**2.9 The covariances and correlation coefficient functions using random square waves 2.9.1 The turbulent flux of the scalar** *F*

The turbulent scalar flux, denoted by *F* , is defined as the mean product between scalar fluctuations (*f*) and velocity fluctuations (߱)

$$
\dot{F} = \overline{\alpha f} \tag{26}
$$

Thus *f* in equation (2) is the turbulent flux of *F* along *z*. The statistical correlation between ߱ and *f* is given by the correlation coefficient function, *r*, defined as

$$r = \frac{\overline{\alpha \cdot f}}{\sqrt{\overline{\alpha \cdot ^2}} \sqrt{f^2}} \tag{27}$$

*r* is a function of *z*, and 0 1 *r* . As it is clear from equations (26) and (27), *r* is also the normalized turbulent flux of *F* and reaches a peak amplitude less than or equal to 1.0, a range convenient for the present method, coinciding with the defined functions *n*, ߙ, ߚ, also bounded by 0.0 and 1.0 (as shown in table 1). The present method allows to express *r* as dependent on *n*, the normalized mean profile of *F*.
