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We present a simplified model for linearized perturbations in a fluid withboth differential rotation and differential vorticity. Without the latter themodel reduces to the classical Shearing Sheet used in the description of spiraldensity waves in astrophysical disks. Without the former it reduces to the$\beta$-plane approximation, used in the description of Rossby waves.  Retaining both, our model allows one in general to discuss the couplingbetween 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 ofRossby waves in a differentially rotating disk. We find that their propagationis quenched by differential rotation: after a limited number of oscillations, aRossby wave collapses to a singular vortex, as fluid elements are sheared apartby differential rotation. In a keplerian disk, this number of oscillations isalways lower than one. We also describe how, in a similar manner, a vortex issheared in a very short time.Comment: to appear in A&

On Rossby waves and vortices with differential rotation