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The stability of stratified, rotating systems and the generation of vorticity in the Sun  [PDF]
Steven A. Balbus,Emmanuel Schaan
Physics , 2012, DOI: 10.1111/j.1365-2966.2012.21729.x
Abstract: We examine the linear behavior of three-dimensional Lagrangian displacements in a stratified, shearing background. The isentropic and iso-rotation surfaces of the equilibrium flow are assumed to be axisymmetric, but otherwise fully two-dimensional. Three-dimensional magnetic fields are included in the perturbation equations; however the equilibrium is assumed to be well-described by purely hydrodynamic forces. The model, in principle very general, is used to study the behavior of fluid displacements in an environment resembling the solar convection zone. Some very suggestive results emerge. All but high-latitude displacements align themselves with the observed surfaces of constant angular velocity. The tendency for the angular velocity to remain constant with depth in the bulk of the convective zone, together with other critical features of the rotation profile, emerge from little more than a visual inspection of the governing equation. In the absence of a background axial angular velocity gradient, displacements exhibit no poleward bias, suggesting that solar convection "plays-off" of prexisting shear rather than creates it. We argue that baroclinic vorticity of precisely the right order is generated at the radiative/convective zone boundary due to centrifugal distortion of equipotential surfaces that is not precisely followed by isothermal surfaces. If so, many features of the Sun's internal rotation become more clear, including: i) the general appearance of the tachocline; ii) the extension of differential rotation well into the radiative zone; iii) the abrupt change of morphology of convective zone isorotation surfaces; and iv) the inability of current numerical simulations to reproduce the solar rotation profile without imposed entropy boundary conditions.
On the axisymmetric stability of stratified and magnetized accretion disks  [PDF]
Gopakumar Mohandas,Martin E. Pessah
Physics , 2015,
Abstract: We conduct a comprehensive axisymmetric, local linear mode analysis of a stratified, differentially rotating disk permeated by a toroidal magnetic field which could provide significant pressure support. In the adiabatic limit, we derive a new stability criteria that differs from the one obtained for weak magnetic fields with a poloidal component and reduces continuously to the hydrodynamic Solberg-H{\o}iland criteria. Three fundamental unstable modes are found in the locally isothermal limit. They comprise of overstable: (i) acoustic oscillations, (ii) radial epicyclic (acoustic-inertial) oscillations and (iii) vertical epicyclic (or vertical shear) oscillations. All three modes are present for finite ranges of cooling times but they are each quickly quenched past respective cut-off times. The acoustic and acoustic-inertial overstable modes are driven by the background temperature gradient. When vertical structure is excluded, we find that the radial epicyclic modes appear as a nearly degenerate pair. One of these is the aforementioned acoustic-inertial mode and the other has been previously identified in a slightly different guise as the convective overstability. Inclusion of vertical structure leads to the development of overstable oscillations destabilized by vertical shear but also has the effect of suppressing the radial epicyclic modes. Although our study does not explicitly account for non-ideal effects, we argue that it may still shed light into the dynamics of protoplanetary disk regions where a strong toroidal field generates as a result of the Hall-shear instability.
Novel mechanism for vorticity generation in black-hole accretion disks  [PDF]
Chinmoy Bhattacharjee,Rupam Das,Swadesh M. Mahajan
Physics , 2015, DOI: 10.1103/PhysRevD.91.123005
Abstract: Vorticity generation in accretion disks around Schwarzschild and Kerr black holes is investigated in the context of magnetofluid dynamics derived for both General Relativity (GR), and modified gravity formulations. In both cases, the Kerr geometry leads to a "stronger" generation of vorticity than its Schwarzschild counterpart. Of the two principal sources, the relativistic drive peaks near the innermost stable circular orbit (isco), whereas the baroclinic drive dominates at larger distances. Consequences of this new relativistic vorticity source are discussed in several astrophysical settings.
Vorticity Budget of Weak Thermal Convection in Keplerian disks  [PDF]
Pin-Gao Gu
Physics , 2000, DOI: 10.1086/309328
Abstract: By employing the equations of mean-square vorticity (enstrophy) fluctuations in strong shear flows, we demonstrate that unlike energy production of turbulent vorticity in nonrotating shear flows, the turbulent vorticity of weak convection in Keplerian disks cannot gain energy from vortex stretching/tilting by background shear unless the asscoiated Reynolds stresses are negative. This is because the epicyclic motion is an energy sink of the radial component of mean-square turbulent vorticity in Keplerian disks when Reynolds stresses are positive. Consequently, weak convection cannot be self-sustained in Keplerian flows. This agrees with the results implied from the equations of mean-square velocity fluctuations in strong shear flows. Our analysis also sheds light on the explanation of the simulation result in which positive kinetic helicity is produced by the Balbus-Hawley instability in a vertically stratified Keplerian disk. We also comment on the possibility of outward angular momentum transport by strong convection based on azimuthal pressure perturbations and directions of energy cascade.
A powerful local shear instability in stratified disks  [PDF]
D. Richard,F. Hersant,O. Dauchot,F. Daviaud,B. Dubrulle,J-P. Zahn
Physics , 2001,
Abstract: In this paper, we show that astrophysical accretion disks are dynamically unstable to non-axisymmetric disturbances. This instability is present in any stably stratified anticyclonically sheared flow as soon as the angular velocity decreases outwards. In the large Froude number limit, the maximal growth rate is proportional to the angular rotation velocity, and is independent of the stratification. In the low Froude number limit, it decreases like the inverse of the Froude number, thereby vanishing for unstratified, centrigugally stable flows. The instability is not sensitive to disk boundaries. We discuss the possible significance of our result, and its implications on the turbulent state achieved by the disks. We conclude that this linear instability is one of the best candidates for the source of turbulence in geometrically thin disks, and that magnetic fields can be safely ignored when studying their turbulent state. The relevance of the instability for thick disks or nearly neutrally stratified disks remains to be explored.
Structure, stability and evolution of 3D Rossby vortices in protoplanetary disks  [PDF]
Samuel Richard,Pierre Barge,Stephane Le Dizes
Physics , 2013, DOI: 10.1051/0004-6361/201322175
Abstract: Large-scale persistent vortices are known to form easily in 2D disks via the Rossby wave or the baroclinic instability. In 3D, however, their formation and stability is a complex issue and still a matter of debate. We study the formation of vortices by the Rossby wave instability in a stratified inviscid disk and describe their three dimensional structure, stability and long term evolution. Numerical simulations are performed using a fully compressible hydrodynamical code based on a second order finite volume method. We assume a perfect gas law and a non-homentropic adiabatic flow.The Rossby wave instability is found to proceed in 3D in a similar way as in 2D. Vortices produced by the instability look like columns of vorticity in the whole disk thickness; the small vertical motions are related to a weak inclination of the vortex axis appearing during the development of the RWI. Vortices with aspect ratios larger than 6 are unaffected by the elliptical instability. They relax to a quasi-steady columnar structure which survives hundred of rotations while slowly migrating inward toward the star at a rate that reduces with the vortex aspect ratio. Vortices with a smaller aspect ratio are by contrast affected by the elliptic instability. Short aspect ratio vortices are completely destroyed in a few orbital periods. Vortices with an intermediate aspect ratio are partially destroyed by the elliptical instability in a region away from the mid-plane where the disk stratification is sufficiently large. Elongated Rossby vortices can survive a large number of orbital periods in protoplanetary disks in the form of vorticity columns. They could play a significant role in the evolution of the gas and the gathering of the solid particles to form planetesimals or planetary cores, a possibility that receives a renewed interest with the recent discovery of a particle trap in the disk of Oph IRS48.
Stratified disks are locally stable  [PDF]
Jeremy Goodman Steven A. Balbus
Physics , 2001,
Abstract: Notwithstanding recent claims by Richard et al., there is no linear hydrodynamic instability of axisymmetrically stable disks in the local limit. We prove this by means of an exact stability analysis of an unbounded incompressible flow having constant stratification and constant shear.
Potential vorticity dynamics in the framework of disk shallow-water theory: I. The Rossby wave instability  [PDF]
O. M. Umurhan
Physics , 2010, DOI: 10.1051/0004-6361/201015210
Abstract: The Rossby wave instability in astrophysical disks is as a potentially important mechanism for driving angular momentum transport in disks. We aim to 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. 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. 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. 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.
Potential vorticity dynamics in the framework of disk shallow-water theory: II. Mixed Barotropic-Baroclinic Instability  [PDF]
O. M. Umurhan
Physics , 2012, DOI: 10.1051/0004-6361/201218803
Abstract: We extend exploration of potential vorticity instabilities in cold astrophysical disks whose mean states are baroclinic. In particular, we seek to demonstrate the potential existence of traditional baroclinic instabilities of meteorological studies in a simplified two-layer Philips Disk Model. Each disk layer is of constant but differing densities. The resulting mean azimuthal velocity profile shows a variation in the vertical direction implying that the system is baroclinic in the mean state. The stability of the system is treated in the context of disk shallow water theory wherein azimuthal disturbances are much longer than the corresponding radial or vertical scales. The normal-mode problem is solved numerically using two different methods. The results of a symmetric single layer barotropic model is considered and it is found that instability persists for models in which the potential vorticity profiles are not symmetric, consistent with previous results. The instaiblity is interpreted in terms of interacting Rossby waves. For a two layer system in which the flow is fundamentally baroclinic we report here that instability takes on the form of mixed barotropic-baroclinic type: instability occurs but it qualitatively follows the pattern of instability found in the barotropic models. Instability arises because of the phase locking and interaction of the Rossby waves between the two layers. The strength of the instability weakens as the density contrast between layers increases. (For full abstract see article.)
Sensitivity of the Magnetorotational Instability to the shear parameter in stratified simulations  [PDF]
Farrukh Nauman,Eric G. Blackman
Physics , 2014, DOI: 10.1093/mnras/stu2226
Abstract: The magnetorotational instability (MRI) is a shear instability and thus its sensitivity to the shear parameter $q = - d\ln\Omega/d\ln r $ is of interest to investigate. Motivated by astrophysical disks, most (but not all) previous MRI studies have focused on the Keplerian value of $ q=1.5$. Using simulation with 8 vertical density scale heights, we contribute to the subset of studies addressing the the effect of varying $q$ in stratified numerical simulations. We discuss why shearing boxes cannot easily be used to study $q>2$ and thus focus on $q<2$. As per previous simulations, which were either unstratified or stratified with a smaller vertical domain, we find that the $q$ dependence of stress for the stratified case is not linear, contrary to the Shakura-Sunyaev model. We find that the scaling agrees with \cite{1996MNRAS.281L..21A} who found it to be proportional to the shear to vorticity ratio $q/(2-q)$. We also find however, that the shape of the magnetic and kinetic energy spectra are relatively insensitive to $q$ and that the ratio of Maxwell stress to magnetic energy ratio also remains nearly independent of $q$. This is consistent with a theoretical argument in which the rate of amplification of the azimuthal field depends linearly on $q$ and the turbulent correlation time $\tau$ depends inversely on $q$. As such, we measure the correlation time of the turbulence and find that indeed it is inversely proportional to $q$.
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