Since a few decades it is known that in the interior of the clear convective boundary layer (CBL) the vertical flux of the (virtual) potential temperature flows counter the mean vertical gradient. This behaviour impedes a straightforward eddy diffusivity modeling of the vertical flux. From experience it is known that in the CBL the entrainment buoyancy flux is always about -0.2 times the surface buoyancy flux. In this investigation we will use a passive scalar equation which is linear in the concentration field. We study the ratio between the entrainment flux and the surface flux, and aim to systematically explore for which flux-ratio the vertical scalar flux is not down the mean vertical gradient. To this end we have performed a Large-Eddy Simulation of a clear convective boundary layer (CBL) on a large horizontal domain (25.6x.25.6x1.6 km, 256x256x80 grid points). Linearity of the transport equation for the scalar allows the use of the principle of superposition of variables, and the inclusion of a top-down and bottom-up scalar therefore makes that a scalar with any flux ratio can be reconstructed.

It is found that only for a limited range of flux ratios (-2 < r < 0) the vertical flux is countergradient. The vertical height interval where countergradient fluxes are found depends only on the flux ratio.

In addition we apply the principle of the superposition of variables to derive a new formula to parameterize the vertical flux. It contains a 'classical' downgradient term, but also a correction term. Besides the dynamics, the latter contains just one free parameter, namely the flux-ratio, which is rather simple to compute even in a coarse numerical model.

Lastly, we have computed the vertically integrated variance tendency (T) as a function of the flux-ratio. For a given surface flux, it is found that if the entrainment flux is about -0.2 times the surface flux, T is minimal and about zero.