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Abstract: Abstract A nonlinear evolution equation is derived for surface gravity waves correct up to fourth order in wave steepness in the presence of a pycnocline of finite thickness in a fluid domain of infinite depth. Using this nonlinear evolution equation, the stability analysis is performed for a uniform-wave train solution. It is observed that pycnocline of finite thickness has less stabilizing influence than thin pycnocline. The maximum growth rate of instability becomes less due to presence of pycnocline, and it increases as the thickness of the pycnocline increases. It is revealed that the maximum growth rate of instability decreases as the density differences across the pycnocline increases and also decreases as the depth of the pycnocline increases. The maximum growth rate of instability becomes higher for smaller values of the angle between the direction of the space variation of the amplitudes and the direction of propagation of the wave. Also, it is observed that the unstable region increases with the decrease in the depth of the pycnocline. PubDate: 2022-04-12

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Abstract: Abstract The focus here is upon the generalized Korteweg–de Vries equation, $$\begin{aligned} u_t + u_x + \frac{1}{p} \left( u^p \right) _x +u_{xxx} \, = \, 0, \end{aligned}$$ where \(p = 2, 3, \ldots \) . When \(p \ge 5\) , it is thought that the equation is not globally well posed in time for \(L_2\) -based Sobolev class data. Various numerical simulations carried out by multiple research groups indicate that solutions can blowup in finite time for large, smooth initial data. This is known to be the case in the critical case \(p = 5\) , but remains a conjecture for supercritical values of p. Studied here are methods for controlling this potential blow up. Several candidates are put forward; the addition of dissipation or of higher order dispersion are two obvious candidates. However, these apparently can only work for a limited range of nonlinearities. However, the introduction of high frequency temporal oscillations appear to be more effective. Both temporal oscillation of the nonlinearity and of the boundary condition in an initial-boundary-value configuration are considered. The bulk of the discussion will turn around this prospect in fact. PubDate: 2022-04-11

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Abstract: Abstract Understanding the droplet cloud and spray dynamics is important for the study of the ocean surface and marine boundary layer. The role that the wave energy and the type of wave breaking play in the resulting distribution and dynamics of droplets are yet to be understood. The aim of this work was to generate violent plunging breakers in the laboratory and analyze the spray production post-breaking, i.e. after the crest of the wave impacts in the free surface. The droplet sizes and their dynamics were measured with imaging techniques and the effect of different wind speeds on the droplet production was also considered. It was found that the mean radius increases with the wave energy content and the number of large droplets (radius > 1 mm) in the vertical direction increases with the presence of wind. Furthermore, the normalized distribution of droplet sizes is consistent with the distribution of ligament-mediated spray formation. Also, indications of turbulence affecting the droplet dynamics at wind speeds of 5 m/s were found. The amount of large droplets (radius > 1 mm) found in this work was larger than reported in field studies. PubDate: 2022-03-31

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Abstract: Abstract If a wavemaker at one end of a water-wave tank oscillates with a particular frequency, time series of downstream surface waves typically include that frequency along with its harmonics (integer multiples of the original frequency). This behavior is common for the propagation of weakly nonlinear waves with a narrow band of frequencies centered around the dominant frequency such as in the evolution of ocean swell, pulse propagation in optical fibers, and Bose-Einstein condensates. Presented herein are measurements of the amplitudes of the first and second harmonic bands from four surface water wave laboratory experiments. The Stokes expansion for small-amplitude surface water waves provides predictions for the amplitudes of the second and higher harmonics given the amplitude of the first harmonic. Similarly, the derivations of the NLS equation and its generalizations (models for the evolution of weakly nonlinear, narrow-banded waves) provide predictions for the second and third harmonic bands given amplitudes of the first harmonic band. We test the accuracy of these predictions by making two types of comparisons with experimental measurements. First, we consider the evolution of the second harmonic band while neglecting all other harmonic bands. Second, we use explicit Stokes and generalized NLS formulas to predict the evolution of the second harmonic band using the first harmonic data as input. Comparisons of both types show reasonable agreement, though predictions obtained from dissipative generalizations of NLS consistently outperform the conservative ones. Finally, we show that the predictions obtained from these two methods are qualitatively different. PubDate: 2022-02-15 DOI: 10.1007/s42286-022-00055-7

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Abstract: Abstract We study the spectral stability of small-amplitude periodic traveling waves of the Ostrovsky equation. We prove that these waves exhibit spectral instabilities arising from a collision of pair of non-zero eigenvalues on the imaginary axis when subjected to square-integrable perturbations on the whole real line. We also list all such collisions between pairs of eigenvalues on the imaginary axis and do a Krein signature analysis. PubDate: 2022-01-12 DOI: 10.1007/s42286-021-00054-0

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Abstract: Abstract In this article, we provide formulations of energy flux and radiation stress consistent with the scaling regime of the Korteweg–de Vries (KdV) equation. These quantities can be used to describe the shoaling of cnoidal waves approaching a gently sloping beach. The transformation of these waves along the slope can be described using the shoaling equations, a set of three nonlinear equations in three unknowns: the wave height H, the set-down \({\bar{\eta }}\) and the elliptic parameter m. We define a numerical algorithm for the efficient solution of the shoaling equations, and we verify our shoaling formulation by comparing with experimental data from two sets of experiments as well as shoaling curves obtained in previous works. PubDate: 2021-10-25 DOI: 10.1007/s42286-021-00053-1

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Abstract: Abstract In this contribution, we prove that small amplitude, resonant harmonic, spatially periodic traveling waves (Wilton ripples) exist in a family of weakly nonlinear PDEs which model water waves. The proof is inspired by that of Reeder and Shinbrot (Arch. Rat. Mech. Anal. 77:321–347, 1981) and complements the authors’ recent, independent result proven by a perturbative technique (Akers and Nicholls 2021). The method is based on a Banach Fixed Point Iteration and, in addition to proving that this iteration has Wilton ripples as a fixed point, we use it as a numerical method for simulating these solutions. The output of this numerical scheme and its performance are evaluated against a quasi-Newton iteration. PubDate: 2021-08-05 DOI: 10.1007/s42286-021-00052-2

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Abstract: Abstract We investigate energy transfer of air–water interactions and develop a numerical method that captures its temporal variability and generates and tracks the short waves that form in the water surface as a result of the air–water turbulence. We solve a novel system of balance equations derived from the Navier–Stokes equations known as moment field equations. The main advantage of our approach is that we do not assume a priori that the stochastic random variables that quantify the turbulent energy transfer between air and water are Gaussian. We generate non-conservative multifractal measures of turbulent energy transfer using a recursive integration process and a self-affine velocity kernel. The kernel exactly satisfies the (duration limited) kinetic equation for waves as well as invariant scaling properties of the Navier–Stokes equations. This allows us to derive source terms for the moment field equations using a turbulent diffusion operator. The operator quantifies energy transfer along a space time path associated with pressure instabilities in the air–sea interface and transfers the statistical shape (or fractal dimension) of the atmosphere to the wind-sea. Because we use observational data to begin the recursive integration process, the ocean–atmosphere interaction is inherently built into the model. Numerical results from application of our methods to air–sea turbulence off the coast of New Jersey and New York indicate that our methods produce measures of turbulent energy transfer that match theory and observation, and, correspondingly, significant wave heights and average wave periods predicted by our model qualitatively match buoy data. PubDate: 2021-07-01 DOI: 10.1007/s42286-021-00048-y

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Abstract: Abstract In this paper, we derive new shallow asymptotic models for the free boundary plasma-vacuum problem governed by the magnetohydrodynamic equations which are vital when describing large-scale processes in flows of astrophysical plasma. More precisely, we present the magnetic analogue of the 2D Green–Naghdi equations for water waves under a weak magnetic pressure assumption in the presence of weakly sheared vorticity and magnetic currents. Our method is inspired by ideas for hydrodynamic flows developed in Castro and Lannes (2014) to reduce the three-dimensional dynamics of the vorticity and current to a finite cascade of two dimensional equations which can be closed at the precision of the model. PubDate: 2021-07-01 DOI: 10.1007/s42286-021-00050-4

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Abstract: Abstract In 1895, Korteweg and de Vries (Philos Mag 20:20, 1895) studied an equation describing the motion of waves using the assumptions of long wavelength and small amplitude. Two implicit assumptions which they also made were irrotational and inviscid fluids. Comparing experiment and observation seems to suggest that these two assumptions are well justified. This paper removes the assumption of irrotationality in the case of electrohydrodynamics with an assumption of globally constant vorticity in the fluid. A study of the effect of vorticity on wave profiles and amplitudes is made revealing some unusual features. The velocity potential is an important variable in irrotational flow; the vertical component of velocity takes place of this variable in our analysis. This allows the bypassing of the Burns condition and also demonstrates that waves exist even for negative values of the vorticity. The linear and weakly nonlinear models are derived. PubDate: 2021-07-01 DOI: 10.1007/s42286-020-00043-9

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Abstract: Abstract The generation of wind waves at the surface of an established underlying vertically sheared water flow, of constant vorticity, is considered. A particular attention is paid to the role of the vorticity in water on wind-wave generation in finite depth. The present theoretical results are compared with experimental data obtained by Young and Verhagen (Coast Eng 29:47–78, 1996), in the shallow Lake George (Australia), and the least squares fit of these data by Young (Coast Eng 32:181–195, 1997). It is shown that without vorticity in water, there is a deviation between theory and experimental data. However, a good agreement between the theory and the fit of experimental data is obtained when negative vorticity is taken into account. Furthermore, it is shown that the amplitude growth rate increases with vorticity and depth. A limit to the wave energy growth, corresponding to the vanishing of the growth rate, is obtained. The corresponding limiting wave age is derived in a closed form showing its explicit dependence on vorticity and depth. The limiting wave age is found to increase with both vorticity and depth. PubDate: 2021-07-01 DOI: 10.1007/s42286-021-00049-x

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Abstract: Abstract One hundred years ago, Nekrasov published the widely cited paper (Nekrasov in Izvestia Ivanovo-Voznesensk Politekhn Inst 3:52–65, 1921), in which he derived the first of his two integral equations describing steady periodic waves on the free surface of water. We examine how Nekrasov arrived at these equations and his approach to investigating their solutions. In connection with this, Nekrasov’s life after 1917 is briefly outlined, in particular, how he became a prisoner in Stalin’s Gulag. Further results concerning Nekrasov’s equations and related topics are surveyed. PubDate: 2021-06-24 DOI: 10.1007/s42286-021-00051-3

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Abstract: Abstract Motivated by the analysis of the propagation of internal waves in a stratified ocean, we consider in this article the incompressible Euler equations with variable density in a flat strip, and we study the evolution of perturbations of the hydrostatic equilibrium corresponding to a stable vertical stratification of the density. We show the local well-posedness of the equations in this configuration and provide a detailed study of their linear approximation. Performing a modal decomposition according to a Sturm–Liouville problem associated with the background stratification, we show that the linear approximation can be described by a series of dispersive perturbations of linear wave equations. When the so-called Brunt–Vaisälä frequency is not constant, we show that these equations are coupled, hereby exhibiting a phenomenon of dispersive mixing. We then consider more specifically shallow water configurations (when the horizontal scale is much larger than the depth); under the Boussinesq approximation (i.e., neglecting the density variations in the momentum equation), we provide a well-posedness theorem for which we are able to control the existence time in terms of the relevant physical scales. We can then extend the modal decomposition to the nonlinear case and exhibit a nonlinear mixing of different nature than the dispersive mixing mentioned above. Finally, we discuss some perspectives such as the sharp stratification limit that is expected to converge towards two-fluid systems. PubDate: 2021-04-01 DOI: 10.1007/s42286-020-00041-x

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Abstract: Abstract We consider the Isobe–Kakinuma model for water waves, which is obtained as the system of Euler–Lagrange equations for a Lagrangian approximating Luke’s Lagrangian for water waves. We show that the Isobe–Kakinuma model also enjoys a Hamiltonian structure analogous to the one exhibited by V. E. Zakharov on the full water wave problem and, moreover, that the Hamiltonian of the Isobe–Kakinuma model is a higher order shallow water approximation to the one of the full water wave problem. PubDate: 2021-04-01 DOI: 10.1007/s42286-020-00025-x

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Abstract: Abstract We use a Hamiltonian normal form approach to study the dynamics of the water wave problem in the small-amplitude long-wave regime (KdV regime). If \(\mu \) is the small parameter corresponding to the inverse of the wave length, we show that the normal form at order \(\mu ^5\) consists of two decoupled equations: one describing right going waves and the other describing left going waves. Each of these equations is integrable: it is a linear combination of the first three equations in the KdV hierarchy. At order \(\mu ^7\) , we find nontrivial terms coupling the two counter-propagating waves. PubDate: 2021-04-01 DOI: 10.1007/s42286-020-00032-y

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Abstract: Abstract We put forward a solution to the initial boundary value (IBV) problem for the nonlinear shallow water system in inclined channels of arbitrary cross section by means of the generalized Carrier–Greenspan hodograph transform (Rybkin et al. in J Fluid Mech, 748:416–432, 2014). Since the Carrier–Greenspan transform, while linearizing the shallow water system, seriously entangles the IBV in the hodograph plane, all previous solutions required some restrictive assumptions on the IBV conditions, e.g., zero initial velocity, smallness of boundary conditions. For arbitrary non-breaking initial conditions in the physical space, we present an explicit formula for equivalent IBV conditions in the hodograph plane, which can readily be treated by conventional methods. Our procedure, which we call the method of data projection, is based on the Taylor formula and allows us to reduce the transformed IBV data given on curves in the hodograph plane to the equivalent data on lines. Our method works equally well for any inclined bathymetry (not only plane beaches) and, moreover, is fully analytical for U-shaped bays. Numerical simulations show that our method is very robust and can be used to give express forecasting of tsunami wave inundation in narrow bays and fjords. PubDate: 2021-04-01 DOI: 10.1007/s42286-020-00042-w

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Abstract: Abstract We consider the motion of ideal incompressible fluid with free surface. We analyzed the exact fluid dynamics through the time-dependent conformal mapping \(z=x+iy=z(w,t)\) of the lower complex half plane of the conformal variable w into the area occupied by fluid. We established the exact results on the existence vs. nonexistence of the pole and power law branch point solutions for \(1/z_w\) and the complex velocity. We also proved the nonexistence of the time-dependent rational solution of that problem for the second- and the first-order moving pole. PubDate: 2021-04-01 DOI: 10.1007/s42286-020-00040-y

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Abstract: Abstract It is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent. PubDate: 2021-01-06 DOI: 10.1007/s42286-020-00046-6

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Abstract: Abstract This paper is devoted to the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the general question of proving Morawetz inequalities. We continue the analysis initiated in our previous work, where we have established local energy decay estimates for gravity waves. Here we add surface tension and prove a stronger estimate with a local regularity gain, akin to the smoothing effect for dispersive equations. Our main result holds globally in time and holds for genuinely nonlinear waves, since we are only assuming some very mild uniform Sobolev bounds for the solutions. Furthermore, it is uniform both in the infinite depth limit and the zero surface tension limit. PubDate: 2020-11-02 DOI: 10.1007/s42286-020-00044-8