The prevalence and consequences of convection perpendicular to the plane of accretion disks have been discussed for several decades. Recent simulations combining convection and the magnetorotational instability have given fresh impetus to the debate, as the interplay of the two processes can enhance angular momentum transport, at least in the optically thick outburst stage of dwarf novae. In this thesis we seek to isolate and understand the most generic features of disk convection, and so undertake its study in both hydrodynamical and magnetohydrodynamical models.

In the first part of this thesis we investigate hydrodynamic convection in disks. First, we investigate the linear phase of the instability, obtaining estimates of the growth rates both semi-analytically, using one-dimensional spectral computations, as well as analytically, using WKBJ methods. Next we perform three-dimensional, vertically stratified, shearing box simulations with the conservative, finite-volume code PLUTO, both with and without explicit diffusion coefficients. We find that hydrodynamic convection can, in general, drive outward angular momentum transport, a result that we confirm with ATHENA, an alternative finite-volume code. Moreover, we establish that the sign of the angular momentum flux is sensitive to the diffusivity of the numerical scheme. Finally, in sustained convection, whereby the system is continuously forced to an unstable state, we observe the formation of various coherent structures, including large-scale and oscillatory convective cells, zonal flows, and small vortices.

In the second part of this thesis we investigate magnetohydrodynamic convection in disks. First, we perform inviscid, three-dimensional, vertically stratified zero-net-flux MHD shearing box simulations with the conservative, finite-volume code PLUTO without explicit cooling. We find that MRI heating and weak cooling facilitated through advection of material across the vertical boundaries alone results in a convectively stable disk structure. Next we explore the interaction between convection and the MRI in controlled numerical experiments in which we employ an explicit height-dependent cooling prescription and explicit uniform resistivity. We find two characteristic outcomes of the interaction between the two instabilities: MRI-dominated and MRI/convective cycles. In particular we find that MRI/convective cycles lead to alternating phases of convection-dominated quiescence (during which the turbulent transport is weak) and MRI-dominated outbursts. During these outbursts angular momentum transport is enhanced by nearly an order of magnitude. Thus convection and the MRI do not generally interact in an additive manner, though they can certainly interact in non-trivial ways. In addition we find that convection in the non-linear phase takes the form of large-scale and oscillatory convective cells, reproducing a key result of our hydrodynamic investigations in a self-consistent manner, and demonstrating that these structures are a generic feature of turbulent convection in astrophysical disks.

In the final part of this thesis (which is independent of the first two) we investigate the stress-pressure relationship in disks. The stresses accompanying MRI turbulence are related to the pressure in the disk, and have been shown to increase and decrease with the pressure. We examine the time lag associated with this dependence and discuss its implications for thermal instability.