Internal and surface wave production in a stratified fluid

The Statewide California Earthquake Center (SCEC) sequence of earthquake and aseismic slip (SEAS) group has regularly developed and published benchmarks along the years to follow recent development in the modeling of sequences of aseismic slip and earthquakes, as well as progress in numerical methods. These benchmarks, as well as the results from the different groups are publicly available at: https://strike.scec.org/cvws/seas/. It provides both reference solutions for code verification and a framework for systematic comparison of different modeling approaches. Every group is welcomed to join this Community driven comparison.Recent efforts have focused on adding physics to better reproduce earthquake cycle by considering 2-dimensional fault (Jiang et al., 2022), improve our understanding of fluid injection processes (Lambert et al., 2025), and the effect of free surface and dipping fault (Erickson et al., 2023).To follow up these developments, the SCEC SEAS group is designing two new benchmarks, that will be released to the community soon:[1] A generalization of our benchmark about fluid injection (BP6) from a 2-dimensional domain to 3-dimensional domain. In this benchmark pore fluid diffuses along a 2-dimensional fault, modifying the effective normal traction through one-way hydromechanical coupling. The fluid is injected for 10 hours on a rate strengthening faut and then shut off. The benchmark is designed to admit an analytical formulation for pore fluid diffusion while avoiding numerical singularities that may occurred with point source injection. For this reason, fluid is injected along a Gaussian profile.[2] An updated version of our previous dipping fault benchmark (BP3), in a two-dimensional medium with a free surface. Previous version assumed that the normal traction along the fault was constant. This is obviously a strong assumption, because the normal traction should increase with depth. However, this has proven to be difficult to simulate numerically as the system is going stiffer with lower normal traction. This benchmark therefore aims at providing a more realistic simulation of a dipping fault with a free surface by introducing depth dependent normal traction while also testing the ability of different numerical code to circumvent the problem of stiffness. This benchmark will be a joint benchmark with CRESCENT (Cascadia Region Earthquake Science Center).This contribution will present the design of these forthcoming benchmarks and will provide an opportunity for the community to discuss about future benchmarks and directions for SEAS code comparison efforts.

The interaction between turbulence and internal waves in a stratified fluid with a shear flow is considered on the basis of perturbation theory permitting a description of internal wave damping by turbulence. The results show that the behavior of the damping rate for internal waves is rather anomalous for the long-wave range: it remains finite for such waves but may decrease to very small quantities in a restricted range of wavenumbers, depending on the mode structure. Detailed computations are made for the first three modes of the internal waves. The effects of anisotropy of the turbulent fluxes are also considered. For the case of constant buoyancy frequency an exact solution is obtained of the linearized semiempirical equations valid even for a strong wave damping. Estimates show that the damping by the upper turbulent layer may be an effective energy sink for long internal waves in the ocean.

Wind farm-induced atmospheric gravity waves have been the subject of recent research as they can impact wind farm performance. Pressure variations associated with gravity waves can contribute to the global blockage effect and wind farm wake recovery. Therefore, accurate numerical simulation of flow fields, where wind-farm-induced gravity waves may be produced, is important. Three main considerations in such simulations are the overall domain size, the use of Rayleigh damping near domain boundaries to dampen gravity waves, and advection damping at the inlet to prevent spurious oscillations. Often these considerations are treated ad hoc rather than systematically. This work aims to test and extend the systematic modelling of internal gravity waves proposed in a preliminary investigation to modelling of both internal and trapped gravity waves. The preliminary study identifies the length scales to set the domain and damping layer sizes and the time scale to configure the Rayleigh damping coefficient but under linearly stratified conditions. Large eddy simulations of flow through a wind farm canopy are performed under conventionally neutral boundary layer (CNBL) conditions to test the validity of proposed setups for CNBL conditions. Background atmospheric parameters, such as Froude number (Fr), inversion height (Hi ), and inversion layer Froude number (Fri ) control most of the atmospheric gravity wave characteristics. We validated for CBNL conditions that the effective wavelengths of the internal gravity waves are the correct length scale to configure the domain size and damping layer thickness. Likewise, the optimum damping coefficient to dampen the internal gravity waves relates to the free atmosphere’s buoyancy frequency or buoyant perturbations’ time scale. We infer that the damping coefficient in the inversion layer may relate to the inversion buoyancy frequency to effectively dampen the trapped gravity waves. Moreover, the advection damping length is linked to the horizontal wavelength of the trapped gravity waves in the inversion layer to prevent spurious waves at the inlet by retaining wave energy accumulation.

<p>The interaction of small-scale turbulence with internal and surface waves is an urgent problem of hydrology and oceanology. In particular, this issue is especially important for the properties of the upper layer of the ocean and the inland waters.</p><p>Small-scale processes that exist against the background of average profiles of various hydrophysical quantities (temperature, velocity, density, and large-scale currents caused, in particular, by wind forcing) are usually nonlinear and therefore effectively interact with each other. We consider some aspects of the interaction of internal waves and turbulence in the upper layer of the ocean and inland waters within the framework of the semi-empirical theory of turbulence in a stratified fluid. The model used in this study takes into account  mutual transformation of the kinetic and potential energies of turbulent fluctuations [Ostrovsky&Troitskaya, 1987; Zilitinkevich et al., 2013]. The effects of amplification and maintenance of turbulence by low-frequency and high-frequency internal waves, quasi-stationary distributions of turbulent energy in the presence of a shear caused by a low-frequency internal wave are investigated; the role of the transformation of energies on the indicated processes is analyzed.</p><p>A modification of the k-epsilon mixing scheme is also proposed, which removes the limitation on the existence of turbulence at large values of the gradient Richardson number. Within the framework of the modification, the parameterization of the Prandtl number is used, which makes it possible to take into account the influence of density stratification and velocity shear on mixing processes.</p><p>A numerical study of the influence of vertical mixing schemes on the transfer processes of biochemical fields in an internal reservoir was also carried out. The modified scheme was implemented into a three-dimensional model of thermo-hydrodynamics and biochemistry of an inland water body [Gladskikh et al., 2021], and a series of numerical experiments was conducted.</p><p>The work was supported by the RFBR (20-05-00776; 20-05-00322; 21-05-52005), and by Moscow Center of Fundamental and Applied Mathematics (agreement with the Ministry of Science and Higher Education 075-15-2019-1621).</p><p>Ostrovsky LA, Troitskaya YuI (1987) A model of turbulent transfer and dynamics of turbulence in a stratified shear flow. Izv Akad Nauk SSSR, Fiz Atmos Okeana. 3:101–104.<br>Zilitinkevich SS, Elperin T, Kleeorin N, Rogachevskii I, Esau I (2013) A hierarchy of Energyand Flux-Budget (EFB) turbulence closure models for stably-stratified geophysical flow. Boundary-Layer Meteorol. 146:341–373<br>Gladskikh DS, Stepanenko VM, Mortikov EV (2021) The Effect of the Horizontal Dimensions of Inland Water Bodies on the Thickness of the Upper Mixed Layer. Water Resour 48:226–234</p>

A tank experiment is being conducted to explore sound propagation in a continuously stratified ocean containing internal gravity waves that produce fluctuations in acoustic signals. The tank is filled with water of increasing salinity with increasing depth in the fluid; the variation in salinity results in a 150 m/s variation of the sound speed over the 50 cm-depth fluid. Sound arrival times determined in three-dimensional hydrophone scans map out the sound diffraction pattern in the stratified fluid. Good agreement is found between the measurements and results obtained from wavefront modeling by ray acoustics. Sound fluctuations produced by internal waves are measured for several acoustic track locations and at different phases of the internal waves that radiate from a mechanical wave generator. Internal wave fields measured by particle image velocimetry provide input for a parabolic equation model, which is used to predict the sound fluctuations for comparison with measurements. This research is designed to improve the understanding and modeling of sound propagation and scattering by internal waves in the realistic oceans. [Research support: The 2013-14 ASA F. V. Hunt Postdoctoral Research Fellowship (L. Zhang) and ONR MURI Grant N000141110701 (WHOI).]

Damped internal wave beams in stratified fluids have long been known to generate strong mean flows through a mechanism analogous to acoustic streaming. While the role of viscous boundary layers in acoustic streaming has been thoroughly addressed, it remains largely unexplored in the case of internal waves. Here we compute the mean flow generated close to an undulating wall that emits internal waves in a viscous, linearly stratified two-dimensional Boussinesq fluid. Using a quasi-linear approach, we demonstrate that the form of the boundary conditions dramatically impacts the generated boundary streaming. In the no-slip scenario, the early-time Reynolds stress divergence within the viscous boundary layer is much stronger than within the bulk while also driving flow in the opposite direction. Whatever the boundary condition, boundary streaming is however dominated by bulk streaming at larger time. Using a Wentzel–Kramers–Brillouin approach, we investigate the consequences of adding boundary streaming effects to an idealised model of wave–mean flow interactions known to reproduce the salient features of the quasi-biennial oscillation. The presence of wave boundary layers has a quantitative impact on the flow reversals.

The present article is a theoretical study of the combined propagation of acoustic and internal waves of finite amplitude in a (horizontally, say) stratified fluid. The linear theory of small amplitude waves predicts that the two waves (i.e., the fluctuations in acoustic pressure and in vertical particle velocity or displacement) are in general coupled. The waves are decoupled whenever the typical wavenumbers are large compared with the inverse of the scale height for the stratification. Furthermore, a horizontally propagating acoustic wave with zero vertical particle displacement (or velocity), when this is possible, is always decoupled from the internal waves, according to linear theory. In a weakly nonlinear theory, however, the waves may be coupled. A time independent radiation pressure is formed within a standing, horizontally directed acoustic wave, which produces a stationary vertical displacement of the particles. Experimental evidence of this effect has been reported elsewhere [1].

Infrasonic waves propagate at long ranges through atmospheric ducts resulting from the stratification of atmospheric properties. In order to study the infrasonic wave propagation, we resort to atmospheric specification combining Numerical Weather Prediction and climatological models. However, these models do not describe small‐scale variability such as perturbations associated to the presence of internal gravity waves. These waves play an important role in the atmospheric dynamic by transferring momentum to the mean flow at critical levels and at wave‐breaking altitudes. In this study we intend to describe the interaction of infrasonic waves with internal gravity waves in order to understand the extended wave trains observed in broadband infrasound signals. We use a numerical model for the propagation of internal waves to generate realistic perturbations of the background atmospheric states. By using a linear full‐wave model of infrasound propagation, our goal is to ultimately relate infrasound characteristics to internal waves properties. We apply those numerical models to different atmospheric mean states including gravity wave perturbations and compare the simulations to infrasound signals recorded from rocket motor fuel elimination events at the Utah Test and Training Range (UTTR). Our results, based on an intensive simulation of infrasound waveforms, show that combining atmospheric background with gravity wave propagation model is relevant to explain, to first order, the behavior of the infrasound field at the distance of the first stratospheric arc. In particular we obtain a better match between pulse duration and peak‐to‐peak amplitude along the stratospheric arc. Such study supports the use of gravity wave simulation in conjunction with infrasound modeling in order to improve gravity wave parameterization in Numerical Weather Forecasting.

This study evaluates tsunami hazards in the Marmara Sea from possible worst-case tsunami scenarios that are from submarine earthquakes and landslides. In terms of fault-generated tsunamis, seismic ruptures can propagate along the North Anatolian Fault (NAF), which has produced historical tsunamis in the Marmara Sea. Based on the past studies, which consider fault-generated tsunamis and landslide-generated tsunamis individually, future scenarios are expected to generate tsunamis, and submarine landslides could be triggered by seismic motion. In addition to these past studies, numerical modeling has been applied to tsunami generation and propagation from combined earthquake and landslide sources. In this study, tsunami hazards are evaluated from both individual and combined cases of submarine earthquakes and landslides through numerical tsunami simulations with a grid size of 90 m for bathymetry and topography data for the entire Marmara Sea region and validated with historical observations from the 1509 and 1894 earthquakes. This study implements TUNAMI model with a two-layer model to conduct numerical tsunami simulations, and the numerical results show that the maximum tsunami height could reach 4.0 m along Istanbul shores for a full submarine rupture of the NAF, with a fault slip of 5.0 m in the eastern and western basins of the Marmara Sea. The maximum tsunami height for landslide-generated tsunamis from small, medium, and large of initial landslide volumes (0.15, 0.6, and 1.5 km3, respectively) could reach 3.5, 6.0, and 8.0 m, respectively, along Istanbul shores. Possible tsunamis from submarine landslides could be significantly higher than those from earthquakes, depending on the landslide volume significantly. These combined earthquake and landslide sources only result in higher tsunami amplitudes for small volumes significantly because of amplification within the same tsunami amplitude scale (3.0–4.0 m). Waveforms from all the coasts around the Marmara Sea indicate that other residential areas might have had a high risk of tsunami hazards from submarine landslides, which can generate higher tsunami amplitudes and shorter arrival times, compared to Istanbul.

Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden–Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of nonconservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are important in designing the numerical scheme. A numerical method is designed with simplicity and minimal computational cost as the main design principles. Numerical tests demonstrate that no catastrophic effects are introduced when hyperbolicity is lost, and the scheme can represent propagating discontinuities without introducing spurious oscillations.

The quantity and accuracy of satellite-geodetic measurements have increased over time, revolutionizing the monitoring of tectonic processes. Global Navigation Satellite System (GNSS) and satellite radar signals provide observations beyond ground deformation, including how earthquake and tsunami processes affect variations in the ionosphere. Here, we study the Hunga Tonga Hunga Ha’apai (HTHH) volcanic eruption 2022 and its associated tsunami propagation with the analysis GNSS derived Total Electron Content (TEC), Synthetic Aperture Radar (SAR) Sentinel-1 data, complemented with tide gauge observations. We utilize GNSS sites data within a ~ 5000 km radius from the volcanic eruption for estimating the ionospheric perturbation as Vertical TEC. We give evidence on the detection of acoustic gravity, internal gravity, and atmospheric Lamb waves signatures in the TEC perturbation. In particular, the internal gravity waves that concentrated in the southwest of Tonga, directly correlates with the observed tsunami propagation direction as accounted by the tide gauge measurements. However, the acoustic gravity wave signature in the TEC is dominant in the north direction suggesting a surface deformation, which could be verified using Sentinel-1A SAR amplitude data. The analysis presented herein shows that within 5 h of the volcanic eruption, the central part of the HTHH island landscape disappeared with the biggest explosion. The unprecedented detail resolved by integrating satellite data yields previously unknown details of the deformation of the 2022 HTHH volcano eruption.

Optical visualization of internal gravity waves in stratified fluid

Gravity currents are of high interest both for their relevance in natural scenarios and because varying horizontal buoyancy presents a canonical problem in fluid mechanics [Huppert, “Gravity currents: A personal perspective,” J. Fluid Mech. 554, 299–322 (2006)]. In this paper, attention is directed to gravity currents with a full-depth lock release propagating into a linearly stratified ambient fluid. For the case of an unstratified ambient, similarity solutions are known to capture the evolving height profile of the gravity current. We will compare this solution class with numerical data from high fidelity simulations. The presence of ambient stratification (quantified by the stratification intensity, S) introduces internal gravity waves that interact with the propagating current head, which will inhibit Kelvin–Helmholtz billows, decelerate current propagation, and smooth the shape of the current head. We perform direct numerical simulations of planar two- and three-dimensional gravity currents released into stratified ambient fluid of varying S and analyze the gravity current kinematics. Our analysis complements existing findings from performed laboratory and numerical experiments [Dai et al., “Gravity currents propagating at the base of a linearly stratified ambient,” Phys. Fluids 33, 066601 (2021)] that show a stratified ambient modifies the current front velocity. Previous literature employed has inconsistent Reynolds numbers and boundary conditions, complicating interpretations. In the present numerical campaign, a closer analysis clarifies influence of the top boundary condition choice on formation and structure of the internal gravity waves. While acknowledging there is no available choice for a high-accuracy simplified numerical representation of a free-surface, a family of profiles for internal wave formation emerges varying with buoyancy Reynolds number and top boundary condition selection. The subsequent results appraise similarity solutions for the distribution of the heavy fluid in the gravity current. Our results show that for unstratified and low stratification ambient fluid, height profiles permit a similarity solution but higher values of S are less amenable; these profiles suggest a continuing time dependency on the traveling internal wave.

We study the weakly non linear solutions of theDubreil-Jacotin—Long elliptic equation in a strip, which describes two dimensional gravity internal waves propagating steadily in a stratified fluid. In the neighborhood of the first critical value of the Froude number, the center manifold theorem ensures that small solutions are parametrized by two coordinates which verify a system of nonlinear ordinary differential equations. We compute numerically the coefficients of the normal form of this reduced system for a three parameters family of stratifications and show that the quadratic coefficient (the most important) may become small. In that case, nonusual waves such as fronts can propagate. The last part of our work studies the case when a smooth stratification converges towards a piecewise constant profile having one discontinuity. We observe formally that the small waves which propagate at the interface of two homogeneous fluids are limits at leading order of waves travelling in the region where the smooth density varies rapidly.

On an internal wave equation describing a stratified fluid with finite depth