The longitudinal, vertical and transverse components of the time dependent wind velocity vector at a given location can be expressed as a sum of a constant term and a time dependent function with zero mean:
The functions , and can be assumed to represent stationary random processes at least during a time interval of a few minutes. In general the longitudinal turbulent component is the most significant with respect to the response of a structure, and of a telescope in particular. A measure of the correlation of at different time instants separated by a time interval is given by the autocorrelation function
where is the variance of .
is equal to unity for and vanishes for .
One may then define a length scale
which is a measure of the average size of the turbulent eddies in the mean flow direction. Vertical and lateral turbulence length scales are similarly defined.
The power spectrum may be obtained as the Fourier transform of the autocorrelation function. The form of the velocity power spectrum in the inertial domain was obtained by Kolmogorov as:
This expression is valid for . Various empirical expressions which account also for lower frequencies have been proposed for use in response computations procedures and building codes: a form which is very convenient for parametric wind response computations is the Von Karman spectrum
with x =
Note that the peak frequency of the spectrum can be evaluated from the length scale as