Thesis Defense |
A self-consistent weak turbulence closure is constructed for the second-order correlations governing energetic dynamics in a collisionless ITG fluid model. In the collisionless limit, the governing equations exhibit parity–time reversal symmetry, resulting in centrosymmetric correlation evolution operators. The resulting saturated states are characterized by equipartition of energy between stable and unstable eigenmodes at each wavenumber. These results highlight the central role of zonal-flow–catalyzed nonlinear energy transfer from unstable to stable modes and suggest a mechanism by which turbulent heat flux can vanish despite the presence of linear instability. Numerical simulations confirm the predictions of the closure theory.
A novel weak turbulence closure technique is then developed to calculate two-point, two-time correlation functions and power spectra in systems of instability-driven turbulence. Application of this framework to zonal flows yields a complex nonlinear decorrelation frequency that governs both oscillatory behavior and temporal decay of the two-time energy spectrum. The resulting power spectra demonstrate that nonlinear interactions strongly modify zonal flow dynamics, imparting wavenumber-dependent oscillation frequencies absent at the linear level.
Weak turbulence closure theory is further extended to fourth-order statistics to investigate the Dimits shift through the lens of intermittent turbulence. A new closure method is developed to compute the growth rate of a fourth-order cumulant associated with energy fluctuations. The resulting evolution equation predicts the development of intermittent, non-Gaussian statistics at large spatial scales below the nonlinear critical gradient, and their decay above it. These predictions are validated through comparisons with numerical simulations, including measurements of probability distribution functions, kurtosis, and entropy of the turbulent heat flux. The emergence and suppression of intermittency are traced to qualitative changes in nonlinear mode coupling between stable and unstable eigenmodes across the nonlinear threshold.
Overall, this work demonstrates that weak turbulence closure theory provides a unified framework for describing saturation, temporal correlations, and intermittency in ITG turbulence, and offers new insight into the dynamical mechanisms underlying zonal-flow–regulated transport and the Dimits regime.