Instability Of Gravity Waves Research Articles

A Simple Multicloud Parameterization for Convectively Coupled Tropical Waves. Part I: Linear Analysis

AbstractRecent observational analysis reveals the central role of three multicloud types, congestus, stratiform, and deep convective cumulus clouds, in the dynamics of large-scale convectively coupled Kelvin waves, westward-propagating two-day waves, and the Madden–Julian oscillation. A systematic model convective parameterization highlighting the dynamic role of the three cloud types is developed here through two baroclinic modes of vertical structure: a deep convective heating mode and a second mode with low-level heating and cooling corresponding respectively to congestus and stratiform clouds. A systematic moisture equation is developed where the lower troposphere moisture increases through detrainment of shallow cumulus clouds, evaporation of stratiform rain, and moisture convergence and decreases through deep convective precipitation. A nonlinear switch is developed that favors either deep or congestus convection depending on the relative dryness of the troposphere; in particular, a dry troposphere with large convective available potential energy (CAPE) has no deep convection and only congestus clouds. The properties of the multicloud model parameterization are tested by linearized analysis in a two-dimensional setup with no rotation with constant sea surface temperature. In particular, the present study reveals new mechanisms for the large-scale instability of moist gravity waves with features resembling observed convectively coupled Kelvin waves in realistic parameter regimes without any effect of wind-induced surface heat exchange (WISHE). A detailed dynamical analysis for the linear waves is given herein and idealized nonlinear numerical simulations are reported in a companion paper. A maximum congestus heating leads during the dry phase of the wave. It is followed by an increase of the boundary layer θe, that is, CAPE, and lower troposphere moistening that precondition the upper troposphere for the next deep convective episode. In turn, deep convection consumes CAPE and removes moisture, thus yielding the dry episode.

On the superharmonic instability of surface gravity waves on fluid of finite depth

The linear stability of two-dimensional surface gravity waves on fluid of finite depth is investigated for superharmonic disturbances. For this problem, Zufiria & Saffman suggested that an exchange of stability occurs when the total wave energy becomes stationary as a function of wave speed for fixed 'Bernoulli constant'. In defining the potential energy of the above total wave energy, the surface displacement was measured from the quiescent surface with the same 'Bernoulli constant'. We have re-examined this problem both analytically and numerically

On the role of optimal perturbations in the instability of monochromatic gravity waves

Motivated by the useful new insights from optimal-perturbation theory into the onset of turbulence in other fields singular vectors (SVs) in stable and unstable gravity waves have been determined within the framework of the Boussinesq equations on an f plane. The difference between the dynamics of normal modes (NMs) and SV is characterized by a time invariance in the comparative role of the various possible exchange processes between NM and basic wave, while SV can have a highly time-dependent structure, allowing a more efficient energy exchange over a finite time. Both inertia-gravity waves (IGWs) and high-frequency gravity waves (HGWs) have been considered. At Reynolds numbers typical for the middle to upper mesosphere IGW admit rapid nonmodal growth even when no unstable NMs exist. SV energy growth within one Brunt-Vaisala period can cover two orders of magnitude, suggesting the possibility of turbulence onset under conditions where this would not be predicted by a NM analysis. HGWs show a dependence of short-term optimal growth on the direction of propagation of the perturbation with respect to the wave which is, at weak to moderate wave amplitudes, quite different from that of NM but reproduced in ensemble integrations from random initial perturbations. Their SVs are sharply peaked pulses with negligible group velocity which are repeatedly excited as the rapidly propagating wave passes over them. The transition of these to the leading NM, which is not moving with respect to the wave and which is typically broader in structure, is very slow, so that in many cases the turbulence onset via local perturbations of a gravity wave might be more appropriately described using optimal-perturbation theory. This might contribute to a better understanding of the often observed occurrence of thin turbulent layers in the middle atmosphere.

Simulations of crescent water wave patterns on finite depth

A numerical study of the instabilities of Stokes waves on finite depth has been carried out using an efficient fully nonlinear method [D. Clamond and J. Grue, “A fast method for fully nonlinear water-wave computations,” J. Fluid Mech. 447, 337 (2001)]. First, attention is given to five-wave instabilities with k0h=O(1), k0 being the wavenumber and h the depth. Both instabilities leading to breaking and instabilities leading to recurrence are studied, yielding considerably different patterns than on infinite depth. Higher-order instabilities are exemplified, for the first time, by simulations of six- and seven-wave instabilities. Simulations of interactions between four- and five-wave instabilities show that a classical modulational instability can destabilize a three-dimensional perturbation causing crescent waves to appear, in accordance with the hypothesis of [M.-Y. Su and A. W. Green, “Coupled two- and three-dimensional instabilities of surface gravity waves,” Phys. Fluids 27, 2595 (1984)]. Also, a recurrent five-wave instability can boost the energy in a four-wave instability.

Radiating Instabilities of Internal Inertio-Gravity Waves

The vertical radiation of local convective and shear instabilities of internal inertio-gravity waves is examined within linear stability theory. A steady, plane-parallel Boussinesq flow with vertical profiles of horizontal velocity and static stability resembling an internal inertio-gravity wave packet without mean vertical shear is used as the dynamical framework. The influence of primary wave frequency and amplitude, as well as orientation and horizontal wavenumber of the instability on vertical radiation, is discussed. Considerable radiation occurs at small to intermediate instability wavenumbers for basic-state gravity waves with high to intermediate frequencies and moderately convectively supercritical amplitudes. Radiation is then strongest when the horizontal wave vector of the instability is aligned parallel to the horizontal wave vector of the basic-state gravity wave. These radiating modes are essentially formed by shear instability. Convective instabilities, that occur at large instability wavenumbers or strongly convectively supercritical amplitudes, as well as shear instabilities of low-frequency basic-state gravity waves, are nonradiating, trapped in the region of instability. The radiation of an instability is found to be related to the existence of critical levels, a radiating mode being characterized by the absence of critical levels outside the region of instability of the primary wave.

Landau damping and coherent structures in narrow-banded 1+1 deep water gravity waves.

We study the modulational instability in surface gravity waves with random phase spectra. Starting from the nonlinear Schrödinger equation and using the Wigner-Moyal transform, we study the stability of the narrow-banded approximation of a typical wind-wave spectrum, i.e., the JONSWAP spectrum. By performing numerical simulations of the nonlinear Schrödinger equation we show that in the unstable regime, the nonlinear stage of the modulational instability is responsible for the formation of coherent structures. Furthermore, a Landau-type damping, due to the incoherence of the waves, whose role is to provide a stabilizing effect against the modulational instability, is both analytically and numerically discussed.

Philip Gerald Drazin

Philip Gerald Drazin

The criterion of gravity wave instability induced by photochemistry in summer polar mesopause region

This paper studies the effect of photochemistry on the gravity wave instability in summer polar mesopause region. The calculation method of the effects of eddy viscosity, conductivity and eddy diffusion of chemical species on the gravity wave instability induced by photochemistry are studied. The critical wavelength of the instability is given in this paper. The influences of some parameters on it are discussed. The study shows that the gravity wave instability induced by photochemistry is sensitive to the temperature and atomic oxygen profiles.

Instability of MHD-modified interfacial gravity waves revisited

We reveal the basic mechanism of instability of the two-layer conductive fluid system carrying a normal current and exposed to a uniform external magnetic field. This process is a reflection of a MHD-modified interfacial gravity wave from the boundary. Due to special boundary conditions, the reflection coefficient turns out to be greater than 1 for some directions of the wave propagation. We consider two cases: reflection of a monochromatic plane wave from the plane boundary and reflection of rotating waves in a circular geometry. We believe that the proposed mechanism gives a new understanding of the instability formation in the system ‘liquid metal–electrolyte’ type.  2001 Elsevier Science B.V. All rights reserved.

The gravity wave instability induced by photochemistry in summer polar mesopause region

The effect of diabatic process due to the photochemical heating and cooling on the gravity wave propagation in middle atmosphere is studied. A linear gravity wave model which considers the diabatic process is established. The unstable region and the growth rate of the gravity wave caused by photochemistry are calculated. And the comparison between the model and the adiabatic gravity wave theory of pure dynamics is made. The results indicate that the photochemical heating process can induce the instability of gravity wave at mesopause. The intensity of the instability becomes stronger as the temperature decreases. The temperature feature and the altitude characteristics of the instability are consistent with the observation. Therefore, the instability of the gravity wave induced by photochemistry may be an important mechanism in polar mesopause region in summer.

On the Breaking of Internal Waves in the Ocean

Abstract The onset of instability in monochromatic two-dimensional internal near-inertial gravity waves propagating in an ocean of constant buoyancy frequency and no mean shear is examined for increasing values of the wave steepness s, the product of the wave amplitude, and the vertical wavenumber of the waves. Stability of disturbances to the quasi-steady flow depends on the minimum Richardson number of the flow in the direction of the disturbance vector. The minimum Richardson numbers both of the quasi-steady x-directed flow, Jx (that is, in the horizontal direction of wave propagation), and of the transverse y-directed flow, Jy, may be <1/4 for steepness, s < 1, provided that σ/f is sufficiently close to unity, where σ is the wave frequency and f the Coriolis frequency. For waves of increasing steepness but fixed frequency, it is found that the minimum Richardson number of flow in the y direction, Jy, becomes less than 1/4 before those of flows in other directions, suggesting that disturbances in the y...

Nonlinear Energy Transfer from Semidiurnal Barotropic Motion to Near-Inertial Baroclinic Motion

A theoretical investigation into the subharmonic parametric instability of internal gravity waves to progressive surface waves in a continuously stratified medium is presented. A perturbation and multiple scales analysis is utilized and is carried out to the third order. Recent parallel efforts for a two-layer model have shown that surface waves can resonate, exponentially in time, two oblique internal wave trains whose frequencies are nearly subharmonic to that of the surface wave. This analysis is generalized in the current paper to allow for continuous variation of density with depth, as well as planetary rotation. Specific attention is focused on the energy transfer from the semidiurnal barotropic tide to near-inertial baroclinic motion. At low latitudes, linear analysis demonstrates that such a resonant triad indeed exists, suggesting that this mechanism may play a role in the development of the inertial peak seen in many internal wave spectra. At the second order, evolution equations are obtained that demonstrate that the internal waves initially grow exponentially in time. Finally, when the analysis is carried to the third order, it is shown that an equilibrium exists. Nonlinear detuning of the internal wave frequencies results in the internal wave amplitudes reaching a steady-state value, which is readily calculated.

Dust-acoustic gravity vortices in a nonuniform dusty atmosphere

A pair of nonlinear equations for dust-acoustic gravity waves in a nonuniform multicomponent dusty gas is obtained. The possibility of dust-acoustic gravity wave instability and the formation of vortices at the marginal stability point are discussed. The results can have relevance to the generation of waves and coherent nonlinear structures in low-temperature ionospheric as well as in astrophysical dusty plasmas.

Toward a Unified Theory of Gravity Wave Stability

Abstract This paper reveals relationships among linear instabilities of internal gravity waves often supposed to be independent. Using a Floquet analysis of a monochromatic wave propagating in a uniformly stratified background without shear, which accounts for finite wave amplitude, spatial and temporal periodicity, tilted phase planes, and 3D disturbances, it is demonstrated that the dominant instabilities of overturning waves are the large-wave-amplitude manifestations of resonant and slantwise instabilities of small amplitude waves, and that they possess no threshold amplitudes. An energy budget analysis examines the relation of parametric instabilities at large wave amplitude to vertical dynamic and static instabilities; however, the instability characteristics for propagating waves are very different from those inferred by analogy to Kelvin–Helmholtz instability and Benard convection in simpler backgrounds. At small amplitudes, resonant instabilities rely on horizontal or slantwise gradients of wave ...

Three-Dimensional Instabilities of Capillary Gravity Waves of Permanent Form near the Fourth Harmonic Resonance

Three-dimensional instabilities of two-dimensional periodic capillary gravity waves of permanent form near the fourth harmonic resonance are investigated numerically. It is confirmed that the unstable regions appear in the neighborhood of the linear resonance curves of sum interactions associated with a fundamental mode and its fourth harmonic. No unstable regions overlap in the wave number plane of disturbances. Two regions are frequently reconnected.

Crest instabilities of gravity waves. Part 3. Nonlinear development and breaking

The ‘almost-highest wave’, which is the asymptotic form of the flow in a steep irrotational water wave of less than the limiting height, was recently shown to be unstable to infinitesimal disturbances (see Longuet-Higgins &amp; Cleaver 1994). It was also shown numerically that the lowest eigenfrequency is asymptotic to that of the lower superharmonic instability of a progressive wave in deep water (Longuet-Higgins, Cleaver &amp; Fox 1994). In the present paper these calculations are revised, indicating the presence of more than one such instability, in agreement with recent calculations on steep periodic and solitary waves (Longuet-Higgins &amp; Tanaka 1997).The nonlinear development of the fastest-growing instability is also traced by a boundary-integral time-stepping method and the initial, linear growth rate is confirmed. The subsequent, nonlinear stages of growth depend as expected on the sign of the initial perturbation. Perturbations of one sign lead to the familiar overturning of the wave crest. Perturbations of the opposite sign lead to a smooth transition of the wave to a lower progressive wave having nearly the same total energy, followed by a return to a wave of almost the initial wave height. This appears to be the beginning of a nonlinear recurrence phenomenon.

Linearization instability of gravity waves?

Gravity waves in irrotational dust spacetimes are characterized by a nonzero magnetic Weyl tensor ${H}_{\mathrm{ab}}.$ In the linearized theory, the divergence of ${H}_{\mathrm{ab}}$ is set to zero. Recently Lesame, Ellis, and Dunsby [Phys. Rev. D 53, 738 (1996)] presented an argument to show that, in the exact nonlinear theory, $\mathrm{}\mathrm{div}H=0$ forces ${H}_{\mathrm{ab}}=0$, thus implying a linearization instability for gravity waves interacting with matter. However a sign error in the equations invalidates their conclusion. Bianchi type V spacetimes are shown to include examples with $\mathrm{}\mathrm{div}H=0\ensuremath{\ne}{H}_{\mathrm{ab}}.$ An improved covariant formalism is used to show that in a generic irrotational dust spacetime the covariant constraint equations are preserved under evolution. It is shown elsewhere that $\mathrm{}\mathrm{div}H=0$ does not generate further conditions.

Instability and breakdown of internal gravity waves. I. Linear stability analysis

We have performed three-dimensional linear stability analysis, based on Floquet theory, to study the stability of finite amplitude internal gravity waves. This analysis has been used to compute instability growth rates over a range of wave amplitudes and propagation angles, especially waves above and below overturning amplitude, and identifies several new characteristics of wave instability. Computation of instability eigenfunctions has allowed us to analyze the energetics of the instability and to clarify the paths of energy transfer from the base wave to the instability. We find that the presence of wave overturning has no qualitative effect on the wave instability, except for the limiting case when the wavenumber vector is vertical. Instabilities which are nearly two-dimensional are closely related to second-order wave–wave interactions. But the three-dimensional instabilities, more prominent at higher wave amplitudes, may be caused by higher order resonance interactions. The energetics of the instabilities range from being shear driven to being driven by ‘‘density gradient’’ production (the potential energy analog of ‘‘shear’’ production); this characteristic is strongly dependent on wave propagation angle and the three-dimensionality of the instability.

Higher-order resonant instabilities of internal gravity waves

We investigate the three-dimensional characteristics of a general class of resonant temporal instabilities of internal gravity waves, in which the disturbance comprises two infinitesimal internal gravity waves satisfying the conditions $\omega_I + \omega_{II} = n\tilde{\omega}$, with growth rates of order $(\vert \tilde{u}\vert \vert \tilde{k}\vert /N)^n$, assuming small dimensional primary-wave velocity amplitude $\vert \tilde{u} \vert$. We derive simple equations for their wave-numbers, frequency, growth rate, and energy budget. Interactions involving more than two disturbance components are shown not to represent distinct families of solutions, but rather to comprise order transitions linking together two or more ‘two-disturbance-component’ solutions of different order n. Unlike n = 1 resonant instabilities, those with n [ges ] 3 can align with the wave shear flow. We calculate the peak growth rate, spanwise wavenumber, and energy budget of shear-aligned resonance, as functions of wave frequency; they extract energy from both the wave shear and buoyancy fields.

The instability of internal gravity waves to localised disturbances

Abstract. The instability of an internal gravity wave due to nonlinear wave-wave interaction is studied theoretically and numerically. Three different aspects of this phenomenon are examined. 1. The influence of dissipation on both the resonant and the nonresonant interactions is analysed using a normal mode expansion of the basic equations. In particular, the modifications induced in the interaction domain are calculated and as a result some modes are shown to be destabilised by dissipation. 2. The evolution of an initial unstable disturbance of finite vertical extent is described as the growth of two secondary wave packets travelling at the same group velocity. A quasi-linear correction to the basic primary wave is calculated, corresponding to a localised amplitude decrease due to the disturbance growth. 3. Numerical experiments are carried out to study the effect of a basic shear on wave instability. It appears that the growing secondary waves can have a frequency larger than that of the primary wave, provided that the shear is sufficient. The instability of waves with large amplitude and long period, such as tides or planetary waves, could therefore be invoked as a possible mechanism for the generation of gravity waves with shorter period in the middle atmosphere.