## 1. Introduction

[2] The parameterization of cumulus convection is a key challenge in improving the performance of Numerical Weather Prediction (NWP) and climate models, in particular in the tropics. The formulation of convection schemes has a large impact on phenomena such as the Madden-Julian Oscillation [*Zhang and Mu*, 2005; *Lin et al.*, 2006], the structure and location of the Intertropical Convergence Zone [*Frey and Latif*, 1997; *Liu et al.*, 2010] and the diurnal cycle over land [*Yang and Slingo*, 2001; *Bechtold et al.*, 2004]. Moreover, the parametric assumptions on the mixing between convective clouds and their environment have been shown to be a main source in climate model uncertainty [*Murphy et al.*, 2004].

[3] Most convection schemes are based on a bulk-plume approach, where the parameterization describes the behavior of the ensemble of clouds that are actively contributing to the transport of heat, moisture and momentum. The intensity and vertical extent of the moist convection is described by the mass-flux*M*_{c} = *ρa*_{c}*w*_{c}, with *a*_{c} denoting the fractional area of the cloud core, *w*_{c} its vertical velocity and *ρ*the density. The behavior of the mass-flux is governed by the (fractional) entrainment and detrainment rates,*ϵ* and *δ*, which represent the mixing of environmental air into the cloud core and the removal of air that no longer contributes to updrafts:

The sensitivity of deep convection to the relative humidity in the free troposphere has been the topic of many recent studies. Both observations [*Holloway and Neelin*, 2009] and Cloud Resolving Model (CRM) simulations [*Derbyshire et al.*, 2004; *Waite and Khouider*, 2010] show that besides stability, a dry environment can be a key inhibiting factor for the development of deep clouds. More detailed Large Eddy Simulation (LES) studies [*De Rooy and Siebesma*, 2008] showed that a dryer environment promotes stronger detrainment rates for shallow cumulus convection, a finding that was more recently confirmed in CRM studies of deep convection [*Derbyshire et al.*, 2011; *De Rooy et al.*, 2012].

[4] Most convection parameterizations fail to show any significant sensitivity to the free tropospheric humidity [*Derbyshire et al.*, 2004]. In an attempt to rectify this, *Bechtold et al.* [2008]proposed to make the entrainment an explicit function of the environmental relative humidity, such that entrainment decreases in a moister environment. This has been shown to improve the performance of the global NWP model used at the European Center for Medium-Range Weather Forecasts (ECMWF).

[5] One existing bulk-plume scheme that intrinsically incorporates the sensitivity of convection to its environment was developed by*Kain and Fritsch* [1990]. The scheme is based on the assumption that at the interface of the cloud core and the environment, a certain mass of cloud core air *m*_{c} mixes with a mass *m*_{e} of air with the properties of the mean environment (the subscripts *c* and *e* will be used throughout this article). The mixtures that result can be described in terms of a mixing ratio *χ*, which indicates the fraction of environmental air in the mixture.

[6] The key idea behind the Kain-Fritsch scheme is that positively buoyant mixtures entrain into the cloud core, whereas mixtures that become negatively buoyant due to evaporative cooling (having*χ* > *χ*_{cr}) are detrained. Assuming all mixing ratios occur with equal probability *p*(*χ*), the entrainment and detrainment rates can be analytically expressed in terms of *χ*_{cr} [*Bretherton and McCaa*, 2004]:

Here, *λ* is a length scale for the mixing of the cloud core with its environment.

[7] *De Rooy and Siebesma* [2008, equation (A9)] showed that *χ*_{cr} increases with higher relative humidity and with higher buoyancy excess of the cloud core, which in turn depends on the stability. A fundamental problem with the formulation (equation (2)) is that it predicts a decreasing entrainment rate (which results in larger moisture excesses) in a drier environment, in clear disagreement with the CRM/LES results and proposed parameterization discussed above.

[8] In the current work, we will systematically determine the relative dependency of entrainment, detrainment and the intensity moist convection in general on humidity, stability and the critical mixing ratio *χ*_{cr}. This will be done by a massive sensitivity study of 90 three-dimensional Large Eddy Simulations of deep convection.