An affine model of the dynamics of astrophysical discs

Abstract

Thin astrophysical discs are very often modelled using the equations of twodimensional 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.

A simplified model for the secular dynamics of eccentric discs and applications to planet–disc interactions

“…Finally, it is of interest to compare our planet -eccentric disc model with planet -inclined disc models studied by many authors. The theory of warped discs in the presence of an inclined perturber is as complex as that of eccentric discs, if not more so (see, e.g., Papaloizou & Pringle 1983;Papaloizou & Lin 1995;Ogilvie 1999;Lubow & Ogilvie 2000;Ogilvie 2001;Lubow & Ogilvie 2001;Ogilvie 2006;Ogilvie & Latter 2013;Foucart & Lai 2014;Ogilvie 2018). However, if we assume that the disc warp is small and that the disc is able to maintain coherent rigid-body-like nodal precession -both assumptions can be justified in many situations, then the evolution equations governing an inclined planet -disc system become very simple, and can lead to many novel applications (see, e.g., Batygin 2012;Lai 2014;Zanazzi & Lai 2018a,b,c).…”

On the equations of warped disc dynamics

“…Finally, we will show how our set of equations can be intuitively interpreted using the affine tilted slab picture of Ogilvie (2018), thereby resolving the apparent "first glance misconception" mentioned above.…”

Non-linear dynamics of hydrodynamic tori as a model of oscillations and bending waves in astrophysical discs

“…Formally this is a coupled system of 6 non-linear, second order differential equations which govern the evolution of the flow. These can in fact also be derived as a specific sub-case of the general Lagrangian framework for the affine motion of discs as developed by Ogilvie (2018) and discussed in appendix A.…”

“…More recently, Ogilvie (2018) has developed an an affine model of the dynamics of thin discs, which incorporates vertical degrees of freedom wherein the evolution is described at each radial location by the time-dependent affine transformation of a fluid column. We can show that our ring model is a special case of this more general theory.…”

“…Our ring model presents a special case of the affine model for warped discs as developed by Ogilvie (2018), hereafter OL18. Indeed, our Lagrangian formalism can be derived from this theory upon introducing some simplifying assumptions about the reference state and the nature of the linear Jacobian mapping.…”