27 results
A novel (2+1)-dimensional nonlinear evolution equation for weakly stratified free-surface boundary layers
- Joseph O. Oloo, Victor I. Shrira
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- Journal:
- Journal of Fluid Mechanics / Volume 973 / 25 October 2023
- Published online by Cambridge University Press:
- 23 October 2023, A40
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To get an insight into the dynamics of the oceanic surface boundary layer we develop an asymptotic model of the nonlinear dynamics of linearly decaying three-dimensional long-wave perturbations in weakly stratified boundary-layer flows. Although in nature the free-surface boundary layers in the ocean are often weakly stratified due to solar radiation and air entrainment caused by wave breaking, weak stratification has been invariably ignored. Here, we consider an idealized hydrodynamic model, where finite-amplitude three-dimensional perturbations propagate in a horizontally uniform unidirectional weakly stratified shear flow confined to a boundary layer adjacent to the water surface. Perturbations satisfy the no-stress boundary condition at the surface. They are assumed to be long compared with the boundary-layer thickness. Such perturbations have not been studied even in a linear setting. By exploiting the assumed smallness of nonlinearity, wavenumber, viscosity and the Richardson number, on applying triple-deck asymptotic scheme and multiple-scale expansion, we derive in the distinguished limit a novel essentially two-dimensional nonlinear evolution equation, which is the main result of the work. The equation represents a generalization of the two-dimensional Benjamin–Ono equation modified by the explicit account of viscous effects and new dispersion due to weak stratification. It describes perturbation dependence on horizontal coordinates and time, while its vertical structure, to leading order, is given by an explicit analytical solution of the linear boundary value problem. It shows the principal importance of weak stratification for three-dimensional perturbations.
Apparent singularities of the finite-depth Zakharov equation
- Paolo Pezzutto, Victor I. Shrira
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- Journal:
- Journal of Fluid Mechanics / Volume 972 / 10 October 2023
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- 04 October 2023, A35
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The description of weakly nonlinear water-wave evolution over a horizontal bottom by the integro-differential Zakharov equation, because of utilising the underlying Hamiltonian structure, has many advantages over direct use of the Euler equations. However, its application to finite-depth situations is not straightforward since, in contrast to the deep-water case, the kernels governing the four-wave interactions are singular, as well as the kernels in the canonical transformation that removes non-resonant interactions from the original equations of motion. At the singularities, these kernels are finite but not unique. The issue of how to use the Zakharov equation for finite depth and whether it is possible at all was debated intensely in the literature for decades but remains outstanding. Here we show that the absence of a limit of the kernels at the singularities is inconsequential, since in the equations of motion it is only the integral that matters. By applying the definition of the Dirac-$\delta$, we show that all the integrals involving a trivial manifold singularity are evaluated uniquely. Therefore, the Zakharov evolution equation and the nonlinear canonical transformation are only apparently singular. The findings are validated by application to examples where predictions based on the Zakharov equation are compared with known solutions obtained from the Euler equations.
Can edge waves be generated by wind?
- Victor I. Shrira, Alex Sheremet, Yulia I. Troitskaya, Irina A. Soustova
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- Journal:
- Journal of Fluid Mechanics / Volume 934 / 10 March 2022
- Published online by Cambridge University Press:
- 14 January 2022, A16
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Edge waves, the infragravity waves trapped by near-shore topography, are important in morphodynamics and flooding on mildly sloping beaches. Edge waves are usually generated by swell via triad interactions. Here, we examine the possibility that edge waves might be also generated directly by wind. By processing data from the SandyDuck’97 near-shore experiment, we show that pronounced directional asymmetry of edge waves does occur in nature, apparently unrelated to the direction of swells and along-shore currents. These observations exhibit edge waves propagating in the downwind direction under moderate wind against the along-shore currents, while swell is incident nearly normally to the shoreline, which strongly suggests generation of edge waves by wind. We examine theoretically possible mechanisms of edge-wave excitation by wind. We show that the ‘maser’ mechanism suggested by Longuet-Higgins (Proc. R. Soc. Lond. A, vol. 311, issue 1506, 1969b, pp. 371–389) in the context of excitation of free water waves is effective under favourable conditions: nonlinearly interacting random short wind-forced waves create a viscous shear stress on the water surface with the variation of stress being phase linked to edge waves, which allows self-excitation of a coherent edge wave. The model we put forward is based upon the kinetic equation for short wind waves propagating on the inhomogeneous current due to an edge wave. The model needs a dedicated experiment for validation. Analysis of plausible alternative mechanisms of generation via Miles’ critical layer and via the viscous shear stresses induced by the edge wave in the air revealed no instability in the consideration confined to the main mode and constant slope bathymetry.
On the physical mechanism of front–back asymmetry of non-breaking gravity–capillary waves
- Alexander Dosaev, Yuliya I. Troitskaya, Victor I. Shrira
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- Journal:
- Journal of Fluid Mechanics / Volume 906 / 10 January 2021
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- 13 November 2020, A11
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In nature, the wind waves of the gravity–capillary range are noticeably skewed forward. The salient feature of such waves is a characteristic pattern of capillary ripples on their crests. The train of these ‘parasitic capillaries’ is not symmetric with respect to the crest, it is localised on the front slope and decays towards the trough. Although understanding the gravity–capillary waves front–back asymmetry is important for remote sensing and, potentially, for wave–wind interaction, the physical mechanisms causing this asymmetry have not been identified. Here, we address this gap by extensive numerical simulations of the Euler equations employing the method of conformal mapping for two-dimensional potential flow and taking into account wave generation by wind and dissipation due to molecular viscosity. On examining the role of various factors contributing to the wave profile front–back asymmetry: wind forcing, viscous stresses and the Reynolds stresses caused by ripples, we found, in the absence of wave breaking, the latter to be by far the most important. It is the lopsided ripple distribution which leads to the noticeable fore–aft asymmetry of the mean wave profile. We also found how the asymmetry depends on wavelength, steepness, wind, viscosity and surface tension. The results of the model are discussed in the context of the available experimental data on asymmetry of gravity–capillary waves in both the breaking and non-breaking regimes. A reasonable agreement of the model with the data has been found for the regime without breaking or microbreaking.
Upper-ocean Ekman current dynamics: a new perspective
- Victor I. Shrira, Rema B. Almelah
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- Journal:
- Journal of Fluid Mechanics / Volume 887 / 25 March 2020
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- 28 January 2020, A24
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The work examines upper-ocean response to time-varying winds within the Ekman paradigm. Here, in contrast to the earlier works we assume the eddy viscosity to be both time and depth dependent. For self-similar depth and time dependence of eddy viscosity and arbitrary time dependence of wind we find an exact general solution to the Navier–Stokes equations which describes the dynamics of the Ekman boundary layer in terms of the Green’s function. Two basic scenarios (a periodic wind and an increase of wind ending up with a plateau) are examined in detail. We show that accounting for the time dependence of eddy viscosity is straightforward and that it substantially changes the ocean response, compared to the predictions of the models with constant-in-time viscosity. We also examine the Stokes–Ekman equations taking into account the Stokes drift created by surface waves with an arbitrary spectrum and derive the general solution for the case of a linearly varying with depth eddy viscosity. Stability of transient Ekman currents to small-scale perturbations has never been examined. We find that the Ekman currents evolving from rest quickly become unstable, which breaks down the assumed horizontal uniformity. These instabilities proved to be sensitive to the model of eddy viscosity, they have small (${\sim}10^{2}~\text{m}$) spatial scales and can be very fast compared to the inertial period, which suggests spikes of dramatically enhanced mixing localized in the vicinity of the water surface. This picture is incompatible with the Ekman paradigm and thus prompts radical revision of the Ekman-type models.
Spectral evolution of weakly nonlinear random waves: kinetic description versus direct numerical simulations
- Sergei Y. Annenkov, Victor I. Shrira
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- Journal:
- Journal of Fluid Mechanics / Volume 844 / 10 June 2018
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- 12 April 2018, pp. 766-795
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Kinetic equations are widely used in many branches of science to describe the evolution of random wave spectra. To examine the validity of these equations, we study numerically the long-term evolution of water wave spectra without wind input using three different models. The first model is the classical kinetic (Hasselmann) equation (KE). The second model is the generalised kinetic equation (gKE), derived employing the same statistical closure as the KE but without the assumption of quasistationarity. The third model, which we refer to as the DNS-ZE, is a direct numerical simulation algorithm based on the Zakharov integrodifferential equation, which plays the role of the primitive equation for a weakly nonlinear wave field. It does not employ any statistical assumptions. We perform a comparison of the spectral evolution of the same initial distributions without forcing, with/without a statistical closure and with/without the quasistationarity assumption. For the initial conditions, we choose two narrow-banded spectra with the same frequency distribution and different degrees of directionality. The short-term evolution ($O(10^{2})$ wave periods) of both spectra has been previously thoroughly studied experimentally and numerically using a variety of approaches. Our DNS-ZE results are validated both with existing short-term DNS by other methods and with available laboratory observations of higher-order moment (kurtosis) evolution. All three models demonstrate very close evolution of integral characteristics of the spectra, approaching with time the theoretical asymptotes of the self-similar stage of evolution. Both kinetic equations give almost identical spectral evolution, unless the spectrum is initially too narrow in angle. However, there are major differences between the DNS-ZE and gKE/KE predictions. First, the rate of angular broadening of initially narrow angular distributions is much larger for the gKE and KE than for the DNS-ZE, although the angular width does appear to tend to the same universal value at large times. Second, the shapes of the frequency spectra differ substantially (even when the nonlinearity is decreased), the DNS-ZE spectra being wider than the KE/gKE ones and having much lower spectral peaks. Third, the maximal rates of change of the spectra obtained with the DNS-ZE scale as the fourth power of nonlinearity, which corresponds to the dynamical time scale of evolution, rather than the sixth power of nonlinearity typical of the kinetic time scale exhibited by the KE. The gKE predictions fall in between. While the long-term DNS show excellent agreement with the KE predictions for integral characteristics of evolving wave spectra, the striking systematic discrepancies for a number of specific spectral characteristics call for revision of the fundamentals of the wave kinetic description.
Trapped waves on jet currents: asymptotic modal approach
- Victor I. Shrira, Alexey V. Slunyaev
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- Journal of Fluid Mechanics / Volume 738 / 10 January 2014
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- 02 December 2013, pp. 65-104
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An asymptotic theory of surface waves trapped on vertically uniform jet currents is developed as a first step towards a systematic description of wave dynamics on oceanic jet currents. It has been shown that in a linear setting an asymptotic separation of vertical and horizontal variables, which underpins the modal description of the wave field on currents, is possible if either the current velocity is small compared to the wave celerity or the current width is large compared to the wavelength along the current. The scheme developed enables us to obtain solutions as an asymptotic series with any desired accuracy. The initially three-dimensional problem is reduced to solving one-dimensional equations with the lateral and vertical dependence being prescribed by the corresponding modal structure. To leading order in current magnitude to wave celerity, the boundary value problem specifying the modes and eigenvalues reduces to classical Sturm–Liouville type based upon the one-dimensional stationary Schrödinger equation. The modes, both trapped and ‘passing-through’, form a complete orthogonal set. This makes the modal description of waves on currents a mathematically attractive alternative to the approaches currently adopted. Properties of trapped eigenmodes and their dispersion relations are examined both for broad currents of arbitrary magnitude, where the modes are not orthogonal, and for weak currents, where the modes are orthogonal. Several model profiles for which nice analytical solutions of the leading-order boundary value problem are known were used to get an insight. The asymptotic solutions proved not only to capture qualitative behaviour well but also to provide a good quantitative description even for unrealistically strong and narrow currents. The results are discussed for various oceanic currents, with particular attention paid to the Agulhas Current, for which specific estimates were derived. For typical dominant wind waves and swell, all oceanic-jet-type currents are weak and, correspondingly, the developed asymptotic scheme based upon one-dimensional stationary Schrödinger equation for modes applies.
On the highest non-breaking wave in a group: fully nonlinear water wave breathers versus weakly nonlinear theory
- Alexey V. Slunyaev, Victor I. Shrira
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- Journal:
- Journal of Fluid Mechanics / Volume 735 / 25 November 2013
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- 23 October 2013, pp. 203-248
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In nature, water waves usually propagate in groups and the open question about the characteristics of the highest possible wave in a group is of significant theoretical and practical interest. We examine the problem of the highest non-breaking wave in a wave group by direct numerical simulations of the exact Euler equations. The main aim of the study is twofold: (i) to describe the highest wave in a group in fully nonlinear setting and find its dependence on parameters; (ii) to examine correspondence between the exact breather solutions of weakly nonlinear analytic theory based on the integrable nonlinear Schrödinger (NLS) equation and their strongly nonlinear analogues. In contrast to weakly nonlinear models the very notion of the highest wave is ill-defined: the maximal crest elevation, the maximal trough-to-crest height and the deepest trough all occur at close but different moments; correspondingly, we have to speak about distinctively different extreme waves. In the simulations small initial perturbation of a uniform wave train were prescribed in a way ensuring that the initial perturbation excites a single breather-type modulation. The ensuing growth results in higher wave magnitudes and takes longer time to develop compared with the NLS theory. The maxima of crest elevation noticeably exceed their weakly nonlinear analogues. The wave with the highest crest differs significantly from the unmodulated wave: the local wavelength contracts considerably, the crest becomes noticeably higher; the vicinity of the crest of such an extreme wave is close to that of the limiting Stokes periodic wave. Thus, the shape of the maximal crest wave is almost universal, i.e. it practically does not depend on the way the wave group evolved, or even whether there was initially more than one group. The evolution of a single NLS breather has been shown to have a qualitatively similar but quantitatively quite different analogue in the fully nonlinear setting. The one-to-one mapping of the NLS breather solutions onto fully nonlinear ones has been constructed. The fully nonlinear breathers are found to be robust, which provides grounds for applying the results for developing short-term deterministic forecasting of rogue waves.
Large-time evolution of statistical moments of wind–wave fields
- Sergei Y. Annenkov, Victor I. Shrira
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- Journal:
- Journal of Fluid Mechanics / Volume 726 / 10 July 2013
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- 11 June 2013, pp. 517-546
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We study the long-term evolution of weakly nonlinear random gravity water wave fields developing with and without wind forcing. The focus of the work is on deriving, from first principles, the evolution of the departure of the field statistics from Gaussianity. Higher-order statistical moments of elevation (skewness and kurtosis) are used as a measure of this departure. Non-Gaussianity of a weakly nonlinear random wave field has two components. The first is due to nonlinear wave–wave interactions. We refer to this component as ‘dynamic’, since it is linked to wave field evolution. The other component is due to bound harmonics. It is non-zero for every wave field with finite amplitude, contributes both to skewness and kurtosis of gravity water waves and can be determined entirely from the instantaneous spectrum of surface elevation. The key result of the work, supported both by direct numerical simulation (DNS) and by the analysis of simulated and experimental (JONSWAP) spectra, is that in generic situations of a broadband random wave field the dynamic contribution to kurtosis is small in absolute value, and negligibly small compared with the bound harmonics component. Therefore, the latter dominates, and both skewness and kurtosis can be obtained directly from the instantaneous wave spectra. Thus, the departure of evolving wave fields from Gaussianity can be obtained from evolving wave spectra, complementing the capability of forecasting spectra and capitalizing on the existing methodology. We find that both skewness and kurtosis are significant for typical oceanic waves; the non-zero positive kurtosis implies a tangible increase of freak wave probability. For random wave fields generated by steady or slowly varying wind and for swell the derived large-time asymptotics of skewness and kurtosis predict power law decay of the moments. The exponents of these laws are determined by the degree of homogeneity of the interaction coefficients. For all self-similar regimes the kurtosis decays twice as fast as the skewness. These formulae complement the known large-time asymptotics for spectral evolution prescribed by the Hasselmann equation. The results are verified by the DNS of random wave fields based on the Zakharov equation. The predicted asymptotic behaviour is shown to be very robust: it holds both for steady and gusty winds.
Non-steady columnar motions in rotating stratified Boussinesq fluids: exact Lagrangian and Eulerian description
- Evsei I. Yakubovich, Victor I. Shrira
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- Journal of Fluid Mechanics / Volume 691 / 25 January 2012
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- 05 December 2011, pp. 417-439
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This paper aims to narrow the gap between the Lagrangian and Eulerian descriptions of rotating stratified fluids. To this end, without loss of generality the primitive Lagrangian equations with arbitrary oriented time-dependent rotation and arbitrary stable stratification have been simplified and made more amenable for analysis. The bulk of the work is concerned with developing in parallel exact Lagrangian and Eulerian descriptions of a particular interesting class of motions of rotating stratified incompressible Boussinesq fluids: the vertically uniform columnar motions. The Lagrangian description is confined to ideal fluids, while the Eulerian one includes viscosity and diffusivity. Assuming the rotation axis to be parallel to gravity, with the rotation rate being an arbitrary function of time, and the buoyancy frequency to be constant, it is found that for vertically uniform motions there is always an exact split into horizontal and vertical subsystems. Evolution of the horizontal velocities and displacements is governed by the classical equations of two-dimensional incompressible hydrodynamics, only slightly modified by accounting for the variable rotation rate. These equations are independent of stratification and vertical motions. The Coriolis term is potential and can be incorporated into pressure. The vertical motions represent a manifestation of packets of inertia–gravity waves with strictly horizontal wavevectors, and are exactly described by linear equations independently of the wave amplitudes. They do not depend on rotation, either constant or variable. The wavepackets do not interact with each other or with horizontal motions. For ideal fluids or those with Rayleigh friction there are explicit solutions describing these motions for arbitrary initial conditions. The Cauchy problem for the columnar motions in ideal fluids is found to be well posed. Thus there is a natural extension of well-studied two-dimensional incompressible hydrodynamics which retains the property of the absence of vortex stretching: all two-dimensional flows could be ‘dressed up’ by adding appropriate vertical motions of a rotating stratified fluid. All the columnar motions could be described in such a way. The examined columnar motions exist under arbitrary relations between the parameters of rotation and stratification and, in particular, without rotation. In the limit of strong rotation one recovers the results known in the literature, in particular, under additional assumptions of small amplitude and steadiness of motions the solutions describe the classical Taylor–Proudman columns.
Inertia-gravity waves beyond the inertial latitude. Part 1. Inviscid singular focusing
- VICTOR I. SHRIRA, WILLIAM A. TOWNSEND
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- Journal:
- Journal of Fluid Mechanics / Volume 664 / 10 December 2010
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- 15 October 2010, pp. 478-509
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The paper is concerned with analytical study of inertia-gravity waves in rotating density-stratified ideal fluid confined in a spherical shell. It primarily aims at clarifying the possible role of these motions in deep ocean mixing. Recently, it was found that on the ‘non-traditional’ β-plane inertia-gravity internal waves can propagate polewards beyond their inertial latitude, where the wave frequency equals the local Coriolis parameter, by turning into subinertial modes trapped in the narrowing waveguides around the local minima of buoyancy frequency N. The behaviour of characteristics was established: wave horizontal and vertical scales decrease as the wave advances polewards and tend to zero at a latitude corresponding to an attractor of characteristics. However, the basic questions about wave evolution, its quantitative description and the possibility of its reflection from the critical latitude remain open. The present work addresses these issues by studying the linear inviscid evolution of finite bandwidth wavepackets on the ‘non-traditional’ β-plane past the inertial latitude for generic oceanic stratification. Beyond the inertial latitude, the wave field is confined in narrowing waveguides of three distinct generic types around different local minima of the buoyancy frequency. In the oceanic context, the widest is adjacent to the flat bottom, the thinnest is the upper mixed layer, and the middle one is located between the seasonal and main thermocline. We find explicit asymptotic solutions describing the wave field in the WKB approximation. As a byproduct, the conservation of wave action principle is explicitly formulated for all types of internal waves on the ‘non-traditional’ β-plane. The wave velocities and vertical shear tend to infinity and become singular at the attractor latitude or its vicinity for both monochromatic and finite bandwidth packets. We call this phenomenon singular focusing. These WKB solutions are shown to remain valid up to singularity for the bottom and mid-ocean waveguides. The main conclusion is that even in the inviscid setting the wave evolution towards smaller and smaller horizontal and vertical scales is irreversible: there is no reflection. For situations typical of deep ocean, a simultaneous increase in wave amplitude and decrease of vertical scale causes a sharp increase of vertical shear, which may lead to wave breaking and increased mixing.
Can bottom friction suppress ‘freak wave’ formation?
- VIACHESLAV V. VORONOVICH, VICTOR I. SHRIRA, GARETH THOMAS
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- Journal of Fluid Mechanics / Volume 604 / 10 June 2008
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- 14 May 2008, pp. 263-296
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The paper examines the effect of the bottom stress on the weakly nonlinear evolution of a narrow-band wave field, as a potential mechanism of suppression of ‘freak’ wave formation in water of moderate depth. Relying upon established experimental studies the bottom stress is modelled by the quadratic drag law with an amplitude/bottom roughness-dependent drag coefficient. The asymptotic analysis yields Davey–Stewartson-type equations with an added nonlinear complex friction term in the envelope equation. The friction leads to a power-law decay of the spatially uniform wave amplitude. It also affects the modulational (Benjamin–Feir) instability, e.g. alters the growth rates of sideband perturbations and the boundaries of the linearized stability domains in the modulation wavevector space. Moreover, the instability occurs only if the amplitude of the background wave exceeds a certain threshold. Since the friction is nonlinear and increases with wave amplitude, its effect on the formation of nonlinear patterns is more dramatic. Numerical experiments show that even when the friction is small compared to the nonlinear term, it hampers formation of the Akhmediev/Ma-type breathers (believed to be weakly nonlinear ‘prototypes’ of freak waves) at the nonlinear stage of instability. The specific predictions for a particular location depend on the bottom roughness ks in addition to the water depth and wave field characteristics.
On radiating solitons in a model of the internal wave–shear flow resonance
- VYACHESLAV V. VORONOVICH, IGOR A. SAZONOV, VICTOR I. SHRIRA
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- Journal of Fluid Mechanics / Volume 568 / 10 December 2006
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- 10 November 2006, pp. 273-301
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The work concerns the nonlinear dynamics of oceanic internal waves in resonance with a surface shear current. The resonance occurs when the celerity of the wave matches the mean flow speed at the surface. The evolution of weakly nonlinear waves long compared to the thickness of the upper mixed layer is found to be described by two linearly coupled equations (a linearized intermediate long wave equation and the Riemann wave equation). The presence of a pseudodifferential operator leads to qualitatively new features of the wave dynamics compared to the previously studied case of shallow water. The system is investigated primarily by means of numerical analysis. It possesses a variety of both periodic and solitary wave stationary solutions, including ‘delocalized solitons’ with a localized core and very small non-decaying oscillatory tails (throughout the paper we use the term ‘soliton’ as synonymous with ‘solitary wave’ and do not imply any integrability of the system). These ‘solitons’ are in linear resonance with infinitesimal waves, which in the evolutionary problem normally results in radiative damping. However, the rate of the energy losses proves to be so small, that these delocalized radiating solitons can be treated as quasi-stationary, that is, effectively, as true solitons at the characteristic time scales of the system. Moreover, they represent a very important class of intermediate asymptotics in the evolution of initial localized pulses. A typical pulse evolves into a sequence of solitary waves of all kinds, including the ‘delocalized’ ones, plus a decaying train of periodic waves. The remarkable feature of this evolution is that of all the products of the pulse fission (in a wide range of parameters of the initial pulse) the radiating solitons have by far the largest amplitudes. We argue that the radiating solitons acting as intermediate asymptotics of initial-value problems are a generic phenomenon not confined to the particular model under consideration.
Role of non-resonant interactions in the evolution of nonlinear random water wave fields
- SERGEI YU ANNENKOV, VICTOR I. SHRIRA
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- Journal:
- Journal of Fluid Mechanics / Volume 561 / 25 August 2006
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- 09 August 2006, pp. 181-207
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We present the results of direct numerical simulations (DNS) of the evolution of nonlinear random water wave fields. The aim of the work is to validate the hypotheses underlying the statistical theory of nonlinear dispersive waves and to clarify the role of exactly resonant, nearly resonant and non-resonant wave interactions. These basic questions are addressed by examining relatively simple wave systems consisting of a finite number of wave packets localized in Fourier space. For simulation of the long-term evolution of random water wave fields we employ an efficient DNS approach based on the integrodifferential Zakharov equation. The non-resonant cubic terms in the Hamiltonian are excluded by the canonical transformation. The proposed approach does not use a regular grid of harmonics in Fourier space. Instead, wave packets are represented by clusters of discrete Fourier harmonics.
The simulations demonstrate the key importance of near-resonant interactions for the nonlinear evolution of statistical characteristics of wave fields, and show that simulations taking account of only exactly resonant interactions lead to physically meaningless results. Moreover, exact resonances can be excluded without a noticeable effect on the field evolution, provided that near-resonant interactions are retained. The field evolution is shown to be robust with respect to the details of the account taken of near-resonant interactions. For a wave system initially far from equilibrium, or driven out of equilibrium by an abrupt change of external forcing, the evolution occurs on the ‘dynamical’ time scale, that is with quadratic dependence on nonlinearity $\varepsilon$, not on the $O(\varepsilon^{-4})$ time scale predicted by the standard statistical theory. However, if a wave system is initially close to equilibrium and evolves slowly in the presence of an appropriate forcing, this evolution is in quantitative accordance with the predictions of the kinetic equation. We suggest a modified version of the kinetic equation able to describe all stages of evolution.
Although the dynamic time scale of quintet interactions $\varepsilon^{-3}$ is smaller than the kinetic time scale $\varepsilon^{-4}$, they are not included in the existing statistical theory, and their effect on the evolution of wave spectra is unknown. We show that these interactions can significantly affect the spectrum evolution, although on a time scale much larger than $O(\varepsilon^{-4})$. However, for waves of high but still realistic steepness $\varepsilon\,{\sim}\,0.25$, the scales of evolution are no longer separated. By tracing the evolution of high statistical moments of the wave field, we directly verify one of the main assumptions used in the derivation of the kinetic equation: the quasi-Gaussianity of the wave holds throughout the evolution, both with and without accounting for quintet interactions.
The conclusions are not confined to water waves and are applicable to a generic weakly nonlinear dispersive wave field with prohibited triad interactions.
Manifestations of bottom topography on the ocean surface: the physical mechanism for large scales
- Victor I. Shrira, Sergei Yu. Annenkov
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- Journal of Fluid Mechanics / Volume 308 / 10 February 1996
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- 26 April 2006, pp. 313-340
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The paper is a first attempt at theoretical investigation of the experimentally observed enigmatic phenomenon of surface manifestations of bottom topography reproducing well the image of the relief despite several kilometres of ocean depth. Both satellite observations and direct measurements have been repeatedly reported in the last two decades. We suggest a possible mechanism for these manifestations in a large scale range (of the order 5 × 101–103 km), based on the hydrodynamic theory of quasi-geostrophic stratified flow over topography on a β-plane.
The classical theories of quasi-geostrophic flow over topography on a β-plane do not include vertical shear, and it is well-known that the disturbance caused by topography cannot reach the surface of a stratified ocean unless the stratification or current velocity is unrealistic. The new element changing the situation qualitatively is the taking into account of the influence of near-bottom and near-surface boundary layers, where flow velocities, velocity gradients and stratification can significantly exceed the corresponding values for the flow in the main body. The asymptotic solution derived shows the considerable increase of the normal mode amplitude towards the boundaries. Thus, this specific distortion of the eigenmode structure results in effective forcing of the modes by topography and, on the other hand, leads to pronounced disturbances in the fields of near-surface characteristics. The mechanism effectiveness is demonstrated by the fact that the surface disturbance amplitude normally significantly exceeds the corresponding value for the barotropic current equal to the maximum of the shear flow. A remarkable feature of the solution is that the Green's function is strongly localized in the horizontal plane for a wide range of relevant parameters, thus leading to the close resemblance of surface patterns and bottom relief. To get a better understanding of the quantitative characteristics of the mechanism, the dependence of the effect on the parameters of an N-layer model was studied in detail. The amplification of the surface manifestations due to the presence of boundary layers can reach several orders of magnitude and thus make the manifestations easily observable. The surface temperature anomalies due to topography were estimated and found to be observable under favourable conditions.
On the evaluation of barotropic-baroclinic instability parameters of zonal flows on a beta-plane
- Vladimir G. Gnevyshev, Victor I. Shrira
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- Journal:
- Journal of Fluid Mechanics / Volume 221 / December 1990
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- 26 April 2006, pp. 161-181
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The paper is concerned with the problem of the linear stability of an arbitrary inviscid zonal flow on a β-plane. Based on the analysis of integral relations following from the linear boundary-value problem, new evaluations, considerably more exact than the previously known ones, of the parameter region of unstable disturbances are derived. Some new relations among these bounds are established.
Surface waves on shear currents: solution of the boundary-value problem
- Victor I. Shrira
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- Journal:
- Journal of Fluid Mechanics / Volume 252 / July 1993
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- 26 April 2006, pp. 565-584
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We consider a classic boundary-value problem for deep-water gravity-capillary waves in a shear flow, composed of the Rayleigh equation and the standard linearized kinematic and dynamic inviscid boundary conditions at the free surface. We derived the exact solution for this problem in terms of an infinite series in powers of a certain parameter e, which characterizes the smallness of the deviation of the wave motion from the potential motion. For the existence and absolute convergence of the solution it is sufficient that e be less than unity.
The truncated sums of the series provide approximate solutions with a priori prescribed accuracy. In particular, for the short-wave instability, which can be interpreted as the Miles critical-layer-type instability, the explicit approximate expressions for the growth rates are derived. The growth rates in a certain (very narrow) range of scales can exceed the Miles increments caused by the wind.
The effect of thin boundary layers on the dispersion relation was also investigated using an asymptotic procedure based on the smallness of the product of the layer thickness and wavenumber. The criterion specifying when and with what accuracy the boundary-layer influence can be neglected has been derived.
On two approaches to the problem of instability of short-crested water waves
- Sergei I. Badulin, Victor I. Shrira, Christian Kharif, Mansour Ioualalen
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- Journal:
- Journal of Fluid Mechanics / Volume 303 / 25 November 1995
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- 26 April 2006, pp. 297-326
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The work is concerned with the problem of the linear instability of symmetric short-crested water waves, the simplest three-dimensional wave pattern. Two complementary basic approaches were used. The first, previously developed by Ioualalen & Kharif (1993, 1994), is based on the application of the Galerkin method to the set of Euler equations linearized around essentially nonlinear basic states calculated using the Stokes-like series for the short-crested waves with great precision. An alternative analytical approach starts with the so-called Zakharov equation, i.e. an integro-differential equation for potential water waves derived by means of an asymptotic procedure in powers of wave steepness. Both approaches lead to the analysis of an eigenvalue problem of the type {\rm det}|{\boldmath A}-\gamma{\boldmath B}|=0 where A and B are infinite square matrices. The first approach should deal with matrices of quite general form although the problem is tractable numerically. The use of the proper canonical variables in our second approach turns the matrix B into the unit one, while the matrix A gets a very specific ‘nearly diagonal’ structure with some additional (Hamiltonian) properties of symmetry. This enables us to formulate simple necessary and sufficient a priori criteria of instability and to find instability characteristics analytically through an asymptotic procedure avoiding a number of additional assumptions that other authors were forced to accept.
A comparison of the two approaches is carried out. Surprisingly, the analytical results were found to hold their validity for rather steep waves (up to steepness 0.4) for a wide range of wave patterns. We have generalized the classical Phillips concept of weakly nonlinear wave instabilities by describing the interaction between the elementary classes of instabilities and have provided an understanding of when this interaction is essential. The mechanisms of the relatively high stability of short-crested waves are revealed and explained in terms of the interaction between different classes of instabilities. A helpful interpretation of the problem in terms of an infinite chain of interacting linear oscillators was developed.
Nonlinear dynamics of vorticity waves in the coastal zone
- Victor I. Shrira, Vyacheslav V. Voronovich
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- Journal:
- Journal of Fluid Mechanics / Volume 326 / 10 November 1996
- Published online by Cambridge University Press:
- 26 April 2006, pp. 181-203
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Vorticity waves are wave-like motions occurring in various types of shear flows. We study the dynamics of these motions in alongshore shear currents in situations where it can be described within weakly nonlinear asymptotic theory. The principal mechanism of vorticity waves can be interpreted as potential vorticity conservation with the background vorticity gradient provided both by the mean current shear and the variation of depth. Under the assumption that the mean potential vorticity distibution is monotonic in the cross-shore direction, the nonlinear stage of the dynamics of weakly nonlinear vorticity waves, long in comparison with the current cross-shore scale, is found to be governed by an evolution equation of the generalized Benjamin–Ono type. The dispersive terms are given by an integro-differential operator with the kernel determined by the large-scale cross-shore depth and current dependence. The derived equations form a wide new class of nonlinear evolution equations. They all tend to the Benjamin–Ono equation in the short-wave limit, while in the long-wave limit their asymptotics depend on the specific form of the depth and current profiles. For a particular family of model bottom profiles the equations are ‘intermediate’ between Benjamin–Ono and Korteweg–de Vries equations, but are distinct from the Joseph intermediate equation. Solitary-wave solutions to the equations for these depth profiles are found to decay exponentially. Taking into account coastline inhomogeneity or/and alongshore depth variations adds a linear forcing term to the evolution equation, thus providing an effective generation mechanism for vorticity waves.
A model of water wave ‘horse-shoe’ patterns
- Victor I. Shrira, Sergei I. Badulin, Christian Kharif
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- Journal:
- Journal of Fluid Mechanics / Volume 318 / 10 July 1996
- Published online by Cambridge University Press:
- 26 April 2006, pp. 375-405
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The work suggests a simple qualitative model of the wind wave ‘horse-shoe’ patterns often seen on the sea surface. The model is aimed at explaining the persistent character of the patterns and their specific asymmetric shape. It is based on the idea that the dominant physical processes are quintet resonant interactions, input due to wind and dissipation, which balance each other. These processes are described at the lowest order in nonlinearity. The consideration is confined to the most essential modes: the central (basic) harmonic and two symmetric oblique satellites, the most rapidly growing ones due to the class II instability. The chosen harmonics are phase locked, i.e. all the waves have equal phase velocities in the direction of the basic wave. This fact along with the symmetry of the satellites ensures the quasi-stationary character of the resulting patterns.
Mathematically the model is a set of three coupled ordinary differential equations for the wave amplitudes. It is derived starting with the integro-differential formulation of water wave equations (Zakharov's equation) modified by taking into account small (of order of quartic nonlinearity) non-conservative effects. In the derivation the symmetry properties of the unperturbed Hamiltonian system were used by taking special canonical transformations, which allow one exactly to reduce the Zakharov equation to the model.
The study of system dynamics is focused on its qualitative aspects. It is shown that if the non-conservative effects are neglected one cannot obtain solutions describing persistent asymmetric patterns, but the presence of small non-conservative effects changes drastically the system dynamics at large times. The main new feature is attractive equilibria, which are essentially distinct from the conservative ones. For the existence of the attractors a balance between nonlinearity and non-conservative effects is necessary. A wide class of initial configurations evolves to the attractors of the system, providing a likely scenario for the emergence of the long-lived three-dimensional wind wave patterns. The resulting structures reproduce all the main features of the experimentally observed horse-shoe patterns. In particular, the model provides the characteristic ‘crescent’ shape of the wave fronts oriented forward and the front-back asymmetry of the wave profiles.