  Simulations of core-collapse supernovae (CCSNe) result in successful
explosions once the neutrino luminosity exceeds a critical curve, and recent
simulations indicate that turbulence further enables explosion by reducing this
critical neutrino luminosity. We propose a theoretical framework to derive this
result and take the first steps by deriving the governing mean-field equations.
Using Reynolds decomposition, we decompose flow variables into background and
turbulent flows and derive self-consistent averaged equations for their
evolution. As basic requirements for the CCSN problem, these equations
naturally incorporate steady-state accretion, neutrino heating and cooling,
non-zero entropy gradients, and turbulence terms associated with buoyant
driving, redistribution, and dissipation. Furthermore, analysis of
two-dimensional (2D) CCSN simulations validate these Reynolds-averaged
equations, and we show that the physics of turbulence entirely accounts for the
differences between 1D and 2D CCSN simulations. As a prelude to deriving the
reduction in the critical luminosity, we identify the turbulent terms that most
influence the conditions for explosion. Generically, turbulence equations
require closure models, but these closure models depend upon the macroscopic
properties of the flow. To derive a closure model that is appropriate for
CCSNe, we cull the literature for relevant closure models and compare each with
2D simulations. These models employ local closure approximations and fail to
reproduce the global properties of neutrino-driven turbulence. Motivated by the
generic failure of these local models, we propose an original model for
turbulence which incorporates global properties of the flow. This global model
accurately reproduces the turbulence profiles and evolution of 2D CCSN
simulations.
