Two-dimensional Gravity Waves Research Articles

Two-dimensional gravity waves at low regularity I: Energy estimates

This article represents the first installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on sharp cubic energy estimates. Precisely, we introduce and develop the techniques to prove a new class of energy estimates, which we call balanced cubic estimates . This yields a key improvement over the earlier cubic estimates of Hunter–Ifrim–Tataru (2016), while preserving their scale-invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using any Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness, drastically improving earlier results obtained by Alazard–Burq–Zuily (2014, 2018), Hunter–Ifrim–Tataru (2016) and Ai (2019).

General solution for linear 2D anisotropic surface gravity water waves on uniform current

Measuring and modeling ocean waves is a routine task for oceanographers and marine engineers. Accounting for current effects on waves is less common; however, such effects can be non-trivial, particularly for strong currents. Recent research has also found that currents are increasing with global temperatures. Thus, accounting for currents is of potential importance to a growing subset of real-world environments. One significant impact of currents is their modification of dispersion. It is known that, in contrast to dispersion without current, wave–current dispersion is multivalued and anisotropic, meaning it varies with direction and can yield multiple solutions for the same frequency. Strictly speaking, a complete analysis ought to account for all dispersion solutions. This study presents a general wave field solution for linear steady-state two-dimensional surface gravity waves on current. In contrast to existing formulations, it accounts for the complete set of dispersion solutions. The multiple solutions correspond to linearly independent (same frequency) spatial modes with no analogy in models without current. To fully determine the complete temporal and spatial features of a wave field, one must determine the amplitudes of all spatial modes. This can be achieved through the solution of an inverse problem, provided one has sufficient initial conditions. The significance of accounting for all spatial modes is demonstrated through examples. It is shown that a single time-series measurement is insufficient for determining the spatial features of a monochromatic (single frequency) wave. Rather, additional spatial information must be introduced to render the problem well-posed. The theory and methodology presented can be used to improve wave models and measurements.

Linear Time-Invariant Models of a Large Cumulus Ensemble

Abstract Methods in system identification are used to obtain linear time-invariant state-space models that describe how horizontal averages of temperature and humidity of a large cumulus ensemble evolve with time under small forcing. The cumulus ensemble studied here is simulated with cloud-system-resolving models in radiative–convective equilibrium. The identified models extend steady-state linear response functions used in past studies and provide accurate descriptions of the transfer function, the noise model, and the behavior of cumulus convection when coupled with two-dimensional gravity waves. A novel procedure is developed to convert the state-space models into an interpretable form, which is used to elucidate and quantify memory in cumulus convection. The linear problem studied here serves as a useful reference point for more general efforts to obtain data-driven and interpretable parameterizations of cumulus convection.

Two-dimensional gravity waves at low regularity II: Global solutions

This article represents the second installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on global solutions for small and localized data. Such solutions have been proved to exist in Ionescu– Pusateri [Invent. Math. 199 (2015)], Alazard–Delort [Ann. Sci. Éc. Norm. Supér. (4) 48 (2015)], Hunter et al. [Comm. Math. Phys. 346 (2016)], and Ifrim–Tataru [Bull. Soc. Math. France 144 (2016)] in much higher regularity. Our goal in this paper is to improve these results and prove global well-posedness under minimal regularity and decay assumptions for the initial data. One key ingredient here is represented by the balanced cubic estimates in our first paper. Another is the nonlinear vector field Sobolev inequalities, an idea first introduced by Ifrim–Tataru [Ann. Sci. Éc. Norm. Supér. (4) 52 (2019)] in the context of the Benjamin–Ono equations.

Two-dimensional internal gravity wave beam instability. Linear theory and subcritical instability

The instability of propagating internal gravity waves (IGWs) is of long-standing interest in geophysical fluid dynamics since breaking IGWs exchange energy and momentum with the large-scale flow and hence they support the large-scale circulation. In this study a low-order IGW beam model is used to delineate both linear and so called non-modal transient instability. In the first part of the study, linear normal mode instability of a wave beam consisting of two finite-amplitude plane monochromatic IGWs with the same frequency and parallel wave vectors of different magnitude is investigated using the Galerkin method. It is concluded that the wave beam is linearly more unstable than its constituent plane waves, taken separately. The degree of instability increases with the separation of the constituent waves in the wave number space, that is, with the wave beam concentration in the physical space. The narrower a wave beam is, the more linearly unstable it is. In its turn, transient instability typically occurs for linearly stable flows or before linear instability can set in (subcritical instability) if the governing system matrix is non-normal. In the second part of the paper, first the non-normality of the linear system matrix of the wave beam model is examined by computing measures like the Henrici number, the pseudospectrum, and the range of the matrix. Subsequently, the robustness of the transient growth is studied when the initial condition for optimal growth is randomly perturbed. It is concluded that for full randomisation, in particular, shallow wave beams can show subcritical growth when entering a turbulent background field. Such growing and eventually breaking wave beams might add turbulence to existing background turbulence that originates from other sources of instability. However, the robustness of transient growth for wave beam perturbations depends strongly on the strength of randomisation of the initial conditions, the beam angle and the perturbation wavelength.

On the breaking inception of unsteady water wave packets evolving in the presence of constant vorticity

Abstract

Spatial Form of a Hamiltonian Dysthe Equation for Deep-Water Gravity Waves

An overview of a Hamiltonian framework for the description of nonlinear modulation of surface water waves is presented. The main result is the derivation of a Hamiltonian version of Dysthe’s equation for two-dimensional gravity waves on deep water. The reduced problem is obtained via a Birkhoff normal form transformation which not only helps eliminate all non-resonant cubic terms but also yields a non-perturbative procedure for surface reconstruction. The free surface is reconstructed from the wave envelope by solving an inviscid Burgers’ equation with an initial condition given by the modulational Ansatz. Particular attention is paid to the spatial form of this model, which is simulated numerically and tested against laboratory experiments on periodic groups and short-wave packets. Satisfactory agreement is found in all these cases.

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.

Nonlinear reflection of a two-dimensional finite-width internal gravity wave on a slope

The nonlinear reflection of a finite-width plane internal gravity wave incident onto a uniform slope is addressed, relying on the inviscid theory of Thorpe (J. Fluid Mech., vol. 178, 1987, pp. 279–302) for pure plane waves. The aim of this theory is to determine the conditions under which the incident and the reflected waves form a resonant triad with the second-harmonic wave resulting from their interaction. Thorpe’s theory leads to an indeterminacy of the second-harmonic wave amplitude at resonance. In waiving this indeterminacy, we show that the latter amplitude has a finite behaviour at resonance, increasing linearly from the slope. We investigate the influence of background rotation and find similar results with a weaker growth rate. We then adapt the theory to the case of an incident plane wave of finite width. In this case, nonlinear interactions are confined to the area where the incident and reflected finite-width waves superpose, implying that the amplitude of the second-harmonic wave is bounded at resonance. We find good agreement with the results of numerical simulations in a vertical plane as long as the dissipated power of the incident and reflected waves remain smaller than the power transferred to the second-harmonic wave. This is the case for small slope angles. As the slope angle increases, the focusing of the reflected wave enhances viscous effects and dissipation eventually dominates over nonlinear transfer. We finally discuss the relevance of laboratory experiments to assess the validity of the theoretical results.

Sampling Errors in Observed Gravity Wave Momentum Fluxes from Vertical and Tilted Profiles

Observations from radiosondes or from vertically pointing remote sensing profilers are often used to estimate the vertical flux of momentum due to gravity waves. For planar, monochromatic waves, these vertically integrated fluxes are equal to the phase averaged flux and equivalent to the horizontal averaging used to deduce momentum flux from aircraft data or in numerical models. Using a simple analytical solution for two-dimensional hydrostatic gravity waves over an isolated ridge, it is shown that this equivalence does not hold for mountain waves. For a vertical profile, the vertically integrated flux estimate is proportional to the horizontally integrated flux and decays with increasing distance of the profile location from the mountain. For tilted profiles, such as those obtained from radiosonde ascents, there is a further sampling error that increases as the trajectory extends beyond the localised wave field. The same sampling issues are seen when the effects of the Coriolis force on the gravity waves are taken into account. The conclusion of this work is that caution must be taken when using radiosondes or other vertical profiles to deduce mountain wave momentum fluxes.

Evolution and Stability of Two-Dimensional Anelastic Internal Gravity Wave Packets

The weakly nonlinear evolution, stability, and overturning of horizontally and vertically localized internal gravity wave packets is examined for a nonrotating, anelastic atmosphere that is stationary in the absence of waves. The weakly nonlinear evolution is examined through the derivation of their wave-induced mean flow, which is used to formulate a nonlinear Schrödinger equation. The induced flow is manifest as a long, hydrostatic, bow wake-like disturbance, whose flow direction transitions from positive on the leading flank of the wave packet to negative on the trailing flank of the wave packet. As such, two-dimensional wave packets are always modulationally unstable. This instability results in enhanced amplitude growth confined to either the leading or trailing flank. Hence, when combined with anelastic growth predicted by linear theory, we anticipate two-dimensional waves will overturn either somewhat below or just above the heights predicted by linear theory. Numerical solutions of the Schrödinger equation are compared with the results of fully nonlinear simulations to establish the validity of the weakly nonlinear theory. Actual wave overturning heights are determined quantitatively from a range of fully nonlinear simulations.

Nonlinear water waves in shallow water in the presence of constant vorticity: A Whitham approach

Two-dimensional nonlinear gravity waves travelling in shallow water on a vertically sheared current of constant vorticity are considered. Using Euler equations, in the shallow water approximation, hyperbolic equations for the surface elevation and the horizontal velocity are derived. Using Riemann invariants of these equations, that are obtained analytically, a closed-form nonlinear evolution equation for the surface elevation is derived. A dispersive term is added to this equation using the exact linear dispersion relation. With this new single first-order partial differential equation, vorticity effects on undular bores are studied. Within the framework of weakly nonlinear waves, a KdV-type equation and a Whitham equation with constant vorticity are derived from this new model and the effect of vorticity on solitary waves and periodic waves is considered. Furthermore, within the framework of the new model and the Whitham equation a study of the effect of vorticity on the breaking time of dispersive waves and hyperbolic waves as well is carried out.

Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth

This paper describes an efficient algorithm for computing steady two-dimensional surface gravity waves in irrotational motion. The algorithm complexity is $O(N\log N)$, $N$ being the number of Fourier modes. This feature allows the arbitrary precision computation of waves in arbitrary depth, i.e. it works efficiently for Stokes, cnoidal and solitary waves, even for quite large steepnesses, up to approximately 99 % of the maximum steepness for all wavelengths. In particular, the possibility to compute very long (cnoidal) waves accurately is a feature not shared by other algorithms and asymptotic expansions. The method is based on conformal mapping, the Babenko equation rewritten in a suitable way, the pseudo-spectral method and Petviashvili iterations. The efficiency of the algorithm is illustrated via some relevant numerical examples. The code is open source, so interested readers can easily check the claims, use and modify the algorithm.

Recent advances on the global regularity for irrotational water waves.

We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions. We begin by introducing the free boundary Euler equations and discussing the local existence of solutions using the paradifferential approach. We then describe in a unified framework, using the Eulerian formulation, global existence results for three- and two-dimensional gravity waves, and our joint result (with Deng and Pausader) on global regularity for the gravity-capillary model in three dimensions. We conclude this review with a short discussion about the formation of singularities and give a few additional references to other interesting topics in the theory.This article is part of the theme issue 'Nonlinear water waves'.

New exact relations for steady irrotational two-dimensional gravity and capillary surface waves.

Steady two-dimensional surface capillary-gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained.This article is part of the theme issue 'Nonlinear water waves'.

Linear-shear-current modified Schrödinger equation for gravity waves in finite water depth.

A nonlinear Schrödinger equation for the propagation of two-dimensional surface gravity waves on linear shear currents in finite water depth is derived. In the derivation, linear shear currents are assumed to be a linear combination of depth-uniform currents and constant vorticity. Therefore, the equation includes the combined effects of depth-uniform currents and constant vorticity. Next, using the equation, the properties of the modulational instability of gravity waves on linear shear currents are investigated. It is showed that shear currents significantly modify the modulational instability properties of weakly nonlinear waves. Furthermore, the influence of linear shear currents on Peregrine breather which can be seen as a prototype of freak waves is also studied. It is demonstrated that depth-uniform opposing currents can reduce the breather extension in both the time and spatial domain in intermediate water depth, but following currents has the adverse impact, indicating that a wave packets with freak waves formed on following currents contain more hazardous waves in finite water depth. However, the corresponding and coexisting vorticity can counteract the influence of currents. Additionally, if the water depth is deep enough, shear currents have negligible effect on the characteristics of Peregrine breathers.

Stabilization of gravity water waves

This paper is devoted to the stabilization of the incompressible Euler equation with free surface. We study the damping of two-dimensional gravity waves by an absorbing beach where the water-wave energy is dissipated by using the variations of the external pressure.

Steady periodic rotational gravity waves with negative surface tension

Consideration in this paper is the two-dimensional steady periodic rotational gravity waves with negative surface tension. Local curves of small amplitude solutions of the resulting problem are obtained by using the Crandall–Rabinowitz local bifurcation theory. By means of the global bifurcation theory combined with the Schauder theory of elliptic equations with the Venttsel boundary conditions, the curves of small amplitude solutions is extended to the global continuum of solutions. Furthermore, it is shown that those waves are necessarily symmetric about the crest under the assumption that their surface profiles are monotonic between troughs and crests and locally strictly monotonic near the troughs.

Generation of a mean flow by an internal wave

We experimentally study the generation of a mean flow by a two-dimensional progressive internal gravity wave. Due to the viscous damping of the wave, a non-vanishing Reynolds stress gradient forces a mean flow. When the forcing amplitude is low, the wave amplitude is proportional to the forcing and the mean flow is quadratic in the forcing. When the forcing amplitude is large, the mean flow decreases the wave amplitude. This feedback saturates both the wave and the mean flow. The profiles of the mean flow and the wave are compared with a one-dimensional analytical model. Decreasing the forcing frequency leads to a wave and a mean flow localized on a smaller height, in agreement with the model.

Experimental determination of radiated internal wave power without pressure field data

We present a method to determine, using only velocity field data, the time-averaged energy flux \documentclass[12pt]{minimal}\begin{document}$\left<\bm {J}\right>$\end{document}J and total radiated power P for two-dimensional internal gravity waves. Both \documentclass[12pt]{minimal}\begin{document}$\left<\bm {J}\right>$\end{document}J and P are determined from expressions involving only a scalar function, the stream function ψ. We test the method using data from a direct numerical simulation for tidal flow of a stratified fluid past a knife edge. The results for the radiated internal wave power given by the stream function method agree to within 0.5% with results obtained using pressure and velocity data from the numerical simulation. The results for the radiated power computed from the stream function agree well with power computed from the velocity and pressure if the starting point for the stream function computation is on a solid boundary, but if a boundary point is not available, care must be taken to choose an appropriate starting point. We also test the stream function method by applying it to laboratory data for tidal flow past a knife edge, and the results are found to agree with the direct numerical simulation. The supplementary material includes a Matlab code with a graphical user interface that can be used to compute the energy flux and power from two-dimensional velocity field data.