The SAO and Annual Disturbances

In Chapter 11 of the book Mathematical GeoEnergy, we model the QBO of equatorial stratospheric winds, but only touch on the related cycle at even higher altitudes, the semi-annual oscillation (SAO). The figure at the top of a recent post geometrically explains the difference between SAO and QBO — the basic idea is that the SAO follows the solar tide and not the lunar tide because of a lower atmospheric density at higher altitudes. Thus, the heat-based solar tide overrides the gravitational lunar+solar tide and the resulting oscillation is primarily a harmonic of the annual cycle.

Figure 1 : The SAO modeled with the GEM software fit to 1 hPa data along the equator
Continue reading

Double-Sideband Suppressed-Carrier Modulation vs Triad

An intriguing yet under-reported finding concerning climate dipole cycles is the symmetry in power spectra observed. This was covered in a post on auto-correlations. The way that this symmetry reveals itself is easily explained by a mirror-folding about one-half some selected carrier frequency, as shown in Fig. 1 below.

Figure 1 : ENSO amplitude spectrum is mirror-folded about 1/2 the annual frequency.
Both the data and model align with their mirrored counterparts, as seen in highlighted box.
Continue reading

Characterizing Wavetrains

In Chapter 11 and Chapter 12 of the book we characterize deterministic and stochastic variability in waves. While reviewing the presentations at last week’s EGU meeting, one study covered some of the same ground [1] and was worth a more detailed look. The distribution of stratospheric wind wave energy collected by Nastrom [2] (shown below) that we model in Chapter 11 is apparently still not completely understood.

From Mathematical Geoenergy, Chap 11
Continue reading

Length of Day II

This is a continuation from the previous Length of Day post showing how closely the ENSO forcing aligns to the dLOD forcing.

Ding & Chao apply an AR-z technique as a supplement to Fourier and Max Entropy spectral techniques to isolate the tidal factors in dLOD

The red data points are the spectral values used in the ENSO model fit.

The top panel below is the LTE modulated tidal forcing fitted against the ENSO time series. The lower panel below is the tidal forcing model over a short interval overlaid on the dLOD/dt data.

That’s all there is to it — it’s all geophysical fluid dynamics. Essentially the same tidal forcing impacts both the rotating solid earth and the equatorial ocean, but the ocean shows a lagged nonlinear response as described in Chapter 12 of the book. In contrast, the solid earth shows an apparently direct linear inertial response. Bottom line is that if one doesn’t know how to do the proper GFD, one will never be able to fit ENSO to a known forcing.

Triad Waves

In Chapter 12 of the book, we discuss tropical instability waves (TIW) of the equatorial Pacific as the higher wavenumber (and higher frequency) companion to the lower wavenumber ENSO (El Nino /Southern Oscillation) behavior. Sutherland et al have already published several papers this year that appear to add some valuable insight to the mathematical underpinnings to the fluid-mechanical relationship.

“It is estimated that globally 1 TW of power is transferred from the lunisolar tides to internal tides[1]. The action of the barotropic tide over bottom topography can generate vertically propagating beams near the source. While some fraction of that energy is dissipated in the near field (as observed, for example, near the Hawaiian Ridge [2]), most of the energy becomes manifest as low-mode internal tides in the far field where they may then propagate thousands of kilometers from the source [3]. An outstanding question asks how the energy from these waves ultimately cascades from large to small scale where it may be dissipated, thus closing this branch of the oceanic energy budget. Several possibilities have been explored, including dissipation when the internal tide interacts with rough bottom topography, with the continental slopes and shelves, and with mean flows and eddies (for a recent review, see MacKinnon et al. [4]). It has also been suggested that, away from topography and background flows, internal modes may be dissipated due to nonlinear wave-wave interactions including the case of triadic resonant instability (hereafter TRI), in which a pair of “sibling” waves grow out of the background noise field through resonant interactions with the “parent” wave”

see reference [2]
Continue reading

Filtering of Climate Data

One of the frustrating aspects of climatology as a science is in the cavalier treatment of data that is often shown, and in particular through the potential loss of information through filtering. A group of scientists at NASA JPL (Perigaud et al) and elsewhere have pointed out how constraining it is to remove what are considered errors (or nuisance parameters) in time-series by assuming that they relate to known tidal or seasonal factors and so can be safely filtered out and ignored. The problem is that this is only appropriate IF those factors relate to an independent process and don’t also cause non-linear interactions with the rest of the data. So if a model predicts both a linear component and non-linear component, it’s not helpful to eliminate portions of the data that can help distinguish the two.

As an example, this extends to the pre-mature filtering of annual data. If you dig enough you will find that NINO3.4 data is filtered to remove the annual data, and that the filtering is over-zealous in that it removes all annual harmonics as well. Worse yet, the weighting of these harmonics changes over time, which means that they are removing other parts of the spectrum not related to the annual signal. Found in an “ensostuff” subdirectory on the site:

Continue reading

Lemming/Fox Dynamics not Lotka-Volterra

Appendix E of the book contains information on compartmental models, of which resource depletion models, contagion growth models, drug delivery models, and population growth models belong to.

undefinedOne compartmental population growth model, that specified by the Lotka-Volterra-type predator-prey equations, can be manipulated to match a cyclic wildlife population in a fashion approximating that of observations. The cyclic variation is typically explained as a nonlinear resonance period arising from the competition between the predators and their prey. However, a more realistic model may take into account seasonal and climate variations that control populations directly. The following is a recent paper by wildlife ecologist H. L. Archibald who has long been working on the thesis that seasonal/tidal cycles play a role (one paper that he wrote on the topic dates back to 1977! ).

Continue reading

Stratospheric Sudden Warming

Chapter 11 of the book describes a model for the QBO of stratospheric equatorial winds. The stratified layers of the atmosphere reveal different dependencies on the external forcing depending on the altitude, see Fig 1.

Figure 1 : At high altitudes, only the sun’s annual cycle impacts the stratospheric as a semi-annual oscillation (SAO). Below that the addition of the lunar nodal cycle forces the QBO. The earth itself shows a clear wobble with the lunar cycle interacting with the annual.

Well above these layers are the mesosphere, thermosphere, and ionosphere. These are studied mainly in terms of space physics instead of climate but they do show tidal interactions with behaviors such as the equatorial electrojet [1].

The behaviors known as stratospheric sudden warmings (SSW) are perhaps a link between the lower atmospheric behaviors of equatorial QBO and/or polar vortex and the much higher atmospheric behavior comprising the electrojet. Papers such as [1,2] indicate that lunar tidal effects are showing up in the SSW and that is enhancing characteristics of the electrojet. See Fig 2.

Figure 2 : During SSW events, a strong modulation of period ~14.5 days emerges, close to the lunar fortnightly period as seen in these spectrograms. Taken from ref [2] and see quote below for more info.

“Wavelet spectra of foEs during two SSW events exhibit noticeable enhanced 14.5‐day modulation, which resembles the lunar semimonthly period. In addition, simultaneous wind measurements by meteor radar also show enhancement of 14.5‐day periodic oscillation after SSW onset.”

Tang et al [2]

So the SSW plays an important role in ionospheric variations, and the lunar tidal effects emerge as the higher atmospheric density of a SSW upwelling becomes more sensitive to lunar tidal forcing. That may be related to how the QBO also shows a dependence on lunar tidal forcing due to its higher density.


  1. Siddiqui, T. A. Relationship between lunar tidal enhancements in the equatorial electrojet and stratospheric wind anomalies during stratospheric sudden warmings. (2020). Originally presented at AGU 2018 Fall Meeting
  2. Tang, Q., Zhou, C., Liu, Y. & Chen, G. Response of Sporadic E Layer to Sudden Stratospheric Warming Events Observed at Low and Middle Latitude. Journal of Geophysical Research: Space Physics e2019JA027283 (2020).

Australia Bushfire Causes

The Indian Ocean Dipole (IOD) and the El Nino Southern Oscillation (ENSO) are the primary natural climate variability drivers impacting Australia. Contrast that to AGW as the man-made driver. These two categories of natural and man-made causes form the basis of the bushfire attribution discussion, yet the naturally occurring dipoles are not well understood. Chapter 12 of the book describes a model for ENSO; and even though IOD has similarities to ENSO in terms of its dynamics (a CC of around 0.3) the fractional impact of the two indices is ultimately responsible for whether a temperature extreme will occur in a region such as Australia (not to mention other indices such as MJO and SAM).  

Continue reading