An affine model of the dynamics of astrophysical discs

TL;DR

This paper introduces an affine model for thin astrophysical discs that extends traditional 2D hydrodynamics by incorporating deformations, warps, and internal shearing motions, derived from Hamilton's Principle.

Contribution

The novel affine model captures complex disc deformations and internal motions without relying on small-amplitude assumptions, improving upon standard 2D hydrodynamic models.

Findings

Reproduces linear theories of warped and eccentric discs.

Conserves energy and potential vorticity in non-planar configurations.

Applicable to more general disc dynamics beyond secular approximations.

Abstract

Thin astrophysical discs are very often modelled using the equations of two-dimensional hydrodynamics. We derive an extension of this model that describes more accurately the behaviour of a thin disc in the absence of self-gravity, magnetic fields and complex internal motions. The ideal fluid theory is derived directly from Hamilton's Principle for a three-dimensional fluid after making a specific approximation to the deformation gradient tensor. We express the equations in Eulerian form after projection on to a reference plane. The disc is thought of as a set of fluid columns, each of which is capable of a time-dependent affine transformation, consisting of a translation together with a linear transformation in three dimensions. Therefore, in addition to the usual two-dimensional hydrodynamics in the reference plane, the theory allows for a deformation of the midplane (as occurs in warped discs) and for the internal shearing motions that accompany such deformations. It also allows for the vertical expansions driven in non-circular discs by a variation of the vertical gravitational field around the horizontal streamlines, or by a divergence of the horizontal velocity. The equations of the affine model embody conservation laws for energy and potential vorticity, even for non-planar discs. We verify that they reproduce exactly the linear theories of three-dimensional warped and eccentric discs in a secular approximation. However, the affine model does not rely on any secular or small-amplitude assumptions and should be useful in more general circumstances.