Long-term Varia�on of the Coupling between Solar Proxies: Coupled Oscillators Approach Anton Savos�anov, Alexander Shapoval, Mikhail Shnirman NRU Higher School of Economics, Laboratory of Complex Systems Modelling and Control Ins�tute of Earthquake Predic�on Theory and Mathema�cal Geophysics Moscow, Russia July 9, 2019
Introduction: cyclic scenarios and the meridional fmow Reconstructions in Kuramoto model Advances: time-dependent coupling and Kuramoto’s applicability Advances: model misinterpretation — van der Poll-Duffjng case 2 Outline
Hazra et al, 2014 Idea Independently of physical background one should try to reproduce phenomena of the solar dynamo with models of interacting oscillators. 3 Intro: Oscilla�ng Trend in Solar Ac�vity
Boning et al, 2017: GONG reconstructions 2004-2012 meridional fmow per hemisphere are assumed; debate; an oscillator; one should consider their interactions (couplings). 4 Intro: Meridional Flow and Cells Structure • Three cells of the • Their placement is up to π d=3π π • Each cell is associated with
5 natural frequencies Oscillators are synchronized: The coupling is symmetrical, through the middle layer Relationship between the inner and outer layers is organized There-layer structure of the meridional fmow Assumptions coupling Kuramoto Model of the Coupled Oscillators Oscillators’ Modelling: Kuramoto � � Oscillators: X i ( t ) = A i ( t ) sin Ω i t + ϕ i ; θ i = Ω i t + ϕ i are phases � ˙ sin ( θ j − θ i ) , θ i = + Ω i are frequencies ω i κ ij ���� ���� j
5 natural frequencies Oscillators are synchronized: The coupling is symmetrical, through the middle layer Relationship between the inner and outer layers is organized Assumptions coupling Kuramoto Model of the Coupled Oscillators Oscillators’ Modelling: Kuramoto � � Oscillators: X i ( t ) = A i ( t ) sin Ω i t + ϕ i ; θ i = Ω i t + ϕ i are phases � ˙ sin ( θ j − θ i ) , θ i = + Ω i are frequencies ω i κ ij ���� ���� j • There-layer structure of the meridional fmow
5 natural frequencies Oscillators are synchronized: The coupling is symmetrical, through the middle layer Assumptions coupling Kuramoto Model of the Coupled Oscillators Oscillators’ Modelling: Kuramoto � � Oscillators: X i ( t ) = A i ( t ) sin Ω i t + ϕ i ; θ i = Ω i t + ϕ i are phases � ˙ sin ( θ j − θ i ) , θ i = + Ω i are frequencies ω i κ ij ���� ���� j • There-layer structure of the meridional fmow • Relationship between the inner and outer layers is organized
5 natural frequencies Oscillators are synchronized: through the middle layer Assumptions coupling Kuramoto Model of the Coupled Oscillators Oscillators’ Modelling: Kuramoto � � Oscillators: X i ( t ) = A i ( t ) sin Ω i t + ϕ i ; θ i = Ω i t + ϕ i are phases � ˙ sin ( θ j − θ i ) , θ i = + Ω i are frequencies ω i κ ij ���� ���� j • There-layer structure of the meridional fmow • Relationship between the inner and outer layers is organized • The coupling is symmetrical, κ ij = κ ji
5 natural frequencies through the middle layer Assumptions coupling Kuramoto Model of the Coupled Oscillators Oscillators’ Modelling: Kuramoto � � Oscillators: X i ( t ) = A i ( t ) sin Ω i t + ϕ i ; θ i = Ω i t + ϕ i are phases � ˙ sin ( θ j − θ i ) , θ i = + Ω i are frequencies ω i κ ij ���� ���� j • There-layer structure of the meridional fmow • Relationship between the inner and outer layers is organized • The coupling is symmetrical, κ ij = κ ji • Oscillators are synchronized: ˙ θ i − ˙ θ j = 0
6 is computed over the period, say, and represented, f. i., by solar proxies ISSN and aa Oscillators’ Modelling: Reconstruc�on in Kuramoto • Two oscillators X 1 ( t ) and X 2 ( t ) given by the toroidal and poloidal magnetic fjelds θ ( t ) = 2∆ ω − κ ( t ) sin θ ( t ) , where θ = θ 1 − θ 2 . ˙ • Single equation: � � Corr ( X 1 ( t ) , Y 1 ( t )) • Correlation C 1 ( t ) = T = 10 . 75 years with the sliding windows • Estimate ϕ of the phase θ : ϕ ( t ) = arccos C 1 ( t ) , Blanter et al (2014) θ = 0 ⇒ 2∆ ω − κ ( t ) sin ϕ ( t ) ≈ 0 ˙ • Synchronization • The estimate of the coupling 2∆ ω κ ( t ) = sin ϕ ( t )
Kuramoto Model with Three Oscillators Blanter et al. 2018 top (in red) and bottom (in blue) oscillators in the northern and southern hemispheres Faster velocity of the surface cell and lower velocity of the deep cell The opposite regime: the 1920s, the late 1960s, and the 2000s– 7 A Reconstruc�on of Natural Frequencies frequencies ω of
Kuramoto Model with Three Oscillators Blanter et al. 2018 top (in red) and bottom (in blue) oscillators in the northern and southern hemispheres The opposite regime: the 1920s, the late 1960s, and the 2000s– 7 A Reconstruc�on of Natural Frequencies frequencies ω of • Faster velocity of the surface cell and lower velocity of the deep cell
Kuramoto Model with Three Oscillators Blanter et al. 2018 top (in red) and bottom (in blue) oscillators in the northern and southern hemispheres 7 A Reconstruc�on of Natural Frequencies frequencies ω of • Faster velocity of the surface cell and lower velocity of the deep cell • The opposite regime: the 1920s, the late 1960s, and the 2000s–
Time-dependency allowed. Derived regularities relaxation time of reconstructed coupling into complete reconstruction failure (fjg.) probability depends on autocorrelation and deterministic coupling) 8 Kuramoto Reconstruc�on: �me-dependent coupling • We move from constant couplings κ to time-dependent κ ( t ) ; • Earlier there were no synchronisation if ∆ ω > κ ; now temporarily breaks are • rapid changes in initial coupling results in long • long breaks of the synchronisation inequality results • Adding AR(1) noise could lead to this failure (the
Courtesy of Andrés Muñoz–Jaramillo Savostianov et al., submitted, revised and went to the farm where all old papers go Example for solar data procedure applied to polar faculae data; extracted by quasi-stationary assumption; the coupling is reconstructed one; 9 Kuramoto Reconstruc�on: �me-dependent coupling • Reconstruction • Initial coupling
Courtesy of Andrés Muñoz–Jaramillo Savostianov et al., submited, revised and went to the farm where all old papers go Example for solar data (except the 20th cycle) is successful; AR(1) noise, the 20th cycle has a high probability of reconstruction failure. 10 Kuramoto Reconstruc�on: �me-dependent coupling • Reconstruction • With addition of
Lopes et al. 2015 Balthasar van der Pol 11 Oscillators’ modelling: van der Poll-Duffjng Oscillator • Lack of solar physical background in case of Kuramoto oscillators; • More comprehensive (no sine-like) form of solar cycles; • Can be derived from MHD equations (+ assumption of axysymmetry) d 2 B φ φ − 1) dB φ + ω 2 B φ + µ (3 ξB 2 dt − λB 3 φ = 0 dt 2
12 Coupled van der Poll-Duffjng relation per solar cycle correlation/coupling relation for VPD model correlation/coupling Idea From Single VPD to Coupled VPD � x + (1 − ∆ ω ) x + βx 3 − µ ( ˙ x − ( λ 1 − x 2 ) ˙ ¨ x − ˙ y ) = 0 y − ( λ 2 − y 2 ) ˙ y + (1 + ∆ ω ) y + βy 3 − µ ( ˙ ¨ y − ˙ x ) = 0 20 21 18 17 16 23 12 22 14 19 15 24 11 13 0.4 and • Numerically establish 0.2 0.4 ) / • Extract VPD coupling from 0.2 0.0 ( 0.0 0.2 0.4 0.6 0.8 Correlation, = 0.2 = 0.1 = 0.05 • Reconstruct with Kuramoto
13 Results Models’ Misinterpreta�on Bias in Reconstruc�on 0 10 / Coupling ratio, 1 9 × 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Cycle number = 0.2 = 0.1 • Reconstruction is not entirely catastrophic (error does not exceed ∼ 10 % ) • The 20th cycle remains to be anomaly • Proper way to choose the natural frequency level ( ∆ ω ) is crucial
Thank you for attention Contact: a.s.savostyanov@gmail.com Contact someone older: abshapoval@gmail.com
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