Magnetized stratified rotating shear waves

Destabilizing resonances of precessing inertia-gravity waves

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.

Energy partition, scale by scale, in magnetic Archimedes Coriolis weak wave turbulence

Magnetic Archimedes Coriolis (MAC) waves are omnipresent in several geophysical and astrophysical flows such as the solar tachocline. In the present study, we use linear spectral theory (LST) and investigate the energy partition, scale by scale, in MAC weak wave turbulence for a Boussinesq fluid. At the scale k^{-1}, the maximal frequencies of magnetic (Alfvén) waves, gravity (Archimedes) waves, and inertial (Coriolis) waves are, respectively, V_{A}k,N, and f. By using the induction potential scalar, which is a Lagrangian invariant for a diffusionless Boussinesq fluid [Salhi et al., Phys. Rev. E 85, 026301 (2012)PLEEE81539-375510.1103/PhysRevE.85.026301], we derive a dispersion relation for the three-dimensional MAC waves, generalizing previous ones including that of f-plane MHD "shallow water" waves [Schecter et al., Astrophys. J. 551, L185 (2001)AJLEEY0004-637X10.1086/320027]. A solution for the Fourier amplitude of perturbation fields (velocity, magnetic field, and density) is derived analytically considering a diffusive fluid for which both the magnetic and thermal Prandtl numbers are one. The radial spectrum of kinetic, S_{κ}(k,t), magnetic, S_{m}(k,t), and potential, S_{p}(k,t), energies is determined considering initial isotropic conditions. For magnetic Coriolis (MC) weak wave turbulence, it is shown that, at large scales such that V_{A}k/f≪1, the Alfvén ratio S_{κ}(k,t)/S_{m}(k,t) behaves like k^{-2} if the rotation axis is aligned with the magnetic field, in agreement with previous direct numerical simulations [Favier et al., Geophys. Astrophys. Fluid Dyn. (2012)] and like k^{-1} if the rotation axis is perpendicular to the magnetic field. At small scales, such that V_{A}k/f≫1, there is an equipartition of energy between magnetic and kinetic components. For magnetic Archimedes weak wave turbulence, it is demonstrated that, at large scales, such that (V_{A}k/N≪1), there is an equipartition of energy between magnetic and potential components, while at small scales (V_{A}k/N≫1), the ratio S_{p}(k,t)/S_{κ}(k,t) behaves like k^{-1} and S_{κ}(k,t)/S_{m}(k,t)=1. Also, for MAC weak wave turbulence, it is shown that, at small scales (V_{A}k/sqrt[N^{2}+f^{2}]≫1), the ratio S_{p}(k,t)/S_{κ}(t) behaves like k^{-1} and S_{κ}(k,t)/S_{m}(k,t)=1.

Instability in stratified accretion flows under primary and secondary perturbations

We consider horizontal linear shear flow (shear rate denoted by Λ) under vertical uniform rotation (ambient rotation rate denoted by Ω(0)) and vertical stratification (buoyancy frequency denoted by N) in unbounded domain. We show that, under a primary vertical velocity perturbation and a radial density perturbation consisting of a one-dimensional standing wave with frequency N and amplitude proportional to w(0)sin(ɛNx/w(0))≈ɛNx(≪1), where x denotes the radial coordinate and ɛ a small parameter, a parametric instability can develop in the flow, provided N(2)>8Ω(0)(2Ω(0)-Λ). For astrophysical accretion flows and under the shearing sheet approximation, this implies N(2)>8Ω(0)(2)(2-q), where q=Λ/Ω(0) is the local shear gradient. In the case of a stratified constant angular momentum disk, q=2, there is a parametric instability with the maximal growth rate (σ(m)/ɛ)=3√[3]/16 for any positive value of the buoyancy frequency N. In contrast, for a stratified Keplerian disk, q=1.5, the parametric instability appears only for N>2Ω(0) with a maximal growth rate that depends on the ratio Ω(0)/N and approaches (3√[3]/16)ɛ for large values of N.

Nonmodal growth of the magnetorotational instability

We analyze the linear growth of the magnetorotational instability (MRI) in the short-time limit using nonmodal methods. Our findings are quite different from standard results, illustrating that shearing wave energy can grow at the maximum MRI rate -dΩ/dlnr for any choice of azimuthal and vertical wavelengths. In addition, by comparing the growth of shearing waves with static structures, we show that over short time scales shearing waves will always be dynamically more important than static structures in the ideal limit. By demonstrating that fast linear growth is possible at all wavelengths, these results suggest that nonmodal linear physics could play a fundamental role in MRI turbulence.

Instability of subharmonic resonances in magnetogravity shear waves

We study analytically the instability of the subharmonic resonances in magnetogravity waves excited by a (vertical) time-periodic shear for an inviscid and nondiffusive unbounded conducting fluid. Due to the fact that the magnetic potential induction is a Lagrangian invariant for magnetohydrodynamic Euler-Boussinesq equations, we show that plane-wave disturbances are governed by a four-dimensional Floquet system in which appears, among others, the parameter ɛ representing the ratio of the periodic shear amplitude to the vertical Brunt-Väisälä frequency N(3). For sufficiently small ɛ and when the magnetic field is horizontal, we perform an asymptotic analysis of the Floquet system following the method of Lebovitz and Zweibel [Astrophys. J. 609, 301 (2004)]. We determine the width and the maximal growth rate of the instability bands associated with subharmonic resonances. We show that the instability of subharmonic resonance occurring in gravity shear waves has a maximal growth rate of the form Δ(m)=(3√[3]/16)ɛ. This instability persists in the presence of magnetic fields, but its growth rate decreases as the magnetic strength increases. We also find a second instability involving a mixing of hydrodynamic and magnetic modes that occurs for all magnetic field strengths. We also elucidate the similarity between the effect of a vertical magnetic field and the effect of a vertical Coriolis force on the gravity shear waves considering axisymmetric disturbances. For both cases, plane waves are governed by a Hill equation, and, when ɛ is sufficiently small, the subharmonic instability band is determined by a Mathieu equation. We find that, when the Coriolis parameter (or the magnetic strength) exceeds N(3)/2, the instability of the subharmonic resonance vanishes.