Stability of rotating stratified shear flow: an analytical study

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 .

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.

Wave-vortex mode coupling in neutrally stable baroclinic flows

Rotating stratified flows in thermal wind balance are at the center of geophysical fluid dynamics. Recently, endeavors were put on studying the linear response of such flows to potential vorticity perturbations. It has been shown that the initial potential vorticity (PV) distribution is fundamental and is responsible for important transient growth of the perturbation and gravity-wave generation. Using Pfeiffer's theorem [J. Differ. Equat. 11, 145 (1972)], we give the mathematical demonstration of the stability of asymmetric perturbations k1≠0 of a uniform, unbounded flow in thermal wind balance. Incidentally, we prove that both the wave mode (that corresponds to a vanishing PV) and the vortex mode (corresponding to a nonzero PV) are stable. The emphasis is put on the nontrivial behavior of inertia-gravity waves (IGWs) when deformed by a background shear. In particular, we show that in the linear limit, sheared inertia-gravity waves asymptotically oscillate at the inertial waves frequency, but their amplitude is sensitive to shear, stratification, and rotation. Last, we study the development of the IGWs dynamics considering isotropic initial conditions. Computations indicate that both the vortex mode and the wave mode generate IGWs, but the energy of the IGWs generated by the vortex mode is more important than the energy of the IGWs generated by the wave mode. It is also found that, at large times, the energy of the IGWs generated by the vortex mode increases as the ratio kv/kh (initial vertical wavenumber over horizontal wavenumber) increases (like kv(2)/kh(2)), while the energy of the IGWs generated by the wave mode oscillates in function of kv/kh.

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.

Nonlinear dynamics and anisotropic structure of rotating sheared turbulence

Homogeneous turbulence in rotating shear flows is studied by means of pseudospectral direct numerical simulation and analytical spectral linear theory (SLT). The ratio of the Coriolis parameter to shear rate is varied over a wide range by changing the rotation strength, while a constant moderate shear rate is used to enable significant contributions to the nonlinear interscale energy transfer and to the nonlinear intercomponental redistribution terms. In the destabilized and neutral cases, in the sense of kinetic energy evolution, nonlinearity cannot saturate the growth of the largest scales. It permits the smallest scale to stabilize by a scale-by-scale quasibalance between the nonlinear energy transfer and the dissipation spectrum. In the stabilized cases, the role of rotation is mainly nonlinear, and interacting inertial waves can affect almost all scales as in purely rotating flows. In order to isolate the nonlinear effect of rotation, the two-dimensional manifold with vanishing spanwise wave number is revisited and both two-component spectra and single-point two-dimensional energy components exhibit an important effect of rotation, whereas the SLT as well as the purely two-dimensional nonlinear analysis are unaffected by rotation as stated by the Proudman theorem. The other two-dimensional manifold with vanishing streamwise wave number is analyzed with similar tools because it is essential for any shear flow. Finally, the spectral approach is used to disentangle, in an analytical way, the linear and nonlinear terms in the dynamical equations.

Magnetized stratified rotating shear waves

We present a spectral linear analysis in terms of advected Fourier modes to describe the behavior of a fluid submitted to four constraints: shear (with rate S), rotation (with angular velocity Ω), stratification, and magnetic field within the linear spectral theory or the shearing box model in astrophysics. As a consequence of the fact that the base flow must be a solution of the Euler-Boussinesq equations, only radial and/or vertical density gradients can be taken into account. Ertel's theorem no longer is valid to show the conservation of potential vorticity, in the presence of the Lorentz force, but a similar theorem can be applied to a potential magnetic induction: The scalar product of the density gradient by the magnetic field is a Lagrangian invariant for an inviscid and nondiffusive fluid. The linear system with a minimal number of solenoidal components, two for both velocity and magnetic disturbance fields, is eventually expressed as a four-component inhomogeneous linear differential system in which the buoyancy scalar is a combination of solenoidal components (variables) and the (constant) potential magnetic induction. We study the stability of such a system for both an infinite streamwise wavelength (k(1) = 0, axisymmetric disturbances) and a finite one (k(1) ≠ 0, nonaxisymmetric disturbances). In the former case (k(1) = 0), we recover and extend previous results characterizing the magnetorotational instability (MRI) for combined effects of radial and vertical magnetic fields and combined effects of radial and vertical density gradients. We derive an expression for the MRI growth rate in terms of the stratification strength, which indicates that purely radial stratification can inhibit the MRI instability, while purely vertical stratification cannot completely suppress the MRI instability. In the case of nonaxisymmetric disturbances (k(1) ≠ 0), we only consider the effect of vertical stratification, and we use Levinson's theorem to demonstrate the stability of the solution at infinite vertical wavelength (k(3) = 0): There is an oscillatory behavior for τ > 1+|K(2)/k(1)|, where τ = St is a dimensionless time and K(2) is the radial component of the wave vector at τ = 0. The model is suitable to describe instabilities leading to turbulence by the bypass mechanism that can be relevant for the analysis of magnetized stratified Keplerian disks with a purely azimuthal field. For initial isotropic conditions, the time evolution of the spectral density of total energy (kinetic + magnetic + potential) is considered. At k(3) = 0, the vertical motion is purely oscillatory, and the sum of the vertical (kinetic + magnetic) energy plus the potential energy does not evolve with time and remains equal to its initial value. The horizontal motion can induce a rapid transient growth provided K(2)/k(1)>>1. This rapid growth is due to the aperiodic velocity vortex mode that behaves like K(h)/k(h) where k(h)(τ)=[k(1)(2) + (K(2) - k(1)τ)(2)](1/2) and K(h) =k(h)(0). After the leading phase (τ > K(2)/k(1)>>1), the horizontal magnetic energy and the horizontal kinetic energy exhibit a similar (oscillatory) behavior yielding a high level of total energy. The contribution to energies coming from the modes k(1) = 0 and k(3) = 0 is addressed by investigating the one-dimensional spectra for an initial Gaussian dense spectrum. For a magnetized Keplerian disk with a purely vertical field, it is found that an important contribution to magnetic and kinetic energies comes from the region near k(1) = 0. The limit at k(1) = 0 of the streamwise one-dimensional spectra of energies, or equivalently, the streamwise two-dimensional (2D) energy, is then computed. The comparison of the ratios of these 2D quantities with their three-dimensional counterparts provided by previous direct numerical simulations shows a quantitative agreement.

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

Linear magnetohydrodynamic instabilities are studied analytically in the case of unbounded inviscid and electrically conducting flows that are submitted to both rotation and precession with shear in an external magnetic field. For given rotation and precession the possible configurations of the shear and of the magnetic field and their interplay are imposed by the "admissibility" condition (i.e., the base flow must be a solution of the magnetohydrodynamic Euler equations): we show that an "admissible" basic magnetic field must align with the basic absolute vorticity. For these flows with elliptical streamlines due to precession we undertake an analytical stability analysis for the corresponding Floquet system, by using an asymptotic expansion into the small parameter ε (ratio of precession to rotation frequencies) by a method first developed in the magnetoelliptical instabilities study by Lebovitz and Zweibel [Astrophys. J. 609, 301 (2004)]10.1086/420972. The present stability analysis is performed into a suitable frame that is obtained by a systematic change of variables guided by symmetry and the existence of invariants of motion. The obtained Floquet system depends on three parameters: ε , η (ratio of the cyclotron frequency to the rotation frequency) and χ=cos α, with α being a characteristic angle which, for circular streamlines, ε=0, identifies with the angle between the wave vector and the axis of the solid body rotation. We look at the various (centrifugal or precessional) resonant couplings between the three present modes: hydrodynamical (inertial), magnetic (Alfvén), and mixed (magnetoinertial) modes by computing analytically to leading order in ε the instabilities by estimating their threshold, growth rate, and maximum growth rate and their bandwidths as functions of ε, η, and χ. We show that the subharmonic "magnetic" mode appears only for η>square root of 5/2 and at large η (>>1) the maximal growth rate of both the "hydrodynamic" and magnetic modes approaches ε/2, while the one of the subharmonic "mixed" mode approaches zero.