A note on the closure in Lilly-type inversion models

Closure conditions in models of lifted inversions are usually based on the buoyancy term in the equation of turbulent energy. The paper discusses this type of closure and compares the condition r = constant (r is the ratio of integral dissipation over integral production of turbulent energy, both due to buoyancy forces; the integral is taken over the whole mixed layer) with the frequently used condition of k = constant (k represents the negative ratio of minimum over maximum flux of virtual static energy). It is shown that in the dry case r = k². In the wet case including condensation within a St- or Sc-deck, however, it is not possible to set up such a simple general relationship. The problem is illustrated by some model case studies. Finally it is shown that essential characteristics of the inversions are not very sensitive to changes in r or k. This leads to the conclusion that a relatively rough closure assumption suffices for most purposes.