Surface Shallow Water Research Articles

Vortex Motion on the Surface of Shallow and Deep Water

The attenuation of vortex motion on the surface of shallow and deep water is investigated. The vortex flow was formed by two mutually perpendicular waves excited by plungers at a frequency of 6 Hz (wavelength λ = 5.6 cm) on the surface of water measuring 70 × 70 cm and depth h. After reaching a stationary state in the vortex system, the wave pumping was switched off, and the attenuation of the surface flow was recorded. On the surface of shallow water, λ/2π ≈ h, when the characteristic size of the vortices L exceeds the depth of the liquid, L h, the time dependence of the energy of the vortex flow Е(t) is described by an exponential function at all pumping levels, and the dependence of the enstrophy Ф(t) = Ω2(t) has an exponential dependence only in the range of wave vectors 0–0.3 см–1. On the surface of deep water λ/2π h, L ≈ h dependence Е(t) and Ф(t) are far from exponential in all ranges of wave numbers. At high pumping levels, the dependencies Е(t) and Ф(t) are nonmonotonic, which can be attributed to the influence of volumetric flows.

Behavioral response of megafauna to boat collision measured via animal-borne camera and IMU

Overlap between marine megafauna and maritime activities is a topic of global concern. Basking sharks (Cetorhinus maximus; CM) are listed as Globally Endangered under the IUCN, though reported sightings appear to be increasing in Ireland. While such trends in the region are welcome, increasing spatiotemporal overlap between CM and numerous water users poses an increased risk of boat strikes to the animals. To demonstrate the risk and impact of boat strikes on marine megafauna, we present camera-enabled animal-borne inertial measurement unit (IMU) data from a non-lethal boat strike on a CM within a proposed National Marine Park in Ireland. We tagged a ~7-m female CM in County Kerry, Ireland, which was struck by a boat ~6 h after tag deployment. Comparison of pre-strike data with 4 h of video and ~7.5 h of IMU data following the boat strike provides critical insight into the animal’s response. While the CM reacted momentarily with an increase in activity and swam to the seafloor, it quickly reduced its overall activity (i.e., overall dynamic body acceleration, tailbeat cycles, tailbeat amplitude, and vertical velocity) for the remainder of the deployment. Notably, the animal also ceased feeding for the duration of the video and headed towards deep offshore waters, which is in stark contrast to the pre-strike period where the animal was consistently observed feeding along the surface in shallow coastal water. This work provides insight into a CM’s response to acute injury and highlights the need for appropriate protections to mitigate risks for marine megafauna.

Vortex Motion on the Surface of Shallow and Deep Water

Vortex Motion on the Surface of Shallow and Deep Water

Explorations of certain nonlinear waves of the Boussinesq and Camassa–Holm equations using physics-informed neural networks

The Boussinesq and Camassa–Holm equations are, respectively, used to describe the bidirectional and unidirectional motions of small-amplitude waves on shallow water surfaces, but their wave dynamics remain a challenge for almost all conventional numerical methods. In this paper, we use physics-informed neural networks (PINNs), a mesh-free deep learning method, to accurately predict the soliton (peakon) interaction or rogue wave behaviours of both equations with only a few initial and boundary data, providing a method for solving certain numerically unstable fluid systems or extreme wave solutions of regular fluid systems. It is revealed that by decomposing both of the equations into lower-order coupled systems, one can obtain higher-precision wave behaviours, especially for the Camassa–Holm equation. To retrieve certain unknown hydraulic parameters based on the observed wave data, e.g. the water depth, we use a multiple PINN method and stably identify the dynamic parameters in the Boussinesq equation through the association of localized soliton and rogue wave solutions. Furthermore, we compare the PINNs with conventional high-precision time-splitting Fourier spectral (TSFS) method and find that to achieve the same split feature of the Y-shapedsoliton of the Boussinesq equation, PINNs require only one-third of the initial and boundary data of the TSFS method.

Soliton Solution of the Nonlinear Time Fractional Equations: Comprehensive Methods to Solve Physical Models

In this paper, we apply two different methods, namely, the G′G-expansion method and the G′G2-expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation in the conformable sense. The mentioned nonlinear partial differential equations (NPDEs) arise in diverse physical applications such as ion sound waves in plasma and waves on shallow water surfaces. There exist multiple wave solutions to many NPDEs and researchers are interested in analytical approaches to obtain these multiple wave solutions. The multi-exp-function method (MEFM) formulates a solution algorithm for calculating multiple wave solutions to NPDEs and at the end of paper, we apply the MEFM for calculating multiple wave solutions to the (2 + 1)-dimensional equation.

A mathematical fractional model of waves on Shallow water surfaces: The Korteweg-de Vries equation

<abstract><p>The homotopy perturbation transform method was examined in the present research to address the nonlinear time-fractional Korteweg-de Vries equations using a nonsingular kernel fractional derivative that Caputo-Fabrizio recently developed. We devoted our research to the nonlinear time-fractional Korteweg-de Vries equation and certain associated phenomena because of some physical applications of this equation. The results are significant and necessary for illuminating a range of physical processes. This paper considered an innovative method and fractional operator in this context to obtain satisfactory approximations to the provided issues. To solve nonlinear time-fractional Korteweg-de Vries equations, we first considered the Yang transform of the Caputo-Fabrizio fractional derivative. In order to confirm the applicability and efficacy of the provided method, we took into consideration two cases of the nonlinear time-fractional Korteweg-de Vries equation. He's polynomials were useful in order to manage nonlinear terms. In this method, the outcome was calculated as a convergent series, and it was demonstrated that the homotopy perturbation transform method solutions converge to the exact solutions. The main benefit of the suggested method was that it offered solutions with a high degree of precision while requiring minimal computation. Graphs were also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both examples outcomes were examined using diagrams and numerical data. These graphs showed how the approximated solution's graph and the precise solution's graph eventually converged as the non-integer order gets closer to integer order. When $ \varsigma = 1 $, several numerical comparisons were conducted with the exact solutions. The numerical simulation was offered to illustrate the efficiency and reliability of the proposed approach. In addition, the behavior of the provided solutions was explained using a number of fractional orders. The theoretical analysis matched with the findings obtained using the current technique, and the suggested technique can be extended to tackle many higher-order nonlinear dynamics problems.</p></abstract>

EXACT TRAVELING WAVE SOLUTION OF GENERALIZED (4+1)-DIMENSIONAL LOCAL FRACTIONAL FOKAS EQUATION

In this paper, within the scope of the local fractional derivative theory, the (4+1)-dimensional local fractional Fokas equation is researched. The study of exact solutions of high-dimensional nonlinear partial differential equations plays an important role in understanding complex physical phenomena in reality. In this paper, the exact traveling wave solution of generalized functions is analyzed defined on Cantor sets in high-dimensional integrable systems. The results of non-differentiable solutions in different cases are numerically simulated when the fractal dimension is equal to [Formula: see text] = ln 2/ln 3. The results show that the exact solution of the local fractional Fokas equation represents the fractal waves on the shallow water surface. Through numerical simulation, we find that the exact solution of the local fractional Fokas equation can describe the fractal waves and waves characteristics of shallow water surface. It also shows that the study of traveling wave solutions of nonlinear local fractional equations has important significance in mathematical physics.

THE EXACT TRAVELING WAVE SOLUTIONS OF LOCAL FRACTIONAL GENERALIZED HIROTA–SATSUMA COUPLED KORTEWEG–DE VRIES EQUATIONS ARISING IN INTERACTION OF LONG WAVES

Wave–wave interaction occurs in the propagation deformation of nonlinear long waves in shallow-water. In order to further study the propagation mechanism of shallow-water long waves interaction, the exact traveling wave solutions of the local fractional generalized Hirota–Satsuma coupled Korteweg–de Vries (HS-KdV) equations defined by the Cantor sets are obtained. The non-differentiable solutions with fixed fractal dimension and different propagation velocity are discussed. The results indicate that the exact solutions of the local fractional generalized HS-KdV equations characterize the interaction of fractal long waves with different dispersion relations on shallow-water surfaces.

Tropical cyclone effects enhance limnetic plankton trophic‐level relationships in a subtropical oligotrophic freshwater ecosystem

Abstract Tropical cyclones (TCs), as natural extreme weather events, alter plankton and hydrological environments, affecting the stability of biological processes in freshwater ecosystems, and such TC effects vary with water depths. Previous studies have found increased phytoplankton biomass resulting from TC effects has been observed, leading to potential strong grazing of zooplankton and enhanced plankton trophic‐level relationships. However, this remains understudied, particularly under in situ conditions. Using a zooplankton to phytoplankton (ZB/PB) ratio to represent the plankton trophic‐level relationship, we estimated the ZB/PB ratios at various depth intervals, including surface (2 m depth) and euphotic (depths between 0 and 20 m) depths and depth layer 0–50 m (depths between 0 and 50 m), in a subtropical deep oligotrophic freshwater ecosystem from 2012 to 2015 to understand how TC effects would influence changes in the ZB/PB ratio variations. TCs affected the surface and euphotic ZB/PB ratios but not those at the depth layer 0–50 m. The TC durations had an initially negative and then positive impact on the surface ZB/PB ratio, indicating that slow‐moving TCs might restructure surface plankton trophic‐level relationships. The water temperature and nutrient dynamics during the TC weeks showed the highest correlations with the ZB/PB ratios at the surface and euphotic depths. The combined environmental effects influenced the ZB/PB ratios at the surface and euphotic depths during the TC weeks, with 65.1% and 72.2% of the total variations explained in the multivariate regressions, respectively. There were greater impacts of TCs in shallow water (surface and euphotic depth) than in deep water. Aquatic food chains may be unexpectedly vulnerable to natural extreme weather events, such as TCs, and continuous assessments of food chain dynamics are necessary to better manage potential risks from natural extreme weather events in freshwater ecosystems.

Computational Analysis of Fractional-Order KdV Systems in the Sense of the Caputo Operator via a Novel Transform

The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy perturbation approach, and the Adomian decomposition method in the present investigation. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. The fractional derivative is regarded in the Caputo meaning. These approaches apply straightforward steps through symbolic computation to provide a convergent series solution. Different nonlinear time-fractional KdV systems are used to test the effectiveness of the suggested techniques. The symmetry pattern is a fundamental feature of the KdV equations and the symmetrical aspect of the solution can be seen from the graphical representations. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Additionally, the system’s approximative solution is illustrated graphically. The results show that these techniques are extremely effective, practically applicable for usage in such issues, and adaptable to other nonlinear issues.

Novel Computations of the Time-Fractional Coupled Korteweg–de Vries Equations via Non-Singular Kernel Operators in Terms of the Natural Transform

In the present research, we establish an effective method for determining the time-fractional coupled Korteweg–de Vries (KdV) equation’s approximate solution employing the fractional derivatives of Caputo–Fabrizio and Atangana–Baleanu. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. Some theoretical physical features of quantum mechanics are also explained by the KdV model. Many investigations have been conducted on this precisely solvable model. Numerous academics have proposed new applications for the generation of acoustic waves in plasma from ions and crystal lattices. Adomian decomposition and natural transform decomposition techniques are combined in the natural decomposition method (NDM). We first apply the natural transform to examine the fractional order and obtain a recurrence relation. Second, we use the Adomian decomposition approach to the recurrence relation, and then, using successive iterations and the initial conditions, we can establish the series solution. We note that the proposed fractional model is highly accurate and valid when using this technique. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Two examples are given to illustrate how the technique performs. Tables and 3D graphs display the best current numerical and analytical results. The suggested method provides a series form solution, which makes it quite easy to understand the behavior of the fractional models.

Estimation of seabed properties from ocean ambient noise in shallow water: A review

Ocean ambient noise is the sound persisting in the ocean, which includes contributions from natural and anthropogenic sources, such as rain, wind, shipping, and marine mammals. Multiple interactions of the sound with the surface and bottom in shallow water affect the spatial characteristics of ambient noise. Thus, the spatial noise properties, coherence, and directionality of ambient noise depend on the characteristics of the shallow water environment, such as depth, sound speed profile, and seabed sediment properties. Acoustic properties of seafloor sediments can be extracted from ambient noise based on these noise properties using various types of noise data and geo-acoustic inversion methods. This paper presents a review of literature concerning investigations into available methods, the implementations of these methods, and their verification and validation. The efficacy and the applicability of these inversion methods for a wide range of ocean noise data are also examined in this review.

Semi-Analytic Approach to Solving Rosenau-Hyman and Korteweg-De Vries Equations Using Integral Transform

In this research, we proposed the fusing of Elzaki transform and projected differential transform (PDTM) to obtain an analytical or approximate solution of the Rosenau-Hyman and Korteweg-de Vries equations which respectively govern pattern formation in liquid drops and model of waves on shallow water surfaces. The results obtained presented in tables and graphs showed better efficiency, accuracy, and convergence of the method to handle Rosenau-Hyman and Korteweg-de Vries equations when compared to other methods in the literature.
 Keywords: Rosenau-Hyman Equation; Korteweg-de Vries equation; Elzaki Projected differential transform method; Semi-analytic approach

Erratum to: Features of the Generation of Vortex Motion by Waves on the Surface of Shallow and Deep Water

An Erratum to this paper has been published: https://doi.org/10.1134/S1027451023010457

Features of the Generation of Vortex Motion by Waves on the Surface of Shallow and Deep Water

Features of the Generation of Vortex Motion by Waves on the Surface of Shallow and Deep Water

New Soliton Solutions of Time-Fractional Korteweg–de Vries Systems

Model construction for different physical situations, and developing their solutions, are the major characteristics of the scientific work in physics and engineering. Korteweg–de Vries (KdV) models are very important due to their ability to capture different physical situations such as thin film flows and waves on shallow water surfaces. In this work, a new approach for predicting and analyzing nonlinear time-fractional coupled KdV systems is proposed based on Laplace transform and homotopy perturbation along with Caputo fractional derivatives. This algorithm provides a convergent series solution by applying simple steps through symbolic computations. The efficiency of the proposed algorithm is tested against different nonlinear time-fractional KdV systems, including dispersive long wave and generalized Hirota–Satsuma KdV systems. For validity purposes, the obtained results are compared with the existing solutions from the literature. The convergence of the proposed algorithm over the entire fractional domain is confirmed by finding solutions and errors at various values of fractional parameters. Numerical simulations clearly reassert the supremacy and capability of the proposed technique in terms of accuracy and fewer computations as compared to other available schemes. Analysis reveals that the projected scheme is reliable and hence can be utilized with other kernels in more advanced systems in physics and engineering.

Soliton solution of coupled Korteweg-de Vries equation by quintic UAH tension B-spline differential quadrature method

The concern of this paper is to acquire the numerical solution of Coupled Korteweg-de Vries (CKdV) equation describing the behaviour of shallow water surfaces in terms of solitons. A new regime quintic Uniform Algebraic Hyperbolic (QUAH) tension B-spline with a very well-known method called Differential Quadrature Method (DQM) is used. Over the past few years, the idea of consuming DQM to solve partial differential equations numerically has acknowledged much consideration all over the science community. Initially, the QUAH tension B-spline is used as a basis function in DQM to solve the weighting coefficients. After calculating weighting coefficients, the basis function converts the Initial boundary value problem into an ordinary differential equation, which is further solved by using a strong preserving Runge-Kutta 43 regime. Then, to check the accuracy and amendment of the used method, different test problems are solved numerically as well as error norms L2 and L∞ are calculated. Calculated results are shown graphically and in tabular form for quick and easy access. An obtained result gives the insight that the proposed technique is effective and efficient for elucidating the variety of complex partial differential equations.

Analysis of the Magnetic Signature of Surface Waves Measured in a Laboratory Experiment

Abstract A magnetic signature is created by secondary magnetic field fluctuations caused by the phenomenon of seawater moving in Earth’s magnetic field. A laboratory experiment was conducted at the Surge Structure Atmosphere Interaction (SUSTAIN) facility to measure the magnetic signature of surface waves using a differential method: a pair of magnetometers, separated horizontally by one-half wavelength, were placed at several locations on the outer tank walls. This technique significantly reduced the extraneous magnetic distortions that were detected simultaneously by both sensors and additionally doubled the magnetic signal of surface waves. Accelerometer measurements and local gradients were used to identify magnetic noise produced from tank vibrations. Wave parameters of 4-m-long waves with a 0.56-Hz frequency and a 0.1-m amplitude were used in this experiment. Freshwater and saltwater experiments were completed to determine the magnetic difference generated by the difference in conductivity. Tests with an empty tank were conducted to identify the noise of the facility. When the magnetic signal was put through spectral analysis, it showed the primary peak at the wave frequency (0.56 Hz) and less pronounced higher-frequency harmonics, which are caused by the nonlinearity of shallow water surface waves. The magnetic noise induced by the wavemaker and related vibrations peaked around 0.3 Hz, which was removed using filtering techniques. These results indicate that the magnetic signature produced by surface waves was an order of magnitude larger than in traditional model predictions. The discrepancy may be due to the magnetic permeability difference between water and air that is not considered in the traditional model.

Statistical analysis and hybrid modeling of high-frequency underwater acoustic channels affected by wind-driven surface waves.

High frequency is a solution to high data-rate underwater acoustic communications. Extensive studies have been conducted on high-frequency (>40 kHz) acoustic channels, which are strongly susceptible to surface waves. The corresponding channel statistics related to acoustic communications, however, still deserve systematic investigation. Here, an efficient channel modeling method based on statistical analysis is proposed. Three wind-associated environmental models are integrated into this hybrid model. The Texel-Marsen-Arsole spectral model is adopted to generate a three-dimensional shallow-water surface, which affects the Doppler shifts of large-scale paths. Small-scale micropaths are statistically analyzed and modeled according to the measured channels. The Hall-Novarini model is adopted to simulate the refraction and attenuation caused by wind-generated bubbles. An existing wind-generated noise model is applied to calculate the noise spectrum. The proposed model has been validated by the at-sea measurements collected in the Gulf of Mexico in 2016 and 2017. This model can be used to further analyze the channels at different carrier frequencies, bandwidths, and wind speeds for certain transmission conditions.

Reflectionless wave propagation on shallow water with variable bathymetry and current. Part 2

We show that in the linear approximation there are three classes of reflectionless wave propagation on a surface of shallow water in the channel with spatially varying depth, width and current speed. Two of these classes have been described in our previous paper (Churilov & Stepanyants, J. Fluid Mech., vol. 931, 2022, A15), and the third one was discovered recently and is described here. The general analysis of the problem shows that, within the approach used in both of our papers, these three classes apparently exhaust all possible cases of exact solutions of the problem considered. We show that the reflectionless flow can be global for certain conditions, i.e. it can exist on the entire $x$ -axis. There are also reflectionless flows which exist only on limited intervals of the $x$ -axis.