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 \(1 - \xi \), where \(\xi = h/H\), 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.