Magnetohydrodynamic instabilities in rotating and precessing sheared flows: an asymptotic analysis

Nonlinear Effects on the Precessional Instability in Magnetized Turbulence

By means of direct numerical simulations (DNS), we study the impact of an imposed uniform magnetic field on precessing magnetohydrodynamic homogeneous turbulence with a unit magnetic Prandtl number. The base flow which can trigger the precessional instability consists of the superposition of a solid-body rotation around the vertical ( x 3 ) axis (with rate Ω ) and a plane shear (with rate S = 2 ε Ω ) viewed in a frame rotating (with rate Ω p = ε Ω ) about an axis normal to the plane of shear and to the solid-body rotation axis and under an imposed magnetic field that aligns with the solid-body rotation axis ( B ‖ Ω ) . While rotation rate and Poincaré number are fixed, Ω = 20 and ε = 0.17 , the B intensity was varied, B = 0.1 , 0.5 , and 2.5 , so that the Elsasser number is about Λ = 0.1 , 2.5 and 62.5 , respectively. At the final computational dimensionless time, S t = 2 ε Ω t = 67 , the Rossby number Ro is about 0.1 characterizing rapidly rotating flow. It is shown that the total (kinetic + magnetic) energy ( E ) , production rate ( P ) due the basic flow and dissipation rate ( D ) occur in two main phases associated with different flow topologies: (i) an exponential growth and (ii) nonlinear saturation during which these global quantities remain almost time independent with P ∼ D . The impact of a "strong" imposed magnetic field ( B = 2.5 ) on large scale structures at the saturation stage is reflected by the formation of structures that look like filaments and there is no dominance of horizontal motion over the vertical (along the solid-rotation axis) one. The comparison between the spectra of kinetic energy E ( κ ) ( k ⊥ ) , E ( κ ) ( k ⊥ , k ‖ = 1 , 2 ) and E κ ) ( k ⊥ , k ‖ = 0 ) at the saturation stage reveals that, at large horizontal scales, the major contribution to E ( κ ) ( k ⊥ ) does not come only from the mode k ‖ = 0 but also from the k ‖ = 1 mode which is the most energetic. Only at very large horizontal scales at which E ( κ ) ( k ⊥ ) ∼ E 2 D ( κ ) ( k ⊥ ) , the flow is almost two-dimensional. In the wavenumbers range 10 ≤ k ⊥ ≤ 40 , the spectra E ( κ ) ( k ⊥ ) and E ( κ ) ( k ⊥ , k ‖ = 0 ) respectively follow the scaling k ⊥ − 2 and k ⊥ − 3 . Unlike the velocity field the magnetic field remains strongly three-dimensional for all scales since E 2 D ( m ) ( k ⊥ ) ≪ E ( m ) ( k ⊥ ) . At the saturation stage, the Alfvén ratio between kinetic and magnetic energies behaves like k ‖ − 2 for B k ‖ / ( 2 ε Ω ) < 1 .

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.

Elliptical flow instability in a conducting fluid triggered by an external magnetic field

We demonstrate that arbitrarily weak magnetic field may cause violent instability of an anticyclonic, recirculating flow with uniform mean angular velocity. This magnetohydrodynamic instability would trigger turbulence in the cores of vortices where neither centrifugal, exchange instability, nor magnetorotational instability is effective. In the accretion disk vortices this can be an important mechanism of enhanced outward transport of angular momentum.