A shallow-water theory for annular sections of Keplerian Disks

Context. A scaling argument is presented that leads to a shallow water theory of non-axisymmetric disturbances in annular sections of thin Keplerian disks. Aims. To develop a theoretical construction that will aid in physically understanding the relationship of known twodimensional vortex dynamics to their three-dimensional counterparts in Keplerian disks. Methods. Using asymptotic scaling arguments varicose disturbances of a Keplerian disk are considered on radial and vertical scales consistent with the height of the disk while the azimuthal scales are the full 2π angular extent of the disk. For simplicity perturbations are assumed to be homentropic according to a polytropic equation of state. The timescales considered are long compared to the local disk rotation time. Results. The scalings lead to dynamics which are radially geostrophic and vertically hydrostatic. It follows that a potential vorticity quantity emerges and is shown to be conserved in a Lagrangian sense. Uniform potential vorticity linear solutions are explored and the theory is shown to contain an incarnation of the strato-rotational instability under channel flow conditions. Linearized solutions of a single defect on an infinite domain is developed and is shown to support a propagating Rossby edgewave. Linear non-uniform potential vorticity solutions are also developed and are shown to be similar in some respects to the dynamics of strictly two-dimensional inviscid flows. The relationship of the scalings and some of the resulting dynamics are considered with respect to other approximations employed in the literature. Based on the framework of this theory, arguments based on geophysical notions are advanced in support of the assertion that the strato-rotational instability is barotropic in nature. Extensions of this formalism are also proposed. Conclusions. The shallow water formulation achieved by the asymptotic theory developed here opens a new approach to studying disk dynamics.