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-meaned u 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. In addition, write w as 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 Californiav
Small minor axis
Oriented ~40º from East
Slope ~ 0.84

6. a b c d

7. Fourier Coefficients

8. S+ S-

9. Fortnightly
(0.068 cpd) S+ S-

10. S+ S-

11. where:

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)