Nonlinear Dynamics: Is Climate Chaotic?

Posted by: chaamjamal on: May 25, 2018

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DEPENDENCE: A SIMPLE WAY TO UNDERSTAND CHAOS IN TIME SERIES DATA.

Figure 1: Screen grab from the Youtube video “Forklift causes whole warehouse to collapse“. An example of dependence, nonlinear dynamics and chaos. The chaos we see here is the creation of a series of dependencies.

Link to video:  https://www.youtube.com/watch?v=pkrMgMoR0RU

Figure 3: Demonstration of chaotic behavior of Hurst persistence in time series data. A simple demonstration of dependency

1. The video in Figure 2 plots the same time series twice – in red and in blue. In the red line, the events in the time series evolve independently with no dependence or persistence. Such independence is an important assumption in OLS (ordinary least squares) regression. In the blue line the events in the time series evolve as a chaotic system with dependence and persistence. A small dependence has been inserted as a 1% tendency for persistence. Without persistence rising and falling tendencies are always equal with 50% probability at each event. The result of that is seen in the red line where the ups and downs are not visible because they are small.
2. The 1% persistence in the blue line means that if the prior change was positive, the probabilities change from 50%/50% to 51%/49% favoring a positive change for the next event. If it is positive again in the next event, the probabilities are changed to 52%/48% but if it is negative they change back to 50%/50%. The probabilities keep changing to favor the direction of change in the prior event. This behavior is called persistence and it is very common in nature, particularly in surface temperature.
3. When you play the video you will see the blue line take various shapes and trends both rising and falling mostly staying very close to the Gaussian red line. BUT, it is capable of sudden departures from the Gaussian to form what appears to be patterns such as rising and falling trends. These shapes do not imply cause and effect phenomena. They are the random behavior of a chaotic system.
4. All of these shapes are representations of the same underlying phenomenon. The differences among these curves have no interpretation because they represent randomness. The human instinct to look for causes for unusual patterns derives from Darwinian survival but it leads us astray when we study chaotic systems.
5. The shapes and trends the blue line forms may be found to be statistically significant if the system is assumed to be deterministic and the violations of OLS assumptions are ignored; but that statistical significance is meaningless. Yet this kind of analysis is common. The conclusions they imply in terms of the phenomena of nature have no interpretation because they are the product of violations of assumptions.
6. Climate variables at decadal time scales are are known to be chaotic and understood as Internal Climate Variability: [LINK]and therefore, not all climate events contain information about cause and effect phenomena particularly so at brief time scales.That nature is chaotic is well understood. These three books present that argument with data, examples, and case studies: EA Jackson, Exploring Nature’s Dynamics; C Letellier, Chaos in Nature; Cushing et al, Chaos in Ecology.
7. References to chaos in nature are also found in the climate change literature. These two excellent papers by Timothy Palmer are a good introduction to the subject of chaos in climate data: “A nonlinear dynamical perspective on climate prediction” Journal of Climate 12.2 (1999): 575-591, and “Predicting uncertainty in forecasts of weather and climate” Reports on Progress in Physics 63.2 (2000): 71.
8. A common method of detecting fractal/chaotic behavior in long time series of field data is to compute the so called Hurst exponent “H” of the time series as a way of detecting persistence in the data.  Persistence implies that the the data in the time series do not evolve independently but contain a dependence on prior values such that prior changes tend to persist into the next time slice. This method was first described by Harold Edwin Hurst in 1951 in the only paper he ever published which is still cited hundreds of times a year every year ( Hurst, H.E. (1951). “Long-term storage capacity of reservoirs”. Transactions of American Society of Civil Engineers).
9. The Hurst exponent method of detecting non-linear dynamics in time series data is used in climate change research. This trend in climate science has been led by the very charismatic and controversial hydrologist Demetris Koutsoyiannis, Professor of Hydrology, National Technical University of Athens. Demetris finds that many hydrology time series behaviors that climate science ascribes to emissions can be explained in terms of non-linear dynamics.
10. Here are some examples from the literature of  Hurst persistence analysis of climate data: Cohn, Timothy etal “Nature’s style: Naturally trendy.” Geophysical Research Letters 32.23 (2005) (by “naturally trendy” Tim means that things that look like trends are actually randomness); Weber, Rudolf etal. “Spectra and correlations of climate data from days to decades.” Journal of Geophysical Research: Atmospheres 106.D17 (2001): 20131-20144; Koutsoyiannis, Demetris. “Climate change, the Hurst phenomenon, and hydrological statistics.” Hydrological Sciences Journal 48.1 (2003): 3-24; Markonis, Yannis, “Climatic variability over time scales spanning nine orders of magnitude: Connecting Milankovitch cycles with Hurst–Kolmogorov dynamics.” Surveys in Geophysics 34.2 (2013): 181-207; Pelletier, Jon etal. “Long-range persistence in climatological and hydrological time series: analysis, modeling and application to drought hazard assessment.” Journal of Hydrology 203.1-4 (1997): 198-208; Rybski, Diego, et al. “Long‐term persistence in climate and the detection problem.” Geophysical Research Letters 33.6 (2006).
11. Much of the empirical work in climate science is presented in terms of time series of field data (field data means data made by nature over which the researcher has no control – as opposed to data collected in controlled experiments). Information about climate contained in these data is usually gathered by researchers in terms of OLS (ordinary least squares) regression analysis. Surprisingly, even at the highest levels of climate research little or no attention is paid to the assumptions of OLS analysis which include for example the assumption that the data in the time series evolve as I.I.D, (independent identically distributed). The stationarity assumption further enforces the requirement that the distribution must not change as the time series evolves.
12. More information on regression analysis may be found in a companion post on SPURIOUS CORRELATIONS. A good reference for regression analysis of time series data is “Time Series Analysis: Forecasting and Control, Wiley 2015, by George Box & Gwilym Jenkins. Another is Time Series Analysis and Its Applications with Examples, Springer 2017, by Robert Shumway. These authors have shown that when time series analysis goes awry you can bet it has to do with violations of OLS assumptions.
13. In the analysis of temperature, sea level rise, precipitation, solar activity, and ozone depletion listed below, the time series is tested for OLS violations by computing the Hurst exponent H. The theoretical neutral value with no serial dependence is H=0.5 but it has been shown that the neutral value for comparison in empirical research needs to be adjusted for the specific sub-sampling strategy used in the estimation of H. Therefore, this estimation is carried out twice – once with the data and again with a Monte Carlo simulation of the data that generates a corresponding IID/stationary series.
14. The two values of H are then compared. If the difference between the two values of H is not statistically significant, we conclude that there is no evidence of Hurst behavior in the data and OLS regression results may therefore be interpreted in terms of the phenomena under study. However if the value of H in the data is greater than the value of H in the IID simulation, then we can conclude that the data contain Fractal/Chaotic behavior by virtue of Hurst persistence and that therefore OLS results may not be interpreted strictly in terms of the phenomena under study. Nonlinear dynamics must be considered. A list of these studies appears below with links to the freely downloadable full text.

1. Ozone Depletion: Mean global total ozone is estimated as the latitudinally weighted average of total ozone measured by the TOMS and OMI satellite mounted ozone measurement devices for the periods 1979-1992 and 2005-2015 respectively. The TOMS dataset shows an OLS depletion rate of 0.65 DU per year on average in mean monthly ozone from January 1979 to December 1992. The OMI dataset shows an OLS accretion rate of 0.5 DU per year on average in mean monthly ozone from January 2005 to December 2015. The conflicting and inconsequential OLS trends may be explained in terms of the random variability of nature and violations of OLS assumptions that can create the so called Hurst phenomenon. These findings are inconsistent with the Rowland-Molina theory of ozone destruction by anthropogenic chemical agents because the theory implies continued and dangerous depletion of total ozone on a global scale until the year 2040. [FULL TEXT]
2. Global Warming: A study of daily mean temperature data from five USCRN stations in the sample period 1/1/2005-3/31/2016 shows that the seasonal cycle can be captured with significantly greater precision by dividing the year into smaller parts than calendar months. The enhanced precision greatly reduces vestigial patterns in the deseasonalized and detrended residuals. Rescaled Range analysis of the residuals indicates a violation of the independence assumption of OLS regression. The existence of dependence, memory, and persistence in the data is indicated by high values of the Hurst exponent. The results imply that decadal and even multi-decadal OLS trends in USCRN daily mean temperature may be spurious.  [FULL TEXT]
3. Precipitation: Rescaled range analysis of precipitation in the sample period 1893-2014 for ten USHCN stations in five states of the USA does not provide evidence of dependence, long term memory, or persistence in the time series. All of the observed Hurst exponents of precipitation are indicative of Gaussian randomness. Therefore, multi-decadal and non-periodic drought and flood events observed at some of these stations are more likely to be irregular cyclical phenomena of nature than the random effects of persistence and long term memory in the data. [FULL TEXT]
4. Global Warming: Trends in time series data estimated with OLS linear regression may be tested with a robust procedure that is less sensitive to influential observations and violations of regression assumptions. The test consists of comparing the average age of data tritiles. If the higher tritiles are newer a rising trend is indicated for the sample period. If the higher tritiles are older a declining trend is indicated. If neither of these conditions is met, no sustained trend in the sample period may be inferred from the data. Daily temperature data from selected USHCN and USCRN stations are used to demonstrate the utility of the proposed methodology. [FULL TEXT]
5. Solar Activity: It is shown that the time series of sunspot counts may be represented as the sum of a regular cyclical process and a random Hurst process. In the 2375-month study period 1/1818-11/2015, the optimal cyclical components of mean monthly sunspot counts consist of a short wave function with a period of 131 months and a long wave function in which the amplitude of the short wave undergoes a 100-year cycle. The residuals of this model, though random, exhibit properties of the Hurst phenomenon in which dependence, memory, and persistence generate apparent patterns out of randomness. The findings imply that not all patterns in the empirical record of sunspot counts contain useful information because some patterns represent random behavior. [FULL TEXT]
6. Global Warming: The deseasonalized monthly mean surface temperature time series for Nuuk, Greenland in the 148-year sample period 1866-2013 shows a statistically significant OLS warming trend of 0.1C per decade. Rescaled range analysis of the deseasonalized and detrended residuals reveals a high value of the Hurst exponent indicative of memory, dependence, and persistence in the time series. A robust test for trends described in a previous paper (Munshi, 2015) indicates that the observed OLS trend in the Nuuk data is spurious and therefore possibly an artifact of dependence and persistence. [FULL TEXT]
7. Global Warming: High values of the Hurst exponent of H=0.66±0.05 for deseasonalized monthly mean surface temperatures in the sample period 1850-2015 suggest persistence and long term memory in the temperature time series. Such Hurst phenomena have been observed in the stochastic processes of nature in the area of hydrology (Hurst) and also in the proxy record of annual mean surface temperature at a millennial time scale (Barnett). Our study suggests that these patterns may also exist in deseasonalized monthly means of the measured temperature record in the post industrial era, a period that is normally associated with global warming and climate change. [FULL TEXT]
8. Global Warming: A month by month trend analysis at an annual time scale of the daily mean Central England Temperature (CET) series 1772-2016 shows a general warming trend for most autumn and winter months. These trends are usually described in terms of anthropogenic global warming (AGW). OLS diagnostics reveal anomalies in the data having to do with asymmetry, non-linearity, and serial Hurst dependence in the series of generational trends in a 30-year moving window. Therefore, the phenomena of nature that generated this temperature series are best understood in terms of nonlinear patterns within the sample period rather than a single linear OLS trend-line across the whole of the sample period. [FULL TEXT]
9. Precipitation: Month by month analysis of precipitation in England and Wales 1766-2016 is carried out at an annual time scale for all twelve calendar months. No evidence of dependence, long term memory, or persistence is found. All twelve of the observed Hurst exponents of precipitation are found to be H≈0.5 indicative of Gaussian randomness. Therefore, non-periodic clusters of flood years observed in this area are more likely to be irregular cyclical phenomena of nature than effects of the Hurst phenomenon.  [FULL TEXT]

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CHAOS THEORY IN CLIMATE SCIENCE: A BIBLIOGRAPHY

1. Zeng, Xubin, Roger A. Pielke, and R. Eykholt. “Chaos theory and its applications to the atmosphere.” Bulletin of the American Meteorological Society 74.4 (1993): 631-644.  A brief overview of chaos theory is presented, including bifurcations, routes to turbulence, and methods for characterizing chaos. The paper divides chaos applications in atmospheric sciences into three categories: new ideas and insights inspired by chaos, analysis of observational data, and analysis of output from numerical models. Based on the review of chaos theory and the classification of chaos applications, suggestions for future work are given.
2. Marotzke, Jochem. “Abrupt climate change and thermohaline circulation: Mechanisms and predictability.” Proceedings of the National Academy of Sciences 97.4 (2000): 1347-1350.  The ocean’s thermohaline circulation has long been recognized as potentially unstable and has consequently been invoked as a potential cause of abrupt climate change on all timescales of decades and longer. However, fundamental aspects of thermohaline circulation changes remain poorly understood. [LINK TO FULL TEXT PDF]
3. Rial, Jose A., and C. A. Anaclerio. “Understanding nonlinear responses of the climate system to orbital forcing.” Quaternary Science Reviews 19.17-18 (2000): 1709-1722.  Frequency modulation (FM) of the orbital eccentricity forcing may be one important source of the nonlinearities observed in δ¹⁸O time series from deep-sea sediment cores (J.H. Rial (1999a) Pacemaking the lce Ages by frequency modulation of Earth’s orbital eccentricity. Science 285, 564–568). Here we present further evidence of frequency modulation found in data from the Vostok ice core. Analyses of the 430,000-year long, orbitally untuned, time series of CO₂, deuterium, aerosol and methane, suggest frequency modulation of the 41 kyr (0.0244 kyr⁻¹) obliquity forcing by the 413 kyr-eccentricity signal and its harmonics. Conventional and higher-order spectral analyses show that two distinct spectral peaks at ∼29 kyr (0.034 kyr⁻¹) and ∼69 kyr (0.014 kyr⁻¹) and other, smaller peaks surrounding the 41 kyr obliquity peak are harmonically (nonlinearly) related and likely to be FM-generated sidebands of the obliquity signal. All peaks can be closely matched by the spectrum of an appropriately built theoretical FM signal. A preliminary model, based on the classic logistic growth delay differential equation, reproduces the longer period FM effect and the familiar multiply peaked spectra of the eccentricity band. Since the FM effect appears to be a common feature in climate response, finding out its cause may help understand climate dynamics and global climate change.
4. Ashkenazy, Yosef, et al. “Nonlinearity and multifractality of climate change in the past 420,000 years.” Geophysical research letters 30.22 (2003).  Evidence of past climate variations are stored in polar ice caps and indicate glacial‐interglacial cycles of ∼100 kyr. Using advanced scaling techniques we study the long‐range correlation properties of temperature proxy records of four ice cores from Antarctica and Greenland. These series are long‐range correlated in the time scales of 1–100 kyr. We show that these time series are nonlinear for time scales of 1–100 kyr as expressed by temporal long‐range correlations of magnitudes of temperature increments and by a broad multifractal spectrum. Our results suggest that temperature increments appear in clusters of big and small increments—a big (positive or negative) climate change is most likely followed by a big (positive or negative) climate change and a small climate change is most likely followed by a small climate change.
5. Rial, Jose A. “Abrupt climate change: chaos and order at orbital and millennial scales.” Global and Planetary Change 41.2 (2004): 95-109.  Successful prediction of future global climate is critically dependent on understanding its complex history, some of which is displayed in paleoclimate time series extracted from deep-sea sediment and ice cores. These recordings exhibit frequent episodes of abrupt climate change believed to be the result of nonlinear response of the climate system to internal or external forcing, yet, neither the physical mechanisms nor the nature of the nonlinearities involved are well understood. At the orbital (10⁴–10⁵ years) and millennial scales, abrupt climate change appears as sudden, rapid warming events, each followed by periods of slow cooling. The sequence often forms a distinctive saw-tooth shaped time series, epitomized by the deep-sea records of the last million years and the Dansgaard–Oeschger (D/O) oscillations of the last glacial. Here I introduce a simplified mathematical model consisting of a novel arrangement of coupled nonlinear differential equations that appears to capture some important physics of climate change at Milankovitch and millennial scales, closely reproducing the saw-tooth shape of the deep-sea sediment and ice core time series, the relatively abrupt mid-Pleistocene climate switch, and the intriguing D/O oscillations. Named LODE for its use of the logistic-delayed differential equation, the model combines simplicity in the formulation (two equations, small number of adjustable parameters) and sufficient complexity in the dynamics (infinite-dimensional nonlinear delay differential equation) to accurately simulate details of climate change other simplified models cannot. Close agreement with available data suggests that the D/O oscillations are frequency modulated by the third harmonic of the precession forcing, and by the precession itself, but the entrained response is intermittent, mixed with intervals of noise, which corresponds well with the idea that the climate operates at the edge between chaos and order. LODE also predicts a persistent ∼1.5 ky oscillation that results from the frequency modulated regional climate oscillation.
6. Huybers, Peter, and Carl Wunsch. “Obliquity pacing of the late Pleistocene glacial terminations.” Nature 434.7032 (2005): 491.  The 100,000-year timescale in the glacial/interglacial cycles of the late Pleistocene epoch (the past ∼700,000 years) is commonly attributed to control by variations in the Earth’s orbit¹. This hypothesis has inspired models that depend on the Earth’s obliquity (∼ 40,000 yr; ∼40 kyr), orbital eccentricity (∼ 100 kyr) and precessional (∼ 20 kyr) fluctuations^2,3,4,5, with the emphasis usually on eccentricity and precessional forcing. According to a contrasting hypothesis, the glacial cycles arise primarily because of random internal climate variability^6,7,8. Taking these two perspectives together, there are currently more than thirty different models of the seven late-Pleistocene glacial cycles⁹. Here we present a statistical test of the orbital forcing hypothesis, focusing on the rapid deglaciation events known as terminations^10,11. According to our analysis, the null hypothesis that glacial terminations are independent of obliquity can be rejected at the 5% significance level, whereas the corresponding null hypotheses for eccentricity and precession cannot be rejected. The simplest inference consistent with the test results is that the ice sheets terminated every second or third obliquity cycle at times of high obliquity, similar to the original proposal by Milankovitch¹². We also present simple stochastic and deterministic models that describe the timing of the late-Pleistocene glacial terminations purely in terms of obliquity forcing.
7. Tziperman, Eli, Carl Wunsch. “Consequences of pacing the Pleistocene 100 kyr ice ages by nonlinear phase locking to Milankovitch forcing.” Paleoceanography 21.4 (2006).:    The consequences of the hypothesis that Milankovitch forcing affects the phase (e.g., termination times) of the 100 kyr glacial cycles via a mechanism known as “nonlinear phase locking” are examined. Phase locking provides a mechanism by which Milankovitch forcing can act as the “pacemaker” of the glacial cycles. Nonlinear phase locking can determine the timing of the major deglaciations, nearly independently of the specific mechanism or model that is responsible for these cycles as long as this mechanism is suitably nonlinear. A consequence of this is that the fit of a certain model output to the observed ice volume record cannot be used as an indication that the glacial mechanism in this model is necessarily correct. Phase locking to obliquity and possibly precession variations is distinct from mechanisms relying on a linear or nonlinear amplification of the eccentricity forcing. Nonlinear phase locking may determine the phase of the glacial cycles even in the presence of noise in the climate system and can be effective at setting glacial termination times even when the precession and obliquity bands account only for a small portion of the total power of an ice volume record. Nonlinear phase locking can also result in the observed “quantization” of the glacial period into multiples of the obliquity or precession periods.
8. Eisenman, Ian, Norbert Untersteiner, and J. S. Wettlaufer. “On the reliability of simulated Arctic sea ice in global climate models.” Geophysical Research Letters 34.10 (2007).  While most of the global climate models (GCMs) currently being evaluated for the IPCC Fourth Assessment Report simulate present‐day Arctic sea ice in reasonably good agreement with observations, the intermodel differences in simulated Arctic cloud cover are large and produce significant differences in downwelling longwave radiation. Using the standard thermodynamic models of sea ice, we find that the GCM‐generated spread in longwave radiation produces equilibrium ice thicknesses that range from 1 to more than 10 meters. However, equilibrium ice thickness is an extremely sensitive function of the ice albedo, allowing errors in simulated cloud cover to be compensated by tuning of the ice albedo. This analysis suggests that the results of current GCMs cannot be relied upon at face value for credible predictions of future Arctic sea ice.
9. Frank, Patrick, and John McCarthy. “A climate of belief.” Skeptic 14.1 (2008): 22-30. The claim that anthropogenic CO2 is responsible for the current warming of Earth climate is scientifically insupportable because climate models are unreliable by Patrick Frank “He who refuses to do arithmetic is doomed to talk nonsense.” — John McCarthy “The latest scientific data confirm that the earth’s climate is rapidly changing. … The cause? A thickening layer of carbon dioxide pollution, mostly from power plants and automobiles, that traps heat in the atmosphere. … *A+verage U.S. temperatures could rise another 3 to 9 degrees by the end of the century … Sea levels will rise, *and h+eat waves will be more frequent and more intense. Droughts and wildfires will occur more often. Disease-carrying mosquitoes will expand their range. And species will be pushed to extinction.” So says the National Resources Defense Council,2 with agreement by the Sierra Club,3 Greenpeace,4 National Geographic,5 the US National Academy of Sciences,6 and the US Congressional House leadership.7 Concurrent views are widespread,8 as a visit to the internet or any good bookstore will verify. Since at least the 1995 Second Assessment Report, the UN Intergovernmental Panel on Climate Change (IPCC) has been making increasingly assured statements that human-produced carbon dioxide (CO2) is influencing the climate, and is the chief cause of the global warming trend in evidence since about 1900. The current level of atmospheric CO2 is about 390 parts per million by volume (ppmv), or 0.039% by volume of the atmosphere, and in 1900 was about 295 ppmv. If the 20th century trend continues unabated, by about 2050 atmospheric CO2 will have doubled to about 600 ppmv. This is the basis for the usual “doubled CO2” scenario. Doubled CO2 is a bench-mark for climate scientists in evaluating greenhouse warming. Earth receives about 342 watts per square meter (W/m2 ) of incoming solar energy, and all of this energy eventually finds its way back out into space. However, CO2 and other greenhouse gasses, most notably water vapor, absorb some of the outgoing energy and warm the atmosphere. This is the greenhouse effect. Without it Earth’s average surface temperature would be a frigid -19°C (-2.2 F). With it, the surface warms to about +14°C (57 F) overall, making Earth habitable.9 With more CO2, more outgoing radiant energy is absorbed, changing the thermal dynamics of
   the atmosphere. All the extra greenhouse gasses that have entered the atmosphere since 1900, including CO2, equate to an extra 2.7 W/m2 of energy absorption by the atmosphere.10 This is the worrisome greenhouse effect. On February 2, 2007, the IPCC released the Working Group I (WGI) “Summary for Policymakers” (SPM) report on Earth climate,11 which is an executive summary of the science supporting the predictions quoted above. The full “Fourth Assessment Report” (4AR) came out in sections during 2007.  [LINK TO FULL TEXT PDF]
10. Huybers, Peter John. “Pleistocene glacial variability as a chaotic response to obliquity forcing.” (2009).  The mid-Pleistocene Transition from 40 ky to ~100 ky glacial cycles is generally characterized as a singular transition attributable to scouring of continental regolith or a long-term decrease in atmospheric CO2 concentrations. Here an alternative hypothesis is suggested, that Pleistocene glacial variability is chaotic and that transitions from 40 ky to ~100 ky modes of variability occur spontaneously. This alternate view is consistent with the presence of ~80 ky glacial cycles during the early Pleistocene and the lack of evidence for a change in climate forcing during the mid-Pleistocene. A simple model illustrates this chaotic scenario. When forced at a 40 ky period the model chaotically transitions between small 40 ky glacial cycles and larger 80 and 120 ky cycles which, on average, give the ~100 ky variability.
11. Dima, Mihai, and Gerrit Lohmann. “Conceptual model for millennial climate variability: a possible combined solar-thermohaline circulation origin for the~ 1,500-year cycle.” Climate Dynamics 32.2-3 (2009): 301-311.  Dansgaard-Oeschger and Heinrich events are the most pronounced climatic changes over the last 120,000 years. Although many of their properties were derived from climate reconstructions, the associated physical mechanisms are not yet fully understood. These events are paced by a ~1,500-year periodicity whose origin remains unclear. In a conceptual model approach, we show that this millennial variability can originate from rectification of an external (solar) forcing, and suggest that the thermohaline circulation, through a threshold response, could be the rectifier. We argue that internal threshold response of the thermohaline circulation (THC) to solar forcing is more likely to produce the observed DO cycles than amplification of weak direct ~1,500-year forcing of unknown origin, by THC. One consequence of our concept is that the millennial variability is viewed as a derived mode without physical processes on its characteristic time scale. Rather, the mode results from the linear representation in the Fourier space of nonlinearly transformed fundamental modes.
12. Dijkstra, Henk A. Nonlinear climate dynamics. Cambridge University Press, 2013.