A Physical–Mathematical Approach to Climate Change Effects Through Stochastic Resonance
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climate Article A Physical–Mathematical Approach to Climate Change Effects through Stochastic Resonance Maria Teresa Caccamo 1 and Salvatore Magazù 2,3,* 1 Consiglio Nazionale delle Ricerche (CNR)—Istituto per i Processi Chimico-Fisici (IPCF), Viale Ferdinando Stagno D’Alcontres n◦37, S. Agata, 98166 Messina, Italy; [email protected] 2 Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina, Viale Ferdinando Stagno D’Alcontres n◦31, S. Agata, 98166 Messina, Italy 3 Istituto Nazionale di Alta Matematica “F. Severi”—INDAM—Gruppo Nazionale per la Fisica Matematica—GNFM, P.le Aldo Moro 5, 00185 Roma, Italy * Correspondence: [email protected] Received: 22 November 2018; Accepted: 24 January 2019; Published: 27 January 2019 Abstract: The aim of this work is to study the effects induced by climate changes in the framework of the stochastic resonance approach. First, a wavelet cross-correlation analysis on Earth temperature data concerning the last 5,500,000 years is performed; this analysis confirms a correlation between the planet’s temperature and the 100,000, 41,000, and 23,000-year periods of the Milankovitch orbital cycles. Then, the stochastic resonance model is invoked. Specific attention is given to the study of the impact of the registered global temperature increase within the stochastic model. Further, a numerical simulation has been performed, based on: (1) A double-well potential, (2) an external periodic modulation, corresponding to the orbit eccentricity cycle, and (3) an increased value of the global Earth temperature. The effect of temperature increase represents one of the novelties introduced in the present study and is determined by downshifting the interaction potential used within the stochastic resonance model. The numeric simulation results show that, for simulated increasing values of the global temperature, the double-well system triggers changes, while at higher temperatures (as in the case of the absence of a global temperature increase although with a different threshold) the system goes into a chaotic regime. The wavelet analysis allows characterization of the stochastic resonance condition through the evaluation of the signal-to-noise ratio. On the basis of the obtained findings, we hypothesize that the global temperature increase can suppress, on a large time scale corresponding to glacial cycles, the external periodic modulation effects and, hence, the glacial cycles. Keywords: stochastic resonance model; climate change effects; temperature increasing; simulation 1. Introduction Climate occurs as a result of processes operating on multiple scales, some of which are slow, as in the Earth’s glacial cycles, and others of which are fast, such as daily weather fluctuations [1]. Glacial cycles, on the order of tens of thousands of years, start with a gradual temperature decrease, which gives rise to an increase of sea ice, polar cap volume, and total global area occupied by ice, increasing Earth’s overall albedo and the amount of the sun’s energy reflected away from the Earth. With less energy entering the Earth’s system, temperatures decrease further, creating positive feedback. Furthermore, as more ocean water is converted to ice, the overall water volume of the oceans decreases, allowing continental submerged portions to emerge [2–5]. The equilibrium global average temperature reached by this process represents a lower stationary regime of the overall climate during that period. Glacial ages end with a temperature increase reversing the water transfer from the cryosphere back to Climate 2019, 7, 21; doi:10.3390/cli7020021 www.mdpi.com/journal/climate Climate 2019, 7, x FOR PEER REVIEW 2 of 16 stationary regime. On the other hand, climate variations can give rise to harsh conditions that are of interest for the study of extremophile organisms [10–13] for biotechnological applications [14–22]. ClimateTemperature2019, 7, 21 determinations performed at the Vostok scientific station (Figure 1) revealed2 of 16a Climatestrong 2019, correlation7, x FOR PEER with REVIEW the Milankovitch cycles [23–32], and are shown in Figure 2 2 of 16 stationarythe oceans regime. [6–9 ].On Equilibrium the other temperaturehand, climate at thesevariatio timesns representscan give rise an upperto harsh stationary conditions regime. that On are of the other hand, climate variations can give rise to harsh conditions that are of interest for the study of interest for the study of extremophile organisms [10–13] for biotechnological applications [14–22]. extremophile organisms [10–13] for biotechnological applications [14–22]. Temperature determinations performed at the Vostok scientific station (Figure 1) revealed a Temperature determinations performed at the Vostok scientific station (Figure1) revealed a strong strongcorrelation correlation with with the Milankovitch the Milankovitch cycles cycles [23–32], [23–32], and are and shown are in shown Figure 2in. Figure 2 Figure 1. Temperature as a function of time during the last 5,500,000 years [31]. In particular, the cycles correspond to: (1) an eccentricity variation of the Earth’s orbit around the Sun over a period of about 100,000 years; (2) a variation in the inclination of the Earth’s axis from a minimum of 21°55′ to a maximum of 24°20′, over a period of approximately 41,000 years; and (3) a variation of the Earth’s axis during a double-conic motion, called solar-precession, over a period of about 23,000 years [33–41]. FigureFigure 1. Temperature 1. Temperature as as a afunction function of time duringduring the the last last 5,500,000 5,500,000 years years [31]. [31]. In particular, the cycles correspond to: (1) an eccentricity variation of the Earth’s orbit around the Sun over a period of about 100,000 years; (2) a variation in the inclination of the Earth’s axis from a minimum of 21°55′ to a maximum of 24°20′, over a period of approximately 41,000 years; and (3) a variation of the Earth’s axis during a double-conic motion, called solar-precession, over a period of about 23,000 years [33–41]. FigureFigure 2. 2. ThreeThree Milankovitch Milankovitch cycles. cycles. SeveralIn particular, mathematical the cycles procedures correspond are to: reported (1) an eccentricity in the literature variation to ofcorrelate the Earth’s in order orbit to around compare the differentSun over data a period sets through of about a 100,000 spectral years; comparison. (2) a variation Wavelet in transform the inclination (WT) ofanalysis the Earth’s is powerful axis from tool a forminimum analysing of 21variations◦550 to a maximumof spectral ofpower 24◦20 within0, over given a period data of series. approximately By decomposing 41,000 years; data andsets (3)into a waveletvariation components, of the Earth’s it axisis possible during to a double-coniccompare the dominant motion, called signal solar-precession, spectral modes overas well a period as their of correlationabout 23,000 degree. years [ 33In– 41particular,]. in order to highlight the correlation between the registered temperaturesSeveral mathematical of Figure 1 and procedures the mentioned are reported Milankovitch in the literature periodicities, to correlate in this in work order the to comparewavelet cross-correlationdifferent data sets (XWT) through approach a spectralFigure has comparison. been 2. Three applied Milankovitch Wavelet [42–44]. transform Furthermore, cycles. (WT) analysis these glacial is powerful cycles toolcan forbe treatedanalysing as variations a slow process of spectral or power low-frequency within given clim dataate series. “signal”. By decomposing At this geological data sets scale, into wavelet daily temperaturecomponents,Several mathematical fluctuations it is possible procedures can to compare be thought are the dominantreportedof as a fa signalinst process,the spectralliterature or modesa much to correlate as higher-frequency well as in their order correlation to signal compare differentsuperimposeddegree. data In sets particular, on through the glacial in a order spectral signal. to highlight comparison.Because the the correlation variation Wavelet in betweentransform solar radiation the (WT) registered over analysis the temperatures Milankovitch is powerful of tool for analysingcyclesFigure alone1 and variations is the not mentioned able of to spectral justify Milankovitch temperature power within periodicities, variation given from indata this interglacial series. work theBy age waveletdecomposing to glacial cross-correlation age, data in 1980, sets into Roberto(XWT) approach Benzi introduced has been appliedthe idea [ 42that–44 the]. Furthermore, high-frequency these temperature glacial cycles fluctuations can be treated were as a aweak slow wavelet components, it is possible to compare the dominant signal spectral modes as well as their periodicprocess orinput low-frequency and a source climate of “noise” “signal”. on the Atglacial this geologicalsignal, creating scale, a dailysmall temperatureresonance that fluctuations amplified correlation degree. In particular, in order to highlight the correlation between the registered thecan signal be thought of the oflower as a glacial fast process, frequency. or a muchHe termed higher-frequency this hypothesis signal “stochas superimposedtic resonance” on the (SR) glacial [45– temperatures52].signal. Figure Because of3 shows Figure the variationa typical1 and curv inthe solar ementioned of radiation output performance overMilankovitch the Milankovitch as a periodicities,function cycles of input