The total velocity field of the atmosphere is variable in both space and time, composed of a “mean” value and variations from it. The mean wind is a problem primarily for navigation and guidance and is not of interest here. We eliminate it by choosing as our reference frame the atmosphere – fixed frame Fa (see Sec. 4.2.4) relative to which the mean motion is zero. Let the velocity of the air relative to FA at position r = [x1x2x3]T and time t be

u(r, t) = [игиги^т (13.2,1)

Then ufv, t) are random functions of space and time, i. e. we have to deal with the statistics of a random vector function of four variables (x1; x2, x3, t).

Associated with any given point r and time t there is а З X 3 correlation matrix (second-order tensor)

r) = <u{(r, t)u}(v + t + r)> (13.2,2)|

As indicated, it is the ensemble average of the product of ut at r and t with Uj at the different point r + and the later time t + r. The associated four­dimensional Fourier integral is the 3×3 matrix of four-dimensional spectrum functions

The inverse relation for Fourier integrals gives oo

, со)еі(П-?+0>г> dOx d0.2 dQз dm

— co

The functions Bu and di} serve (together with the assumption of normality) to describe the needed statistics of the turbulence. From them all the pertinent results can be derived (see Sec. 2.6); a principle objective of re­search into atmospheric turbulence is to ascertain their forms, and how their parameters depend on meteorological conditions, terrain, etc.

f 0, r) should not be confused with the time-delayed correlation measured by a fixed instrument in a flow passing it at a mean speed U.