I've had trouble scanning the literature for an answer in the case of convective instabilities. In the case of Navier Stokes turbulence, this has been well characterized beyond the multifractal spectrum, in order to positively evaluate Richardson's picture.
For more details, have a look at e.g. A. Arneodo, S. Manneville, J. F. Muzy and S. G. Roux, Phil. Trans. R. Soc. Lond. A (1999) 357, 2415–2438.