Instability Of Gravity Waves Research Articles

The Excitation of Radiating Waves and Kelvin-Helmholtz Instabilities by the Gravity Wave-Critical Level Interaction

Abstract The gravity wave-critical level interaction is found to excite both radiating waves and Kelvin-Helmholtz instabilities through nonlinear interactions near the critical level. Radiating waves are forced directly by perturbations in the harmonies of the incident gravity wave and Kelvin-Helmholtz instabilities, once excited through nonlinear interactions, grow on the unstable velocity shears created by the incident wave. Results are presented which demonstrate that radiating waves can significantly increase the wave-action and momentum flux which is found above a critical level and that Kelvin-Helmholtz instabilities are responsible for stabilizing the induced unstable velocity shears. Finally, the implications of these results for the atmosphere and the oceans are discussed.

The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics

In this paper we embark on a calculation of all the normal-mode perturbations of nonlinear, irrotational gravity waves as a function of the wave steepness. The method is to use as coordinates the stream-function and velocity potential in the steady, unperturbed wave (seen in a reference frame moving with the phase speed) together with the time t. The dependent quantities are the cartesian displacements and the perturbed stream function at the free surface. To begin we investigate the ‘superharmonics’, i.e. those perturbations having the same horizontal scale as the fundamental wave, or less. When the steepness of the fundamental is small, the normal modes take the form of travelling waves superposed on the basic nonlinear wave. As the steepness increases the frequency of each perturbation tends generally to be diminished. At a steepness ak ≈ 0.436 it appears that the two lowest modes coalesce and an instability arises. There is evidence that this critical steepness corresponds precisely with the steepness at which the phase velocity is a maximum, considered as a function of ak. The calculations are facilitated by the discovery of some new identities between the coefficients in Stokes’s expansion for waves of finite amplitude.

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

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

The occurrence of parametric instabilities in finite-amplitude internal gravity waves

The parametric instability of a plane internal gravity wave is considered. When the two-dimensional equations of vorticity and mass conservation are linearized in the disturbance quantities, partial differential equations with periodic coefficients result. Substitution of a perturbation of the form dictated by Floquet theory into these equations yields compatibility conditions which, when evaluated numerically, give the curves of neutral stability and constant disturbance growth rate. These results reveal that, for an internal wave of even infinitesimal amplitude, disturbance waves can begin to grow in amplitude. Moreover, these parametric instabilities are shown to reduce to the classical case of the nonlinear resonant interaction in the limit of vanishingly small basic-state amplitude. The fact that these unstable disturbances can exist for an internal wave of any amplitude suggests that this phenomenon may be an important mechanism for extracting energy from an internal gravity wave.

On the instability of Helmholtz velocity profiles in stably stratified fluids when a lower boundary is present

We investigate the stability of a Helmholtz velocity profile in a stably stratified fluid when a lower boundary is present. In addition to the traditional Kelvin-Helmholtz interfacial instability we find an infinitude of unstable internal gravity waves, the most unstable of which bear a close resemblance to disturbances observed in connection with clear air turbulence. It is shown that the gravity wave instabilities result from the ability of an unstable shear layer to overreflect such waves (reflected amplitudes are greater than incident amplitudes). Overreflection at the shear layer is constantly fed by total reflection from the lower boundary. The growth rate of gravity wave instabilities depends markedly on the distance between the shear layer and the ground. A mechanism is suggested whereby gravity wave instability can lead to long-lived almost neutral shear layers.

Resonant trapped gravity waves and turbulent patches in an inversion layer

Instability of resonant trapped gravity waves is proposed as the mechanism responsible for the creation of turbulent patches in the inversion capping a well developed convective layer. The plume circulation of the convective layer results in a periodic forcing at the base of the inversion layer; trapped gravity waves in resonance with such forcing are considered. Good agreement with experimental results is obtained.

Parametric instability of gravity waves on the surface of deep water

We propose to investigate the characteristics of the parametric generation of gravity waves on the surface of a body of deep water. The threshold conditions for the onset of generation are determined, and the results are compared with the experimental data. The singularities of the excitation of parametric oscillations in a resonator are noted.

Wave-induced eddy diffusion coefficients in the upper atmosphere of Mars

A theory and method previously used to calculate terrestrial eddy diffusion coefficients due to instabilities in internal gravity waves have been extended to obtain wave-induced eddy diffusion coefficients in the upper atmosphere of Mars. If the Martian atmosphere is relatively dry (water vapor mixing ratio much less than .001), the effects of radiative damping are minimal for all but the longest-period waves. For greater concentrations of water vapor the effects of radiative damping are increased, but in any event it is reasonable to expect wave-induced turbulence, with eddy diffusion coefficients of the order of 10 to the 7th sq cm/sec in the Martian upper atmosphere.

Correction [to “Eddy diffusion coefficients due to instabilities in internal gravity waves”

Journal of Geophysical Research (1896-1977)Volume 78, Issue 1 p. 335-336 Free Access Correction [to “Eddy diffusion coefficients due to instabilities in internal gravity waves”] C. O. Hines, C. O. HinesSearch for more papers by this author C. O. Hines, C. O. HinesSearch for more papers by this author First published: 1 January 1973 https://doi.org/10.1029/JA078i001p00335Citations: 7AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article. References Hines, C. O., Eddy diffusion coefficients due to instabilities in internal gravity waves, J. Geophys. Res., 75, 3937– 3939, 1970. Hodges Jr., R. R., Eddy diffusion coefficients due to instabilities in internal gravity waves, J. Geophys. Res., 74, 4087– 4090, 1969. Pitteway, M. L. V., C. O. Hines, The viscous damping of atmospheric gravity waves, Can. J. Phys., 41, 1935– 1948, 1963. Citing Literature Volume78, Issue1Space Physics1 January 1973Pages 335-336 ReferencesRelatedInformation

An instability of internal gravity 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.

Eddy diffusion coefficients due to instabilities in internal gravity waves

Eddy diffusion coefficients due to instabilities in internal gravity waves

Resonant Internal Wave Interactions

INTERNAL waves in the ocean are of critical importance both to the large scale oceanic energy and momentum balance and to smaller scale oceanic processes. Most discussions of the physics of these waves concentrate on linear waves of infinitesimal amplitude; however, both the breaking of internal waves and the generation and decay of their oceanic spectra involve nonlinear processes. We describe here the preliminary results of an experimental and theoretical investigation of the effects of non-linearity on a resonant pair of discrete internal waves. The only other resonant interaction experiments known to the authors are (1) careful definitive measurements of initial trends for a very weak third-order surface gravity wave interaction1, and (2) experiments which relate an instability of internal gravity waves in a two-fluid system to a theorized second-order resonance2. Our objective was to generate two waves theoretically capable of resonant interactions and to show, by measuring elementary definitive properties, that a second-order resonance does, in fact, occur in the fluid.

Eddy diffusion coefficients due to instabilities in internal gravity waves

Internal gravity waves in the upper atmosphere tend to grow in amplitude with increasing height to maintain continuous vertical transport of wave energy. The growth of amplitude of a wave is limited by convective instabilities that must form at the height where the temperature oscillation becomes great enough to include a superadiabatic region. This turbulence attenuates the wave by eddy transport of heat and viscous stress through the wave structure, essentially maintaining a constant amplitude of the wave at greater altitudes. Eddy diffusion coefficients of the order of 107 cm²/sec are readily produced by typical gravity waves near the mesopause. Since an average coefficient of only 106 cm²/sec is sufficient to explain the thermal structure of the mesopause region, sporadic turbulence produced by instabilities in gravity waves is probably an important process of the upper atmosphere.

Atmospheric gravity wave instability?

Recently Hines [1965] in a review of atmospheric gravity wave theory invited the attention of wave theorists from other fields. This note is a direct result and is a by-product of other work in plasma waves. The intent of this note is only to point out an interesting feature of an equation given by Hines and treated somewhat on the same footing as a dispersion equation. From the analysis given by Martyn [1950] using a particular variable φ, the following differential equation is obtained (in Hines' notation) for waves in a nonisothermal atmosphere ((1)) ((2)) with C = velocity of sound. g = acceleration due to gravity. γ = ratio of specific heats. ωa = γg/2C, the acoustic cutoff frequency. ωB = [γ - 1 + γ (∂H/∂Z)]1/2(g/C)the modified Brunt-Vaisala frequency. H = scale height = C2/γg = KT/Mg. kx = horizontal wave number. Z = vertical distance.

Secondary instability in steady gravity waves

A stratified stream may be stable when flowing horizontally and yet develop a Gortler type of instability in gravity waves when the flow is curved. The equation for the generation of secondary vorticity (rotation round the streamlines) may be used to study this, and we show that the curved flow in two-dimensional gravity waves could become unstable in real cases. This is, therefore, one possible mechanism for the generation of ‘clear-air turbulence’ in mountain waves.

Secondary instability in steady gravity waves

AbstractA stratified stream may be stable when flowing horizontally and yet develop a Görtler type of instability in gravity waves when the flow is curved. The equation for the generation of secondary vorticity (rotation round the streamlines) may be used to study this, and we show that the curved flow in two‐dimensional gravity waves could become unstable in real cases. This is, therefore, one possible mechanism for the generation of ‘clear‐air turbulence’ in mountain waves.