Upscale energy transfer in thick turbulent fluidlayers

Flows in natural fluid layers are often forced simultaneously at scales smaller and much larger than the depth. For example, the Earth’s atmospheric flows are powered by gradients of solar heating: vertical gradients cause three-dimensional (3D) convection whereas horizontal gradients drive planetary scale flows. Nonlinear interactions spread energy over scales1,2. The question is whether intermediate scales obtain their energy from a large-scale 2D flow or from a small-scale 3D turbulence. The paradox is that 2D flows do not transfer energy downscale whereas 3D turbulence does not support an upscale transfer. Here we demonstrate experimentally how a large-scale vortex and small-scale turbulence conspire to provide for an upscale energy cascade in thick layers. We show that a strong planar vortex suppresses vertical motions, thus facilitating an upscale energy cascade. In a bounded system, spectral condensation into a box-size vortex provides for a self-organized planar flow that secures an upscale energy transfer. Turbulence in thin layers is quasi-two-dimensional and supports an inverse energy cascade3, as has been confirmed in many experiments in electrolytes4–6 and soap films7,8. In bounded systems the inverse cascade may lead to a spectral condensation, that is, the formation of a flow coherent over the entire domain4–6. One expects that in thick layers the flow is 3D and there is no inverse energy cascade. Indeed, as has been demonstrated in 3D numerical modelling, when the layer thickness, h, exceeds half the forcing scale, lf, h/lf > 0.5, the onset of vertical motions destroys the quasi-two-dimensionality of the turbulence and stops the upscale energy transfer9,10. In this Letter we report new laboratory studies of turbulence in layers that show that a large-scale horizontal vortex, either imposed externally or generated by spectral condensation in turbulence, suppresses vertical motions in thick layers. This leads to a robust inverse energy cascade even in thick layers with h/lf>0.5. In our experiments, turbulence is generated by the interaction of a large number of electromagnetically driven vortices11–13. The d.c. electric current flowing through a conducting fluid layer interacts with the spatially variable vertical magnetic field. The field is produced by an array of 900 magnets placed beneath the fluid cell, the size of which is 0.3× 0.3m2. The flow is visualized using seeding particles, which are suspended in the fluid, illuminated using a horizontal laser slab, and filmed from above. Particle image velocimetry (PIV) is used to derive the turbulent velocity fields. To visualize the vertical flows, a vertical laser slab is used and the particle motion is filmed from the side. To quantify the velocity fluctuations in 3D, a defocusing PIV technique has been developed that allows all three velocity components to bemeasured14. We use two different configurations: (1) a single layer of electrolyte on a solid bottom, and (2) a layer of electrolyte on top of another layer of a non-conducting heavier liquid,