When we conjecture that both fluctuations have the same scaling laws, namely , we recover the Kolmogorov scaling for the field increments
In the real space turbulence properties can be described using either the probability distribution functions (PDFs hereafter) of increments, or the longitudinal structure functions, which represents nothing but the higher order moments of the field. Disregarding the magnetic field, in a purely fully developed fluid turbulence, this is defined as . These quantities, in the inertial range, behave as a power law , so that it is interesting to compute the set of scaling exponent . Using, from a phenomenological point of view, the scaling for field increments (see Equation (22)), it is straightforward to compute the scaling laws . Then results to be a linear function of the order .
When we assume the scaling law , we can compute the high-order moments of the structure functions for increments of the Elsässer variables, namely , thus obtaining a linear scaling , similar to usual fluid flows. For Gaussianly distributed fields, a particular role is played by the second-order moment, because all moments can be computed from . It is straightforward to translate the dimensional analysis results to Fourier spectra. The spectral property of the field can be recovered from , say in the homogeneous and isotropic case
where is the wave vector, so that in the inertial range where Equation (26) is verifiedKolmogorov spectrum (see Equation (23)) is largely observed in all experimental investigations of turbulence, and is considered as the main result of the K41 phenomenology of turbulence (Frisch, 1995). However, spectral analysis does not provide a complete description of the statistical properties of the field, unless this has Gaussian properties. The same considerations can be made for the spectral pseudo-energies , which are related to the order structure functions .
© Max Planck Society and the author(s)