Transverse Instability of Stokes Waves at Finite Depth

We investigate the Benjamin–Feir (or modulational) instability of Stokes waves, i.e. small-amplitude, one-dimensional periodic gravity waves of permanent form and constant velocity, in water of finite and infinite depth. We develop a perturbation method to describe to high-order accuracy the unstable spectral elements associated with this instability, obtained by linearizing Euler's equations about the small-amplitude Stokes waves. These unstable elements form a figure-eight curve centred at the origin of the complex spectral plane, which is parametrized by a Floquet exponent. Our asymptotic expansions of this figure-eight are in excellent agreement with numerical computations as well as recent rigorous results by Bertiet al.(Full description of Benjamin–Feir instability of Stokes waves in deep water, 2021, arXiv:2109.11852) and Bertiet al.(Benjamin–Feir instability of Stokes waves in finite depth, 2022, arXiv:2204.00809). From our expansions, we derive high-order estimates for the growth rates of the Benjamin–Feir instability and for the parametrization of the Benjamin–Feir figure-eight curve with respect to the Floquet exponent. We are also able to compare the Benjamin–Feir and high-frequency instability spectra analytically for the first time, revealing three different regimes of the Stokes waves, depending on the predominant instability.

A new mathematical formulation for the realization of nonlinear wave profiles and its nonlinear solution proce- dure, based on the Banach fixed-point theorem, is proposed. To apply the formulation, a nonlinear equation for the Stokes wave in a finite depth was derived, and some numerical solu- tions are given. A numerical study showed that the proposed iteration method, based on linear progressive wave potential only, enabled us to realize the Stokes nonlinear wave profiles in a finite depth. The nonlinear strategy of iteration has a very fast convergence rate, i.e., only about 6-10 iterations are re- quired to obtain a numerically converged solution.

Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth {mathtt h} is larger than a critical threshold texttt{h}_{scriptscriptstyle {textsc {WB}}}approx 1.363 . In this paper, we completely describe, for any finite value of mathtt h >0 , the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent mu is turned on. We prove, in particular, the existence of a unique depth texttt{h}_{scriptscriptstyle {textsc {WB}}}, which coincides with the one predicted by Whitham and Benjamin, such that, for any 0< mathtt h < texttt{h}_{scriptscriptstyle {textsc {WB}}}, the eigenvalues close to zero are purely imaginary and, for any mathtt h > texttt{h}_{scriptscriptstyle {textsc {WB}}}, a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent. As {mathtt h} rightarrow texttt{h}_{scriptscriptstyle {textsc {WB}}}^{, +} the “8” collapses to the origin of the complex plane. The complete bifurcation diagram of the spectrum is not deduced as in deep water, since the limits texttt{h}rightarrow +infty (deep water) and mu rightarrow 0 (long waves) do not commute. In finite depth, the four eigenvalues have all the same size mathcal {O}(mu ), unlike in deep water, and the analysis of their splitting is much more delicate, requiring, as a new ingredient, a non-perturbative step of block-diagonalization. Along the whole proof, the explicit dependence of the matrix entries with respect to the depth texttt{h} is carefully tracked.

Various classes of steady and unsteady dark solitary waves (DSWs) are known to exist in modulation equations for water waves in finite depth. However, there is a class of steady DSWS of the full water-wave problem which are missed by the classical modulation equations such as the Hasimoto-Ono, Benney-Roskes, and Davey-Stewartson. These steady DSWs, recently discovered by Bridges and Donaldson, are pervasive in finite depth, arise through secondary criticality of Stokes gravity waves, and are synchronized with the Stokes wave. In this paper, the role of DSWs in modulation equations for water waves is reappraised. The intrinsic unsteady nature of existing modulation equations filters out some interesting solutions. On the other hand, the geometry of DSWs in modulation equations is very similar to the full water wave problem and these geometrical properties are developed. A model equation is proposed which illustrates the general nature of the emergence of steady DSWs due to wave-generated mean flow coupled to a periodic wave. Although the existing modulation equations are intrinsically unsteady, it is shown that there are also important shortcomings when one wants to use them for stability analysis of DSWs.

For numerical simulation of structure-wave interaction, the wave generation with high accuracy is prime to analyze the wave loads and motions of the structure. Based on the fifth-order Stokes theory, a two-dimensional viscous wave flume, which was modeled using the commercial CFD solver ANSYS-FLUENT, was applied to the generation and propagation of regular waves in finite water depth. With the user-defined function provided by the solver, the momentum source term and boundary condition, which are used for the wave generation and dissipation, were developed to ensure the accuracy of wave simulation with large steepness. In addition, the wave flume was separated into two regions, which are governed by the laminar model and turbulent model, respectively. The separation of laminar and turbulent regions can alleviate the side effect of turbulence on the accuracy of wave generation. In order to validate the present method, the regular wave propagating with different steepness in finite water depth were simulated. The numerical results were in good agreement with the theoretical ones. The study showed that the present method was effective for the simulation of Stokes wave in finite water depth, especially effective to improve the numerical accuracy in case of large wave steepness.

We consider the two-dimensional problem for steady water waves with vorticity on water of finite depth. While neglecting the effects of surface tension we construct connected families of large amplitude periodic waves approaching a limiting wave, which is either a solitary wave, the highest solitary wave, the highest Stokes wave or a Stokes wave with a breaking profile. In particular, when the vorticity is nonnegative we prove the existence of highest Stokes waves with an included angle of 120^circ . In contrast to previous studies, we fix the Bernoulli constant and consider the wavelength as a bifurcation parameter, which guarantees that the limiting wave has a finite depth. In fact, this is the first rigorous proof of the existence of extreme Stokes waves with vorticity on water of finite depth. Aside from the existence of highest waves, we provide a new result about the regularity of Stokes waves of arbitrary amplitude (including extreme waves). Furthermore, we prove several new facts about steady waves, such as a lower bound for the wavelength of Stokes waves, while also eliminating a possibility of the wave breaking for waves with non-negative vorticity.

The motion of turbulent Stokes waves on a finite constant depth fluid with a rough bed is considered. First and second order turbulent boundary layer equations are solved numerically for a range of roughness parameters, and from the solutions are calculated the mass transport velocity profiles and attenuation coefficients. A new mechanism of turbulent mass transport is found which predicts a reduction and reversal of drift velocity in shallow water in agreement with experimental observations under turbulent conditions. This transpires because the second order Stokes wave motion, in a turbulent boundary layer, can directly influence the mass transport velocity by mode coupling interactions between different second order Fourier modes of oscillation. It is also found that the Euler contribution due to the radiation stress of the first order motion is reduced to half of it's corresponding laminar value as a consequence of the velocity squared stress law. The attenuation is found to be of inverse algebraic type with the reciprocal wave height varying linearly with either distance or time. The severe wave height restriction applicable to the Longuet-Higgins [4] solution is shown not to apply to progressive waves on a finite constant depth of fluid. The existence of sand bars on sloping beaches exposed to turbulent waves is predicted.

The second approximation to cnoidal waves is compared with the third approximation for Stokes waves of permanent form in water of finite depth. The comparison clearly indicates that cnoidal wave theory should not be applied to finite amplitude waves if their wavelengths are shorter than 5 times the depth. It is shown how the limiting heights of cnoidal waves are also related to the vanishing of the pressure gradient near the wave crest. The third approximation to Stokes waves in finite water depths is verified by the use of the classical small-perturbation expansion method which is best suited for small wave amplitudes. For finite amplitude waves the series expansion in terms of the infinitesimal-wave parameter is found to be most suitable for wavelengths shorter than 8 times the depth.

A numerical study of the instabilities of Stokes waves on finite depth has been carried out using an efficient fully nonlinear method [D. Clamond and J. Grue, “A fast method for fully nonlinear water-wave computations,” J. Fluid Mech. 447, 337 (2001)]. First, attention is given to five-wave instabilities with k0h=O(1), k0 being the wavenumber and h the depth. Both instabilities leading to breaking and instabilities leading to recurrence are studied, yielding considerably different patterns than on infinite depth. Higher-order instabilities are exemplified, for the first time, by simulations of six- and seven-wave instabilities. Simulations of interactions between four- and five-wave instabilities show that a classical modulational instability can destabilize a three-dimensional perturbation causing crescent waves to appear, in accordance with the hypothesis of [M.-Y. Su and A. W. Green, “Coupled two- and three-dimensional instabilities of surface gravity waves,” Phys. Fluids 27, 2595 (1984)]. Also, a recurrent five-wave instability can boost the energy in a four-wave instability.

The Hindmarsh instability theory of drumlin formation is applied to the study of interfacial instabilities, which may arise when ice flows viscously over deformable sediments. Here, the analytic form of this theory is extended to the case where the ice is Newtonian viscous and of finite depth, and where the basal till can be both sheared by the ice and squeezed by basal effective pressure gradients: previous authors assumed infinitely deep ice, based on the assumption that the developing waveforms had wavelength much less than ice depth. The previous infinite depth theory only allowed transverse instabilities to occur, and these have been associated with the formation of ribbed moraine; one of the purposes of extending the analysis to finite depth is to see whether three-dimensional instabilities, which might be associated with the formation of drumlins or mega-scale glacial lineations, can occur: we find that they do not. A second purpose is to calculate under what circumstances the infinite depth theory provides accurate prediction of bedform development in ice of finite depth d i . We find that this is the case if the waveforms have a wavelength less than approximately 1.2 d i . Finally, the finite depth theory allows us to compute, for the first time, the response of the ice surface to the developing unstable bedforms. We find that this response is rapid, and we give explicit recipes for the surface perturbation transfer functions in terms of the perturbations to the basal stress and the basal topography.

For Stokes waves in finite depth within the neighbourhood of the Benjamin–Feir stability transition, there are two families of periodic waves, one modulationally unstable and the other stable. In this paper we show that these two families can be joined by a heteroclinic connection, which manifests in the fluid as a travelling front. By shifting the analysis to the setting of Whitham modulation theory, this front is in wavenumber and frequency space. An implication of this jump is that a permanent frequency downshift of the Stokes wave can occur in the absence of viscous effects. This argument, which is built on a sequence of asymptotic expansions of the phase dynamics, is confirmed via energetic arguments, with additional corroboration obtained by numerical simulations of a reduced model based on the Benney–Roskes equation.

Fourth order evolution equations and stability analysis for stokes waves on arbitrary water depth

We prove high-frequency modulational instability of small-amplitude Stokes waves in deep water under longitudinal perturbations, providing the first isola of unstable eigenvalues branching off from \mathrm{i}\frac{3}{4} . Unlike the finite depth case this is a degenerate problem and the real part of the unstable eigenvalues has a much smaller size than in finite depth. By a symplectic version of Kato theory, we reduce to search the eigenvalues of a 2\times 2 Hamiltonian and reversible matrix which has eigenvalues with nonzero real part if and only if a certain analytic function is not identically zero. In deep water, we prove that the Taylor coefficients up to order three of this function vanish, but not the fourth-order one.

Periodic water waves of permanent form travelling at constant speed, the so-called Stokes waves, are studied in water of fixed finite depth using methods previously used in water of infinite depth. We apply our methods to waves of varying steepness over a range of fixed depths in order to determine how a number of physical quantities related to the waves change as the steepness of the waves increases. Finally, we examine the complex singularities outside of their domain of definition when the waves are considered as a function of a conformal variable.

Energy properties of regular water waves over horizontal bottom with increasing nonlinearity