Simulation Of Surface Waves Research Articles

GWSM4C: A global wave surrogate model for climate simulation based on a convolutional architecture

Accurate and rapid ocean surface wave simulations are essential for navigation safety, marine activity, and climate change research. However, traditional numerical wave models incur substantial computational costs, especially in long-term simulations for climatology research. Deep-learning methods have shown promise in enabling faster simulations with fewer computational resources. However, the cumulative error in current deep-learning wave models limits their use as surrogate component models in climate modeling. In this work, a deep-learning model, named the Global Wave Surrogate Model for Climate simulation (GWSM4C), was developed based on a convolutional architecture, which accounts for the physical processes (wind waves and swells) of waves. Because GWSM4C can automatically generate initial conditions for each simulation moment depending only on historical wind fields, the accumulation of errors from previous simulated wave states can be avoided. The experimental results demonstrated that simulated global significant wave height produced by GWSM4C exhibited a correlation coefficient of 0.94 and a root-mean-square error of 0.21 m compared with the traditional numerical wave model. Furthermore, the generalization test of the GWSM4C indicates that it is feasible to use GWSM4C for long-term wave simulation in climatology-related studies or to integrate it into a coupled climate model as a surrogate wave component model.

Cube2sph : A toolkit enabling flexible and accurate continental-scale seismic wave simulations using the SPECFEM3D_Cartesian package

To enable flexible and accurate seismic wave simulations at continental scales (10°−60°) based on the spectral-element method using the open-source SPECFEM3D_Cartesian package, we develop a toolkit, Cube2sph, that allows the generation of customized spherical meshes that account for the Earth’s curvature. This toolkit enables the usage of the perfectly matched layer (PML) absorbing boundary condition even when the artificial boundaries do not align with the coordinate axes. A series of numerical experiments are presented to validate the effectiveness of this toolkit. From these numerical experiments, we conclude that (1) continental-scale seismic wave simulations, especially surface wave simulations, can be more efficiently performed without the loss of accuracy by truncating the mesh at an appropriate depth, (2) curvilinear-grid PML can be used to effectively suppress artificial reflections for seismic wave simulations at continental scales, and (3) the Earth’s spherical geometry needs to be accurately meshed in order to obtain accurate simulation results for study regions larger than 8°.

A toy model for damped water waves

We consider a toy model for a damped water waves system in a domain [Formula: see text]. The toy model is based on the paradifferential formulation of the water waves system derived in the work of Alazard–Burq–Zuily [On the water-waves equations with surface tension, Duke Math. J. 158 (2011) 413–499]. The form of damping we consider is a modified sponge layer proposed for the [Formula: see text] water waves system by Clamond et al. in [An efficient model for three-dimensional surface wave simulations. Part II: Generation and absorption, J. Comput. Phys. 205 (2005) 686–705]. We show that, in the case of small Cauchy data, solutions to the toy model exhibit a quadratic lifespan. This is done via proving energy estimates with Klainerman vector fields.

A generating absorbing boundary condition for simulating wave interaction with maritime structures in current or at forward speed

The lack of suitable boundary conditions in practical surface wave simulations with maritime structures in current or at forward speed may cause energy in the computational domain to accumulate due to spurious wave reflection. The common way to prevent wave reflection is to use passive wave absorbers, such as damping zones or relaxation zones, which requires larger domains at the cost of computational effort. Our goal is to derive a local generating absorbing boundary condition (GABC) for long-crested irregular waves on top of a mean flow, using the flow to model the forward speed of a structure such as a ship. Earlier work has demonstrated that a local GABC for free surface waves has a performance similar to passive wave absorbers, but at a reduced computational effort. New in the present work is that we extend, verify and validate the GABC in the presence of a nonzero mean flow. The GABC is designed to be accurate for a range of wave components in irregular sea states, with the resulting reflection coefficients for each component lower than a chosen value, say 5%. Having used potential flow theory for its derivation means that the boundary should not be placed at the exact location where wave breaking is expected, such as very close to the structure in the domain, or in the surf zone in coastal modeling. For the application with ships in this article that does not pose a limitation. The performance is demonstrated for a range of dimensionless wave number between 0 and 6. Such a boundary condition is obtained through a rational approximation of the linear dispersion relation with a mean flow, in combination with vertical derivatives of the solution variables along the boundary. Local linearization means that the GABC incorrectly considers bound, nonlinear wave components to be freely propagating wave components. Bound components, however, tend to have smaller amplitudes and do not appear to affect performance for the considered cases. Results of simulations with regular and irregular waves, on top of flows with different magnitudes and directions, are found to agree with the theory. The main source of differences is the implementation of the second derivate in the GABC near the free surface. Simulations of a Wigley hull at forward speed in irregular waves are compared to an experiment that was conducted specifically for validating the ABC. The data of the experiment are available as open data through doi: 10.4121/21320604. The comparison between simulation and experiment demonstrates that the GABC with a mean flow can be applied not only for theoretical simulations with propagating waves, but also for more practical applications with a structure in the domain.

Simulation of Nonlinear Free Surface Waves using a Fixed Grid Method

The simulation of nonlinear surface waves is of significant importance in safety studies of fluid containers and reservoirs. In this paper, nonlinear free surface flows are simulated using a fixed grid method which employs local exponential basis functions (EBFs). Assuming the flow to be inviscid and irrotational, the velocity potential Laplace’s equation is spatially discretized and solved by considering the nonlinear Bernoulli’s equation for irrotational flow as the boundary condition on the free surface. The nonlinear boundary conditions are imposed through a semi-implicit iterative time marching. The fixed grid feature of the method, based on a Lagrangian description of fluid flow, allows for retaining the portion of the discretization performed in the first time step for the bulk of the fluid. Thus, the portion which pertains to the regions near the moving boundaries is reprocessed during the time marching. The accuracy and efficiency of the existing solution is shown by simulating various problems such as liquid sloshing induced by external excitation of the reservoir or initial deformed shape of liquid, seiche phenomena and solitary wave propagation in a basin with constant depth or with a step, and comparing the results with those which are analytically available or those from available codes such as Abaqus. The proposed method shows far better stability of the results when compared with those of Abaqus which sometimes exhibit divergence after a relatively large number of time steps. For instance, in the propagation of the considered solitary wave in an infinite-like domain problem, the wave height is calculated by the maximum error of 1.6% and 9% using the present method and Abaqus, respectively.

Finite-difference method for modeling the surface wave propagation with surface topography in anisotropic-viscoelastic media

The growth of urbanization makes it increasingly important to accurately investigate underground spaces. Surface-wave-based inversion methods are becoming popular due to their cost-effectiveness and efficiency in reconstructing underground space. However, these methods may produce inaccurate results in the presence of media anisotropy or/and viscoelasticity. To investigate the influence of media anisotropy or viscoelasticity on surface waves, it is crucial to develop an efficient and accurate forward modeling method in anisotropic-viscoelastic (AV) media. The finite-difference (FD) method is a widely used forward modeling method in surface-wave-based inversion methods. However, implementing the free-surface boundary condition and representing non-flat topography can be challenging. Considering the challenges described above, this study proposes a simple and efficient FD method for modeling surface-wave propagation. The proposed method employs a parameter-modified strategy to implement the stress-free condition by modifying the model parameters near the (non-) flat free-surface boundary. A staircase discretization strategy is then used to represent the non-flat free surface. To suppress the staircase diffractions caused by the staircase discretization strategy, an independent wavefield superposition is adopted with modeling results of different parameter configurations. Compared with conventional FD methods, the proposed method can reduce computational costs while accurately simulating surface waves. Numerical experiments have demonstrated the accuracy and feasibility of the proposed method. The proposed method provides a theoretical basis for surface-wave-based inversion methods and contributes to the development of accurate underground space investigation.

PIC simulations of stable surface waves on a subcritical fast magnetosonic shock front

We study with particle-in-cell (PIC) simulations the stability of fast magnetosonic shocks. They expand across a collisionless plasma and an orthogonal magnetic field that is aligned with one of the directions resolved by the 2D simulations. The shock speed is 1.6 times the fast magnetosonic speed when it enters a layer with a reduced density of mobile ions, which decreases the shock speed by up to 15% in 1D simulations. In the 2D simulations, the density of mobile ions in the layer varies sinusoidally perpendicularly to the shock normal. We resolve one sine period. This variation only leads to small changes in the shock speed evidencing a restoring force that opposes a shock deformation. As the shock propagates through the layer, the ion density becomes increasingly spatially modulated along the shock front and the magnetic field bulges out where the mobile ion density is lowest. The perturbed shock eventually reaches a steady state. Once it leaves the layer, the perturbations of the ion density and magnetic field oscillate along its front at a frequency close to the lower-hybrid frequency; the shock is mediated by a standing wave composed of obliquely propagating lower-hybrid waves. We perform three 2D simulations with different box lengths along the shock front. The shock front oscillations are aperiodically damped in the smallest box with the fastest variation of the ion density, strongly damped in the intermediate one, and weakly damped in the largest box. The shock front oscillations perturb the magnetic field in a spatial interval that extends by several electron skin depths upstream and downstream of the shock front and could give rise to Whistler waves that propagate along the shock’s magnetic field overshoot. Similar waves were observed in hybrid and PIC simulations and by the MMS satellite mission.

Two-phase flow simulations of surface waves in wind-forced conditions

The paper is devoted to two-phase flow simulations and investigates the ability of a diffusive interface Cahn–Hilliard volume-of-fluid model to capture the dynamics of the air–sea interface at geophysically relevant Reynolds numbers. It employs a hybrid filtered/averaging improved detached eddy simulation method to model turbulence and utilizes a continuum model to account for surface tension if the diffuse interface is under-resolved by the grid. A numerical wind-wave tank is introduced, and results obtained for two known wind-wave conditions are analyzed in comparison to experimental data at matched Reynolds numbers. The focus of the comparison is on both time-averaged and wave-coherent quantities, and includes pressure, velocity as well as modeled and resolved Reynolds stresses. In general, numerical predictions agree well with the experimental measurements and reproduce many wave-dependent flow features. Reynolds stresses near the water surface are found to be especially important in modulating the critical layer height. It is concluded that the diffusive interface approach proves to be a promising method for future studies of air–sea interface dynamics in geophysically relevant flows.

Finite-difference modeling with topography using 3D viscoelastic parameter-modified free-surface condition

The free-surface boundary condition is a crucial aspect in the numerical modeling of (visco)elastic wave equations, especially when using a finite-difference (FD) method in the presence of surface topography. The parameter-modified method is a widely used approach to solve this problem. In this regard, the adaptive Poisson’s ratio parameter-modified method has proven to be effective in accurately simulating seismic surface waves within the FD discretization framework. Based on the equivalent medium theory, vacuum approximation, and mathematical limit, we develop a viscoelastic parameter-modified (VPM) method for the implementation of the free-surface boundary condition in the 3D viscoelastic wave equation. Our approach modifies the viscoelastic constitutive relation and density at the free surface and provides a formulation in terms of displacement and stress. We determine that our VPM method is more general than the viscoelastic stress-image method as it includes the latter as a limit case when the Poisson’s ratio equals zero. The presented free-surface method is represented implicitly within the FD grid, and we provide implementation details when simulating surface waves with topography in the standard staggered-grid FD scheme. We support our theory’s feasibility and accuracy through numerical examples.

Spatiotemporal distributions of air-sea CO2 flux modulated by windseas in the Southern Indian Ocean

The Southern Indian Ocean is a major reservoir for rapid carbon exchange with the atmosphere, plays a key role in the world’s carbon cycle. To understand the importance of anthropogenic CO2 uptake in the Southern Indian Ocean, a variety of methods have been used to quantify the magnitude of the CO2 flux between air and sea. The basic approach is based on the bulk formula—the air-sea CO2 flux is commonly calculated by the difference in the CO2 partial pressure between the ocean and the atmosphere, the gas transfer velocity, the surface wind speed, and the CO2 solubility in seawater. However, relying solely on wind speed to measure the gas transfer velocity at the sea surface increases the uncertainty of CO2 flux estimation. Recent studies have shown that the generation and breaking of ocean waves also significantly affect the gas transfer process at the air-sea interface. In this study, we highlight the impact of windseas on the process of air-sea CO2 exchange and address its important role in CO2 uptake in the Southern Indian Ocean. We run the WAVEWATCH III model to simulate surface waves in this region over the period from January 1st 2002 to December 31st 2021. Then, we use the spectral partitioning method to isolate windseas and swells from total wave fields. Finally, we calculate the CO2 flux based on the new semiempirical equation for gas transfer velocity considering only windseas. We found that after considering windseas’ impact, the seasonal mean zonal flux (mmol/m2·d) increased approximately 10%-20% compared with that calculated solely on wind speed in all seasons. Evolution of air-sea net carbon flux (PgC) increased around 5.87%-32.12% in the latest 5 years with the most significant seasonal improvement appeared in summer. Long-term trend analysis also indicated that the CO2 absorption capacity of the whole Southern Indian Ocean gradually increased during the past 20 years. These findings extend the understanding of the roles of the Southern Indian Ocean in the global carbon cycle and are useful for making management policies associated with marine environmental protection and global climatic change mitigation.

Application of Non-Reflective Boundary Conditions in Three-Dimensional Numerical Simulations of Free-Surface Flow Problems

This paper considers the issue of using non-reflective boundaries for surface wave simulations within the framework of three-dimensional Navier–Stokes equations. We distinguish a wave damping approach among the known implementations of non-reflective boundary conditions in surface wave simulations. The approach employs a sponge boundary layer to dampen incident waves. In this paper, we describe implementations of this approach on unstructured meshes. We also present the calibration of free parameters, the values of which control wave damping efficiency and the amplitude of reflected waves. Comparison of the results obtained at different types of distribution for the free parameter was conducted. The implemented wave damping approach was tested by simulating a solitary wave propagating in a water tank. We demonstrate the use of damping non-reflective boundary conditions for the case of a wave traveling across the surface of a real body of water near Sakhalin Island while considering its bathymetry.

Design of a higher-order nodal-line semimetal in a spring-shaped acoustic topological crystal

The concept of higher-order topology has prompted the development of topological insulators, and a series of theoretical and experimental schemes have been proposed and realized to demonstrate higher-order hinge or corner states. While these ideas thrive in topological semimetals such as higher-order Dirac or Weyl semimetals, other degeneracy types such as nodal lines or nodal loop semimetals still remain to be fully exploited. In this work we consider a setup that hosts an acoustic springlike higher-order nodal-line semimetal protected by a screw-rotational symmetry. The underlying sandwich architecture is engineered with different coupling modes between different layers to construct a specific space group configuration, which enables the emergence of higher-order nodal lines in momentum space. Numerical calculations of the band dispersion show that nodal lines in opposite directions result in linked double hinge arcs. A corresponding topological invariant index is introduced to characterize the different topological phases. Furthermore, a propagation simulation of hinge and surface waves in a multilayer sample is performed that matches well with theoretical computations. Our work thus presents a higher-order nodal-line semimetal in an acoustic crystal and may receive practical applications in manufacturing sonic devices.

Coupling 3D Liquid Simulation with 2D Wave Propagation for Large Scale Water Surface Animation Using the Equivalent Sources Method

AbstractThis paper proposes a method for simulating liquids in large bodies of water by coupling together a water surface wave simulator with a 3D Navier‐Stokes simulator. The surface wave simulation uses the equivalent sources method (ESM) to efficiently animate large bodies of water with precisely controllable wave propagation behavior. The 3D liquid simulator animates complex non‐linear fluid behaviors like splashes and breaking waves using off‐the‐shelf simulators using FLIP or the level set method with semi‐Lagrangian advection.We combine the two approaches by using the 3D solver to animate localized non‐linear behaviors, and the 2D wave solver to animate larger regions with linear surface physics. We use the surface motion from the 3D solver as boundary conditions for 2D surface wave simulator, and we use the velocity and surface heights from the 2D surface wave simulator as boundary conditions for the 3D fluid simulation. We also introduce a novel technique for removing visual artifacts caused by numerical errors in 3D fluid solvers: we use experimental data to estimate the artificial dispersion caused by the 3D solver and we then carefully tune the wave speeds of the 2D solver to match it, effectively eliminating any differences in wave behavior across the boundary. To the best of our knowledge, this is the first time such a empirically driven error compensation approach has been used to remove coupling errors from a physics simulator.Our coupled simulation approach leverages the strengths of each simulation technique, animating large environments with seamless transitions between 2D and 3D physics.

A high-order spectral method for effective simulation of surface waves interacting with an internal wave of large amplitude

In this study, we propose a new method for simulating complex surface waves interacting with a large amplitude internal solitary wave. Our model is based on a high-order spectral method for surface waves with the bottom boundary conditions computed from an internal wave solver. The convergence of the model is first tested using an internal wave case without surface waves. We then perform simulations of a canonical nonlinear wave interaction case involving two surface wave components and a prescribed internal wave component. The initial wave energy growth rate of the resonant wave component is found to agree with the analytical value predicted by the perturbation theory. We also use the model to simulate the interaction between surface waves and a weakly nonlinear internal wave, and the result is nearly identical to that calculated using a two-layer model. Finally, we show the application of our model in a multiscale nested modeling framework, where the internal wave parameters are extracted from the mesoscale simulation using a nonhydrostatic ocean model. The evolution of the spatial variations of the surface roughness is captured and the surface wave orbital velocity also changes in space due to the surface motions induced by the internal wave. Our phase-resolved model provides a computationally efficient tool for simulating complex surface wave fields in the background of internal waves of large amplitudes.

Mathematical modeling of stress waves under concentrated vertical action in the form of a triangular pulse: Lamb’s problem

The aim of the work. The problem of numerical simulation of longitudinal, transverse and surface waves on the free surface of an elastic half-plane is considered. Methods. To solve the non-stationary dynamic problem of elasticity theory with initial and boundary conditions, the finite element method in displacements was used. Using the finite element method in displacements, a linear problem with initial and boundary conditions was led to a linear Cauchy problem. A quasiregular approach to solving a system of second-order linear ordinary differential equations in displacements with initial conditions and to approximating the area under study is proposed. The method is based on the schemes: point, line and plane. The study area is divided by spatial variables into triangular and rectangular finite elements of the first order. According to the time variable, the study area is divided into linear end elements with two nodal points. The Fortran-90 algorithmic language was used in the development of the software package. Results. Some information is given about numerical modeling of elastic stress waves in an elastic half-plane with a concentrated wave action in the form of a Delta function. The estimated area under study has 12 008 001 nodal points. A system of equations consisting of 48 032 004 unknowns is solved. The change of elastic contour stress on the free surface of the half-plane at different points is shown. The amplitude of Rayleigh surface waves is significantly greater than the amplitudes of longitudinal, transverse, and other waves with a concentrated vertical action in the form of a triangular pulse on the surface of an elastic half-plane. After surface Rayleigh waves, a dynamic process is observed in the form of standing waves.

Full Waveform Tomography to Address Challenges with Surface Wave Dispersion Information Caused by Significant Stochastic Variability of Subsurface Stiffness

Surface wave methods have increased in popularity as a means to acquire rapid and accurate shear wave velocity ( VS) profiles for engineering applications. Use of the multichannel analysis of surface waves (MASW) method, in particular, has proliferated due to multiple factors, including the ease with which strong signal-to-noise ratio can be achieved and the increased dispersion resolution offered by multichannel acquisitions. However, typical MASW processing to extract dispersion information assumes the surface waves propagate through a layered model. Errors can arise when significant lateral variability is present in the underlying stratigraphy as encountered in certain geologic settings such as residual deposits. This study investigated the effects of such variability on the dispersion information acquired with MASW. In particular, a spatially-correlated Gaussian random field was used to model the natural fluctuations in stiffness introduced by depositional processes, which differs from the approach in other studies where more specific anomalous features have been explored. Numerical modeling was subsequently performed to simulate surface wave propagation in the representative geotechnical site condition. The recovered surface waves were used to develop a subsurface stiffness profile using a dispersion-based pseudo-2D MASW approach and a tomographic approach using full waveform inversion (FWI). The results demonstrate that considerable natural spatial variability significantly complicates interpretation of dispersion information in two primary ways: (1) uncertainty can arise regarding what the dispersion curve exactly quantifies since none of the underlying VSprofiles nor the average VSprofile are obtained; and (2) the dispersion images exhibit evidence of false depth-related dispersion information indicative of multiple “fundamental” modes from the superposition of multiple stratigraphic units. The FWI procedure that bypasses extraction of dispersion information was found to better recover the underlying subsurface conditions when compared to the pseudo-2D MASW results at the cost of additional computational efforts.

Плановая модель волновой гидродинамики с дисперсионным соотношением повышенной точности. II. Четвертый, шестой и восьмой порядки

Построена полностью нелинейная слабо дисперсионная модель волновой гидродинамики четвертого порядка длинноволновой аппроксимации. За скорость в модели взята усредненная по глубине горизонтальная составляющая скорости трехмерного течения. Учтена подвижность дна. Выполненная модификация модели обеспечивает шестой и восьмой порядки точности аппроксимации дисперсионного соотношения трехмерной модели потенциальных течений. In the numerical simulation of medium-length surface waves in the framework of nonlinear dispersive (NLD) models, an increased accuracy of reproducing the characteristics of the simulated processes is required. A number of works (Kirby (2016), e.g.) describe approaches to improve the known NLD-models. In particular, NLD-models of the fourth order of the long-wave approximation have been proposed and, based on a comparison of numerical results with experimental data, their high accuracy has been demonstrated (Ataie-Ashtiani and Najafi-Jilani (2007); Zhou and Teng (2010)). In these new models, the horizontal component of the velocity vector of the threedimensional (FNPF-) model of potential flows at a certain surface located between the bottom and the free boundary was chosen as the velocity vector. The result was a very cumbersome form of equations. In addition, the laws of conservation of mass and momentum do not hold for these models. The main result of this work is the derivation of a two-parameter fully nonlinear weakly dispersive (mSGN4) model of the fourth order of the long-wave approximation, which is a generalization of the well-known Serre-Green-Naghdi (SGN) second order model. In the derivation, the velocity averaged over the thickness of the liquid layer was used. The assumption about the potentiality of the three-dimensional flow was used only at the stage of closing the model. The movement of the bottom is taken into account. For the derived model, the law of conservation of mass is satisfied, and the law of conservation of total momentum is satisfied in the case of a horizontal stationary bottom. The equations of the mSGN4-model are invariant under the Galilean transformation and are presented in a compact form similar to the equations of gas dynamics. The dispersion relation of the mSGN4-model has the fourth order of accuracy in the long wave region and satisfactorily approximates the dispersion relation of the FNPF-model in the short wave region. Moreover, with a special choice of the values of the model parameters, an increased accuracy of approximating the dispersion relation of the FNPF-model at long waves (sixth or eighth order) is achieved. Analysis of the deviations of the values of the phase velocity of the mSGN4 model from the values of the “reference” speed of the FNPF model in the entire wavelength range showed that the most preferable is the mSGN4 model with the parameter values corresponding to the Pad’e approximant (2,4).

Simulation of scattered seismic surface waves on mountainous onshore areas: Understanding the “ground roll energy cone”

The accurate simulation of seismic surface waves on complex land areas requires elastic models with realistic near-surface parameters. The SEAM Phase II Foothills model, proposed by the SEG Advanced Modeling (SEAM) Corporation, is one of the most comprehensive efforts undertaken by the geophysics community to understand complex seismic wave propagation in foothills areas. However, while this model includes a rough topography, alluvial sediments, and complex geologic structures, synthetic data from the SEAM consortium do not reproduce the qualitative characteristics of the scattering energy that is generally interpreted as the “ground roll energy cone” on shot records of real data. To simulate the scattering, a near-surface elastic model in mountainous areas ideally must include the following three elements: (1) rough topography and bedrock, (2) low-velocity layer, and (3) small-scale heterogeneities (size approximately Rayleigh wavelength). The SEAM Foothills model only includes element (1) and, to a lesser extent, element (2). We represent a heterogeneous near surface as a random medium with two parameters: the average size of the heterogeneities and fractional fluctuation. A parametric analysis shows the influence of each parameter on the synthetic data and how similar it is compared to real data acquired in a foothills area in Colombia. We perform the analysis in the shot gather panel and dispersion image. Our study shows that it is necessary to include the low-velocity layer and small-scale distributed heterogeneities in the shallow part of the SEAM model to get synthetic data with realistic scattered surface-wave energy.

Traction open boundary condition for incompressible, turbulent, single- or multi-phase flows, and surface wave simulations

In simulations, artificial boundaries need to be introduced to limit the size of computational domains and thereby lower computational cost. At these boundaries, flow variables need to be calculated in a way that will not induce any perturbation of the interior solution, which poses a great challenge in incompressible flows. In this paper, we demonstrate the potential of a new traction open boundary condition to address the classical problems encountered in simulations with open boundary conditions: backflow instability, wave reflections, and confinement caused by the proximity of the outlet. This novel boundary treatment, based on a Lagrangian estimation of the traction in the outlet section coupled to a stabilization term, is shown to provide accuracy and stability for turbulent, single- or multi-phase flows, test cases. Using a simulation of surface gravity waves, we show that if special care is given to the computation of the estimated traction, it is possible to get a fully non-reflective open boundary condition.

Mutual Interaction between Surface Waves and Langmuir Circulations Observed in Wave-Resolving Numerical Simulations

AbstractWave-resolving simulations of monochromatic surface waves and Langmuir circulations (LCs) under an idealized condition are performed to investigate the dynamics of wave–current mutual interaction. When the Froude number (the ratio of the friction velocity of wind stress imposed at the surface and wave phase speed) is large, waves become refracted by the downwind jet associated with LCs and become amplitude modulated in the crosswind direction. In such cases, the simulations using the Craik–Leibovich (CL) equation with a prescribed horizontally uniform Stokes drift profile are found to underestimate the intensity of LCs. Vorticity budget analysis reveals that horizontal shear of Stokes drift induced by the wave modulation tilts the wind-driven vorticity to the downwind direction, intensifying the LCs that caused the waves to be modulated. Such an effect is not reproduced in the CL equation unless the Stokes drift of the waves modulated by LCs is prescribed. This intensification mechanism is similar to the CL1 mechanism in that the horizontal shear of the Stokes drift plays a key role, but it is more likely to occur because the shear in this interaction is automatically generated by the LCs whereas the shear in the CL1 mechanism is retained only when a particular phase relation between two crossing waves is kept locked for many periods.