On Vertically Global, Horizontally Local Models for Astrophysical Disks
  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.
