Intermittent onset of turbulence and control of extreme events
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1 Intermittent onset of turbulence and control of extreme events Sergio Roberto Lopes, Paulo P. Galúzio Departamento de Física UFPR São Paulo 03 de Abril 2014 Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
2 Outline 1 Motivation 2 Some theoretical results for a simple map 3 UDV and on-o intermittency 4 The nonlinear Schrödinger equation - NLSE 5 Results General dynamics of the NLSE Intermittency Control of extreme events 6 Conclusions Conclusions Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
3 Conteúdo 1 Motivation 2 Some theoretical results for a simple map 3 UDV and on-o intermittency 4 The nonlinear Schrödinger equation - NLSE 5 Results General dynamics of the NLSE Intermittency Control of extreme events 6 Conclusions Conclusions Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
4 Motivation So much more to know... Science 309, 5731 (2005), Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
5 Plasma Bursts Antar, G. Y., Counsell, G., Yu, Y., Labombard, B., Devynck, P. Universality of intermittent convective transport in the scrape-o layer of magnetically conned devices. Physics of Plasmas 10, 2 (2003), 419? Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
6 Conteúdo 1 Motivation 2 Some theoretical results for a simple map 3 UDV and on-o intermittency 4 The nonlinear Schrödinger equation - NLSE 5 Results General dynamics of the NLSE Intermittency Control of extreme events 6 Conclusions Conclusions Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
7 Some theoretical results for a simple map x n+1 = (1 ɛ)rx n (1 x n ) + ɛw φ n+1 = 1 2π px n sin (2πφ n ) S(x) = x 3 / x 2 3/2 K(x) = x 4 / x 2 2 Galuzio, P., Lopes, S., dos Santos Lima, G., Viana, R., Benkadda, M. Evidence of determinism for intermittent convective transport in turbulence processes. Physica A: Statistical Mechanics and its Applications 402, 0 (2014), 8 13.? Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
8 Conteúdo 1 Motivation 2 Some theoretical results for a simple map 3 UDV and on-o intermittency 4 The nonlinear Schrödinger equation - NLSE 5 Results General dynamics of the NLSE Intermittency Control of extreme events 6 Conclusions Conclusions Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
9 UDV and on-o intermittency Time distribution of the laminar states (the time between two ejection events): { τ ν τ pequeno; P (τ) e γτ τ grande. (1) Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
10 Conteúdo 1 Motivation 2 Some theoretical results for a simple map 3 UDV and on-o intermittency 4 The nonlinear Schrödinger equation - NLSE 5 Results General dynamics of the NLSE Intermittency Control of extreme events 6 Conclusions Conclusions Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
11 The nonlinear Schrödinger equation iψ t Ψ xx g Ψ 2 Ψ = 0 (2) Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
12 The nonlinear Schrödinger equation iψ t Ψ xx g Ψ 2 Ψ = 0 (2) Periodic boundary conditions: Ψ(x, t) = Ψ(x + L, t) External energy source periodic forcing ɛe iω2t ; linear damping term γ. iψ t Ψ xx g Ψ 2 Ψ = ɛe iω2t iγψ. (3) Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
13 The nonlinear Schrödinger equation iψ t Ψ xx g Ψ 2 Ψ = 0 (2) Periodic boundary conditions: Ψ(x, t) = Ψ(x + L, t) External energy source periodic forcing ɛe iω2t ; linear damping term γ. iψ t Ψ xx g Ψ 2 Ψ = ɛe iω2t iγψ. (3) Variable changing : ψ(x, t) = Ψ(x, t)e iω2 t iψ t ψ xx ( g ψ 2 Ω 2) ψ = ɛ iγψ. (4) Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
14 The Schrödinger nonlinear equation Mass M = 1 L L 2 L 2 ψ 2 dx, (5) Energy E = 1 L L 2 L 2 ( ψ x 2 + g 2 ψ 4 Ω 2 ψ 2 ) dx, (6) Hamiltonian H = 1 L L 2 L 2 ( ψ x 2 g 2 ψ 4 + Ω 2 ψ 2 ɛ(ψ + ψ ) ) dx, (7) Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
15 The Schrödinger nonlinear equation Mass M = 1 L L 2 L 2 ψ 2 dx, (5) Energy Integral E = 1 ( L of the 2 ψ 2 motion L x + g when ) ɛ = γ = 0 2 ψ 4 Ω 2 ψ 2 dx, (6) L 2 Hamiltonian H = 1 L L 2 L 2 ( ψ x 2 g 2 ψ 4 + Ω 2 ψ 2 ɛ(ψ + ψ ) ) dx, (7) Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
16 Conteúdo 1 Motivation 2 Some theoretical results for a simple map 3 UDV and on-o intermittency 4 The nonlinear Schrödinger equation - NLSE 5 Results General dynamics of the NLSE Intermittency Control of extreme events 6 Conclusions Conclusions Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
17 General dynamics ɛ = 0.10 Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
18 General dynamics ɛ = 0.21 Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
19 General dynamics ɛ = 0.20 Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
20 General dynamics ɛ = 0.30 Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
21 General dynamics ɛ = 0.30 Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
22 General dynamics ɛ = 0.30 Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
23 General dynamics ɛ = 0.44 Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
24 General dynamics ɛ = 0.40 Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
25 Lyapunov exponents N/2 (n + 1) 2 φ n 2 N 2 = n=0 N/2 φ n 2 n=0 (8) Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
26 Intermittency ɛ = 0.3
27 Conditional average ɛ = 0.3 Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
28 Inter-Bursts distributions ɛ = 0.3 Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
29 Positive fraction of nite time Lyapunov exponents FTLE for ɛ = 0.10(a n ) ɛ = 0.30(b n ) ɛ = 0.45(c n ) Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
30 Positive fraction of nite time Lyapunov exponents FTLE for ɛ = 0.10(a n ) ɛ = 0.30(b n ) ɛ = 0.45(c n ) Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
31 Low dimension chaotic attractor Notation: φ n (t) ξ n Longitudinal components: ξ 0, ξ 1,..., ξ N 1 Chaotic attractor: A Transversal components: ξ N, ξ N +1,..., ξ N Burst: condition ξ N > η Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
32 Low dimension chaotic attractor Notation: φ n (t) ξ n Longitudinal components: ξ 0, ξ 1,..., ξ N 1 Chaotic attractor: A Transversal components: ξ N, ξ N +1,..., ξ N Burst: condition ξ N > η Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
33 2D PDFs of the projections the phase space Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
34 2D PDFs of the projections the phase space Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
35 2D PDFs of the projections the phase space Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
36 Control of extreme events Perturbation: (ξ 0, ξ 1 ) max(ρe,ρ A ) = (0.60, 0.20) d 0.005; Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
37 Control of extreme events Perturbation: (ξ 0, ξ 1 ) max(ρe,ρ A ) = (0.60, 0.20) d 0.005; Harmonic signal: Wave vector number equal to 2π/L; Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
38 Control of extreme events Perturbation: (ξ 0, ξ 1 ) max(ρe,ρ A ) = (0.60, 0.20) d 0.005; Harmonic signal: Wave vector number equal to 2π/L; Amplitude: G(µ = 0, σ = ); Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
39 Control of extreme events Perturbation: (ξ 0, ξ 1 ) max(ρe,ρ A ) = (0.60, 0.20) d 0.005; Harmonic signal: Wave vector number equal to 2π/L; Amplitude: G(µ = 0, σ = ); Local dissipation: In order to make sure the system energy will stay constant. Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
40 Control of extreme events Perturbation: (ξ 0, ξ 1 ) max(ρe,ρ A ) = (0.60, 0.20) d 0.005; Harmonic signal: Wave vector number equal to 2π/L; Amplitude: G(µ = 0, σ = ); Local dissipation: In order to make sure the system energy will stay constant. Interval of time for the perturbation: 3.7% of the integration time. Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
41 Control of extreme events Perturbation: (ξ 0, ξ 1 ) max(ρe,ρ A ) = (0.60, 0.20) d 0.005; Harmonic signal: Wave vector number equal to 2π/L; Amplitude: G(µ = 0, σ = ); Local dissipation: In order to make sure the system energy will stay constant. Interval of time for the perturbation: 3.7% of the integration time. Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
42 Conteúdo 1 Motivation 2 Some theoretical results for a simple map 3 UDV and on-o intermittency 4 The nonlinear Schrödinger equation - NLSE 5 Results General dynamics of the NLSE Intermittency Control of extreme events 6 Conclusions Conclusions Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
43 Conclusions Intermittent transition temporal chaos Turbulence; Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
44 Conclusions Intermittent transition temporal chaos Turbulence; Fourier spectrum of the system in the high level energy state is consistent with a turbulent ux; Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
45 Conclusions Intermittent transition temporal chaos Turbulence; Fourier spectrum of the system in the high level energy state is consistent with a turbulent ux; time distribution of laminar states is a signature of on-o intermittency; Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
46 Conclusions Intermittent transition temporal chaos Turbulence; Fourier spectrum of the system in the high level energy state is consistent with a turbulent ux; time distribution of laminar states is a signature of on-o intermittency; Due to UDV the problem has localized unstable regions; Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
47 Conclusions Intermittent transition temporal chaos Turbulence; Fourier spectrum of the system in the high level energy state is consistent with a turbulent ux; time distribution of laminar states is a signature of on-o intermittency; Due to UDV the problem has localized unstable regions; The control makes possible to prevent a great number of extreme events in the system. Lopes, S. R. (Física UFPR) Intermittent onset of turbulence and control... SP 03/04/ / 24
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