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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:

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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+

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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 - )

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

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

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abcdabcd

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Fourier Coefficients

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S+S+ S-S-

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

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S+S+ S-S-

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where:

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

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Ellipticity = minor / major

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Examples: Miles Sundermeyer notes (U MASS)

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Examples: Miles Sundermeyer notes (U MASS)

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Examples: Miles Sundermeyer notes (U MASS)

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Examples: Miles Sundermeyer notes (U MASS)

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