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This paper studies Gardner equation, which represents long nonlinear internal waves. The collocation method based on B-splines is applied to the equation. The stability of the proposed numerical scheme is analyzed by using von Neumann theory. To observe some physical properties of long nonlinear internal waves, three test problems which contain the propagation of solitary waves, the interaction of solitary waves and evolution of solitons are considered. Also, the effect of nonlinearity on physical problems is investigated. In order to see this effect clearly, the same parameters are used during the computation for different degrees of nonlinearity in each problem.

In this paper, a five-order Korteweg–de Vries (KdV) equation is studied, which is used to describe the nonlinear phenomena in the fluids, especially those of the surface gravity waves and internal waves in the stratified fluids. (a) Via the symbolic calculation, this KdV equation cannot pass the Painlevé test without any constraint conditions. By virtue of the ansatz method, bell-shape and kink soliton solutions of this KdV equation are attained. (b) Via the bilinear method, multisoliton solutions of this KdV equation are obtained under some constraint conditions. Propagation and interaction of the multisoliton are discussed. Soliton interaction is elastic, that is to say, they have no effect on each other’s amplitude and speed except for phase shift. We hope that our results will be useful for experimental studies of surface gravity waves and internal waves since the coefficients of this KdV equation are all expressed in terms of physical constants, depths, and densities of the fluid.

A number of physical mechanisms give rise to confined linear wave systems whose spatial structure is governed by a hyperbolic equation. These lack the discrete set of regular eigenmodes that are found in classical wave systems governed by an elliptic equation. In most 2D hyperbolic cases the discrete eigenmodes are replaced by a continuous spectrum of wave fields that possess a self-similar spatial structure and have a (point, line or planar) singularity in the interior. These singularities are called wave attractors because they form the attracting limit set of an iterated nonlinear map, which is employed in constructing exact solutions of this hyperbolic equation. While this is an inviscid, ideal fluid result, observations support the physical relevance of wave attractors by showing localization of wave energy onto their predicted locations. It is shown that in 3D, wave attractors may co-exist with a regular kind of trapped wave. Wave attractors are argued to be of potential relevance to fluids that are density-stratified, rotating, or subject to a magnetic field (or a combination of these) all of which apply to geophysical media.

Following the global strategy introduced recently by Bona, Lannes and Saut in Ref. 7, we derive here in a systematic way, and for a large class of scaling regimes, asymptotic models for the propagation of internal waves at the interface between two layers of immiscrible fluids of different densities, under the rigid lid assumption, the presence of surface tension and with uneven bottoms. The full (Euler) model for this situation is reduced to a system of evolution equations posed spatially on ℝ^d, d = 1, 2, which involve two nonlocal operators. The different asymptotic models are obtained by expanding the nonlocal operators and the surface tension term with respect to suitable small parameters that depend variously on the amplitude, wavelengths and depth ratio of the two layers. Furthermore, the consistency of these asymptotic systems with the full Euler equations is established.

Numerical solutions are given for a parabolic approximation to the acoustic wave equation at 75 Hz in two and three spatial dimensions to determine if azimuthal coupling of the field significantly affects horizontal coherence. Coupling is a small effect at 4000 km in the presence of internal gravity waves. This implies that accurate solutions are possible using computations from uncoupled vertical slices through the field. The shape of horizontal coherence is inconsistent with shapes given by two theories. Estimates of horizontal coherence at 4000 km and 25, 50, and 75 Hz are 10, 2, and 1 km respectively.

Numerical solutions are given for a parabolic approximation to the acoustic wave equation at 100 and 150 Hz in two and three spatial dimensions to determine if azimuthal coupling in the horizontal coordinate significantly affects horizontal correlation in the presence of internal gravity waves in the sea. Coupling is a small effect at 4000 km. This implies that accurate solutions are possible using computations from uncoupled vertical slices. Shape of horizontal correlation is inconsistent with shapes given by two theories. Estimates of horizontal correlation at 4000 km and 100 and 150 Hz are about 1 km and 0.5 km respectively.

A nonlocal spectral formulation of classic water waves is derived, and its connection to the classic water wave equations and the Dirichlet–Neumann operator is explored. The nonlocal spectral formulation is also extended to a two-fluid system with a free interface, from which long wave asymptotic reductions are obtained. Of particular interest is an asymptotically distinguished (2 + 1)-dimensional generalization of the intermediate long wave equation, which includes the Kadomtsev–Petviashvili equation and the Benjamin–Ono equation as limiting cases. Lump-type solutions to this (2 + 1)-dimensional ILW equation are obtained, and the speed versus amplitude relationship is shown to be linear in the shallow, intermediate, and deep water regimes.

In this paper, we establish local existence of solutions of a variant of a system derived by Choi and Camassa [Weakly nonlinear internal waves in a two-fluids system, J. Fluid Mech.313 (1996) 83–103] to describe the propagation of an internal wave at the interface of two immiscible fluids with constant densities. We also present a numerical solver to approximate the solutions of the Cauchy problem.

The deformation of nonlinear internal waves (NLIW) from depression to elevation waves during the shoaling process has been attributed to three key factors: wave amplitude, mixed-layer depth and water depth. This study examined the critical conditions for wave deformation by means of five strings of thermistors deployed at the Dongsha Atoll in the northern South China Sea (SCS). During a two-day experiment, four wave trains consisting of 78 solitons at five sites were identified. The parameters related to soliton transformation were formulated by applying the extended Korteweg–de Vries (eKdV) theory. The results indicated that the soliton energy dispersed dramatically along the sharp bottom slope, likely as a result of bottom friction and turbulence mixing. Large-amplitude depression waves were observed to transform into elevation waves before the thickness of the lower layer became equal to that of the upper layer. The ratio of the wave amplitude to the lower-layer thickness was found to be a good indicator of wave deformation. The critical conditions for the transition of the internal waves (IWs) occurred at the ratio approximately equal to $0.66\pm 0.2$. This means that a bottom-trapped elevation wave could form before passing through the theoretical critical point. The solitons felt the bottom and deformed when their amplitudes approached half of the lower-layer thickness. The solitons existed in the form of elevation waves when the waves were in contact with the bottom.

In this paper we construct small amplitude periodic internal waves traveling at the boundary region between two rotational and homogeneous fluids with different densities. Within a period, the waves we obtain have the property that the gradient of the stream function associated to the fluid beneath the interface vanishes, on the wave surface, at exactly two points. Furthermore, there exists a critical layer which is bounded from above by the wave profile. Besides, we prove, without excluding the presence of stagnation points, that if the vorticity function associated to each fluid in part is real-analytic, bounded, and non-increasing, then capillary-gravity steady internal waves are a priori real-analytic. Our new method provides the real-analyticity of capillary and capillary-gravity waves with stagnation points traveling over a homogeneous rotational fluid under the same restrictions on the vorticity function.

Adiabatic approximation (AP) combined with perturbation theory gives a fast normal-mode solution of temporal coherence for sound field in a two-dimensional deep water with time-varying random internal waves. Internal waves induced mode changes are deduced using the first-order perturbation theory [C. T. Tindle, L. M. O’Driscoll and C. J. Higham, Coupled mode perturbation theory of range dependence, J. Acoust. Soc. Am. 108(1) (2000) 76–83]. And mode perturbations in amplitude are neglected by the adiabatic method with wavenumber perturbations in phase merely considered. The AP expression of temporal coherence function is theoretically identical to the adiabatic transport equation theory [J. A. Colosi, T. K. Chandrayadula, A. G. Voronovich and V. E. Ostashev, Coupled mode transport theory for sound transmission through an ocean with random sound speed perturbations: Coherence in deep water environments, J. Acoust. Soc. Am. 134(4) (2013) 3119–3133]. Numerical results of the adiabatic temporal coherence function for several low frequencies and ranges up to 1000km are calculated. Then the coherence time scales obtained from the calculations are examined by a one-way coupled theory considering forward scattering [A. G. Voronovich, V. E. Ostashev and J. A. Colosi, Temporal coherence of acoustic signals in a fluctuating ocean, J. Acoust. Soc. Am. 129(6) (2011) 3590–3597]. Comparisons demonstrate that the range and frequency dependence of coherence time for both methods are quite close. And this shows good agreement with the well-known inverse frequency and inverse square root range laws. In addition, the internal wave energy dependence of coherence time is also studied.

Crossing internal wave trains are commonly observed in continental shelf shallow water. In this paper, we study the effects of crossing internal wave structures on three-dimensional acoustic ducts with both theoretical and numerical approaches. We show that, depending on the crossing angle, acoustic energy, which is trapped laterally between internal waves of one train, can be either scattered, cross-ducted or reflected by the internal waves in the crossing train. We describe the governing physics of these effects and illustrate them for selected internal wave scenarios using full-field numerical simulations.

Frequency-difference source localization (FDSL) is a relatively new method for locating sources emitting acoustic or electromagnetic waves from recorded data. Estimates of location are produced by matching a nonlinear function of the received data with the same nonlinear function of modeled data. This so-called matched-field approach depends on the accuracy of the modeled field. FDSL exhibits less sensitivity to model errors in some cases of interest. Fluctuations of the speed of sound in the ocean are affected by a wide variety of phenomena and internal gravity waves is one where their spectrum is usually well known. There is no analytical theory for predicting how these fluctuations of sound speed affect the accuracy of FDSL, leaving a numerical evaluation as the alternative for the focus of this paper. Experimental application of FDSL was previously made for an acoustic source of 100 Hz bandwidth surrounding 250 Hz for a 129 km section in the Philippine Sea with data recorded on a long vertical array [Geroski and Dowling, Long-range frequency-difference source localization in the Philippine Sea, J. Acoust. Soc. Am. 146 (2019) 4727–4739.]. The role of internal waves is quantified with a realistic FDSL simulation using models based on the parabolic and ray approximations of the linear acoustic wave equation. The modeled vertical variation of FDSL error is similar to observed while the modeled horizontal error is too small. The complicated software model is automatic but is computationally intensive, with our version needing about 1.5 mo to compute on a PC with 16 cpu cores.

We consider a Boussinesq system describing one-dimensional internal waves which develop at the boundary between two immiscible fluids, and we restrict to its traveling waves. The method which yields explicitly all the elliptic or degenerate elliptic solutions of a given nonlinear, any order algebraic ordinary differential equation is briefly recalled. We then apply it to the fluid system and, restricting in this preliminary report to the generic situation, we obtain all the solutions in that class, including several new solutions.