## Combined Space and Time Spectrums

Hi all. I have a probability problem, beyond my knowledge or brain power!

Background

To give some context, I am looking the effect of turbulence on the design of wind turbine blades. The wind turbine control system can cope with low frequency turbulence, but not high frequency. Hence the following problem...

Normal Distribution Wind Speed
The wind speed distribution due to turbulence for a given mean wind speed can be assumed to be a simple Normal distribution with a standard deviation of 0.2, which is invariant of the mean wind speed.

Frequency Spectrum

However there is also a frequency spectrum, as some turbulence is low frequency gusts, other are much higher frequency.

The normalised frequency spectrum is given by Von Karman:

$\frac{nS_{u}(n)}{\sigma^2_{u}}=\frac{4nL_{2u}/U_{mean}}{(1+70.8*(nL_{2u}/U_{mean})^2)^{(5/6)}}$

$_{u}$ signifies the longitudinal direction (the one I'm interested in)
$L_{2u}$ is a length scale which is (happily) known
$n$ is the frequency

My question:

How can I calculate the probability distribution of the wind speed above a certain frequency?

For frequencies of 1HZ and above I want to be able to have the probability distribution for the instantaneous wind velocity, given the mean wind speed. I can't believe its the same as the general distribution...

Any help would, of course, be gratefully received.