Abstract:
Numerical simulations are presented to study the stability of gaps opened by giant planets in 3D self-gravitating disks. In weakly self-gravitating disks, a few vortices develop at the gap edge and merge on orbital time-scales. The result is one large but weak vortex with Rossby number -0.01. In moderately self-gravitating disks, more vortices develop and their merging is resisted on dynamical time-scales. Self-gravity can sustain multi-vortex configurations, with Rossby number -0.2 to -0.1, over a time-scale of order 100 orbits. Self-gravity also enhances the vortex vertical density stratification, even in disks with initial Toomre parameter of order 10. However, vortex formation is suppressed in strongly self-gravitating disks and replaced by a global spiral instability associated with the gap edge which develops during gap formation.

Abstract:
Numerical simulations of global three-dimensional (3D), self-gravitating discs with a gap opened by an embedded planet are presented. The simulations are customised to examine planetary gap stability. Previous results, obtained by Lin & Papaloizou from two-dimensional (2D) disc models, are reproduced in 3D. These include (i) the development of vortices associated with local vortensity minima at gap edges and their merging on dynamical timescales in weakly self-gravitating discs, (ii) the increased number of vortices as the strength of self-gravity is increased and their resisted merging, and (iii) suppression of the vortex instability and development of global spiral arms associated with local vortensity maxima in massive discs. The vertical structure of these disturbances are examined. In terms of the relative density perturbation, the vortex disturbance has weak vertical dependence when self-gravity is neglected. Vortices become more vertically stratified with increasing self-gravity. This effect is seen even when the unperturbed region around the planet's orbital radius has a Toomre stability parameter ~10. The spiral modes display significant vertical structure at the gap edge, with the midplane density enhancement being several times larger than that near the upper disc boundary. However, for both instabilities the vertical Mach number is typically a few per cent,and on average vertical motions near the gap edge do not dominate horizontal motions.

Abstract:
Vortex formation through the Rossby wave instability (RWI) in protoplanetary discs has been invoked to play a role in planet formation theory, and suggested to explain the observation of large dust asymmetries in several transitional discs. However, whether or not vortex formation operates in layered accretion discs, i.e. models of protoplanetary discs including dead zones near the disc midplane --- regions that are magnetically inactive and the effective viscosity greatly reduced --- has not been verified. As a first step toward testing the robustness of vortex formation in layered discs, we present non-linear hydrodynamical simulations of global 3D protoplanetary discs with an imposed kinematic viscosity that increases away from the disc midplane. Two sets of numerical experiments are performed: (i) non-axisymmetric instability of artificial radial density bumps in viscous discs; (ii) vortex-formation at planetary gap edges in layered discs. Experiment (i) shows that the linear instability is largely unaffected by viscosity and remains dynamical. The disc-planet simulations also show the initial development of vortices at gap edges, but in layered discs the vortices are transient structures which disappear well into the non-linear regime. We suggest that the long term survival of columnar vortices, such as those formed via the RWI, requires low effective viscosity throughout the vertical extent of the disc, so such vortices do not persist in layered discs.

Abstract:
Protoplanetary discs may become dynamically unstable due to structure induced by an embedded giant planet. In this thesis, I discuss the stability of such systems and explore the consequence of instability on planetary migration. I begin with non-self-gravitating, low viscosity discs and show that giant planets induce shocks inside its co-orbital region, leading to a profile unstable to vortex formation around a potential vorticity minimum. This instability is commonly known as the vortex or Rossby wave instability. Vortex-planet interaction lead to episodic phases of migration, which can be understood in the framework of type III migration. I then examine the effect of disc self-gravity on gap stability. The linear theory of the Rossby wave instability is extended to include disc gravity, which shows that self-gravity is effective at stabilising the vortex instability at small azimuthal wavenumber. This is consistent with the observation that more vortices develop with increasing disc mass in hydrodynamic simulations. Vortices in self-gravitating discs also resist merging, and is most simply understood as pair-vortices undergoing mutual horsehoe turns upon encounter. I show that in sufficiently massive discs vortex modes are suppressed. Instead, global spiral instabilities develop which are associated with a potential vorticity maximum at the gap edge. These edge modes can be physically understood as a result of unstable interaction between the gap edge and the exterior disc through gravity. I show the spiral arms can provide a positive torque on the planet, leading to fast migration outwards. I confirm the above results, obtained from razor-thin disc models, persist in three-dimensions.

Abstract:
Numerical calculations of the linear Rossby wave instability (RWI) in global three-dimensional (3D) disks are presented. The linearized fluid equations are solved for vertically stratified, radially structured disks with either a locally isothermal or polytropic equation of state, by decomposing the vertical dependence of the perturbed hydrodynamic quantities into Hermite and Gegenbauer polynomials, respectively. It is confirmed that the RWI operates in 3D. For perturbations with vertical dependence assumed above, there is little difference in growth rates between 3D and two-dimensional (2D) calculations. Comparison between 2D and 3D solutions of this type suggest the RWI is predominantly a 2D instability and that three-dimensional effects, such as vertical motion, to be interpreted as a perturbative consequence of the dominant 2D flow. The vertical flow around co-rotation, where vortex-formation is expected, is examined. In locally isothermal disks the expected vortex center remains in approximate vertical hydrostatic equilibrium. For polytropic disks the vortex center has positive vertical velocity, whose magnitude increases with decreasing polytropic index $n$.

Abstract:
We describe a new mechanism that leads to the destabilisation of non-axisymmetric waves in astrophysical discs with an imposed radial temperature gradient. This might apply, for example, to the outer parts of protoplanetary discs. We use linear density wave theory to show that non-axisymmetric perturbations generally do not conserve their angular momentum in the presence of a forced temperature gradient. This implies an exchange of angular momentum between linear perturbations and the background disc. In particular, when the disturbance is a low-frequency trailing wave and the disc temperature decreases outwards, this interaction is unstable and leads to the growth of the wave. We demonstrate this phenomenon through numerical hydrodynamic simulations of locally isothermal discs in 2D using the FARGO code and in 3D with the ZEUS-MP and PLUTO codes. We consider radially structured discs with a self-gravitating region which remains stable in the absence of a temperature gradient. However, when a temperature gradient is imposed we observe exponential growth of a one-armed spiral mode (azimuthal wavenumber $m=1$) with co-rotation radius outside the bulk of the spiral arm, resulting in a nearly-stationary one-armed spiral pattern. The development of this one-armed spiral does not require the movement of the central star, as found in previous studies. Because destabilisation by a forced temperature gradient does not explicitly require disc self-gravity, we suggest this mechanism may also affect low-frequency one-armed oscillations in non-self-gravitating discs.

Abstract:
Astrophysical disks with localized radial structure, such as protoplanetary disks containing dead zones or gaps due to disk-planet interaction, may be subject to the non-axisymmetric Rossby wave instability (RWI) that lead to vortex-formation. The linear instability has recently been demonstrated in three-dimensional (3D) barotropic disks. It is the purpose of this study to generalize the 3D linear problem to include an energy equation, thereby accounting for baroclinity in three-dimensions. Linear stability calculations are presented for radially structured, vertically stratified, geometrically-thin disks with non-uniform entropy distribution in both directions. Polytropic equilibria are considered but adiabatic perturbations assumed. The unperturbed disk has a localized radial density bump making it susceptible to the RWI. The linearized fluid equations are solved numerically as a partial differential equation eigenvalue problem. Emphasis on the ease of method implementation is given. It is found that when the polytropic index is fixed and adiabatic index increased, non-uniform entropy has negligible effect on the RWI growth rate, but pressure and density perturbation magnitudes near a pressure enhancement increases away from the midplane. The associated meridional flow is also qualitatively changed from homentropic calculations. Meridional vortical motion is identified in the nonhomentropic linear solution, as well as in a nonlinear global hydrodynamic simulation of the RWI in an initially isothermal disk evolved adiabatically. Numerical results suggest buoyancy forces play an important role in the internal flow of Rossby vortices.

Abstract:
The linear Rossby wave instability (RWI) in global, 3D polytropic discs is revisited with a much simpler numerical method than that previously employed by the author. The governing partial differential equation is solved with finite differences in the radial direction and spectral collocation in the vertical direction. RWI modes are calculated subject to different upper disc boundary conditions. These include free surface, solid boundaries and variable vertical domain size. Boundary conditions that oppose vertical motion increase the instability growth rate by a few per cent. The magnitude of vertical flow throughout the fluid column can be affected but the overall flow pattern is qualitatively unchanged. Numerical results support the notion that the RWI is intrinsically two dimensional. This implies that inconsistent upper disc boundary conditions, such as vanishing enthalpy perturbation, may inhibit the RWI in 3D.

Abstract:
Magneto-rotational instability (MRI) and gravitational instability (GI) are the two principle routes to turbulent angular momentum transport in accretion disks. Protoplanetary disks may develop both. This paper aims to reinvigorate interest in the study of magnetized massive protoplanetary disks, starting from the basic issue of stability. The local linear stability of a self-gravitating, uniformly magnetized, differentially rotating, three-dimensional stratified disk subject to axisymmetric perturbations is calculated numerically. The formulation includes resistivity. It is found that the reduction in the disk thickness by self-gravity can decrease MRI growth rates; the MRI becomes global in the vertical direction, and MRI modes with small radial length scales are stabilized. The maximum vertical field strength that permits the MRI in a strongly self-gravitating polytropic disk with polytropic index $\Gamma=1$ is estimated to be $B_{z,\mathrm{max}} \simeq c_{s0}\Omega\sqrt{\mu_0/16\pi G} $, where $c_{s0}$ is the midplane sound speed and $\Omega$ is the angular velocity. In massive disks with layered resistivity, the MRI is not well-localized to regions where the Elsasser number exceeds unity. For MRI modes with radial length scales on the order of the disk thickness, self-gravity can enhance density perturbations, an effect that becomes significant in the presence of a strong toroidal field, and which depends on the symmetry of the underlying MRI mode. In gravitationally unstable disks where GI and MRI growth rates are comparable, the character of unstable modes can transition smoothly between MRI and GI. Implications for non-linear simulations are discussed briefly.

Abstract:
Several observations of transition discs show lopsided dust-distributions. A potential explanation is the formation of a large-scale vortex acting as a dust-trap at the edge of a gap opened by a giant planet. Numerical models of gap-edge vortices have thus far employed locally isothermal discs, but the theory of this vortex-forming or `Rossby wave' instability was originally developed for adiabatic discs. We generalise the study of planetary gap stability to non-isothermal discs using customised numerical simulations of disc-planet systems where the planet opens an unstable gap. We include in the energy equation a simple cooling function with cooling timescale $t_c=\beta\Omega_k^{-1}$, where $\Omega_k$ is the Keplerian frequency, and examine the effect of $\beta$ on the stability of gap edges and vortex lifetimes. We find increasing $\beta$ lowers the growth rate of non-axisymmetric perturbations, and the dominant azimuthal wavenumber $m$ decreases. We find a quasi-steady state consisting of one large-scale, over-dense vortex circulating the outer gap edge, typically lasting $O(10^3)$ orbits. Vortex lifetimes were found to generally increase with cooling times up to an optimal value, beyond which vortex lifetimes decrease. This non-monotonic dependence is qualitatively consistent with recent studies using strictly isothermal discs that vary the disc aspect ratio.