Turbulent compressible convection in a deep atmosphere .5. Higher order statistical moments for a deeper case

Treating the turbulence statistically to obtain moments of the fluid equations has been a traditional way to describe stellar convection analytically. Associated with this approach, a well-known difficulty is the problem of closure. For practical reasons, hypothetical closures are normally applied to the low-order moments (second, third, or fourth). In this paper, based on the results of a large eddy simulation of a rather deep convection zone (approximately seven pressure scale heights), we examine various forms of algebraic and diffusive-type closures for the second-, third-, and fourth-order moments involving the vertical velocity and temperature fluctuation. Some popular closures are found to perform poorly. On the other hand, we show that the flux of kinetic energy (essentially a third-order moment) can be estimated (in terms of second- and first-order moments) by combining an algebraic approximation in the lower region (including the overshoot region) and a diffusive-type approximation in the upper region of the convection zone.