On standing internal gravity waves of finite amplitude

1968 ◽  
Vol 32 (3) ◽  
pp. 489-528 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Density Gradient ◽  
Gravity Waves ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Constant Density ◽  
Interfacial Wave ◽  
Large Wave ◽  
Fractional Density ◽  
Rectangular Container ◽  
Good Agreement

Two-dimensional internal gravity waves in a rectangular container are examined theoretically and experimentally in (a) fluids which contain a single density discontinuity and (b) fluids in which the density gradient is everywhere continuous. The fractional density difference between the top and bottom of the fluid is small.Good agreement is found between the observed and calculated wave profiles in case (a). Unlike surface standing waves, which tend to sharpen at their crests as the wave amplitude increases, and which eventually break at the crests when fluid accelerations become equal to that of gravity, internal wave crests are found to be flat and exhibit no instability. In the case (a) breaking is found to occur at the nodes of the interfacial wave, where the current shear, generated by the wave itself, is greatest. For sufficiently large wave amplitudes, a disturbance with the form of a vortex but with direction of rotation reversing twice every cycle, grows at the wave node and causes mixing. This instability is found to be followed by the generation of cross-waves, of which two different forms are observed.Several modes of oscillation can be generated and are observed in a fluid with constant density gradient. The wave frequencies and shape are well predicted by theory. The experiments failed to establish any limitation of the possible wave amplitudes.

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

The distortion of short internal waves produced by a long wave, with application to ocean boundary mixing

1989 ◽  
Vol 208 ◽  
pp. 395-415 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Density Gradient ◽  
Internal Waves ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Finite Amplitude ◽  
Constant Density ◽  
Boundary Mixing ◽  
Short Waves ◽  
Long Wave ◽  
Boussinesq Fluid

The propagation of a train of short, small-amplitude, internal waves through a long, finite-amplitude, two-dimensional, internal wave is studied. An exact solution of the equations of motion for a Boussinesq fluid of constant density gradient is used to describe the long wave, and its distortion of the density gradient as well as its velocity field are accounted for in determining the propagation characteristics of the short waves. To illustrate the magnitude of the effects on the short waves, particular numerical solutions are found for short waves generated by an idealized flow induced by a long wave adjacent to sloping, sinusoidal topography in the ocean, and the results are compared with a laboratory experiment. The theory predicts that the long wave produces considerably distortion of the short waves, changing their amplitudes, wavenumbers and propagation directions by large factors, and in a way which is generally consistent with, but not fully tested by, the observations. It is suggested that short internal waves generated by the interaction of relatively long waves with a rough sloping topography may contribute to the mixing observed near continental slopes.

On the shape and breaking of finite amplitude internal gravity waves in a shear flow

1978 ◽  
Vol 85 (1) ◽  
pp. 7-31 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Shear Flow ◽  
Internal Waves ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Mixing Layer ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Depth Interval ◽  
Plane Shear

This paper is concerned with two important aspects of nonlinear internal gravity waves in a stably stratified inviscid plane shear flow, their shape and their breaking, particularly in conditions which are frequently encountered in geophysical applications when the vertical gradients of the horizontal current and the density are concentrated in a fairly narrow depth interval (e.g. the thermocline in the ocean). The present theoretical and experimental study of the wave shape extends earlier work on waves in the absence of shear and shows that the shape may be significantly altered by shear, the second-harmonic terms which describe the wave profile changing sign when the shear is increased sufficiently in an appropriate sense.In the second part of the paper we show that the slope of internal waves at which breaking occurs (the particle speeds exceeding the phase speed of the waves) may be considerably reduced by the presence of shear. Internal waves on a thermocline which encounter an increasing shear, perhaps because of wind action accelerating the upper mixing layer of the ocean, may be prone to such breaking.This work may alternatively be regarded as a study of the stability of a parallel stratified shear flow in the presence of a particular finite disturbance which corresponds to internal gravity waves propagating horizontally in the plane of the flow.

Visualization and measurement of internal waves by ‘synthetic schlieren’. Part 1. Vertically oscillating cylinder

1999 ◽  
Vol 390 ◽  
pp. 93-126 ◽  
Author(s):  
BRUCE R. SUTHERLAND ◽  
STUART B. DALZIEL ◽  
GRAHAM O. HUGHES ◽  
P. F. LINDEN
Keyword(s):  
Density Gradient ◽  
Gravity Waves ◽  
Wave Attenuation ◽  
Time Derivative ◽  
Low Frequency ◽  
High Sensitivity ◽  
Internal Gravity Waves ◽  
Quantitative Agreement ◽  
Gradient Field ◽  
Displacement Distribution

We present measurements of the density and velocity fields produced when an oscillating circular cylinder excites internal gravity waves in a stratified fluid. These measurements are obtained using a novel, non-intrusive optical technique suitable for determining the density fluctuation field in temporally evolving flows which are nominally two-dimensional. Although using the same basic principles as conventional methods, the technique uses digital image processing in lieu of large and expensive parabolic mirrors, thus allowing more flexibility and providing high sensitivity: perturbations of the order of 1% of the ambient density gradient may be detected. From the density gradient field and its time derivative it is possible to construct the perturbation fields of density and horizontal and vertical velocity. Thus, in principle, momentum and energy fluxes can be determined.In this paper we examine the structure and amplitude of internal gravity waves generated by a cylinder oscillating vertically at different frequencies and amplitudes, paying particular attention to the role of viscosity in determining the evolution of the waves. In qualitative agreement with theory, it is found that wave motions characterized by a bimodal displacement distribution close to the source are attenuated by viscosity and eventually undergo a transition to a unimodal displacement distribution further from the source. Close quantitative agreement is found when comparing our results with the theoretical ones of Hurley & Keady (1997). This demonstrates that the new experimental technique is capable of making accurate measurements and also lends support to analytic theories. However, theory predicts that the wave beams are narrower than observed, and the amplitude is significantly under-predicted for low-frequency waves. The discrepancy occurs in part because the theory neglects the presence of the viscous boundary layers surrounding the cylinder, and because it does not take into account the effects of wave attenuation resulting from nonlinear wave–wave interactions between the upward and downward propagating waves near the source.

On the role of density jumps in the reflexion and breaking of internal gravity waves

1975 ◽  
Vol 69 (3) ◽  
pp. 445-464 ◽  
Author(s):  
Donald P. Delisi ◽  
Isidoro Orlanski
Keyword(s):  
Gravity Waves ◽  
Gravitational Instability ◽  
Internal Gravity Waves ◽  
Basic Flow ◽  
Density Jump ◽  
Critical Amplitude ◽  
Horizontal Advection ◽  
Reflected Wave ◽  
Good Agreement

A laboratory experiment is presented which examines the role of density jumps in the reflexion and breaking of internal gravity waves. It is found that the measured phase shift of the reflected wave and the measured amplitude of the density jump are in good agreement with linear theory. Local overturning occurs when wave amplitude becomes large, and there appears to be a critical amplitude above which overturning will occur and below which it will not. The overturning seems to be due to local gravitational instability, caused by the horizontal advection of density. Overturning changes the basic flow field in the region of interaction; and it results in smaller-scale motions.

On the shape of progressive internal waves

1968 ◽  
Vol 263 (1145) ◽  
pp. 563-614 ◽  
Keyword(s):  
Experimental Evidence ◽  
Internal Waves ◽  
Gravity Waves ◽  
Boussinesq Approximation ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Fluid System ◽  
Expansion Technique ◽  
Theoretical Predictions ◽  
Two Fluid

An expansion technique, analogous to that of Stokes in the study of surface waves, is used to investigate the effects of finite amplitude on a progressive train of internal gravity waves. The paper is divided into two main parts, a study of interfacial waves in a two-fluid system and an examination of internal waves in a continuously stratified fluid. Experimental evidence is presented which confirms some of the theoretical predictions. The validity of the Boussinesq approximation is examined and particular examples are taken to illustrate the general results.

The instabilities of gravity waves of finite amplitude in deep water II. Subharmonics

1978 ◽  
Vol 360 (1703) ◽  
pp. 489-505 ◽  
Keyword(s):  
Gravity Waves ◽  
Maximum Growth ◽  
Finite Amplitude ◽  
Wave Steepness ◽  
Maximum Value ◽  
Small Wave ◽  
New Type ◽  
Rates Of Growth ◽  
Good Agreement ◽  
Zero Frequency

Calculation of the normal-mode perturbation of steep irrotational gravity waves, begun in part I, is here extended to a study of the subharmonic perturbations, namely those having horizontal scales greater than the basic wavelength (2π/k). At small wave amplitudes a , it is found that all perturbations tend to become neutrally stable; but as ak increases the perturbations coalesce in pairs to produce unstable modes. These may be identified with the instabilities analysed by Benjamin & Feir (1967) when ak is small. However, as ak increases beyond about 0.346, these modes become stable again. The maximum growth scale of this type of mode in the unstable range is only about 14 % per wave period, which value occurs at ak ≈0.32. At values of ak near 0.41 a new type of instability appears which has initially zero frequency but a much higher growth rate. It is pointed out that this type might be expected to arise at wave amplitudes for which the first Fourier coefficient in the basic wave is at its maximum value, as a function of the wave height. The corresponding wave steepness was found by Schwartz (1974) to be ak = 0.412. A comparison of the calculated rates of growth are in rather good agreement with those observed by Benjamin (1967) in the range 0.07< ak <0.17

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