  The characterisation and the modelling of air concentration distributions in
self-aerated free-surface flows has been subject to sustained research interest
since the 1970s. Recently, a novel two-state formulation of the structure of a
self-aerated flow was proposed by Kramer & Valero [2023 J. Fluid Mech. 966,
A37], which physically explains the air concentration through the weak
interaction of two canonical flow momentum layers, comprising a Turbulent
Boundary Layer (TBL) and a Turbulent Wavy Layer (TWL). The TWL was modelled
using a Gaussian error function, assuming that the most dominant contribution
are wave troughs. Here, it is shown that air bubbles form an integral part of
the TWL, and its formulation is expanded by adopting a superposition principle
of entrapped air (waves) and entrained air (bubbles). Combining the
superposition principle with the two-state formulation, an expression for the
depth-averaged (mean) air concentration is derived, which allows to quantify
the contribution of different physical mechanisms to the mean air
concentration. Overall, the presented concepts help to uncover new flow
physics, thereby contributing fundamentally to our understanding of
self-aerated flows.
