An instability of internal gravity waves

1972 ◽  
Vol 52 (2) ◽  
pp. 393-399 ◽  
Author(s):  
Ronald Smith
Keyword(s):  
Surface Wave ◽  
Internal Wave ◽  
Surface Waves ◽  
Gravity Waves ◽  
Internal Gravity Waves ◽  
Long Waves ◽  
Wave Number ◽  
Short Waves

When water is slightly stratified, internal gravity waves are considerably shorter than surface waves of comparable frequency. Here, this fact is exploited in demonstrating that an internal wave is unstable when it forms part of a resonant triad with a surface wave and another internal wave whose wave number is approximately equal to that of the original internal wave. It is suggested that in a system where there are two classes of waves of comparable frequencies but greatly differing wavelengths the short waves may be expected to generate long waves by this mechanism.

Energy transfer between external and internal gravity waves

1964 ◽  
Vol 19 (3) ◽  
pp. 465-478 ◽  
Author(s):  
F. K. Ball
Keyword(s):  
Energy Transfer ◽  
Internal Wave ◽  
Surface Waves ◽  
Shallow Water ◽  
Gravity Waves ◽  
External Surface ◽  
Internal Gravity Waves ◽  
Liquid System ◽  
Resonant Interactions ◽  
Non Linear

In a two-layer liquid system non-linear resonant interactions between a pair of external (surface) waves can result in transfer of energy to an internal wave when appropriate resonance conditions are satisfied. This energy transfer is likely to be more powerful than similar transfers between external waves. The shallow water case is discussed in detail.

Coupling of surface and internal gravity waves: a mode coupling model

1976 ◽  
Vol 77 (1) ◽  
pp. 185-208 ◽  
Author(s):  
Kenneth M. Watson ◽  
Bruce J. West ◽  
Bruce I. Cohen
Keyword(s):  
Energy Transfer ◽  
Surface Wave ◽  
Internal Wave ◽  
Surface Waves ◽  
Wave Field ◽  
Internal Waves ◽  
Wave Spectrum ◽  
Coupled Model ◽  
Internal Gravity Waves ◽  
Growth Time

A surface-wave/internal-wave mode coupled model is constructed to describe the energy transfer from a linear surface wave field on the ocean to a linear internal wave field. Expressed in terms of action-angle variables the dynamic equations have a particularly useful form and are solved both numerically and in some analytic approximations. The growth time for internal waves generated by the resonant interaction of surface waves is calculated for an equilibrium spectrum of surface waves and for both the Garrett-Munk and two-layer models of the undersea environment. We find energy transfer rates as a function of undersea parameters which are much faster than those based on the constant Brunt-ViiisSila model used by Kenyon (1968) and which are consistent with the experiments of Joyce (1974). The modulation of the surface-wave spectrum by internal waves is also calculated, yielding a ‘mottled’ appearance of the ocean surface similar to that observed in photographs taken from an ERTS1 satellite (Ape1 et al. 1975b).

On the interaction of surface and internal gravity waves: uniformly valid solution by extended stationary phase

1976 ◽  
Vol 74 (4) ◽  
pp. 667-683 ◽  
Author(s):  
B. A. Hughes
Keyword(s):  
Fourier Transform ◽  
Surface Wave ◽  
Dispersion Relation ◽  
Stationary Phase ◽  
Internal Wave ◽  
Gravity Waves ◽  
Surface Current ◽  
Internal Gravity Waves ◽  
Surface Gravity Waves ◽  
Valid Solution

Using simple Fourier transform techniques and extensions to stationary-phase methods, the behaviour of surface gravity waves is determined near triply coalescing roots of the dispersion relation. It is shown that the amplitude of the surface wave is proportional to (∂U/∂x)−¼ at the location of the triple root. Far from the triple root it satisfies conservation of action. The internal wave is modelled simply by its surface current U.Asymptotic orders of magnitude are also given for the case ∂U/∂x = 0 at the triple root.

A conservation law for internal gravity waves

1976 ◽  
Vol 78 (2) ◽  
pp. 209-216 ◽  
Author(s):  
Michael Milder
Keyword(s):  
Internal Wave ◽  
Group Velocity ◽  
Internal Waves ◽  
Gravity Waves ◽  
Conservation Law ◽  
Short Wavelength ◽  
Internal Gravity Waves ◽  
Volume Integral ◽  
Weak Density ◽  
Progressive Waves

The scaled vorticity Ω/N and strain ∇ ζ associated with internal waves in a weak density gradient of arbitrary depth dependence together comprise a quantity that is conserved in the usual linearized approximation. This quantity I is the volume integral of the dimensionless density DI = ½[Ω2/N2 + (∇ ζ)2]. For progressive waves the ‘kinetic’ and ‘potential’ parts are equal, and in the short-wavelength limit the density DI and flux FI are related by the ordinary group velocity: FI = DIcg. The properties of DI suggest that it may be a useful measure of local internal-wave saturation.

Subharmonic resonance of short internal standing waves by progressive surface waves

1996 ◽  
Vol 321 ◽  
pp. 217-233 ◽  
Author(s):  
D. F. Hill ◽  
M. A. Foda
Keyword(s):  
Surface Wave ◽  
Internal Wave ◽  
Surface Waves ◽  
Internal Waves ◽  
Growth Rates ◽  
Evolution Equations ◽  
Standing Waves ◽  
Second Order ◽  
Inviscid Limit ◽  
Initial Growth

Experimental evidence and a theoretical formulation describing the interaction between a progressive surface wave and a nearly standing subharmonic internal wave in a two-layer system are presented. Laboratory investigations into the dynamics of an interface between water and a fluidized sediment bed reveal that progressive surface waves can excite short standing waves at this interface. The corresponding theoretical analysis is second order and specifically considers the case where the internal wave, composed of two oppositely travelling harmonics, is much shorter than the surface wave. Furthermore, the analysis is limited to the case where the internal waves are small, so that only the initial growth is described. Approximate solution to the nonlinear boundary value problem is facilitated through a perturbation expansion in surface wave steepness. When certain resonance conditions are imposed, quadratic interactions between any two of the harmonics are in phase with the third, yielding a resonant triad. At the second order, evolution equations are derived for the internal wave amplitudes. Solution of these equations in the inviscid limit reveals that, at this order, the growth rates for the internal waves are purely imaginary. The introduction of viscosity into the analysis has the effect of modifying the evolution equations so that the growth rates are complex. As a result, the amplitudes of the internal waves are found to grow exponentially in time. Physically, the viscosity has the effect of adjusting the phase of the pressure so that there is net work done on the internal waves. The growth rates are, in addition, shown to be functions of the density ratio of the two fluids, the fluid layer depths, and the surface wave conditions.

scholarly journals Nonlinear interactions among standing surface and internal gravity waves

1974 ◽  
Vol 63 (4) ◽  
pp. 801-825 ◽  
Author(s):  
Terrence M. Joyce
Keyword(s):  
Energy Transfer ◽  
Steady State ◽  
Internal Wave ◽  
Surface Waves ◽  
Viscous Dissipation ◽  
Growth Rates ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Resonant Interaction ◽  
Initial Growth

A laboratory study has been undertaken to measure the energy transfer from two surface waves to one internal gravity wave in a nonlinear, resonant interaction. The interacting waves form triads for which \[ \sigma_{1s} - \sigma_{2s} \pm\sigma_1 = 0\quad {\rm and}\quad \kappa_{1s} - \kappa_2s} \pm \kappa_I = 0; \] σj and κj being the frequency and wavenumber of the jth wave. Unlike previously published results involving single triplets of interacting waves, all waves here considered are standing waves. For both a diffuse, two-layer density field and a linearly increasing density with depth, the growth to steady state of a resonant internal wave is observed while two deep water surface eigen-modes are simultaneously forced by a paddle. Internal-wave amplitudes, phases and initial growth rates are compared with theoretical results derived assuming an arbitrary Boussinesq stratification, viscous dissipation and slight detuning of the internal wave. Inclusion of viscous dissipation and slight detuning permit predictions of steady-state amplitudes and phases as well as initial growth rates. Satisfactory agreement is found between predicted and measured amplitudes and phases. Results also suggest that the internal wave in a resonant triad can act as a catalyst, permitting appreciable energy transfer among surface waves.

scholarly journals Interaction Features of Internal Wave Breathers in a Stratified Ocean

Fluids ◽  
2020 ◽  
Vol 5 (4) ◽  
pp. 205
Author(s):  
Ekaterina Didenkulova ◽  
Efim Pelinovsky
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Gravity Waves ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Vries Equation ◽  
Internal Gravity Waves ◽  
Wave Fields ◽  
Tidal Waves ◽  
Stratified Ocean

Oscillating wave packets (breathers) are a significant part of the dynamics of internal gravity waves in a stratified ocean. The formation of these waves can be provoked, in particular, by the decay of long internal tidal waves. Breather interactions can significantly change the dynamics of the wave fields. In the present study, a series of numerical experiments on the interaction of breathers in the frameworks of the etalon equation of internal waves—the modified Korteweg–de Vries equation (mKdV)—were conducted. Wave field extrema, spectra, and statistical moments up to the fourth order were calculated.

Surface Oscillations of Water in a Rotating Cylindrical Vessel

1930 ◽  
Vol 26 (4) ◽  
pp. 446-452 ◽  
Author(s):  
R. O. Street
Keyword(s):  
Circular Cylinder ◽  
Surface Waves ◽  
Cross Section ◽  
Circular Cross Section ◽  
Cylindrical Vessel ◽  
Long Waves ◽  
Horizontal Velocity ◽  
Uniform Rotation ◽  
Tidal Waves ◽  
Short Waves

This paper is devoted chiefly to the consideration of the surface oscillations of water contained in a vessel in the shape of a circular cylinder with its axis vertical, when the motion is slightly disturbed from a uniform rotation about the axis of the vessel. The work was undertaken with the hope of finding some indication of the effect of the depth of the water in the vessel on the period of the surface waves, and for the purpose a vessel of circular cross-section was naturally chosen. It is shown that a slight change of shape does not affect the periods of the oscillations. The solution of the corresponding problem when the surface oscillations take the form of “long waves” or “tidal waves” is well known, and the present paper deals only with “short waves,” for which the horizontal velocity is not the same at all depths.

On the interaction of surface and internal waves

1972 ◽  
Vol 52 (1) ◽  
pp. 179-191 ◽  
Author(s):  
A. E. Gargettt ◽  
B. A. Hughes
Keyword(s):  
Surface Wave ◽  
Internal Wave ◽  
Surface Waves ◽  
Internal Waves ◽  
Wind Waves ◽  
Phase Speed ◽  
Surface Current ◽  
Propagation Direction ◽  
Wave Crest ◽  
Critical Position

The steady-state interaction between surface waves and long internal waves is investigated theoretically using the radiation stress concepts derived by Longuet-Higgins & Stewart (1964) (or Phillips 1966). It is shown that, over internal wave crests, those surface waves for which cg0cosϕ0 > ci experience a change in direction of propagation towards the line of propagation of the internal waves and their amplitudes are increased. Here cg0 is the surface-wave group speed at U = 0, ϕ0 is the angle between the propagation direction of the surface waves at U = 0 and the propagation direction of the internal waves, and ci is the phase speed of the internal waves. If cg0cos ϕ0 < ci the direction of the surface waves is turned away and their amplitudes are decreased. Over troughs the opposite effects occur.At positions where the local velocity of surface-wave energy transmission measured relative to the internal wave phase velocity is zero, i.e. cg + U − ci = 0, there is a singularity in the energy of the surface waves with resulting infinite amplitudes. It is shown that at these critical positions two wavenumbers which were real and distinct on one side coalesce and become complex on the other. The critical positions are thus shown to be barriers to the propagation of those wave-numbers. It is also shown that there is a critical position representing the coalescence of three wavenumbers. Surface-wave crest configurations are shown for three numerical examples. The frequency and direction of propagation of surface waves that exhibit critical positions somewhere in an internal wave field are shown as a function of the maximum horizontal surface current. This is compared with measurements of wind waves that have been reported elsewhere.

Surface waves on water of non-uniform depth

1958 ◽  
Vol 4 (6) ◽  
pp. 607-614 ◽  
Author(s):  
Joseph B. Keller
Keyword(s):  
Wave Propagation ◽  
Surface Wave ◽  
Surface Waves ◽  
Linear Theory ◽  
Gravity Waves ◽  
Geometrical Optics ◽  
Surface Wave Propagation ◽  
Special Cases

Gravity waves occur on the surface of a liquid such as water, and the manner in which they propagate depends upon its depth. Although this dependence is described in principle by the equations of the ‘exact linear theory’ of surface waves, these equations have not been solved except in some special cases. Therefore, oceanographers have been unable to use the theory to describe surface wave propagation in water whose depth varies in a general way. Instead they have employed a simplified geometrical optics theory for this purpose (see, for example, Sverdrup & Munk (1944)). It has been used very successfully, and consequently various attempts, only partially successful, have been made to deduce it from the exact linear theory. It is the purpose of this article to present a derivation which appears to be satisfactory and which also yields corrections to the geometrical optics theory.

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