not only observed scaling exponents but also a previously unobserved scaling regime that emerges when microbes subsist on the minimum energy flux required for survival. The resulting picture suggests that the carbon cycle's age-dependent dynamics are analogous to the slow approach to equilibrium in disordered systems. The impact of these slow dynamics is profound: They preclude complete oxidation of organic carbon in sediments, thereby freeing molecular oxygen to accumulate in the atmosphere.

Presence or absence of stabilizing Earth system feedbacks on different time scales.

The question of how Earth's climate is stabilized on geologic time scales is important for understanding Earth's history, long-term consequences of anthropogenic climate change, and planetary habitability. Here, we quantify the typical amplitude of past global temperature fluctuations on time scales from hundreds to tens of millions of years and use it to assess the presence or absence of long-term stabilizing feedbacks in the climate system. On time scales between 4 and 400 ka, fluctuations fail to grow with time scale, suggesting that stabilizing mechanisms like the hypothesized "weathering feedback" have exerted dominant control in this regime. Fluctuations grow on longer time scales, potentially due to tectonically or biologically driven changes that make weathering act as a climate forcing and a feedback. These slower fluctuations show no evidence of being damped, implying that chance may still have played a nonnegligible role in maintaining the long-term habitability of Earth.

Rate-induced collapse in evolutionary systems.

Recent work has highlighted the possibility of 'rate-induced tipping', in which a system undergoes an abrupt transition when a perturbation exceeds a critical rate of change. Here, we argue that this is widely applicable to evolutionary systems: collapse, or extinction, may occur when external changes occur too fast for evolutionary adaptation to keep up. To bridge existing theoretical frameworks, we develop a minimal evolutionary-ecological model showing that rate-induced extinction and the established notion of 'evolutionary rescue' are fundamentally two sides of the same coin: the failure of one implies the other, and vice versa. We compare the minimal model's behaviour with that of a more complex model in which the large-scale dynamics emerge from the interactions of many individual agents; in both cases, there is a well-defined threshold rate to induce extinction, and a consistent scaling law for that rate as a function of timescale. Due to the fundamental nature of the underlying mechanism, we suggest that a vast range of evolutionary systems should in principle be susceptible to rate-induced collapse. This would include ecosystems on all scales as well as human societies; further research is warranted.

Oxidative metabolisms catalyzed Earth's oxygenation.

The burial of organic carbon, which prevents its remineralization via oxygen-consuming processes, is considered one of the causes of Earth's oxygenation. Yet, higher levels of oxygen are thought to inhibit burial. Here we propose a resolution of this conundrum, wherein Earth's initial oxygenation is favored by oxidative metabolisms generating partially oxidized organic matter (POOM), increasing burial via interaction with minerals in sediments. First, we introduce the POOM hypothesis via a mathematical argument. Second, we reconstruct the evolutionary history of one key enzyme family, flavin-dependent Baeyer-Villiger monooxygenases, that generates POOM, and show the temporal consistency of its diversification with the Proterozoic and Phanerozoic atmospheric oxygenation. Finally, we propose that the expansion of oxidative metabolisms instigated a positive feedback, which was amplified by the chemical changes to minerals on Earth's surface. Collectively, these results suggest that Earth's oxygenation is an autocatalytic transition induced by a combination of biological innovations and geological changes.

The Balance of Nature: A Global Marine Perspective.

The ancient idea of the balance of nature continues to influence modern perspectives on global environmental change. Assumptions of stable biogeochemical steady states and linear responses to perturbation are widely employed in the interpretation of geochemical records. Here, we review the dynamics of the marine carbon cycle and its interactions with climate and life over geologic time, focusing on what the record of past changes can teach us about stability and instability in the Earth system. Emerging themes include the role of amplifying feedbacks in producing past carbon cycle disruptions, the importance of critical rates of change in the context of mass extinctions and potential Earth system tipping points, and the application of these ideas to the modern unbalanced carbon cycle. A comprehensive dynamical understanding of the marine record of global environmental disruption will be of great value in understanding the long-term consequences of anthropogenic change.

Asymmetry of extreme Cenozoic climate-carbon cycle events.

The history of Earth's climate and carbon cycle is preserved in deep-sea foraminiferal carbon and oxygen isotope records. Here, we show that the sub-million-year fluctuations in both records have exhibited negatively skewed non-Gaussian tails throughout much of the Cenozoic era (66 Ma to present), suggesting an intrinsic asymmetry that favors "hyperthermal-like" extreme events of abrupt global warming and oxidation of organic carbon. We show that this asymmetry is quantitatively consistent with a general mechanism of self-amplification that can be modeled using stochastic multiplicative noise. A numerical climate-carbon cycle model in which the amplitude of random biogeochemical fluctuations increases at higher temperatures reproduces the data well and can further explain the apparent pacing of past extreme warming events by changes in orbital parameters. Our results also suggest that, as anthropogenic warming continues, Earth's climate may become more susceptible to extreme warming events on time scales of tens of thousands of years.

Routes to global glaciation.

Theory and observation suggest that Earth and Earth-like planets can undergo runaway low-latitude glaciation when changes in solar heating or in the carbon cycle exceed a critical threshold. Here, we use a simple dynamical-system representation of the ice-albedo feedback and the carbonate-silicate cycle to show that glaciation is also triggered when solar heating changes faster than a critical rate. Such 'rate-induced glaciations' remain accessible far from the outer edge of the habitable zone, because the warm climate state retains long-term stability. In contrast, glaciations induced by changes in the carbon cycle require the warm climate state to become unstable, constraining the kinds of perturbations that could have caused global glaciation in Earth's past. We show that glaciations can occur when Earth's climate transitions between two warm stable states; this property of the Earth system could help explain why major events in the development of life have been accompanied by glaciations.

Characteristic disruptions of an excitable carbon cycle.

The history of the carbon cycle is punctuated by enigmatic transient changes in the ocean's store of carbon. Mass extinction is always accompanied by such a disruption, but most disruptions are relatively benign. The less calamitous group exhibits a characteristic rate of change whereas greater surges accompany mass extinctions. To better understand these observations, I formulate and analyze a mathematical model that suggests that disruptions are initiated by perturbation of a permanently stable steady state beyond a threshold. The ensuing excitation exhibits the characteristic surge of real disruptions. In this view, the magnitude and timescale of the disruption are properties of the carbon cycle itself rather than its perturbation. Surges associated with mass extinction, however, require additional inputs from external sources such as massive volcanism. Surges are excited when [Formula: see text] enters the oceans at a flux that exceeds a threshold. The threshold depends on the duration of the injection. For injections lasting a time [Formula: see text] y in the modern carbon cycle, the threshold flux is constant; for smaller [Formula: see text], the threshold scales like [Formula: see text] Consequently the unusually strong but geologically brief duration of modern anthropogenic oceanic [Formula: see text] uptake is roughly equivalent, in terms of its potential to excite a major disruption, to relatively weak but longer-lived perturbations associated with massive volcanism in the geologic past.

Mineral protection regulates long-term global preservation of natural organic carbon.

The balance between photosynthetic organic carbon production and respiration controls atmospheric composition and climate1,2. The majority of organic carbon is respired back to carbon dioxide in the biosphere, but a small fraction escapes remineralization and is preserved over geological timescales3. By removing reduced carbon from Earth's surface, this sequestration process promotes atmospheric oxygen accumulation2 and carbon dioxide removal1. Two major mechanisms have been proposed to explain organic carbon preservation: selective preservation of biochemically unreactive compounds4,5 and protection resulting from interactions with a mineral matrix6,7. Although both mechanisms can operate across a range of environments and timescales, their global relative importance on 1,000-year to 100,000-year timescales remains uncertain4. Here we present a global dataset of the distributions of organic carbon activation energy and corresponding radiocarbon ages in soils, sediments and dissolved organic carbon. We find that activation energy distributions broaden over time in all mineral-containing samples. This result requires increasing bond-strength diversity, consistent with the formation of organo-mineral bonds8 but inconsistent with selective preservation. Radiocarbon ages further reveal that high-energy, mineral-bound organic carbon persists for millennia relative to low-energy, unbound organic carbon. Our results provide globally coherent evidence for the proposed7 importance of mineral protection in promoting organic carbon preservation. We suggest that similar studies of bond-strength diversity in ancient sediments may reveal how and why organic carbon preservation-and thus atmospheric composition and climate-has varied over geological time.

Symmetric rearrangement of groundwater-fed streams.

Streams shape landscapes through headward growth and lateral migration. When these streams are primarily fed by groundwater, recent work suggests that their tips advance to maximize the symmetry of the local Laplacian field associated with groundwater flow. We explore the extent to which such forcing is responsible for the lateral migration of streams by studying two features of groundwater-fed streams in Bristol, Florida: their confluence angle near junctions and their curvature. First, we find that, while streams asymptotically form a 72° angle near their tips, they simultaneously exhibit a wide 120° confluence angle within approximately 10 m of their junctions. We show that this wide angle maximizes the symmetry of the groundwater field near the junction. Second, we argue that streams migrate laterally within valleys and present a new spectral analysis method to relate planform curvature to the surrounding groundwater field. Our results suggest that streams migrate laterally in response to fluxes from the surrounding groundwater table, providing evidence of a new mechanism that complements Laplacian growth at their tips.

Thresholds of catastrophe in the Earth system.

The history of the Earth system is a story of change. Some changes are gradual and benign, but others, especially those associated with catastrophic mass extinction, are relatively abrupt and destructive. What sets one group apart from the other? Here, I hypothesize that perturbations of Earth's carbon cycle lead to mass extinction if they exceed either a critical rate at long time scales or a critical size at short time scales. By analyzing 31 carbon isotopic events during the past 542 million years, I identify the critical rate with a limit imposed by mass conservation. Identification of the crossover time scale separating fast from slow events then yields the critical size. The modern critical size for the marine carbon cycle is roughly similar to the mass of carbon that human activities will likely have added to the oceans by the year 2100.

A free-boundary model of diffusive valley growth: theory and observation.

Valleys that form around a stream head often develop characteristic finger-like elevation contours. We study the processes involved in the formation of these valleys and introduce a theoretical model that indicates how shape may inform the underlying processes. We consider valley growth as the advance of a moving boundary travelling forward purely through linearly diffusive erosion, and we obtain a solution for the valley shape in three dimensions. Our solution compares well to the shape of slowly growing groundwater-fed valleys found in Bristol, Florida. Our results identify a new feature in the formation of groundwater-fed valleys: a spatially variable diffusivity that can be modelled by a fixed-height moving boundary.

Exact solution for the Poisson field in a semi-infinite strip.

The Poisson equation is associated with many physical processes. Yet exact analytic solutions for the two-dimensional Poisson field are scarce. Here we derive an analytic solution for the Poisson equation with constant forcing in a semi-infinite strip. We provide a method that can be used to solve the field in other intricate geometries. We show that the Poisson flux reveals an inverse square-root singularity at a tip of a slit, and identify a characteristic length scale in which a small perturbation, in a form of a new slit, is screened by the field. We suggest that this length scale expresses itself as a characteristic spacing between tips in real Poisson networks that grow in response to fluxes at tips.

Path selection in the growth of rivers.

River networks exhibit a complex ramified structure that has inspired decades of studies. However, an understanding of the propagation of a single stream remains elusive. Here we invoke a criterion for path selection from fracture mechanics and apply it to the growth of streams in a diffusion field. We show that, as it cuts through the landscape, a stream maintains a symmetric groundwater flow around its tip. The local flow conditions therefore determine the growth of the drainage network. We use this principle to reconstruct the history of a network and to find a growth law associated with it. Our results show that the deterministic growth of a single channel based on its local environment can be used to characterize the structure of river networks.

Mechanisms for mechanical trapping of geologically sequestered carbon dioxide.

Carbon dioxide (CO2) sequestration in subsurface reservoirs is important for limiting atmospheric CO2 concentrations. However, a complete physical picture able to predict the structure developing within the porous medium is lacking. We investigate theoretically reactive transport in the long-time evolution of carbon in the brine-rock environment. As CO2 is injected into a brine-rock environment, a carbonate-rich region is created amid brine. Within the carbonate-rich region minerals dissolve and migrate from regions of high-to-low concentration, along with other dissolved carbonate species. This causes mineral precipitation at the interface between the two regions. We argue that precipitation in a small layer reduces diffusivity, and eventually causes mechanical trapping of the CO2. Consequently, only a small fraction of the CO2 is converted to solid mineral; the remainder either dissolves in water or is trapped in its original form. We also study the case of a pure CO2 bubble surrounded by brine and suggest a mechanism that may lead to a carbonate-encrusted bubble owing to structural diffusion.

Hidden cycle of dissolved organic carbon in the deep ocean.

Marine dissolved organic carbon (DOC) is a large (660 Pg C) reactive carbon reservoir that mediates the oceanic microbial food web and interacts with climate on both short and long timescales. Carbon isotopic content provides information on the DOC source via δ(13)C and age via Δ(14)C. Bulk isotope measurements suggest a microbially sourced DOC reservoir with two distinct components of differing radiocarbon age. However, such measurements cannot determine internal dynamics and fluxes. Here we analyze serial oxidation experiments to quantify the isotopic diversity of DOC at an oligotrophic site in the central Pacific Ocean. Our results show diversity in both stable and radio isotopes at all depths, confirming DOC cycling hidden within bulk analyses. We confirm the presence of isotopically enriched, modern DOC cocycling with an isotopically depleted older fraction in the upper ocean. However, our results show that up to 30% of the deep DOC reservoir is modern and supported by a 1 Pg/y carbon flux, which is 10 times higher than inferred from bulk isotope measurements. Isotopically depleted material turns over at an apparent time scale of 30,000 y, which is far slower than indicated by bulk isotope measurements. These results are consistent with global DOC measurements and explain both the fluctuations in deep DOC concentration and the anomalous radiocarbon values of DOC in the Southern Ocean. Collectively these results provide an unprecedented view of the ways in which DOC moves through the marine carbon cycle.

Methanogenic burst in the end-Permian carbon cycle.

The end-Permian extinction is associated with a mysterious disruption to Earth's carbon cycle. Here we identify causal mechanisms via three observations. First, we show that geochemical signals indicate superexponential growth of the marine inorganic carbon reservoir, coincident with the extinction and consistent with the expansion of a new microbial metabolic pathway. Second, we show that the efficient acetoclastic pathway in Methanosarcina emerged at a time statistically indistinguishable from the extinction. Finally, we show that nickel concentrations in South China sediments increased sharply at the extinction, probably as a consequence of massive Siberian volcanism, enabling a methanogenic expansion by removal of nickel limitation. Collectively, these results are consistent with the instigation of Earth's greatest mass extinction by a specific microbial innovation.

Bifurcation dynamics of natural drainage networks.

As water erodes a landscape, streams form and channellize the surficial flow. In time, streams become highly ramified networks that can extend over a continent. Here, we combine physical reasoning, mathematical analysis and field observations to understand a basic feature of network growth: the bifurcation of a growing stream. We suggest a deterministic bifurcation rule arising from a relationship between the position of the tip in the network and the local shape of the water table. Next, we show that, when a stream bifurcates, competition between the stream and branches selects a special bifurcation angle α=2π/5. We confirm this prediction by measuring several thousand bifurcation angles in a kilometre-scale network fed by groundwater. In addition to providing insight into the growth of river networks, this result presents river networks as a physical manifestation of a classical mathematical problem: interface growth in a harmonic field. In the final sections, we combine these results to develop and explore a one-parameter model of network growth. The model predicts the development of logarithmic spirals. We find similar features in the kilometre-scale network.

Bifurcation dynamics of natural drainage networks.

As water erodes a landscape, streams form and channellize the surficial flow. In time, streams become highly ramified networks that can extend over a continent. Here, we combine physical reasoning, mathematical analysis and field observations to understand a basic feature of network growth: the bifurcation of a growing stream. We suggest a deterministic bifurcation rule arising from a relationship between the position of the tip in the network and the local shape of the water table. Next, we show that, when a stream bifurcates, competition between the stream and branches selects a special bifurcation angle alpha = 2pi/5. We confirm this prediction by measuring several thousand bifurcation angles in a kilometre-scale network fed by groundwater. In addition to providing insight into the growth of river networks, this result presents river networks as a physical manifestation of a classical mathematical problem: interface growth in a harmonic field. In the final sections, we combine these results to develop and explore a one-parameter model of network growth. The model predicts the development of logarithmic spirals. We find similar features in the kilometre-scale network.

Shape and dynamics of seepage erosion in a horizontal granular bed.

We investigate erosion patterns observed in a horizontal granular bed resulting from seepage of water motivated by observation of beach rills and channel growth in larger scale land forms. Our experimental apparatus consists of a wide rectangular box filled with glass beads with a narrow opening in one of the side walls from which eroded grains can exit. Quantitative data on the shape of the pattern and erosion dynamics are obtained with a laser-aided topography technique. We show that the spatial distribution of the source of groundwater can significantly impact the shape of observed patterns. An elongated channel is observed to grow upstream when groundwater is injected at a boundary adjacent to a reservoir held at constant height. An amphitheater (semicircular) shape is observed when uniform rainfall infiltrates the granular bed to maintain a water table. Bifurcations are observed as the channels grow in response to the groundwater. We further find that the channels grow by discrete avalanches as the height of the granular bed is increased above the capillary rise, causing the deeper channels to have rougher fronts. The spatiotemporal distribution of avalanches increase with bed height when partial saturation of the bed leads to cohesion between grains. However, the overall shape of the channels is observed to remain unaffected indicating that seepage erosion is robust to perturbation of the erosion front.