The evolution of the mixed layer during a clear day can be described with a slab model. The model equations have to be closed by a parameterization of the turbulent kinetic energy budget. Several possibilities for this parameterization have been proposed. In order to assess the practical applicability of these models for the atmosphere, field experiments were carried out on ten clear days in 1977 and 1978. Within the accuracy of the measurements the mixed-layer height in fully convective conditions (at noon on clear days) is well predicted taking a constant heat flux ratio -0~~ = 0.2&v,. In the early morning hours mechanical entrainment is also important. Good overall results are obtained with the entrainment formulation -Bw,, = 0.2&v, + Su?T/gh. Only large differences in the entrainment coefficients lead to significantly different results. Making the entrainment model more complex does not lead to substantial improvement. The mean potential temperature in the mixed layer is reproduced within 0.5 "C. This temperature is insensitive to the choice of a particular entrainment formulation and depends more on the surface heat input and the temperature gradient in the stable air aloft.

The parameterization of penetrative convection and other cases of turbulent entrainment by the atmospheric boundary layer is reviewed in this paper. The conservation equations for a one-layer model of entrainment are straightforward; all modeling problems arise in the context of the parameterization of various terms in the budget of turbulent kinetic energy. There is no consensus in the literature on the parameterization of shear production and of dissipation. Unfortunately, field experiments are not sufl%iently accurate to guide the selection of suitable hypotheses. Carefully designed laboratory experiments are. needed to settle the problems that remain.

We discuss the structure and evolution of a cloud-free atmospheric boundary layer (ABL) during daytime over land. starting from a shallow ABL at sunrise and developing into a deep ABL with strong convection in the afternoon. The structure of the turbulence in the lower half of a convective ABL capped by an inversion is reasonably well understood. Less is known about the details of the turbulence in higher regions affected by entrainment, because of the difficulty in taking turbulence measurements there. For the evolution in time of the height of the ABL and its mean potential temperature mixed-layer models have been developed that give satisfactory agreement with observations. It has been shown that for many practical applications accurate knowledge of forcing functions and boundary conditions is more important than a refinement of the entrainment hypothesis. Observations show that the assumption of well-mixedness of first-order moments of conservative variables is not valid for all quantities. A simple similarity relation for the inclusion of the effect of entrainment on the shape of the vertical profiles is given.

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An improved first-order closure approximation is developed for the non-local 'transilient turbulence' parameterization. Instead of using Richardson numbers, this improved approach uses non-local approximations to the shear, buoyancy, storage, and dissipation terms of the turbulence kinetic energy equation to parameterize the turbulent mixing potential between every combination of grid points in a I-D model of the atmosphere. The original .(n* -n) degrees of freedom associated with the independent transihent matrix coefficients for a model of n grid points is thus reduced to four degrees of freedom associated with the four free p~~eters.The resulting parameterization is applied to three consecutive case-study days of boundary-layer data acquired near the Cabauw tower in The Netherlands. The tirst day is used for sensitivity tests to select the best values of the four free parameters. The remaining two days, used as independent tests, demonstrate that realistic entraining mixed layers and nocturnal boundary layers form in the model without expiici~y paramete~2ing such boundary layers. Sim~ations are also presented for two idealized cases: 'dry' stratocumulus-induced convection and a neutral boundary layer.