Mean-field transport equations and energy theorem for plasma edge turbulent transport

This paper establishes a mean-field equation set and an energy theorem to provide a theoretical basis in view of the development of self-consistent, physics-based turbulent transport models for mean-field transport codes. A rigorous averaging procedure identifies the exact form of the perpendicular turbulent fluxes which are modelled by ad hoc diffusive terms in mean-field transport codes, next to other closure terms which are not commonly considered. Earlier work suggested that the turbulent $E\times B$ particle and heat fluxes, which are thus identified to be important closure terms, can be modelled to reasonable accuracy using the kinetic energy in the $E\times B$ velocity fluctuations ($k_{E}$). The related enstrophy led to further modelling improvements in an initial study, although further analysis is required. To support this modelling approach, transport equations are derived analytically for both quantities. In particular, an energy theorem is established in which the various source and sink terms of $k_{E}$ are shown to couple to mean-field and turbulent parallel kinetic energy, kinetic energy in the other perpendicular velocity components, the thermal energy and the magnetic energy. This provides expressions for the interchange, drift-wave and Reynolds stress terms amongst others. Note that most terms in these energy equations are in turn closure terms. It is suggested to evaluate these terms using reference data from detailed turbulence code simulations in future work.