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We review our current knowledge of thermal and viscous instabilities in accretion discs around compact objects. We begin with classical disc models based on analytic viscosity prescriptions, discussing physical uncertainties and exploring…

Magnetic effects have long been considered a possible factor in stabilizing the outer regions of active galactic nuclei (AGN) accretion disks against gravitational instability (GI). However, the computational demands of testing this…

We study general semilinear scalar-field equations on the real line with variable coefficients in the linear terms. These coefficients are uniformly small, but slowly decaying, perturbations of a constant-coefficient operator. We are…

Using a matched asymptotic expansion we analyze the two-dimensional, near- critical reflection of a weakly nonlinear, internal gravity wave from a sloping boundary in a uniformly stratified fluid. Taking a distinguished limit in which the…

A new type of instability that is expected to drive magnetohydrodynamic (MHD) turbulence from a purely toroidal magnetic field in an accretion disk is presented. It is already known that in a differentially rotating system, the uniform…

Protoplanetary discs should exhibit a weak vertical variation in their rotation profiles. Typically this `vertical shear' issues from a baroclinic effect driven by the central star's radiation field, but it might also arise during the…

MHD in protostellar discs is modified by the Hall current when the ambipolar diffusion approximation breaks down. Here I examine the Balbus-Hawley (magnetorotational) instability of a weak, vertical magnetic field within a weakly-ionized…

We present global 2D and 3D simulations of self-gravitating magnetized tori. We used the 2D calculations to demonstrate that the properties of the MRI are not affected by the presence of self-gravity: MHD turbulence and enhanced angular…

Accretion disks in close binary systems are subject to a tidally driven parametric instability which leads to the growth of internal waves near the outer edges of such disks (Goodman 1993). These waves are important in understanding the…

We consider an unregularized optimal control problem subject to the steady-state Navier-Stokes equations. We derive the existence of optimal solutions and prove first- and second-order optimality conditions. To approximate solutions to the…

This research was stimulated by the recent studies of damping solutions in dynamically stable spherical stellar systems. Using the simplest model of the homogeneous stellar medium, we discuss nontrivial features of stellar systems. Taking…

We explore the linear stability of astrophysical discs exhibiting vertical shear, which arises when there is a radial variation in the temperature or entropy. Such discs are subject to a "vertical-shear instability", which recent nonlinear…

In certain models of conformal gravity, the propagation of gravitational waves is governed by a fourth order scalar partial differential equation. We study the initial value problem for a generalization of this equation, and derive a…

We consider local, stratified, numerical models of isothermal accretion disks. The novel feature of our treatment is that radial extent L_x and azimuthal extent L_y satisfy H << L_x, L_y << R, where H is the scale height and R is the local…

Magnetized disk winds and wind-driven accretion are an essential and intensively studied dispersion mechanism of protoplanetary disks. However, the stability of these mechanisms has yet to be adequately examined. This paper employs…

The conditions that limit the possible excitation of ideal MHD axisymmetric ballooning modes in thin accretion disks are discussed. As shown earlier by Coppi and Coppi (2001), these modes are well-localized in the vertical direction but…

We present a numerical investigation of three-dimensional, short-wavelength linear instabilities in Kelvin-Helmholtz (KH) vortices in homogeneous and stratified environments. The base flow, generated using two-dimensional numerical…

We study the stability of gaps opened by a giant planet in a self-gravitating protoplanetary disc. We find a linear instability associated with both the self-gravity of the disc and local vortensity maxima which coincide with gap edges. For…

A new class of disk MHD equilibrium solutions is described, which is valid within the standard local (``shearing sheet'') approximation scheme. These solutions have the following remarkable property: velocity streamlines and magnetic lines…

Fragmentation of rotating gaseous systems via gravitational instability is believed to be a crucial mechanism in several astrophysical processes, such as formation of planets in protostellar discs, of molecular clouds in galactic discs, and…