Rescaling Transforms for Local Models of Spherical Flows

Abstract: Previously we developed a local model for a spherically contracting/expanding gas cloud that can be used to study turbulence and small scale instabilities in such flows. In this work we generalise the super-comoving variables used in studies of cosmological structure formation to our local spherical flow model, which make it significantly easier to derive analytical solutions and analyse the interactions of more complex flows with the background. We show that a wide class of solutions to the local spherical flow model can be obtained via a mapping from the corresponding solutions in regular Cartesian flows. The rescaling of time in the transformation results in a modification of the linear instabilities that can occur in spherical flows, causing them to have a time dependent growth rate in the physical time coordinate, and can prevent slower instabilities from operating. Finally, we show that the small scale flows in isotropic contraction/expansion can be mapped directly to Cartesian, inviscid, incompressible hydrodynamics, meaning that one expects a form of rescaled Kolmogorov-turbulence at the small scale of isotropically contracting/expanding flows.