The north Atlantic oscillation (NAO) is a most erratic climate index, often showing full cycles spanning less than a year. It is intimidating to consider that the raw NAO time-series can even be modelled, as it is often characterized as a weather precursor/indicator for Europe . Four years ago, I posted this to the ATTP blog:

This was fit applying the usual LTE model: a semi-annual stroboscopic sample of a LOD-derived forcing, integrated and then an LTE modulation applied. The semi-annual impulses were allowed to bleed through as they seem to capture the fast cycling well. The LTE modulation is approximately a winding=1 for each + or – semi-annual excursion. One can see the 3.8-year aliased cycling of the principal Mf tidal factor in the middle forcing panel, and an overall 18.6-year envelope.

On recreating this fit the past few days, I couldn’t duplicate the exact same configuration from the repo, but came close: (note in all the following that the training and validation legend is reversed in the upper right regression panel)

In both cases, the same prominent 3.8-year cycles in forcing (with 18.6-year envelope) were observed, along with essentially the same LTE modulation. The algorithm behind the fit is to perturb the calibrated LOD tidal factors enough so they will start to align with the NAO time series, especially synching the phases since timing is so crucial. In general, it’s difficult to achieve the same height of excursions, but the running windowed correlation (bottom panel with GREEN curve) shows a uniform accounting of the full cycling. Below is a DTW metric trained fit, which does a dynamic time warp on the time-axis, thus relaxing the alignment on each excursion. The improvement is subtle.

The alignment with the original LOD calibration was shifted by approximately 2-years (see below), but this could also be due to derivative adjustments. It’s not clear which order of forcing that the NAO is responding to — is it just the original dLOD/dt or some higher order acceleration? A higher derivative would generate a lead factor, i.e. a sin(t) derivative would turn into a cos(t) lead term.

Different perspective

Shorter interval

The NAO used above covers years after 1950 and stops short of 2018. On Climate Explorer: Monthly time series one can find other NAO variants that cover a wider timespan interval. In the plot below, the NAO is extended back to 1880 and forward to nearly present day. The back extrapolation is not the best but the forward beyond 2015 is excellent. The fact that the LTE modulation in the middle right panel still shows a strong low-order winding indicates that the fundamental patter remains across the entire timespan but is structurally sensitive to the fitted parameters.

Refitting by including the back-extrapolated interval prior to 1953 does not significantly reduce the correlation in the excluded post-2015 interval.

Structural sensitivity in this class of models is significant. Think about how tides with daily to monthly variation need to be calibrated to effects that evolve over decades. This will take time and effort to improve, likely with input from others that can contribute subtleties that I have missed. Yet, doing this kind of deep fitting exercise is at odds with those that have been indoctrinated to a weather mindset of predicting only what’s in store for the immediate future.