PaperNo Access

Inverse Problem in the Kuramoto Model with a Phase Lag: Application to the Sun

Elena Blanter

http://orcid.org/0000-0002-5946-0357

IEPT RAS, Profsoyuznaya str. 84/32, 117997, Moscow, Russia

National Research University Higher School of Economics, Myasnitskaya Ulitsa, 20, 101000, Moscow, Russia

E-mail Address: [email protected]

Search for more papers by this author

and

Mikhail Shnirman

IEPT RAS, Profsoyuznaya str. 84/32, 117997, Moscow, Russia

National Research University Higher School of Economics, Myasnitskaya Ulitsa, 20, 101000, Moscow, Russia

Search for more papers by this author

https://doi.org/10.1142/S0218127420501655Cited by:2 (Source: Crossref)

Abstract

We solve the inverse problem in the Kuramoto model with three nonidentical oscillators and a phase lag. The model represents a three-cell-in-depth radial profile of the solar meridional circulation in each solar hemisphere and describes the synchronization between the two components (toroidal and poloidal) of the solar magnetic field. We reconstruct natural frequencies of the top and the bottom oscillators from the evolution of their phases when the oscillators are phase-locked. The phase-locking allows to solve the inverse problem when the phase of the middle oscillator is not available. We present the exact solution and investigate its stability. The model reveals a crucial role of the phase lag in the inverse problem solution. We apply the model to the reconstruction of the deep meridional circulation of the Sun and discuss peculiarities of its evolution in terms of solar dynamo.

Keywords:

References

Acebron, J. A., Bonilla, L. L., Vicente, C. J. P. & Ritort, F. [2005] “ The Kuramoto model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77, 137. Crossref, Web of Science, Google Scholar
Belucz, B., Dikpati, M. & Forgács-Dajka, E. [2015] “ A Babcock–Leighton solar dynamo model with multi-cellular meridional circulation in advection- and diffusion-dominated regimes,” Astrophys. J. 806, 169. Crossref, Web of Science, Google Scholar
Bick, C., Goodfellow, M., Laing, C. R. & Martens, E. A. [2019] “Understanding the dynamics of biological and neural oscillator networks through mean-field reductions: A review,” arXiv e-prints, arXiv:1902.05307. Google Scholar
Blanter, E., Le Mouël, J.-L., Shnirman, M. & Courtillot, C. [2014] “ Kuramoto model of nonlinear coupled oscillators as a way for understanding phase synchronization: Application to solar and geomagnetic indices,” Solar Phys. 289, 4309–4333. Crossref, Web of Science, Google Scholar
Blanter, E., Le Mouël, J.-L., Shnirman, M. & Courtillot, C. [2016] “ Kuramoto model with nonsymmetric coupling reconstructs variations of the solar-cycle period,” Solar Phys. 291, 1003–1023. Crossref, Web of Science, Google Scholar
Blanter, E., Le Mouël, J.-L., Shnirman, M. & Courtillot, C. [2017] “ Reconstruction of the North–South solar asymmetry with a Kuramoto model,” Solar Phys. 292, 54. Crossref, Web of Science, Google Scholar
Blanter, E., Le Mouël, J.-L., Shnirman, M. & Courtillot, C. [2018] “ Long term evolution of solar meridional circulation and phase synchronization viewed through a symmetrical Kuramoto model,” Solar Phys. 293, 134. Crossref, Web of Science, Google Scholar
Brun, A. S., Miesch, M. S. & Toomre, J. [2011] “ Modeling the dynamical coupling of solar convection with the radiative interior,” Astrophys. J. 742, 79. Crossref, Web of Science, Google Scholar
Charbonneau, P. [2014] “ Solar dynamo theory,” Ann. Rev. Astron. Astrophys. 52, 251–290. Crossref, Web of Science, Google Scholar
Chen, R. & Zhao, J. [2017] “ A comprehensive method to measure solar meridional circulation and the center-to-limb effect using time-distance helioseismology,” Astrophys. J. 849, 144. Crossref, Web of Science, Google Scholar
Choudhuri, A. R. [1995] “ The solar dynamo with meridional circulation,” Astron. Astrophys. 303, L29. Web of Science, Google Scholar
DeVille, L. & Ermentrout, B. [2016] “ Phase-locked patterns of the Kuramoto model on 3-regular graphs,” Chaos 26, 094820. Crossref, Web of Science, Google Scholar
Dörfler, F. & Bullo, F. [2011] “ On the critical coupling for Kuramoto oscillators,” SIAM J. Appl. Dyn. Syst. 10, 1070. Crossref, Web of Science, Google Scholar
Dörfler, F. & Bullo, F. [2014] “ Synchronization in complex networks of phase oscillators: A survey,” Automatica 50, 1539–1564. Crossref, Web of Science, Google Scholar
Earl, M. G. & Strogatz, S. H. [2003] “ Synchronization in oscillator networks with delayed coupling: A stability criterion,” Phys. Rev. E 67, 036204. Crossref, Web of Science, Google Scholar
Elaeva, M. S., Blanter, E. M. & Shapoval, A. B. [2020] “Inverse problem in the Kuramoto model with nonidentical coupling inspired by the solar dynamo,” Chaos (submitted). Google Scholar
Featherstone, N. A. & Miesch, M. S. [2015] “ Meridional circulation in solar and stellar convection zone,” Astrophys. J. 804, 67. Crossref, Web of Science, Google Scholar
Hathaway, D. H. & Upton, L. [2014] “ The solar meridional circulation and sunspot cycle variability,” J. Geophys. Res. 119, 3316–3324. Crossref, Web of Science, Google Scholar
Hathaway, D. H. [2015] “ The solar cycle,” Living Rev. Solar Phys. 12, 4. Crossref, Web of Science, Google Scholar
Hazra, G., Karak, B. B. & Choudhuri, A. R. [2014] “ Is a deep one-cell meridional circulation essential for the flux transport solar dynamo?” Astrophys. J. 782, 93. Crossref, Web of Science, Google Scholar
Hong, H. & Strogatz, S. H. [2011] “ Kuramoto model of coupled oscillators with positive and negative coupling parameters: An example of conformist and contrarian oscillators,” Phys. Rev. Lett. 106, 054102. Crossref, Web of Science, Google Scholar
Hong, H. & Strogatz, S. H. [2012] “ Mean-field behavior in coupled oscillators with attractive and repulsive interactions,” Phys. Rev. E 85, 056210. Crossref, Web of Science, Google Scholar
Hung, C. P., Jouve, L., Brun, A. S., Fournier, A. & Talagrand, O. [2015] “ Estimating the deep solar meridional circulation using magnetic observations and a dynamo model: A variational approach,” Astrophys. J. 814, 151. Crossref, Web of Science, Google Scholar
Hung, C. P., Brun, A. S., Fournier, A., Jouve, L., Talagrand, O. & Zakari, M. [2017] “ Variational estimation of the large-scale time-dependent meridional circulation in the sun: Proofs of concept with a solar mean field dynamo model,” Astrophys. J. 849, 160. Crossref, Web of Science, Google Scholar
Javaraiah, J. [2017] “ Will solar cycles 25 and 26 be weaker than cycle 24?” Solar Phys. 292, 172. Crossref, Web of Science, Google Scholar
Kitchatinov, L. L. [2016] “ Meridional circulation in the sun and stars,” Geomagnet. Aeron. 56, 945–951. Crossref, Web of Science, Google Scholar
Komm, R. W., Howard, R. F. & Harvey J. W. [1993] “ Meridional flow of small photospheric magnetic features,” Solar Phys. 147, 207–223. Crossref, Web of Science, Google Scholar
Komm, R., González Hernández, I., Howe, R. & Hill, F. [2015] “ Solar-cycle variation of subsurface meridional flow derived with ring-diagram analysis,” Solar Phys. 290, 3113–3136. Crossref, Web of Science, Google Scholar
Kuramoto, Y. [1975] “ Self-entrainment of a population of coupled nonlinear oscillators,” Int. Symp. Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, Vol. 39 (Springer), pp. 420–422. Crossref, Google Scholar
Kuznetsov, A. P. & Sedova, Y. V. [2014] “ Low-dimensional discrete Kuramoto model: Hierarchy of multifrequency quasiperiodicity regimes,” Int. J. Bifurcation and Chaos 24, 1430022-1–10. Link, Web of Science, Google Scholar
Lohe, M. A. [2017] “ The WS transform for the Kuramoto model with distributed amplitudes, phase lag and time delay,” J. Phys. A: Math. Gen. 50, 505101. Crossref, Google Scholar
Mandal, K., Hanasoge, S. M., Rajaguru, S. P. & Antia, H. M. [2018] “ Helioseismic inversion to infer the depth profile of solar meridional flow using spherical born kernels,” Astrophys. J. 863, 39. Crossref, Web of Science, Google Scholar
Muñoz-Jaramillo, A., Sheeley, N. R., Jr.,, Zhang, J. & DeLuca, E. E. [2012] “ Calibrating 100 years of polar faculae measurements: Implications for the evolution of the heliospheric magnetic field,” Astrophys. J. 753, 146. Crossref, Web of Science, Google Scholar
Ott, E. & Antonsen, T. M. [2008] “ Low dimensional behavior of large systems of globally coupled oscillators,” Chaos 18, 037113. Crossref, Web of Science, Google Scholar
Pikovsky, A. [2018] “ Reconstruction of a random phase dynamics network from observations,” Phys. Lett. A 382, 147. Crossref, Web of Science, Google Scholar
Rodrigues, F. A., Peron, T. K., Ji, P. & Kurths, J. [2016] “ The Kuramoto model in complex networks,” Phys. Rep. 610, 1. Crossref, Web of Science, Google Scholar
Sakaguchi, H. & Kuramoto, Y. [1986] “ A soluble active rotater model showing phase transitions via mutual entertainment,” Prog. Theor. Phys. 76, 576–581. Crossref, Web of Science, Google Scholar
Savostyanov, A., Shapoval, A. & Shnirman, M. [2020] “ The inverse problem for the Kuramoto model of two nonlinear coupled oscillators driven by applications to solar activity,” Physica D 401, 132160. Crossref, Web of Science, Google Scholar
Strogatz, S. H. [2000] “ From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators,” Physica D 143, 1–20. Crossref, Web of Science, Google Scholar
Yeung, M. K. S. & Strogatz, S. H. [1999] “ Time delay in the Kuramoto model of coupled oscillators,” Phys. Rev. Lett. 82, 648–651. Crossref, Web of Science, Google Scholar
Zhao, J., Bogart, R. S., Kosovichev, A. G., Duvall, T. L., Jr., & Hartlep, T. [2013] “ Detection of equatorward meridional flow and evidence of double-cell meridional circulation inside the sun,” Astrophys. J. Lett. 774, L29. Crossref, Web of Science, Google Scholar