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.

A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid

1967 ◽  
Vol 28 (1) ◽  
pp. 1-16 ◽  
Author(s):  
D. E. Mowbray ◽  
B. S. H. Rarity
Keyword(s):  
Experimental Investigation ◽  
Internal Waves ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Internal Gravity Waves ◽  
Forced Oscillations ◽  
Two Dimensional ◽  
Impulsive Motion ◽  
Schlieren System ◽  
Stratified Liquid

Experiments were conducted to test the linear theory of internal gravity waves produced in a stably stratified liquid by the forced oscillations and the initial impulsive motion of a two-dimensional stationary disturbance. The measurements of the wave configuration in a medium whose density increased linearly with depth were made by means of a Toepler-schlieren system. The agreement between observation and prediction was found to be good.

scholarly journals Generalized function treatment of the Alfvén-gravity wave problem

1983 ◽  
Vol 6 (2) ◽  
pp. 395-402
Author(s):  
L. Debnath ◽  
K. Vajravelu
Keyword(s):  
Steady State ◽  
Gravity Waves ◽  
Function Method ◽  
Radiation Condition ◽  
Physical Significance ◽  
Generalized Function ◽  
Wave Problem ◽  
Physical Interest ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid

A study is made of the steady-state Alfvén-gravity waves in an inviscid incompressible electrically conducting fluid with an interface due to a harmonically oscillating pressure distribution acting on the interface. The generalized function method is employed to solve the problem in the fluid of infinite, finite and shallow depth. A unique solution of physical interest is derived by imposing the Sommerfeld radiation condition at infinity. Several limiting cases of physical interest are obtained from the present analysis. The physical significance of the solutions and their limiting cases are discussed.

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 Observations of Near-Inertial Internal Gravity Waves Radiating from a Frontal Jet

2013 ◽  
Vol 43 (6) ◽  
pp. 1225-1239 ◽  
Author(s):  
Matthew H. Alford ◽  
Andrey Y. Shcherbina ◽  
Michael C. Gregg
Keyword(s):  
Gravity Waves ◽  
Three Dimensional ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Shear Layers ◽  
Adjustment Process ◽  
Vertical Wavelength ◽  
The North ◽  
Inertial Wave ◽  
The North Pacific

Abstract Shipboard ADCP and towed CTD measurements are presented of a near-inertial internal gravity wave radiating away from a zonal jet associated with the Subtropical Front in the North Pacific. Three-dimensional spatial surveys indicate persistent alternating shear layers sloping downward and equatorward from the front. As a result, depth-integrated ageostrophic shear increases sharply equatorward of the front. The layers have a vertical wavelength of about 250 m and a slope consistent with a wave of frequency 1.01f. They extend at least 100 km south of the front. Time series confirm that the shear is associated with a downward-propagating near-inertial wave with frequency within 20% of f. A slab mixed layer model forced with shipboard and NCEP reanalysis winds suggests that wind forcing was too weak to generate the wave. Likewise, trapping of the near-inertial motions at the low-vorticity edge of the front can be ruled out because of the extension of the features well south of it. Instead, the authors suggest that the wave arises from an adjustment process of the frontal flow, which has a Rossby number about 0.2–0.3.

Scattering of internal tides by irregular bathymetry of large extent

2014 ◽  
Vol 747 ◽  
pp. 481-505 ◽  
Author(s):  
Yile Li ◽  
Chiang C. Mei
Keyword(s):  
Gravity Waves ◽  
Method Of Characteristics ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Internal Gravity Waves ◽  
Internal Tides ◽  
Two Dimensional ◽  
One Dimensional ◽  
Stratified Ocean ◽  
Stationary Random Function

AbstractWe present an analytical theory of scattering of tide-generated internal gravity waves in a continuously stratified ocean with a randomly rough seabed. Based on a linearized approximation, the idealized case of constant mean sea depth and Brunt–Väisälä frequency is considered. The depth fluctuation is assumed to be a stationary random function of space, characterized by small amplitude and a correlation length comparable to the typical wavelength. For both one- and two-dimensional topographies the effects of scattering on the wave phase over long distances are derived explicitly by the method of multiple scales. For one-dimensional topography, numerical results are compared with Bühler & Holmes-Cerfon (J. Fluid Mech., vol. 678, 2011, pp. 271–293), computed by the method of characteristics. For two-dimensional topography, new results are presented for both statistically isotropic and anisotropic cases.

Trapped internal gravity waves in a geostrophic boundary current

1993 ◽  
Vol 247 ◽  
pp. 205-229
Author(s):  
Hong Ma
Keyword(s):  
Gravity Waves ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Combined Effects ◽  
Kelvin Wave ◽  
Boundary Current ◽  
Trapping Mechanism ◽  
Rossby Radius Of Deformation ◽  
Wave Guides ◽  
Internal Kelvin Wave

The effect of a geostrophic boundary current on internal gravity waves is studied with a reduced-gravity model. We found that the boundary current not only modifies the coastal Kelvin wave, but also forms wave guides for short internal gravity waves. The combined effects of current shear, the boundary, and the slope of the interface create the trapping mechanism. These trapped internal gravity waves appear as groups of discrete zonal modes. They have wavelengths comparable to or shorter than the internal Rossby radius of deformation. Their phase speeds are close to that of the internal Kelvin wave. However, they can propagate both in, or opposite to, the direction of the Kelvin wave. The results of the present work suggest the possibility of finding an energetic internal gravity wave phenomenon with near-inertial frequency in a broad geostrophic boundary current.

A non-linear investigation of critical levels for internal atmospheric gravity waves

1971 ◽  
Vol 50 (3) ◽  
pp. 545-563 ◽  
Author(s):  
R. J. Breeding
Keyword(s):  
Steady State ◽  
Linear Theory ◽  
Gravity Waves ◽  
Incident Wave ◽  
Critical Level ◽  
Internal Gravity Waves ◽  
Mean Flow ◽  
Non Linear ◽  
Energy And Momentum ◽  
Almost All

The behaviour of internal gravity waves near a critical level is investigated by means of a transient two dimensional finite difference model. All the important non-linear, viscosity and thermal conduction terms are included, but the rotational terms are omitted and the perturbations are assumed to be incompressible. For Richardson numbers greater than 2·0 the interaction of the incident wave and the mean flow is largely as predicted by the linear theory–very little of the incident wave penetrates through the critical level and almost all of the wave's energy and momentum are absorbed by changes in the original wind. However, these changes in the wind are centred above the critical level, so that the change in the wind has only a small effect on the height of the critical level. For Richardson numbers less than 2·0 and greater than 0·25 a significant fraction of the incident wave is reflected, part of which could have been predicted by the linear theory. For these stable Richardson numbers a steady state is apparently reached where the maximum wind change continues to grow slowly, but the minimum Richardson number and wave magnitudes remain constant. This condition represents a balance between the diffusion outward of the added momentum and the rate at which it is absorbed. For Richardson numbers less than 0·25, over-reflexion, predicted from the linear theory, is observed, but because the system is dynamically unstable no over-reflecting steady state is ever reached.

Numerical Investigation of Mechanisms Underlying Oceanic Internal Gravity Wave Power-Law Spectra

2020 ◽  
Vol 50 (9) ◽  
pp. 2713-2733
Author(s):  
Yulin Pan ◽  
Brian K. Arbic ◽  
Arin D. Nelson ◽  
Dimitris Menemenlis ◽  
W. R. Peltier ◽  
...  
Keyword(s):  
Power Law ◽  
Gravity Waves ◽  
High Frequency ◽  
Internal Gravity Wave ◽  
Numerical Data ◽  
Field Measurements ◽  
Internal Gravity Waves ◽  
Collision Integral ◽  
Turbulence Theory ◽  
Nonlinear Interactions

AbstractWe consider the power-law spectra of internal gravity waves in a rotating and stratified ocean. Field measurements have shown considerable variability of spectral slopes compared to the high-wavenumber, high-frequency portion of the Garrett–Munk (GM) spectrum. Theoretical explanations have been developed through wave turbulence theory (WTT), where different power-law solutions of the kinetic equation can be found depending on the mechanisms underlying the nonlinear interactions. Mathematically, these are reflected by the convergence properties of the so-called collision integral (CL) at low- and high-frequency limits. In this work, we study the mechanisms in the formation of the power-law spectra of internal gravity waves, utilizing numerical data from the high-resolution modeling of internal waves (HRMIW) in a region northwest of Hawaii. The model captures the power-law spectra in broad ranges of space and time scales, with scalings ω−2.05±0.2 in frequency and m−2.58±0.4 in vertical wavenumber. The latter clearly deviates from the GM76 spectrum but is closer to a family of induced-diffusion-dominated solutions predicted by WTT. Our analysis of nonlinear interactions is performed directly on these model outputs, which is fundamentally different from previous work assuming a GM76 spectrum. By applying a bicoherence analysis and evaluations of modal energy transfer, we show that the CL is dominated by nonlocal interactions between modes in the power-law range and low-frequency inertial motions. We further identify induced diffusion and the near-resonances at its spectral vicinity as dominating the formation of power-law spectrum.

scholarly journals On the appearance of internal wave attractors due to an initial or parametrically excited disturbance

2013 ◽  
Vol 714 ◽  
pp. 283-311 ◽  
Author(s):  
Janis Bajars ◽  
Jason Frank ◽  
Leo R. M. Maas
Keyword(s):  
Gravity Waves ◽  
Normal Modes ◽  
Internal Gravity Waves ◽  
Harmonic Oscillators ◽  
Two Dimensional ◽  
Boussinesq Model ◽  
Resonant Modes ◽  
The Waves ◽  
Shaped Domain ◽  
Parametric Forcing

AbstractIn this paper we solve two initial value problems for two-dimensional internal gravity waves. The waves are contained in a uniformly stratified, square-shaped domain whose sidewalls are tilted with respect to the direction of gravity. We consider several disturbances of the initial stream function field and solve both for its free evolution and for its evolution under parametric excitation. We do this by developing a structure-preserving numerical method for internal gravity waves in a two-dimensional stratified fluid domain. We recall the linearized, inviscid Euler–Boussinesq model, identify its Hamiltonian structure, and derive a staggered finite difference scheme that preserves this structure. For the discretized model, the initial condition can be projected onto normal modes whose dynamics is described by independent harmonic oscillators. This fact is used to explain the persistence of various classes of wave attractors in a freely evolving (i.e. unforced) flow. Under parametric forcing, the discrete dynamics can likewise be decoupled into Mathieu equations. The most unstable resonant modes dominate the solution, forming wave attractors.

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