Energetics of mixing for the filling box and the emptying-filling box

The mixing efficiency of a plume in a filling box and an emptying-filling box is calculated for both transient and steady states. The mixing efficiency of a plume in a filling box in an asymptotic state is 1/2, independent of the details of this state or how the plume is modelled. The mixing efficiency of a plume in an emptying-filling box in steady state is 1-ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 - \xi $$\end{document}, where ξ=h/H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi = h/H$$\end{document}, the depth of the ambient layer h normalised by the height of the box H. A deeper mixed layer therefore corresponds to a higher mixing efficiency. These results shed light on the interpretation of mixing efficiencies of open and closed systems.