In pursuit of structures in protoplanetary disks

This brief summarizes work devoted to studying the linear stability of baroclinic protoplanetary Keplerian disks. This work largely builds on the foundation and fundamental results obtained by Barranco, Marcus & Umurhan (2000) (hereafter BMU) which presented a model to describe structures in weakly baroclinic Keplerian disks. Interest in protoplanetary disks has grown since the discovery of the first of what are now dozens of extrasolar planets around F and G type stars (Marcy 1996, Perryman 2000). The apparent ubiquity of planets suggests that the nebular disks (or protoplanetary disks), out of which these stars are born, must be susceptible to the development of structures. However, theoretical work (see references in BMU) argues that purely pressure-supported gaseous barotropic Keplerian disks do not undergo any known type of hydrodynamic instabilities. The basis for this claim is that the strong rotation in Keplerian shear flows manages to suppress any of the usual instabilities seen in laboratory shear flows such as the Taylor-Couette and Kelvin-Helmholtz instabilities (Drazin & Reid, 1982). Some investigators (e.g. Shu et al., 1993) have gone so far as to claim that protoplanetary disks are featureless objects totally bereft of vortical structures or long-lasting stable patterns. To counter this assertion, some investigators have suggested that the protoplanetary disk is sufficiently magnetized to undergo a magneto-rotational instability as first explored by Chandresekhar (1956) using linear analysis and later demonstrated numerically by Balbus, Hawley & Stone (1996) for fully nonlinear flows. However, Desch (2000) has indicated that disks suspected to be around solar type stars will be too cool and weakly ionized such that local dust particles in short time will sweep up the ionization in the gas, leaving the fluid nearly neutral. What little ionization is left in the disk is too little to allow strong coupling to an external magnetic field. Thus this mechanism is unlikely to play an important role in the formation of planets at the radii where the large gas giants are observed to reside. The point of view taken by BMU is that a weakly baroclinic Keplerian disk will produce the type of shear flow that will promote hydrodynamical instabilities. The assumption was that the added component of vertical shear driven by baroclinic effects generates coherent structures in an otherwise featureless protoplanetary disk. To isolate this effect they derived a number of asymptotic reductions of the 3D Euler equations in cylindrical geometry with a central mass source. The work here outlines the linear theory of these reduced equations in order to: