Local models of astrophysical discs

TL;DR

This paper provides a rigorous derivation of local models for astrophysical gaseous discs, clarifying assumptions, limitations, and the suitability of different approximations for studying small-scale phenomena.

Contribution

It introduces a hierarchy of locally derived models with explicit assumptions, improving the physical consistency and applicability of astrophysical disc simulations.

Findings

Hierarchical local models with explicit assumptions

Analysis of energy conservation issues in anelastic approximation

Discussion on the suitability of models for different flow regimes

Abstract

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.