H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, Mass., 1972) pp. 98–99.
V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), pp. 40–58.
V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Washington, D.C., 1971; available from National Technical Information Service, Springfield, Va. 22161), pp. 46–102.
R. E. Hufnagel, "Propagation through atmospheric turbulence," in The Infrared Handbook (U.S. Government Printing Office, Washington, D.C., 1978), Chap. 6, pp. 6-1–6-56.
S. F. Clifford, "The classical theory of wave propagation in a turbulent medium," in Topics in Applied Physics—Laser Beam Propagation in the Atmosphere (Springer-Verlag, New York, 1978), Vol. 25, Chap. 2, pp. 9–43.
J. C. Wyngaard, "On surface-layer turbulence," in Workshop on Micrometeorology (American Meteorological Society, Boston, Mass., 1972), Chap. 3, pp. 101–149.
The data in Figs. 1–4 are based on a log-normal average of 50-m spatial segments. An optical device performs a longer linear spatial average that has a different mean. This linear mean x¯ is obtained from the log-normal mean and standard deviation y¯, σy by x¯ = exp(y¯ + σy/2). For the standard deviations of Figs. 1–4, this corresponds to a multiplicative factor of about about 1.8 increase in the mean. See, for example, J. Aitchison and J. Brown, The Log Normal Distribution (Cambridge U. Press, New York, 1957), Chap. 2, pp. 7–9.
Figure 1 shows a z-1.16 dependence for Cn 2, whereas most theories, such as Eq. (8), have a z-4/3 dependence for the unstable, midday convective period. The z-1.16 form we observe appears to be the result of the climatological averaging process, where Cn 2 data for several days are combined. Two effects occur that reduce the exponential coefficient. First, the inversion height has been neglected in compiling Fig. 1, and, from Fig. 15, Cn 2 increases as the inversion is approached. Second, the strength of the wind speed influences the thickness of the surface layer, which is of the order of a few meters to tens of meters. The high wind shear within the surface boundary layer forces Cn 2 toward a z-2/3 dependence (see Refs. 16 and 17). Depending on the wind speed, we have observed both the z-2/3 and z-4/3 forms. Thus, above the surface boundary layer, the z-4/3 dependence is the most representative form for the change in Cn 2 with altitude during the day up to about one half of the inversion height.
A known wind-speed dependence discussed in Refs. 16 and 17 is ignored in Fig. 10. The significance of the wind-speed term is evident, in part, in the standard deviation of Fig. 10.
L. G. McAllister et al., "Acoustic sounding—a new approach to the study of atmospheric structure," Proc. IEEE 57, 579–587 (1969).
F. F. Hall, Jr., "Temperature and wind structure studies by acoustic echo-sounding," in Remote Sensing of the Troposphere (U.S. Government Printing Office, Washington, D.C., 1972), Chap. 18, pp. 18-1–18-26.
W. D. Neff, "An observational and numerical study, calibration techniques of the atmospheric boundary layer overlying the East Antarctic ice sheet," Ph.D. thesis (University of Colorado, Boulder, Colo., 1980).
T. Beer, Atmospheric Waves (Halsted, New York, 1974), pp. 54–86.
C. Fein, ARPA Maui Observatory Station, Maui, Hawaii, personal communication, September 20, 1979.