Convective stability in magnetic stars

Dynamical stability of a static axisymmetrical magnetic star with respect to high-order modes of oscillation is investigated by means of the energy method, neglecting the Eulerian perturbation of gravity. The magnetic field is assumed to be continuous across the surface of the star and its first-order spatial derivatives, but it may have both toroidal and poloidal components.The second variation of the potential energy is written in a way which, in the case of apurely toroidal field, and for axisymmetrical and non-axisymmetrical modes, yields Tayler's local stability criteria which are necessary and sufficient conditions for convective stability, and in the case of ageneral field yields a single local stability criterion, which is a sufficient condition for convective stability.     

Bounds on the ideal MHD stability of line-tied 2-D coronal magnetic fields

A tractable method for investigating the linear stability of line-tied 2-D coronal magnetic fields is introduced. It is based on the Bernstein et al. (1958) energy principle and can be applied to non-isothermal equilibria with gravity, having a translational invariance. The perturbed potential energy integral is manipulated to produce either necessary conditions for stability to localized modes or sufficient conditions for stability to global modes. Each condition only requires the solution of a set of ordinary differential equations, integrated along the magnetic field lines. The tests are employed to two different classes of equilibria. A linear force-free field is shown to be completely stable, regardless of the shear. The role of pressure gradients, footpoint displacements, line-tying and stratification on an isothermal magneto-hydrostatic equilibrium is assessed.     

Stability of line-tied 1-d coronal loops: Significance of an extended Suydam criterion

The stability of coronal magnetic loops is investigated with the influence of the dense photosphere (line-tying) included. The stability method, based on the Finite Fourier Series method developed by Einaudi and Van Hoven (1981, 1983), is applied to two different equilibria and the approximate critical conditions for the onset of different azimuthal instabilities are investigated. It is shown that, for nearly force-free loops, the extended Suydam criterion, obtained by De Bruyne and Hood (1989) for localized modes, predicts the existence of a global kink instability when a localized mode is just destabilized. For loops with substantial gas pressure gradients it is the localized modes that are destabilized first of all and the extended Suydam criterion gives the necessary and sufficient conditions for an instability. In this latter case, the instability threshold for the kink mode is quite close to the localized mode threshold. Finally, it is shown that the growth times of the instabilities are comparable to the Aflvén travel times along the loop when the extended Suydam criterion is violated.     

Simple tests for the ideal MHD stability of line-tied coronal magnetic fields

Methods for investigating the stability of line-tied, cylindrically-symmetric magnetic fields are presented. The energy method is used and the perturbed potential energy integral is manipulated to produce simple tests that predict either stability to general coronal disturbances or instability to localized modes, both satisfying photospheric line-tying. Using these tests the maximum amount of magnetic energy, that can be stored in the coronal magnetic field prior to an instability, can be estimated. The tests are applied to four different classes of equilibria and results are obtained for both arcade and loop geometries.     

The effect of a poloidal magnetic field on the linear and adiabatic oscillations of a polytropen=3

The frequencies of the linear and adiabatic oscillations of a gaseous polytrope with a poloidal magnetic field are determined with the aid of a perturbation method. The influence of the poloidal magnetic field on the different types of spheroidal oscillation modes is discussed. The poloidal magnetic field generally strengthens the stability of the oscillation modes and this effect is the largest in the case of the non-radialp-modes.     

球状星团中低光度红巨星的脉动稳定性

利用一种非定常的恒星非局部对流理论,对球状星团中低光度的红巨星进行了线性非绝热脉动理论计算．结果表明,对所有温度高于约5400 K模型的基音到4阶泛音都是脉动稳定的．随着恒星光度的增大,低阶泛音也变得脉动不稳定．对中低光度的红巨星,脉动稳定性非常低,接近中性稳定．因此他们将是不变星或非常小振幅的短周期变星(P<2天)．     

上部主序星非径向脉动不稳定模

根据非局部和非定常恒星对流理论,计算了质量3～30 M_⊙恒星演化模型低球谐阶(l=1～4)p4～g39非径向模的线性非绝热脉动,研究了对流对非径向脉动的影响.结果表明,对流对恒星非径向脉动稳定性有不可忽略的影响.然而假若不关注单个恒星和单个振动模的细节,从总体上粗略地看,理论的脉动不稳定区的大致位置和范围并未受到对流的明显影响.     

Stability of power-law discs — II. The global spiral modes

This paper reports on the in-plane normal modes in the self-consistent and the cut-out power-law discs. Although the cut-out discs are remarkably stable to bisymmetric perturbations, they are very susceptible to one-armed modes. For this harmonic, there is no inner Lindblad resonance, thus removing a powerful stabilizing influence. A physical mechanism for the generation of the one-armed instabilities is put forward. Incoming trailing waves are reflected as leading waves at the inner cut-out, thus completing the feedback for the swing-amplifier. Growing three-armed and four-armed modes occur only at very low temperatures. However, neutral m  = 3 and m  = 4 modes are possible at higher temperatures for some discs. The rotation curve index β has a marked effect on stability. For all azimuthal wavenumbers, any unstable modes persist to higher temperatures and grow more vigorously if the rotation curve is rising (β < 0) than if the rotation curve is falling (β > 0). If the central regions or outer parts of the disc are carved out more abruptly, any instabilities become more virulent. The self-consistent power-law discs possess a number of unusual stability properties. There is no natural time-scale in the self-consistent disc. If a mode is admitted at some pattern speed and growth rate, then it must be present at all pattern speeds and growth rates. Our analysis — although falling short of a complete proof — suggests that such a two-dimensional continuum of non-axisymmetric modes does not occur and that the self-consistent power-law discs admit no global non-axisymmetric normal modes whatsoever. Without reflecting boundaries or cut-outs, there is no resonant cavity and no possibility of unstable growing modes. The self-consistent power-law discs certainly admit equi-angular spirals as neutral modes, together with a one-dimensional continuum of growing axisymmetric modes.     

Attenuation of small-amplitude oscillations in a prominence–corona model with a transverse magnetic field

Observations show that small-amplitude prominence oscillations are usually damped after a few periods. This phenomenon has been theoretically investigated in terms of non-ideal magnetoacoustic waves, non-adiabatic effects being the best candidates to explain the damping in the case of slow modes. We study the attenuation of non-adiabatic magnetoacoustic waves in a slab prominence embedded in the coronal medium. We assume an equilibrium configuration with a transverse magnetic field to the slab axis and investigate wave damping by thermal conduction and radiative losses. The magnetohydrodynamic equations are considered in their linearised form and terms representing thermal conduction, radiation and heating are included in the energy equation. The differential equations that govern linear slow and fast modes are numerically solved to obtain the complex oscillatory frequency and the corresponding eigenfunctions. We find that coronal thermal conduction and radiative losses from the prominence plasma reveal as the most relevant damping mechanisms. Both mechanisms govern together the attenuation of hybrid modes, whereas prominence radiation is responsible for the damping of internal modes and coronal conduction essentially dominates the attenuation of external modes. In addition, the energy transfer between the prominence and the corona caused by thermal conduction has a noticeable effect on the wave stability, radiative losses from the prominence plasma being of paramount importance for the thermal stability of fast modes. We conclude that slow modes are efficiently damped, with damping times compatible with observations. On the contrary, fast modes are less attenuated by non-adiabatic effects and their damping times are several orders of magnitude larger than those observed. The presence of the corona causes a decrease of the damping times with respect to those of an isolated prominence slab, but its effect is still insufficient to obtain damping times of the order of the period in the case of fast modes.     

Unstable Non-radial Modes of Upper Main-sequence Stars†

Using a non-local time-dependent theory of convection, the linear stability is analyzed for low-degree (l = 1 − 4) g39-p4 modes of the evolutionary sequences of stars with masses from 3M_ to 30M_. The influence of convection on the pulsation stability of the non-radial modes is studied. The results show that this influence is not negligible. However, if the details of the single star and the single mode are not concerned, the location and scope of instability strip are not affected significantly by convection.     

Local stellar stability

The traditional linearized stellar-structure equations, which provide information on the global stability of a stellar equilibrium state, are shown to engender a class of continuous modes, if the eigenfunctions are sought in a generalized function space. To endow the intuitive concept of Local Stellar Statility with a precise mathematical meaning, we endeavour to link this notion to the family of continuous spectra. We indicate a simple numerical method to compute the eigenfunctions associated with these modes, and we illustrate our approach by briefly analysing the continuous spectra connected with the radial and nonradial dynamical stability problem.     

Unstable non-radial modes in radial pulsators: theory and an example

We study the possibility of the excitation of non-radial oscillations in classical pulsating stars. The stability of an RR Lyrae model is examined through non-adiabatic non-radial calculations. We also explore stability in the presence of non-linear coupling between radial and non-radial modes of nearly identical frequency.   In our model, a large number of unstable low-degree (ℓ = 1,2) modes have frequencies in the vicinity of unstable radial mode frequencies. The growth rates of such modes, however, are considerably smaller than those of the radial modes. We also recover an earlier result that at higher degrees (ℓ = 6–12) there are modes trapped in the envelope with growth rates similar to those of radial modes.   Subsequently, monomode radial pulsation of this model is considered. The destabilizing effect of the 1:1 resonance between the radial mode and nearby non-radial modes of low degrees is studied, with the assumption that the excited radial mode saturates the linear instability of all other modes. The instability depends on the radial mode amplitude, the frequency difference, the damping rate of the non-radial mode, and the strength of the non-linear coupling between the modes considered. At the pulsation amplitudes typical for RR Lyrae stars, the instability of the monomode radial pulsation and the concomitant resonant excitation of some non-radial oscillation modes is found to be very likely.     

The paradox of the scale-free discs

Scale-free discs have no preferred length or time-scale. The question has been raised whether such discs have a continuum of unstable linear modes or perhaps no unstable modes at all. We resolve this paradox by analysing the particular case of a gaseous, isentropic disc with a completely flat rotation curve (the Mestel disc) exactly . The heart of the matter is this: what are the correct boundary conditions to impose at the origin or central cusp? We argue that the linear stability problem is ill-posed and that similar ambiguities may afflict general disc models with power-law central cusps. From any finite radius, waves reach the origin after finite time but with logarithmically divergent phase. Instabilities exist, but their pattern speeds depend upon an undetermined phase with which waves are reflected from the origin. For any definite choice of this phase, there is an infinite but discrete set of growing modes. The ratio of growth rate to pattern speed is independent of the central phase. This ratio is derived in closed form for non-self-gravitating normal modes and is shown to agree with approximate results obtained from the shearing sheet in the short-wavelength limit. This provides the first exact, analytically solved stability analysis for a differentially rotating disc. For self-gravitating normal modes, the ratio of growth rate to pattern is found numerically by solving recurrence relations in Mellin-transform space.     

Bounds on the stability of 3D magnetic equilibria in the solar corona

A necessary and a sufficient condition are derived for the ideal magnetohydrodynamic stability of any 3D magnetohydrostatic equilibrium using the energy method and incorporating photospheric line-tying. The theory is demonstrated by application to a simple class of theoretical 3D equilibria. The main thrust of the method is the formulation of the stability conditions as two sets of ordinary differential equations together with appropriate boundary conditions which may be numerically integrated along tied field lines one at a time. In the case of the shearless fields with non-negligible plasma pressure treated here the conditions for stability arenecessary and sufficient. The method employs as a trial function a destabilizing ballooning mode, of large wave number vector perpendicular to the equilibrium field lines. These modes may not be picked up in a solution of the full partial differential equations which arise from a direct treatment of the problem.     

On the effect of turbulent anisotropy on pulsation stability of stars

Within the framework of a non-local time-dependent stellar convection theory, we study in detail the effect of turbulent anisotropy on stellar pulsation stability. The results show that anisotropy has no substantial influence on pulsation stability of g modes and low-order(radial order n_r 5) p modes.The effect of turbulent anisotropy increases as the radial order increases. When turbulent anisotropy is neglected, most high-order(n r 5) p modes of all low-temperature stars become unstable. Fortunately,within a wide range of the anisotropic parameter c_3, stellar pulsation stability is not sensitive to the specific value of c_3. Therefore it is safe to say that calibration errors of the convective parameter c_3 do not cause any uncertainty in the calculation of stellar pulsation stability.     

Trapped gravity waves and the five-minute oscillations of the solar atmosphere

The various modes of hydrodynamic waves are considered for a model of the solar atmosphere which is based on the Bilderberg model and includes the effects of ionization. The atmosphere forms a potential well for internal gravity waves, since the stability is low at the base (near the convection) and low again in the region of partial ionization in the chromosphere. Calculations show that there are two resonant (trapped) modes of internal gravity waves for horizontal wavelengths based on the scale of the granulation. The properties of these modes are in close agreement with the two modes of oscillation observed by Frazier (1968). Trapped acoustic modes are found to have periods too short to account for the observations.Presently Visiting Fellow, Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colo.     

Stability of a steady vertical flow in a viscous fluid

The one-dimensional non-linear equations for a viscous fluid with finite thermal conductivity are solved to get an exact solution for a steady vertical flow. The stability of such a steady flow is examined to find that the viscosity has a very pronounced stabilizing influence on convective and acoustic modes.     

Global shearing modes of galactic disks

A new scheme has been devised to calculate discrete unstable global shearing spiral modes of gaseous disk models of galaxies. The scheme makes use of the Legendre expansion of eigen functions and the problem of stability analysis is reduced to an eigenvalue problem of an infinite matrix. The spiral patterns of these shearing wave solutions of linearized equations change their form in the course of time due to the differential rotation of the equilibrium disk. These shearing wave solutions are presumed to have intermediate characteristics between so-called density-waves and material arms. Comparison between these shearing modes and the non-shearing normal modes for a series of disk models is presented.     

Surface trapping and leakage of low-frequency g modes in rotating early-type stars – II. Global analysis

A global analysis of the surface trapping of low-frequency non-radial g modes in rotating early-type stars is undertaken within the Cowling, adiabatic and traditional approximations. The dimensionless pulsation equations governing these modes are reviewed, and the boundary conditions necessary for solution of the equations are considered; in particular, an outer mechanical boundary condition, which does not enforce complete wave trapping at the stellar surface, is derived and discussed in detail. The pulsation equations are solved for a 7-M_⊙ model star over a range of rotation rates, using a numerical approach.
The results of the calculations confirm the findings of the preceding paper in the series: modes with eigenfrequencies below a cut-off cannot be fully trapped within the star, and exhibit leakage in the form of outwardly propagating waves at the surface. The damping rates resulting from leakage are calculated for such 'virtual' modes, and found to be appreciably larger than typical growth rates associated with opacity-driven pulsation. Furthermore, it is demonstrated that the surface perturbations generated by virtual modes are significantly changed from those caused by fully trapped modes; the latter result suggests differences in the line-profile variations exhibited by these two types of mode.
The findings are discussed in the context of the 53 Per, SPB and pulsating Be classes of variable star. Whilst wave leakage will probably not occur for overstable g modes in the 53 Per and slowly rotating SPB stars, the adoption of the new outer mechanical boundary condition may still affect the pulsational stability of these systems. Wave leakage for overstable modes remains a possibility in Be stars and the more rapidly rotating SPB stars.