rotary spectra
DESCRIPTION
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 + , + ) - PowerPoint PPT PresentationTRANSCRIPT
Rotary SpectraSeparate 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:
nnnnn
nnnnn
)tsin(d)tcos(c)t(v
)tsin(b)tcos(a)t(u
)tsin(d)tcos(ci)tsin(b)tcos(a ivuw
)tsin()idb()tcos()ica(
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:
)tsin(i)AA()tcos()AA(
)tsin(i)tcos(A)tsin(i)tcos(A eAeAw titi
idbAAiicaAA
)bc(idaA
)bc(idaA
2121
dibAAicaAA
)tsin()idb()tcos()ica(w
dibAAicaAA
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.
2
122
2122
2121
bcdaA
bcdaA
Major axis = (A++ A-)
minor axis = (A+- A-)
)bc(idaA
)bc(idaA
2121
21
21
:phase
:norientatio
dabctan
dabctan
1
1
where:
Major axis = (A++ A-)
minor axis = (A+- A-)
tN
AS
tNAS
2
2
and the components of the rotary spectrum:
La Paz Lagoon, Gulf of California
Small minor axisOriented ~40º from EastSlope ~ 0.84
uv
abcd
nnnnn
nnnnn
)tsin(d)tcos(c)t(v
)tsin(b)tcos(a)t(u
2/2/
2/2/
)/2sin(2)/2cos(2
)/2cos(2)/2cos(2
N
nn
N
nn
N
nn
N
nn
NnjvN
dNnjuN
b
NnjvN
cNnjuN
a
22
22
dc
ba
Fourier Coefficients
tN
AS
tNAS
2
2
2
122
2122
2121
bcdaA
bcdaA
S+ S-
S+
S-
Fortnightly(0.068 cpd)
S+S-
21 :norientatio
dabctan
dabctan
1
1
where:
Major axis = (A++ A-)
minor axis = (A+- A-)
SSSSr
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)