Instabilities of finite-width internal wave beams: from Floquet analysis to PSI

• Published in: Journal of fluid mechanics Vol. 913
• Main Authors: Fan, Boyu, Akylas, T.R.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 19.02.2021
• Subjects: Advection; Amplitude; Amplitudes; Broadband; Components; Computational fluid dynamics; Eigenvalues; Fluid mechanics; Fluids; Frequency spectra; Frequency spectrum; Frequency stability; Gravity waves; Growth rate; Instability; Internal waves; JFM Papers; Laboratories; Perturbation; Photosystem I; Reynolds number; Stability; Stability analysis; Velocity; Viscosity; Wavelengths; Width; internal waves
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2020.1172

The parametric subharmonic instability (PSI) of finite-width internal gravity wave beams is revisited using a formal linear stability analysis based on Floquet theory. The Floquet stability eigenvalue problem is studied asymptotically in the limit where PSI arises, namely for a small-amplitude beam of frequency $\omega$ subject to fine-scale perturbations under nearly inviscid conditions. It is found that, apart from the two dominant subharmonic perturbation components with frequency $\omega /2$, PSI also involves two smaller components with frequency $3\omega /2$, which affect the instability growth rate and were ignored in the earlier models for PSI by Karimi & Akylas (J. Fluid Mech., vol. 757, 2014, pp. 381–402) and Karimi & Akylas (Phys. Rev. Fluids, vol. 2, 2017, 074801). After accounting for these components, the revised PSI models are in excellent agreement with numerical solutions of the Floquet eigenvalue problem. The Floquet stability analysis also reveals that PSI is restricted to a finite range of perturbation wavenumbers: as the perturbation wavenumber is increased (for fixed beam amplitude), higher-frequency components eventually come into play due to the advection of the perturbation by the underlying wave beam, so the components at $\omega /2$ no longer dominate. By adopting a frame riding with the wave beam, this advection effect is factored out and it is shown that small-amplitude beams that are not generally susceptible to PSI may develop an essentially inviscid instability with broadband frequency spectrum.