Magneto-gravity-elliptic instability. (English) Zbl 1520.76030
Summary: Magneto-gravity-elliptic instability is addressed here considering an unbounded strained vortex (with constant vorticity \(2\varOmega\) and with ellipticity parameter \(\varepsilon\)) of a perfectly conducting fluid subjected to a uniform axial magnetic field (with Alfvén velocity scaled from the basic magnetic field \(B)\) and an axial stratification (with a constant Brunt-Väisälä frequency \(N\)). Such a simple model allows us to formulate the stability problem as a system of equations for disturbances in terms of Lagrangian Fourier (or Kelvin) modes which is universal for wavelengths of the perturbation sufficiently small with respect to the scale of variation of the basic velocity gradients. It can model localised patches of elliptic streamlines which often appear in some astrophysical flows (stars, planets and accretion discs) that are tidally deformed through gravitational interaction with other bodies. In the limit case where the flow streamlines are exactly circular (\(\varepsilon =0\)), there are fast and slow magneto-inertia-gravity waves with frequencies \(\omega_{1, 2}\) and \(\omega_{3, 4}\), respectively. Under the effect of finite ellipticity, the resonant cases of these waves, \(\omega_i-\omega_j = n\varOmega\) (\(i \neq j\)) (\(n\) being an integer), can become destabilising. The maximal growth rate of the subharmonic instability (related to the resonance of order \(n = 2\)) is determined by extending the asymptotic method by
N. R. Lebovitz and E. Zweibel [Astrophys. J. 609, 301–312 (2004; doi:10.1086/420972)]. The domains of the \((k_0B/\varOmega, N/\varOmega)\) plane for which this instability operates are identified (\(1/k_0\) being a characteristic length scale). We demonstrate that the \(N\rightarrow 0\) limit is, in fact, singular (discontinuous). The axial stable stratification enhances the subharmonic instability related to the resonance between two slow modes because, at large magnetic field strengths, its maximal growth rate is twice that found in the case without stratification.
MSC:
| 76E25 | Stability and instability of magnetohydrodynamic and electrohydrodynamic flows |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 76M45 | Asymptotic methods, singular perturbations applied to problems in fluid mechanics |
Keywords:
Boussinesq magneto-hydrodynamic equations; parametric instability; subharmonic instability; resonance; Floquet-Lyapunov theorem; asymptotic method, axial stable stratification; constant vorticityReferences:
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