Finite-amplitude Internal Gravity Wave Research Articles

Reflection of an internal gravity wave beam off a horizontal free-slip surface

The reflection of a planar finite-amplitude internal gravity wave beam off a free-slip flat horizontal surface is investigated numerically in a uniformly stratified Boussinesq fluid. Nonlinear effects such as mean currents and harmonics are observed in the wave reflection zone. Mean currents form a stationary, vertically oscillatory, layered structure under the free-slip reflecting surface. The vertical wavelength of the mean-flow layers equals half of the vertical wavelength of the reflecting wave. An empirical predictive model for the steady-state mean flow strength, based on the degree of wave nonlinearity and hydrostaticity, is proposed and subsequently compared to the weakly nonlinear theory by Tabaei et al. [J. Fluid Mech. 526, 217–243 (2005)10.1017/S0022112004002769]. Very strong agreement between simulation results and theory is observed for all waves considered, suggesting although weakly nonlinear in its formulation, the Tabaei et al. theory is valid for the full range of finite amplitudes for which a wave remains stable. Both propagating and evanescent superharmonics are observed, and for waves with steepnesses of O(5%), parametric subharmonic instabilities can occur in the later stages of the reflection process. When a subsurface mixed layer is incorporated into the simulations, the mean currents at the middle of the underlying pycnocline are similar in structure and magnitude to their uniformly-stratified counterparts.

Two-dimensional evolution equation of finite-amplitude internal gravity waves in a uniformly stratified fluid

We derive a fully nonlinear evolution equation that can describe the two-dimensional motion of finite-amplitude long internal waves in a uniformly stratified three-dimensional fluid of finite depth. The derived equation is the two-dimensional counterpart of the evolution equation obtained by Grimshaw and Yi [J. Fluid Mech. 229, 603 (1991)]. In the small-amplitude limit, our equation is reduced to the celebrated Kadomtsev-Petviashvili equation.

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.

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 ...

Two‐dimensional instability of finite amplitude internal gravity wave packets near a critical level

A high‐resolution two‐dimensional numerical model is used to simulate the propagation of finite amplitude internal wave packets into a mean shear flow that varies slowly in space. For moderate packet amplitudes the interaction is well described by weakly nonlinear asymptotic theory. At higher amplitude, however, a region develops near the critical level in which nonlinearity dominates, the wave packet becomes unstable, and the wavelike motion breaks down into smaller scales. The form of the observed internal wave breakdown is unexpected in that convectively unstable density gradients persist for many buoyancy periods. The eventual rapid transition, from large‐amplitude wavelike motion to a more complicated flow, is triggered by an instability driven by the intensified wave shear and not by a convective breakdown of the unstable stratification. This illustrates that unstable stratification does not necessarily lead to convective instability in two dimensions.

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...

On parametric instabilities of finite-amplitude internal gravity waves

The equations describing parametric instabilities of a finite-amplitude internal gravity wave in an inviscid Boussinesq fluid are studied numerically. By improving the numerical approach, discarding the concept of spurious roots and considering the whole range of directions of the Floquet vector, Mied's work is generalized to its full complexity. In the limit of large disturbance wavenumbers, the unstable disturbances propagate in the directions of the two infinite curve segments of the related resonant-interaction diagram. They can therefore be classified into two families which are characterized by special propagation directions. At high wavenumbers the maximum growth rates converge to limits which do not depend on the direction of the Floquet vector. The limits are different for both families; the disturbance waves propagating at the smaller angle to the basic gravity wave grow at the larger rate.

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

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.

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.

Nonlinear aspects of an internal gravity wave co-existing with an unstable mode associated with a Helmholtz velocity profile

Recently Lindzen (1974) has proposed a model of a shear-layer instability which allows unstable modes to co-exist with radiating internal gravity waves. The model is an infinite, continuously stratified, Boussinesq fluid, with a simple jump discontinuity in the velocity profile. Linear stability theory shows that the model is stable for wavenumbers k such that k2 < N2/2U2, where N is the Brunt—Väisälä frequency and 2U is the change in velocity across the discontinuity. For N2/2U2 < k2 < N2/U2 an unstable mode may co-exist with an internal gravity wave. This paper examines the weakly nonlinear aspects of this model for wavenumbers k close to the critical wavenumber N/2½U. An equation governing the evolution of the amplitude of the interfacial displacement is derived. It is shown that the interface may support a stable finite amplitude internal gravity wave.