A Local Model for the Spherical Collapse/Expansion Problem

Abstract

Spherical flows are a classic problem in astrophysics which are typically studied from a global perspective. However, much like with accretion discs, there are likely many instabilities and small scale phenomena which would be easier to study from a local perspective. For this purpose, we develop a local model for a spherically contracting/expanding gas cloud, in the spirit of the shearing box, $\beta$-plane and expanding box models which have had extensive use in studies of accretion discs, planets and stellar winds respectively. The local model consists of a, spatially homogeneous, periodic box with a time varying aspect ratio, along with a scale factor (analogous to that in FRW/Newtonian cosmology) relating the box coordinates to the physical coordinates of the global problem. We derive a number of symmetries and conservation laws exhibited by the local model. Some of these reflect symmetries of the periodic box, modified by the time dependant geometry, while others are local analogues for symmetries of the global problem. The energy, density and vorticity in the box also generically increase(/decrease) as a consequence of the collapse(/expansion). We derive a number of nonlinear solutions, including a local analogue of uniform density zonal flows, which grow as a consequence of angular momentum conservation. Our model is closely related to the accelerated expanding box model of Tenerani \& Velli and is an extension of the isotropic model considered by Robertson \& Goldreich.

arXiv comment20 pages, 6 figures, accepted for publication in MNRAS

References in corpus (19)

• Cosmological Non-Linearities as an Effective Fluid
• Gravitational collapse of magnetized clouds. I. Ideal MHD accretion flow
• Adiabatic Heating of Contracting Turbulent Fluids
• Local and global dynamics of warped astrophysical discs
• Local and global dynamics of eccentric astrophysical discs
• Local models of astrophysical discs
• Evolving waves and turbulence in the outer corona and inner heliosphere: the accelerating expanding box
• Viscous overstability and eccentricity evolution in three-dimensional gaseous discs
• The Local Instability of Steady Astrophysical Flows with non Circular Streamlines with Application to Differentially Rotating Disks with Free Eccentricity
• Propagation of Alfvén waves in the expanding solar wind with the fast-slow stream interaction
• HFQPOs and discoseismic mode excitation in eccentric, relativistic discs. II. Magnetohydrodynamic simulations
• Local numerical simulations of warped discs
• Quasi-periodic oscillations, trapped inertial waves and strong toroidal magnetic fields in relativistic accretion discs
• Conservation of Total Wave Action in the Expanding Solar Wind
• Exact solutions of the aspherical axisymmetric gravo-magnetic condensation
• Semi-analytical homologous solutions of the gravo-magnetic contraction
• Hydrodynamics of warps in the local model of astrophysical discs
• A new hydrodynamic spherical accretion exact solution and its quasi-spherical perturbations
• On Nonshearing Magnetic Configurations in Differentially Rotating Disks