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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.

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Any help would be greatly appreciated! Thanks

Will Watson
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  • you can start by taking a look at my answer here - https://earthscience.stackexchange.com/questions/12353/centrifugal-force-in-the-navier-stokes-equations/16767#16767 –  Apr 26 '21 at 15:40
  • Can you provide a source for (1) and (2) ? –  Apr 26 '21 at 16:09
  • 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 - https://climate.ucdavis.edu/ATM121/AtmosphericDynamics-Chapter03-Part02-BalancedFlow.pdf –  Apr 26 '21 at 16:24
  • Hi for reference the first equation comes from

    Rochette 2006, "A primer of the ageostrophic wind," (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.663.2094&rep=rep1&type=pdf)

    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" https://www.sciencedirect.com/topics/earth-and-planetary-sciences/ageostrophic-flow

     – Will Watson Apr 26 '21 at 21:19

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