  We analyse the entrainment and energetics of equal and opposite axisymmetric
turbulent air plumes in a vertically confined space at a Rayleigh number of
$1.24\times 10^{7}$ using theory and direct numerical simulation. On domains of
sufficiently large aspect ratio, the steady state consists of turbulent plumes
penetrating an interface between two layers of approximately uniform buoyancy.
As described by Baines and Turner (J. Fluid Mech. vol. 37, 1969, pp. 51-80),
upon penetrating the interface the flow in each plume becomes forced and
behaves like a constant-momentum jet, due to a reduction in its mean buoyancy
relative to the local environment. To observe the behaviour of the plumes we
partition the domain into sub-domains corresponding to each plume. Domains of
relatively small aspect ratio produce a single primary mean-flow circulation
between the sub-domains that is maintained by entrainment into the plumes. At
larger aspect ratios the mean flow between the sub-domains bifurcates,
indicating the existence of a secondary circulation within each layer
associated with entrainment into the jets. To study the flow's energetics we
use a local definition of available potential energy (APE). For plumes with
Gaussian velocity and buoyancy profiles, the theory we develop suggests that
the kinetic energy dissipation is split equally between the jets and the plumes
and, collectively, accounts for almost half of the input of APE at the
boundaries. In contrast, $1/4$ of the APE dissipation and background potential
energy (BPE) production occurs in the jets, with the remaining $3/4$ occurring
in the plumes. These bulk theoretical predictions agree with observations of
BPE production from simulations to within $1\%$ and form the basis of a
similarity solution that models the vertical dependence of APE dissipation and
BPE production.
