On the Linear and Nonlinear Dynamics of the Hasegawa-Mima Equation

Abstract

Understanding turbulence in plasmas is essential for studying transport properties in magnetically confined fusion devices. This thesis studies the Hasegawa-Mima (HM) model, which describes the evolution of electrostatic potential fluctuations in a plasma. First, we derive the HM model from first principles using fundamental conservation equations. Analytical studies of the HM model are then performed to investigate the role of the linear term and its effect on the system dynamics. In particular, this study explores the evolution of an initial electrostatic potential perturbation initialized as white noise. The effect of the linear term on the dynamics is highlighted through its dependence on the density gradient. The steepness of the density profile is varied, therefore, assessing the impact of the linear term on the overall dynamics. Then, numerical simulations of the HM model are conducted using FreeFEM++. The simulation results are then analyzed in light of the analytical results. The re sults show that an increase in the amplitude of the electrostatic potential occurs in the gradient region, and there is no spreading of the turbulence beyond that region