Instability Of Gravity Waves Research Articles

The instability of internal gravity waves to localised disturbances

The instability of internal gravity waves to localised disturbances

Long-Wavelength Gravity Wave Instabilities: A Comparison of the Jeffreys Drag Mechanism with the Shear Instability

Abstract A simple two-layer model of atmospheric flow over a rough surface is used to compare the Jeffreys drag mechanism with shear flow instability. Both function as sources of internal gravity waves, but the analysis indicates that the Jeffreys drag mechanism is the more effective source of long-wavelength disturbances.

Crest instabilities of gravity waves. Part 2. Matching and asymptotic analysis

In a previous study (Longuet-Higgins & Cleaver 1994) we calculated the stability of the flow near the crest of a steep, irrotational wave, the ‘almost-highest’ wave, considered as an isolated wave crest. In the present paper we consider the modification of this inner flow when it is matched to the flow in the rest of the wave, and obtain the normal-mode perturbations of the modified inner flow. It is found that there is just one exponentially growing mode. Its rate of growth β is a decreasing function of the matching parameter ε and hence a decreasing function of the wave steepness ak. When compared numerically to the rates of growth of the lowest superharmonic instability in a deep-water wave as calculated by Tanaka (1983) it is found that the present theory provides a satisfactory asymptote to the previously calculated values of the growth rate. This suggests that the instability of the lowest superharmonic is essentially due to the flow near the crest of the wave.

Crest instabilities of gravity waves. Part 1. The almost-highest wave

It is shown theoretically that the crest of a steep, irrotational gravity wave, considered in isolation, is unstable. There exists just one basic mode of instability, whose exponential rate of growth β equals 0.123(g / R)½, where g denotes gravity and R is the radius of curvature at the undisturbed crest. A volume of water near the crest is shifted towards the forward face of the wave; the ‘toe’ of the instability is at a horizontal distance 0.45R ahead of the crest. The instability may represent the initial stage of a spilling breaker. On small scales, the ‘toe’ may be a source of parasitic capillary waves.

The formation of layered structures by parametric instability of finite-amplitude gravity waves

VHF Doppler radars occasionally indicate the presence of almost monochromatic long-period gravity waves in the mesosphere. The radar echo intensity occurs in the form of nearly horizontally extended layers which are assumed to be generated by enhanced turbulence although the Richardson numbers are significantly larger than unity, excluding both shear and static instabilities as primarymechanisms for transition to turbulence. In good agreement with these observations, the theory of parametric instability of finite-amplitude monochromatic gravity waves predicts fast growing anisotropic small-scale disturbances whose size is limited by molecular dissipation. The disturbances form turbulent layers moving with the phase velocity of the primary wave. They are “frozen” in the primary wave flow (Taylor's hypothesis) and thus confirm the general assumption that the fluid velocity can be obtained from the Doppler shift of backscattered radar echoes.

Two- and three-dimensional parametric instabilities in finite-amplitude internal gravity waves

Abstract The linearized stability equations of a monochromatic finite-amplitude gravity wave in a Boussinesq fluid are solved for two-dimensional perturbations and three-dimensional perturbations propagating in the plane normal to the primary wave plane and the primary wave fronts. In statically stable primary waves, two-dimensional instability modes grow fastest. They can be divided into two classes: a continuous spectrum of small-scale modes and an isolated mode. In the limit of vanishing primary wave amplitudes, the small-scale modes reduce to second-order resonant wave interactions whereas the underlying wave interaction mechanism of the isolated mode remains uncertain. In statically unstable primary waves, three-dimensional instability modes grow fastest and are either fundamental or subharmonic. They reduce to resonant wave interactions of orders higher than three in the limit of zero primary wave amplitudes. The underlying interaction mechanisms of the fundamental modes also include vortical modes ...

On the role of parametric instability of internal gravity waves in atmospheric radar observations

In the presence of a monochromatic finite‐amplitude internal gravity wave, many properties of tropospheric and mesospheric radar observations can be explained consistently in terms of parametric instability. At small primary wave amplitudes, the fastest growing instability modes can be divided into two classes. In one class, the instability modes have scales comparable to that of the primary wave leading to a broadening of the wave number spectrum. The second class consists of small‐scale instability modes with a continuous wave number spectrum so that the primary wave can be “seen” by radars even in the absence of conventional instability mechanisms such as shear or static instability. The small‐scale instability modes generated by long‐period primary waves propagate almost vertically and form extended layers moving with the phase velocity of the primary wave. Thus, in accordance with observational results, the signal power of backscattered radar signals is strongest at near vertical antenna beam directions and shows a layered structure. The small‐scale instability modes further satisfy Taylor's frozen turbulence field hypothesis so that the Doppler shift of scattered radar signals yields the space and time dependent fluid velocity of the primary wave. At sufficiently large primary wave amplitudes, there is in particular an isolated fast growing instability mode which has a frequency close to the mean Väisälä‐Brunt frequency and is probably due to shear instability. It may give rise to a cascade of successive instability modes that can explain organized substructures in layers of enhanced tropospheric UHF radar returns and the frequent occurrence of short‐period evanescent waves in mesospheric VHF radar observations which cannot be attributed to Kelvin‐Helmholtz instability of the mean shear flow.

Sideband damping of water waves over a soft bed

The Benjamin & Feir sideband instability of gravity waves over a rigid bed is extended here to the case of wave propagation over a compliant bed. The bed is assumed elastic with inhomogeneous properties, following some vertical stratification profile. The elastic stiffness at the mudline is assumed very small, as is typically the case for upper-stratum gel-like marine mud, so that its effect is assumed comparable in magnitude with that of wave nonlinearity. It is then shown that through the sideband instability, the carrier gravity wave would lose energy to sideband oscillations. These sidebands will be shown to be contaminated with small-amplitude, but very short elastic shear waves. These shear waves will interact with the viscous boundary layer at the bed-water interface, and thus significantly enhance the viscous damping of wave energy in the boundary layer.

Deep-water gravity wave instabilities at large steepness

An extension of earlier computations on deep-water gravity wave instabilities up to a wave steepness of 0.434 is performed to compare the relative strength of class I versus class II instabilities. The three-dimensional class II, m=1 instabilities are still predominant. However, as the steepness increases, the difference between the growth rates of two-dimensional and three-dimensional perturbations decreases rapidly and becomes negligible around a 0.43 wave steepness.

Energy computations for evolution of class I and II instabilities of Stokes waves

The modified Zakharov equation is used to study the coupled evolution of class I and class II instabilities of surface gravity waves on infinitely deep water. In contrast to single class (I or II) evolution, the coupled behaviour is non-periodic. Except for the very steep waves a dominance of class I modes over those of class II is observed. Energy calculations show that the Hamiltonian of the wave field considered is nearly constant. Thus the Zakharov and the modified Zakharov equations represent consistent approximations of the original water-wave problem.

Bifurcation and instability in gravity waves

Calculations of the two-dimensional normal-mode perturbations of gravity waves on deep water (Longuet-Higgins, Proc. R. Soc. Lond . A 360, 471-488 (1978 a ); Longuet-Higgins, Proc. R. Soc. Lond . A 360, 489-505 (1978 b )) are here extended to values of the wave steepness ak as high as 0.43. This is achieved ( a ) by using a new method to evaluate the coefficients in Stokes’s series to high order, and ( b ) by rearrangement of the matrix equations so as to reduce the order by half. The behaviour of the normal-mode frequencies σ n in the range 0.35 < ak < 0.43 is clarified. Subharmonics of the form n = ( l / m , 2 - l / m ) Where l and m are integers and l < 2 m are shown to combine in pairs to form type II instabilities with relatively high rates of growth. For these modes, the critical values of ak at which σ vanishes correspond precisely to bifurcation points. In the special case l / m = ½ the two modal curves coincide. The family of frequency curves is bounded by the lowest superharmonic ( n = 2). It is verified that this mode becomes unstable when ak = 0.4292, corresponding to the lowest maximum of the energy density E . The boundaries of the parameter regions for instabilities of both type I and type II are determined.

Coupled two- and three-dimensional instabilities of surface gravity waves

The results of the experimental investigation of the coupling of two- and three-dimensional instabilities of surface gravity waves with initial wave steepness of 0.09≤a0k0≤0.20 are reported. The essentially two-dimensional instabilities (type I) are observed to cause sufficient wave train modulations to trigger the predominantly three-dimensional instabilities (type II). Type II, in turn, limits the growth of the type I and leads to wave breaking and the directional spreading of the wave energy. Hence, type I and II instabilities interact strongly during the evolution of wave trains with moderate initial steepness.

Observations indicating parametric instabilities in internal gravity waves at thermospheric heights

Abstract Theoretical studies predict a parametric instability of finite-amplitude internal gravity waves which hitherto has been observed only in laboratory experiments. The occurrence of this process in the atmosphere is of basic interest because finite-amplitude gravity waves, which are almost ubiquitous especially at upper atmospheric heights, would produce unstable flows even at large Richardson numbers. Maximum entropy power spectra of a strong internal gravity wave in the thermosphere, which was generated by a volcanic eruption and detected on records of the Doppler shift of high-frequency radio waves, in fact show good agreement with the spectra of synthetic Doppler records obtained from a calculated unstable gravity wave. The frequencies and wavenumbers observed in the gravity wave domain satisfy in particular the theoretically predicted resonance conditions. The observed Doppler records also show two significant lines in the acoustic domain which probably result from a nonlinear interaction with ...

Parametric instabilities of internal gravity waves in boussinesq fluids with large reynolds numbers

Abstract Previous studies of parametric instabilities of a finite-amplitude internal gravity wave in dissipationless Boussinesq fluids predict the largest growth rates for perturbations with vanishingly small wavelengths, leading to the practical question of which perturbations are most unstable in the presence of molecular dissipation. At high Reynolds numbers, the amplitude of the gravity wave decays on spatial and temporal scales which are large compared to the wavelength and period respectively. Therefore, the interaction equations describing the growth or decay of small perturbations can be solved by a conventional expansion procedure yielding a leading term in the form of a Floquet solution. Numerical calculations show that the competitive processes of parametric growth and viscous decay produce instabilities with maximum growth rates at wavelengths which may be similar or even longer than the wavelength of the basic gravity wave. Two transformations leading to a significant reduction of computation...

Instabilities in a Stratified Fluid Having One Critical Level. Part I: Results

We consider instabilities in a stratified Boussinesq fluid with basic velocity Uo(z). We initiate this study by considering the detailed stability properties of a profile where Uo(z) is constant in a top and bottom layer and varies linearly with z in an intermediate layer. For an infinite fluid, we find, in addition to Kelvin-Helmholtz instabilities similar to those found by Miles and Howard (1964), an unstable gravity wave mode propagating energy away from exactly one side of the shear zone if and only if the Richardson number in the shear zone is less than 0.1164±0.0001. With a lower boundary present, we find gravity wave instabilities propagating above as well as below the shear zone can exist only when the Richardson number is less than 0. 116. The ability of a shear zone to sustain instabilities propagating energy to infinity on one side is hence only present when the Richardson number is less than 0.116, whether or, not the ground is present. With a lower boundary present, we also find gravity wave instabilities evanescent above the shear zone for Richardson numbers up to 0.2499. After an exhaustive search involving varying the height of the shear zone above the ground and the wavenumber k, we conclude that, for any Richardson number less than ¼, the growth rates of gravity wave instabilities can be as much as, but not greater than, 26% of the maximal growth rates for Kelvin-Helmholtz instabilities having a critical level with the same Richardson number as the gravity wave instabilities have. Even though the relative importance of gravity wave instabilities can be significantly greater than found by other investigators (e.g., Davis and Peltier, 1976), the largest growth rates are still associated with Kelvin-Helmholtz instabilities. In order to understand the basis for this behavior we will analyze, in subsequent parts, both types of instability in terms of wave overreflection. This will enable us to show that the relative importance of Kelvin-Helmholtz and gravity wave instabilities is by no means universal.

Instabilities in a Stratified Fluid Having One Critical Level. Part III: Kelvin-Helmholtz Instabilities as Overreflected Waves

Abstract We show that Kelvin-Helmholtz instabilities result from overreflected, vertically propagating, vorticity waves. Such waves exist in regions of large vorticity gradients. Just as with gravity wave instabilities, growth rates may be inferred from the quantization and overreflection of such waves. Moreover, overreflection concepts offer convenient insights into why Kelvin-Helmholtz instabilities, when they exist, tend to grow faster than gravity wave instabilities.

Instabilities in a Stratified Fluid Having One Critical Level. Part II: Explanation of Gravity Wave Instabilities Using the Concept of Overreflection

We show that there is a one-to-one correspondence between the values of the parameters Ri (Richardson number at the critical level, where mean flow equals phase speed), k (horizontal wavenumber) and cr (phase speed) for which gravity wave instabilities in the presence of a rigid lower boundary were found in Part I and the parameters Ri, k and cr for which overreflection of a neutral wave incident on the shear zone from below would occur. A simple formula involving the reflection coefficient, vertical group velocity of the mode, and the ground to shear layer distance can give quantitative estimates of the growth rate that are especially accurate for small growth rates. The maximum growth rates and corresponding reflection coefficients can be parameterized in a more physically meaningful manner in terms of the Richardson number at the critical level than in terms of the minimum Richardson number in the background wind profile.

Instabilities of finite-amplitude gravity waves on water of finite depth

A study of the instabilities of finite-amplitude periodic gravity waves has extended recent deep-water results to finite depth. Results are presented for three depths, one greater and two less than the depth for stabilization of two-dimensional long-wave perturbations. The results include those instabilities found by perturbation methods as well as some new types of instability.

The instability of atmospheric gravity waves through wave‐wave interactions

We investigate the possibility of a propagating primary wave decaying into secondary waves through three‐wave interactions. Such a possibility exists in an inviscid isothermal atmosphere when the vertical wavelength is small compared to the scale height. Initially, the amplitude of the secondary waves grows exponentially, and this growth time decreases exponentially with height. This means that as the primary wave propagates upward its energy is continuously transfered to the secondary waves. This transfer becomes increasingly rapid with increasing height. It is shown that this process can take place even when the local Richardson number is more than 100, indicating dynamic stability.

Modulation of the amplitude of steep wind waves

Lake & Yuen (1978) have suggested that in very steep wind waves the modulation-frequency of the wave amplitude may correspond to the frequency of the fastest-growing subharmonic instability of a uniform train of waves whose amplitude equals the mean wave amplitude $\overline{a}$. The approximate theory of Benjamin & Feir (1967) gives this frequency as $(\overline{a}k)f_d$, where κ is the wavenumber and fd the frequency of the unperturbed waves. This expression applies strictly only to very small values of the wave steepness $\overline{a}k$.More recently (Longuet-Higgins 1978) the present author calculated accurately all the normal-mode instabilities of steep gravity waves on deep water. In this note these calculations are used to determine the frequency of the fastest-growing sub-harmonic instabilities precisely. When compared with the experimental data of Lake & Huen, these frequencies show even closer agreement.