Question on the derivation of a prognostic equation for the mean concentration (Assume horizontal homogeneity in X) In deriving the prognostic equation for the mean quantity of pollutant C, as shown in my solution below:

Am I right in that I do not need to separate T (temperature) into its mean and turbulent parts (like $T = \overline{T}$ + t’)? If I am right, is it because T does not depend on anything in this equation?
When assuming horizontal homogeneity in the x-direction, this means that the change in c with respect to x (dc/dx = 0) is zero. So does that leave the final prognostic equation with no advective terms since I aligned U to be in X also?


Thanks for any help!
In the question, T was not specifically mentioned to be a constant. So by also expanding $T = \overline{T} + t’$, we end up with the following answer. Where, the covariance term containing $\overline{c’t’}$ reflects that turbulence still plays a role in determining the mean quantity of C.