Energy transfer among coupled oscillators representing a ballooning mode in a toroidally rotating tokamak

Abstract

We have numerically solved the coupled ordinary differential equations (ODEs) representing the ballooning modes in toroidally rotating tokamaks. The ODEs were derived by the eigenfunction expansion of the ballooning mode, of which the governing equation was originally derived as a partial differential equation (PDE). Generally, the truncation of the coupled ODEs to a finite number can generate essential differences between the solutions to the ODEs and to the corresponding PDE, because the truncation reduces the degree of freedom of the motion of the oscillators, or the expansion coefficients. The comparison between the solutions shows that the regularized eigenfunctions used for the expansion, developed by Furukawa and Tokuda (2005 Phys. Rev. Lett.94, 175001), efficiently confine the motion of the oscillators in the phase space with a small number of dimensions; the solution to the truncated ODEs can approximate well the ballooning mode obtained as a solution to the corresponding PDE within a finite time.