Air super-saturation controls nucleation and growth of cloud particles and therefore determines the formation of rain. Models, which calculate cloud microphysics, have to correctly determine super-saturation.

This study analyses the methods used to calculate the mean super-saturation and the time dependent super-saturation. The analytical method is the first method that has been studied and is hereby presented. The present work was motivated by the following questions:

1. Is it correct to keep the super-saturation constant in simulations for an entire time step?

2. How to find the equilibrium super-saturation?

3. What is the temporal variability of super-saturation in the clouds?

4. How to determine the super-saturation integral for a time step of the simulation?

Removal of excess water vapor so that there is no super-saturation at the end of each time step of the model usually gives an over-estimation of the latent heat exchanged with the system during phase changes. The energy balance is not the only one affected by saturation. The activation of condensation nuclei and ice nuclei is very sensitive to the value of super-saturation in clouds.

To introduce the temporal variation of saturation into the models is, therefore, necessary.

In the work presented two different cases arise: the time step is smaller than the saturation relaxation time and the time step is larger than the saturation relaxation time.

The solution of the differential equations of saturation for warm and cold clouds should be calculated from the integral of saturation in the case where the time step is smaller than the saturation relaxation time. In the case where the time step is longer than the relaxation time, the model should calculate the equilibrium saturation.

In order to confirm our assumption, we present some results obtained with the Cloud Resolving Model using the integral of the saturation ratio versus the results given by the same model which cuts the saturation at 100% at the end of which time step.