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.

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 parametric instabilities of finite-amplitude internal gravity waves

1982 ◽  
Vol 119 ◽  
pp. 367-377 ◽  
Author(s):  
J. Klostermeyer
Keyword(s):  
Gravity Wave ◽  
Internal Gravity Wave ◽  
Maximum Growth ◽  
Numerical Approach ◽  
Internal Gravity Waves ◽  
Resonant Interaction ◽  
Finite Amplitude ◽  
Interaction Diagram ◽  
Parametric Instabilities ◽  
Boussinesq Fluid

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.

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.

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

1976 ◽  
Vol 76 (1) ◽  
pp. 65-84 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Wave ◽  
Unstable Mode ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Linear Stability Theory ◽  
Jump Discontinuity ◽  
Weakly Nonlinear ◽  
Boussinesq Fluid

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.

Mean motions and impulse of a guided internal gravity wave packet

1973 ◽  
Vol 60 (4) ◽  
pp. 801-811 ◽  
Author(s):  
Michael E. Mcintyre
Keyword(s):  
Wave Packet ◽  
Gravity Wave ◽  
Electromagnetic Waves ◽  
Internal Gravity Wave ◽  
Single Frequency ◽  
Buoyancy Frequency ◽  
Mean Fields ◽  
The Mean ◽  
Boussinesq Fluid ◽  
The Waves

Second-order mean fields of motion and density are calculated for the two-dimensional problem of an internal gravity wave packet (the waves are predominantly of a single frequency o and wavenumberk) propagating as a wave-guide mode in an inviscid, diffusionless Boussinesq fluid of constant buoyancy frequencyN, confined between horizontal boundaries. (The same mathematical analysis applies to the formally identical problem for inertia waves in a homogeneous rotating fluid.)To leading order the mean motions turn out to be zero outside the wave packet, which consequently possesses a well-defined fluid impulse [Iscr ]. This is directed horizontally, and is given in magnitude and sense by\[ {\cal I} = \alpha{\cal M};\quad\alpha = \frac{2c_{\rm g}(c-c_{\rm g})(c+2c_{\rm g})}{c^3-4c^3_{\rm g}}. \]Here [Mscr ] is the so-called ‘wave momentum’, defined as wave energy divided by horizontal phase velocity c ≡ ω/k, andcg=c(N2–ω2)/N2, the group velocity.If the wave packet is supposed generated by a horizontally towed obstacle, [Mscr ] appears as the total fluid impulse, but of this a portion [Mscr ]-[Iscr ] in general propagates independently away from the wave packet in the form of long waves. When the wave packet itself is totally reflected by a vertical barrier immersed in the fluid, the time-integrated horizontal force on the barrier equals 2 [Iscr ] (and not 2 [Mscr ] as might have been expected from a naive analogy with the radiation pressure of electromagnetic 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.

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