# 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:

1. 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?
2. 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!

• What happened to $\overline{c’T’}$? Oct 26, 2020 at 15:17
• ^That is, what makes you think that temperature is uniform? Oct 26, 2020 at 16:03
• Well, actually the question does not state this. I may have overlooked the statement that specifically says that the decomposition of C depends on T. Thanks! Oct 27, 2020 at 1:58

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.