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.
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