Nonlinear Internal Gravity Waves Research Articles

Solitary wave solutions for the (1+1)-dimensional nonlinear Kakutani–Matsuuchi model of internal gravity waves

The Kakutani–Matsuuchi (KM) equation provides a mathematical framework for studying the dynamics of extended nonlinear internal gravity waves in a stratified fluid medium, including important phenomena such as solitary wave propagation. This study explores various analytical and semi-analytical methods to obtain accurate solutions for the (1＋1)-dimensional KM model.The primary objective of this research is to derive new solitary wave and periodic wave solutions, and subsequently validate these analytical techniques by comparing them with solutions obtained through the Adomian decomposition method. The KM model governs the dynamics of nonlinear waves and exhibits significant connections with other nonlinear equations, notably the (KdV) equation. The analytical methods, particularly the Khater II and auxiliary equation sub-equation methods, lead to closed-form solutions by reducing the partial differential equation to a system of ordinary differential equations. Conversely, the Adomian method provides a semi-analytical solution expressed as an infinite series.Solitary and periodic traveling wave solutions are systematically derived using the aforementioned analytical techniques. To verify their accuracy, solutions obtained through Adomian decomposition are computed recursively and compared. The results highlight the effectiveness of the analytical methods in discovering new exact solutions, thereby enhancing our understanding of the physical phenomena described by the KM equation. These solutions contribute to a more nuanced understanding of the properties and behaviors of internal waves, thereby enriching our overall knowledge in this field. Consequently, this research further develops analytical solution techniques tailored for nonlinear wave models prevalent in fluid dynamics.

Nonlinear internal gravity waves in the atmosphere: Rogue waves, breathers and dark solitons

We study nonlinear internal gravity waves (IGWs) in the atmosphere. The reductive perturbation method is used to derive a system of two-dimensional nonlinear equations for the envelope of velocity stream function and the mean flow. In the one-dimensional case, we obtain a nonlinear Schrödinger (NLS) equation corresponding to both horizontal and vertical propagation of IGWs. Depending on the characteristic wavelengths, the NLS equation is focusing or defocusing. In the focusing case, non-stationary solutions in the form of the Peregrine soliton, the Akhmediev breather and the Kuznetsov–Ma breather are considered as potential candidates for the modeling of rogue waves in the atmosphere. In the defocusing case, stationary nonlinear IGWs are considered in the form of nonlinear periodic waves and dark solitons.

Effects of nonlinear internal gravity waves on normal-incident reflection measurements of seafloor sediments.

Sonar data acquired by sub-bottom profilers and echosounder systems are widely used to estimate geoacoustic properties of marine sediments. However, the uncertainty of the seabed property estimates caused by water-column variability may limit the application. In this paper, the acoustic focusing and defocusing effects of nonlinear internal gravity waves on normal-incident acoustic reflection measurements are studied. The experiment data were collected in the South China Sea from two transceiver moorings located at two different sites, one of which contained strong nonlinear internal waves (NIWs), while another site did not. The observed reflection intensity variation at the internal wave site varied up to 10 dB. On the other hand, the bottom reflections at the other site without internal waves were stable, and a seafloor sediment sample collected there was analyzed to validate the sediment type inferred from bottom loss. Numerical simulations using ray-tracing and parabolic equation models confirmed the cause of this intensity fluctuation by the acoustic focusing and defocusing of NIWs. This study eventually showed that NIWs may induce a significant bias for geoacoustic property estimates from seabed reflection coefficients.

Coupled effects of nonlinear internal gravity waves and seabed properties on underwater sound propagation

Underwater sound propagation can be affected by strong sound speed gradients induced by nonlinear internal waves in the ocean. Meanwhile, geoacoustic properties of the seabed control acoustic reflections. Experimental data collected at the shelfbreak on the northeast of the South China Sea were analyzed to investigate the joint effects. Both two-dimensional (2D) and three-dimensional (3D) numerical sound propagation models with realistic seafloor and oceanographic inputs have been established to study the nonlinear internal wave effects observed on hydrophone vertical line array moorings. Compared to the 3D model, the prediction errors and uncertainties of the 2D scheme neglecting the transversal/azimuth energy propagation are discussed. In addition, the impact of sediment properties on 3D propagation effects affecting underwater soundscapes in the area is investigated. [Work supported by the Office of Naval Research.]

Effects of nonlinear internal gravity waves on normal-incident reflection measurements of marine sediments

Sonar data acquired by sub-bottom profilers and multi-beam echosounder systems are widely used to estimate geoacoustic properties of marine sediments. However, the uncertainty of the seabed property estimates caused by water column variability may limit the application. In this study, we focus on the acoustic focusing and defocusing effects of nonlinear internal waves on normal-incident reflection measurements in the South China Sea. The field experiment was conducted using a transceiver mooring consisted of a sound source and a four channel vertical hydrophone array. The variation of acoustic signal reflections from sub-bottom layers during three days at the same site is observed. The effects of the nonlinear internal waves passing through the transceiver mooring are identified. The impact on the sub-bottom property estimates is investigated. [Work supported by the Office of Naval Research.]

Observations of acoustic noise bursts accompanying nonlinear internal gravity waves on the continental shelf off New Jersey.

Anomalously large, transient fluctuations of acoustical noise intensity, up to 4-5 orders of magnitude above the background, were observed with single-hydrophone receiver units (SHRUs) and on the L-shaped horizontal and vertical line array of hydrophones (HVLA) in the Shallow Water 2006 experiment on the continental shelf off New Jersey. Here, temporal and spatial properties of these noise bursts are investigated. As tidally generated nonlinear internal waves (NIWs) move across the site of the experiment from the shelf break toward the coast, they form trains of localized, soliton-like waves with up to 25-35 m displacement of isopycnal surfaces. The NIW trains consecutively cross the positions of five SHRUs and HVLA that are located about 5-8 km from each other along a line perpendicular to the coast. The noise bursts were observed when a NIW train passed through locations of the corresponding acoustic receivers. Turbulence of the water flow, saltation, and bedload of marine sediments were the dominant causes of the acoustic noise bursts caused by NIWs at different frequency bands. On near-bottom hydrophones, the most energetic part of the observed noise bursts is attributed to collisions of suspended sediment particles with each other, the sensor, and the seafloor.

Physical mechanisms of generation of the noise bursts associated with nonlinear internal tides in shallow water

Recent paper of authors [J. Acoust. Soc. Am. 145(3, Pt. 2), 1670 (2019)] reported observations of repeated increase in acoustic noise intensity during passage of trains of nonlinear internal gravity waves. Periodic increases of pressure fluctuations by up to 35–40 dB in the frequency band from 10 to 5000 Hz were recorded on multiple near-bottom hydrophones during Shallow Water 2006 experiment. The present paper extends the earlier analysis to a more diverse set of nonlinear internal wave events by exploiting the acoustic data obtained at sites with a wider range of water depths and over a longer observation period. Three distinct physical mechanisms are identified, which are responsible for increased noise in different frequency bands. The observed noise bursts are attributed to turbulence of the water flow past hydrophones, sediment transport, and sediment saltation. Correlation between the velocity of the internal tide-induced near-bottom currents and noise intensity at low and mid-frequencies is investigated. [Work supported by ONR, NSF, and BSF.]

Using ambient noise to evaluate the acoustic Green’s function for a passive geoacoustic inversion in a dynamic shallow-water environment

Cross-correlation functions (CCFs) of ambient and shipping noise recorded by two hydrophones approximate the deterministic Green’s function and contain information about the propagation environment. This paper employs the data collected in the Shallow Water 2006 experiment on the New Jersey continental shelf to investigate the factors that affect the emergence of approximate Green’s functions from noise CCF estimates and accuracy of the approximation. One month-long continuous records of noise obtained by moored single-hydrophone receivers are analyzed. Hydrophones are located in 80–100-m deep water at distances of several kilometers from each other. Rapid variations of the water sound speed profile, which are primarily due to propagation of trains of strong nonlinear internal gravity waves, limit useful noise-averaging time. Available water temperature data are used to guide the selection of time windows for noise averaging and improve CCF evaluation and retrieval of information on seafloor properties. Various approaches to coherent stacking of CCFs are compared. Time warping transform is applied to the resultant noise CCF to extract dispersion curves of acoustic normal modes. The results of the geoacoustic inversion based on the passively measured dispersion curves are compared with the earlier results obtained using controlled sound sources. [Work supported by NSF and BSF.]

On mean flow generation due to oblique reflection of internal waves at a slope

AbstractSmall‐amplitude expansions are utilized to discuss the mean flow induced by the reflection of a weakly nonlinear internal gravity wave beam at a uniform rigid slope, in the case where the beam planes of constant phase meet the slope at an arbitrary direction, not necessarily parallel to the isobaths, and the flow cannot be taken as two dimensional. Along the vertical, the Eulerian mean flow, due to such an oblique reflection, is equal and opposite to the Stokes drift so the Lagrangian mean flow vanishes, similar to a two‐dimensional reflection. The horizontal Eulerian mean flow, however, is controlled by the mean potential vorticity (PV) and the corresponding Lagrangian mean flow is generally nonzero, in contrast to two‐dimensional flow where PV identically vanishes. For an oblique reflection, furthermore, viscous dissipation can trigger generation of horizontal mean flow via irreversible production of mean PV, a phenomenon akin to streaming.

Onset of non-linear internal gravity waves in intermediate-mass stars

Internal gravity waves (IGW) propagate in the radiation zones of all stars. During propagation, their amplitudes are affected by two main features: radiative diffusion and density stratification. We have studied the implications of these two features on waves travelling within the radiative zones of non-rotating stars with stellar parameters obtained from the one-dimensional stellar evolution code, MESA. As a simple measure of induced wave dynamics, we define a criterion to see if waves can become non-linear and if so, under what conditions. This was done to understand the role IGW may play in angular momentum transport and mixing within stellar interiors. We find that the IGW generation spectrum, convective velocities, and the strength of density stratification all play major roles in whether waves become non-linear. With increasing stellar mass, there is an increasing trend in non-linear wave energies. The trends with different metallicities and ages depend on the generation spectrum.

A non-hydrostatic non-Boussinesq algorithm for free-surface ocean modelling

A three-mode time-split algorithm is proposed to simulate non-hydrostatic ocean processes in a natural way: fast pressure adjustments via acoustic waves are explicitly represented. The full set of compressible Navier–Stokes equations is integrated for a free-surface ocean with three time-steps respectively associated to the internal, external and compressible (non Boussinesq) modes. The resulting three-mode algorithm can be implemented either in a fully compressible configuration or in a “pseudo-compressible” configuration by artificially lowering acoustic-wave velocities to reduce computational costs. The present description of the three-mode algorithm focuses in the first place on the simulation of non-hydrostatic processes in free-surface ocean models.Several test simulations related to linear and non-linear acoustic, surface and internal gravity waves are studied in details with the proposed three-mode algorithm.

3D underwater sound propagation in coastal environments

Underwater sound propagation in a coastal ocean can be influenced by a variety of physical oceanographic conditions that are unique in the coastal and estuarine environment, such as salt wedge fronts, river-plume induced nonlinear internal gravity waves, strong tidal currents, surf zone waves, etc. The confined boundaries in the area can also produce reflection, diffraction and even scattering of sound due to shoaling seafloor. The morphological dynamics in the area is complex and results in corrugated and channeled bathymetry, which may cause 3D ducting and trapping of sound. In this talk, we will present our numerical study of 3D underwater sound propagation in Long Island Sound, a tidal estuary of the Atlantic Ocean. The Unstructured Grid Finite Volume Community Ocean Model (FVCOM) will be utilized to simulate physical oceanographic processes and produce water-column environmental input to the sound propagation model. A bathymetric database from high-resolution multibeam surveys will also be used to in...

The internal wave prediction component of the Integrated Ocean Dynamics and Acoustics Project

The accuracy of ocean sound propagation modeling depends on properly representing the environment. The water column portion of this has phenomena and features that are time- and space-dependent, covering a huge range of scales. Nonlinear internal gravity waves (nonlinear internal waves, NIW) in shallow areas are important features that are moving, evolving and anisotropic, and whose effects can be handled properly only with 3D sound modeling. Thus, 3D NIW field predictions would be needed for comprehensive modeling. One challenging part of our project to make acoustic condition forecasts from available data involves making this 3D NIW field prediction using data-assimilating regional models that do not faithfully handle NIW. Our methods for extracting internal tide signals from the models, analyzing their propagation into regions where they transform in reality to NIW (but not in the models), and predicting NIW conditions in these regions are explained here. Outstanding challenges such as how to parameterize internal-tide coupled mode propagation on slopes and how to model highly nonlinear crossing NIW groups will be presented. The question of what constitutes an effective NIW prediction for acoustic purposes will be addressed.

Incidence and reflection of internal waves and wave-induced currents at a jump in buoyancy frequency

Abstract. Weakly nonlinear internal gravity waves are treated in a two-layer fluid with a set of nonlinear Schrodinger equations. The layers have a sharp interface with a jump in buoyancy frequency approximately modeling the tropopause. The waves are periodic in the horizontal but modulated in the vertical and Boussinesq flow is assumed. The equation governing the incident wave packet is directly coupled to the equation for the reflected packet, while the equation governing transmitted waves is only coupled at the interface. Solutions are obtained numerically. The results indicate that the waves create a mean flow that is strong near and underneath the interface, and discontinuous at the interface. Furthermore, the mean flow has an oscillatory component that can contaminate the wave envelope and has a vertical wavelength that decreases as the wave packet interacts with the interface.

Dynamics of IGW and traveling ionospheric disturbances in regions with sharp gradients of the ionospheric parameters

The dynamics of the 2D solitary nonlinear internal gravity waves (IGW), as well as traveling ionospheric disturbances (TID) of the electron density excited by them at heights of the ionosphere F-region, for conditions close to those of the F-layer assuming that the source of initial perturbation has the pulse character is studied analytically and numerically. On a level with general case the rather interesting applications when the sharp gradients of the ionospheric parameters are the functions of space coordinates and time, namely the IGW and TID dynamics in the frontal regions of the solar terminator and solar eclipse are considered. The results obtained describe the dynamical structure, evolution and transformation of the IGW and TID at heights of the ionosphere F-layer including its strongly heterogeneous regions.

Sheared stably stratified turbulence and large-scale waves in a lid driven cavity

We investigated experimentally stably stratified turbulent flows in a lid driven cavity with a non-zero vertical mean temperature gradient in order to identify the parameters governing the mean and turbulent flows and to understand their effects on the momentum and heat transfer. We found that the mean velocity patterns (e.g., the form and the sizes of the large-scale circulations) depend strongly on the degree of the temperature stratification. In the case of strong stable stratification, the strong turbulence region is located in the vicinity of the main large-scale circulation. We detected the large-scale nonlinear oscillations in the case of strong stable stratification which can be interpreted as nonlinear internal gravity waves. The ratio of the main energy-containing frequencies of these waves in velocity and temperature fields in the nonlinear stage is about 2. The amplitude of the waves increases in the region of weak turbulence (near the bottom wall of the cavity), whereby the vertical mean temperature gradient increases.

Dust devil generation

The equations describing axi-symmetric nonlinear internal gravity waves in an unstable atmosphere are derived. A hydrodynamic model of a dust devil generation mechanism in such an atmosphere is investigated. It is shown that in an unstably stratified atmosphere the convective plumes with poloidal motion can grow exponentially. Furthermore, it is demonstrated that these convective plumes in an atmosphere with weak large scale toroidal motion are unstable with respect to three-dimensional dust devil generation.

Rolls of the internal gravity waves in the Earth's atmosphere

Abstract. The effect of the wind shear on the roll structures of nonlinear internal gravity waves (IGWs) in the Earth's atmosphere with the finite vertical temperature gradients is investigated. A closed system of equations is derived for the nonlinear dynamics of the IGWs in the presence of temperature gradients and sheared wind. The solution in the form of rolls has been obtained. The new condition for the existence of such structures was found by taking into account the roll spatial scale, the horizontal speed and wind shear parameters. We have shown that the roll structures can exist in a dynamically unstable atmosphere.

An experimental study of the free evolution of rotating, nonlinear internal gravity waves in a two-layer stratified fluid

AbstractThe temporal evolution of nonlinear large-scale internal gravity waves, in a two-layer flow affected by background rotation, is studied via laboratory experiments conducted in a cylindrical tank, mounted on a rotating turntable. The internal wave field is excited by the relaxation of an initial forced tilt of the density interface ($\eta _{i}$), which generates internal waves, such as Kelvin and Poincaré waves, in response to rotation effects. The behaviour of $\eta _{i}$, in the shore region, is analysed in terms of the background rotation and the nonlinear steepening of the basin-scale waves. The results show that the degeneration of the fundamental Kelvin wave into a solitary-type wave packet is caused by nonlinear steepening and it is influenced by the background rotation. In addition, the physical scales of the leading solitary-type wave are closer to Korteweg–de Vries theory as the rotation increases. Moreover, the nonlinear interaction between the Kelvin wave and the Poincaré wave can transfer energy to higher or lower frequencies than the frequency of the fundamental Kelvin wave, as a function of the background rotation. In particular, a specific normal mode in the off-shore region could be energized by this interaction. Finally, the bulk decay rate of the fundamental Kelvin wave, $\tau _{dk}$, was investigated. The results exhibit that $\tau _{dk}$ is concordant with the Ekman damping time scale when there is no evidence of steepening in the basin-scale waves. However, as nonlinear processes increase, $\tau _{dk}$ shows a strong decrease. In this context, the nonlinear processes play an important role in the decay of the fundamental Kelvin wave, via the energy radiation to other modes. The results reported demonstrate that the background rotation and nonlinear processes are essential aspects in understanding the degeneration and the decay of large-scale internal gravity waves on enclosed basins.

Self-organization of IGW structures in an inhomogeneous ionosphere: 2. Nonlinear vortex structures

The generation and further nonlinear dynamics of internal gravity wave (IGW) structures in a dissipative ionosphere in the presence of an inhomogeneous zonal wind (shear flow) have been studied. The effectiveness of the IGW amplification mechanism during the interaction with an inhomogeneous zonal wind is analyzed based on the corresponding model system of nonlinear dynamic equations constructed in (Aburjania et al., 2013). It has been indicated that IGWs effectively obtain the shear flow energy at the initial linear evolution stage and substantially (by an order of magnitude) increase their amplitude and, correspondingly, energy. The nonlinear self-localization mechanism starts operating with increasing amplitude, and the process terminates with the self-organization of nonlinear solitary strongly localized vortex structures. A new degree of system freedom and the disturbance evolution trend in a medium with a shear flow appear in such a way. Nonlinear IGW structures can be a purely monopoly vortex, a transverse vortex chain, and/or a longitudinal vortex path against the background of an inhomogeneous zonal wind, depending on the shear flow velocity profile. The accumulation of such vortices in the ionospheric medium can generate a strongly turbulent state.