Non-linear fluid dynamics of eccentric discs

A new theory of eccentric accretion discs is presented. Starting from the b asic fluid-dynamical equations in three dimensions, I derive the fundamenta l set of one-dimensional equations that describe how the mass, angular mome ntum and eccentricity vector of a thin disc evolve as a result of internal stresses and external forcing. The analysis is asymptotically exact in the limit of a thin disc, and allows for slowly varying eccentricities of arbit rary magnitude. The theory is worked out in detail for a Maxwellian viscoel astic model of the turbulent stress in an accretion disc. This generalizes the conventional alpha viscosity model to account for the non-zero relaxati on time of the turbulence, and is physically motivated by a consideration o f the nature of magnetohydrodynamic turbulence. It is confirmed that circul ar discs are typically viscously unstable to eccentric perturbations, as fo und by Lyubarskij, Postnov & Prokhorov, if the conventional alpha viscosity model is adopted. However, the instability can usually be suppressed by in troducing a sufficient relaxation time and/or bulk viscosity. It is then sh own that an initially uniformly eccentric disc does not retain its eccentri city as had been suggested by previous analyses. The evolutionary equations should be useful in many applications, including understanding the origin of planetary eccentricities and testing theories of quasi-periodic oscillat ions in X-ray binaries.