Surface Gravity Waves In Water Research Articles

Interaction of vortices with a progressive surface wave

Interaction of submerged vortices with a progressive surface wave is investigated by the finite-difference numerical solution of Navier-Stokes equations. The progressive wave is the surface gravity water wave in a finite depth. The initial vortex model is Oseen vortex. The numerical computations show that a special pattern of the wave surface may be observed by the interaction from the submerged vortices. The influences of Froude number, the initial geometric configuration of vortices, and the amplitude, initial phase of surface wave on the wave pattern are discussed.

STRUCTURES GENERATED BY MODEL NONLINEAR HAMILTONIANS BASED ON OCEAN WAVES

When surface gravity waves in deep water are driven to a balance between energy input and dissipation, it has been observed that the spectrum of wave-height correlations exhibits power-law behavior consistent with the interpretation that over a range of length scales the surface is a self-affine fractal. Zakharov and L’vov long ago invoked a Kolmogorov cascade hypothesis to describe the nonlinear modal transfer of energy which might explain this observation, even in the absence of driving and dissipation. If their hypothesis is correct, a purely autonomous Hamiltonian system would be capable of driving a surface to fractal roughness, in analogy to dynamical system chaos. In this report I summarize a body of recent work aimed at testing the Z-L hypothesis through numerical integration of Zakharov’s Hamiltonian equations. I investigate both scaling and the existence of postulated emergent dynamical times which scale with modenumber. The results are mixed, in ways not easily abstracted. They lead to hard questions regarding the roles of driving and dissipation, and the process of equilibration of the waveheight statistical distribution, which cannot be answered at our present state of knowledge. But I hope that the form of the questions will prove stimulating.

Dynamical time scales generated by a model nonlinear Hamiltonian.

Zakharov and L'vov long ago invoked a Kolmogorov cascade hypothesis to describe the modal transfer of energy for surface gravity waves in deep water. Within this context the cascade hypothesis requires the existence of dynamical time scales in the evolution of the mode amplitudes which scale with wave number. The purpose of this Letter is to study the temporal evolution of individual modes, with the aim of identifying these dynamical time scales. The main result reported here is that mode-dependent time scales \ensuremath{\tau}(k)\ensuremath{\sim}${\mathit{k}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$ can indeed be identified, with exponent \ensuremath{\beta} related to the exponent \ensuremath{\alpha} which describes the spectrum of the waveheight autocorrelation function.

Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth

The radiative energy transfer for partially coherent water waves is analysed. For linear gravity waves in water of constant depth, the coherence properties are described within the framework of optical coherence theory. Exact energy transfer relations are derived and compared to similar relations for acoustical compression waves. On the basis of previous results for non-dispersive waves, an exact, geometrical description of the radiative energy transfer for dispersive water waves is derived. The theory reduces to the classical theory of radiative energy transfer within a quasihomogeneous wave model. The results are discussed in relation to the applications of the classical description for local wave energy prediction.

Universal aspects of the Hasselmann nonlinear interaction

The asymptotic spectrum of the wave-height autocorrelation is of power-law form for surface gravity waves in deep water. Physical oceanographers have utilized Kolmogorov-type scaling arguments within transport theory in order to predict the exponent of the power law once the wave statistics become stationary. I advance the hypothesis that the asymptotics can be achieved using Hamilton's equations correct to third order with an arbitrary nonsingular interaction term which has the symmetry, degree of homogeneity, and quasilocality in k space of the full Hasselmann interaction term. Examples which support the hypothesis are presented, and its further implications are discussed.

Measurements of the Directional Spectra of Shallow Water Waves Using the Maximum Entropy Principle and a Single Ocean Bottom Seismometer

Abstract A new method for measurement of the directional spectra of surface gravity (water) waves is presented here. Measurements are made at a single point by orthogonally mounted seismometers buried 0.5 meters below the seabed surface and a pressure sensor resting on the seafloor. The maximum entropy principle, which uses entropy as an index of uncertainty in the directional distribution, is used to find the directional distribution function for the waves traveling through the sediments as measured by the seismometers. This function is combined with the frequency spectra of the ocean surface waves to form the directional spectra of the surface gravity waves. Single point measurements are advantageous due to the relative case of implementation and expense compared to arrays, as well as for measurement of non-uniform wave fields. Due to attenuation of shorter wavelengths with depth, this procedure is limited to shallow or intermediate depths. Field data was collected on the New Jersey Shelf and the result...

A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water

The stability of a train of nonlinear surface gravity waves in deep water in the presence of wind blowing over water is considered. An evolution equation is derived for the wave envelope that is correct to fourth order in the wave steepness. The importance of the fourth-order term in the evolution equation was pointed out by Dysthe [Proc. R. Soc. London Ser. A 369, 105 (1979)]. From this evolution equation the expressions for the maximum growth rate of the instability and the frequency at marginal stability are derived, and graphs are plotted for those two expressions against the wave steepness.

Very low frequency Ocean Bottom ambient seismic noise and coupling on the shallow continental shelf

Sources of very low frequency (0.01 to 1.0 Hz) ambient seismic noise in the shallow (<100 m) water continental margin sediments are investigated using Ocean Bottom Seismometers (OBS). The predominant seismic motions are found to be due to surface gravity (water) waves and water-sediment interface waves. Actual experimental measurements of seabed acceleration and hydrodynamic pressure are given, including side by side comparisons between buried and plate-mounted OBS units. OBS-sediment resonant effects are found to be negligible at the low frequencies under investigation. Wherever there exists relative motion between the seabed and the water, however, an exposed OBS is subject to ‘added mass’ forces that cause it to move with the water rather than the sediments. Calculations based on measured seabed motions show that a neutral density, buried seismometer has superior sediment coupling charactersitics to any exposed OBS design.

A note on breaking waves

Some simple general properties of wave breaking are deduced from the known behaviour of surface gravity waves in deep water, on the assumption that breaking occurs in association with wave groups. In particular we derive equations for the time interval, ז, between the onset of breaking of successive waves: ז ═ T / |1 – ( c ⋅ c g )/ c 2 |, and for the propagation vector c b (referred to as the ‘wave-breaking vector’) of the position at which breaking, once initiated, will proceed: c b ═ c (1 – c ⋅ c g / c 2 )+ c g . Here c is the phase velocity, and c g the group velocity, of waves of period T . Interfacial waves, internal gravity waves, inertial waves and planetary waves are considered as particular examples. The results apply not only to wave breaking, but to the movement of any property (e. g. fluid acceleration, gradient Richardson number) that is carried through a medium in association with waves. One application is to describe the distribution, in space and time, of regions of turbulent mixing, or transitional phenomena, in the oceans or atmosphere.

Friction-Induced Roll Motion in Short-Crested Surface Gravity Waves

Induced streaming due to deep water surface gravity waves propagating at oblique angles to each other is studied theoretically on the basis of a Lagrangian description. The ocean is slightly viscous, and the primary waves are maintained at constant amplitude by a suitably adjusted small wind stress distribution at the surface. The induced secondary motion in the nonrotating case consists of parallel rolls with axes aligned along the wave propagation direction, and a horizontally undulating Stokes drift. Surface convergence in the roll motion occurs at lines through nodal points of the primary wave system, and downwelling occurs below them. The surface value of the undulating Stokes drift has a minimum at these nodal points if the angle between the crossing waves is less than 76.4°. If this angle is larger than 76.4°, the Stokes drift at the surface has a maximum here. The roll motion described in the present paper is discussed in connection with the basis for the recent theoretical development of Langmuir circulations. Finally, a solution for the steady, horizontally averaged drift current in a rotating ocean is presented.

Notes on long-crested water waves

Fully three-dimensional surface gravity waves in deep water are investigated in the limit in which the length of the wave crests become long. We describe an analytic solution to fourth order in wave steepness, which matches onto known short-crested wave solutions on the one hand and onto the well-known two-dimensional progressivewave solution on the other. In the progressive-wave limit a particular solution in which the wave crests are semi-finite is given to sixth-order accuracy. These solutions are part of a more general set of solutions which are found from a nonlinear Schrodinger equation.

Irrotational, progressive surface gravity waves near the limiting height

Numerical solutions of irrotational, progressive surface gravity waves in water of a constant depth are obtained by means of an iterative method. Our results suggest that waves with the surface slope angle greater thanπ/6 may exist. The calculated phase velocity of deep water waves near the wave steepness 0.14 is significantly smaller than the value given by the Stokes' fourth approximation.