<p>Markov equations are used to solve complex nonlinear differential equations that are not easily solvable. In the context of climate data, these equations can provide insights into patterns and trends in temperature changes over time. Temperature fluctuations are the backbone of climate change, and due to their nonlinear nature, applying novel approaches is vital. This paper investigates the nonlinear dynamics of Earth’s surface temperature data from NASA’s Goddard Institute for Space Studies (GISS). Through the application of Markov analysis to the GISS temperature data and a thorough investigation of key transition states, we can explore the potential for formulating a robust equation that effectively captures the dynamics of Earth’s temperature fluctuations. Here, we investigate the Markovian nature of the temperature data via Chapman-Kolmogorov (CK) verification. The Markov time interval is determined by the CK equation as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_15149_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">\((t_M = 2)\)</EquationSource> </InlineEquation>. This indicates that for time intervals larger than 2 months, temperature fluctuations can be considered a Markov process. By recognizing the Markov characteristic time, we derive the Fokker–Planck equation for the temperature fluctuations. The drift and diffusion coefficients in this equation demonstrate linear and quadratic trends with respect to the fluctuations, respectively. The drift coefficient is given by <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_15149_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="181" /> </InlineMediaObject> <EquationSource Format="TEX">\((D^{(1)} = -(0.47 \pm 0.02)T)\)</EquationSource> </InlineEquation>, which shows a degree of correlation among the temperature data. The quadratic nature of the diffusion coefficient, represented as <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_15149_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="174" /> </InlineMediaObject> <EquationSource Format="TEX">\((D^{(2)} = (0.21 \pm 0.03)T^2)\)</EquationSource> </InlineEquation>, implies that at higher temperatures, the influence of random fluctuations increases significantly, leading to multiaffinity and nonlinearity in the scaling exponents of the temperature data. Finally, the governed Langevin equation related to the dynamics of the GISS temperature fluctuations is derived, providing a complex theoretical framework with practical applications and significance in advancing climate research.</p>

Markov analysis and the Langevin equation of the Earth surface temperature data

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2025年8月11日

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Maedeh Lak,
Sakineh Hosseinabadi,
Amir Ali Masoudi

摘要

Markov equations are used to solve complex nonlinear differential equations that are not easily solvable. In the context of climate data, these equations can provide insights into patterns and trends in temperature changes over time. Temperature fluctuations are the backbone of climate change, and due to their nonlinear nature, applying novel approaches is vital. This paper investigates the nonlinear dynamics of Earth’s surface temperature data from NASA’s Goddard Institute for Space Studies (GISS). Through the application of Markov analysis to the GISS temperature data and a thorough investigation of key transition states, we can explore the potential for formulating a robust equation that effectively captures the dynamics of Earth’s temperature fluctuations. Here, we investigate the Markovian nature of the temperature data via Chapman-Kolmogorov (CK) verification. The Markov time interval is determined by the CK equation as \((t_M = 2)\) . This indicates that for time intervals larger than 2 months, temperature fluctuations can be considered a Markov process. By recognizing the Markov characteristic time, we derive the Fokker–Planck equation for the temperature fluctuations. The drift and diffusion coefficients in this equation demonstrate linear and quadratic trends with respect to the fluctuations, respectively. The drift coefficient is given by \((D^{(1)} = -(0.47 \pm 0.02)T)\) , which shows a degree of correlation among the temperature data. The quadratic nature of the diffusion coefficient, represented as \((D^{(2)} = (0.21 \pm 0.03)T^2)\) , implies that at higher temperatures, the influence of random fluctuations increases significantly, leading to multiaffinity and nonlinearity in the scaling exponents of the temperature data. Finally, the governed Langevin equation related to the dynamics of the GISS temperature fluctuations is derived, providing a complex theoretical framework with practical applications and significance in advancing climate research.