On Vertically Global, Horizontally Local Models for Astrophysical Disks

TL;DR

This paper introduces the vertically global shearing box (VGSB), a new local modeling framework for astrophysical disks that accounts for height-dependent angular frequency, extending the traditional shearing box approach to more complex baroclinic equilibria.

Contribution

The paper develops explicit equations for the VGSB, enabling implementation in MHD codes and allowing for height-dependent rotation profiles in local disk models.

Findings

VGSB can model non-axisymmetric barotropic disks with height-independent angular frequency.

VGSB accommodates height-dependent angular frequency in baroclinic disks.

Application to vertical shear and magnetorotational instabilities demonstrates the framework's utility.

Abstract

Disks with a barotropic equilibrium structure, for which the pressure is only a function of the density, rotate on cylinders in the presence of a gravitational potential, so that the angular frequency of such a disk is independent of height. Such disks with barotropic equilibria can be approximately modeled using the shearing box framework, representing a small disk volume with height-independent angular frequency. If the disk is in baroclinic equilibrium, the angular frequency does generally depend on height, and it is thus necessary to go beyond the standard shearing box approach. In this paper, we show that given a global disk model, it is possible to develop approximate models that are local in horizontal planes without an expansion in height with shearing-periodic boundary conditions. We refer to the resulting framework as the vertically global shearing box (VGSB). These models can be non-axisymmetric for globally barotropic equilibria but should be axisymmetric for globally baroclinic equilibria. We provide explicit equations for this VGSB which can be implemented in standard magnetohydrodynamic codes by generalizing the shearing-periodic boundary conditions to allow for a height-dependent angular frequency and shear rate. We also discuss the limitations that result from the radial approximations that are needed in order to impose height-dependent shearing periodic boundary conditions. We illustrate the potential of this framework by studying a vertical shear instability and examining the modes associated with the magnetorotational instability.