Atmospheric gravity wave instability? (original) (raw)
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European Journal of Applied Mathematics, 2005
The wave equation (u t + uu x) x − u = 0 is a model for shallow water waves with Coriolis force, sound waves in a bubbly liquid and more generally "is the canonical asymptotic equation for genuinely nonlinear waves that are nondispersive as their wavelength tends to zero" in the words of Hunter[12], who has previously studied it. This "Hunter's" equation has steadily-translating, spatially periodic solutions which exist only when c ≤ c limit. The limiting wave ("parabolic wave") is exactly given by piecewise quadratic polynomials in x with a discontinuous slope at the crest. We show that near the limit, the travelling waves ("paraboloidal waves") can be approximated by matched asymptotic expansions: the inner solution rounds off the point while the outer solution, valid over most of the spatial domain, is to lowest order just the parabolic wave. In the opposite limit of small amplitude, we derive a Fourier-and-powers-ofamplitude ("Stokes' series"). We show that this is remarkably accurate even very close to the limiting wave and converges to the limiting wave for unit amplitude. We demonstrate also that the Fourier pseudospectral method gives first order convergence even for the slope-discontinuous parabolic wave.
Journal of the Mechanics and Physics of Solids, 2020
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On some localized waves described by the extended KdV equation
Comptes Rendus …, 2005
The influence of higher-order nonlinear terms on the shape of solitary waves is studied for mechanical systems governed by a generalization of the 5th order Korteweg-de Vries equation. New localized travelling wave with intrinsic oscillations (not breathers) is shown to arise from arbitrary initial pulse thanks only to the higher-order quadratic nonlinearity, while cubic nonlinearity is responsible for the formation of so-called 'fat' solitary wave. To cite this article: A.V. Porubov et al., C. R. Mecanique 333 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.
Comparisons between model equations for long waves
Journal of Nonlinear Science, 1991
Considered here are model equations for weakly nonlinear and dispersive long waves, which feature general forms of dispersion and pure power nonlinearity. Two variants of such equations are introduced, one of Korteweg-de Vries type and one of regularized long-wave type. It is proven that solutions of the pure initial-value problem for these two types of model equations are the same, to within the order of accuracy attributable to either, on the long time scale during which nonlinear and dispersive effects may accumulate to make an order-one relative difference to the wave profiles. Key words, long wave models, Korteweg-de Vries-type equations, regularized longwave equations, dispersion relations Earlier Theory and Rationale To understand more precisely the results in view, and to grasp their importance, it is worthwhile to briefly review the theory developed earlier for the special case of the Korteweg-de Vries equation itself in Bona et al. (1983). Among the many assumptions that come to the fore in deriving models like those in (1.1) and (1.2) are that the wave motions in question have small amplitude and large wavelength. Letting a and A, respectively, denote typical, scaled, nondimensionalized values of these quantities, the assumption is that both a and A-l are small. However, in order that nonlinear and dispersive effects be balanced, these two small quantities must be related. In the case of the Korteweg-de Vries equation where p = 1 and re(k) = k 2, so that (1.1) and (1.2) take the forms ~Tt + rlx + rlrlx + rlxxx = 0 (1.4)
An empirical modification to linear wave theory
An empirical modification to the linear theory equation for the celerity of gravity waves is presented. The modified equation reduces to the generally accepted expression for solitary waves as shallow water conditions are reached but retains its usual form for deep water. In the region where cnoidal theory is most valid, the modified equation yields celerities in reasonable agreement with those predicted for cnoidal waves.
Acoustic-gravity waves in the atmosphere: from Zakharov equations to wave-kinetics
Physica Scripta
We develop a wave-kinetic description of acoustic-gravity (AG) waves in the atmosphere. In our paper the high frequency spectrum of waves is described as a gas of quasi-particles. Starting from the Zakharov-type of equations, where coupling between fast and slow density perturbations is considered, we derive the corresponding wave-kinetic equations, written in terms of an appropriate Wigner function. This provides an alternative description for the nonlinear interaction between the two dispersion branches of the AG waves.
Nonlinear Rayleigh-Taylor instabilities, atmospheric gravity waves and equatorial spread F
Journal of Geophysical Research, 1993
Although it is generally accepted that equatorial spread F (ESF) is due to nonlinear evolution of the Rayleigh-Taylor instability, a number of important properties of the process remain unexplained. In particular, we investigate two as yet unexplained features of ESF: the common dominance of very large scale features (> 20 kin) and their large amplitude. Although associated for years with spread F we show here for the first time that atmospheric gravity waves can initiate the Rayleigh-Taylor instability. In agreement with other analytical theories and computer simulations we find that the initiated instability will be saturated by nonlinear coupling of unstable modes to damped modes if the amplitude of the seed gravity wave is small. However, if the amplitude of the seed gravity wave is large enough, the relative plasma density perturbations can reach 50% or more, implying essentially no saturation. The required initial amplitudes are not unreasonable. In the latter case, significant enhancements and depletions of the plasma density occur within several hundred seconds. The analysis presented here demonstrates the possibility that large-scale spread F is triggered by gravity waves with the Rayleigh-Taylor instability as a soume of amplification. In a separate calculation we also show that large-amplitude plasma perturbations can be produced by an explosive instability of the Rayleigh-Taylor modes. It is found that the condition for explosive growth of the Rayleigh-Taylor modes can be satisfied in the ionospheric F region. We propose that the explosive mode coupling of the Rayleigh-Taylor instability is a possible mechanism for production of large-amplitude bottomside sinusoidal structures. 1. INTRODUCTION Equatorial spread F irregularities exhibit a broad range of scale sizes from global scales down to wavelengths as short as 10 cm. As suggested by Kelley [1985], the wave number spectrum can be divided into four distinct wavelength regimes: long (>20 km), intermediate (20 km-100 m), transitional (100-10 m) and short (<10 m). The driving mechanism of the intermediate scale irregularities is thought to be the generalized Rayleigh-Taylor instability. The smaller scale disturbances are either the drift modes which are produced in the steep density gradient regions set up by the Rayleigh-Taylor instability [Ossakow, 1981] or damped modes coupled to large-scale through a three-wave process [Hysell, 1992]. The sources of intermediate and smaller structures in equatorial spread F are thus fairly well understood. However, when viewed from the perspective of a satellite sensor at F peak altitudes, the peak in the fluctuation strength lies well into the long wavelength portion of the spectrum (•, >20 km) where the linear growth rate of the Rayleigh-Taylor instability is small [Kelley and Hysell, 1991]. One purpose of this paper is to explain this long-standing discrepancy between experiment and theory. A second purpose of the paper is to explain the large amplitude of the bottomside sinusoidal structures [Valladares et al., 1983]. An example showing how bottomside undulations break into full scale ESF is presented in Figure 1. The plot is a backscatter map made with the Altair radar on the Island of Kwajalein while scanning in an east-west plane. The radar line of sight is nearly perpendicular to the magnetic field but far enough off-perpendicular that coherent echoes are suppressed. IOn leave from Wuhan Institute of Physics, the Chinese Academy of Sciences, Wuhan.
Instability of variable media to long waves with odd dispersion relations
Communications in Mathematical Sciences, 2006
The instability of variable media to a broad class of long waves having dispersion relations that are an odd function of wavenumber is examined. For Hamiltonian media, new necessary conditions for the existence and structure of global modes are obtained. For non-Hamiltonian media, an analysis of the complex WKB branch points yields explicit expressions for the frequency and structure of the global modes, which manifest as spatially oscillatory wave packets or smooth envelope structures. These distinct modes and their locations within the media can be predicted by simply examining the local convergence or divergence of the group velocity in the long wave limit.