Mixing of a passive scalar in magnetically forced two-dimensional turbulence

Abstract

More than 35 years ago, Batchelor discussed from a theoretical point of view the spatial power spectrum E θ (k) of a weakly diffusing impurity mixed by a turbulentfluid flow. Under plausible assumptions including the random straining of fluid elements, E θ (k) is expected to scale as k −1 for a range of wave numbers k beyond the cutoff of the energy spectrum, followed by a diffusive tail at a wave number determined by the rate of strain and the diffusivity. Some experiments over the years appear to support this conclusion, while others do not. We have investigated this issue experimentally in a quasi-two-dimensional turbulent flow, established in a thin buoyant layer that is electromagnetically forced by an array of permanent magnets beneath the cell. The wave number cutoff of the velocity field is established by particle imagevelocimetry, and the fluctuations are determined to be homogeneous and isotropic. To study mixing, a dye solution is introduced steadily at one side of the buoyant layer, and mixed fluid is extracted at the other until the fluctuations become steady. The time-averaged spatial power spectrum of the dye concentration distribution is measured for several distinct forcing configurations, including both regular and random magnet arrays; the latter produces convincingly isotropic concentration fluctuations. We find that E θ (k) falls strongly below k −1 at wave numbers lower than expected from theory. The results do not appear to depend significantly on the scalar injection parameters or on the magnet arrangement. Periodic forcing at higher viscosity leads to chaotic rather than turbulent flows, with little change in E θ (k). The observations can be explained in part by the intermittency characteristic of these two-dimensional flows, where the stretching of fluid elements is localized in space and time.