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Dynamical instabilities in disc-planet interactions

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.