Train Of Gravity Waves Research Articles

Modified nonlinear Schrödinger equation for gravity waves with the influence of wind, currents, and dissipation

A new modified nonlinear Schrödinger (MNLS) equation is derived for gravity waves with the presence of wind, dissipation, and shear currents in finite water depth. Horizontal surface currents are assumed stationary and slowly varying spatially. Using the MNLS equation, the modulational instability (MI) of deep-water gravity wave trains affected by wind and dissipation is considered. It was demonstrated that the modulational perturbation of waves is unstable or becomes unstable after several wave periods, whereas the dissipation will suppress the MI. Then, a new theoretical formula for predicting the maximum amplitude is derived to take into account the effect of vorticity, dissipation, and wind. The effect of dissipation becomes significant in strong currents, while wind can increase the height amplification. Furthermore, an explicit analytical Peregrine breather (PB) solution that considers the effect of vorticity, dissipation, and wind is presented. Opposing currents and winds will increase the height of PB. However, following currents and dissipation have opposite effects. The effects of the shear current, dissipation, and wind on nondimensional maximum amplitudes during the evolution of the Akhmediev breather are similar to PB solution.

Ideal planar fluid flow over a submerged obstacle: Review and extension

A classical problem in free-surface hydrodynamics concerns flow in a channel, when an obstacle is placed on the bottom. Steady-state flows exist and may adopt one of three possible configurations, depending on the fluid speed and the obstacle height; perhaps the best known has an apparently uniform flow upstream of the obstacle, followed by a semiinfinite train of downstream gravity waves. When time-dependent behaviour is taken into account, it is found that conditions upstream of the obstacle are more complicated, however, and can include a train of upstream-advancing solitons. This paper gives a critical overview of these concepts, and also presents a new semianalytical spectral method for the numerical description of unsteady behaviour. doi:10.1017/S1446181121000341

IDEAL PLANAR FLUID FLOW OVER A SUBMERGED OBSTACLE: REVIEW AND EXTENSION

AbstractA classical problem in free-surface hydrodynamics concerns flow in a channel, when an obstacle is placed on the bottom. Steady-state flows exist and may adopt one of three possible configurations, depending on the fluid speed and the obstacle height; perhaps the best known has an apparently uniform flow upstream of the obstacle, followed by a semiinfinite train of downstream gravity waves. When time-dependent behaviour is taken into account, it is found that conditions upstream of the obstacle are more complicated, however, and can include a train of upstream-advancing solitons. This paper gives a critical overview of these concepts, and also presents a new semianalytical spectral method for the numerical description of unsteady behaviour.

ANALYTICAL SOLUTIONS OF COUPLED NONLINEAR SCHRODINGER EQUATIONS FOR TWO NON LINEARLY INTERACTING STOKES WAVE TRAINS OVER INFINITE DEPTH

An attempt to find the exact analytical solutions of the two coupled nonlinear Schrodinger equations of 3rd order occurring from the oblique interaction of two capillary gravity wave trains in the case of crossing sea states in deep water is the main premise of the present paper. The solutions obtained here are due to the nonlinear interaction of two Stokes wave trains in one spatial dimension. Graphs have been plotted to investigate the influence of capillarity on the amplitudes of such wave trains. From 3D figures it has been observed that the capillarity has diminishing influence on the amplitudes of the either wave packet.

Numerical investigation of third-order resonant interactions between two gravity wave trains in deep water

Intense resonant interactions between two deep-water gravity waves can lead to the bending and then splitting of crests and troughs. The curvatures of the crests and troughs increase with an increase of the primary wave's steepness and the resonant wave's propagation distance. For primary waves with small steepness, the crest and trough lines will eventually split, as long as the propagation distance is sufficiently large. The crest line is split into a longer segment with a larger wave crest and a shorter one with a smaller wave crest.

Numerical study of meteotsunamis driven by atmospheric gravity waves in coastal waters of Buenos Aires Province, Argentina

The generation and propagation of meteorological tsunamis (meteotsunamis) were numerically investigated in coastal waters of Buenos Aires Province. This study was carried out using a vertically integrated ocean model driven by a theoretical atmospheric forcing. This forcing simulates a train of nondispersive atmospheric gravity waves (AGW) propagating within a bounded area that moves at the speed of the synoptic systems. Firstly, a study case of simultaneous AGW and meteotsunami activity was simulated to validate the implemented methodology. Subsequently, after several numerical experiments, it was obtained that the amplitude, the dominant period, and the direction of propagation of the AGW train were the parameters that have the largest impact on the simulated meteotsunami amplitude. Maximum meteotsunami wave height (0.85 m) was obtained at Punta Rasa (the northern extreme of the coast of Buenos Aires Province) when the AGW reached this location. Numerical outcomes also showed that the meteotsunamis would propagate like ocean edge waves. In these cases, the continuous transference of energy from the atmosphere to the ocean could be possible (Greenspan resonance). Even though the implemented theoretical forcing is realistic, the numerical experiments revealed that some particular issues should be enhanced to better simulate the genesis and propagation of the meteotsunamis in coastal waters of the Buenos Aires Province. These issues are analyzed and discussed in this paper.

Gravity Wave Excitation during the Coastal Transition of an Extreme Katabatic Flow in Antarctica

Abstract The offshore extent of Antarctic katabatic winds exerts a strong control on the production of sea ice and the formation of polynyas. In this study, we make use of a combination of ground-based remotely sensed and meteorological measurements at Dumont d’Urville (DDU) station, satellite images, and simulations with the Weather Research and Forecasting Model to analyze a major katabatic wind event in Adélie Land. Once well developed over the slope of the ice sheet, the katabatic flow experiences an abrupt transition near the coastal edge consisting of a sharp increase in the boundary layer depth, a sudden decrease in wind speed, and a decrease in Froude number from 3.5 to 0.3. This so-called katabatic jump manifests as a turbulent “wall” of blowing snow in which updrafts exceed 5 m s−1. The wall reaches heights of 1000 m and its horizontal extent along the coast is more than 400 km. By destabilizing the boundary layer downstream, the jump favors the trapping of a gravity wave train—with a horizontal wavelength of 10.5 km—that develops in a few hours. The trapped gravity waves exert a drag that considerably slows down the low-level outflow. Moreover, atmospheric rotors form below the first wave crests. The wind speed record measured at DDU in 2017 (58.5 m s−1) is due to the vertical advection of momentum by a rotor. A statistical analysis of observations at DDU reveals that katabatic jumps and low-level trapped gravity waves occur frequently over coastal Adélie Land. It emphasizes the important role of such phenomena in the coastal Antarctic dynamics.

Physical mechanisms of generation of the noise bursts associated with nonlinear internal tides in shallow water

Recent paper of authors [J. Acoust. Soc. Am. 145(3, Pt. 2), 1670 (2019)] reported observations of repeated increase in acoustic noise intensity during passage of trains of nonlinear internal gravity waves. Periodic increases of pressure fluctuations by up to 35–40 dB in the frequency band from 10 to 5000 Hz were recorded on multiple near-bottom hydrophones during Shallow Water 2006 experiment. The present paper extends the earlier analysis to a more diverse set of nonlinear internal wave events by exploiting the acoustic data obtained at sites with a wider range of water depths and over a longer observation period. Three distinct physical mechanisms are identified, which are responsible for increased noise in different frequency bands. The observed noise bursts are attributed to turbulence of the water flow past hydrophones, sediment transport, and sediment saltation. Correlation between the velocity of the internal tide-induced near-bottom currents and noise intensity at low and mid-frequencies is investigated. [Work supported by ONR, NSF, and BSF.]

Experimental Observation of Modulational Instability in Crossing Surface Gravity Wavetrains

The coupled nonlinear Schrödinger equation (CNLSE) is a wave envelope evolution equation applicable to two crossing, narrow-banded wave systems. Modulational instability (MI), a feature of the nonlinear Schrödinger wave equation, is characterized (to first order) by an exponential growth of sideband components and the formation of distinct wave pulses, often containing extreme waves. Linear stability analysis of the CNLSE shows the effect of crossing angle, θ , on MI, and reveals instabilities between 0 ∘ < θ < 35 ∘ , 46 ∘ < θ < 143 ∘ , and 145 ∘ < θ < 180 ∘ . Herein, the modulational stability of crossing wavetrains seeded with symmetrical sidebands is determined experimentally from tests in a circular wave basin. Experiments were carried out at 12 crossing angles between 0 ∘ ≤ θ ≤ 88 ∘ , and strong unidirectional sideband growth was observed. This growth reduced significantly at angles beyond θ ≈ 20 ∘ , reaching complete stability at θ = 30–40 ∘ . We find satisfactory agreement between numerical predictions (using a time-marching CNLSE solver) and experimental measurements for all crossing angles.

Using ambient noise to evaluate the acoustic Green’s function for a passive geoacoustic inversion in a dynamic shallow-water environment

Cross-correlation functions (CCFs) of ambient and shipping noise recorded by two hydrophones approximate the deterministic Green’s function and contain information about the propagation environment. This paper employs the data collected in the Shallow Water 2006 experiment on the New Jersey continental shelf to investigate the factors that affect the emergence of approximate Green’s functions from noise CCF estimates and accuracy of the approximation. One month-long continuous records of noise obtained by moored single-hydrophone receivers are analyzed. Hydrophones are located in 80–100-m deep water at distances of several kilometers from each other. Rapid variations of the water sound speed profile, which are primarily due to propagation of trains of strong nonlinear internal gravity waves, limit useful noise-averaging time. Available water temperature data are used to guide the selection of time windows for noise averaging and improve CCF evaluation and retrieval of information on seafloor properties. Various approaches to coherent stacking of CCFs are compared. Time warping transform is applied to the resultant noise CCF to extract dispersion curves of acoustic normal modes. The results of the geoacoustic inversion based on the passively measured dispersion curves are compared with the earlier results obtained using controlled sound sources. [Work supported by NSF and BSF.]

Efficient computation of two‐dimensional steady free‐surface flows

SummaryWe consider a family of steady free‐surface flow problems in two dimensions, concentrating on the effect of nonlinearity on the train of gravity waves that appear downstream of a disturbance. By exploiting standard complex variable techniques, these problems are formulated in terms of a coupled system of Bernoulli equation and an integral equation. When applying a numerical collocation scheme, the Jacobian for the system is dense, as the integral equation forces each of the algebraic equations to depend on each of the unknowns. We present here a strategy for overcoming this challenge, which leads to a numerical scheme that is much more efficient than what is normally used for these types of problems, allowing for many more grid points over the free surface. In particular, we provide a simple recipe for constructing a sparse approximation to the Jacobian that is used as a preconditioner in a Jacobian‐free Newton‐Krylov method for solving the nonlinear system. We use this approach to compute numerical results for a variety of prototype problems including flows past pressure distributions, a surface‐piercing object and bottom topographies.

Spatial evolution of an initially narrow-banded wave train

Nonlinear evolution of narrow-banded unidirectional gravity wave trains along wave flume is studied both experimentally and numerically. The spatial version of the Zakharov equation serves as the theoretical model. The frequency domains of spatially linearly unstable disturbances to a monochromatic wave, as well as the frequencies of the most linearly unstable modes are determined theoretically as functions of the carrier wave steepness; these results serve as a reference for the following study. The effect of the spectral width on the evolution along the tank is considered for bi-modal initial spectra, as well as for spectra consisting of a carrier wave and two sidebands with small but finite amplitude. Good agreement between the experimental results and numerical simulations is obtained. The variation of the frequency spectra along the tank resulting from nonlinearity, as well as of the maximum envelope and crest height is investigated as a function of the initial conditions. Fermi–Pasta–Ulam recurrence is obtained for frequency spacing between the initial spectral harmonics approximately corresponding to most unstable disturbance. For narrower spectra, the evolution pattern becomes irregular, numerous additional harmonics are generated by nonlinearity; in this process very steep waves can be generated. The relevance of those findings to appearance of rogue waves is discussed.

Two-dimensionality of gravity water flows governed by the equatorial f-plane approximation

We show that gravity wave trains governed by the equatorial f-plane approximation propagate at the free surface of a rotational water flow of constant vorticity vector (Omega _1, Omega _2, Omega _3) over a flat bed only if the flow is two-dimensional. Owing to the presence of Coriolis effects, our result is also true even if the vorticity vector vanishes. This represents a striking difference when compared with the cases without geophysical effects discussed in Constantin (Europhys Lett 86:29001, 2009, Eur J Mech 30:12–16; 2011) and Martin (J Math Fluid Mech 2016. doi:10.1007/s00021-016-0306-1), where the conclusion about the two-dimensionality of the flow was possible under the assumption of constant nonvanishing vorticity vector. Another upshot is that the only nonzero component of the vorticity that may not vanish is Omega _2, that is, the one pointing in the horizontal direction orthogonal to the direction of wave propagation.

Fourth Order Nonlinear Evolution Equation For Interfacial Gravity Waves In The Presence Of Air Flowing Over Water And A Basic Current Shear

A fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves as first pointed out by Dysthe (1979) is derived for gravity waves propagating at the interface of two superposed fluids of infinite depth in the presence of air flowing over water and a basic current shear. A stability analysis is then made for a uniform Stokes gravity wave train. Graphs are plotted for the maximum growth rate of instability and for wave number at marginal stability against wave steepness for different values of air flow velocity and basic current shears. Significant deviations are noticed from the results obtained from the third order evolution equation, which is the nonlinear Schrödinger equation.

Stability analysis from fourth order evolution equation for counter-propagating gravity wave packets in the presence of wind flowing over water

Fourth order nonlinear evolution equations are derived for two counter-propagating surface gravity wave packets in deep water in the presence of wind flowing over water. The resulting equations are asymptotically exact and nonlocal. Stability analysis is made for a uniform standing surface gravity wave train for longitudinal perturbation on the basis of these equations. Graphs are plotted for maximum growth rate of instability and for wave number at marginal stability against wave steepness for some different values of dimensionless wind velocity. Significant deviations are noticed between the results obtained from third order and fourth order nonlinear evolution equations. This paper has an application in rough waves. References T. Brooke Benjamin and J. E. Feir. The disintegration of wave trains on deep water part 1. theory. Journal of Fluid Mechanics , 27:417–430, 1967. doi:10.1017/S002211206700045X Sudebi Bhattacharyya and K. P. Das. Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline. The ANZIAM Journal , 39:214–229, 1997. doi:10.1017/S033427000000881X K. P. Das. On evolution equations for a three-dimensional surface gravity wave packet in a two-layer fluid. Wave Motion , 8(2):191–204, 1986. doi:10.1016/0165-2125(86)90024-7 A. Davey and K. Stewartson. On three-dimensional packets of surface waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 338(1613):101–110, 1974. doi:10.1098/rspa.1974.0076 Suma Debsarma and K. P. Das. Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water. The ANZIAM Journal , 43:513–524, 2002. doi:10.1017/S1446181100012116 A. K. Dhar and K. P. Das. A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water. Physics of Fluids A: Fluid Dynamics , 2(5), 1990. A. K. Dhar and K. P. Das. Fourth-order nonlinear evolution equation for two Stokes wave trains in deep water. Physics of Fluids A: Fluid Dynamics , 3(12), 1991. A. K. Dhar and K. P. Das. Stability analysis from fourth order evolution equation for small but finite amplitude interfacial waves in the presence of a basic current shear. The ANZIAM Journal , 35:348–365, 1994. doi:10.1017/S0334270000009346 V. D. Djordjevic and L. G. Redekopp. On two-dimensional packets of capillary-gravity waves. Journal of Fluid Mechanics , 79:703–714, 1977. doi:10.1017/S0022112077000408 J. C. Dungey and W. H. Hui. Nonlinear energy transfer in a narrow gravity-wave spectrum. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 368(1733):239–265, 1979. doi:10.1098/rspa.1979.0126 K. B. Dysthe. Note on a modification to the nonlinear schrodinger equation for application to deep water waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 369(1736):105–114, 1979. doi:10.1098/rspa.1979.0154 Tetsu Hara and Chiang C. Mei. Frequency downshift in narrow banded surface waves under the influence of wind. Journal of Fluid Mechanics , 230:429–477, 1991. doi:10.1017/S002211209100085X Tetsu Hara and Chiang C. Mei. Wind effects on the nonlinear evolution of slowly varying gravity-capillary waves. Journal of Fluid Mechanics , 267:221–250, 1994. doi:10.1017/S0022112094001163 S. J. Hogan. The fourth-order evolution equation for deep-water gravity-capillary waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 402(1823):359–372, 1985. doi:10.1098/rspa.1985.0122 Peter A. E. M. Janssen. On a fourth-order envelope equation for deep-water waves. Journal of Fluid Mechanics , 126:1–11, 1983. doi:10.1017/S0022112083000014 M. S. Longuet-Higgins. The instabilities of gravity waves of finite amplitude in deep water. I. superharmonics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 360(1703):471–488, 1978. doi:10.1098/rspa.1978.0080 M. S. Longuet-Higgins. The instabilities of gravity waves of finite amplitude in deep water II. subharmonics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 360(1703):489–505, 1978. doi:10.1098/rspa.1978.0081 R. D. Pierce and E. Knobloch. On the modulational stability of traveling and standing water waves. Physics of Fluids , 6(3), 1994. Michael Stiassnie. Note on the modified nonlinear Schrodinger equation for deep water waves. Wave Motion , 6(4):431–433, 1984. doi:10.1016/0165-2125(84)90043-X M. A. Weissman. Nonlinear wave packets in the Kelvin–Helmholtz instability. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 290(1377):639–681, 1979. doi:10.1098/rsta.1979.0019 V. E. Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied Mechanics and Technical Physics , 9(2):190–194, 1968. doi:10.1007/BF00913182

Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wavetrains

AbstractInternal gravity wavetrains in continuously stratified fluids are generally unstable as a result of resonant triad interactions which, in the inviscid limit, amplify short-scale perturbations with frequency equal to one half of that of the underlying wave. This so-called parametric subharmonic instability (PSI) has been studied extensively for spatially and temporally monochromatic waves. Here, an asymptotic analysis of PSI for time-harmonic plane waves with locally confined spatial profile is made, in an effort to understand how such wave beams differ, in regard to PSI, from monochromatic plane waves. The discussion centres upon a system of coupled evolution equations that govern the interaction of a small-amplitude wave beam with short-scale subharmonic wavepackets in a nearly inviscid uniformly stratified Boussinesq fluid. For beams with general localized profile, it is found that triad interactions are not strong enough to bring about instability in the limited time that subharmonic perturbations overlap with the beam. On the other hand, for quasi-monochromatic wave beams whose profile comprises a sinusoidal carrier modulated by a locally confined envelope, PSI is possible if the beam is wide enough. In this instance, a stability criterion is proposed which, under given flow conditions, provides the minimum number of carrier wavelengths a beam of small amplitude must comprise for instability to arise.

Meteotsunami in the Great Lakes and on the Atlantic coast of the United States generated by the “derecho” of June 29–30, 2012

Tsunami-like intense sea-level oscillations, associated with atmospheric activity (meteorological tsunamis), are common in the Great Lakes and on the East Coast of the United States. They are generated by various types of atmospheric disturbances including hurricanes, frontal passages, tornados, trains of atmospheric gravity waves, and derechos. “Derecho” is a rapidly moving line of convectively induced intense thunder storm fronts producing widespread damaging winds and squalls. The derecho of June 29–30, 2012 devastatingly propagated from western Iowa to the Atlantic coast, passing more than 1,000 km and producing wind gusts up to 35 m/s. This derecho induced pronounced seiche oscillations in Lake Michigan, Chesapeake Bay, and along the US Atlantic coast. Sea-level records from the updated National Oceanic and Atmospheric Administration (NOAA) tide gauge network, together with the NOAA and automated surface-observing system air pressure and wind records, enabled us to examine physical properties and temporal/spatial variations of the generated waves. Our findings indicate that the generation mechanisms of extreme seiches in the basins under study are significantly different: energetic winds play the main role in seiche formation in Chesapeake Bay; atmospheric pressure disturbances are most important for the Atlantic coast; and the combined effect of pressure oscillations and wind is responsible for pronounced events in the Great Lakes. The “generation coefficient,” which is the ratio of the maximum observed sea-level height and the height of air pressure disturbance, was used to map the sea-level response and to identify “hot spots” for this particular event, i.e., harbors and bays with amplified seiche oscillations. The Froude number, Fr = U/c, where U is the speed of the atmospheric disturbance and c is the long-wave speed, is the key parameter influencing the water response to specific atmospheric disturbances; the maximum response was found for those regions and disturbance parameters for which Fr ~1.0.

A comparison of WRF model simulations with SAR wind data in two case studies of orographic lee waves over the Eastern Mediterranean Sea

Two case studies of intense lee waves generated by the interaction of the atmospheric flow with the orography over an area in the eastern Mediterranean Sea, located east of Crete and south of Rhodes islands, are analyzed using the Weather Research and Forecasting (WRF) model. The model has been implemented in two different configurations, using distinct boundary layer parametrization schemes, the Yonsei University (YSU) and the Mellor–Yamada–Janjic (MYJ) schemes, and three 2-way nested domains, the inner one covering the area of interest with a horizontal resolution of 1km. The model-derived wind fields have been compared with the sea surface wind extracted from Envisat Advanced Synthetic Aperture Radar (ASAR) images, from which the wind has been retrieved using a methodology based on the two-dimensional continuous wavelet transform.In the first case, the trains of atmospheric gravity waves, individuated by the signature in the ASAR image, extend downstream for more than 150km and are realistically simulated in the two model configurations, although some appreciable differences are present. The wavelength spectra of the wind speed from ASAR and the two WRF runs have been calculated along different transects in the area affected by atmospheric gravity waves. The agreement is satisfactory, and the difference in wavelengths among the different wave trains generated by the orographic obstacles is well represented in the model runs. Among the two simulations, the YSU run performs better than MYJ. The latter result is consistent with the fact that the YSU experiment reproduces the observed wind profile and the Scorer parameter profile better than the MYJ run.The second case study is briefly analyzed to test the generality of the results of the first event. In this case, part of the wave energy is distributed at larger scales, due to the interaction of the flow with the Peloponnese. Again, both model configurations are able to reproduce correctly the wave patterns, but in this case the difference between the two model runs is negligible.

Gravity‐wave‐induced perturbations in marine stratocumulus

AbstractWe discuss the role of atmospheric gravity waves in modulating cloud radiative and dynamical properties over the southeast Pacific. Satellite imagery and satellite‐retrieved cloud properties during October 2008 illustrate three distinct episodes of horizontal propagation of gravity wave trains across the large‐scale stratocumulus (Sc) cloud deck capping the local marine boundary layer. In one period, 7–9 October 2008, the waves modulated cloud‐top‐height by up to 400 m peak‐to‐trough, propagating perpendicular to the synoptic boundary layer flow with phase speed 15.3 m s−1, period ∼1 h and horizontal wavelength 55 km. The gravity waves were observed to be non‐dispersive. These waves were first evident in the cloud deck near 30°S, 85°W during a 24 h period beginning at midday on 7 October 2008, and propagated northeastward toward the Peruvian coast for the following 48 h. During this time they induced both reversible and non‐reversible changes in cloud‐radiative and cloud‐dynamic properties, such that areas of clear sky developed in the troughs of passing wave‐fronts. These pockets of open cells persisted long after the passage of the gravity waves, advecting northwestward with the background wind. Using the analysis fields of the European Centre for Medium‐Range Weather Forecasts in conjunction with infrared and microwave satellite imagery, we show that these gravity waves emerged from a disturbed subtropical jet stream. The radiant of the waves was coincident in all cases with centres of large negative residuals in nonlinear balance, suggesting that geostrophic readjustment of sharply divergent flows associated with the disturbed jet provided a source for the wave energy. Conversely, gravity waves were not observed in more quiescent jet conditions. This case study highlights the important and irreversible effects that gravity waves propagating in the troposphere can have on cloud radiative properties (and hence surface radiation budgets) over a very wide area. It also highlights the importance of synoptic influence on Sc‐covered marine boundary layers. Copyright © 2012 Royal Meteorological Society

Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability

The Benjamin-Feir instability describes the instability of a uniform oscillatory wave train in an irrotational flow subject to small perturbation of wave number, amplitude and frequency. Their instability analysis is based on the perturbation around the second order Stokes wave which satisfies the dynamic and kinematic free-surface boundary conditions up to the second order. In the same irrotational flow and perturbation framework of the Benjamin-Feir analysis, the perturbation in the present paper is around a nonlinear oscillatory wave train which solves exactly the dynamic free-surface boundary condition and satisfies the kinematic free-surface boundary condition up to the third order. It is shown that the nonlinear oscillatory wave train is stable with respect to the perturbation when the irrotational flow involves small Rayleigh energy dissipation.