Meridional circulation in the atmospheres of uniformly rotating stars of early spectral type

It has been recognised since 1925 (Eddington:''The Internal Constitution of the Stars" (C.U.P. 1930, 285) that von Zeipel's paradox for uniformly rotating stars could be resolved if a large- scale circulation were set up in meridian planes. Investigations since then have shown, amongst other things, that the velocity of the circulation currents is inversely proportional to the density. Thus, even though the currents are very slow deep inside a star, they become very fast near the surface. If simple, zero density boundary conditions are applied at the surface, there is a formal singularity there,Although the surface layers of non-rotating stars are now understood in considerable detail, the same cannot be said for rotating stars. It appears that a detailed theory of the surface layers must take circulation into account. The main purpose of this thesis is to develop such a theory, with particular emphasis on the removal of the surface singularity.This singularity must arise from the neglect of some important physical factor. It has normally been assumed that viscous and inertial forces are negligible, and this assumption must clearly be questioned when the theory predicts very large velocities, However, a preliminary investigation by the author suggested that this assumption is valid arbitrarily near the surface if the rotation speed is slow enough. An assumption which is certainly invalid whatever the rotation speed is that the photon mean free path is short near the surface. That assumption is implicit in the use of an equation for the radiative flux of a form normally used only in the theory of stellar interiors. Accordingly, a theory of the surface layers has been developed in thesis which uses the non-local radiative transfer equation appropriate to the theory of stellar atmospheres. It is found that, although the use of a non-local transfer equation does remove the formal singularity at the surface, the circulation speeds near the surface are still unrealistically large, when the assumption that viscous and inertial forces can be neglected is re-examined, it is found that, although inertial forces do become important near the surface, these forces are not sufficient to damp the speed of the flow. However, the circulation violates a stability criterion based on the Richardson number (see, for example, L, Prandtl, Essentials of Fluid Dynamics, Blackie 1952), and the flow becomes turbulent in a thin surface layer. Turbulence sets in when the flow speeds are of the order of the speed of sound, and turbulent viscosity then acts to prevent the speeds from further increasing, A qualitative model of the turbulent surface layer has been developed, on the basis of order-of-magnitude estimates. Although no detailed prediction is given for the emergent flux, it is concluded that the commonly used von Zeipel gravity-darkening cannot be correct when a turbulent layer is present