observed i theoretical then R = log C + m log z (2.4) Hence by plotting R versus log z, the value of the exponent m can be measured. The size of m determines to what extent the observed height dependence of the high frequency spectral levels follows the height dependence predicted from the semi-empirical dimensional arguments for the surface layer. 2.3.1 Predicted Height Dependence of the Spectral Levels -5/3 The high frequency spectral levels (in the k region) can be related to the surface fluxes and z/L (as in the similarity theory) using equations (2.1) and (2.2) and other theoretical and empirical results. In equation (2.1) the Kolmogorov constant, o( ( , was taken as equal to 0.55 following the results of McBean et al (1971) and Paquin and Pond (1971); C/ 2 and Ot^ were taken as 4/3 Of, . There is much discrepency between the observed values of |9 , in equation (2.2). Paquin and Pond (1971) found Q =0.4 based on measurements of structure functions; Wyngaard and Cote (1971) found ^ = 0.41 from an energy balance; a direct measurement of ^ by Boston (1970) gave a value of ^ = 0.8. Since 16 Boston s value was a directly measured value, g =0.8 was adopted for use in the \"theoretical\" spectral levels in this analysis. From the Navier-Stokes equation and the heat equation expressions can be derived for £ and £ e (see for example, Lumley and Panofsky (1964)): (2.5) = w ' T - D 9 (2.6) where primed quantities are fluctuations, and D and D & are residual terms. The residual terms include the effects of horizontal transports, pressure terms and the vertical divergences of the vertical transports. Previous energy balance studies close to the surface (Mordukhovich and Tsvang (1966), McBean et al (1971) and Wyngaard and Cote (1971) have shown that under unstable conditions close to the surface the above residual terms are not small, particularly the vertical divergence terms, ~^7T ( w' e') in (2.5) and - -r- (wT ) in (2.6), where e is the turbulent kinetic a Z 2 2 2 energy (per unit mass) given by e •= 1/2 ( u' + v' + w' ). Since detailed accurate profiles of wind and temperature were not measured, empirical expressions for iT//3 z and ~i9 / 3 z as functions of stability have been used in the calculation of the \"theoretical\" spectral levels to be compared to the observed spectral levels at large z . Expressions for 3U/ d z and j6/a z There are several empirical expressions for the stability , 17 dependence of the non-dimensional wind shear

h -1/3 does not have the z dependence expected from dimensional arguments. The observed spectral levels will be used to test which of these two formulations leads to the observed height dependence at large z. The predicted spectral levels for temperature and velocity using equations (2.1) through (2.9) can be written, in,)-(2.13) and I X2 U - 1 KM (2.14) where D is the normalized form of the residual term D in equation (2.5). For the calculation of \"theoretical\" spectral levels to be compared with the observed levels (see equation (2.3)), the remainder terms D*and D e will be ignored and the Keyps profiles, equations (2.9) and (2.10), with = ^ » will be used. Since there is a net transport of kinetic energy and mean square temperature fluctuations out of the surface layer and into the layer near the inversion, then the divergence of the transport terms - 211 z (w'e') and -3/3z (w'T ) must change sign between the surface layer and the inversion. Hence in the middle layer of the PBL, neglecting the vertical divergences of the 19 vertical transports may not cause a large error in the \"theoretical\" spectral levels. It is also seen that the choice of the Keyps rather than the Businger-Dyer representation for the case of large z will be of l i t t l e consequence for the velocity spectral levels, (in equation (2.13)), since the term involving <|> will be small. 2.3.2 Predicted Height Dependences of the Variances According to similarity theory the height dependences of the variances of temperature, (T^ . and vertical velocity, 0\"^ , when normalized, are representable in terms of universal functions of z/L . From dimensional arguments based on empirical results for the surface layer, (see for example, Lumley and Panofsky (1964, p. 133ff.). 0*w « Z for neutral conditions 0~- ^ * for near-neutral conditions and 01 oc Z r The observed height dependencies of [Tw and Q\" will be compared to the above dependencies. A l l turbulent statistics such as variances,turbulent fluxes and spectral levels are height dependent. In the following sections each particular parameter will be examined in turn. The summary of the gross statistics of the measurements made on July 14 and July 16 are presented in Tables I and II. The statistics are evaluated by summing the observed spectral densities over wavenumbers greater than some lower limit, k^ , as shown in the Tables. Detailed discussion of the statistics are carried for free convection - z / L ^ l . ) T A B L E I A T E M P E R A T U R E A N D V E R T I C A L V E L O C I T Y S T A T I S T I C S . J U L Y 1 6 A I R C R A F T S T A T I S T I C S W'TV [o 2 a 2 ] 1 / 2 w T H E I G H T L E G [ m ] L . 4 x l 0 6 1 0 305 228 1 5 2 1 2 2 92 6 1 f l (1 (1 (1 a TOWER S T A T I S T I C S 9 2 m 4 8 m 3 . 5 m