Uncertainties In Boundary Conditions Research Articles

A Consistent Procedure Using Response Surface Methodology to Identify Stiffness Properties of Connections in Machine Tools.

Accurate finite element models of mechanical systems are fundamental resources to perform structural analyses at the design stage. However, uncertainties in material properties, boundary conditions, or connections give rise to discrepancies between the real and predicted dynamic characteristics. Therefore, it is necessary to improve these models in order to achieve a better fit. This paper presents a systematic three-step procedure to update the finite element (FE) models of machine tools with numerous uncertainties in connections, which integrates statistical, numerical, and experimental techniques. The first step is the gradual application of fractional factorial designs, followed by an analysis of the variance to determine the significant variables that affect each dynamic response. Then, quadratic response surface meta-models, including only significant terms, which relate the design parameters to the modal responses are obtained. Finally, the values of the updated design variables are identified using the previous regression equations and experimental modal data. This work demonstrates that the integrated procedure gives rise to FE models whose dynamic responses closely agree with the experimental measurements, despite the large number of uncertainties, and at an acceptable computational cost.

Flood mapping under uncertainty: a case study in the Canadian prairies

In Canada, approaches for hazard mapping involve using a hydraulic model to generate a flood extent map that distinguishes between inundated and non-inundated areas in a deterministic way. The authors adapt a probabilistic approach to obtain flood hazard maps by accounting for uncertainty in boundary conditions and calibration parameters of a hydrodynamic model, while considering extent, probability, and depth of inundation as important flood indicators. The Qu’Appelle River in the Canadian prairies is considered as a case study to demonstrate the benefits of a probabilistic approach to obtain flooding indicators. The probability and depth of inundation are obtained for all locations in the floodplain, and their uncertainties are evaluated. A sensitivity analysis to determine factors affecting the extent of flooding at multiple locations along the river is also carried out. Further, the criterion to distinguish between flooded and non-flooded areas is modified and its subsequent effect on the flooding indicators is also evaluated. Results show low variability in depth and probability of inundation at most locations along the floodplain. It is also observed that the influence of the roughness parameters on the flooding extents is higher in the steeper stretches of the river, compared to the flatter stretches. Another component of the presented work focusses on evaluating an existing topography-based hazard map for Canada that classifies the country into different levels based on severity of flooding, by comparing it with the probabilistic hazard map obtained in this study. The comparison indicates a good level of agreement between both maps and highlights the reliability of using the topography-based hazard map where hydraulic modeling is infeasible.

Dual-stage uncertainty modeling and evaluation for transient temperature effect on structural vibration property

In practical engineering, the changing structural temperature distribution has been considered as an important factor influencing the structural vibration property. By treating the uncertainties in material properties, boundary conditions and external loads as interval variables, this paper proposes a dual-stage uncertainty analysis framework to evaluate the variation of structural natural frequency associated with transient non-uniform temperature distribution. Based on the subinterval dividing strategy, a modified interval vertex method is firstly proposed to predict the uncertain temperature field. As the material mechanical properties are temperature dependent, the components with different temperatures are consequently employed to discrete the structural model. Then by using the Taylor series, an interval perturbation method is developed to solve the generalized eigenvalue problem for the natural frequency prediction. By introducing the traditional Monte Carlo simulations as reference, a 3D plate structure with uncertain parameters is adopted to verify the proposed method.

Stochastic multi-objective optimisation of the cure process of thick laminates

A stochastic multi-objective cure optimisation methodology is developed in this work and applied to the case of thick epoxy/carbon fibre laminates. The methodology takes into account the uncertainty in process parameters and boundary conditions and minimises the mean values and standard deviations of cure time and temperature overshoot. Kriging is utilised to construct a surrogate model of the cure substituting Finite Element (FE) simulation for computational efficiency reasons. The surrogate model is coupled with Monte Carlo and integrated into a stochastic multi-objective optimisation framework based on Genetic Algorithms. The results show a significant reduction of about 40% in temperature overshoot and cure time compared to standard cure profiles. This reduction is accompanied by a reduction in variability by about 20% for both objectives. This highlights the opportunity of replacing conventional cure schedules with optimised profiles achieving significant improvement in both process efficiency and robustness.

Uncertainty quantification of a generic scramjet engine using a probabilistic collocation and a hybrid approach

A hybrid probabilistic model was developed and used to investigate a generic scramjet engine at flight conditions (Mach 8, 30 km altitude). To assess the robustness of the design, uncertainties in boundary conditions and modelling were considered. The full-system model is composed of three parts, which were modelled separately: inlet with isolator, combustion chamber and nozzle. The inlet and the nozzle were resolved using two-dimensional calculations solving the Reynolds averaged Navier Stokes equations, while the combustion chamber modelling was based on an one-dimensional stream tube approach. Flight properties and injection parameters were considered aleatoric in nature. Uncertainties introduced by different averaging approaches at the model interface were presumed to be epistemic. The uncertainties were propagated using probabilistic collocation if feasible and Monte Carlo simulation if necessary. The more recent probabilistic collocation approach allows a reduction of the necessary number of evaluations. Hence, a more detailed model can be employed. However, as this polynomial chaos approach is only valid for smooth transformations, it fails to predict singular hazardous events, e.g. discontinuities such as thermal choking of the combustor. To consider these events, a Monte Carlo simulation had to be applied. The hybrid model and the full probabilistic collocation model showed equivalent results. But the hybrid model acquired additional information at overall lower calculation expanses.

Investigation on the Subcooled Boiling in Vertical Pipe With Uncertainties From Boundary Conditions by Using FLUENT

Subcooled boiling flow taking place in the reactor system plays a critical role in the safety of nuclear power plants. It has been studied by experiments and system codes in the past decades. Now subcooled boiling can be predicted with CFD code based on the Eulerian two-fluid model, with the development in the computational technology and the understanding in the mechanism of two-phase flow. The published works on the validation of CFD code for two-phase flow were carried out based on the deterministic analysis by comparing the calculated and experimental nominal inputs and outputs, which is not sufficient for code validation since it didn’t consider the inevitable uncertainty in the experiment measurements. In the current work, subcooled boiling was predicted by using a CFD code, FLUENT, with consideration of the uncertainties of boundary conditions. The resultant parameters with uncertainties were compared to the experiment data for validation purpose. Confidence intervals of the two-phase parameters were predicted. Besides, correlations between the boundary conditions and the outputs were analyzed.

Uncertainty Quantification Study of CTF for the OECD/NEA LWR Uncertainty Analysis in Modeling Benchmark

This work describes the results of a quantitative uncertainty analysis of the thermal-hydraulic subchannel code for nuclear engineering applications, Coolant Boiling in Rod Arrays-Three Field (COBRA-TF). CTF is used, which is a version of COBRA-TF developed in cooperation between the Consortium for Advanced Simulation of Light Water Reactors and North Carolina State University. Four steady-state cases from Phase II Exercise 3 of the Organisation for Economic Co-operation and Development/Nuclear Energy Agency Light Water Reactor Uncertainty Analysis in Modeling (UAM) Benchmark are analyzed using the statistical analysis tool, Design Analysis Kit for Optimization and Terascale Applications (Dakota). The input parameters include boundary condition, geometry, and modeling uncertainties, which are selected using a sensitivity study and then defined based on expert judgment. A forward uncertainty quantification method with Latin hypercube sampling (LHS) is used, where the sample size is based on available computational resources.The means and standard deviations of thermal-hydraulic quantities of interest are reported, as well as the Spearman rank correlation coefficients between the inputs and outputs. The means and standard deviations are accompanied by their respective standard errors, and the correlation coefficients are tested for statistical significance. The quantities of interest include void fractions, temperatures, and pressure drops. The predicted uncertainty in all parameters remains relatively low for all quantities of interest. The dominant sources of uncertainty are identified. For cases based on experiments, two different validation metrics are used to quantify the difference between measured and predicted void fractions.The results compare well with past studies, but with a number of improvements: the use of an updated CTF input deck using the current UAM specification and the most recent version of CTF, the use of an LHS method, an analysis of standard errors for the statistical results, and a quantitative comparison to experimental data.Though the statistical uncertainty analysis framework presented herein is applied to thermal-hydraulic analyses, it is generally applicable to any simulation tool. Given a specified amount of computational resources, it can be used to quantify statistical significance through the use of fundamental statistical analyses. This is in contrast with the prevailing methods in nuclear engineering, which provide a sample size necessary to achieve a specified level of statistical certainty.

Impact of uncertainties in outflow boundary conditions on the predictions of hemodynamic simulations of ascending thoracic aortic aneurysms

The impact of outflow boundary conditions on the results of numerical simulations of the flow inside ascending thoracic aortic aneurysms is investigated. The adopted outflow conditions are based on the lumped three-element Windkessel model, in which it is assumed that the pressure at the outflow is related to the flow rate and to a terminal constant pressure, through an electric circuit analogue that has a proximal resistance in series with a parallel arrangement of a capacitance and of a distal resistance. The values of the Windkessel model parameters must be a-priori specified. The impact on the numerical simulation results of uncertainties in the values of the Windkessel model parameters is quantified here through a stochastic approach. The propagation of the considered uncertainties is evaluated, in particular, for the instantaneous and time-averaged wall shear stress. The generalized Polynomial Chaos is used as a surrogate model to obtain continuous response surfaces of the quantities of interest in the parameter space starting from a few deterministic simulations. A patient-specific geometry, obtained through in-vivo imaging, is considered. The analysis is carried out for both rigid and compliant vessel walls. Our results show that, although the uncertainties in the selected outflow parameters may give significant variability of the instantaneous shear stresses in regions characterized by flow recirculation or large streamline curvature, the impact on cycle-averaged shear stresses is moderate. Taking into account the wall compliance seems to reduce the impact of the uncertainties in the outflow parameters compared to the rigid case.

Effects of Uncertainties in Electric Field Boundary Conditions for Ring Current Simulations

AbstractPhysics‐based simulation results can vary widely depending on the applied boundary conditions. As a first step toward assessing the effect of boundary conditions on ring current simulations, we analyze the uncertainty of cross‐polar cap potentials (CPCP) on electric field boundary conditions applied to the Rice Convection Model‐Equilibrium (RCM‐E). The empirical Weimer model of CPCP is chosen as the reference model and Defense Meteorological Satellite Program CPCP measurements as the reference data. Using temporal correlations from a statistical analysis of the “errors” between the reference model and data, we construct a Monte Carlo CPCP discrete time series model that can be generalized to other model boundary conditions. RCM‐E simulations using electric field boundary conditions from the reference model and from 20 randomly generated Monte Carlo discrete time series of CPCP are performed for two large storms. During the 10 August 2000 storm main phase, the proton density at 10 RE at midnight was observed to be low (< 1.4 cm−3) and the observed disturbance Dst index is bounded by the simulated Dst values. In contrast, the simulated Dst values during the recovery phases of the 10 August 2000 and 31 August 2005 storms tend to underestimate systematically the observed late Dst recovery. This suggests a need to improve the accuracy of particle loss calculations in the RCM‐E model. Application of this technique can aid modelers to make efficient choices on either investing more effort on improving specification of boundary conditions or on improving descriptions of physical processes.

The PMIP4 contribution to CMIP6 – Part 4: Scientific objectives and experimental design of the PMIP4-CMIP6 Last Glacial Maximum experiments and PMIP4 sensitivity experiments

Abstract. The Last Glacial Maximum (LGM, 21 000 years ago) is one of the suite of paleoclimate simulations included in the current phase of the Coupled Model Intercomparison Project (CMIP6). It is an interval when insolation was similar to the present, but global ice volume was at a maximum, eustatic sea level was at or close to a minimum, greenhouse gas concentrations were lower, atmospheric aerosol loadings were higher than today, and vegetation and land-surface characteristics were different from today. The LGM has been a focus for the Paleoclimate Modelling Intercomparison Project (PMIP) since its inception, and thus many of the problems that might be associated with simulating such a radically different climate are well documented. The LGM state provides an ideal case study for evaluating climate model performance because the changes in forcing and temperature between the LGM and pre-industrial are of the same order of magnitude as those projected for the end of the 21st century. Thus, the CMIP6 LGM experiment could provide additional information that can be used to constrain estimates of climate sensitivity. The design of the Tier 1 LGM experiment (lgm) includes an assessment of uncertainties in boundary conditions, in particular through the use of different reconstructions of the ice sheets and of the change in dust forcing. Additional (Tier 2) sensitivity experiments have been designed to quantify feedbacks associated with land-surface changes and aerosol loadings, and to isolate the role of individual forcings. Model analysis and evaluation will capitalize on the relative abundance of paleoenvironmental observations and quantitative climate reconstructions already available for the LGM.

Prediction of Lake-Effect Snow Using Convection-Allowing Ensemble Forecasts and Regional Data Assimilation

Abstract Lake-effect snow (LES) is a cold-season mesoscale convective phenomenon that can lead to significant snowfall rates and accumulations in the Great Lakes region of the United States. While limited-area numerical weather prediction models have shown skill in prediction of warm-season convective storms, forecasting the sharp nature of LES precipitation timing, intensity, and location is difficult because of model error and initial and boundary condition uncertainties. Ensemble forecasting can incorporate and quantify some sources of forecast error, but ensemble design must be considered. This study examines the relative contributions of forecast uncertainties to LES forecast error using a regional convection-allowing data assimilation and ensemble prediction system. Ensembles are developed using various methods of perturbations to simulate a long-lived and high-precipitation LES event in December 2013, and forecast performance is evaluated using observations including those from the Ontario Winter Lake-Effect Systems (OWLeS) campaign. Model lateral boundary conditions corresponding to weather conditions beyond the Great Lakes region play an influential role in LES precipitation forecasts and their uncertainty, as evidenced by ensemble spread, particularly at lead times beyond one day. A strong forecast dependence on regional initial conditions was shown using data assimilation. This sensitivity impacts the timing and intensity of predicted precipitation, as well as band location and orientation assessed with an object-based verification approach, giving insight into the time scales of practical predictability of LES. Overall, an assimilation-cycling convection-allowing ensemble prediction system could improve future lake-effect snow precipitation forecasts and analyses and can help quantify and understand sources of forecast uncertainty.

Numerical investigation of the impact of branching vessel boundary conditions on aortic hemodynamics

Abstract CFD has gained significant attention as a tool to model aortic hemodynamics. However, obtaining accurate patient-specific boundary conditions still poses a major challenge and represents a major source of uncertainties, which are difficult to quantify. This study presents an attempt to quantify these uncertainties by comparing 14 patient-specific simulations of the aorta (reference method), each exhibiting stenosis, against simulations using the same geometries without the branching vessels of the aortic arch (simplified method). Results were evaluated by comparing pressure drop along the aorta, secondary flow degree (SFD) and surface-averaged wall shear stress (WSS) for each patient. The comparison shows little difference in pressure drop between the two methods (simplified-reference) with the mean difference being 1.2 mmHg (standard deviation: 3.0 mmHg). SFD and WSS, however, show striking differences between the methods: SFD downstream of the stenosis is on average 61 % higher in the simplified cases, while WSS is on average 3.0 Pa lower in the simplified cases. Although unphysiological, the comparison of both methods gives an upper bound for the error introduced by uncertainties in branching vessel boundary conditions. For the pressure drop this error appears to be remarkably low, while being unacceptably high for SFD and WSS.

Impact of dynamical regionalization on precipitation biases and teleconnections over West Africa

West African societies are highly dependent on the West African Monsoon (WAM). Thus, a correct representation of the WAM in climate models is of paramount importance. In this article, the ability of 8 CMIP5 historical General Circulation Models (GCMs) and 4 CORDEX-Africa Regional Climate Models (RCMs) to characterize the WAM dynamics and variability is assessed for the period July-August-September 1979–2004. Simulations are compared with observations. Uncertainties in RCM performance and lateral boundary conditions are assessed individually. Results show that both GCMs and RCMs have trouble to simulate the northward migration of the Intertropical Convergence Zone in boreal summer. The greatest bias improvements are obtained after regionalization of the most inaccurate GCM simulations. To assess WAM variability, a Maximum Covariance Analysis is performed between Sea Surface Temperature and precipitation anomalies in observations, GCM and RCM simulations. The assessed variability patterns are: El Nino-Southern Oscillation (ENSO); the eastern Mediterranean (MED); and the Atlantic Equatorial Mode (EM). Evidence is given that regionalization of the ENSO–WAM teleconnection does not provide any added value. Unlike GCMs, RCMs are unable to precisely represent the ENSO impact on air subsidence over West Africa. Contrastingly, the simulation of the MED–WAM teleconnection is improved after regionalization. Humidity advection and convergence over the Sahel area are better simulated by RCMs. Finally, no robust conclusions can be determined for the EM–WAM teleconnection, which cannot be isolated for the 1979–2004 period. The novel results in this article will help to select the most appropriate RCM simulations to study WAM teleconnections.

Finite element model updating considering boundary conditions using neural networks

A novel technique to evaluate the bridge boundary condition using neural networks is proposed. It can be used to establish a more accurate finite element (FE) model considering the behaviors of boundary conditions. In the proposed method, the aging and constraining effect of the boundary condition is represented by an artificial rotational spring at each support. A relationship between the responses of the bridge and the rotational spring constant is analytically investigated. This relationship can be used to estimate the rotational spring constant of the bridge using neural networks. The proposed method was verified through laboratory tests and field tests on a steel girder bridge. The proposed method can estimate the bridge boundary conditions directly from the actual behaviors of bridge supports, and this can effectively reduce the uncertainty of boundary conditions in FE model updating.

Updating of aerodynamic reduced order models generated using computational fluid dynamics

Reduced order models of computational fluid dynamics codes have been developed to decrease computational costs; however, each reduced order model has a limited range of validity based on the data used in its construction. Further, like the computational fluid dynamics from which it is derived, such models exhibit differences from experimental data due to uncertainty in boundary conditions and numerical accuracy. Model updating provides the opportunity to use small amounts of additional data to modify the behaviour of a reduced order model, which means that the range of validity of the reduced order model can be extended. Whilst here computational fluid dynamics data have been used for updating, the approach offers the possibility that experimental data can be used in future. In this work, the baseline reduced order models are constructed using the Eigensystem realisation algorithm and the steps used to update these models are given in detail. The methods developed are then applied to remove the effects of wind tunnel walls and to include viscous effects.

A precipiton‐based approach to model hydro‐sedimentary hazards induced by large sediment supplies in alluvial fans

AbstractMountain ranges are frequently subjected to mass wasting events triggered by storms or earthquakes and supply large volumes of sediment into river networks. Besides altering river dynamics, large sediment deliveries to alluvial fans are known to cause hydro‐sedimentary hazards such as flooding and river avulsion. Here we explore how the sediment supply history affects hydro‐sedimentary river and fan hazards, and how well can it be predicted given the uncertainties on boundary conditions. We use the 2D morphodynamic model Eros with a new 2D hydrodynamic model driven by a sequence of flood, a sediment entrainment/transport/deposition model and a bank erosion law. We first evaluate the model against a natural case: the 1999 Mount Adams rock avalanche and subsequent avulsion on the Poerua river fan (West Coast, New Zealand). By adjusting for the unknown sediment supply history, Eros predicts the evolution of the alluvial riverbed during the first post‐landslide stages within 30 cm. The model is subsequently used to infer how the sediment supply volume and rate control the fan aggradation patterns and associated hazards. Our results show that the total injected volume controls the overall levels of aggradation, but supply rates have a major control on the location of preferential deposition, avulsion and increased flooding risk. Fan re‐incision following exhaustion of the landslide‐derived sediment supply leads to sediment transfer and deposition downstream and poses similar, but delayed, hydro‐sedimentary hazards. Our results demonstrate that 2D morphodynamics models are able to capture the full range of hazards occurring in alluvial fans including river avulsion aggradation and floods. However, only ensemble simulations accounting for uncertainties in boundary conditions (e.g., discharge history, initial topography, grain size) as well as model realization (e.g., non‐linearities in hydro‐sedimentary processes) can be used to produce probabilistic hazards maps relevant for decision making. Copyright © 2017 John Wiley & Sons, Ltd.

Effects of different boundary conditions on the simulation of groundwater flow in a multi-layered coastal aquifer system (Taranto Gulf, southern Italy)

The evaluation of the accuracy or reasonableness of numerical models of groundwater flow is a complex task, due to the uncertainties in hydrodynamic properties and boundary conditions and the scarcity of good-quality field data. To assess model reliability, different calibration techniques are joined to evaluate the effects of different kinds of boundary conditions on the groundwater flow in a coastal multi-layered aquifer in southern Italy. In particular, both direct and indirect approaches for inverse modeling were joined through the calibration of one of the most uncertain parameters, namely the hydraulic conductivity of the karst deep hydrostratigraphic unit. The methodology proposed here, and applied to a real case study, confirmed that the selection of boundary conditions is among the most critical and difficult aspects of the characterization of a groundwater system for conceptual analysis or numerical simulation. The practical tests conducted in this study show that incorrect specification of boundary conditions prevents an acceptable match between the model response to the hydraulic stresses and the behavior of the natural system. Such effects have a negative impact on the applicability of numerical modeling to simulate groundwater dynamics in complex hydrogeological situations. This is particularly important for management of the aquifer system investigated in this work, which represents the only available freshwater resource of the study area, and is threatened by overexploitation and saltwater intrusion.

Quantifying local rainfall dynamics and uncertain boundary conditions into a nested regional‐local flood modeling system

AbstractInflow discharge and outflow stage estimates for hydraulic flood models are generally derived from river gauge data. Uncertainties in the measured inflow data and the neglect of rainfall‐runoff contributions to the modeled domain downstream of the gauging locations can have a significant impact on these estimated “whole reach” inflows and consequently on flood predictions. In this study, a method to incorporate rating curve uncertainty and local rainfall‐runoff dynamics into the predictions of a reach‐scale flood model is proposed. The methodology is applied to the July 2007 floods of the River Severn in UK. Discharge uncertainty bounds are generated applying a nonparametric local weighted regression approach to stage‐discharge measurements for two gauging stations. Measured rainfall downstream from these locations is used as input to a series of subcatchment regional hydrological model to quantify additional local inflows along the main channel. A regional simplified‐physics hydraulic model is then applied to combine these contributions and generate an ensemble of discharge and water elevation time series at the boundaries of a local‐scale high complexity hydraulic model. Finally, the effect of these rainfall dynamics and uncertain boundary conditions are evaluated on the local‐scale model. Accurate prediction of the flood peak was obtained with the proposed method, which was only possible by resolving the additional complexity of the extreme rainfall contributions over the modeled area. The findings highlight the importance of estimating boundary condition uncertainty and local rainfall contributions for accurate prediction of river flows and inundation at regional scales.

Novel reliability-based optimization method for thermal structure with hybrid random, interval and fuzzy parameters

• New reliability model with hybrid random, interval and fuzzy parameters. • Reliability is defined by novel interval ranking strategy and integral operation. • Extra safety information in the transition state is fully utilized. • Cut levels of different fuzzy variables are assumed to be independent. • Subinterval vertex method is presented for extrema of interval limit state function. In this paper, novel reliability-based optimization model and method are proposed for thermal structure design with random, interval and fuzzy uncertainties in material properties, external loads and boundary conditions. Random variables are used to quantify the probabilistic uncertainty with sufficient sample data; whereas, interval variables and fuzzy variables are adopted to model the non-probabilistic uncertainty associated with objective limited information and subjective expert opinions, respectively. Using the interval ranking strategy, the level-cut limit state function is precisely quantified to represent the safety state. The eventual safety possibility is derived based on multiple integral, where the cut levels of different fuzzy variables are considered to be independent. Then a hybrid reliability-based optimization model is established with considerable computational cost caused by three-layer nested loop. To improve the computational efficiency, a subinterval vertex method is presented to replace the inner-loop and middle-loop. Comparing numerical results with traditional reliability model, a mono-objective example and a multi-objective example are provided to demonstrate the feasibility of proposed method for hybrid reliability analysis and optimization in practical engineering.

Constraints on ocean carbonate chemistry and pCO2 in the Archaean and Palaeoproterozoic

One of the great problems in the history of Earth’s climate is how to reconcile evidence for liquid water and habitable climates on early Earth with the Faint Young Sun predicted from stellar evolution models. Possible solutions include a wide range of atmospheric and oceanic chemistries, with large uncertainties in boundary conditions for the evolution and diversification of life and the role of the global carbon cycle in maintaining habitable climates. Increased atmospheric CO_2 is a common component of many solutions, but its connection to the carbon chemistry of the ocean remains unknown. Here we present calcium isotope data spanning the period from 2.7 to 1.9 billion years ago from evaporitic sedimentary carbonates that can test this relationship. These data, from the Tumbiana Formation, the Campbellrand Platform and the Pethei Group, exhibit limited variability. Such limited variability occurs in marine environments with a high ratio of calcium to carbonate alkalinity. We are therefore able to rule out soda ocean conditions during this period of Earth history. We further interpret this and existing data to provide empirical constraints for carbonate chemistry of the ancient oceans and for the role of CO_2 in compensating for the Faint Young Sun.