The role of different nonlinearities and potential vorticity conservation in two-dimensional fluid ITG models

Abstract

The nature of turbulent energy cascade of simple two-dimensional fluid models of ion temperature gradient driven turbulence is studied in detail. Notably, it is observed that a minimal two-field model of toroidal ITG, behaves qualitatively differently with or without the diamagnetic nonlinearity. In its absence, the zonal flows always dominate and the system never reaches a high-transport state. In contrast, when this term is included, zonal flows dominate only near marginality, while away from it, an inverse cascade with high levels of transport is observed, requiring large-scale dissipation (hypoviscosity) to saturate and hyperviscosity to regularize small scale instability associated with this nonlinear term. However introducing such a term, together with the existence of the curvature term, breaks potential vorticity conservation, which is one of the key symmetries of drift-wave turbulence. This can be remedied by considering a more complete model that retains higher-order terms in the pressure equation. This form of the model, with four nonlinearities, conserves potential vorticity and behaves similarly to the original model, requiring hyperviscosity to saturate since the added nonlinearity generates small scale instability also for the pressure equation. To characterize the roles of potential vorticity conservation and higher-order terms in the pressure equation, the behavior of both the standard ITG system, and the potential vorticity conserving system, is studied by analyzing their spectra, turbulent cascades, and sensitivity to viscosity. Finally, the direction of turbulent cascade due to the different nonlinearities (i.e. the diamagnetic nonlinearity in particular) is investigated by examining their contributions to the spectral energy transfer and by considering the triadic instability assumption for each of these nonlinearities separately.