Bivariate frequency analysis of nonstationary low‐flow series based on the time‐varying copula

Many studies have analysed the nonstationarity in single hydrological variables due to changing environments. Yet, few researches have been done to investigate how the dependence structure between different individual hydrological variables is affected by changing environments. To investigate how the reservoirs have altered the dependence structure between river flows at different locations on the Hanjiang River, a time‐varying copula model, which takes the nonstationarity in the marginal distribution and/or the time variation in dependence structure between different hydrological series into consideration, is presented in this paper to perform a bivariate frequency analysis for the low‐flow series from two neighbouring hydrological gauges. The time‐varying moments model with either time or reservoir index as explanatory variables is applied to build the time‐varying marginal distributions of the two low‐flow series. It's found that both marginal distributions are nonstationary, and the reservoir index yields better performance than the time index in describing the nonstationarities in the marginal distributions. Then, the copula with the dependence parameter expressed as a function of either time or reservoir index is applied to model the variable dependence between the two low‐flow series. The copula with reservoir index as the explanatory variable of the dependence parameter has a better fitting performance than the copula with the constant or the time‐trend dependence parameter. Finally, the effect of the time variation in the joint distribution on three different types of joint return periods (i.e. AND, OR and Kendall) of low flows at two neighbouring hydrological gauges is presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Climate‐informed low‐flow frequency analysis using nonstationary modelling

Stationarity is often assumed for frequency analysis of low flows in water resources management and planning. However, many studies have shown that flow characteristics, particularly the frequency spectrum of extreme hydrologic events, were modified by climate change and human activities. Thus, the conventional frequency analysis that fails to consider the nonstationary characteristics may lead to costly design. The analysis presented in this paper was based on the more than 100 years of daily flow data from the Yichang gauging station 44 km downstream of the Three Gorges Dam. The Mann–Kendall trend test under the scaling hypothesis showed that the annual low flows had a significant monotonic trend, whereas an abrupt change point was identified in 1936 by the Pettitt test. The climate‐informed low‐flow frequency analysis and the divided and combined method were employed to account for the impacts from related climate variables and nonstationarities in annual low flows. Without prior knowledge of the probability density function for the gauging station, six distribution functions including the generalized extreme values (GEV), Pearson Type III, Gumbel, Gamma, Lognormal and Weibull distributions have been tested to find the best fit, in which the local likelihood method is used to estimate the parameters. Analyses show that GEV had the best fit for the observed low flows. This study has also shown that the climate‐informed low‐flow frequency analysis is able to exploit the link between climate indices and low flows, which would account for the dynamic feature for reservoir management and provide more accurate and reliable designs for infrastructure and water supply. Copyright © 2014 John Wiley & Sons, Ltd.     

Regional analysis of low flow using L-moments for Dongjiang basin,South China

Abstract

Dongjiang water has been the key source of water supplies for Hong Kong and its neighbouring cities in the Pearl River Delta in South China since the mid-1960s. Rapid economic development and population growth in this region have caused serious concerns over the adequacy of the quantity and quality of water withdrawn from the Dongjiang River in the future. Information on the magnitude and frequency of low flows in the basin is needed for planning of water resources at present and in the near future. The L-moment method is used to analyse the regional frequency of low flows, since recent studies have shown that it is superior to other methods that have been used previously, and is now being adopted by many organizations worldwide. In this study, basin-wide analysis of low flows is conducted for Dongjiang basin using five distributions: generalized logistic, generalized extreme value, lognormal, Pearson type III and generalized Pareto. Each of these has three parameters estimated by the L-moment method. The discordancy index and homogeneity testing show that 14 out of the 16 study sites belong to a homogenous region; these are used for further analysis. Based on the L-moment ratios diagram, the Hosking and Wallis goodness-of-fit statistical criterion and the L-kurtosis criterion, the three-parameter lognormal distribution is identified as the most appropriate distribution for the homogeneous study region. The regional low-flow estimates for each return period are obtained using the index flood procedure. Examination of the observed and simulated low flows by regional frequency analysis shows a good agreement in general, and the results may satisfy practical application. Furthermore, the regional low-flow relationship between mean annual 7-day low flows and basin area is developed using linear regression, providing a simple and effective method for estimation of low flows of desired return periods for ungauged catchments.     

Perennial springs provide information to predict low flows in mountain basins

A new method for estimating low flows in ungauged rivers from minimum discharge of perennial springs is proposed. This spring-based approach (SBA) is tested in 21 catchments from the northern Apennines, Italy. First, the hydrogeological behaviour of each geological formation and superficial deposit is related to the spatial distribution and discharge of perennial springs in a test area using a Bayesian approach, weight of evidence (WoE). Second, the observed river flow exceeded for 95% of the observation period is related to the type of geological formations outcropping within the catchment. Finally, the q95 low flows are estimated from the WoE weights. The SBA performance is assessed by leave-one-out cross-validation and compared with the results of a multiple regression (MR) model that accounts for selected catchment characteristics, but no springs. The results show that the SBA outperforms MR. The better performance of the SBA may be related to its ability to capture bedrock characteristics, which are the main controls of low flows in the study area.     

Identification of uncertainty in low flow frequency analysis using Bayesian MCMC method

This study employs the Bayesian Markov Chain Monte Carlo (MCMC) method with the Metropolis–Hastings algorithm and maximum likelihood estimation (MLE) using a quadratic approximation of the likelihood function for the evaluation of uncertainties in low flow frequency analysis using a two‐parameter Weibull distribution. The two types of prior distributions, a non‐data‐based distribution and a data‐based distribution using regional information collected from neighbouring stations, are used to establish a posterior distribution. Eight case studies using the synthetic data with a sample size of 100, generated from two‐parameter Weibull distribution, are performed to compare with results of analysis using MLE and Bayesian MCMC. Also, Bayesian MCMC and MLE are applied to 36 years of gauged data to validate the efficiency of the developed scheme. These examples illustrate the advantages of Bayesian MCMC and the limitations of MLE based on a quadratic approximation. From the point of view of uncertainty analysis, Bayesian MCMC is more effective than MLE using a quadratic approximation when the sample size is small. In particular, Bayesian MCMC method is more attractive than MLE based on a quadratic approximation because the sample size of low flow at the site of interest is mostly not enough to perform the low flow frequency analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

Bivariate flood frequency analysis using the copula function: a case study of the Litija station on the Sava River

As an alternative to the commonly used univariate flood frequency analysis, copula frequency analysis can be used. In this study, 58 flood events at the Litija gauging station on the Sava River in Slovenia were analysed, selected based on annual maximum discharge values. Corresponding hydrograph volumes and durations were considered. Different bivariate copulas from three families were applied and compared using different statistical, graphical and upper tail dependence tests. The parameters of the copulas were estimated using the method of moments with the inversion of Kendall's tau. The Gumbel–Hougaard copula was selected as the most appropriate for the pair of peak discharge and hydrograph volume (Q‐V). The same copula was also selected for the pair hydrograph volume and duration (V‐D), and the Student‐t copula was selected for the pair of peak discharge and hydrograph duration (Q‐D). The differences among most of the applied copulas were not significant. Different primary, secondary and conditional return periods were calculated and compared, and some relationships among them were obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

The impacts of climate change on local hydrology and low flow frequency in the Geum River Basin,Korea

The climate sensitive analysis of potential climate change on streamflow has been conducted using a hydrologic model to identify hydrologic variability associated with climate scenarios as a function of perturbed climatic variables (e.g. carbon dioxide, temperature, and precipitation). The interannual variation of water resources availability as well as low flow frequency driven by monsoonal time shifts have been investigated to evaluate the likelihood of droughts in a changing climate. The results show that the timing shift of the monsoon window associated with future climate scenarios clearly affect annual water yield change of ? 12 and ? 8% corresponding to 1‐month earlier and 1‐month later monsoon windows, respectively. Also, a more severe low flow condition has been predicted at 0·03 m³/s as opposed to the historic 7Q10 flow of 1·54 m³/s given at extreme climate scenarios. Copyright © 2011 John Wiley & Sons, Ltd.     

Analysis of nonstationarity in low flow in the Loess Plateau of China

An approach for nonstationary low‐flow frequency analysis is developed and demonstrated on a dataset from the rivers on the Loess Plateau of China. Nonstationary low‐flow frequency analysis has drawn significant attention in recent years by establishing relationships between low‐flow series and explanatory variables series, but few studies have tested whether the time‐varying moments of low flow can be fully described by the time‐varying moments of the explanatory variables. In this research, the low‐flow distributions are analytically derived from the 2 basic explanatory variables—the recession duration and the recession coefficient—with the assumption that the recession duration and recession coefficient variables follow exponential and gamma distributions, respectively; the derived low‐flow distributions are applied to test whether the time‐varying moments of explanatory variables can explain the nonstationarities found in the low‐flow variable. The effects of ecosystem construction measures, that is, check dam, terrace, forest, and grassland, on the recession duration and recession coefficient are further discussed. Daily flow series from 11 hydrological stations from the Loess Plateau are used and processed with a moving average technique. Low‐flow data are extracted following the pit under threshold approach. Six of the 11 low‐flow series show significant nonstationarities at the 5% significance level, and the trend curves of the moments of low flow are in close agreement with the curves estimated from the derived distribution with time‐dependent moments of the recession duration and time‐constant moments of the recession coefficient. It is indicated that the nonstationarity in the low‐flow distribution results from the nonstationarity in the recession duration in all 6 cases, and the increase in the recession duration is resulted from large‐scale ecosystem constructions rather than climate change. The large‐scale ecosystem constructions are found to have more influence on the decrease in streamflow than on the increase in watershed storage, thus resulting in the reduction of low flow. A high return period for the initial fixed design value decreases dramatically with an increasing recession duration.     

Past,present and future of a meandering river in the Bolivian Amazon basin

Field observations on small rivers of the Amazon basin are less common due to their remote location and difficult accessibility. Here we show, through remote sensing analysis and field works, the planform evolution and riverbed topography of a small river located in the upper foreland Amazon basin, the Ichilo River. By tracking planform changes over 30 years, we identified the factors that control meander migration rates in the Ichilo River: cutoffs, climate and human interventions. The data suggest that neck cutoffs are the main controls in the Ichilo River, with an annual density of 0.022 cutoffs/km. In addition, climate controls have been identified in the form of high-precipitation events that may have promoted cutoffs, an increase in meander migration rate and channel widening. The width distribution of the Ichilo River is well represented by general extreme value and inverse Gaussian distributions. The spatiotemporal variability of meandering migration rates in the Ichilo River is analysed in two locations where neck cutoffs are expected. Analysing the distance across the neck in these two points, we predict the occurrence of a new cutoff. The combined methodology of bathymetric surveys and structure from motion photogrammetry shows us the Ichilo riverbed topography and banks at high resolution, where two scour holes were identified. Finally, we discuss the impact of planform changes of the Ichilo River on communities that are established along its riverbanks.     

Temporal and spatial variability of extreme river flow quantiles in the Upper Vistula River basin,Poland

Temporal and spatial variability in extreme quantile anomalies of seasonal and annual maximum river flows was studied for 41 gauging stations at rivers in the Upper Vistula River basin, Poland. Using the quantile perturbation method, the temporal variability in anomalies was analysed. Interdecadal oscillating components were extracted from the series of anomalies using the Hilbert‐Huang transform method. Period length, part of variance of each component, and part of unexplained variance were assessed. Results show an oscillating pattern in the temporal occurrence of extreme flow quantiles with clusters of high values in the 1960–1970s and since the late 1990s and of low values in the 1980s and at the beginning of the 1990s. The anomalies show a high variability on the right bank of the Upper Vistula River basin during the summer season with the highest values in catchments located in the western and south‐western parts of the basin. River flow extreme quantiles were found to be associated with large‐scale climatic variables from the regions of the North Atlantic Ocean, Scandinavia, Eastern Europe, Asia, and, to a lesser extent, the Pacific Ocean. Similarities between temporal variability of river flows and climatic factors were revealed. Results of the study are important for flood frequency analysis because a long observation period is necessary to capture clusters of high and low river flows.     

Predictability of river flow and suspended sediment transport in the Mississippi River basin: a non‐linear deterministic approach

As the Mississippi River plays a major role in fulfilling various water demands in North America, accurate prediction of river flow and sediment transport in the basin is crucial for undertaking both short‐term emergency measures and long‐term management efforts. To this effect, the present study investigates the predictability of river flow and suspended sediment transport in the basin. As most of the existing approaches that link water discharge, suspended sediment concentration and suspended sediment load possess certain limitations (absence of consensus on linkages), this study employs an approach that presents predictions of a variable based on history of the variable alone. The approach, based on non‐linear determinism, involves: (1) reconstruction of single‐dimensional series in multi‐dimensional phase‐space for representing the underlying dynamics; and (2) use of the local approximation technique for prediction. For implementation, river flow and suspended sediment transport variables observed at the St. Louis (Missouri) station are studied. Specifically, daily water discharge, suspended sediment concentration and suspended sediment load data are analysed for their predictability and range, by making predictions from one day to ten days ahead. The results lead to the following conclusions: (1) extremely good one‐day ahead predictions are possible for all the series; (2) prediction accuracy decreases with increasing lead time for all the series, but the decrease is much more significant for suspended sediment concentration and suspended sediment load; and (3) the number of mechanisms dominantly governing the dynamics is three for each of the series. Copyright © 2005 John Wiley & Sons, Ltd.     

Simulation of river flow in Britain under climate change: Baseline performance and future seasonal changes

Climate change is likely to manifest in river flow changes across the globe, which could have wide-ranging consequences for society and the natural environment. A number of previous studies used the UK Climate Projections 2009 (UKCP09) to investigate the potential impacts on river flows in Britain, but these projections were recently updated by the release of UKCP18, thus there is a need to update flow studies. Here, the UKCP18 Regional (12 km) projections are applied using a national-scale grid-based hydrological model, to investigate potential future changes in seasonal mean river flows across Great Britain. Analysis of hydrological model performance using baseline climate model data (1980–2010) shows relatively good agreement with use of observation-based data, particularly after application of a monthly precipitation bias-correction. Analysis of seasonal mean flow changes for two future time-slices (2020–2050 and 2050–2080) suggests large decreases in summer flows across the country (median −45% by 2050–2080), but possible increases in winter flows (median 9% by 2050–2080), especially in the north and west. Information on the potential range of flow changes using the latest projections is necessary to develop appropriate adaptation strategies, and comparisons with previous projections can help update existing plans, although such comparisons are often not straightforward.     

Performance of flood frequency pooling analysis in a low CV context

Abstract

Pooling of flood data is widely used to provide a framework to estimate design floods by the Index Flood method. Design flood estimation with this approach involves derivation of a growth curve which shows the relationship between X_T and the return period T, where X_T ?=?Q_T /Q_I and Q_I is the index flood at the site of interest. An implicit assumption with the Index Flood procedure of pooling analysis is that the X_T –T relationship is the same at all sites in a homogeneous pooling group, although this assumption would generally be violated to some extent in practical cases, i.e. some degree of heterogeneity exists. In fact, in only some cases is the homogeneity criterion effectively satisfied for Irish conditions. In this paper, the performance of the index-flood pooling analysis is assessed in the Irish low CV (coefficient of variation) hydrology context considering that heterogeneity is taken into account. It is found that the performance of the pooling method is satisfactory provided there are at least 350 station years of data included. Also it is found that, in a highly heterogeneous group, it is more desirable to have many sites with short record lengths than a smaller number of sites with long record lengths. Increased heterogeneity decreases the advantage of pooling group-based estimation over at-site estimation. Only a heterogeneity measure (H1) less than 4.0 can render the pooled estimation preferable to that obtained for at-site estimation for the estimation of 100-year flood. In moderately to highly heterogeneous regions it is preferable to conduct at-site analysis for the estimation of 100-year flood if the record length at the site concerned exceeds 50.

Editor Z.W. Kundzewicz; Associate editor A. Carsteanu

Citation Das, S. and Cunnane, C., 2012. Performance of flood frequency pooling analysis in a low CV context. Hydrological Sciences Journal, 57 (3), 433–444.     

The effects of ice cover on flow characteristics in a subarctic meandering river

The effects of ice cover on flow characteristics in meandering rivers are still not completely understood. Here, we quantify the effects of ice cover on flow velocity, the vertical and spatial flow distribution, and helical flow structure. Comparison with open‐channel low flow conditions is performed. An acoustic doppler current profiler (ADCP) is used to measure flow from up to three meander bends, depending on the year, in a small sandy meandering subarctic river (Pulmanki River) during two consecutive ice‐covered winters (2014 and 2015). Under ice, flow velocities and discharges were predominantly slower than during the preceding autumn open‐channel conditions. Velocity distribution was almost opposite to theoretical expectations. Under ice, velocities reduced when entering deeper water downstream of the apex in each meander bend. When entering the next bend, velocities increased again together with the shallower depths. The surface velocities were predominantly greater than bottom/riverbed velocities during open‐channel flow. The situation was the opposite in ice‐covered conditions, and the maximum velocities occurred in the middle layers of the water columns. High‐velocity core (HVC) locations varied under ice between consecutive cross‐sections. Whereas in ice‐free conditions the HVC was located next to the inner bank at the upstream cross‐sections, the HVC moved towards the outer bank around the apex and again followed the thalweg in the downstream cross‐sections. Two stacked counter‐rotating helical flow cells occurred under ice around the apex of symmetric and asymmetric bends: next to the outer bank, top‐ and bottom‐layer flows were towards the opposite direction to the middle layer flow. In the following winter, no clear counter‐rotating helical flow cells occurred due to the shallower depths and frictional disturbance by the ice cover. Most probably the flow depth was a limiting factor for the ice‐covered helical flow circulation, similarly, the shallow depths hinder secondary flow in open‐channel conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

Morphology and sedimentology of a complex debris flow in a clay‐shale basin

Coupling morphological, sedimentological, and rheological studies to numerical simulations is of primary interest in defining debris‐flow hazard on alluvial fans. In particular, numerical runout models must be carefully calibrated by morphological observations. This is particularly true in clay‐shale basins where hillslopes can provide a large quantity of poorly sorted solid materials to the torrent, and thus change both the mechanics of the debris flow and its runout distance. In this context, a study has been completed on the Faucon stream (southeastern French Alps), with the objectives of (1) defining morphological and sedimentological characteristics of torrential watersheds located in clay‐shales, and (2) evaluating through a case study the scouring potential of debris flows affecting a clay‐shale basin. Morphological surveys, grain‐size distributions and petrographic analyses of the debris‐flow deposits demonstrate the granular character of the flow during the first hectometre, and its muddy character from there to its terminus on the debris fan. These observations and laboratory tests suggest that the contributing areas along the channel have supplied the bulk of the flow material. Copyright © 2005 John Wiley & Sons, Ltd.     

Regression‐based approach to low flow prediction in the Maryland Piedmont region under joint climate and land use change

We examine the low flow records for six urbanized watersheds in the Maryland Piedmont region and develop regression equations to predict annual minimum low flow events. The effects of both future climate (based on precipitation and temperature projections from two climate models: Hadley and the Canadian Climate Centre (CCC)) and land use change are incorporated to illustrate possible future trends in low flows. A regression modelling approach is pursued to predict the minimum annual 7‐day low flow estimates for the proposed future scenarios. A regional regression model was calibrated with between 10 and 50 years of daily precipitation, daily average temperature, annual imperviousness, and the daily observed flow time‐series across six watersheds. Future simulations based on a 55 km² urbanizing watershed just north of Washington, DC, were performed. When land use and climate change were employed singly, the former predicted no trends in low flows and the latter predicted significant increasing trends under Hadley and no trends under CCC. When employed jointly, however, low flows were predicted to decrease significantly under CCC, whereas Hadley predicted no significant trends in low flows. Antecedent precipitation was the most influential predictor on low flows, followed by urbanization. Copyright © 2006 John Wiley & Sons, Ltd.     

Flood frequency under the influence of trends in the Pearl River basin,China: changing patterns,causes and implications

Using a nonstationary flood frequency model, this study investigates the impact of trends on the estimation of flood frequencies and flood magnification factors. Analysis of annual peak streamflow data from 28 hydrological stations across the Pearl River basin, China, shows that: (1) northeast parts of the West and the North River basins are dominated by increasing annual peak streamflow, whereas decreasing trends of annual peak streamflow are prevailing in other regions of the Pearl River basin; (2) trends significantly impact the estimation of flood frequencies. The changing frequency of the same flood magnitude is related to the changing magnitude or significance/insignificance of trends, larger increasing frequency can be detected for stations with significant increasing trends of annual peak streamflow and vice versa, and smaller increasing magnitude for stations with not significant increasing annual peak streamflow, pointing to the critical impact of trends on estimation of flood frequencies; (3) larger‐than‐1 flood magnification factors are observed mainly in the northeast parts of the West River basin and in the North River basin, implying magnifying flood processes in these regions and a higher flood risk in comparison with design flood‐control standards; and (4) changes in hydrological extremes result from the integrated influence of human activities and climate change. Generally, magnifying flood regimes in the northeast Pearl River basin and in the North River basin are mainly the result of intensifying precipitation regime; smaller‐than‐1 flood magnification factors along the mainstream of the West River basin and also in the East River basin are the result of hydrological regulations of water reservoirs. Copyright © 2014 John Wiley & Sons, Ltd.     

Evaluation of basin storage–discharge sensitivity in Taiwan using low‐flow recession analysis

Sensitivity analysis of the hydrological behaviour of basins has mainly focused on the correlation between streamflow and climate, ignoring the uncertainty of future climate and not utilizing complex hydrological models. However, groundwater storage is affected by climatic change and human activities. The streamflow of many basins is primarily sourced from the natural discharge of aquifers in upstream regions. The correlation between streamflow and groundwater storage has not been thoroughly discussed. In this study, the storage–discharge sensitivity of 22 basins in Taiwan was investigated by means of daily streamflow and rainfall data obtained over more than 30 years. The relationship between storage and discharge variance was evaluated using low‐flow recession analysis and a water balance equation that ignores the influence of rainfall and evapotranspiration. Based on the obtained storage–discharge sensitivity, this study explored whether the water storage and discharge behaviour of the studied basins is susceptible to climate change or human activities and discusses the regional differences in storage–discharge sensitivity. The results showed that the average storage–discharge sensitivities were 0.056 and 0.162 mm^?1 in the northern and southern regions of Taiwan, respectively. In the central and eastern regions, the values were both 0.020 mm^?1. The storage–discharge sensitivity was very high in the southern region. The regional differences in storage–discharge sensitivity with similar climate conditions are primarily due to differences in aquifer properties. Based on the recession curve, other factors responsible for these differences include land utilization, land coverage, and rainfall patterns during dry and wet seasons. These factors lead to differences in groundwater recharge and thus to regional differences in storage–discharge sensitivity.     

Changes in stream flow regime in headwater catchments of the Yellow River basin since the 1950s

The headwater catchments of the Yellow River basin generate over 35% of the basin's total stream flow and play a vital role in meeting downstream water resources requirements. In recent years the Yellow River has experienced significant changes in its hydrological regime, including an increased number of zero‐flow days. These changes have serious implications for water security and basin management. We investigated changes in stream flow regime of four headwater catchments since the 1950s. The rank‐based non‐parametric Mann–Kendall test was used to detect trends in annual stream flow. The results showed no significant trend for the period 1956 to 2000. However, change‐point analysis showed that a significant change in annual stream flow occurred around 1990, and hence the stream‐flow data can be divided into two periods: 1956–1990 and 1991–2000. There was a considerable difference in average annual stream flow between the two periods, with a maximum reduction of 51%. Wet‐season rainfall appears to be the main factor responsible for the decreasing trend in annual stream flow. Reductions in annual stream flow were associated with decreased interannual variability in stream flow. Seasonal stream flow distribution changed from bimodal to unimodal between the two periods, with winter stream flow showing a greater reduction than other seasons. Daily stream flow regime represented by flow duration curves showed that all percentile flows were decreased in the second period. The high flow index (Q₅/Q₅₀) reduced by up to 28%, whereas the reduction in the low flow index (Q₉₅/Q₅₀) is more dramatic, with up to 100% reduction. Copyright © 2006 John Wiley & Sons, Ltd.     

The structure of separated flow past a circular cylinder in a rotating frame

Abstract

This paper examines the detailed E ^1/4-layer structure of separated flow past a circular cylinder in a low-Rossby-number rotating fluid as the Ekman number E tends to zero. This structure is based on an initial proposal by Page (1987) but with some modifications in response to further evidence, outlined both in this paper and elsewhere, on the behaviour of E ^1/4-layer flows in this context. Numerical calculations for flow in an E ^1/4 shear layer along the separated free streamline are described and the mass flux from this layer is then used to calculate the higher-order flow within the separation bubble. The flow structure is found to have two forms, depending on the value of the O(1) parameter λ, and these are compared with results from published “Navier-Stokes” type calculations for the flow at small but finite values of E.