Sea-Level Height as a proxy for ENSO

Sea-level height has several scales. At the daily scale it represents the well-known lunisolar tidal cycle. At a multi-decadal, long-term scale it represents behaviors such as global warming. In between these two scales is what often appears to be noisy fluctuations to the untrained eye. Yet it’s fairly well-accepted [1] that much of this fluctuation is due to the side-effects of alternating La Nina and El Nino cycles (aka ENSO, the El Nino Southern Oscillation), as represented by measures such as NINO34 and SOI.

To see how startingly aligned this mapping is, consider the SLH readings from Ft. Denison in Sydney Harbor. The interval from 1980 to 2012 is shown below, along with a fit used recently to model ENSO.

(click to expand chart)

I chose a shorter interval to somewhat isolate the trend from a secular sea-level rise due to AGW. The last point is 2012 because tide gauge data collection ended then.

As cross-validation, this fit is extrapolated backwards to show how it matches the historic SOI cycles

Much of the fine structure aligns well, indicating that intrinsically the dynamics behind sea-level-height at this scale are due to ENSO changes, associated with the inverted barometer effect. The SOI is essentially the pressure differential between Darwin and Tahiti, so the prevailing atmospheric pressure occurring during varying ENSO conditions follows the rising or lowering Sydney Harbor sea-level in a synchronized fashion. The change is 1 cm for a 1 mBar change in pressure, so that with the SOI extremes showing 14 mBar variation at the Darwin location, this accounts for a 14 cm change in sea-level, roughly matching that shown in the first chart. Note that being a differential measurement, SOI does not suffer from long-term secular changes in trend.

Yet, the unsaid implication in all this is that not only are the daily variations in SLH due to lunar and solar cyclic tidal forces, but so are these monthly to decadal variations. The longstanding impediment is that oceanographers have not been able to solve Laplace’s Tidal Equations that reflected the non-linear character of the ocean’s response to the long-period lunisolar forcing. Once that’s been analytically demonstrated, we can observe that both SLH and ENSO share essentially identical lunisolar forcing (see chart below), arising from that same common-mode linked mechanism.

Many geographically located tidal gauge readings are available from the Permanent Service for Mean Sea Level (PSMSL) repository so I can imagine much can be done to improve the characterization of ENSO via SLH readings.


[1] F. Zou, R. Tenzer, H. S. Fok, G. Meng and Q. Zhao, “The Sea-Level Changes in Hong Kong From Tide-Gauge Records and Remote Sensing Observations Over the Last Seven Decades,” in IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 14, pp. 6777-6791, 2021, doi: 10.1109/JSTARS.2021.3087263.

Highly-Resolved Models of NAO and AO Indices

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.

(click on any image to expand)

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! )