Inverse problem in the Kuramoto model with a phase lag: application to the Sun. (English) Zbl 1450.85001
Summary: 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.
MSC:
| 85A15 | Galactic and stellar structure |
| 78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |
| 35R30 | Inverse problems for PDEs |
| 78A25 | Electromagnetic theory (general) |
Keywords:
solar magnetic field; Kuramoto model; nonidentical oscillators; delay; inverse problem; solar meridional circulationReferences:
| [1] | 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. |
| [2] | 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. |
| [3] | 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. · Zbl 1448.92011 |
| [4] | 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. |
| [5] | 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. |
| [6] | 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. |
| [7] | 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. |
| [8] | Brun, A. S., Miesch, M. S. & Toomre, J. [2011] “ Modeling the dynamical coupling of solar convection with the radiative interior,” Astrophys. J.742, 79. |
| [9] | Charbonneau, P. [2014] “ Solar dynamo theory,” Ann. Rev. Astron. Astrophys.52, 251-290. |
| [10] | 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. |
| [11] | Choudhuri, A. R. [1995] “ The solar dynamo with meridional circulation,” Astron. Astrophys.303, L29. |
| [12] | DeVille, L. & Ermentrout, B. [2016] “ Phase-locked patterns of the Kuramoto model on 3-regular graphs,” Chaos26, 094820. · Zbl 1382.34039 |
| [13] | Dörfler, F. & Bullo, F. [2011] “ On the critical coupling for Kuramoto oscillators,” SIAM J. Appl. Dyn. Syst.10, 1070. · Zbl 1242.34096 |
| [14] | Dörfler, F. & Bullo, F. [2014] “ Synchronization in complex networks of phase oscillators: A survey,” Automatica50, 1539-1564. · Zbl 1296.93005 |
| [15] | Earl, M. G. & Strogatz, S. H. [2003] “ Synchronization in oscillator networks with delayed coupling: A stability criterion,” Phys. Rev. E67, 036204. |
| [16] | 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). |
| [17] | Featherstone, N. A. & Miesch, M. S. [2015] “ Meridional circulation in solar and stellar convection zone,” Astrophys. J.804, 67. |
| [18] | Hathaway, D. H. & Upton, L. [2014] “ The solar meridional circulation and sunspot cycle variability,” J. Geophys. Res.119, 3316-3324. |
| [19] | Hathaway, D. H. [2015] “ The solar cycle,” Living Rev. Solar Phys.12, 4. |
| [20] | 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. |
| [21] | 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. |
| [22] | Hong, H. & Strogatz, S. H. [2012] “ Mean-field behavior in coupled oscillators with attractive and repulsive interactions,” Phys. Rev. E85, 056210. |
| [23] | 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. |
| [24] | 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. |
| [25] | Javaraiah, J. [2017] “ Will solar cycles 25 and 26 be weaker than cycle 24?” Solar Phys.292, 172. |
| [26] | Kitchatinov, L. L. [2016] “ Meridional circulation in the sun and stars,” Geomagnet. Aeron.56, 945-951. |
| [27] | Komm, R. W., Howard, R. F. & Harvey, J. W. [1993] “ Meridional flow of small photospheric magnetic features,” Solar Phys.147, 207-223. |
| [28] | 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. |
| [29] | Kuramoto, Y. [1975] “ Self-entrainment of a population of coupled nonlinear oscillators,” Int. Symp. Mathematical Problems in Theoretical Physics, , Vol. 39 (Springer), pp. 420-422. · Zbl 0335.34021 |
| [30] | Kuznetsov, A. P. & Sedova, Y. V. [2014] “ Low-dimensional discrete Kuramoto model: Hierarchy of multifrequency quasiperiodicity regimes,” Int. J. Bifurcation and Chaos24, 1430022-1-10. · Zbl 1300.37014 |
| [31] | 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. · Zbl 1385.34047 |
| [32] | 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. |
| [33] | 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. |
| [34] | Ott, E. & Antonsen, T. M. [2008] “ Low dimensional behavior of large systems of globally coupled oscillators,” Chaos18, 037113. · Zbl 1309.34058 |
| [35] | Pikovsky, A. [2018] “ Reconstruction of a random phase dynamics network from observations,” Phys. Lett. A382, 147. · Zbl 1381.05067 |
| [36] | Rodrigues, F. A., Peron, T. K., Ji, P. & Kurths, J. [2016] “ The Kuramoto model in complex networks,” Phys. Rep.610, 1. · Zbl 1357.34089 |
| [37] | Sakaguchi, H. & Kuramoto, Y. [1986] “ A soluble active rotater model showing phase transitions via mutual entertainment,” Prog. Theor. Phys.76, 576-581. |
| [38] | 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 D401, 132160. · Zbl 1453.34025 |
| [39] | Strogatz, S. H. [2000] “ From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators,” Physica D143, 1-20. · Zbl 0983.34022 |
| [40] | Yeung, M. K. S. & Strogatz, S. H. [1999] “ Time delay in the Kuramoto model of coupled oscillators,” Phys. Rev. Lett.82, 648-651. |
| [41] | 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. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
