1. 1 00/XXXX © Crown copyright The Unified Model Cloud Scheme. Damian Wilson, Met Office.

2. 2 00/XXXX © Crown copyright A PDF cloud scheme T L =T - L/c p l q T =q+l q T = q sat (T L )  = q T - q sat (T L ) l = q T -q sat (T) = q T - q sat (T L + L/c p l) l = q T - [ q sat (T L ) +  L/c p l] where  is the chord gradient l = a L [q T - q sat (T L )] where a L = [1+  L/c p ] -1 This formulation is only valid if condensation is rapid, hence no supersaturation.

3. 3 00/XXXX © Crown copyright Mathematical formulation l = a L [ q T - q sat (T L ) ] Write in terms of a gridbox mean and fluctuation. l  a L [ - q sat ( )] + a L [ q T ’ -  T L ’ ] l  Q c + s If we know the distribution G(s) of ‘s’ in a gridbox then we can integrate across the distribution to find C and.

4. 4 00/XXXX © Crown copyright The Smith parametrization s G(s) s=-Q c bsbs We parametrize G(s) to be triangular with a width given by b s = a L [1-RH c ]q sat ( ) When -Q C =b s we have -a L [ - q sat ( )] = a L [1- RH c ]q sat ( ) or = RH c. Cloudy Clear

5. 5 00/XXXX © Crown copyright Implementation Values of C and l can be solved analytically. But at what temperature should a L be calculated? This problem arises from the linear approximation we made earlier. We need a form of ‘average’ a L. We calculate  and as the gradient of the chord between T, q sat (T) and T L, q sat (T L ). This means that there is an iteration to find the value of (but not necessary for C).

6. 6 00/XXXX © Crown copyright Important issues s= –Q c Cloudy What is the width of the PDF? What is the skewness? What is the shape of the PDF? Is the PDF adequately described by simple parameters? Analysis should be performed in the ‘s’ framework. s=a L [ q T ’ -  T L ’ ] How does the PDF change with time, C, …?

7. 7 00/XXXX © Crown copyright Consequences of PDF shape. QcQc C 1 0 0.5 Larger RH c Triangular shape Top-hat shape Skewed b s = f(C) The shape of the PDF determines how C and vary with Q c. Remember, though, that it is also possible for the shape to change with time or as a function of C or the physical process which is occuring.

8. 8 00/XXXX © Crown copyright Summary The Met Office cloud scheme uses a q T and T L framework to formulate a PDF description of cloud fraction and condensation. Other cloud schemes can be presented similarly using q T and T L ideas. It relies on the assumption of instantaneous condensation and evaporation. The resulting behaviour of cloud fraction and condensate depends critically on how the shape of the PDF is parametrized. Is the shape sensible?