Axisymmetric Stability Research Articles

Testing the locality of transport in self-gravitating accretion discs

In this paper, we examine the issue of characterizing the transport associated with gravitational instabilities in relatively cold discs, discussing in particular the conditions under which it can be described within a local, viscous framework. We present the results of global, three-dimensional, smoothed particle hydrodynamics simulations of self-gravitating accretion discs, in which the disc is cooled using a simple parametrization for the cooling function. Our simulations show that the disc settles in a ‘self-regulated’ state, where the axisymmetric stability parameter Q≈ 1 and where transport and energy dissipation are dominated by self-gravity. We have computed the gravitational stress tensor and compared our results with expectations based on a local theory of transport. We find that, as long as the disc mass is smaller than 0.25M★ and the aspect ratio H/R≲ 0.1, transport is determined locally, thus allowing for a viscous treatment of the disc evolution.

Local Axisymmetric Diffusive Stability of Weakly Magnetized, Differentially Rotating, Stratified Fluids

We study the local stability of stratified, differentially rotating fluids to axisymmetric perturbations in the presence of a weak magnetic field and of finite resistivity, viscosity, and heat conductivity. This is a generalization of the Goldreich-Schubert-Fricke (GSF) double-diffusive analysis to the magnetized and resistive, triple-diffusive case. Our fifth-order dispersion relation admits a novel branch that describes a magnetized version of multidiffusive modes. We derive necessary conditions for axisymmetric stability in the inviscid and perfect-conductor (double-diffusive) limits. In each case, rotation must be constant on cylinders and angular velocity must not decrease with distance from the rotation axis for stability, irrespective of the relative strength of viscous, resistive, and heat diffusion. Therefore, in both double-diffusive limits, solid-body rotation marginally satisfies our stability criteria. The role of weak magnetic fields is essential to reach these conclusions. The triple-diffusive situation is more complex, and its stability criteria are not easily stated. Numerical analysis of our general dispersion relation confirms our analytic double-diffusive criteria but also shows that an unstable double-diffusive situation can be significantly stabilized by the addition of a third, ostensibly weaker, diffusion process. We describe a numerical application to the Sun's upper radiative zone and establish that it would be subject to unstable multidiffusive modes if moderate or strong radial gradients of angular velocity were present.

Kinematic, stress, and hardening analysis in finite double slip

A kinematic, stress, and hardening analysis of finite double slip in fcc crystals under axial loading is presented. The relative amounts of slip in classic 1925 experiments by Taylor and Elam (determined analytically in [Int. J. Plasticity 9 (1993) 159–179] by comparing theoretical and experimental cones of unextended directions), together with load-extension data and other measurements, are used to calculate resolved shear stress vs. slip curves and assess predictions of several finite distortional hardening theories. In particular, a new hardening rule is introduced that gives very close agreement with the anisotropic experimental results in double slip yet is consistent with the axisymmetric deformation, lattice stability, and isotropic hardening that are characteristic of fcc and bcc crystals in high symmetry axial-load orientations involving 6- and 8-fold slip.

Perturbed solutions of fixed boundary MHD equilibria

In this study, the fixed boundary plasma MHD equilibrium problem is solved by the finite element method; then, by perturbing the flux at the plasma boundary nodes, linear formulae are derived linking the variation of several plasma parameters of interest to the variation of the currents flowing in the external circuits. On the basis of these formulae it is shown how it is possible to efficiently solve two central problems in plasma engineering, namely (1) the optimization of the currents in a given set of coils necessary to maintain a specified equilibrium configuration and (2) the derivation of a linear dynamic model describing the plasma axisymmetric displacement (n = 0 mode) about a given magnetic configuration. A case study—based on the ITER reference equilibrium magnetic configuration at burn—is analysed both in terms of equilibrium currents optimality as well as axisymmetric stability features. The results obtained by these formulae are also compared with the predictions of a non-linear free boundary code and of a linear, dynamic model. As shown, the formulae derived here are in good agreement with such predictions, confirming the validity of the present approach.

Axisymmetric stability criterion for two gravitationally coupled singular isothermal discs

Using the two-fluid formalism with the polytropic approximation, we examine the axisymmetric stability criterion for a composite system of gravitationally coupled stellar and gaseous singular isothermal discs (SIDs). Both SIDs are taken to be infinitely thin and are in a self-consistent steady background rotational equilibrium with power-law surface mass densities ( r - 1 ) and flat rotation curves. Recently, Lou & Shen derived exact solutions for both axisymmetric and non-axisymmetric stationary perturbations in such a composite SID system and proposed the D s criterion of stability for axisymmetric perturbations. Here, for axisymmetric perturbations, we derive and analyse the time-dependent Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) dispersion relation to study stability properties. By introducing a dimensionless stellar SID rotation parameter D s , defined as the ratio of the constant stellar rotation speed V s to the constant stellar velocity dispersion a s , one can readily determine the axisymmetric stability D s criterion numerically by identifying a stable range of D s . Those systems which rotate too slowly (collapse) or too rapidly (ring fragmentation) are unstable. We found that the stable range of D 2 s depends on the mass ratio δ of the gaseous SID to the stellar SID and on the square of the ratio β of the stellar velocity dispersion (which mimics the sound speed) to the gaseous isothermal sound speed. An increment of either δ or β or both will diminish the stable range of D 2 s. The WKBJ results of instabilities provide physical explanations for the stationary configurations derived by Lou & Shen. It is feasible to introduce an effective Q parameter for a composite SID system. The closely relevant theoretical studies of Elmegreen, Jog and Shu et al. are discussed. A study of composite partial SID system reveals that an axisymmetric dark matter halo will promote the stability of the composite SID system against axisymmetric disturbances. Potential applications to disc galaxies, circumnuclear discs around nuclei of galaxies and protostellar discs are briefly discussed.

Black Hole Accretion Disks on the Edge

The local axisymmetric stability of hydrodynamical and magnetized, nearly-Keplerian gaseous accretion disks around non-rotating black holes is examined in the vicinity of the classical marginally-stable orbit (at radii ~ R_ms). An approximate Paczynski-Wiita pseudo-Newtonian potential is used. Hydrodynamical disks are linearly unstable inside a radius which differs slightly from the classical R_ms value because of finite pressure and radial stratification effects. Linear stresses associated with unstable hydrodynamical modes vanish exactly at the radius of marginal stability and are generally positive inside of that radius. When a magnetic field is introduced, however, the concept of radius of marginal stability becomes largely irrelevant because there are linearly unstable magneto-rotational modes everywhere. Associated linear stresses are positive and continuous across the region of hydrodynamical marginal stability, even for large-scale "hydro-like" modes subject only to weak magnetic tension. This conclusion is valid for arbitrarily thin disks (in ideal MHD) and it does not require a large-scale "radially-connecting" magnetic field. Results on hydrodynamical diskoseismic modes trapped in deep relativistic potential wells should be revised to account for the short-wavelength Alfven-like behavior of inertio-gravity waves in magnetized disks.

Interchange Method in Compressible Magnetized Couette Flow: Magnetorotational and Magnetoconvective Instabilities

We obtain the general forms of the axisymmetric stability criteria in a magnetized compressible Couette flow using an energy variational principle, the so-called interchange or Chandrasekhar s met hod, which we applied successfully in the incompressible case. This formulation accounts for the simultaneous presence of gravity, rotation, a toroidal magnetic field, a weak axial magnetic field, entropy gradients, and density gradients in the initial equilibrium state. The power of the method lies in its simplicity which allows us to derive extremely compact and physically clear expressions for the relevant stability criteria despite the inclusion of so many physical effects. In the implementation of the method, all the applicable conservation laws are explicitly taken into account during the variations of a quantity with dimensions of energy which we call the free energy function. As in the incompressible case, the presence of an axial field invalidates the conservation laws of angular momentum and azimuthal magnetic flux and introduces instead isorotation and axial current conservation along field lines. Our results are therefore markedly different depending on whether an axial magnetic field is present, and generalize in two simple expressions all previously known, partial stability criteria for the appearance of magnetorotational instability. Furthermore, the coupling between magnetic tension and buoyancy and its influence to the dynamics of nonhomoentropic magnetized flows becomes quite clear from our results. In the limits of plane-parallel atmospheres and homoentropic flows, our formulation easily recovers the stability criteria for suppression of convective and Parker instabilities, as well as some related special cases studied over 40 years ago by Newcomb and Tserkovnikov via laborious variational techniques.

Axisymmetric dynamic stability of sandwich circular plates

The axisymmetric dynamic stability of sandwich circular plates with a constrained damping layer treatment subjected to a periodic uniform radial loading along the outer edge of the host plate is studied by the discrete layer annular finite element method. The viscoelastic material with frequency dependent properties in the damping layer is assumed to be incompressible, and the extensional and shear moduli are described by complex quantities. The regions of dynamic instability are determined by Bolotin’s method while two eigenvalue problem systems with frequency dependent parameters are formed. The modified complex eigensolution method is employed to solve the eigenvalue problems. Numerical results are shown that the constrained damping layer attached to circular plates tends to stabilize the circular plate system. The effects of various parameters, such as types of viscoelastic material response, treatment thickness and area, on dynamic stability are also investigated.

The Magnetorotational Instability in a Collisionless Plasma

We consider the linear axisymmetric stability of a differentially rotating collisionless plasma in the presence of a weak magnetic field; we restrict our analysis to wavelengths much larger than the proton Larmor radius. This is the kinetic version of the magnetorotational instability explored extensively as a mechanism for magnetic field amplification and angular momentum transport in accretion disks. The kinetic calculation is appropriate for hot accretion flows onto compact objects and for the growth of very weak magnetic fields, where the collisional mean free path is larger than the wavelength of the unstable modes. We show that the kinetic instability criterion is the same as in MHD, namely that the angular velocity decrease outward. However, nearly every mode has a linear kinetic growth rate that differs from its MHD counterpart. The kinetic growth rates also depend explicitly on β, i.e., on the ratio of the gas pressure to the pressure of the seed magnetic field. For β ~ 1 the kinetic growth rates are similar to the MHD growth rates, while for β 1 they differ significantly. For β 1, the fastest growing mode has a growth rate ≈ Ω for a Keplerian disk, larger than its MHD counterpart; there are also many modes whose growth rates are negligible, β-1/2Ω Ω. We provide a detailed physical interpretation of these results and show that gas pressure forces, rather than just magnetic forces, are central to the behavior of the magnetorotational instability in a collisionless plasma. We also discuss the astrophysical implications of our analysis.

Magnetorotational instability of dissipative Couette flow

Axisymmetric stability of viscous resistive magnetized Couette flow is re-examined, with the emphasis on flows that would be hydrodynamically stable according to Rayleigh's criterion: opposing gradients of angular velocity and specific angular momentum. In this regime, magnetorotational instabilities (MRI) may occur. Previous work has focused on the Rayleigh-unstable regime. To prepare for an experimental study of MRI, which is of intense astrophysical interest, we solve for global linear modes in a wide gap with realistic dissipation coefficients. Exchange of stability appears to occur through marginal modes. Velocity eigenfunctions of marginal modes are nearly singular at conducting boundaries, but magnetic eigenfunctions are smooth and obey a fourth-order differential equation in the inviscid limit. The viscous marginal system is of tenth order; an eighth-order approximation previously used for Rayleigh-unstable modes does not permit MRI. Peak growth rates are insensitive to boundary conditions. They are predicted with surprising accuracy by WKB methods even for the largest-scale mode. We conclude that MRI is achievable under plausible experimental conditions using easy-to-handle liquid metals such as gallium.

A new approach to the problem of modes in the Mestel disk

We examine the modes admitted by the Mestel disk, a disk with a globally flat rotation curve. In contrast to previous analyses of this problem by Zang ([CITE]) and Evans & Read ([CITE], [CITE]), we approximate the orbits to obtain almost closed expressions for the kernel of the integral equation governing the behaviour of the modes. Otherwise we, like them, follow Kalnajs' programme to simultaneously solve the Boltzmann and Poisson equations. We investigate the modes admitted by both the self-con sis tent and a cut-out Mestel disk, the difference being that in the latter, a part of the matter in the disk is immobilised. This breaks the self-similarity and produces a pronouncedly different picture, both technically and in terms of the disk properties. The self-consistent disk is governed by a Cauchy integral equation, the cut-out disk by an integral equation that can be treated as a Fredholm equation of the second kind. In general, our approximation reproduces the results of the previous works remarkably well, yielding quantities mostly within 5% of the values reported by Zang and Evans & Read and thus also the basic result that in a "standard" cut-out disk, only one-armed modes are unstable at the limit of axisymmetric stability. In the self-consistent disk, relatively compact expressions for the kernel allow an intuitive understanding of most of the properties of neutral (non-rotating, non-growing) modes there. We finally show that self-consistent Mestel disks do not admit growing or rotating modes in this sort of stellar-dynamical analysis.

The local axisymmetric instability criterion in a thin, rotating, multicomponent disc

Purely gravitational perturbations are considered in a thin rotating disc composed of gas and several stellar components. The dispersion relation for the axisymmetric density waves propagating through the disc is found and the criterion for the local axisymmetric stability of the whole system is formulated. In the appropriate limit of two-component gas we confirm the findings of Jog & Solomon and extend consideration to the case when one component is collisionless. Gravitational stability of the Galactic disc in the solar neighbourhood based on the multicomponent instability condition is explored using recent measurements of the stellar composition and kinematics in the local Galactic disc obtained by the Hipparcos satellite.

Electrohydrodynamic behaviour of a drop subjected to a steady uniform electric field at finite electric Reynolds number

The electrohydrodynamic flow associated with a fluid drop in an electric field is a consequence of the tangential electric stress at the fluid interface. The tangential viscous stress due to the electrohydrodynamic flow arises to just balance the tangential electric stress at the fluid interface so that the traction boundary condition is satisfied. Influenced by both the local electric stress and viscous stress, the drop interface may exhibit various shapes. The presence of fluid flow also leads to charge convection phenomena. The relative significance of the charge convection effect is usually measured in terms of the electric Reynolds number, Re E , defined as the ratio of the timescales of charge convection by flow and that for charge relaxation by ohmic conduction. This work presents a quantitative analysis of the charge convection effects in a framework of the leaky dielectric model at finite Re E , which has not been considered in previous investigations. Axisymmetric steady flows driven by an applied uniform electric field about a deformable fluid drop suspended in an immiscible fluid are studied by computational means of the Galerkin finite–element method with supplementary asymptotic analysis. The results of finite–element computations are in general agreement with the prediction by the asymptotic analysis for spherical drops at vanishingly small Re E . A common effect of charge convection is found to reduce the intensity of electrohydrodynamic flow. As a consequence, oblate drops are predicted to be less deformed in an electric field when charge convection is taken into account. The prolate drops are often associated with an equator–to–pole flow, which convects charges toward the poles to form a charge distribution resembling that in a highly conducting drop immersed in an insulating medium. Therefore, charge convection tends to enhance the prolate drop deformation. In many cases, charge convection effects are found to be significant even at apparently small Re E , corresponding to the charge relaxation time–scale about 10 −3 s, suggesting that many experimental results reported in the previous publications could have been influenced by charge convection effects.

Combined effect of disk inequality and axial gravity on axisymmetric liquid bridge stability

The stability of an axisymmetric liquid bridge between unequal circular disks in an axial gravity field is examined for all possible values of the liquid volume and disk separation. The parameter defining the disk inequality is K, the ratio between the radii of the smaller and larger disks. Both axisymmetric and nonaxisymmetric perturbations are considered. The parameter space chosen to delimit the stability regions is the Λ-V plane. Here, Λ is the slenderness (ratio of the disk separation to the mean diameter, 2r0, of the two support disks), and V is the relative volume (ratio of the actual liquid volume to the volume of a cylinder with a radius equal to r0). Wide ranges of the Bond number and the ratio K are considered. Emphasis is given to previously unexplored parts of the stability boundaries. In particular, we examine the maximum volume stability limit for bridges of arbitrary Λ and the minimum volume stability limit for small Λ bridges. The maximum volume stability limit was found to have two distinct properties: large values of the critical relative volume at small Λ, and the possibility that stability is lost to axisymmetric perturbations at small values of K. For a set of K, the maximum Bond number beyond which stability of the bridge is no longer possible for any combination of V and Λ is determined. In addition, the maximum value of the actual liquid volume of a stable bridge that can be held between given disks for all possible disk separations was examined for fixed Bond number. It is found that this volume decreases as K decreases and (depending on the sign of the Bond number) tends to the critical volume of a sessile or pendant drop attached to the larger disk.

A model equation for axisymmetric stability of small-gap parallel-plate flows

We consider a model equation derived in [18]for the linear stability of a torsional flow of an Oldroyd-B fluid in a parallel-plate device. Axisymmetry, small-gap and zero Reynolds numbers are assumed. The equation is not separable in the radial and vertical variables, but has the advantage that its eigenmodes depend only on the aspect ratio. Once evaluated, the eigenmodes are used to calculate the onset conditions for any retardation parameter and Deborah number. The onset conditions and streamfunction contours are compared with the normal-mode analysis of [14]which assumed radially localized disturbances and the computational results on the full equations [1]. Two facets of the model equation are discussed; first, how well the onset conditions and eigenmodes compare with those of the case where the radially localized assumption is made and secondly, how they approximate those of the full equations.

The Axisymmetric Stability of the Milky Way

The problem of axisymmetric Jeans stability of a stellar disk was essentially completely solved in the classic paper by Toomre (1964). While his analysis was strictly local, the stability criterion it yielded has been found to be remarkably reliable in global studies; I review earlier results and give a further example in Sellwood (1995, hereafter S95). Toomre (1974) concluded that the MW is safely stable to axisymmetric Jeans modes.

Stability of a Stretching Capillary Jet with Parabolic Axial Velocity

The linear axisymmetric stability of a stretching capillary jet with parabolic axial velocity is investigated, numerically. Since the basic flow is time-dependent, the stability analysis is reduced to an initial-value problem of ordinary differential equations, instead of an eigenvalue problem for normal modes. It is shown that a parabolic velocity profile has a remarkable stabilizing effect.

An effective Q parameter for two-fluid instabilities in spiral galaxies

The equations for the two-fluid instability developed by Jog & Solomon are reduced to a form in which a single Qeff parameter determines the state of gravitational stability. This parameter has the correct values in the limits of either fluid alone and is a natural extension of the usual one-fluid Q parameter to the two-fluid case. Finite thickness effects are included. Variations in Qeff with radius for typical galaxies are studied. The results suggest that the axisymmetric stability properties of a galactic disc can be estimated from the pure gas stability parameter, Qg, only when Qg≪Qs, the pure-star parameter; otherwise Qeff should be used. Implications for star-formation thresholds and hidden gas at large galactic radii are discussed. If the dark matter required to give the Galactic rotation curve were in the form of in-plane gas with a normal velocity dispersion, then the outer disc would be very unstable.

Equilibrium Shapes and Stability of Nonconducting Pendant Drops Surrounded by a Conducting Fluid in an Electric Field

The shapes and stability of pendant drops in the presence of an electric field is a classical problem in capillarity. This problem has been studied in great detail by numerous investigators when the drops are either perfect conductors or nonconductors and the surrounding fluid is a nonconductor. In this paper, the axisymmetric equilibrium shapes and stability of a nonconducting drop hanging from a nonconducting nozzle that is immersed in a perfectly conducting ambient fluid, a problem that has heretofore not been considered in the literature, are determined by solving the free boundary problem comprised of the Young-Laplace equation for drop shape and an integral equation for the electric field distribution. Here the free boundary problem is discretized by a hybrid technique in which the Young-Laplace equation is solved by the finite element method and the electrostatic problem is solved by the boundary element method. An electrode in the form of a metal rod or an annulus is placed inside and coaxial with the lube whose tip is located a distance H1 above or below the tube outlet. The electric field is generated by connecting the rod or the annulus electrode to a source of high voltage at potential P and grounding the ambient conducting fluid that surrounds the drop and the tube. When the force due to surface (interfacial) tension is large compared to those due to gravity and electric field, equilibrium drop shapes are sections of spheres and can be parametrized by a parameter -1 ≤ D ≤ 1: D = 0 corresponds to a hemisphere, as D → 1 the drop approaches a sphere, and as D → -1 the drop becomes vanishingly small. The results show that equilibrium families of fixed drop volume, or fixed D, lose stability at turning points with respect to the applied potential, where P = P*. Detailed computations reveal the importance of varying the drop size and geometric factors such as the location of the tip of the electrode H1, electrode thickness W, and, in the case of the annulus electrode, the inner radius of the annulus RΛ on the value of P*. Except for very skinny drops, i.e., D → -1, the computed profiles of drops at their limits of stability imply that nonconducting drops in a conducting ambient fluid should become unstable by pinching off near the contact line. This finding agrees with recent experiments on such drops, but stands in marked contrast to previous studies on conducting and nonconducting drops immersed in a nonconducting ambient fluid where the drops become unstable by developing conical tips from which a jet issues. Very skinny drops, however, take on a dog-boned appearance at their limits of stability regardless of whether the drop is a conductor or a nonconductor immersed in a nonconducting ambient fluid or a nonconductor immersed in a conducting ambient fluid.

Axisymmetric dynamic stability of an annular plate subject to pulsating conservative radial loads

The equations of motion of an annular plate subject to prescribed timedependent radial compressive loads are formulated based on Hamilton's principle and the assumed mode method. The effects of sinusoidal perturbations in respect of the radial loads are then examined using Bolotin's method. The respective regions of instability for axisymmetric deformations are determined by converting the resulting equations of motion to the standard form of a generalized eigenvalue problem. Instability regions are presented for various combinations of average value and amplitude of the sinusoidal perturbations of the radial loads and also for various edge conditions for an isotropic annular plate. The unstable regions are found to be insensitive to changes in the average in-plane load when the annular plate is loaded only on the inner edge. The unstable regions for the case where the plate is loaded on both edges are similar to the case where the plate is loaded only on the outer edge.