Stellar evolution with rotation. III. Meridional circulation with MU -gradients and non-stationarity

Models of stellar evolution with rotation are very much needed in order to interpret recent stellar properties, in particular for massive stars. For that we proceed to a detailed investigation of some critical physical processes in rotating stars: 1. The equation expressing the transport of angular momentum by circulation and diffusion is revised to also account for expansion and contraction in non-stationary models. 2. We examine the current expressions for the velocity U(r) of the meridional circulation, also taking into account mu -gradients and horizontal turbulence. We show that there are no ``mu -currents'' but just meridional currents which can be sizeably influenced by the gradients of mean molecular weight mu . A proper account of the mu -gradients may reduce U(r) by one or two orders of magnitude with respect to current expressions. 3. While the usual expressions for the meridional circulation would predict an infinite velocity at the edge of a radiative and semiconvective zone and an inverted circulation in a semiconvective zone, the present developments give a continuity of the solutions for the circulation. 4. The approximation of a stationary circulation, which is no longer valid after the main-sequence phase, is also removed and the case of a general equation of state is considered. We notice that in the stationary regime the horizontal fluctuations of mu represent some fixed fraction of the vertical mu -gradient. To first order, this fraction is not dependent on rotation, because the building of horizontal fluctuations by the circulation is compensated by the smoothing due to horizontal turbulence.