Fractal iso-level sets in high-Reynolds-number scalar turbulence

We study the fractal scaling of iso-levels sets of a passive scalar mixed by three-dimensional homogeneous and isotropic turbulence at high Reynolds numbers. The Schmidt number is unity. A fractal box-counting dimension $D_F$ can be obtained for iso-levels below about $3$ standard deviations of the scalar fluctuation on either side of its mean value. The dimension varies systematically with the iso-level, with a maximum of about $8/3$ for the iso-level at the mean; this maximum dimension also follows as an upper bound from the geometric measure theory. We interpret this result to mean that mixing in turbulence is always incomplete. A unique box-counting dimension for all iso-levels results when we consider the spatial support of the steep cliffs of the scalar conditioned on local strain; that unique dimension is about $4/3$.