This paper investigates the weakly nonlinear isotropic bidirectional Benney–Luke (BL) equation, which is used to describe oceanic surface and internal waves in shallow water, with a particular focus on soliton dynamics. Using the Whitham modulation theory, we derive the modulation equations associated with the BL equation that describe the evolution of soliton amplitude and slope. By analysing rarefaction waves and shock waves within these modulation equations, we derive the Riemann invariants and modified Rankine–Hugoniot conditions. These expressions help characterise the Mach expansion and Mach reflection phenomena of bent and reverse bent solitons. We also derive analytical formulae for the critical angle and the Mach stem amplitude, showing that as the soliton speed is in the vicinity of unity, the results from the BL equation align closely with those of the Kadomtsev–Petviashvili (KP) equation. Corresponding numerical results are obtained and show excellent agreement with theoretical predictions. Furthermore, as a far-field approximation for the forced BL equation – which models wave and flow interactions with local topography – the modulation equations yield a slowly varying similarity solution. This solution indicates that the precursor wavefronts created by topography moving at subcritical or critical speeds take the shape of a circular arc, in contrast to the parabolic wavefronts observed in the forced KP equation.

The stability and dynamics of solitary waves propagating along the surface of an inviscid ferrofluid jet in the absence of gravity are investigated analytically and numerically. For the axisymmetric geometry, the problem is shown to be a conservative system with total energy as the Hamiltonian; however, one of the canonical variables differs from those in the classic water-wave problem in the Cartesian coordinate system. The Dirichlet–Neumann operator appearing in the kinetic energy is then expanded as a Taylor series, described in homogeneous powers of the surface displacement. Based on the further analysis of the Dirichlet–Neumann operator, a systematic procedure is proposed to derive reduced model equations of multiple scales in various asymptotic limits from the full Euler equations in the Hamiltonian/Lagrangian framework. Particularly, a fully dispersive model arising from retaining terms valid up to the quartic order in the series expansion of the kinetic energy, which results in quadratic and cubic algebraic nonlinearities in Hamilton's equations and henceforth is abbreviated as the cubic full-dispersion model, is proposed. By comparing bifurcation curves and wave profiles of various types of axisymmetric solitary waves among different model equations, the cubic full-dispersion model is found to agree well with the full Euler equations, even for waves of considerably large amplitudes. The stability properties of axisymmetric solitary waves subjected to longitudinal disturbances are verified with the newly proposed model. Our analytical results, consistent with Saffman's theory, indicate that in the axisymmetric cylindrical system, the stability exchange subjected to superharmonic perturbations also occurs at the stationary point of the speed-energy bifurcation curve. A series of numerical experiments for the stability and dynamics of solitary waves are performed via the numerical time integration of the model equation, and collision interactions between stable solitary waves show non-elastic features.

Evolutions of internal waves of different modes, particularly mode 1 and mode 2, passing over variable bathymetry are investigated based on a new numerical scheme. The problem is idealized as interfacial waves propagating on two interfaces of a three-layer density stratified fluid system with large-amplitude bottom topography. The Dirichlet-to-Neumann operators are introduced to reduce the spatial dimension by one and to adapt the three-layer system and significant topographic effects. However, for simplicity, nonlinear interactions between interfaces are neglected. Numerical techniques such as the Galerkin approximation, proven effective in previous works, are applied to save computational costs. Shoaling of linear waves on an uneven bottom is first studied to validate the proposed formulation and the corresponding numerical scheme. Then, for two-dimensional numerical experiments, mode transition phenomena excited by locally confined bottom obstacles and quickly varying topographies, including the Bragg resonance, mode-2 excitation, wave homogenization, etc., are investigated. In three-dimensional simulations, internal wave refraction by a Luneberg lens is considered, and good agreement is found in comparison with the ray theory. Finally, in the limiting case, when the top layer can be negligible (for example, a gas layer of extremely small density), the problem is reduced to a two-and-a-half-layer fluid system, where an interface and a surface are unknown free boundaries. In this situation, the surface signature of an internal wave is simulated and verified by introducing the realistic bathymetry of the Strait of Gibraltar and qualitatively compared with the satellite image.

This paper focuses on simulating turbulent flow over propagating waves by solving the full Navier–Stokes equations in a moving frame. A careful comparison of flow statistics with previous experimental and numerical results demonstrates, to some extent, the rationality of simplifying wind waves as turbulent flow over moving wave boundaries. The phase-averaging method is then applied to investigate the momentum and energy transfers between turbulent wind and waves propagating at slow, intermediate and fast speeds. The results suggest that the dominant mechanism for producing Reynolds shear stress (RSS) and turbulent kinetic energy (TKE) is related to the wave age. Slow waves produce RSS and TKE similar to a two-dimensional shear turbulence. However, a fast wave enhances the streamwise Reynolds normal stress, the windward side's negative RSS and the gradient of both streamwise and vertical velocities, leading to additional RSS and TKE productions that can be ignored under the slow wave regimes. A strengthening wave–turbulence exchange is also found for fast waves. The intermediate wave can be regarded as a transitional condition determining this change.

In this paper, a high-level Green–Naghdi (HLGN) model for large-amplitude internal waves in a two-layer fluid system, where the upper-fluid layer is of finite depth and the lower-fluid layer is of infinite depth, is developed under the rigid-lid free-surface approximation. The equations of the present HLGN model follow Euler's equations under the sole assumption that the horizontal and vertical velocity distributions along the vertical column are presented by known shape functions for each layer. The linear dispersion relations of the HLGN model for different levels are presented and compared with those obtained by other strongly nonlinear models for deep water, including the fully nonlinear models that include the dispersion effects $O(\mu )$ (where $\mu$ is the ratio of the upper-fluid layer depth to a typical wavelength) derived by Choi & Camassa (Phys. Rev. Lett., vol. 77, 1996, pp. 1759–1762) and $O(\mu ^2)$ derived by Debsarma et al. (J. Fluid Mech., vol. 654, 2010, pp. 281–303). It is shown that the HLGN model has a wider application range than other models. Solutions of travelling large-amplitude internal solitary waves in the absence and presence of background shear-current are then investigated by using the HLGN model. For the no-current cases, results obtained by the HLGN model show better agreement with Euler's solution on wave profile, velocity profile at the maximum interface displacement and wave speed compared with those obtained by other models. For the background shear-current cases, results obtained by the HLGN model also show good agreement with those obtained by solving the Dubreil-Jacotin–Long equation.

Transitional separating flow induced by a rectangular plate subjected to uniform incoming flow at Reynolds number (based on the incoming velocity and half plate height) 2000 is investigated using direct numerical simulation. The objective is to unveil the long-lasting mystery of low-frequency flapping motion (FM) in flow separation. At a fixed streamwise-vertical plane or from the perspective of previous experimental studies using pointwise or planar measurements, FM manifests as a low-frequency periodic switching between low and high velocities covering the entire separation bubble. The results indicate that in three-dimensional space, FM reflects an intricate evolution of streamwise elongated streaky structures under the influence of separated shear layer and mean flow reversal. The FM is an absolute instability, and is initiated through a lift-up mechanism boosted by mean flow deceleration near the crest of the separating streamline. At this particular location, the shear bends the vortex filament abruptly, so that one end is vertically struck into the first half of the separation bubble, whereas the other end is extended in the streamwise direction in the second half of the separation bubble. These two ends of vortex filament are mutually sustained and also stretched by the vertical acceleration and streamwise acceleration in the first and second halves of the separation bubble, respectively. This process periodically switches the low-velocity (or high-velocity) streaky structure to a high-velocity (or low-velocity) streaky structure encompassing the entire separation bubble, and thus flaps the separated shear layer up and down in the vertical direction. A ‘large vortex’ shedding manifests when the streaky structure switches signs. This large vortex is fundamentally different from the spanwise vortex shedding residing in the shear layer originated from the Kelvin–Helmholtz instability and successive vortex amalgamation. It is also believed that the three-dimensional evolution of streaky structures in the form of FM is applicable for both geometry- and pressure-induced separating flows.

Considering straw resource utilization and air pollution prevention, straw return has been commonly practiced in China. However, the practicability of plenty straw return in an emerging maize–rice rotation and their effects on soil C and N pools have not been extensively investigated. This study has been conducted to examine the effects of straw return on soil nutrients, soil functional C and N fractions, and then to figure out their relationships with yield and N use efficiency. Two treatments of straw return (S2Nck) and without straw return (S0Nck) were compared in 3-year field experiment, and subplots without N application were added in their respective plots in the third year. The results showed that, relative to the control (S0Nck), straw return significantly increased soil mineralized nitrogen (Nmin), available P, and exchange K content by 11.7%, 41.1%, and 17.4% averaged across 3-year experiments, respectively. Straw return substantially increased soil dissolved organic C, microbial biomass C, and microbial biomass N content by 73.0%, 25.2%, and 36.8%, respectively. Furthermore, straw return markedly increased C and N retention in particulate organic matter in microaggregates (iPOM) and mineral associated organic matter within microaggregates (intra-SC), but significantly reduced in free mineral associated organic matter (free-SC) fraction. The structural equation modeling analysis showed that yield and the partial factor productivity of N were positively correlated with labile and slow soil C and N fractions. Consequently, straw incorporation significantly increased grain yields of maize by 14.7% and rice by 15.1%. The annual potential reduction proportion in fertilizer-N induced by straw return was estimated to be 25.7% in the third year. This study suggests that the incorporation of straws is an effective way to enhance soil nutrients and regulate soil C and N pools to improve crop production and has the potential to reduce N fertilizer application under maize–rice rotation in subtropical regions.

The present paper simplifies the naturally formed dunes (riverbeds) as large-scale three-dimensional staggered wavy walls to investigate the features of the accompanying secondary flows and streamwise vortices via large-eddy simulation. A comparison between the swirling strength and the mean velocities suggests where a secondary flow induces upwash or downwash motions. Moreover, we propose a pseudo-convex wall mechanism to interpret the directionality of the secondary flow. The centrifugal instability criterion is then used to reveal the generation of the streamwise vortices. Based on these analytical results, we found that the streamwise vortices are generated in the separation and reattachment points on both characteristic longitudinal–vertical and horizontal cross-sections, which is related to the curvature effect of the turbulent shear layer. Furthermore, the maximum Görtler number characterized by the ratio of centrifugal to viscous effects suggests that, for fixed ratio of spanwise- to streamwise-wavelength cases, the strongest centrifugal instability occurring on the longitudinal–vertical cross-section gradually dominates with the increases in amplitude. A similar trend for the cases with varied spanwise wavelength can also be found. It is also found that the streamwise vortices are generated more readily via transverse flow around the crest near the separation and reattachment points when the ratio of spanwise- to streamwise-wavelength equals 1.

The strongly nonlinear Miyata–Choi–Camassa model under the rigid lid approximation (MCC-RL model) can describe accurately the dynamics of large-amplitude internal waves in a two-layer fluid system for shallow configurations. In this paper, we apply the MCC-RL model to study the internal waves generated by a moving body on the bottom. For the case of the moving body speed $U=1.1c_{0}$, where ${c_0}$ is the linear long-wave speed, the accuracy of the MCC-RL results is assessed by comparing with Euler's solutions, and very good agreement is observed. It is found that when the moving body speed increases from $U=0.8c_{0}$ to $U=1.241c_{0}$, the amplitudes of the generated internal solitary waves in front of the moving body become larger. However, a critical moving body speed is found between $U=1.241c_{0}$ and $U=1.242c_{0}$. After exceeding this critical speed, only one internal wave right above the body is generated. When the moving body speed increases from $U=1.242c_{0}$ to $U=1.5c_{0}$, the amplitudes of the internal waves become smaller.

In this study, we investigate the differences between two transient, three-dimensional, thermomechanically coupled ice-sheet models, namely, a first-order approximation model (FOM) and a ‘full’ Stokes ice-sheet model (FSM) under the same numerical framework. For all numerical experiments, we take the FSM outputs as the reference values and calculate the mean relative errors in the velocity and temperature fields for the FOM over 100 years. Four different boundary conditions (ice slope, geothermal heat flux, basal topography and basal sliding) are tested, and by changing these parameters, we verify the thermomechanical behavior of the FOM and discover that the velocity and temperature biases of the FOM generally increase with increases in the ice slope, geothermal heat flux, undulation amplitude of the ice base, and with the existence of basal sliding. In addition, the model difference between the FOM and FSM may accumulate over time, and the spatial distribution patterns of the relative velocity and temperature errors are in good agreement.

Interfacial waves between two superimposed dielectric fluid layers under a horizontal electric field are investigated from asymptotic and numerical aspects. The fluid is taken to be inviscid, incompressible and non-conducting in each layer. The competing forces resulting from gravity, surface tension and electric field are all considered. A systematic procedure is proposed to derive model equations of multiple scales in various possible limits from the electrified Euler equations in the framework of the Zakharov–Craig–Sulem formulation. Based on thorough analyses of the Dirichlet–Neumann operators in the long-wave approximation, classic weakly nonlinear models – including the Boussinesq-type system, the Korteweg–de Vries (KdV) equation and its variants, and the Benjamin-type equation – are obtained under different scaling assumptions. In addition, strongly nonlinear models (without the smallness assumption on the wave amplitude) in the Benjamin and Barannyk–Papageorgiou–Petropoulos regimes are derived. In these models, the electric effects are shown to produce dispersive regularisations of long and short waves. A modified boundary integral equation method is developed to compute solitary waves in the original electrified Euler equations. Through comparisons with solitary-wave solutions in the Euler equations, it is found that in various asymptotic regimes, weakly nonlinear models are in overall good agreement when wave amplitudes are small. In contrast, the range of validity of the strongly nonlinear model is much broader. It is shown that the horizontal electric field plays a significant role in the physical system: it expands the range of parameters for the existence of progressive waves, changes the qualitative characteristics of solitary waves and leads to a new type of solitary wave, namely the KdV–wavepacket mixed type.

In the context of three-dimensional oceanic internal waves, taking topographic effects into account, a modified Benney–Luke equation is proposed for describing internal wave–wave interactions on a sloping bottom. The derived equation is characterised by isotropy and bi-directional propagation, which are absent in the widely used Kadomtsev–Petviashvili equation. Indeed, these disparities are confirmed by numerical results of the diffraction of a truncated internal solitary wave and the evolution of a partially bent solitary wave. However, a good agreement between the numerical results of the modified Benney–Luke equation and those of the primitive equations confirms the validity of our simplified model. Because the stratification in a realistic ocean environment is usually continuous, in contrast to the assumption of a sharp density discontinuity used here, to maintain the kinematical equivalence, a layering scheme for determining the density and thickness of each layer from a continuous stratification is proposed. In addition, the occasionally observed but rarely examined X-shaped internal wave–wave interactions are shown to feature novel wave patterns, where topographic effects modulate the propagation speed, amplitude and waveform.

The findings regarding the associations between red meat, fish and poultry consumption, and the metabolic syndrome (Mets) have been inconclusive, and evidence from Chinese populations is scarce. A cross-sectional study was performed to investigate the associations between red meat, fish and poultry consumption, and the prevalence of the Mets and its components among the residents of Suzhou Industrial Park, Suzhou, China. A total of 4424 participants were eligible for the analysis. A logistic regression model was used to estimate the OR and 95 % CI for the prevalence of the Mets and its components according to red meat, fish and poultry consumption. In addition, the data of our cross-sectional study were meta-analysed under a random effects model along with those of published observational studies to generate the summary relative risks (RR) of the associations between the highest v. lowest categories of red meat, fish and poultry consumption and the Mets and its components. In the cross-sectional study, the multivariable-adjusted OR for the highest v. lowest quartiles of consumption was 1·23 (95 % CI 1·02, 1·48) for red meat, 0·83 (95 % CI 0·72, 0·97) for fish and 0·93 (95 % CI 0·74, 1·18) for poultry. In the meta-analysis, the pooled RR for the highest v. lowest categories of consumption was 1·20 (95 % CI 1·06, 1·35) for red meat, 0·88 (95 % CI 0·81, 0·96) for fish and 0·97 (95 % CI 0·85, 1·10) for poultry. The findings of both cross-sectional studies and meta-analyses indicated that the association between fish consumption and the Mets may be partly driven by the inverse association of fish consumption with elevated TAG and reduced HDL-cholesterol and, to a lesser extent, fasting plasma glucose. No clear pattern of associations was observed between red meat or poultry consumption and the components of the Mets. The current findings add weight to the evidence that the Mets may be positively associated with red meat consumption, inversely associated with fish consumption and neutrally associated with poultry consumption.

This study gives insights into the interfacial instability of water droplets through a combination of laboratory experiments, numerical simulations and analytical modelling. An experiment is conducted in a narrow gap between two plates to model the two-dimensional cylindrical geometry, and a pulsed laser beam is focused inside a water droplet to generate a cavitation bubble. Three distinct characteristics of droplet deformation can be observed: (i) splashing; (ii) ventilating; and (iii) a stable state. In addition, an analytical model considering the Rayleigh–Taylor instability and bubble oscillation is developed based on the assumption that fluid is inviscid and incompressible. The analytical model is solved to obtain a phase diagram describing three distinct phenomena. Two dimensionless parameters, $\hat {\eta }_{ins}$ and $\hat {\eta }_{sta}$, are used to determine the boundaries between different regimes. The parameter $\hat {\eta }_{ins}$ is defined by the ratio of the perturbation amplitude to the difference between the droplet and bubble radii, whereas the other parameter $\hat {\eta }_{sta}$ is defined by the ratio of the perturbation amplitude to the initial droplet radius. Interfacial instability is induced when the perturbed droplet surface penetrates the bubble boundary, namely, $\rvert \hat {\eta }_{ins} \rvert \geq 1$. Splashing and ventilating phenomena occur for $\hat {\eta }_{ins}\geq 1$ and $\hat {\eta }_{ins}\leq -1$, respectively. A stable state occurs when droplet fragmentation does not appear despite small-amplitude perturbations for $\hat {\eta }_{sta}\leq 0.1$. There is a transition zone between the ventilating and stable state, which is bounded by $\hat {\eta }_{ins} = -1$ and $\hat {\eta }_{sta} = 0.1$. Finally, the phase diagram is verified by the experimental and numerical results.