Revisiting earlier modeling of the North Atlantic Oscillation (NAO) and Arctic Oscillation (AO) indices with the benefit of updated analysis approaches such as negative entropy. These two indices in particular are intimidating because to the untrained eye they appear to be more noise than anything deterministically periodic. Whereas ENSO has periods that range from 3 to 7 years, both NAO and AO show rapid cycling often on a faster-than-annual pace. The trial ansatz in this case is to adopt a semi-annual forcing pattern and synchronize that to long-period lunar factors, fitted to a Laplace’s Tidal Equation (LTE) model.
Start with candidate forcing time-series as shown below, with a mix of semi-annual and annual impulses modulating the primarily synodic/tropical lunar factor. The two diverge slightly at earlier dates (starting at 1880) but the NAO and AO instrumental data only begins at the year 1950, so the two are tightly correlated over the range of interest.
The intensity spectrum is shown below for the semi-annual zone, noting the aliased tropical factors at 27.32 and 13.66 days standing out.
The NAO and AO pattern is not really that different, and once a strong LTE modulation is found for one index, it also works for the other. As shown below, the lowest modulation is sharply delineated, yet more rapid than that for ENSO, indicating a high-wavenumber standing wave mode in the upper latitudes.
The model fit for NAO (data source) is excellent as shown below. The training interval only extended to 2016, so the dotted lines provide an extrapolated fit to the most recent NAO data.
Same for the AO (data source), the fit is also excellent as shown below. There is virtually no difference in the lowest LTE modulation frequency between NAO and AO, but the higher/more rapid LTE modulations need to be tuned for each unique index. In both cases, the extrapolations beyond the year 2016 are very encouraging (though not perfect) cross-validating predictions. The LTE modulation is so strong that it is also structurally sensitive to the exact forcing.
Both NAO and AO time-series appear very busy and noisy, yet there is very likely a strong underlying order due to the fundamental 27.32/13.66 day tropical forcing modulating the semi-annual impulse, with the 18.6/9.3 year and 8.85/4.42 year providing the expected longer-range lunar variability. This is also consistent with the critical semi-annual impulses that impact the QBO and Chandler wobble periodicity, with the caveat that group symmetry of the global QBO and Chandler wobble forcings require those to be draconic/nodal factors and not the geographically isolated sidereal/tropical factor required of the North Atlantic.
It really is a highly-resolved model potentially useful at a finer resolution than monthly and that will only improve over time.
(as a sidenote, this is much better attempt at matching a lunar forcing to AO and jet-stream dynamics than the approach Clive Best tried a few years ago. He gave it a shot but without knowledge of the non-linear character of the LTE modulation required he wasn’t able to achieve a high correlation, achieving at best a 2.4% Spearman correlation coefficient for AO in his Figure 4 — whereas the models in this GeoenergyMath post extend beyond 80% for the interval 1950 to 2016! )