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

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 (Pm) is lowered. On the other hand, calculations for laboratory experiments show that saturation is achieved through modification of the background shear for Pm << 1. Guided by the results of calculations appropriate for laboratory experiments when Pm is very low, the axisymmetric 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. It is found that the emergence and stability of the MRI is sensitive to the boundary conditions adopted. Channel modes do not appear to be stabilized through any modification 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.Comment: To Appear in Astronomy and Astrophysic

The Equations of Magnetoquasigeostrophy

The dynamics contained in magnetized layers of exoplanet atmospheres are important to understand in order to characterize what observational signatures they may provide for future observations. It is important to develop a framework to begin studying and learning the physical processes possible under those conditions and what, if any, features contained in them may be observed in future observation missions. The aims of this study is to formally derive, from scaling arguments, a manageable reduced set of equations for analysis, i.e. a magnetic formulation of the equations of quasigeostrophy appropriate for a multi-layer atmosphere. The main goal is to provide a simpler theoretical platform to explore the dynamics possible within confined magnetized layers of exoplanet atmospheres. We primarily use scaling arguments to derive the reduced equations of "magnetoquasigeostrophy" which assumes dynamics to take place in an atmospheric layer which is vertically thin compared to its horizontal scales. The derived equation set retains features existing in standard shallow-water magnetohydrodynamic equations but are absent in more classical derivations of the quasi-geostrophic limit, namely, the non-divergence of the in-plane components of the magnetic field. We liken this non-divergence of the in-plane magnetic fields as indicative of a quantity whose behaviour mimics a two-dimensional "pseudo"-magnetic monopole source. We also find, using the same scaling argument procedures, appropriate limits of the fundamental parameters of the system which yield reduced equations describing the flow dynamics primarily characterized by magnetostrophic balance. The standard scaling arguments employed here show how traditional magnetized quasigeostrophic equations connect to their magnetized shallow water forms. The equations derived are amenable to analysis using well-known techniques.Comment: 13 pages. Under consideration for publication in Astronomy and Astrophysic

Hydrodynamic response of rotationally supported flows in the Small Shearing Box model

The hydrodynamic response of the inviscid small shearing box model of a midplane section of a rotationally supported astrophysical disk is examined. An energy functional ${\cal E}$ is formulated for the general nonlinear problem. It is found that the fate of disturbances is related to the conservation of this quantity which, in turn, depends on the boundary conditions utilized: ${\cal E}$ is conserved for channel boundary conditions while it is not conserved in general for shearing box conditions. Linearized disturbances subject to channel boundary conditions have normal-modes described by Bessel Functions and are qualitatively governed by a quantity $\Sigma$ which is a measure of the ratio between the azimuthal and vertical wavelengths. Inertial oscillations ensue if $\Sigma >1$ - otherwise disturbances must in general be treated as an initial value problem. We reflect upon these results and offer a speculation.Comment: 6 pages, resubmitted to Astronomy and Astrophysics, shortened with references adde

Non-exponential hydrodynamical growth in density-stratified thin Keplerian discs

The short time evolution of three dimensional small perturbations is studied. Exhibiting spectral asymptotic stability, thin discs are nonetheless shown to host intensive hydrodynamical activity in the shape of non modal growth of initial small perturbations. Two mechanisms that lead to such behavior are identified and studied, namely, non-resonant excitation of vertically confined sound waves by stable planar inertia-coriolis modes that results in linear growth with time, as well as resonant coupling of those two modes that leads to a quadratic growth of the initial perturbations. It is further speculated that the non modal growth can give rise to secondary strato-rotational instabilities and thus lead to a new route to turbulence generation in thin discs

Linear dynamics of weakly viscous accretion disks: A disk analog of Tollmien-Schlichting waves

This paper discusses new perspectives and approaches to the problem of disk dynamics where, in this study, we focus on the effects of viscous instabilities influenced by boundary effects. The Boussinesq approximation of the viscous large shearing box equations is analyzed in which the azimuthal length scale of the disturbance is much larger than the radial and vertical scales. We examine the stability of a non-axisymmetric potential vorticity mode, i.e. a PV-anomaly. in a configuration in which buoyant convection and the strato-rotational instability do not to operate. We consider a series of boundary conditions which show the PV-anomaly to be unstable both on a finite and semi-infinite radial domains. We find these conditions leading to an instability which is the disk analog of Tollmien-Schlichting waves. When the viscosity is weak, evidence of the instability is most pronounced by the emergence of a vortex sheet at the critical layer located away from the boundary where the instability is generated. For some boundary conditions a necessary criterion for the onset of instability for vertical wavelengths that are a sizable fraction of the layer's thickness and when the viscosity is small is that the appropriate Froude number of the flow be greater than one. This instability persists if more realistic boundary conditions are applied, although the criterion on the Froude number is more complicated. The unstable waves studied here share qualitative features to the instability seen in rotating Blasius boundary layers. The implications of these results are discussed. An overall new strategy for exploring and interpreting disk instability mechanisms is also suggested.Comment: Accepted for publication in Astronomy and Astrophysics. 18 pages. This version 3 with corrected style fil

A shallow-water theory for annular sections of Keplerian Disks

A scaling argument is presented that leads to a shallow water theory of non-axisymmetric disturbances in annular sections of thin Keplerian disks. To develop a theoretical construction that will aid in physically understanding the relationship of known two-dimensional vortex dynamics to their three-dimensional counterparts in Keplerian disks. 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 $2\pi$ angular extent of the disk. The scalings lead to dynamics which are radially geostrophic and vertically hydrostatic. It follows that 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 is developed and is shown to support a propagating Rossby edgewave. Linear non-uniform potential vorticity solutions are also developed and are shown to be similar in some respects to the dynamics of strictly two-dimensional inviscid flows. 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. The shallow water formulation achieved by the asymptotic theory developed here opens a new approach to studying disk dynamics.Comment: Accepted (July 21, 2008), now in final for