Small radial perturbations of magnetic polytropes

1997

The authors study the equilibrium of polytropes in a poloidal magnetic field. The magnetic field is the solution of a nonhomogeneous Euler equation for which the authors are able to give the general solution. It depends on the Lane-Emden function. The authors use the approximate analytic solution of the Lane-Emden equation proposed by Liu, to write it explicitly. For the

Astrophysics and Space Science, 1976

The frequencies of the linear and adiabatic oscillations of Prendergast's model are determined with the aid of a perturbation method. The influence of the magnetic field on the frequencies of the different types of spheroidal oscillation modes is discussed. H = KR2[2?(x)L x2 •lr --xl dxd-' 7 (1 -fl2)1/21 o d-fl ~ (1 -fl2)1/21~] 9 (1) 9

Monthly Notices of the Royal Astronomical Society, 2012

The linear stability of thin vertically isothermal density-stratified Keplerian discs in toroidally dominated magnetic fields is treated by asymptotic expansions in the small aspect ratio of the discs. The discs are found to be spectrally stable. The great variety of possible initial conditions leads to three regimes of non-exponential growth of perturbations, which are classified according to different relative levels of the in-plane and axial perturbed velocities. The first two regimes of instability are characterized by the decoupling of the magnetosonic (MS) and inertia-Coriolis (IC) modes, as well as by algebraic temporal growth of the perturbations, which are driven by either MS or IC modes (hereafter MS and IC regimes of instability, respectively). The third, mixed IC-MS regime of non-exponential, non-algebraic growth is due only to non-axisymmetric perturbations. The latter regime is characterized by high radial and azimuthal wavenumbers, and growth time of the order of tens of rotating periods. The mixed IC-MS regime most likely exhibits the maximal growth as compared with the IC and MS regimes. In the first two regimes of instability the compressible MS mode plays a principal role either as the driver of the growth or the driven growing mode, while the mixed IC-MS regime is described by the Boussinesq approximation for incompressible fluid. The latter is obtained as a natural limit of the expansion scheme. The presence of magnetic field in the mixed IC-MS regime may drastically increase the growth rates of the perturbations as compared with the pure hydrodynamic system.

2016

The linear stability of thin vertically-isothermal density-stratified Keplerian discs in toroidally-dominated magnetic fields is treated by asymptotic expansions in the small aspect ratio of the discs. The discs are found to be spectrally stable. The great variety of possible initial conditions leads to three regimes of non-exponential growth of perturbations, which are classified according to different relative levels of the in-plane and axial perturbed velocities. The first two regimes of instability are characterized by the decoupling of the magneto-sonic (MS) and inertia-Coriolis (IC) modes, as well as by algebraic temporal growth of the perturbations, which are driven by either MS or IC modes (hereafter MS-and IC-regimes of instability, respectively). The third, mixed IC-MS regime of non-exponential, non-algebraic growth is due only to nonaxisymmetric perturbations. The latter regime is characterized by high radial and azimuthal wavenumbers, and growth time of the order of tens of rotating periods. The mixed IC-MS regime most likely exhibits the maximal growth as compared with the IC-and MS-regimes. In the first two regimes of instability the compressible MS mode plays a principal role either as the driver of the growth or the driven growing mode, while the mixed IC-MS regime is described by the Bousinesq approximation for incompressible fluid. The latter is obtained as a natural limit of the expansion scheme. The presence of magnetic field in the mixed IC-MS regime may drastically increase the growth rates of the perturbations as compared with the pure hydrodynamic system.

Solar Physics, 2007

In this paper we study non-axisymmetric oscillations of thin twisted magnetic tubes taking the density variation along the tube into account. We use the approximation of the zero-beta plasma. The magnetic field outside the tube is straight and homogeneous, however it is twisted inside the tube. We assume that the azimuthal component of the magnetic field is proportional to the distance from the tube axis, and that the tube is only weakly twisted, i.e. the ratio of the azimuthal and axial components of the magnetic field is small. Using the asymptotic analysis we show that the eigenmodes and eigenfrequencies of the kink and fluting oscillations are described by a classical Sturm-Liouville problem for a second order ordinary differential equation. The main result is that the twist does not affect the kink mode.

2012

The linear instability of thin, vertically-isothermal Keplerian discs, under the influence of axial magnetic field is investigated. Solutions of the stability problem are found explicitly by asymptotic expansions in the small aspect ratio of the disc. It is shown that the perturbations are decoupled into in-plane and vertical modes. Exact expressions for the growth rates as well as the number of unstable modes are derived. Those are the discrete counterpart of the continuous infinite homogeneous cylinder magnetorotational (MRI) spectrum. In addition, a weakly nonlinear analysis of the MRI is performed. It is shown that near the instability threshold the latter is saturated by the stable magnetoacoustic modes.

We consider the flow of an electrically conducting fluid between differentially rotating cylinders, in the presence of an externally imposed toroidal field B 0 (r i /r)ê φ . It is known that the classical, axisymmetric magnetorotational instability does not exist for such a purely toroidal imposed field.

Astronomische Nachrichten, 2012

Stability of thin hot Keplerian discs is investigated asymptotically in small disc's aspect ratio, . The study is carried out in the local approximation for short vertical waves in the disc-thickness scale. Besides the radial rotation shear and the vertical magnetic field, the background configuration is characterized by a vertically near-constant temperature profile with a small vertical gradient. The temperature-gradient term in Ohm's law, which characterizes the thermomagnetic transport is found to be of the order of . The effect of the thermomagnetic transport slightly modifies the conventional magnetorotational instability (MRI), while a new thermomagnetic instability (TMI) emerges in regions of the wavenumber space where MRI is absent. Explicit solutions are obtained for a wide range of values of plasma beta, β, and thermomagnetic transport coefficient, λ. In particular, it is shown for λ 1 that the MRI dominates in weak magnetic fields, β 1, while the TMI is exhibited in strong magnetic fields, β ∼ 1, also with the growth rate of the order of inverse rotation period.

Journal of Fluid Mechanics, 2009

We address the question of stability of the Euler flow with elliptical streamlines in a rotating frame, interacting with uniform external magnetic field perpendicular to the plane of the flow. Our motivation for this study is of astrophysical nature, since many astrophysical objects, such as stars, planets and accretion discs, are tidally deformed through gravitational interaction with other bodies. Therefore, the ellipticity of the flow models the tidal deformations in the simplest way. The joint effect of the magnetic field and the Coriolis force is studied here numerically and analytically in the limit of small elliptical (tidal) deformations (ζ ≪ 1), using the analytical technique developed by Lebovitz & Zweibel (Astrophys. J., vol. 609, 2004, pp. 301-312). We find that the effect of background rotation and external magnetic field is quite complex. Both factors are responsible for new destabilizing resonances as the vortex departs from axial symmetry (ζ ≪ 1); however, just like in the non-rotating case, there are three principal resonances causing instability in the leading order. The presence of the magnetic field is very likely to destabilize the system with respect to perturbations propagating in the direction of the magnetic field if the basic vorticity and the background rotation have opposite signs (i.e. for anticyclonic background rotation). We present the dependence of the growth rates of the modes on various parameters describing the system, such as the strength of the magnetic field (h), the inverse of the Rossby number (R v ), the ellipticity of the basic flow (ǫ) and the direction of propagation of modes (ϑ). Our analytical predictions agree well with the numerical calculations.

Journal of Geophysical Research, 1993

An understanding of fully developed nonlinear MHD phenomena becomes increasingly essential for the interpretation of magnetospheric and interplanetary observations. In a series of four papers we shall study the nonlinear, self-similar evolution of a cylindrical magnetic flux tube with two components of the magnetic field, axial (B z) and azimuthal (B•). In this first paper we restrict ourselves to the case of a plasma of low beta. (Subsequent papers deal with finite beta effects, with dissipation, and with a data example.) Introducing a special class of configurations we call "separable fields," we reduce the problem to an ordinary differential equation. Two cases are to be distinguished: (1) when the total field minimizes on the symmetry axis, the magnetic configuration inexorably collapses, and (2) when, on the other hand, the total field maximizes on the symmetry axis, the magnetic configuration behaves analogously to a nonlinear oscillator. Here we focus on the latter case. The effective Eotential of the motion contains two terms: a strong, repulsive term associated with the gradient of B•/8,r and a weak, restoring term associated with the pinch. We solve the nonlinear differential equation of motion numerically and find that the period of oscillations grows exponentially with the energy of the oscillator. Our treatment emphasizes the role of the force-free configuration as the lowest potential energy state about which the system oscillates. Task 690-010. The Editor thanks B. C. Low and M.D. Moldwin for their assistance in evaluating this paper.