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Benkacem N, Salhi A, Khlifi A, Nasraoui S, Cambon C. Destabilizing resonances of precessing inertia-gravity waves. Phys Rev E 2022; 105:035107. [PMID: 35428057 DOI: 10.1103/physreve.105.035107] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2021] [Accepted: 02/15/2022] [Indexed: 06/14/2023]
Abstract
Instabilities in stratified precessing fluid are investigated. We extend the study by Mahalov [Phys. Fluids A 5, 891 (1993)0899-821310.1063/1.858635] in the stably stratified Boussinesq framework, with an external Coriolis force (with rate Ω_{p}) altering the base flow through the distortion of the circular streamlines of the unperturbed axially stratified rotating columns (with constant vorticity 2Ω.) It is shown that the inviscid part of the modified velocity flow (0,Ωr,-2ɛΩrsinφ) and buoyancy with gradient N^{2}(-2ɛcosφ,2ɛsinφ,1) are an exact solution of Boussinesq-Euler equations. Here (r,φ,z) is a cylindrical coordinate system, with ɛ=Ω_{p}/Ω being the Poincaré number and N the Brunt-Väisälä frequency. The base flow is transformed into a Cartesian coordinate system, and the stability of a superimposed perturbation is studied in terms of Fourier (or Kelvin) modes. The resulting Floquet system for the Fourier modes has three parameters: ɛ, N=N/Ω, and μ, which is the angle between the wave vector k and the solid-body rotation axis in the limit ɛ=0. In this limit, there are inertia-gravity waves propagating with frequency ±ω and the resonant cases are those for which 2ω=nΩ, n being an integer. We perform an asymptotic analysis to leading order in ɛ and characterize the destabilizing resonant case of order n=1 (i.e., the subharmonic instability) which exists and for 0≤N<Ω/2. In this range, the subharmonic instability remains the strongest with a maximal growth rate σ_{m}=[ɛ(5sqrt[15]/8)sqrt[1-4N^{2}]/(4-N^{2})]. Stable stratification acts in such a way as to make the subharmonic instability less efficient, so as it disappears for N≥0.5Ω. The destabilizing resonant cases of order n=2,3,4,5 are investigated in detail by numerical computations. The effect of viscosity on these instabilities is briefly addressed assuming the diffusive coefficients (kinematic and thermal) are equal. Likewise, we briefly investigate the case where N^{2}<0 and show that the instability associated to the mode with k_{3}=0 is the strongest.
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Affiliation(s)
- N Benkacem
- Université de Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire Matériaux Organisation et Propriétés, 2092 Tunis, Tunisie
| | - A Salhi
- Université de Lyon, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR 5509, Ecole Centrale de Lyon, CNRS, UCBL, INSA F-69134 Ecully Cedex, France
- Département de Physique, Faculté des Sciences de Tunis, 2092 Tunis, Tunisia
| | - A Khlifi
- Université de Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire Matériaux Organisation et Propriétés, 2092 Tunis, Tunisie
| | - S Nasraoui
- Université de Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire Matériaux Organisation et Propriétés, 2092 Tunis, Tunisie
- Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong 999077, China
| | - C Cambon
- Université de Lyon, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR 5509, Ecole Centrale de Lyon, CNRS, UCBL, INSA F-69134 Ecully Cedex, France
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