A two-dimensional mixing length theory of convective transport

The helioseismic observations of the internal rotation profile of the Sun raise questions about the two-dimensional (2D) nature of the transport of angular momentum in stars. Here we derive a convective prescription for axisymmetric (2D) stellar evolution models. We describe the small scale motions by a spectrum of unstable linear modes in a Boussinesq fluid. Our saturation prescription makes use of the angular dependence of the linear dispersion relation to estimate the anisotropy of convective velocities. We are then able to provide closed form expressions for the thermal and angular momentum fluxes with only one free parameter, the mixing length. We illustrate our prescription for slow rotation, to first order in the rotation rate. In this limit, the thermodynamical variables are spherically symetric, while the angular momentum depends both on radius and latitude. We obtain a closed set of equations for stellar evolution, with a self-consistent description for the transport of angular momentum in convective regions. We derive the linear coefficients which link the angular momentum flux to the rotation rate ($Lambda$- effect) and its gradient ($alpha$-effect). We compare our results to former relevant numerical work.

Low velocity shocks: signatures of turbulent dissipation in diffuse irradiated gas

We examine the chemical and emission properties of mildly irradiated (G0=1) magnetised shocks in diffuse media (nH=10^2 to 10^4 /cm3) at low to moderate velocities (from 3 to 40 km/s). Results: The formation of some molecules relies on endoergic reactions. In J-shocks, their abundances are enhanced by several orders of magnitude for shock velocities as low as 7 km/s. Otherwise most chemical properties of J-type shocks vary over less than an order of magnitude between velocities from about 7 to about 30 km/s, where H2 dissociation sets in. C-type shocks display a more gradual molecular enhancement as the shock velocity increases. We quantify the energy flux budget (fluxes of kinetic, radiated and magnetic energies) with emphasis on the main cooling lines of the cold interstellar medium. Their sensitivity to shock velocity is such that it allows observations to constrain statistical distributions of shock velocities. We fit various probability distribution functions (PDFs) of shock velocities to spectroscopic observations of the galaxy-wide shock in Stephans Quintet (SQ) and of a Galactic line of sight sampling diffuse molecular gas in Chamaeleon. In both cases, low velocities bear the greatest statistical weight and the PDF is consistent with a bimodal distribution. In the very low velocity shocks (below 5 km/s), dissipation is due to ion-neutral friction which powers H2 low energy transitions and atomic lines. In moderate velocity shocks (20 km/s and above), the dissipation is due to viscous heating and accounts for most of the molecular emission. In our interpretation a significant fraction of the gas on the line of sight is shocked (from 4% to 66%). For example, C+ emission may trace shocks in UV irradiated gas where C+ is the dominant carbon species.