Large swathes of protoplanetary disks are too poorly ionised for the magnetorotational instability to be active. In particular, the cold dense interior regions, between roughly 1 and 10 AU, could be entirely laminar. In the last 10-15 years this state of affairs has renewed interest in hydrodynamical routes to turbulence in these `dead zones', and various weak instabilities, often of a double-diffusive nature, have been uncovered that might operate there. Examples include (a) the vertical shear instability (basically the Goldreich-Schubert-Fricke instability), (b) the convective overstability (related to semiconvection in stars and planets), (c) the subcritical baroclinic instability, and (d) the Rossby wave instability (which one could argue has nothing to do with Rossby waves!).

It is unlikely that any of these instabilities is particular widespread, nor the solution to the question of angular momentum transport, but under certain circumstances they should be important, especially for solid dynamics and the process of planet formation. I also just find their physics inherently intriguing, and so I like to play with them. Perhaps of greatest interest is the potential of all four instabilities mentioned to form zonal flows and vortices on large or small scales. Vortices can accumulate small particles and aid in the vexed business of planet formation.

This area is quite well trodden and so I can't think of a topic that could sustain an entire PhD. Sorry! I might have projects suitable for a MSc though.

Recent papers

*On the survival of zombie vortices in protoplanetary discs (link)
*On the convective overstability in protoplanetary discs (link)
*On the vertical-shear instability in astrophysical discs (link)
*Vortex cycles at the inner edges of dead zones in protoplanetary disks (link)