  Spherical flows are a classic problem in astrophysics which are typically
studied from a global perspective. However, much like with accretion discs,
there are likely many instabilities and small scale phenomena which would be
easier to study from a local perspective. For this purpose, we develop a local
model for a spherically contracting/expanding gas cloud, in the spirit of the
shearing box, $\beta$-plane and expanding box models which have had extensive
use in studies of accretion discs, planets and stellar winds respectively. The
local model consists of a, spatially homogeneous, periodic box with a time
varying aspect ratio, along with a scale factor (analogous to that in
FRW/Newtonian cosmology) relating the box coordinates to the physical
coordinates of the global problem. We derive a number of symmetries and
conservation laws exhibited by the local model. Some of these reflect
symmetries of the periodic box, modified by the time dependant geometry, while
others are local analogues for symmetries of the global problem. The energy,
density and vorticity in the box also generically increase(/decrease) as a
consequence of the collapse(/expansion). We derive a number of nonlinear
solutions, including a local analogue of uniform density zonal flows, which
grow as a consequence of angular momentum conservation. Our model is closely
related to the accelerated expanding box model of Tenerani \& Velli and is an
extension of the isotropic model considered by Robertson \& Goldreich.
