The dynamics of two-dimensional turbulence excited at two scales using electromagnetic forces

Abstract

Several forcing scales can co-exist in nature leading and affecting turbulent flows. This is not critical in three-dimensional systems where only a direct cascade of energy exists, but it is a concern in two dimensions where the direct and inverse cascades lead to different statistical properties of turbulence. The effect of forcing at two different scales on turbulence is studied here using numerical simulation inspired by a recent experiment [L. M. Moubarak and G. Y. Antar, “Dynamics of a two-dimensional flow subject to steady electromagnetic forces,” Exp. Fluids 53, 1627–1636 (2012)] where a thin layer of electrolyte is stirred using electromagnetic forces. The small scale eddies are generated by the Lorentz force near the domain edge while the large scale motion is produced by the magnetic field gradient. We compare the case of one to two forcing scales for steady state turbulence to show that the addition of two forcing scales leads to the onset of turbulence at low Reynolds numbers due to the co-existence and thus the interaction of small and large structures. By determining the k-spectra as well as the energy transfer function, it is established that the dynamics of turbulence change from being dominated by an inverse cascade process, with one forcing scale, to one dominated by a direct cascade process when two scales are present. We believe that these results are important in understanding two- and quasi-two-dimensional turbulence phenomena occurring in nature where several excitation scales co-exist.