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Scalings and Asymptotics of Coherent Vortices in Protoplanetary Disks

Gas that is transported radially inward from the outer edge of an accretion disk and onto a forming central star must be in a nearly Keplerian orbit at all radii. To do this, it must give up part of its angular momentum and energy to the ambient gas, which in turn advects angular momentum outward via a secondary flow. Here, we set up the numerical calculation for computing this flow by obtaining simplified sets of 3D, asymptotic equations that are well-posed and can be computed by the same techniques that are used for the 3D, anelastic Euler equation. The asymptotics allow an easy parameterization of the unknown equations of energy and state and boundary conditions. It is shown analytically that the required mass and angular momentum transport cannot occur if the protoplanetary disk is barotropic. However, a small baroclinicity allows it. Scale analysis shows that if 20% of the protoplanetary disk is filled with vortices, then the required transport can occur with a large enough radially inward mass flux to satisfy the astronomical observations.