scholarly journals Modal and Nonmodal Perturbations of Monochromatic High-Frequency Gravity Waves: Primary Nonlinear Dynamics

2007 ◽  
Vol 64 (6) ◽  
pp. 1977-1994 ◽  
Author(s):  
Ulrich Achatz
Keyword(s):  
Nonlinear Dynamics ◽  
Gravity Waves ◽  
High Frequency ◽  
Normal Modes ◽  
Wave Interactions ◽  
Resonant Wave ◽  
Spatial Dimensions ◽  
Phase Direction ◽  
Parallel Mode ◽  
Boussinesq Fluid

The primary nonlinear dynamics of high-frequency gravity waves (HGWs) perturbed by their most prominent normal modes (NMs) or singular vectors (SVs) in a rotating Boussinesq fluid have been studied by direct numerical simulations (DNSs), with wave scales and values of viscosity and diffusivity characteristic for the upper mesosphere. The DNS is 2.5D in that it has only two spatial dimensions, defined by the direction of propagation of the HGW and the direction of propagation of the perturbation in the plane orthogonal to the HGW phase direction, but describes a fully 3D velocity field. Many results of the more comprehensive fully 3D simulations in the literature are reproduced. So it is found that statically unstable HGWs are subject to wave breaking ending in a wave amplitude with respect to the overturning threshold near 0.3. It is shown that this is a result of a perturbation of the HGW by its leading transverse NM. For statically stable HGWs, a parallel NM has the strongest effect, quite in line with previous results on the predominantly 2D instability of such HGWs. This parallel mode is, however, not the leading NM but a larger-scale pattern, seemingly driven by resonant wave–wave interactions, leading eventually to energy transfer from the HGW into another gravity wave with steeper phase propagation. SVs turn out to be less effective in triggering HGW decay but they can produce turbulence of a strength that is (as that from the NMs) within the range of measured values, however with a more pronounced spatial confinement.

On the instability of an internal gravity wave

1977 ◽  
Vol 356 (1686) ◽  
pp. 411-432 ◽  
Keyword(s):  
Gravity Wave ◽  
Parametric Instability ◽  
Normal Modes ◽  
Internal Gravity Wave ◽  
Marginal Stability ◽  
Wave Interactions ◽  
Resonant Wave ◽  
Physical Conclusion ◽  
Arbitrary Amplitude ◽  
Boussinesq Fluid

The linear stability of an internal gravity wave of arbitrary amplitude in an unbounded stratified inviscid Boussinesq fluid is considered mathematically. The instability is shown to be governed by a Floquet system and treated by a generalization of the method of normal modes. Some properties of the Floquet system, and in particular those of its parametric instability, are analysed. The parametric instability is related to the theory of resonant wave interactions; and the surface of marginal stability in the control space of the amplitude and wavenumbers is shown to be describable by the catastrophe theory of Thom. Finally some results of numerical calculations of the marginal surface are shown. The main physical conclusion is to confirm that the internal gravity wave is unstable always, even when its amplitude is small and so its local Richardson number is large everywhere for all time. It is suggested, by various illustrations and arguments, that the methods developed in this paper are applicable to the instability of many symmetric nonlinear waves.

Resonant wave interactions in a stratified shear flow

1988 ◽  
Vol 190 ◽  
pp. 357-374 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Shear Flow ◽  
Gravity Waves ◽  
Critical Level ◽  
Internal Gravity Waves ◽  
Wave Interactions ◽  
Resonant Wave ◽  
Resonant Interactions ◽  
Stratified Shear Flow ◽  
Weak Resonance ◽  
Background Shear

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

scholarly journals Linear and Weakly Nonlinear Energetics of Global Nonhydrostatic Normal Modes

2019 ◽  
Vol 76 (12) ◽  
pp. 3831-3846 ◽  
Author(s):  
Carlos F. M. Raupp ◽  
André S. W. Teruya ◽  
Pedro L. Silva Dias
Keyword(s):  
Normal Modes ◽  
Physical Space ◽  
Wave Mode ◽  
Finite Amplitude ◽  
Vertical Acceleration ◽  
Wave Interactions ◽  
Weakly Nonlinear ◽  
Resonant Wave ◽  
Zonal Wavenumber ◽  
Resonant Triads

Abstract Here the theory of global nonhydrostatic normal modes has been further developed with the analysis of both linear and weakly nonlinear energetics of inertia–acoustic (IA) and inertia–gravity (IG) modes. These energetics are analyzed in the context of a shallow global nonhydrostatic model governing finite-amplitude perturbations around a resting, hydrostatic, and isothermal background state. For the linear case, the energy as a function of the zonal wavenumber of the IA and IG modes is analyzed, and the nonhydrostatic effect of vertical acceleration on the IG waves is highlighted. For the nonlinear energetics analysis, the reduced equations of a single resonant wave triad interaction are obtained by using a pseudoenergy orthogonality relation. Integration of the triad equations for a resonance involving a short harmonic of an IG wave, a planetary-scale IA mode, and a short IA wave mode shows that an IG mode can allow two IA modes to exchange energy in specific resonant triads. These wave interactions can yield significant modulations in the dynamical fields associated with the physical-space solution with periods varying from a daily time scale to almost a month long.

An experiment on third-order resonant wave interactions

1966 ◽  
Vol 25 (3) ◽  
pp. 417-435 ◽  
Author(s):  
M. S. Longuet-Higgins ◽  
N. D. Smith
Keyword(s):  
Gravity Waves ◽  
Response Curve ◽  
Resonant Interaction ◽  
Third Wave ◽  
Theoretical Discussion ◽  
Wave Interactions ◽  
Third Order ◽  
Large Wave ◽  
The Third ◽  
Resonant Wave

An experiment has been carried out to verify the existence of the resonant interaction between trains of gravity waves, predicted by Phillips (1960). As suggested by Longuet-Higgins (1962), two trains of waves in mutually perpendicular directions were generated in a rectangular wave tank. The ratio σ1/σ2of the wave frequencies was varied (1·4 < σ1/σ2< 2·1). When σ1/σ2[eDot ] 1·7357 it was expected that a resonant interaction would take place, generating a wave of frequency (2σ1−σ2). The amplitude of the third wave was expected to increase almost linearly in the direction of wave propagation. The shape of the response curve as a function of σ1/σ2was also predicted.In the present experiments rather large wave amplitudes had to be used, and the theoretical shape of the response curve was distorted by non-linear detuning. Nevertheless the peak amplitude of the resonant wave was found to increase with distance in very nearly the manner predicted.These experiments were carried out in 1961 but publication was deferred pending a similar but more accurate investigation by McGoldrick, Phillips, Huang & Hodgson (1966). Much of the theoretical discussion given in the present paper is relevant to their work.

Measurements of third-order resonant wave interactions

1966 ◽  
Vol 25 (3) ◽  
pp. 437-456 ◽  
Author(s):  
L. F. Mcgoldrick ◽  
O. M. Phillips ◽  
N. E. Huang ◽  
T. H. Hodgson
Keyword(s):  
Gravity Waves ◽  
Resonant Interaction ◽  
Experimental Demonstration ◽  
Wave Interactions ◽  
Third Order ◽  
The Third ◽  
Resonant Wave ◽  
Interaction Distance ◽  
At Resonance ◽  
Wave Trains

This paper presents the results of experiments on the resonant interaction of gravity waves. Two mutually-orthogonal primary wave trains are generated in a tank and their interaction products studied at various positions on the surface. Under suitable conditions, the growing resonant third-order interaction product is identified; its amplitude is shown to be a linear function of the interaction distance. The band-width of the response decreases with increasing distance, as is characteristic of the phenomenon of resonance. The ratio of the frequencies of the primary waves at resonance is very close to that predicted theoretically; the growth rate of the third component is close to, though about 20% higher than, the predicted value. Conditions far from resonance are also studied; it is found that the growing tertiary wave is absent in this case.These results offer the first unambiguous experimental demonstration of resonant wave interactions.

Theoretical and experimental studies of gravity wave interactions

1967 ◽  
Vol 299 (1456) ◽  
pp. 104-119 ◽  
Keyword(s):  
Gravity Waves ◽  
Experimental Studies ◽  
Internal Gravity Waves ◽  
Stokes Wave ◽  
Wave Interactions ◽  
Principal Characteristics ◽  
Resonant Wave ◽  
Resonant Interactions ◽  
Interaction Distance ◽  
Wave Modes

This paper is concerned with various aspects of the resonant interactions among waves. An experiment was suggested by Longuet-Higgins (1962) to detect this type of interaction among surface waves. This was subsequently performed by Longuet-Higgins & Smith (1966) and by McGoldrick, Phillips, Huang & Hodgson (1966). The results of the two sets of experiments are compared. Together they demonstrate very clearly the principal characteristics of the interaction; the maximum response at resonance and the linear growth with interaction distance, the decrease in band width with interaction distance and the shift of the resonance point that results from the amplitude dispersion. It is shown further that the instability of the Stokes wave, discovered and analysed by Benjamin & Feir, can be described in terms of these interactions and that it is not restricted to purely two dimensional motion. A Stokes wave is unstable to a disturbance containing a pair of wavenumbers defined by any point in the zone just inside the figure-of-eight loop shown in figure 12. Another example of resonant wave interactions is provided by short, internal gravity waves in a stratified fluid with constant Brunt-Väisälä frequency. The interactions among Fourier modes are considered, and it is shown that there arise both free and forced modes. In the latter, the dispersion relation for internal waves is not satisfied; there is no particular relation between wavenumber and frequency. The amplitudes of these are small compared with those of the internal wave modes provided the harmonic mean of the vorticity in the two interacting waves is small compared with the Brunt-Väisälä frequency. The motion then consists of interacting internal gravity waves, whose interaction sets are closed. On the other hand, if the forced components are comparable in magnitude with the wave modes, these interact strongly and indiscriminately; a ‘cascade’, characteristic of turbulence, develops.

An experiment on second-order capillary gravity resonant wave interactions

1970 ◽  
Vol 40 (2) ◽  
pp. 251-271 ◽  
Author(s):  
L. F. McGoldrick
Keyword(s):  
Spatial Variation ◽  
Viscous Dissipation ◽  
Gravity Waves ◽  
Initial Conditions ◽  
Resonant Interaction ◽  
Experimental Measurements ◽  
Wave Interactions ◽  
Resonant Wave ◽  
At Resonance ◽  
Wave Maker

This paper presents the results of a set of detailed experimental measurements on the resonant interaction of capillary-gravity waves for a case in which the entire propagation is in one direction. The influence of viscous attenuation is accounted for in the analysis. The measurements trace the entire spatial variation, or modulation envelope, of the amplitudes of the interacting modes from their inception near a wave-maker to their ultimate extinction through viscous dissipation, in excellent agreement with the theory. This is an unambiguous demonstration that at resonance and for the initial conditions specified at the wave-maker, a wave of uniform profile cannot exist.

The Primary Nonlinear Dynamics of Modal and Nonmodal Perturbations of Monochromatic Inertia–Gravity Waves

2007 ◽  
Vol 64 (1) ◽  
pp. 74-95 ◽  
Author(s):  
Ulrich Achatz
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Instability ◽  
Normal Modes ◽  
Large Degree ◽  
Small Scale ◽  
Turbulent Dissipation ◽  
Static And Dynamic Instability ◽  
Boussinesq Fluid ◽  
Secondary Instabilities ◽  
Elliptically Polarized

Abstract The breaking of an inertia–gravity wave (IGW), initiated by its leading normal modes (NMs) or singular vectors (SVs), and the resulting small-scale eddies are investigated by means of direct numerical simulations of a Boussinesq fluid characterizing the upper mesosphere. The focus is on the primary nonlinear dynamics, neglecting the effect of secondary instabilities. It is found that the structures with the strongest impact on the IGW and also the largest turbulence amplitudes are the NM (for a statically unstable IGW) or short-term SV (statically and dynamically stable IGW) propagating horizontally transversely with respect to the IGW, possibly in agreement with observations of airglow ripples in conjunction with statically unstable IGWs. In both cases these leading structures reduce the IGW amplitude well below the static and dynamic instability thresholds. The resulting turbulent dissipation rates are within the range of available estimates from rocket soundings, even for IGWs at amplitudes low enough precluding NM instabilities. Thus SVs can help explain turbulence occurring under conditions not amenable for the classic interpretation via static and dynamic instability. Because of the important role of the statically enhanced roll mechanism in the energy exchange between IGW and eddies, the turbulent velocity fields are often conspicuously anisotropic. The spatial turbulence distribution is determined to a large degree by the elliptically polarized horizontal velocity field of the IGW.

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