Turbulence near the ground is of the boundary-layer variety (see Fig. 9.36), being variable with height and anisotropic. A model for this case should ideally give the following:

(i) Variation of mean wind with height as function of ground roughness.

(ii) Variation with height of %2, ада2, w32.

(iii) Variation with height of all significant scales.

(iv) The form of the spectrum function 6ip or the correlation function Ri}.

Since the scales are much smaller than at high altitude, it becomes more important to have the two-dimensional spectrum functions 02), which

enable both streamwise and spanwise variations of turbulent velocity to be
taken into account. The anisotropy also leads to the nonvanishing of the correlation R13 = (UjU^ (where xx is in the wind direction and x3 vertical), which is simply related to the turbulent shear stress (Reynolds stress) in the boundary layer.

An interesting fact about low-altitude turbulence is the existence of a gap in the spectrum at a rather useful location. There is considerable evidence to show that the spectrum of wind speed measured by van der Hoven is representative, Fig. 13.7. This is a spectral density of horizontal wind speed taken as a function of time at a fixed point. The gap occurs for periods

Fig. 13.7 Schematic spectrum of wind speed near the ground estimated from a study of van der Hoven (1957) (from ref. 13.6, p. 43).

greater than 6 min, or frequencies less than 10 cycles per hr. The lobe on the right corresponds to the turbulent energy of interest for flight (cf. Fig. 13.6).

The extensive information available on the wind-induced turbulence near the ground—much of it inconclusive and even contradictory—has recently been reviewed in refs. 13.7, 13.8. From these we adopt the following model as a reasonable representation of presently-available information: the turbulence is Gaussian, stationary, and homogeneous w. r.t. horizontal translations; it is anisotropic, but the one-dimensional spectra display isotropic behavior at the highest wave numbers; the turbulence is symmetric w. r.t. vertical planes.