Self-consistent theory of zonal flows in ion temperature gradient turbulence

Abstract

A self-consistent theory of nonlinear zonal flows amidst complex background of ion temperature gradient (ITG) turbulence is presented. Starting with a reactive fluid model, a set of coupled nonlinear equations has been obtained in the form of Zakharov-like equations using the reductive perturbation method. These equations represent dynamical evolution of nonlinearly excited zonal flows and potential fluctuations of ITG turbulence. The derived equations have the potential to provide a qualitative explanation of the evolution of zonal flows and drift wave turbulence and their mutual interaction, which have been observed in recent gyrokinetic simulations [A. Dimits et al., Phys. Plasmas 7, 969 (2000)]. The nonlinear coupling coefficients are studied and show that the excitation of zonal flows is due to a resonance in the energy nonlinearity. The resonance turns out to be sensitive to fluid closure.

On the normalization of transport from ITG Modes

“…The same result was, in fact, obtained in Ref. 13 but there we calculated a zonal flow from the convective nonlinearity in the energy equation. However, the resonance was the same, i.e., the pole in (2d).…”

“…and use (13) to get the flow shear. As it turns out, the present ordering is not completely identical to the ordering in the numerical solution.…”

“…The procedure of the calculations is to calculate the radial electric field in (12) and from there the flow shear from Eq. (13). We have then used the Waltz rule [37][38][39] to obtain the stabilization of the drift waves caused by the flow shear (13).…”

“…In our code we simulate both poloidal and toroidal fluid fluxes which are then inserted into (12) to give the radial electric field. Then the flow shear is calculated from (13). As it turns out, it is usually the poloidal rotation which triggers particularly interesting phenomena like internal and edge transport barriers and Dimits shift.…”

“…We will here recall the mechanism using the Reductive perturbation method. 13,15,17,28 We start with a description of the equations used in simulations of the Dimits shift. The energy equation is…”