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.

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.

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 parametric instabilities of finite-amplitude internal gravity waves

1982 ◽  
Vol 119 ◽  
pp. 367-377 ◽  
Author(s):  
J. Klostermeyer
Keyword(s):  
Gravity Wave ◽  
Internal Gravity Wave ◽  
Maximum Growth ◽  
Numerical Approach ◽  
Internal Gravity Waves ◽  
Resonant Interaction ◽  
Finite Amplitude ◽  
Interaction Diagram ◽  
Parametric Instabilities ◽  
Boussinesq Fluid

The equations describing parametric instabilities of a finite-amplitude internal gravity wave in an inviscid Boussinesq fluid are studied numerically. By improving the numerical approach, discarding the concept of spurious roots and considering the whole range of directions of the Floquet vector, Mied's work is generalized to its full complexity. In the limit of large disturbance wavenumbers, the unstable disturbances propagate in the directions of the two infinite curve segments of the related resonant-interaction diagram. They can therefore be classified into two families which are characterized by special propagation directions. At high wavenumbers the maximum growth rates converge to limits which do not depend on the direction of the Floquet vector. The limits are different for both families; the disturbance waves propagating at the smaller angle to the basic gravity wave grow at the larger rate.

A general method for solving steady-state internal gravity wave problems

1972 ◽  
Vol 56 (4) ◽  
pp. 721-740 ◽  
Author(s):  
D. G. Hurley
Keyword(s):  
Steady State ◽  
Circular Cylinder ◽  
Gravity Waves ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Two Dimensional ◽  
Wave Problem ◽  
Wave Problems ◽  
General Method ◽  
Stratified Liquid

The paper describes a simple but general method for solving 'steady-state’ problems involving internal gravity waves in a stably stratified liquid. Under the assumption that the motion is two-dimensional and that the Brunt-Väisälä frequency is constant, the method is used to re-derive in a very simple way the solutions to problems where the boundary of the liquid is either a wedge or a circular cylinder. The method is then used to investigate the effect that a model of the continental shelf has on an incident train of internal gravity waves. The method involves analytic continuation in the frequency of the disturbance, and may well prove to be effective for other types of wave problem.

Growth and equilibrium of short gravity waves in a wind-wave tank

1977 ◽  
Vol 82 (4) ◽  
pp. 767-793 ◽  
Author(s):  
W. J. Plant ◽  
J. W. Wright
Keyword(s):  
Steady State ◽  
Gravity Wave ◽  
Gravity Waves ◽  
Growth Rates ◽  
Energy Input ◽  
Wind Wave ◽  
Initial Growth ◽  
Wave Tank ◽  
Wind Speeds ◽  
Dominant Wave

Temporal and spatial development of short gravity waves in a linear wind-wave tank has been measured for wind speeds up to 15 m/s using microwave Doppler spectrometry. Surface waves of wavelength 4·1 cm, 9·8 cm, 16·5 cm and 36 cm were observed as a function of fetch, wind speed and wind duration. The waves grew exponentially from inception until they were about 10 dB smaller than their maximum height, and the temporal growth and spectral transport (spatial growth) rates were about equal when the wave amplitude was sufficiently small. The amplitude of a short gravity wave of fixed wavelength was found to decrease substantially at winds, fetches or durations greater than those at which the short gravity wave was approximately the dominant wave; such phenomena are sometimes referred to as overshoot. The dominant short gravity wave was observed to reach a maximum amplitude which depended only on wavelength, showing that wave breaking induced by an augmented wind drift cannot be the primary limitation to the wave height. Waves travelling against the wind were observed for wavelengths of 9·8 cm, 16·5 cm and 36 cm and were shown to be generated by the air flow at low wind speeds.Measured initial growth rates for 16·5 cm and 36 cm waves were greater than expected, suggesting the existence of a growth mechanism in addition to direct transfer from the wind via linear instability of the boundary-layer flow. Initial temporal growth rates and spectral transport rates were compared to yield an experimental determination of the magnitude of the sum of nonlinear interactions and dissipation in short gravity waves. If the steady-state energy input in the neighbourhood of the dominant wave occurs at the measured initial temporal growth rates, then most of the energy input is locally dissipated; relatively little is advected away. Calculated gravitycapillary nonlinear energy transfer rates match those determined from initial growth rates for 9·8 cm waves and the gravity–capillary wave interaction continues to be significant for waves as long as 16·5 cm. For longer waves the gravity–capillary interaction is too small to bring the short gravity wave to a steady state when it is the dominant wave of the wind-wave system.

scholarly journals Sensitivity of internal wave energy distribution over seabed corrugations to adjacent seabed features

2017 ◽  
Vol 824 ◽  
pp. 74-96 ◽  
Author(s):  
Farid Karimpour ◽  
Ahmad Zareei ◽  
Joël Tchoufag ◽  
Mohammad-Reza Alam
Keyword(s):  
Steady State ◽  
Energy Distribution ◽  
Internal Wave ◽  
Internal Waves ◽  
Bragg Reflection ◽  
Vertical Wall ◽  
Internal Gravity Waves ◽  
Destructive Interference ◽  
Complete Disappearance ◽  
Reflected Waves

Here we show that the distribution of energy of internal gravity waves over a patch of seabed corrugations strongly depends on the distance of the patch to adjacent seafloor features located downstream of the patch. Specifically, we consider the steady state energy distribution due to an incident internal wave arriving at a patch of seabed ripples neighbouring (i) another patch of ripples (i.e. a second patch) and (ii) a vertical wall. Seabed undulations with dominant wavenumber twice as large as overpassing internal waves reflect back part of the energy of the incident internal waves (Bragg reflection) and allow the rest of the energy to transmit downstream. In the presence of a neighbouring topography on the downstream side, the transmitted energy from the patch may reflect back; partially if the downstream topography is another set of seabed ripples or fully if it is a vertical wall. The reflected wave from the downstream topography is again reflected back by the patch of ripples through the same mechanism. This consecutive reflection goes on indefinitely, leading to a complex interaction pattern including constructive and destructive interference of multiply reflected waves as well as an interplay between higher mode internal waves resonated over the topography. We show here that when steady state is reached both the qualitative and quantitative behaviour of the energy distribution over the patch is a strong function of the distance between the patch and the downstream topography: it can increase or decrease exponentially fast along the patch or stay (nearly) unchanged. As a result, for instance, the local energy density in the water column can become an order of magnitude larger in certain areas merely based on where the downstream topography is. This may result in the formation of steep waves in specific areas of the ocean, leading to breaking and enhanced mixing. At a particular distance, the wall or the second patch may also result in a complete disappearance of the trace of the seabed undulations on the upstream and the downstream wave field.

Degeneration of resonantly-excited standing internal gravity waves

1971 ◽  
Vol 50 (3) ◽  
pp. 431-448 ◽  
Author(s):  
A. D. McEwan
Keyword(s):  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Resonant Interaction ◽  
Similar Process ◽  
Inertial Oscillation ◽  
Rotating Fluids ◽  
Free Wave ◽  
Geometrical Constraints ◽  
Resonant Triads ◽  
Wave Modes

The factors bringing about the irreversible distortion or degeneration of a continuously forced standing internal gravity wave in a linearly stratified fluid are studied experimentally and theoretically. For a rectangular container there is strong evidence that the process is initiated by the unstable growth, from a subliminal level, of free wave modes forming triads in second-order resonant interaction with the original wave. These free modes grow by de-energizing the original wave and may collectively induce kinematical conditions sufficiently severe to create localized regions of density discontinuity within the fluid, leading to turbulence.Although the possible free wave modes are doubly infinite in number, the geometrical constraints greatly reduce the number of possibilities for resonant triads. In many cases this permits critical wave amplitude to be predicted by consideration of one triad only, and the results are in excellent agreement with experiment.It is speculated that a closely similar process explains observations by Malkus (1968), Aldridge & Toomre (1969), and McEwan (1970) in the analogous context of inertial oscillation of contained rotating fluids.

On PSI Interactions in Internal Gravity Wave Fields and the Decay of Baroclinic Tides

2020 ◽  
Vol 50 (3) ◽  
pp. 751-771 ◽  
Author(s):  
Dirk Olbers ◽  
Friederike Pollmann ◽  
Carsten Eden
Keyword(s):  
Energy Transfer ◽  
Wave Field ◽  
Kinetic Equations ◽  
Internal Gravity Wave ◽  
Circulation Model ◽  
Internal Gravity Waves ◽  
Global Scale ◽  
Seafloor Topography ◽  
Latitude Belt ◽  
Tidal Frequency

AbstractBarotropic tidal oscillations over seafloor topography generate baroclinic tides that may be damped in turn via nonlinear triad interactions with internal gravity waves, fueling the ambient wave field. We derive the kinetic equations for this tidal damping and the energy transfer to the ambient wave field and compute damping times and energy transfer rates for the M2 tide and a Garrett–Munk-like ambient wave field. We show that parametric subharmonic instability (PSI) interactions are important, where the tide interacts resonantly with two background waves, each of half the tidal frequency. PSI is restricted to the latitude belt 28.8°N/S and yields under typical conditions damping times of about 20 days for tides with low vertical wavenumber. Damping times decrease with equivalent mode number j roughly as 1/j2. Outside the critical latitudes PSI is not possible, and damping times are from one to two orders of magnitude larger. The energy transfer to the ambient wave field is concentrated at half the tidal frequency ω at all latitudes within the critical latitude belt. Outside, the transfer is much smaller and peaks at ω + f and N. An estimation of the tidal spectral transfer on the global scale is hampered by insufficient knowledge of the baroclinic energy distribution over the vertical modes. Using results from a numerical circulation model with tidal forcing, we find an energy transfer from the tide to the ambient wave field of typically 0.3 TW, about half of what is currently proposed for the conversion of barotropic to baroclinic energy.

The spectral energy transfer from surface waves to internal waves

1979 ◽  
Vol 92 (2) ◽  
pp. 349-379 ◽  
Author(s):  
Dirk J. Olbers ◽  
Klaus Herterich
Keyword(s):  
Energy Transfer ◽  
Surface Wave ◽  
Internal Wave ◽  
Surface Waves ◽  
Internal Waves ◽  
Strong Dependence ◽  
Spectral Energy ◽  
Model Parameters ◽  
Spatial And Temporal Variability ◽  
Stability Frequency

The generation of internal waves by resonantly interacting surface waves is examined in the framework of spectral scattering theory in the random-phase approximation. Coupling coefficients are derived from Euler's equation of motion for arbitrary stratification. The spectral energy transfer is discussed for deep-water surface waves and a simple three-layer model of the stability frequency. Analytical and numerical evaluation of the transfer integral leads to a parametrization in terms of the basic model parameters. These are the depth, thickness and stability frequency of the thermocline and the scale parameters and bandwidth of the surface wave spectrum. Strong dependence on some of these parameters, in particular the surface wave energy and the ratio of surface and internal wave frequencies, indicates a large spatial and temporal variability of the transfer rate. The transfer to the internal wave field in the oceanic main thermocline is found to be negligible compared with the effect of other processes. High frequency waves in the seasonal thermocline may be generated very efficiently.

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