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
