O. M. Umurhan

Hydrodynamic stability of rotationally supported flows: Linear and nonlinear 2D shearing box results

We present here both analytical and numerical results of hydrodynamic stability investigations of rotationally sup- ported circumstellar flows using the shearing box formalism. Asymptotic scaling arguments justifying the shearing box approxi- mation are systematically derived, showing that there exist two limits which we call small shearing box (SSB) and large shearing box (LSB). The physical meaning of these two limits and their relationship to model equations implemented by previous in- vestigators are discussed briefly. Two dimensional (2D) dynamics of the SSB are explored and shown to contain transiently growing (TG) linear modes, whose nature is first discussed within the context of linear theory. The fully nonlinear regime in 2D is investigated numerically for very high Reynolds (Re) numbers. Solutions exhibiting long-term dynamical activity are found and manifest episodic but recurrent TG behavior and these are associated with the formation and long-term survival of coherent vortices. The life-time of this spatio-temporal complexity depends on the Re number and the strength and nature of the initial disturbance. The dynamical activity in finite Re solutions ultimately decays with a characteristic time increasing with Re. However, for large enough Re and appropriate initial perturbation, a large number of TG episodes recur before any viscous decay begins to clearly manifest itself. In cases where Re = ∞ nominally (i.e. any dissipation resulting only from numerical truncation errors), the dynamical activity persists for the entire duration of the simulation (hundreds of box orbits). Because the SSB approximation used here is equivalent to a 2D incompressible flow, the dynamics can not depend on the Coriolis force. Therefore, three dimensional (3D) simulations are needed in order to decide if this force indeed suppresses nonlinear hydro- dynamical instability in rotationally supported disks in the shearing box approximation, and if recurrent TG behavior can still persist in three dimensions as well - possibly giving rise to a subcritical transition to long-term spatio-temporal complexity.

On the viability of the shearing box approximation for numerical studies of MHD turbulence in accretion disks

Context. Most of our knowledge of the nonlinear development of the magnetorotational instability (MRI) relies on the results of numerical simulations employing the shearing box (SB) approximation. A number of difficulties arising from this approach have recently been pointed out in the literature. Aims. We thoroughly examine the effects of the assumptions made and numerical techniques employed in SB simulations. This is done to clarify and gain a better understanding of those difficulties, in addition to a number of additional serious problems raised here for the first time, and of their impact on the results. Methods. We used analytical derivations and estimates as well as comparative analysis to methods used in the numerical study of turbulence. We performed numerical experiments to support some of our claims and conjectures. Results. The following problems, arising from the (virtually exclusive) use of SB simulations as a tool for the understanding and quantification of the nonlinear MRI development in disks, are analyzed and discussed: (i) inconsistencies in the application of the SB approximation itself; (ii) the limited spatial scale of the SB; (iii) the lack of convergence of most ideal mgnetohydrodynamical (MHD) simulations; (iv) side-effects of the SB symmetry and the non-trivial nature of the linear MRI; and (v) physical artifacts arising from the very small box scale due to periodic boundary conditions. Conclusions. The computational and theoretical challenge posed by the MHD turbulence problem in accretion disks cannot be met by the SB approximation, as it has been used to date. A new strategy to confront this challenge is proposed, based on techniques widely used in numerical studies of turbulent flows - developing (e.g., with the help of local numerical studies) a sub-grid turbulence model and implementing it in global calculations.

Global axisymmetric dynamics of thin viscous accretion disks

The purpose of this paper is to explore the steady state and dynamical behavior of thin, axisymmetric, viscous accretion disks. To facilitate an analytical treatment we replace the energy equation with a general polytropic assumption. The asymptotic expansion of Kluzniak & Kita (2000, Three-dimensional structure of an alpha accretion disk (arXiv:astro-ph/0006266)), which extended the method of Regev (1983, A&A, 126, 146) to a full steady polytropic disk (with n = 3/2), is further developed and implemented for both the steady (for any polytropic index) and time-dependent problems. The spatial form and temporal behavior of selected dynamical disturbances are studied in detail. It is shown that the transient dynamics resulting from initial perturbations on the linearly stable steady state gives rise to substantial growth of perturbations. We identify the initial perturbation space which leads to such transient growth and provide analytical solutions which manifest this behavior three terms (physical causes) responsible for the appearance of transient dynamics are identified. Two depend explicitly on the viscosity while the third one is relevant also for inviscid disks. The main conclusion we draw is that transient dynamics and, in particular, significant perturbation energy amplification occurs in disks on a global scale. We speculate on the possible implications of these findings to accretion disk theory.

Potential vorticity dynamics in the framework of disk shallow-water theory I. The Rossby wave instability

Context. The Rossby wave instability in astrophysical disks is a potentially important mechanism for driving angular momentum transport in disks. Aims. We attempt to more clearly understand this instability in an approximate three-dimensional disk model environment which we assume to be a single homentropic annular layer we analyze using disk shallow-water theory. Methods. We consider the normal mode stability analysis of two kinds of radial profiles of the mean potential vorticity: the first type is a single step and the second kind is a symmetrical step of finite width describing either a localized depression or peak of the mean potential vorticity. Results. For single potential vorticity steps we find there is no instability. There is no instability when the symmetric step is a localized peak. However, the Rossby wave instability occurs when the symmetrical step profile is a depression, which, in turn, corresponds to localized peaks in the mean enthalpy profile. This is in qualitative agreement with previous two-dimensional investigations of the instability. For all potential vorticity depressions, instability occurs for regions narrower than some maximum radial length scale. We interpret the instability as resulting from the interaction of at least two Rossby edgewaves. Conclusions. We identify the Rossby wave instability in the restricted three-dimensional framework of disk shallow water theory. Additional examinations of generalized barotropic flows are needed. Viewing disk vortical instabilities from the conceptual perspective of interacting edgewaves can be useful.

Holmboe modes revisited

A scaling analysis is presented better identifying the conditions in which the Boussinesq approximation may be used to study shear disturbances like that of Holmboe modes. The classic Holmboe normal mode instability is then reanalyzed by including baroclinic effects whose introduction alters the onset of Holmboe’s traveling-wave instability depending on the direction of the propagating modes. Since the introduction of baroclinicity is tantamount to relaxing the Boussinesq assumption, it means that in the presence of shear there is now a vertical variation of the horizontal momentum flux that alters the phase speed and structure of the classic Holmboe modes; the physical source of their broken right-left propagatory symmetry is associated with this physical effect. Furthermore, the regions of parameter space in which Holmboe’s classic analysis predicts there to be nonpropagating double instabilities now supports propagating Holmboe modes when baroclinic effects are included. We also find that a globally co...

A shallow-water theory for annular sections of Keplerian disks

Context. We present a scaling argument that we develop into a shallow water theory of non-axisymmetric disturbances in annular sections of thin Keplerian disks. Aims. We develop a theoretical approach to understand physically the relationship between two-dimensional vortex dynamics that is known and their three-dimensional counterparts in Keplerian disks. Methods. Using asymptotic scaling arguments, varicose disturbances of a Keplerian disk are considered on radial and vertical scales consistent with the height of the disk while the azimuthal scales are the full 2π angular extent of the disk. For simplicity perturbations are assumed to be homentropic according to a polytropic equation of state. The timescales considered are long compared to the local disk rotation time. Results. The scalings relate to dynamics that are radially geostrophic and vertically hydrostatic. A potential vorticity quantity emerges and is shown to be conserved in a Lagrangian sense. Uniform potential vorticity linear solutions are explored and the theory is shown to contain an incarnation of the strato-rotational instability under channel flow conditions. Linearized solutions of a single defect on an infinite domain are developed and shown to support a propagating Rossby edgewave. Linear non-uniform potential vorticity solutions are also developed and shown to be similar in some respects to the dynamics of strictly two-dimensional inviscid flows. The relationship of the scalings and some of the resulting dynamics are considered with respect to other approximations employed in the literature. Based on the framework of this theory, arguments based on geophysical notions are presented to support the assertion that the strato-rotational instability is in a generic class of barotropic/baroclinic potential vorticity instabilities. Extensions of this formalism are also proposed. Conclusions. The shallow water formulation achieved by the asymptotic theory developed here opens a new approach in studying disk dynamics.

Properties of a solid state device with mobile dopants: Analytic analysis for the thin film device

Current-voltage relations, electric field, and charge distribution profiles are calculated for a device in which the dopants are mobile. The thin film limit is discussed. The model solved is restricted to: (a) mobile holes and acceptors, (b) steady state, and (c) metal electrodes which block the ionic current. The solution is expressed as a series expansion in the small parameter δ=L/λD, where L is the sample thickness and λD is a Debye length. The second order of the series expansion is found to vanish, thus the corrections to the leading order appear only in the third term. The approximated analytic solution agrees with numerical results from a previous publication up to the quite high value of δ=1. The leading order in the I-V relations and in the hole distribution is independent of the acceptor motion. This implies that for thin devices of this form any motion of the dopants may be neglected and that dopants need not be limited only to those which exhibit low diffusion constants. Rectification is obse...

On the nature of the hydrodynamic stability of accretion disks

The linear stability of accretion disks is revisited. The governing equations are expanded asymptotically and solved to first order in the expansion parameterdefined by the ratio of the disks vertical thickness to its radial extent. Algebraically growing solutions are found for global perturbations on the radial accretion flow of thin inviscid compressible Keplerian disks. The algebraic temporal behavior is exhibited in the vertical velocities and the thermodynamic variables and has the form t sin Ω0t locally in the disk where Ω0 is the Keplerian rotation rate. The physical implications and relations to the Solberg- Hoiland stability criteria are discussed.

Low magnetic-Prandtl number flow configurations for cold astrophysical disk models: speculation and analysis

Context. Simulations of astrophysical disks in the shearing box that are subject to the magnetorotational instability (MRI) show that activity appears to be reduced as the magnetic Prandtl number P m is lowered. It is therefore important to understand the reasons for this trend, especially if this trend is shown to continue when higher resolution calculations are performed in the near future. Calculations for laboratory experiments show that saturation is achieved through modification of the background shear for P m « 1. Aims. Guided by the results of calculations appropriate for laboratory experiments when P m is very low, the stability of inviscid disturbances in a shearing box model immersed in a constant vertical background magnetic field is considered under a variety of shear profiles and boundary conditions in order to evaluate the hypothesis that modifications of the shear bring about saturation of the instability. Shear profiles q are given by the local background Keplerian mean, q 0 , plus time-independent departures, Q(x), with zero average on a given scale. Methods. The axisymmetric linear stability of inviscid magnetohydrodynamic normal modes in the shearing box is analyzed. Results. (i) The stability/instability of modes subject to modified shear profiles may be interpreted by a generalized Velikhov criterion given by an effective shear and radial wavenumber that are defined by the radial structure of the mode and the form of Q. (ii) Where channel modes occur, comparisons against marginally unstable disturbance in the classical case, Q = 0, shows that all modifications of the shear examined here enhance mode instability. (iii) For models with boundary conditions mimicing laboratory experiments, modified shear profiles exist that stabilize a marginally unstable MRI for Q = 0. (iv) Localized normal modes on domains of infinite radial extent characterized by either single defects or symmetric top-hat profiles for Q are also investigated. If the regions of modified shear are less (greater) than the local Keplerian background, then there are (are no) normal modes leading to the MRI. Conclusions. The emergence and stability of the MRI is sensitive to the boundary conditions adopted. Channel modes do not appear to be stabilized through modifications of the background shear whose average remains Keplerian. However, systems that have non-penetrative boundaries can saturate the MRI through modification of the background shear. Conceptually equating the qualitative results from laboratory experiments to the conditions in a disk may therefore be misleading.

Dynamics in coalescing critical layers

This article continues an exploration of instabilities of jets in two-dimensional, inviscid fluid on the beta-plane. At onset, for particular choices of the physical parameters, the normal modes responsible for instability have critical levels that coalesce along the axis of the jet. Matched asymptotic expansion (critical layer theory) is used to derive a reduced model describing the dynamics of these modes. Because the velocity profile is locally parabolic in the vicinity of the critical levels the dynamics is richer than in standard critical layer problems. The model captures the inviscid saturation of unstable modes, the excitation of neutral Rossby waves, and the decay of disturbances when there are no discrete normal modes. Inviscid saturation occurs when the vorticity distribution twists up into vortical structures that take the form of either a pair of ‘cats eye’ patterns straddling the jet axis, or a single row of vortices. The addition of weak viscosity destroys these cats eye structures and causes the critical layer to spread diffusively. The results are compared with numerical simulations of the governing equations.