Long-time evolution of sequestered CO2_2 in porous media

CO$_2$ sequestration in subsurface reservoirs is important for limiting atmospheric CO$_2$ 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 CO$_2$ 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 concentration 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 CO$_2$. Consequently, only a small fraction of the CO$_2$ 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 CO$_2$ bubble surrounded by brine and suggest a mechanism that may lead to a carbonate-encrusted bubble due to structural diffusion

Scaling of a slope: the erosion of tilted landscapes

We formulate a stochastic equation to model the erosion of a surface with fixed inclination. Because the inclination imposes a preferred direction for material transport, the problem is intrinsically anisotropic. At zeroth order, the anisotropy manifests itself in a linear equation that predicts that the prefactor of the surface height-height correlations depends on direction. The first higher-order nonlinear contribution from the anisotropy is studied by applying the dynamic renormalization group. Assuming an inhomogeneous distribution of soil substrate that is modeled by a source of static noise, we estimate the scaling exponents at first order in \ep-expansion. These exponents also depend on direction. We compare these predictions with empirical measurements made from real landscapes and find good agreement. We propose that our anisotropic theory applies principally to small scales and that a previously proposed isotropic theory applies principally to larger scales. Lastly, by considering our model as a transport equation for a driven diffusive system, we construct scaling arguments for the size distribution of erosion ``events'' or ``avalanches.'' We derive a relationship between the exponents characterizing the surface anisotropy and the avalanche size distribution, and indicate how this result may be used to interpret previous findings of power-law size distributions in real submarine avalanches.Comment: 19 pages, includes 10 PS figures. J. Stat. Phys. (in press

Earth’s carbon cycle: A mathematical perspective

The carbon cycle represents metabolism at a global scale. When viewed through a mathematical lens, observational data suggest that the cycle exhibits an underlying mathematical structure. This review focuses on two types of emerging results: evidence of global dynamical coupling between life and the environment, and an understanding of the ways in which smaller-scale processes determine the strength of that coupling. Such insights are relevant not only to predicting future climate but also to understanding the long-term co-evolution of life and the environment.NASA Astrobiology Institute (NNA08CN84A)NASA Astrobiology Institute (NNA13AA90A)National Science Foundation (U.S.) (OCE-0930866)National Science Foundation (U.S.) (EAR-1338810

Carbon transit through degradation networks

The decay of organic matter in natural ecosystems is controlled by a network of biologically, physically, and chemically driven processes. Decomposing organic matter is often described as a continuum that transforms and degrades over a wide range of rates, but it is difficult to quantify this heterogeneity in models. Most models of carbon degradation consider a network of only a few organic matter states that transform homogeneously at a single rate. These models may fail to capture the range of residence times of carbon in the soil organic matter continuum. Here we assume that organic matter is distributed among a continuous network of states that transform with stochastic, heterogeneous kinetics. We pose and solve an inverse problem in order to identify the rates of carbon exiting the underlying degradation network (exit rates) and apply this approach to plant matter decay throughout North America. This approach provides estimates of carbon retention in the network without knowing the details of underlying state transformations. We find that the exit rates are approximately lognormal, suggesting that carbon flow through a complex degradation network can be described with just a few parameters. These results indicate that the serial and feedback processes in natural degradation networks can be well approximated by a continuum of parallel decay rates.National Science Foundation (U.S.) (Grant EAR-0420592)United States. National Aeronautics and Space Administration (Grant NNA08CN84A

Stochastic equation for the erosion of inclined topography

We present a stochastic equation to model the erosion of topography with fixed inclination. The inclination causes the erosion to be anisotropic. A zero-order consequence of the anisotropy is the dependence of the prefactor of the surface height-height correlations on direction. The lowest higher-order contribution from the anisotropy is studied by applying the dynamic renormalization group. In this case, assuming an inhomogenous distribution of soil material, we find a one-loop estimate of the roughness exponents. The predicted exponents are in good agreement with new measurements made from seafloor topography.Postprint (published version

Inverse method for estimating respiration rates from decay time series

Long-term organic matter decomposition experiments typically measure the mass lost from decaying organic matter as a function of time. These experiments can provide information about the dynamics of carbon dioxide input to the atmosphere and controls on natural respiration processes. Decay slows down with time, suggesting that organic matter is composed of components (pools) with varied lability. Yet it is unclear how the appropriate rates, sizes, and number of pools vary with organic matter type, climate, and ecosystem. To better understand these relations, it is necessary to properly extract the decay rates from decomposition data. Here we present a regularized inverse method to identify an optimally-fitting distribution of decay rates associated with a decay time series. We motivate our study by first evaluating a standard, direct inversion of the data. The direct inversion identifies a discrete distribution of decay rates, where mass is concentrated in just a small number of discrete pools. It is consistent with identifying the best fitting "multi-pool" model, without prior assumption of the number of pools. However we find these multi-pool solutions are not robust to noise and are over-parametrized. We therefore introduce a method of regularized inversion, which identifies the solution which best fits the data but not the noise. This method shows that the data are described by a continuous distribution of rates, which we find is well approximated by a lognormal distribution, and consistent with the idea that decomposition results from a continuum of processes at different rates. The ubiquity of the lognormal distribution suggest that decay may be simply described by just two parameters: a mean and a variance of log rates. We conclude by describing a procedure that estimates these two lognormal parameters from decay data. Matlab codes for all numerical methods and procedures are provided

Geometry of Valley Growth

Although amphitheater-shaped valley heads can be cut by groundwater flows emerging from springs, recent geological evidence suggests that other processes may also produce similar features, thus confounding the interpretations of such valley heads on Earth and Mars. To better understand the origin of this topographic form we combine field observations, laboratory experiments, analysis of a high-resolution topographic map, and mathematical theory to quantitatively characterize a class of physical phenomena that produce amphitheater-shaped heads. The resulting geometric growth equation accurately predicts the shape of decimeter-wide channels in laboratory experiments, 100-meter wide valleys in Florida and Idaho, and kilometer wide valleys on Mars. We find that whenever the processes shaping a landscape favor the growth of sharply protruding features, channels develop amphitheater-shaped heads with an aspect ratio of pi

Statistics of Mars' topography from the Mars Orbiter Laser Altimeter: Slopes, correlations, and physical Models

Data obtained recently by the Mars Orbiter Laser Altimeter (MOLA) were used to study the statistical properties of the topography and slopes on Mars. We find that the hemispheric dichotomy, manifested as an elevation difference, can be described by long baseline tilts but in places is expressed as steeper slopes. The bimodal hypsometry of elevations on Mars becomes unimodal when referenced to the center of figure, contrary to the Earth, for which the bimodality is retained. However, ruling out a model in which the elevation difference is expressed in a narrow equatorial topographic step cannot be done by the hypsometry alone. Mars' slope distribution is longer tailed than those of Earth and Venus, indicating a lower efficiency of planation processes relative to relief-building tectonics and volcanics. We define and compute global maps of statistical estimators, including the interquartile scale, RMS and median slope, and characteristic decorrelation length of the surface. A correspondence between these parameters and geologic units on Mars is inferred. Surface smoothness is distinctive in the vast northern hemisphere plains, where slopes are typically <0.5°. Amazonis Planitia exhibits a variation in topography of <1 m over 35-km baselines. The region of hematite mineralization in Sinus Meridiani is also smooth, with median slopes lower than 0.4°, but does not form a closed basin. The shallower long-wavelength portion of the lowlands' topographic power spectrum relative to the highlands' can be accounted for by a simple model of sedimentation such as might be expected at an ocean's floor. The addition of another process such as cratering is necessary to explain the spectral slope in short wavelengths. Among their application, these MOLA-derived roughness measurements can help characterize sites for landing missions