I. G. Shukhman

An analytical-based method for studying the nonlinear evolution of localized vortices in planar homogenous shear flows

Recent experimental and numerical studies have shown that the interaction between a localized vortical disturbance and the shear of an external base flow can lead to the formation of counter-rotating vortex pairs and hairpin vortices that are frequently observed in wall bounded and free turbulent shear flows as well as in subcritical shear flows. In this paper an analytical-based solution method is developed. The method is capable of following (numerically) the evolution of finite-amplitude localized vortical disturbances embedded in shear flows. Due to their localization in space, the surrounding base flow is assumed to have homogeneous shear to leading order. The method can solve in a novel way the interaction between a general family of unbounded planar homogeneous shear flows and any localized disturbance. The solution is carried out using Lagrangian variables in Fourier space which is convenient and enables fast computations. The potential of the method is demonstrated by following the evolved structures of large amplitude disturbances in three canonical base flows, including simple shear, plane stagnation (extensional) and pure rotation flows, and a general case. The results obtained by the current method for plane stagnation and simple shear flows are compared with the published results. The proposed method could be extended to other flows (e.g. geophysical and rotating flows) and to include periodic disturbances as well.

Notes on the stability threshold for radially anisotropic polytropes

We discuss some contradictions found in the literature concerning the problem of stability of collisionless spherical stellar systems, which are the simplest anisotropic generalization of the well-known polytrope models. Their distribution function F(E, L) is a product of powerlaw functions of the energy E and angular momentum L, i.e. F ∝ L −s (− E) q . On the one hand, calculation of the growth rates in the framework of linear stability theory and N-body simulations shows that these systems become stable when the parameter s characterizing the velocity anisotropy of the stellar distribution is lower than some finite threshold value, s < scrit. On the other hand, Palmer & Papaloizou showed that the instability remains up to the isotropic limit s = 0. Using our method of determining the eigenmodes for stellar systems, we show that the growth rates in weakly radially anisotropic systems are indeed positive, but decrease exponentially as the parameter s approaches zero, i.e. γ ∝ exp(−s∗/s). In fact, for systems with a finite lifetime this means stability.

Effect of angular momentum distribution on gravitational loss-cone instability in stellar clusters around a massive black hole

We study the small perturbations in spherical and thin disc stellar clusters surrounding a massive black hole. Because of the black hole, stars with sufficiently low angular momentum escape from the system through the loss cone. We show that the stability properties of spherical clusters crucially depend on whether the distribution of stars is monotonic or non-monotonic in angular momentum. It turns out that only non-monotonic distributions can be unstable. At the same time, instability in disc clusters is possible for both types of distribution.

On the nature of the radial orbit instability in spherically symmetric collisionless stellar systems

We consider a two-parametric family of radially anisotropic models with non-singular density distribution in the centre. If highly eccentric orbits are locked near the centre, the characteristic growth rate of the instability is much less than the Jeans and dynamic frequencies of the stars (slow modes). The instability occurs only for even spherical harmonics and the perturbations are purely growing (aperiodic). On the contrary, if all orbits nearly reach the outer radius of the sphere, both even and odd harmonics are unstable. Unstable odd modes oscillate having characteristic frequencies of the order of the dynamical frequencies (fast modes). Unstable even harmonics contain a single aperiodic mode and several oscillatory modes, the aperiodic mode being the most unstable. The question of the nature of the radial orbit instability (ROI) is revisited. Two main interpretations of ROI were suggested in the literature. The first one refers to the classical Jeans instability associated with the lack of velocity dispersion of stars in the transverse direction. The second one refers to Lynden-Bells orbital approach to bar formation in disc galaxies, which implies slowness and bi-symmetry of the perturbation. Oscillatory modes, odd spherical harmonics modes, and non-slow modes found in one of the models show that the orbital interpretation is not the only possible.

Temporal evolution of a localized weak vortex in viscous circular shear flows

The evolution of a weak, highly localized, initial vortex in a steady circular flow is addressed. The effect of basic flow curvature, characterized by the ratio between the Coriolis and inertial forces, on the evolution of the initial vortex is studied. It is shown that the vortex remains weak throughout its entire evolution, although in a certain range of orientation angles of the initial vortex plane there occurs a transient algebraic growth of intensity of the vortex characterized by the value of its total enstrophy. Special attention is devoted to comparison of the results obtained with the experimental results reported by Malkiel, Levinski, and Cohen (1999) where the evolution of artificially synthesized localized vortices in a laminar Taylor–Couette flow was investigated. In particular, it was found that the orientation of the plane of vortex localization predicted by the theoretical model differs from the one observed in the experiment. However, the agreement between the theoretical model and exper...

Equilibrium models of radially anisotropic spherical stellar systems with softened central potentials

We study a new class of equilibrium two-parametric distribution functions of spherical stellar systems with a radially anisotropic velocity distribution of stars. The models are less singular counterparts of the so-called generalized polytropes, widely used in the past in works on equilibrium and the stability of gravitating systems. The offered models, unlike the generalized polytropes, have finite density and potential in the centre. The absence of the singularity is necessary for proper consideration of the instability of the radial orbit, which is the most important instability in spherical stellar systems. We provide a comparison of the main observed parameters (i.e. potential, density, anisotropy) predicted by the present models and by other popular equilibrium models.

On the Instability of Weakly Radially Anisotropic Star Clusters

We discuss contradictions existing in the literature in the problem on the stability of collisionless spherical stellar systems, which are the simplest anisotropic generalization of the well-known polytropic models. On the one hand, calculations of the growth rates within the framework of a linear stability theory and N-body simulations suggest that these systems should become stable when the parameter s characterizing the degree of anisotropy of the stellar velocity distribution becomes lower than some critical value scrit > 0. On the other hand, according to Palmer and Papaloizou, the growth rate should be nonzero up to the isotropic limit s = 0. Using our method of determining the eigenmodes of stellar systems, we show that even though the mode growth rates in weakly radially anisotropic systems of this type are nonzero, they are exponentially small, i.e., decrease as γ ∝ exp(−a/s) when s → 0. For slightly radially anisotropic systems with a finite lifetime, this actually implies stability.

Slow modes in stellar systems with nearly harmonic potentials: I. Spoke approximation, radial orbit instability

Using a consistent perturbation theory for collisionless disk-like and spherical star clusters, we construct a theory of slow modes for systems having an extended central region with a nearly harmonic potential due to the presence of a fairly homogeneous (on the scales of the stellar system) heavy, dynamically passive halo. In such systems, the stellar orbits are slowly precessing, centrally symmetric ellipses (2: 1 orbits). Depending on the density distribution in the system and the degree of halo inhomogeneity, the orbit precession can be both prograde and retrograde, in contrast to systems with 1: 1 elliptical orbits where the precession is unequivocally retrograde. In the first paper, we show that in the case where at least some of the orbits have a prograde precession and the stellar distribution function is a decreasing function of angular momentum, an instability that turns into the well-known radial orbit instability in the limit of low angular momenta can develop in the system. We also explore the question of whether the so-called spoke approximation, a simplified version of the slow mode approximation, is applicable for investigating the instability of stellar systems with highly elongated orbits. Highly elongated orbits in clusters with nonsingular gravitational potentials are known to be also slowly precessing 2: 1 ellipses. This explains the attempts to use the spoke approximation in finding the spectrum of slow modes with frequencies of the order of the orbit precession rate. We show that, in contrast to the previously accepted view, the dependence of the precession rate on angular momentum can differ significantly from a linear one even in a narrow range of variation of the distribution function in angular momentum. Nevertheless, using a proper precession curve in the spoke approximation allows us to partially “rehabilitate” the spoke approach, i.e., to correctly determine the instability growth rate, at least in the principal (O(αT−1/2) order of the perturbation theory in dimensionless small parameter αT, which characterizes the width of the distribution function in angular momentum near radial orbits.

Two scenarios of the radial orbit instability in spherically symmetric collisionless stellar systems

The stability of a two-parameter family of radially anisotropic models with a nonsingular central density distribution is considered. Instability takes place at a sufficiently strong radial anisotropy (the so-called radial orbit instability, ROI). We show that the character of instability depends not only on the anisotropy but also on the energy distribution of stars. If this distribution is such that the highly eccentric orbits responsible for the instability are “trapped” in the radial direction near the center, then the instability develops with a characteristic growth time that exceeds considerably the Jeans and dynamical times of the trapped particles. In this case, the instability takes place only for even spherical harmonics and is aperiodic. If, however, almost all of the elongated orbits reach the outer radius of the sphere, then both even and odd harmonics turn out to be unstable. The unstable modes corresponding to odd harmonics are oscillatory in nature with characteristic frequencies of the order of the dynamical ones. The unstable perturbations corresponding to even harmonics contain only one aperiodic mode and several oscillatory modes, with the aperiodic mode being always the most unstable one. Two main interpretations of the ROI available in the literature have been analyzed: the “classical” Jeans instability related to an insufficient stellar velocity dispersion in the transversal direction and the “orbital” approach relying heavily on the analogous Lynden-Bell bar formation mechanism in disk galaxies. The assumptions that the perturbations are slow (compared to the orbital frequency of stars) and that the shape of the perturbed potential is symmetric are inherent integral conditions for the applicability of the latter. Our solutions show that the orbital approach cannot be considered as a universal one.

Radially anisotropic models of collisionless spherical stellar systems without central singularity

A new class of the simplest equilibrium two-parameter distribution functions for spherical stellar systems with a radially anisotropic stellar velocity distribution is investigated. The models under consideration are a less singular counterpart of the so-called generalized polytropes, which in the past were among the most popular models in works on the equilibrium and stability of gravitating systems. In contrast to the well-known generalized polytropes, the proposed models have finite density and potential at the center. The absence of a singularity is necessary for a proper consideration of the radial orbit instability, which is the most important instability of spherical stellar systems. The main observed parameters of the proposed models (potential, density, anisotropy) are compared with those in well-known equilibrium models.