Convection in pulsating stars. I - Nonlinear hydrodynamics. II - RR Lyrae convection and stability

A nonlinear, nonlocal, time-dependent treatment of convection suitable for use in models of cool giant stars is presented. Local conservation equations plus a diffusive transport equation are used to derive the convective hydrodynamic equations for the case in which turbulent pressure, energy, and viscosity cannot be ignored. The effects of convective overshooting, superadiabatic gradients, convection/pulsation interaction, and time dependence enter this treatment in a natural way. Methods of treating turbulent viscosity and acoustic losses are discussed. Also, an efficient computational scheme for computing the derivatives needed for an implicit hydrodynamic code is outlined. Application to RR Lyrae star envelopes will be presented in a companion paper.