Phase Dynamics of Alfven Intermittent Turbulence in the Heliosphere

1. Phase Dynamics of Alfven Intermittent Turbulence in the Heliosphere Abraham C.-L. Chian National Institute for Space Research (INPE), Brazil & Erico L. Rempel, ITA, Brazil

2. Outline • Observation of Alfven intermittent turbulence in the heliosphere • Model of nonlinear phase dynamics of Alfven intermittent turbulence

3. Observation of Alfven intermittent turbulence in the heliosphere

4. Intermittency • Time series displays random regime switching between laminar and bursty periods of fluctuations • Probability distribution function (PDF) displays a non-Gaussian shape due to large-amplitude fluctuations at small scales • Power spectrum displays a power-law behavior Ref.: Burlaga, Interplanetary Magnetohydrodynamics, Oxford U. P. (1995) Biskamp, Magnetohydrodynamic Turbulence, Cambridge U. P. (2003) Lui, Kamide & Consolini, Multiscale Coupling of Sun-Earth Processes, Elsevier (2005)

5. Alfvén intermittency in the solar wind • Time evolution of velocity fluctuations measured by Helios 2, V() = V(t+)-V(t), at 4 different time scales (): • Carbone et al., Solar Wind X, 2003

6. Non-Gaussian PDF for Alfven intermittency in the solar wind measured by Helios 2 Slow streams Fast streams dbt = B(t + t) – B(t) • Sorriso-Valvo et al., PSS, 49, 1193 (2001)

7. Power-law behavior in the power spectrum of Alfvén intermittency in high-speed solar wind Power spectra of outward (solid lines) and inward (dotted lines) propagating Alfvénic fluctuations in high-speed solar wind, indicating power-law behavior • Helios spacecraft (Marsch & Tu, 1990)

8. Chaos Chaotic Attractors and Chaotic Saddles: • Sensitive dependence on initial conditions and system parameters • Aperiodic behavior • Unstable periodic orbits Ref: Lorenz, J. Atm. Sci. (1963) => Lorenz chaotic attractor Chian, Kamide, et al., JGR (2006)=> Alfven chaotic saddle

9. Evidence of chaos in the heliosphere • Chaos in Alfven turbulence in the solar wind Macek & Radaelli, PSS (2001) Macek et al., PRE (2005) • Chaos in solar radio emissions Kurths & Karlicky, SP (1989) Kurths & Schwarz, SSRv (1994) • Chaos in the (AE, AL) auroral indices Vassialiadis et al., GRL (1990) Sharma et al., GRL (1993) Pavlos et al., NPG (1999)

10. Unstable periodic orbits and turbulence • Spatiotemporal chaos can be described in terms of UPOs • Christiansen et al., Nonlinearity 1997 • Identification of an UPO in plasma turbulence in a tokamak experiment • Bak et al, PRL 1999 • Sensitivity of chaotic attractor of a barotropic ocean model to external • influences can be described by UPOs • Kazantsev, NPG, 2001 • Intermittency of a shell model of fluid turbulence is described by an UPO • Kato and Yamada, Phys. Rev. 2003 • Control of chaos in a fluid turbulence by stabilization of an UPO • Kawahara, Phys. Fluids 2005

11. Model of nonlinear phase dynamics of Alfven intermittent turbulence

12. Phase dynamics of MHD turbulence in the solar wind • Geotail magnetic field data shows evidence of phase coherence in MHD waves in the solar wind • Hada, Koga and Yamamoto, SSRv 2003 • Phase coherence of MHD turbulence upstream of the Earth’s bow shock • Koga and Hada, SSRv 2003

13. Two approaches to Alfven chaos • Low-dimensional chaos: Stationary solutions of the derivative nonlinear Schroedinger equation Hada et al., Phys. Fluids 1990 Rempel and Chian, Adv. Space Res. 2005 Chian et al., JGR 2006 • High-dimensional chaos: Spatiotemporal solutions of the Kuramoto-Sivashinsky equation Chian et al., Phys. Rev. E 2002 Rempel et al., Nonlinear Proc. Geophys. 2005 Rempel and Chian, Phys. Rev. E 2005

14. Kuramoto-Sivashinsky equation Phase dynamics of a NL Alfven wave is governed by the Kuramoto-Sivashinsky eqn. (LaQuey et al. PRL 1975, Chian et al. PRE 2002, Rempel and Chian PRE 2005): • is a damping parameter. Assuming periodic boundary conditions: (x,t) = (x+2,t) and expanding  in a Fourier series: we obtain a set of ODE’s for the Fourier modes ak: We seek odd solutions by assuming ak purely imaginary

15. Spatiotemporal phase dynamics of Alfven waves Truncation: N = 16 Fourier modes • Chian et al., PRE (2002) • Rempel and Chian, Phys. Lett. A (2003) • Rempel et al., NPG (2005) • Rempel and Chian, PRE (2005)

16. Chaotic solutions • Chaotic Attractors: - Set of unstable periodic orbits - Positive maximum Lyapunov exponent • Attract all initial conditions in a given neighbourhood (basin of attraction) • Responsible for asymptotic chaos • Chaotic Saddles (Chaotic Non-Attractors): • Set of unstable periodic orbits • Positive maximum Lyapunov exponent • Repel most initial conditions from their neighbourhood, except those on stable manifolds (no basin of attraction) • Responsible for transient chaos

17. Bifurcation Diagram • Rempel and Chian, PRE 71, 016203 (2005).

18. Attractor Merging Crisis

19. Post-Crisis Chaotic Saddles Rempel and Chian, PRE 71, 016203 (2005)

20. Crisis-induced intermittency n = 0.02990 Rempel and Chian, PRE 71, 016203 (2005)

21. Characteristic intermittency time Rempel and Chian, PRE 71, 016203 (2005)

22. BS HILDCAA(High Intensity Long Duration Continuous Auroral Activities) • IMP 8 • Gonzalez, Tsurutani, Gonzalez, SSR 1999 • Tsurutani, Gonzalez, Guarnieri, Kamide, Zhou, Arballo, JASTP (2004)

23. CONCLUSIONS • Observational evidence of chaos and intermittency in the heliosphere • Dynamical systems approach provides a powerfull tool to probe the complex nature of space environment, e.g., Alfven intermittent turbulence • Unstable structures (unstable periodic orbits and chaotic saddles) are the origin of intermittent turbulence • Characteristic intermittency time can be useful for space weather and space climate forecasting

24. Advanced School on Space Environment (ASSE 2006)10-16 September 2006, L’Aquila – ItalyConveners: R. Bruno, A. Chian, Y. Kamide, U. VillanteHandbook of Solar-Terrestrial EnvironmentEditors: Y. Kamide and A. ChianSpringer 2006 WISER mission: ‘linking nations for the peaceful use of the earth-ocean-space environment’ (www.cea.inpe.br/wiser)

25. THANK YOU !