  Local models of gaseous accretion discs have been successfully employed for
decades to describe an assortment of small-scale phenomena, from instabilities
and turbulence, to dust dynamics and planet formation. For the most part, they
have been derived in a physically motivated but essentially ad hoc fashion,
with some of the mathematical assumptions never made explicit nor checked for
consistency. This approach is susceptible to error, and it is easy to derive
local models that support spurious instabilities or fail to conserve key
quantities. In this paper we present rigorous derivations, based on an
asympototic ordering, and formulate a hierarchy of local models
(incompressible, Boussinesq, and compressible), making clear which is best
suited for a particular flow or phenomenon while spelling out explicitly the
assumptions and approximations of each. We also discuss the merits of the
anelastic approximation, emphasising that anelastic systems struggle to
conserve energy unless strong restrictions are imposed on the flow. The
problems encountered by the anelastic approximation are exacerbated by the
disk's differential rotation, but also attend non-rotating systems such as
stellar interiors. We conclude with a defence of local models and their
continued utility in astrophysical research.
