1. Rotary Spectra Separate vector time series (e.g., current or wind data) into clockwise and counter-clockwise rotating circular components. Instead of having two Cartesian components (u, v) we have two circular components (A-,  - ; A+, + ) Suppose we have de-meanedu and v components of velocity, represented by Fourier Series (one coefficient for each frequency): These can be written in complex form (dropping subindices and summation) as:

2. Now write was a sum of clockwise and counter-clockwise rotating components: Remember: e i t= cos(t) + i sin( t) rotates counter-clockwise in the complex plane, and e -i t= cos( t) – i sin( t) rotates clockwise. Equating the coefficients of the cosine and sine parts, we find: A- A+

3. Magnitudes of the rotary components : The -and +components rotate at the same frequency but in opposite directions. → Sometimes they will reinforce each other (pointing in the same direction) and sometimes they will oppose each other (pointing in opposite direction) tending to cancel each other. Major axis = (A++ A-) minor axis = (A+- A-)

4. Major axis = (A++ A-) minor axis = (A+- A-) where: and the components of the rotary spectrum:

5. La Paz Lagoon, Gulf of California Small minor axis Oriented ~40 from East Slope ~ 0.84

6. a b c d

7. Fourier Coefficients

8. S+ S-

9. S+ S-

10. S+ S-

12. Major axis = (A++ A-) minor axis = (A+- A-)

13. Ellipticity = minor / major

14. Examples: Miles Sundermeyer notes (U MASS)

15. Examples: Miles Sundermeyer notes (U MASS)

16. Examples: Miles Sundermeyer notes (U MASS)

17. Examples: Miles Sundermeyer notes (U MASS)