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 e quations has been a traditional way to describe stellar convection ana lytically. Associated with this approach, a well-known difficulty is t he problem of closure. For practical reasons, hypothetical closures ar e 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 height s), we examine various forms of algebraic and diffusive-type closures for the second-, third-, and fourth-order moments involving the vertic al velocity and temperature fluctuation. Some popular closures are fou nd to perform poorly. On the other hand, we show that the flux of kine tic energy (essentially a third-order moment) can be estimated (in ter ms of second- and first-order moments) by combining an algebraic appro ximation in the lower region (including the overshoot region) and a di ffusive-type approximation in the upper region of the convection zone.