A Self-Similar View of Atmospheric Turbulence


  • John A. Dutton,

  • Dennis G. Deaven


The inertial subrange in the energy spectrum appears to be the most consistent feature of atmospheric turbulence at all altitudes. The implications of the fact that amplitudes are related to the 1/3 power of the wavelength are imposed upon the probabilistic structure of the turbulence by B. Mandelbrot's concept of self-similarity. A process with random variable x is self-similar if the variable xh obtained by magnification of the wavelength by h can be represented as a suitable magnification of the amplitude of x so that both x and xh have the same probabilistic structure. If this is the case, and the spectra have power-law behavior S(k) ∼ kv, then the distribution functions for x and xh obey P(x) = Ph(xh) = Ph[h(v − 1)/2x]. The implications for the distribution of increments yT = x(t) − x(t + T) are examined. Observational data from the 150-meter tower at Cape Kennedy and from airborne measurements in the troposphere and stratosphere show that self-similarity gives a remarkably accurate prediction of the behavior of the empirical data in all three cases. The distribution of the increments also reveals an intermittent structure for the turbulence. It is concluded that an attempt to construct a theoretical explanation for the apparently self-similar behavior of atmospheric turbulence may be revealing.