I was always wondering how to do a Fourier series on the LTE modulation without having to do a numerical sort on the ordinal. Well, I should have realized that it's not necessary as long as the input series is discrete. Here is the Fourier amplitude spectrum of the LTE modulation:

![](https://imagizer.imageshack.com/img922/7793/W0B6JJ.png)

This provides a direct representation of the wavenumber dispersion. The largest spike is the primary ENSO dipole and the peak at 15x the ENSO wavenumber lies at the same value as the equatorial Tropical Instability Wave (TIW). The remainder consists of the much higher wavenumber dynamics that apparently generates the almost daily variability in ENSO indices such as SOI. These high wavenumbers are I think due to the details in the step response of the LTE modulation

![](https://imagizer.imageshack.com/img921/9246/Dl9HE4.png)

I have spent some time trying to fit to the daily variations and there may be some hope here. Some variation of long-range order appears to exist based on the fit above, but it may be obscured by other sources of noise.

![daily](https://imagizer.imageshack.com/img924/3492/FRviZi.png)

The daily SOI does not show as clear a Double-Sideband Suppressed Carrier modulation as the monthly time series, which may be the result of the Darwin-Tahiti dipole not capturing the sharp delineation of a higher wavenumber standing wave. In other words, the low wavenumber of the primary ENSO dipole allows uncertainty in the geographic location of the dipole nodes. The hope maybe in trying to force a DSSC symmetry by independently doing the Tahiti and Darwin time series and rejecting points that aren't mirror-symmetry aligned and subsequently combining them.

As with Navier-Stokes, the higher the resolution detail that one seeks in understanding the fluid dynamics the less certainty and more DOF one understandably encounters.

Should add that the inferred LTE modulation from the daily SOI time-series shows a very interesting impulse train that appears due to the TIW. The modulation is inferred as it uses the tidal forcing level as a (sorted) x-coordinate and the actual SOI amplitude is plotted as the amplitude at that coordinate. The impulse train is indicated by the upward arrows below, very regularly spaced. As it is an impulse train, it must have all the higher harmonics of the fundamental LTE TIW modulation. It may have something to do with a specific tidal cycle creating a resonance, and am hoping it's not some computation artifact.

![tiw](https://imagizer.imageshack.com/img923/9632/AWWhpU.png)

![](https://imagizer.imageshack.com/img922/7793/W0B6JJ.png)

This provides a direct representation of the wavenumber dispersion. The largest spike is the primary ENSO dipole and the peak at 15x the ENSO wavenumber lies at the same value as the equatorial Tropical Instability Wave (TIW). The remainder consists of the much higher wavenumber dynamics that apparently generates the almost daily variability in ENSO indices such as SOI. These high wavenumbers are I think due to the details in the step response of the LTE modulation

![](https://imagizer.imageshack.com/img921/9246/Dl9HE4.png)

I have spent some time trying to fit to the daily variations and there may be some hope here. Some variation of long-range order appears to exist based on the fit above, but it may be obscured by other sources of noise.

![daily](https://imagizer.imageshack.com/img924/3492/FRviZi.png)

The daily SOI does not show as clear a Double-Sideband Suppressed Carrier modulation as the monthly time series, which may be the result of the Darwin-Tahiti dipole not capturing the sharp delineation of a higher wavenumber standing wave. In other words, the low wavenumber of the primary ENSO dipole allows uncertainty in the geographic location of the dipole nodes. The hope maybe in trying to force a DSSC symmetry by independently doing the Tahiti and Darwin time series and rejecting points that aren't mirror-symmetry aligned and subsequently combining them.

As with Navier-Stokes, the higher the resolution detail that one seeks in understanding the fluid dynamics the less certainty and more DOF one understandably encounters.

Should add that the inferred LTE modulation from the daily SOI time-series shows a very interesting impulse train that appears due to the TIW. The modulation is inferred as it uses the tidal forcing level as a (sorted) x-coordinate and the actual SOI amplitude is plotted as the amplitude at that coordinate. The impulse train is indicated by the upward arrows below, very regularly spaced. As it is an impulse train, it must have all the higher harmonics of the fundamental LTE TIW modulation. It may have something to do with a specific tidal cycle creating a resonance, and am hoping it's not some computation artifact.

![tiw](https://imagizer.imageshack.com/img923/9632/AWWhpU.png)