This study introduces and evaluates a prognostic description of total specific humidity variance. As nicely introduced by the study, the idea of prognostic equations for the higher moments of total humidity and their use in cloud schemes based on certain PDF families, is around for more than 20 years by now. Nevertheless, new available datasets and parameterizations give enough reason to continue the developments and test new implementations. In the presented study previous work is recaptured and the implementation in the LMDZ model is described. The implementation is mainly oriented on work by Klein et al (2005) to implement the prognostic terms of the variance by taking the different between environmental and detrained air. 

The evaluation is done in SCM mode based on 4-6 LES case studies covering different situations of shallow convection. The evaluation shows similar results for the cloud cover and variance between the new prognostic implementation and an old diagnostic implementation - but improvements for higher order moments as the skewness. 

The study is tackling the important topic of cloud schemes, which didn't see too much progress over the last years to decades and it is following on the interesting development of prognostic PDF schemes. The text is nicely written and the evaluation results are very interesting. 

Main points: 

The source terms for the prognostic variance are based on differences between the mean and variance of environmental vs. detrained air. Especially for the variance this was one of the main issues in implementing those equations as the variance might not be available from convection schemes. This study shows (Fig. 6) that mainly the difference of the mean value is acting as a source and the other terms could be neglected. This is contradicting the original publication (Klein et al. 2005, as also mentioned in l348-351) and sounds a bit like wishful thinking. The authors mention quickly, that this might be due to different sampling strategies. As it seems to be so sensitive and the question if 2 out of 3 source terms (the complicated ones) could be neglected is very important for the implementation, it would be a lot more convincing, if the authors could elaborate on this point and either test different sampling strategies or find a more reasonable argument for this difference between their own and the original publication. 

Figure 1 looks a bit overloaded. And the impression is even increased due to the overlay of legend and plot. I am not having a good idea or brilliant suggestion, but would like to motivate the authors to rethink if there could be an easier way to transport their message for the tuning experiments. 

Figure 2 is showing the representation of the cloud cover. The unit of the left column color bar is mentioned in the text, but it would be nice to also add the unit for the different plots. I assume it is still %. In that case the differences are between 50 and -50%, which sounds quite a lot for me. Would it be possible to elaborate a bit in the text why this difference is so large? And why you don't worry about it? Also considering the very different amount of cloud fraction between the different cases, a relative difference might be more meaningful than an absolute difference.  

There are several versions of bigaussian, bi-Gaussian, and Bigaussian in the text. It would be nice to have that consistent. 

Minor / technical comments:

l6 - asymetry -> asymmetry

l93 - modifief -> modified

l94/95 blanks missing between Ref, Fast and (?) Slow

l115 facilite -> facilitate (?)

l158 closing bracket missing

l166 asymemtry -> asymmetry

l245 very slower (?) - very slow? or slower than?

l310 Her -> Here

Figure 6 - it would be nice to not have the legend overlaying half of one plot.