Spectral stability of nonlinear gravity waves in the atmosphere

Abstract

We apply spectral stability theory to investigate nonlinear gravity waves in the atmosphere. These waves are determined by modulation equations that result from Wentzel-Kramers-Brillouin theory. First, we establish that plane waves, which represent exact solutions to the inviscid Boussinesq equations, are spectrally stable with respect to their nonlinear modulation equations under the same conditions as what is known as modulational stability from weakly nonlinear theory. In contrast to Boussinesq, the pseudoincompressible regime does account for the altitudinal varying background density. Second, we show for the first time that upward-traveling wave fronts solving the inviscid modulation equations, that compare to pseudoincompressible theory, are unconditionally unstable. Both inviscid regimes turn out to be ill-posed as the spectra allow for arbitrarily large instability growth rates. Third, a regularization is found by including dissipative effects. The corresponding traveling wave solutions have localized amplitude and blow up unconditionally by embedded eigenvalue instabilities but the instability growth rate is bounded from above. Additionally, all three types of nonlinear modulation equations are solved numerically to further investigate and illustrate the nature of the analytic stability results. * mark.schlutow@fu-berlin.de

Generalized modulation theory for strongly nonlinear gravity waves in a compressible atmosphere

“…An important effect of the weak nonlinearity is the occurrence of modulational instabilities: plane non-hydrostatic Boussinesq waves become modulationally unstable if the second derivative of the dispersion relation with respect to the vertical wavenumber becomes negative. It was shown in [17] that this holds true even for strongly nonlinear waves of the same kind. In the strongly nonlinear theory, Doppler shift and wave-mean-flow interaction appear to leading order such that the perturbation fields are of the same order of magnitude as the background.…”

“…A boundary condition close to the surface but sufficiently far away to be considered free-slip assuming Λ(0) = 0 can be determined by a periodic mountain ridge with period P and maximum mountain height Hm which we consider to be given constants hereinafter. In terms of the nonlinearity parameter (17) and the no-energy-flux condition (12), we obtain that kx = 2π/P and…”

“…Stream lines as fixed points of f for different q. Sinusoidal stream line of the linear regime where q ≪ 1 (blue line, see(17)). …”

Waves in the gas centrifuge: asymptotic theory and similarities with the atmosphere

“…It was shown in the pioneering work of Grimshaw (1972) that stationary plane gravity waves of large amplitudes destabilize due to modulation. Schlutow, Wahlén & Birken (2019) extended these ideas to classes of travelling wave solutions. Modulational instabilities of a primary wave may even cause the excitation of new, secondary waves, which was proved to be theoretically possible in Schlutow (2019).…”

On strongly nonlinear gravity waves in a vertically sheared atmosphere, Part I: Spectral stability of the refracted wave

“…In fact, the asymptotic expansion of Grimshaw's theory is such that the perturbation advection terms vanish due to the solenoidality of the wind field. Some theoretical investigations on strongly nonlinear effects and their implications with respect to observations and modeling were performed in [23,20].…”