An affine model of the dynamics of astrophysical discs

Abstract

© 2018 The Author(s). Published by Oxford University Press on behalf of the Royal Astronomical Society. Thin astrophysical discs are very often modelled using the equations of 2D 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 3D 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 2D hydrodynamics in the reference plane, the theory allows for a deformation of the mid-plane (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 3D 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.