Rossby Wave Instability of Keplerian Accretion Disks

Abstract

We find a linear instability of non-axisymmetric Rossby waves in a thin non-magnetized Keplerian disk when there is a local maximum in the radial profile of a key function ${\cal L}(r) \equiv {\cal F}(r) S^{2/Γ}(r)$, where ${\cal F}^{-1} = \hat {\bf z}\cdot ({\bf \nabla}\times {\bf v}) /Σ$ is the potential vorticity, $S = P/Σ^Γ$ is the entropy, $Σ$ is the surface mass density, $P$ is the vertically integrated pressure, and $Γ$ is the adiabatic index. We consider in detail the special case where there is a local maximum in the disk entropy profile $S(r)$. This maximum acts to trap the waves in its vicinity if its height to width ratio ${\rm max}(S)/Δr$ is larger than a threshold value. The pressure gradient derived from this entropy variation provides the restoring force for the wave growth. We show that the trapped waves act to transport angular momentum outward. A plausible way to produce an entropy variation is when an accretion disk is starting from negligible mass and temperature, therefore negligible entropy. As mass accumulates by either tidal torquing, magnetic torquing, or Roche-lobe overflow, confinement of heat will lead to an entropy maximum at the outer boundary of the disk. Possible nonlinear developments from this instability include the formation of Rossby vortices and the formation of spiral shocks. What remains to be determined from hydrodynamic simulations is whether or not Rossby wave packets (or vortices) ``hold together'' as they propagate radially inward.