Space Climate 7 Meeting Abstract
Long-term Variation of the Coupling between Solar Proxies: Coupled Oscillators Approach
Anton S Savostianov (NRU Higher School of Economics, Moscow)
Alexander B. Shapoval, NRU Higher School of Economics, Moscow Mikhail M. Shirman, Institute of Earthquake Prediction Theory and Mathematical Geophysics, Moscow
Recent advances in the applications of the Kuramoto model to a wide range of real-life processes require the reconstruction of processes' parameters from observations. Our study explores the inverse problem for the Kuramoto model of two nonlinear oscillators with slowly varying coupling in the form of a single-step function, sine-wave, and auto-regressive process with a view to deriving the basic properties of the reconstruction procedure, that is the connection of the reconstruction efficiency with the coupling strength and estimates of the time it takes for a system to phase-lock. By investigating the de-synchronization of the solar foculae series, which represent signals coming from the northern and southern solar hemispheres, we relate the de-synchronization of the series, which occurred in the early'1960s to the changes in the coupling of the underlying real oscillators.
Kuramoto-like models of coupled oscillators help to understand the unexpected synchronization observed in biological, chemical, physical, and social systems or to design such synchronization, as the solution of the model equations exhibits similarities with real-life processes utilizing the synchronization phenomena. Each oscillator considered by the model tends to go on at its own frequency, whereas the coupling creates a force that synchronizes them.
Blanter et al. [2016, 2017, 2014] applied Kuramoto models with two oscillators to focus on the empirical relationships between components of the solar dynamo. Such applications of the Kuramoto models to the solar dynamo are motivated by the fact that solar activity is one of the most prominent natural examples of quasi-periodic behaviour. The quasi-periodicity is primarily represented by approximately 11-year variation, frequently called ‘solar cycle’. Either the period or the amplitude of the solar cycle evolves within time. Therefore, predicting them still remains a big challenge for researchers [Charbonneau, 2014]. The solar activity variation is also manifested in the fluctuations of decoherence observed in different solar indices [Brajša et al., 2009; Pesnell, 2008; Petrovay, 2010]. Researchers attribute the origin of the fluctuations in the solar activity to the solar dynamo variability, generally observed via the fluctuations in the polar magnetic field generation, the hemispheric asymmetry of the solar magnetic fields, and the meridional circulation strength [Chatterjee et al., 2004; Choudhuri, 2015; Dikpati and Charbonneau, 1999; Hathaway, 2015; Wang, 2017]. Cameron et al. [2017] include the meridional circulation into a flux-transport dynamo model in order to reproduce the changes in the solar cycle length and the migration of the sunspots toward the equator. Donner and Thiel [2007] investigated the phase difference characterizing the asymmetry between northern and southern solar hemispheres and found potentially aperiodic multi-decadal scale variations in the lead/lag relationship between the two hemispheres. According to de Jager et al. [2016], the meridional circulation could affect irregular features of specific solar cycles, especially Solar Cycle 24.
In contrast to [Blanter et al., 2016, 2017, 2014], we establish the coupling between the northern and southern polar magnetic fields of the Sun and assess the long-term variation of this coupling. The polar faclulae observed near the Sun’s north and south poles give a proxy to the corresponding magnetic fields. Although polar faculae have their own intrinsic nature they are related to the general mechanism of the solar dynamo. The both solar faculae time series follow ~ 11 year quasi-cycle. Despite the remote locations of the northern and southern faculae and the absence of visible channels between these locations the phases of the cycle computed with both the time series exhibit similar dynamics [Norton et al., 2015]. The following complex linkage between the polar magnetic fields, which introduces a very rough description of the meridional circulation, can explain their pairing. Each polar field is connected to the corresponding equatorial field through the meridional flow, which transfers the magnetic field from the pole to the equator inside the Sun and from the equator to the pole at the surface, i. e., exhibits a circulation [Zhao et al., 2013]. The equatorial fields can adjoin to each other by such other mechanisms as magnetic diffusion and transequatorialmeridional flow cells. These processes are expected to result in magnetic flux cancellation as the oppositely directed fields come in close proximity and cancel each other out across the magnetic equator late in the solar cycle [Norton et al., 2015]. Accepting this complex and not well understood in detail description, we intend to construct a simple, almost toy-like, model that rigorously defines the coupling of the polar magnetic fields acting in the both hemispheres. The Kuramoto model with two oscillators, where each of them relates to a solar hemisphere, and symmetrical coupling between the oscillators is chosen because of its simplicity. Avoiding the specification of the coupling, we are still going to assess it.
We are the first who estimate the quality of the reconstruction of the time-dependent coupling in the Kuramoto model. The breaking of the quasi-autonomy condition as well as the synchronization inequality can cause a fall in this quality. Each breaking is characterized by the magnitude and duration. We process the four cases considering different families of the genuine couplings. We are able to distinguish errors in the reconstruction related to the procedure itself and to noise in the coupling.
We applied our methodology to the solar faculae time series reconstructing the coupling between them. This observation allows one to speculate about links between the polar faculae and solar dynamo itself. In this study, we successfully reconstructed the coupling between the polar faculae representing the northern and southern hemispheres everywhere except the 20th cycle. Interestingly, the near-equatorial magnetic fields represented by the sunspots also exhibit anomalies in the 20the cycle: the leadership of the hemispheres was changed. The coincidence of the anomalies confirms the relationship between the long-term variations of the polar faculae and solar dyanmo.
Our approach crucially simplifies the mechanism that relates different components of the solar magnetic field, when assuming that the relationship between the polar fields is given by a single quantity, which is the coupling in the Kuramoto model. As a result, the role of those mechanisms - the variations of the meridional circulation or magnetic diffusivity under magnetic quenching are the most natural examples - is not highlighted. Nevertheless, the simplification allows us to derive quantitative estimates of the total coupling between the polar fields, which in turn contributes to the development the solar dynamo theory.
Authors wish to acknowledge the support of the Russian Science Foundation (project No 17-11-01052).
Mode of presentation: oral (Need to be confirmed by the SOC)