Self-consistent modeling of turbulence and transport

We describe an efficient procedure for coupling a turbulent system to a transport equation which evolves the equilibrium fields that drive and are driven by the turbulence. As an example, we apply the procedure to the coupling of turbulence simulations of the two-dimensional Hasegawa–Wakatani equations to a one-dimensional transport equation for the density n. Our coupling scheme uses implicit temporal discretization of the transport equation, rendering it stable for arbitrarily large time steps. This allows the computation of turbulence with self-consistent steady-state equilibrium profiles in a single time step of the transport equations and with a total computational time comparable to that required for the turbulence code alone to reach a statistical steady state with fixed equilibrium profiles. Results are presented for both local and non-local turbulence simulations. In the former, which requires running a separate turbulence simulation for each transport grid cell, the transport flux Γ(x) depends on only local values of n(x) and n′(x); for this case, Γ is expressed using Fick’s law, Γ=−Dn′, with D>0 prescribed by the turbulence. In the non-local simulations, Γ(x) depends on the form of n over the entire domain. For such simulations we present two methods for representing Γ that allow for anomalous flux transport, i.e., regions where the flux flows up the local gradient of n.