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We present a theory for relaxation and transport in phase space for gyrokinetic drift wave turbulence with zonal flow. The interaction between phase space eddys and zonal flows is considered in two different limits, namely for K>>1 and K ≃ 1 where K is the Kubo number. For K>>1, the growth of an isolated coherent phase space structure is calculated, including the associated zonal flow dynamics. For K ≃ 1, mean field relaxation dynamics is considered in the presence of phase space granulations and zonal flows. In both limits, it is shown that the evolution equations for phase space structures are structurally similar to a corresponding Charney-Drazin theorem for zonal momentum balance in a potential vorticity conserving, quasi-geostrophic system. The transport flux in phase space is calculated in the presence of phase space density granulations and zonal flows. The zonal flow exerts a dynamical friction on ion phase space density evolution, which is a fundamentally new zonal flow effect.

On relaxation and transport in gyrokinetic drift wave turbulence with zonal flow