## 1. Introduction

Lateral mixing between convective clouds and their environment represented by entrainment and detrainment are key processes in atmospheric moist convection and the uncertainty of its strength is still a main source in climate model uncertainty (Murphy *et al.*, 2004; Rougier *et al.*, 2009). The strong positive impact of new entrainment and detrainment representation on the predictive skill of numerical weather prediction (NWP) models (Bechtold *et al.*, 2008) shows both the importance and the relative infancy of our knowledge of these processes.

The concept and relevance of entrainment of environmental quiescent air into convective cumulus updraughts was first pointed out by Stommel (1947) and were followed by numerous observational studies of cumulus clouds with aircrafts (e.g. Warner, 1955). In these studies entrainment strength could be determined through the ratio between the measured liquid water in clouds and its adiabatic value. The first quantitative descriptions of entrainment originated from laboratory experiments of thermal plumes (Morton *et al.*, 1956; Turner, 1963) describing an increasing mass flux M with height

where *R* is the radius of the rising plume and is the fractional entrainment coefficient. Many of the early cloud models have adopted this entraining plume model. A distinction between entrainment due to turbulent mixing at the cloud edge and organized inflow of environmental air induced by the increase of the vertical velocity due to buoyancy was first pointed out by Houghton and Cramer (1951) and its relevance has been recently re-emphasized by Holloway and Neelin (2009)

Since the dynamical (organized) and turbulent fractional entrainment rates are by definition positive, they cause the mass flux to increase with height. This is in agreement with dry plumes where entrained air from the environment becomes part of the plume after the mixing process. However, cloudy updraughts can actually also exhibit a decreasing mass flux with height, for instance due to mixing of dry environmental air. The evaporative cooling can actually reduce the updraught area and/or the updraught velocity so that the mass flux can also decrease with height. This so-called detrainment process is, in many respects, the mirror image of entrainment and can also be subdivided into a dynamical and a turbulent part:

Mass flux parametrizations of cumulus convection in NWP and climate models have to take into account the effect of a whole ensemble of clouds rather than a single cloud element. With the exception of the seminal work of Arakawa and Schubert (1974), most mass flux parametrizations employ a so-called bulk approach in which all active cloud elements are represented in one steady-state updraught representing the whole cloud ensemble.

Numerous entrainment and detrainment parametrizations have been proposed for such mass flux bulk schemes. Popular formulations proposed by Tiedtke (1989), Bechtold *et al.* (2008), Nordeng (1994) and Gregory and Rowntree (1990) can be ordered in terms of the right-hand side of (3). Tiedtke (1989) and Nordeng (1994) assume that ϵ_{turb} and δ_{turb} are equal and given by (1), while in Bechtold *et al.* (2008) ϵ_{turb} depends on the saturation specific humidity. Gregory and Rowntree (1990) also propose (1) for ϵ_{turb} but utilize a systematic smaller δ_{turb}. Dynamical entrainment ϵ_{dyn} is based on moisture convergence in Tiedtke (1989), on momentum convergence in Nordeng (1994), on relative humidity in Bechtold *et al.* (2008), and absent in Gregory and Rowntree (1990). Organized detrainment is in general formulated as a massive lateral outflow of mass around the neutral buoyancy level although the precise details differ in the cited parametrizations. In the above-cited parametrizations typically a fixed value of *R* ≃ 500 m for shallow clouds and *R* ≃ 2000 m for deep convection is used.

Another class of entrainment/detrainment parametrizations, that does not distinguish between dynamical and turbulent mixing, is based on the ‘buoyancy sorting’ concept introduced by Raymond and Blyth (1986). This buoyancy sorting concept is transformed into a operational parametrization by Kain and Fritsch (1990). In Kain and Fritsch (1990) an ensemble of mixtures of cloudy and environmental air is formed, where each ensemble member has a different concentration of environmental air. If resulting mixtures are positively buoyant, they remain in the updraught and are part of the entrainment process while negatively buoyant mixtures are rejected from the updraught and are part of the detrainment process. A number of recently proposed shallow cumulus convection schemes (Bretherton *et al.*, 2004; de Rooy and Siebesma, 2008; Neggers *et al.*, 2009) are based on this buoyancy sorting concept. Finally a large number of parametrizations for and (sometimes) δ have been published that are directly or indirectly inspired on large-eddy simulation (LES) results of shallow cumulus convection (e.g. Siebesma, 1998; Grant and Brown, 1999; Neggers *et al.*, 2002; Gregory *et al.*, 2000; Lappen and Randall, 2001).

The objectives of this paper are twofold. Firstly, to create some order in all the proposed parametrizations, general expressions for the dynamical and turbulent entrainment and detrainment rates will be derived. Based on these expressions, a physical picture emerges that resembles Arakawa and Schubert (1974) and Nordeng (1994). Furthermore, through combining budget equations of mass, total water specific humidity and vertical velocity, we will derive analytical expressions for and δ that can be evaluated using LES. The latter expressions show that the entrainment formulations proposed by Nordeng (1994) and Gregory (2001) can be viewed as special cases. The analytical expressions for and δ cannot be used directly as parametrizations as no closure assumptions have been imposed. Secondly, in section 3 we use LES results for different shallow cumulus cases to critically evaluate these analytical expressions. With the help of the expressions, different aspects of and δ can be explained, e.g. the large variability of the detrainment coefficient. Moreover, the expressions are used to evaluate existing parametrization approaches and it is demonstrated how they can serve as a sound physical base for future parametrization developments.