2018/02/28 by Gordon I. Ogilvie
Physics and Astronomy · #Affine transformation #Astrophysical Phenomena and Observations #Astrophysics and Star Formation Studies #Classical mechanics #Eulerian path #Geometry #Gravitation #Gravitational field #Mathematical analysis #Mechanics #Perfect fluid #Physics #Solar and Space Plasma Dynamics #Streamlines, streaklines, and pathlines #Tensor (intrinsic definition) #Theoretical physics #Vortex #Vorticity #astro-ph.SR
paper · pdf · doi:10.1093/mnras/sty588
18 pages, 4 figures, to be published in MNRAS
arxiv created 2018/02/28 · openalex publication_date 2018/03/02 · arxiv updated 2018/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.