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:

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+

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-)

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

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

a b c d

Fourier Coefficients

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)