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:

Now 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- A+A+

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 - )

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

La Paz Lagoon, Gulf of California Small minor axis Oriented ~40º from East Slope ~ 0.84 u v

abcdabcd

Fourier Coefficients

S+S+ S-S-

S+S+ S-S- Fortnightly (0.068 cpd)

S+S+ S-S-

where:

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

Ellipticity = minor / major

Examples: Miles Sundermeyer notes (U MASS)

Examples: Miles Sundermeyer notes (U MASS)

Examples: Miles Sundermeyer notes (U MASS)

Examples: Miles Sundermeyer notes (U MASS)