Lower bounds on the fractal dimension of level sets of advecting passive scalars in turbulent fields are derived, in the limit that the scalar diffusivity kappa goes to zero. The main result is as follows: denote the Holder exponent of the velocity field u by zeta (u), with 0<or= zeta (u)<or=1, and the Holder exponent of the passive scalar (say T) by zeta (T). We derive a lower bound on the dimension D of the level sets of T, D>or=d-1+ zeta (T)+ zeta (u), where d is the dimension of space. The validity of this bound depends on some conditions concerning the limit kappa to 0; when these are satisfied the bound is obtained throughout the range of zeta (u), between the smooth (but random) velocity field with zeta (u)=1 to the extremely rough field with zeta (u)=0. The derivation of the lower bound calls for the introduction of a measure on the level sets and a careful treatment of the singular limit of the scalar diffusivity going to zero. Together with the upper bounds which were derived previously, i.e. D<or=d-1/2+ zeta (u)/2 we discover, when there is no multiscaling, the scaling relation 2 zeta (T)+ zeta (u)=1, which then means that the lower and the upper bounds in fact coincide.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics