Two-dimensional Internal Waves Research Articles

Exploring two-dimensional internal waves: A new three-coupled Davey–Stewartson system and physics-informed neural networks with weight assignment methods

In order to investigate various two-dimensional internal waves in a two-layer finite-water depth fluid with different velocity potentials in each layer, we derive a new (2+1)-dimensional system of three coupled, two-component Davey–Stewartson type equations under the long wave assumption involving complex amplitude and real velocity potentials in the upper and lower layers. It is found that exact solutions of this system can describe internal solitary waves, internal rogue waves, and internal breathers, and more interestingly, it is proposed for the first time that the patterns of changes in the velocity potentials of the upper and lower layers can be used as an indicator for recognizing the occurrence, disappearance, and amplitudes of these internal waves. In view of the application of deep learning techniques in the simulation, prediction and monitoring of internal waves, an improved physical information neural network (PINN) method is introduced to successfully simulate dynamics of internal solitary waves, internal rogue waves, and internal breathers, as well as the corresponding velocity potentials. The effects of no weight assignment and two different weight assignment methods in the loss function on the training results are analyzed in detail, which reveals that better simulating results can be achieved via a fixed-weight loss function for small-amplitude internal waves, while the introduction of weights has little effect on the simulations for larger-amplitude internal waves. In addition, the loss function with adaptive weights performs well in the fast simulation of internal rogue waves and poorly in the learning effect of the velocity potentials, which is contrary for internal solitary waves, but is not applicable in the simulation of internal breathers.

Kinetic Energy Exchanges between a Two-Dimensional Front and Internal Waves

Abstract Fronts and near-inertial waves (NIWs) are energetic motions in the upper ocean that have been shown to interact and provide a route for kinetic energy (KE) dissipation of balanced oceanic flows. In this paper, we study these KE exchanges using an idealized model consisting of a two-dimensional geostrophically balanced front undergoing strain-induced semigeostrophic frontogenesis and internal wave (IW) vertical modes. The front–IW KE exchanges are quantified separately during two frontogenetic stages: an exponential sharpening stage that is characterized by a low Rossby number and is driven by the imposed strain (i.e., mesoscale frontogenesis), followed by a superexponential sharpening stage that is characterized by an Rossby number and is driven by the convergence of the secondary circulation (i.e., submesoscale frontogenesis). It is demonstrated that high-frequency IWs quickly escape the frontal zone and are very efficient at extracting KE from the imposed geostrophic strain field through the deformation shear production (DSP). Part of the extracted KE is then converted to wave potential energy. On the contrary, NIWs remain locked to the frontal zone and readily exchange energy with the ageostrophic frontal circulation. During the exponential stage, NIWs extract KE from the geostrophic strain through DSP and transfer it to the frontal secondary circulation via the ageostrophic shear production (AGSP) mechanism. During the superexponential stage, a newly identified mechanism, convergence production (CP), plays an important role in the NIW KE budget. The CP transfers KE from the convergent ageostrophic secondary circulation to the NIWs and largely cancels out the KE loss due to the AGSP. This CP may explain previous findings of KE transfer enhancement from balanced motions to IWs in frontal regions of realistic ocean models. We provide analytical estimates for the aforementioned energy exchange mechanisms that match well the numerical results. This highlights that the strength of the exchanges strongly depends on the frontal Rossby and Richardson numbers. Significance Statement Fronts with large horizontal density and velocity gradients are ubiquitous in the upper ocean. They are generated by a process known as frontogenesis, which is often initialized by straining motions of mesoscale balanced circulations. Here we examine the energy exchanges between fronts and internal waves in an idealized configuration, aiming to elucidate the mechanisms that can drain energy from oceanic balanced circulations. We identify a new mechanism for energy transfers from the frontal circulation to near-inertial internal waves called convergence production. This mechanism is especially effective during the later stages of frontogenesis when the convergent ageostrophic secondary circulation that develops is strong.

Predicting range-dependent underwater sound propagation from structural sources in shallow water using coupled finite element/equivalent source computations

Predictions of underwater sound propagation (USP) from structural sources in complex shallow-water environments are crucial for underwater navigation, communication, and localization. Modeling range-dependent USP in shallow water remains challenging because structural acoustic radiation is coupled with complex waveguide physics. This paper presents a coupled finite element (FE)/equivalent source (ES) computation scheme for predicting the range-dependent USP from a structural source. The scheme involves coupled vibroacoustic FE/ES analyses and waveguide-field ES computations. The former computes the structural vibration response and reproduces the structural-acoustic radiation at arbitrary spatial positions. The coupled vibroacoustic FE/ES analysis provides the input for the waveguide-field ES computations, which couple the structural-acoustic radiation with the shallow-water environment. A multilayer acoustic–elastic ES method (ESM) is developed to accommodate sound speed inhomogeneities and a range-dependent elastic seabed. Numerical simulations demonstrate the interactions of structural-acoustic radiation with two-dimensional topographies and internal solitary waves. The proposed scheme is extended to three dimensions by combining the coupled vibroacoustic FE/ES analysis with a pre-corrected fast Fourier transform-accelerated ESM. The results validate the proposed scheme and demonstrate its benchmark-quality solutions and high numerical efficiency, suggesting great application potential for optimizing the sonar performance at the preliminary design stage.

On diffraction and oblique interactions of horizontally two-dimensional internal solitary waves

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.

Harmonics from a magic carpet

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Two-dimensional resonant triad interactions in a two-layer system

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Near-field internal wave beams in two dimensions

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The derivation of an isotropic model for internal waves and its application to wave generation

In contrast to two-dimensional oceanic internal waves which have been extensively investigated, there are relatively few theoretical studies on three-dimensional internal waves. The most remarkable theory describing three-dimensional internal waves is the Kadomtsev–Petviashvili (KP) equation. Nevertheless, two shortcomings – unidirectional propagation and anisotropy – limit its application in some general cases. In the current paper, via an asymptotic analysis, we derive an isotropic and bidirectional model, the modified Benney–Luke equation, for a two-layer fluid with bottom topography which can vary both in time and space, therefore tides and currents can be incorporated into the internal wave problem due to the relative motion. In the derivation, the assumption of incompressibility and long waves are invoked and the effects of the Earth’s rotation and dissipation are ignored. Based on this model, the generation of nonlinear internal waves by background currents and barotropic tides flowing over topography are investigated. The resulting equation aims to facilitate dynamical analyses, as well as the interpretation of in-situ observational data and laboratory experimental results.

Flows induced by Coriolis-influenced vertically propagating two-dimensional internal gravity wave packets

In the absence of background rotation, the Eulerian flow induced by two-dimensional internal wave packets is well known to have the structure of long trailing internal waves. We show that rotation makes these waves evanescent and the structure of the flow over the waves in the wave packet qualitatively changes, with implications for the modulational stability of the waves.

Energy budget in internal wave attractor experiments

The current paper presents an experimental study of the energy budget of a two-dimensional internal wave attractor in a trapezoidal domain filled with uniformly stratified fluid. The injected energy flux and the dissipation rate are simultaneously measured from a two-dimensional, two-component, experimental velocity field. The pressure perturbation field needed to quantify the injected energy is determined from the linear inviscid theory. The dissipation rate in the bulk of the domain is directly computed from the measurements, while the energy sink occurring in the boundary layers is estimated using the theoretical expression for the velocity field in the boundary layers, derived recently by Beckebanze et al. (J. Fluid Mech., vol. 841, 2018, pp. 614–635). In the linear regime, we show that the energy budget is closed, in the steady state and also in the transient regime, by taking into account the bulk dissipation and, more importantly, the dissipation in the boundary layers, without any adjustable parameters. The dependence of the different sources on the thickness of the experimental set-up is also discussed. In the nonlinear regime, the analysis is extended by estimating the dissipation due to the secondary waves generated by triadic resonant instabilities, showing the importance of the energy transfer from large scales to small scales. The method tested here on internal wave attractors can be generalized straightforwardly to any quasi-two-dimensional stratified flow.

Interference and transmission of spatiotemporally locally forced internal waves in non-uniform stratifications

Studies of the effects of constructive or destructive interference on the transmission of internal waves through non-uniform stratifications have typically been performed for internal wave fields that are spatiotemporally harmonic. To understand the impacts of spatiotemporal localization, we present a theoretical and experimental study of the transmission of two-dimensional internal waves that are generated by a boundary forcing that is localized in both space and time. The model analysis reveals that sufficient localization leads to the disappearance of transmission peaks and troughs that would otherwise be present for a harmonic forcing. The corresponding laboratory experiments that we perform provide clear demonstration of this effect. Based on the group velocity and angle of propagation of the internal waves, a practical criterion that assesses when the transmission peaks or troughs are evident is obtained.

Two-dimensional fractional algebraic internal solitary waves model and its solution

With the development of theory and advancement of scientific research, fractional calculus has begun to be considered as a method for the study of physical systems. In this work, based on the basic system of equations for internal solitary waves, we have derived two-dimensional Benjamin–Ono equation. Then, the integer-order two-dimensional Benjamin–Ono equation is transformed into the time-fractional Benjamin–Ono equation. To study the properties of the algebraic internal solitary waves, we discuss the conservation laws of the new model. Also, using the Hirota bilinear method, the derived new model is solved. Finally, we explore the characteristics of motion of the algebraic internal solitary waves with the help of the multi-soliton solutions.

On regularizing the strongly nonlinear model for two-dimensional internal waves

To study the evolution of two-dimensional large amplitude internal waves in a two-layer system with variable bottom topography, a new asymptotic model is derived. The model can be obtained from the original Euler equations for weakly rotational flows under the long-wave approximation, without making any smallness assumption on the wave amplitude, and it is asymptotically equivalent to the strongly nonlinear model proposed by Choi and Camassa (1999) [3]. This new set of equations extends the regularized model for one-dimensional waves proposed by Choi et al. (2009) [30], known to be free from shear instability for a wide range of physical parameters. The two-dimensional generalization exhibits new terms in the equations, related to rotational effects of the flow, and possesses a conservation law for the vertical vorticity. Furthermore, it is proved that if this vorticity is initially zero everywhere in space, then it will remain so for all time. This property–in clear contrast with the original strongly nonlinear model formulated in terms of depth-averaged velocity fields–allows us to simplify the model by focusing on the case when the velocity fields involved by large amplitude waves are irrotational. Weakly two-dimensional and weakly nonlinear limits are then discussed. Finally, after investigating the shear stability of the regularized model for flat bottom, the effect of slowly-varying bottom topography is included in the model.

Influence of electric and magnetic properties of bottom rocks on an electromagnetic field induced by surface and internal waves

The electric and magnetic field and electric charges at the medium interface induced by a two-dimensional marine surface and internal waves are considered. The influence of the magnetic permeability and electric conductivity of the bottom rocks on the induced fields is analyzed.

Simulation of two-dimensional internal waves generated by a translating and pitching foil

A code developed on the basis of the flux-difference splitting scheme and the hybrid Cartesian/immersed boundary method is applied for two-dimensional simulation of internal waves generated by a foil that is translating and pitching simultaneously near a material interface. The interface is captured as a moving contact discontinuity without any additional treatment along the interface. An approximate Riemann solver is used to estimate numerical fluxes across the discontinuity. Immersed boundary nodes are distributed within an instantaneous fluid domain on the basis of edges crossing a boundary. Dependent variables are reconstructed at the immersed boundary nodes along local normal lines to the boundary. The present results on the propagation of internal solitary waves generated by the collapse mechanism are compared with other computational results and good agreement is found. The code is validated through comparisons with recent experimental results on the waveform inversion from depression type to elevation type during the interaction between an internal solitary wave and a trapezoidal obstacle. Internal waves generated by a translating and pitching foil are simulated. Grid independence tests of the computed results are carried out. Pairs of traveling vortices are correlated to local sinking or rising at the interface.

Comparison of laboratory and numerically observed scalar fields of an internal wave attractor

Observations of internal gravity wave beams are frequently accompanied by theory that is purely two-dimensional, or two-dimensional numerical models. Although qualitative agreement between such models and laboratory experiments has been demonstrated, quantitative comparison has only been possible in a limited range of cases. Here, we present a quantitative comparison for internal wave attractors in the laboratory and a two-dimensional non-hydrostatic numerical model. To make a closer connection with previous theoretical work, the experimental and numerical results are presented in terms of the streamfunction and density perturbation, rather than the measured velocity and density gradient fields. The streamfunction is commonly used in the two-dimensional descriptions, e.g. to predict spatial patterns found in an enclosed stratified fluid in the laboratory. We demonstrate that, although the laboratory experiment in a narrow tank is only semi-two-dimensional, the flow is well described by two-dimensional internal wave theory and the numerical model reproduces quantitatively comparable attractors. The observed streamfunction field is compared with theoretical predictions, addressing an open question on the form of the streamfunction for internal wave attractor in a trapezoidal domain. The streamfunction has a simple spatial structure with sharp gradients at the attractor separating regions of nearly constant value outside the attractor.

High-order regridding–remapping schemes for continuous isopycnal and generalized coordinates in ocean models

A hierarchy of high-order regridding–remapping schemes for use in generalized vertical coordinate ocean models is presented. The proposed regridding–remapping framework is successfully used in a series of idealized one-dimensional numerical experiments as well as two-dimensional internal wave and overflow test cases. The model is capable of replicating z-, sigma- and isopycnal-coordinate results, among others. Particular emphasis is placed on the design of a continuous isopycnal framework, which is a more general alternative to the layered isopycnal paradigm. Continuous isopycnal coordinates use target interface densities to define layers. In contrast to traditional layered isopycnal models, in which along-layer density gradients vanish, general coordinate approaches must deal with extra terms. For example, the calculation of pressure gradient force is more complicated and must be evaluated carefully. High-order reconstructions within boundary cells are crucial for obtaining sensible results and for reducing spurious diffusion near boundaries. Vertical advection is implicitly embedded in the remapping step and directly benefits from high-order schemes. Volume and all tracers are conserved to machine precision, which is a necessary ingredient for long-term ocean climate modeling. This hybrid vertical coordinate model provides the framework to easily capture the impact of different coordinate systems on dynamics.

A nonhydrostatic numerical model for density stratified flow and its applications

A modular numerical model was developed for simulating density-stratified flow in domains with irregular bottom topography. The model was designed for examining interactions between stratified flow and topography, e.g., tidally driven flow over two-dimensional sills or internal solitary waves propagating over a shoaling bed. The model was based on the non-hydrostatic vorticity-stream function equations for a continuously stratified fluid in a rotating frame. A self-adaptive grid was adopted in the vertical coordinate, the Alternative Direction Implicit (ADI) scheme was used for the time marching equations while the Poisson equation for stream-function was solved based on the Successive Over Relaxation (SOR) iteration with the Chebyshev acceleration. The numerical techniques were described and three applications of the model were presented.

Numerical simulation of a two-dimensional internal wave attractor

Internal (gravity) wave attractors may form in closed containers with boundaries non-parallel and non-normal to the gravity vector. Such attractors have been studied from a theoretical point of view, in laboratory experiments and using linear numerical computations. In the present paper two-dimensional numerical simulations of an internal wave attractor are reported, based upon the nonlinear and non-hydrostatic MIT-gcm numerical code. We first reproduce the laboratory experiment of a wave attractor performed by Hazewinkel et al. (J. Fluid Mech. Vol. 598, 2008 p. 373) and obtain very good agreement with the experimental data. We next propose simple ideas to model the thickness of the attractor. The model predicts that the thickness should scale as the 1/3 power of the non-dimensional parameter measuring the ratio of viscous to buoyancy effects. When the attractor is strongly focusing, the thickness should also scale as the 1/3 power of the spatial coordinate along the attractor. Analysis of the numerical data for two different attractors yields values of the exponent close to 1/3, within 30%. Finally, we study nonlinear effects induced by the attractor.

Beyond ray tracing for internal waves. I. Small-amplitude anelastic waves

We compute the transmission of small-amplitude two-dimensional anelastic internal waves in nonrotating inviscid fluid having arbitrarily specified stratification and background velocity. Whereas stability analyses of the flow involve solving an eigenvalue problem that relates the frequency to horizontal wavenumber, we focus upon the evolution of waves incident from below with independently specified frequency and horizontal wavenumber. A numerical method is developed to ensure that the wave field in the upper domain corresponds only to upward-propagating transmitted waves. Two particular applications are discussed. First, internal waves incident upon a piecewise-linear shear layer are examined and their transmission is computed as a function of the bulk Richardson number Rib and the ratio of the density scale height relative to the depth of the shear layer. The waves transmit partially across critical levels if they coincide with heights where the gradient Richardson number is less than 1/4. Transmission is larger if Rib is smaller. Decreasing the density scale height reduces the frequency and wavenumber range over which internal waves propagate, but this does not significantly affect the magnitude of transmission. Second, internal waves generated by flow over Jan Mayen island are examined. Although the waves are ducted, the waves are found to transmit partially through the top of the duct. The results are used to interpret the discrepancy between predictions of the Fourier-ray tracing model and fully nonlinear numerical simulations of Eckermann et al. [Mon. Weather Rev. 134, 2830 (2006)].