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How do you derive the components of the inertial advective wind in natural coordinates?

The vector form of the inertial advective wind is:

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I have been able to find the components in cartesian coordinates as shown here:

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I want to find the components in natural coordinates (such as shown below), but I have been unsuccessful so far.

enter image description here

Any help would be greatly appreciated! Thanks

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  • $\begingroup$ you can start by taking a look at my answer here - earthscience.stackexchange.com/questions/12353/… $\endgroup$
    – user1066
    Apr 26, 2021 at 15:40
  • $\begingroup$ Can you provide a source for (1) and (2) ? $\endgroup$
    – user1066
    Apr 26, 2021 at 16:09
  • $\begingroup$ I can show how to get Equation (2) but not equation (1). You may have the terms wrong. The second equation should give you the normal component of the ageostrophic wind - climate.ucdavis.edu/ATM121/… $\endgroup$
    – user1066
    Apr 26, 2021 at 16:24
  • $\begingroup$ Hi for reference the first equation comes from Rochette 2006, "A primer of the ageostrophic wind," (citeseerx.ist.psu.edu/viewdoc/…) And the second equation comes from the Encyclopedia of Atmospheric Sciences (2nd edition) volume 5 under the entry on Jet Streaks - a preview with the relevant equation can be found here under the heading "SYNOPTIC METEOROLOGY | Jet Streaks" sciencedirect.com/topics/earth-and-planetary-sciences/… $\endgroup$ Apr 26, 2021 at 21:19

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