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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?

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Thanks for any help!

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    $\begingroup$ What happened to $\overline{c’T’}$? $\endgroup$
    – BarocliniCplusplus
    Oct 26, 2020 at 15:17
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    $\begingroup$ ^That is, what makes you think that temperature is uniform? $\endgroup$
    – BarocliniCplusplus
    Oct 26, 2020 at 16:03
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    $\begingroup$ 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! $\endgroup$
    – meteo_96
    Oct 27, 2020 at 1:58

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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.

enter image description here

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