We explore the linear stability of astrophysical discs exhibiting vertical shear, which arises when there is a radial variation in the temperature or entropy. Such discs are subject to a "vertical-shear instability", which recent nonlinear simulations have shown to drive hydrodynamic activity in the MRI-stable regions of protoplanetary discs. We first revisit locally isothermal discs using the quasi-global reduced model derived by Nelson et al. (2013). This analysis is then extended to global axisymmetric perturbations in a cylindrical domain. We also derive and study a reduced model describing discs with power law radial entropy profiles ("locally polytropic discs"), which are somewhat more realistic in that they possess physical (as opposed to numerical) surfaces. In all cases the fastest growing modes have very short wavelengths and are localised at the disc surfaces (if present), where the vertical shear is maximal. An additional class of modestly growing vertically global body modes is excited, corresponding to destabilised classical inertial waves ("r-modes"). We discuss the properties of both types of modes, and stress that those that grow fastest occur on the shortest available length scales (determined either by the numerical grid or the physical viscous length). This ill-posedness makes simulations of the instability difficult to interpret. We end with some brief speculation on the nonlinear saturation and resulting angular momentum transport.
Vertical Structure and Turbulent Saturation Level in Fully Radiative Protoplanetary Disc Models
利用3D辐射磁流体动力学模拟方法研究了一个大质量（$\varSigma \sim 10000 g cm^{-2}$ at 1 AU）原行星盘模型。盘的垂直结构由MRI引起的湍流加热和辐射冷却之间的平衡自洽地确定。关于垂直结构，可以区分两个不同的区域：一个是气压主导的，光学厚的中面区域，大部分的耗散发生，和一个磁主导的，光学薄的日冕，这是由强冲击为主。在光球层位置，湍流是超音速的（$M \sim 2$），这与以前从YSO光谱拟合得到的结果是一致的。众所周知，MRI诱导湍流模拟中的湍流饱和度确实取决于数值因素，例如数值分辨率和盒子尺寸。然而，通过在不同的分辨率（使用高达64x128x512网格单元）和不同的盒子大小（在垂直方向上高达16个压力刻度高度）下执行一套运行，我们发现，一旦在垂直方向上达到足够的分辨率，饱和度和加热速率都显示出明显的收敛趋势。
