On the photo-gravitational restricted four-body problem with variable mass

This paper deals with the photo-gravitational restricted four-body problem (PR4BP) with variable mass. Following the procedure given by Gascheau (C. R. 16:393–394, 1843) and Routh (Proc. Lond. Math. Soc. 6:86–97, 1875), the conditions of linear stability of Lagrange triangle solution in the PR4BP are determined. The three radiating primaries having masses \(m_{1}\), \(m_{2}\) and \(m_{3}\) in an equilateral triangle with \(m_{2}=m_{3}\) will be stable as long as they satisfy the linear stability condition of the Lagrangian triangle solution. We have derived the equations of motion of the mentioned problem and observed that there exist eight libration points for a fixed value of parameters \(\gamma (\frac{m \ \text{at time} \ t}{m \ \text{at initial time}}, 0<\gamma\leq1 )\), \(\alpha\) (the proportionality constant in Jeans’ law (Astronomy and Cosmogony, Cambridge University Press, Cambridge, 1928), \(0\leq\alpha\leq2.2\)), the mass parameter \(\mu=0.005\) and radiation parameters \(q_{i}, (0< q_{i}\leq1, i=1, 2, 3)\). All the libration points are non-collinear if \(q_{2}\neq q_{3}\). It has been observed that the collinear and out-of-plane libration points also exist for \(q_{2}=q_{3}\). In all the cases, each libration point is found to be unstable. Further, zero velocity curves (ZVCs) and Newton–Raphson basins of attraction are also discussed.     

Turbulence Study Around Bar in a Braided River Model

Water Resources - Turbulent flow is a flow regime which is described by the anarchic property changes. This includes the rapid variation of pressure, high momentum convection and flow velocity in...     

Minimizing the impacts of contaminant intrusion in small water distribution networks through booster chlorination optimization

Contaminant intrusion in a water distribution network (DN) has three basic pre-conditions: source of contaminant (e.g., leaky sewer), a pathway (e.g., water main leaks), and a driving force (e.g., negative pressure). The impact of intrusion can be catastrophic if residual disinfectant (chlorine) is not present. To avoid microbiological water quality failure, higher levels of secondary chlorination doses can be a possible solution, but they can produce disinfectant by-products which lead to taste and odour complaints. This study presents a methodology to identify potential intrusion points in a DN and optimize booster chlorination based on trade-offs among microbiological risk, chemical risk and life-cycle cost for booster chlorination. A point-scoring scheme was developed to identify the potential intrusion points within a DN. It utilized factors such as pollutant source (e.g., sewer characteristics), pollution pathway (water main diameter, length, age, and surrounding soil properties, etc.), consequence of contamination (e.g., population, and land use), and operational factors (e.g., water pressure) integrated through a geographical information system using advanced ArcMap 10 operations. The contaminant intrusion was modelled for E. Coli O156: H7 (a microbiological indicator) using the EPANET-MSX programmer’s toolkit. The quantitative microbial risk assessment and chemical (human health) risk assessment frameworks were adapted to estimate risk potentials. Booster chlorination locations and dosages were selected using a multi-objective genetic algorithm. The methodology was illustrated through a case study on a portion of a municipal DN.     

Cyclic liquefaction screening of sand with non-plastic fines:Critical state approach

There have been significant advances in the application of critical state,CS,in liquefaction potential assessment.This was done by comparing state parameter,j with estimated characteristic cyclic stress ratio,CSR due to an earthquake.A cyclic resistance ratio,CRR curve,which can be determined from cyclic liquefaction tests,separates historical liquefied and non-liquefied data points(j,CSR).On the other hand,the concepts of equivalent granular state parameter,j*,which was developed for sands with fines,can be used in lieu j to provide a unifying framework for characterizing the undrained response of sands with non/low plasticity fines,irrespective of fines content(fc).The present work combines these two propositions,and by merely substituting j*for j into the aforementioned CS approach to capture the influence of fc.A series of static and cyclic triaxial tests were conducted,separately and independently of the concept of j*,for sand with up to fc of 30%.The clean sand was collected from Sabarmati river belt at Ahmedabad city in India which was severely affected during the Bhuj earthquake,2001.The experimental data gave a single relation for CRR and j*which was then used to assess liquefaction potential for a SPT based case study,where fc varies along the depth.The prediction matched with the field observation.     

Sensitivity and uncertainty analysis of mesoscale model downscaled hydro‐meteorological variables for discharge prediction

Precipitation and Reference Evapotranspiration (ETo) are the most important variables for rainfall–runoff modelling. However, it is not always possible to get access to them from ground‐based measurements, particularly in ungauged catchments. This study explores the performance of rainfall and ETo data from the global European Centre for Medium Range Weather Forecasts (ECMWF) ERA interim reanalysis data for the discharge prediction. The Weather Research and Forecasting (WRF) mesoscale model coupled with the NOAH Land Surface Model is used for the retrieval of hydro‐meteorological variables by downscaling ECMWF datasets. The conceptual Probability Distribution Model (PDM) is chosen for this study for the discharge prediction. The input data and model parameter sensitivity analysis and uncertainty estimations are taken into account for the PDM calibration and prediction in the case study catchment in England following the Generalized Likelihood Uncertainty Estimation approach. The goodness of calibration and prediction uncertainty is judged on the basis of the p‐factor (observations bracketed by the prediction uncertainty) and the r‐factor (achievement of small uncertainty band). The overall analysis suggests that the uncertainty estimates using WRF downscaled ETo have slightly smaller p and r values (p= 0.65; r= 0.58) as compared to ground‐based observation datasets (p= 0.71; r= 0.65) during the validation and hence promising for discharge prediction. On the contrary, WRF precipitation has the worst performance, and further research is needed for its improvement (p= 0.04; r= 0.10). Copyright © 2013 John Wiley & Sons, Ltd.     

Entropy,consistency in rainfall distribution and potential water resource availability in Australia

Understanding the variability in monthly rainfall amounts is important for the management of water resources. We use entropy, a measure of variability, to quantify the rainfall variability in Australia. We define the entropy of stable rainfall (ESR) to measure the long‐term average rainfall variability across the months of the year. The stations in northern Australia observe substantially more variability in rainfall distributions and stations in southern Australia observe less variability in rainfall distribution across the months of the year. We also define the consistency index (CI) to compare the distribution of the monthly rainfall for a given year with the long‐term average monthly rainfall distribution. Higher value of the CI indicates the rainfall in the year is consistent with the overall long‐term average rainfall distribution. Areas close to the coastline in northern, southern and eastern Australia observe more consistent rainfall distribution in individual years with the long‐term average rainfall distribution. For the studied stations, we categorize the years into different potential water resource availability on the basis of annual rainfall amount and CI. For almost all Australian rainfall stations, El Niño years have a greater risk of having below median and relatively inconsistent rainfall distribution than La Niña years. The results may be helpful for developing area‐specific water usage strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

Error distribution modelling of satellite soil moisture measurements for hydrological applications

Satellite‐based soil moisture data accuracies are of important concerns by hydrologists because they could significantly influence hydrological modelling uncertainty. Without proper quantification of their uncertainties, it is difficult to optimize the hydrological modelling system and make robust decisions. Currently, the satellite soil moisture data uncertainty has been limited to summary statistics with the validations mainly from the in situ measurements. This study attempts to build the first error distribution model with additional higher‐order uncertainty modelling for satellite soil moisture observations. The methodology is demonstrated by a case study using the Soil Moisture and Ocean Salinity satellite soil moisture observations. The validation is based on soil moisture estimates from hydrological modelling, which is more relevant to the intended data use than the in situ measurements. Four probability distributions have been explored to find suitable error distribution curves using the statistical tests and bootstrapping resampling technique. General extreme value is identified as the most suitable one among all the curves. The error distribution model is still in its infant stage, which ignores spatial and temporal correlations, and nonstationarity. Further improvements should be carried out by the hydrological community by expanding the methodology to a wide range of satellite soil moisture data using different hydrological models. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimation of catchment yield and associated uncertainties due to climate change in a mountainous catchment in Australia

This paper examines the impacts of climate change on future water yield with associated uncertainties in a mountainous catchment in Australia using a multi‐model approach based on four global climate models (GCMs), 200 realisations (50 realisations from each GCM) of downscaled rainfalls, 2 hydrological models and 6 sets of model parameters. The ensemble projections by the GCMs showed that the mean annual rainfall is likely to reduce in the future decades by 2–5% in comparison with the current climate (1987–2012). The results of ensemble runoff projections indicated that the mean annual runoff would reduce in future decades by 35%. However, considerable uncertainty in the runoff estimates was found as the ensemble results project changes of the 5^th (dry scenario) and 95^th (wet scenario) percentiles by ?73% to +27%, ?73% to +12%, ?77% to +21% and ?80% to +24% in the decades of 2021–2030, 2031–2040, 2061–2070 and 2071–2080, respectively. Results of uncertainty estimation demonstrated that the choice of GCMs dominates overall uncertainty. Realisation uncertainty (arising from repetitive simulations for a given time step during downscaling of the GCM data to catchment scale) of the downscaled rainfall data was also found to be remarkably high. Uncertainty linked to the choice of hydrological models was found to be quite small in comparison with the GCM and realisation uncertainty. The hydrological model parameter uncertainty was found to be lowest among the sources of uncertainties considered in this study. Copyright © 2015 John Wiley & Sons, Ltd.     

Parameter uncertainty of the AWBM model when applied to an ungauged catchment

In this study, a quantitative assessment of uncertainty was made in connection with the calibration of Australian Water Balance Model (AWBM) for both gauged and ungauged catchment cases. For the gauged catchment, five different rainfall data sets, 23 different calibration data lengths and eight different optimization techniques were adopted. For the ungauged catchment case, the optimum parameter sets obtained from the nearest gauged catchment were transposed to the ungauged catchments, and two regional prediction equations were used to estimate runoff. Uncertainties were ascertained by comparing the observed and modelled runoffs by the AWBM on the basis of different combinations of methods, model parameters and input data. The main finding from this study was that the uncertainties in the AWBM modelling outputs could vary from ?1.3% to 70% owing to different input rainfall data, ?5.7% to 11% owing to different calibration data lengths and ?6% to 0.2% owing to different optimization techniques adopted in the calibration of the AWBM. The performance of the AWBM model was found to be dominated mainly by the selection of appropriate rainfall data followed by the selection of an appropriate calibration data length and optimization algorithm. Use of relatively short data length (e.g. 3 to 6 years) in the calibration was found to generate relatively poor results. Effects of different optimization techniques on the calibration were found to be minimal. The uncertainties reported here in relation to the calibration and runoff estimation by the AWBM model are relevant to the selected study catchments, which are likely to differ for other catchments. The methodology presented in this paper can be applied to other catchments in Australia and other countries using AWBM and similar rainfall–runoff models. Copyright © 2014 John Wiley & Sons, Ltd.     

Stability analysis of multi-directional anchored rock slope subjected to surcharge and seismic loads

More than one set of anchors with different orientations can be an economical solution to completely stabilize the rock slope. In this note, a general analytical expression for the factor of safety of a multi-directional anchored rock slope (MDARS) against plane failure is derived, incorporating most of the practically occurring destabilizing forces under surcharge and seismic loading conditions. Several special cases of this expression are found to be similar to those reported in the literature. A graphical presentation shows that for any specific inclination of one set of anchors to the normal at failure plane, the variation in the inclination of the second set of anchors to the normal at failure plane, when greater than approximately 60°, does not cause a significant change in the factor of safety of the MDARS.