NONLOCAL TIME-DEPENDENT CONVECTION THEORY

Starting from radiation hydrodynamics, a complete set of dynamical equ ations is derived for the second- and third-order correlation function s of velocity and temperature. Assuming that the fluctuations in turbu lent velocity and temperature obey the normal distribution (Gaussian), following the Millionshchikov assumption, the fourth-order correlatio n functions can be expressed with the second-order correlation functio ns. Anisotropy is carefully considered. It is assumed that the only im portant effect of pressure fluctuation is to restore the isotropy of t urbulence, while all other effects are neglected. In this way, an equa tion of turbulent viscosity very similar to the Stokes viscosity formu la can be set up naturally. Since we adopt an average scheme with weig ht of the gas density for velocity, enthalpy, and extinction, the trea tment of compressibility has been simplified. The Boussinesq approxima tion is no longer needed. The theory is applicable to stellar convecti on even though the density changes by several orders of magnitude acro ss the stellar convection zone. There are two convective parameters, c (1) and Q, which describe the linear size of the energy-containing edd ies 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 cou pled through the emission and absorption of the gas; second, convectio n is coupled with stellar pulsation through Reynolds stress and turbul ent thermal convection. Therefore stellar pulsation, convection, and r adiation are coupled and treated in a consistent way within the curren t theory.