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Given the question below, how would the kinematic momentum flux be at the surface layer?

For letter a, why would it be that way (positive, negative, or 0)?

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Based on the answer below:

I’m not sure if I’ve done it correctly, but this is how I answered it based in your comment below.

  1. First, I assumed that $\overline{u’w’}$ = $\overline{w’u’}$. So is this something like the mixing of horizontal momentum into areas of lower wind speed (as is the case in the surface layer)?

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Instead of just giving you the answer to a textbook problem (which would make SE a version of Chegg that future students would use to cheat on homework), I'll guide you in getting an answer.

  1. Which variable does $\overline{u'w'}$ affect most?
  2. For that variable, identify the surface layer.
  3. Imagine that you are on the line. If you go straight up (that is, $w'>0$) what would the difference be between the $M$ traced out and the graphed $M$? What is the sign of $w' \times u'$?
  4. Do the same as number 3, except start at the top and go down ($w'<0$).
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  • $\begingroup$ Hi! Thanks for your answer. I’m not sure if I got it right. I posted and edit above with some questions. Thanks for your help! $\endgroup$
    – jake_96
    Oct 22 '20 at 5:00
  • $\begingroup$ ... I don't think you did it right. Try thinking of step 3 and step 4 as two different steps- start on the line for both. $\endgroup$ Oct 22 '20 at 14:34
  • $\begingroup$ When you said what the sign of w’ x u’ is, did u mean a cross product? Or something like, for step 3, since w>0 going up is (+) and u in this area would be (-), hence w’u’ < 0. Similarly, for step 4 going down, w’ < 0 and u’ > 0 giving w’u’ <0. So the average or net $\overline{w’u’}$ <0, meaning net momentum flux in this layer is downward? Thanks. $\endgroup$
    – jake_96
    Oct 23 '20 at 2:32
  • $\begingroup$ No, I meant multiplication. You don't have to take an average. If you start on the line and go up. What is the difference between that point and the line? Consider page 52 of Stull. $\endgroup$ Oct 23 '20 at 3:30
  • $\begingroup$ Okay. So going up (w’ >0) , the difference is negative (u < 0), and the sign of w’u’ < 0 [step 3]. Doing the same thing going down (w’ < 0), the difference between the traced line and the surface layer wind is positive (u > 0), making w’u’ < 0 [step 4]. Now, since both steps 3 and 4 returned negative results for w’u’, is this enough to say that flux is towards the ground? $\endgroup$
    – jake_96
    Oct 25 '20 at 8:20

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