Multifractality of fine bubbles in turbulence due to lift

We develop a theory of spatial distribution of fine bubbles in turbulence and confirm it numerically. The bubbles are a few hundred microns in size and the considered value of the turbulence's strength is typical for oceans. The fluid applies a lift force on bubbles that causes bubble clustering by acting in a direction perpendicular to the rise velocity and the horizontal vorticity. We demonstrate that the centripetal attraction of bubbles to the cores of vortices is negligible compared with this effect. The combined action of the lift and the turbulence creates a continuum flow of bubbles distinct from the flow of the fluid and identical to the Peddley-Kessler (PK) model of swimming phytoplankton cells in the limit of strong gravitaxis. This allows one to derive a whole array of results by applying the theory of this model, which is well developed in the limit. The similarity explains the patchiness and clustering of bubbles in regions of downwelling flow; these properties are understood for motile phytoplankton. The bubble clustering decreases the average rise velocity, prolonging the time until the bubble's rupture at the surface. We demonstrate that bubbles form columnar structures characterized by a multifractal distribution with the dimension deficit increasing as the eighth power of the radius. Fractal dimensions are derived from the information dimension given by the Kaplan-Yorke formula. We also provide some results for the shape-dependent PK model. We confirm the theory by direct numerical simulations of the bubbles' motion in Navier-Stokes turbulence.