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A third-order asymptotic solution in Lagrangian description for nonlinear water waves propagating over a sloping beach is derived. The particle trajectories are obtained as a function of the nonlinear ordering parameter 3 and the bottom slope a to the third order of perturbation. A new relationship between the wave velocity and the motions of particles at the free surface profile in the waves propagating on the sloping bottom is also determined directly in the complete Lagrangian framework. This solution enables the description of wave shoaling in the direction of wave propagation from deep to shallow water, as well as the successive deformation of wave profiles and water particle trajectories prior to breaking. A series of experiments are conducted to investigate the particle trajectories of nonlinear water waves propagating over a sloping bottom. It is shown that the present third-order asymptotic solution agrees very well with the experiments.

The environmental fate of polybrominated diphenyl ethers (PBDEs), a group of flame retardants that are considered to be persistent organic pollutants (POPs), around the Zhuoshui River and Changhua County regions of Taiwan was assessed. An investigation into emissions, partitioning, and fate of selected PBDEs was conducted based on the equilibrium constant (EQC) fugacity model developed at Trent University, Canada. Emissions for congeners PBDE 47, PBDE 99, and PBDE 209 to air (4.9–92 × 10−3 kg/h), soil (0.91–17.4 × 10−3 kg/h), and water (0.21–4.04 × 10−3 kg/h), were estimated by modifying previous models on PBDE emission rates by considering both industrial and domestic rates. It was found that fugacity modeling can give a reasonable estimation of the behavior, partitioning, and concentrations of PBDE congeners in and around Taiwan. Results indicate that PBDE congeners have a high affinity for partitioning into sediments then soils. As congener number decreases, the PBDEs then partition more readily into air. As the degree of bromination increases, congeners more readily partition to sediments. Sediments may then act as a long-term source of PBDEs which can be released back into the water column due to resuspension during storm events.Electronic supplementary materialThe online version of this article (doi:10.1007/s11356-016-6428-4) contains supplementary material, which is available to authorized users.

The main purpose of this study is that extending 1D Carrier-Wu-Yeh algorithm and analytical Green's function (AGF) to estimate the arbitrary irregular waveforms induced runup height and the inundation distance, and further builds a pre-calculated runup dataset. In this study, the multiplication and superposition is employed to replace the direct numerical integration. The waveforms are decomposed as numerous Fourier components using fast Fourier transformation. The corresponding mechanical energy can be calculated beforehand and save the results as a database-form. Using this process, the maximum runup height and the inundation distance can be quick calculated after determining the total mechanical energy. Based on application on the real tsunami events, the comparisons show that the present methodology can shorten the computing time in comparison with the direct numerical integration. Moreover, the present approach also applies on real tsunami events, 2004 Indian tsunami and 2011 Tohoku tsunami, to estimate the runup height and the inundation distance. The forecasted results are quite satisfactory in comparison with the field measurement, and it implies that the reasonable accuracy and the computing efficiency are both considered in this study. Keywords：1D Carrier-Wu-Yeh algorithm, analytical Green's function (AGF), 2004 Indian tsunami, 2011 Tohoku tsunami.