The turbulence closure problem for convective boundary layers is considered with the chief aim to advance the understanding and modeling of nonlocal transport due to large-scale semiorganized structures. The key role here is played by third-order moments (fluxes of fluxes). The problem is treated by the example of the vertical turbulent flux of potential temperature. An overview is given of various schemes ranging from comparatively simple countergradient-transport formulations to sophisticated turbulence closures based on budget equations for the second-order moments. As an alternative to conventional "turbulent diffusion parameterization" for the flux of flux of potential temperature, a "turbulent advection plus diffusion parameterization" is developed and diagnostically tested against data from a large eddy simulation. Employing this parameterization, the budget equation for the potential temperature flux provides a nonlocal turbulence closure formulation for the flux in question. The solution to this equation in terms of the Green function is nothing but an integral turbulence closure. In particular cases it reduces to closure schemes proposed earlier, for example, the Deardorff countergradient correction closure, the Wyngaard and Weil transport-asymmetry closure employing the second derivative of transported scalar, and the Berkowicz and Prahm integral closure for passive scalars. Moreover, the proposed Green-function solution provides a mathematically rigorous procedure for the Wyngaard decomposition of turbulence statistics into the bottom-up and top-down components. The Green-function decomposition exhibits nonlinear vertical profiles of the bottom-up and top-down components of the potential temperature flux in sharp contrast to universally adopted linear profiles. For modeling applications, the proposed closure should be equipped with recommendations as to how to specify the temperature and vertical velocity variances and the vertical velocity skewness. © 1999 American Meteorological Society.