On Rossby waves and vortices with differential rotation

Abstract

We present a simplified model for linearized perturbations in a fluid with both differential rotation and differential vorticity. Without the latter the model reduces to the classical Shearing Sheet used in the description of spiral density waves in astrophysical disks. Without the former it reduces to the $β$-plane approximation, used in the description of Rossby waves. Retaining both, our model allows one in general to discuss the coupling between density waves and Rossby waves, resulting in what is known as the ``corotation resonance'' for density waves. Here we derive, as a first application of this model, the properties of Rossby waves in a differentially rotating disk. We find that their propagation is quenched by differential rotation: after a limited number of oscillations, a Rossby wave collapses to a singular vortex, as fluid elements are sheared apart by differential rotation. In a keplerian disk, this number of oscillations is always lower than one. We also describe how, in a similar manner, a vortex is sheared in a very short time.