Nonlocal Time-dependent Convection Theory

Starting from radiation hydrodynamics, a complete set of dynamical equations is derived for the second- and third-order correlation functions of velocity and temperature. Assuming that the fluctuations in turbulent velocity and temperature obey the normal distribution (Gaussian), following the Millionshchikov assumption, the fourth-order correlation functions can be expressed with the second-order correlation functions. Anisotropy is carefully considered. It is assumed that the only important effect of pressure fluctuation is to restore the isotropy of turbulence, while all other effects are neglected. In this way, an equation of turbulent viscosity very similar to the Stokes viscosity formula can be set up naturally. Since we adopt an average scheme with weight of the gas density for velocity, enthalpy, and extinction, the treatment of compressibility has been simplified. The Boussinesq approximation is no longer needed. The theory is applicable to stellar convection even though the density changes by several orders of magnitude across the stellar convection zone. There are two convective parameters, c1 and Q, which describe the linear size of the energy-containing eddies and anisotropy of turbulent convection, respectively. In principle, the equations in the current paper can be applied not only to radial pulsation but to nonradial pulsation of stars as well. As a specific case, we give a complete set of equations for stellar radial pulsation, which possess the following main properties: First, the gas and the radiation field are treated separately, and the two components are coupled through the emission and absorption of the gas; second, convection is coupled with stellar pulsation through Reynolds stress and turbulent thermal convection. Therefore stellar pulsation, convection, and radiation are coupled and treated in a consistent way within the current theory.