1. Modeling the Intermittent Dynamics of Alfvén Waves in the Solar Wind
Abraham C.-L. Chian
National Institute for Space Research (INPE), Brazil
&
Yohsuke Kamide (Nagoya U., Japan), Erico L. Rempel (ITA, Brazil),
Wanderson M. Santana (INPE, Brazil)

2. Outline
Relevance of intermittency and chaos in the solar-terrestrial environment
Modeling the interplanetary Alfvén intermittency driven by chaos
Ref: Chian et al., On the chaotic nature of solar-terrestrial environment: interplanetary Alfvén intermittency, JGR 2006

3. 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 an excess of large- and small-amplitude fluctuations at small scales
Power spectrum displays a power-law behavior

4. Evidence of intermittency in the solar-terrestrial environment
Alfvén intermittency in the solar wind
Bruno et al., ASR (2005)
Bruno & Carbone, (2005)
Intermittency in the Auroral Electrojet (AE) index
Consolini & De Michelis, GRL (1998, 2005)
Intermittency in the earth´s plasma sheet related to bursty bulk flows in the magnetotail
Angelopoulos, Mukai & Kokubun, PP (1999); Voros et al., JGR (2004)

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
Fast streams
Slow 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 & Chaotic Saddles:
Sensitive dependence on initial conditions and system parameters
Aperiodic behavior
Unstable periodic orbits
Lorenz, J. Atm. Sci. (1963): Lorenz chaotic attractor => Weather / Climate
Chian et al., JGR (2006): Alfvén chaotic saddle => Space Weather / Space Climate

9. Chaotic sets Chaotic Attractors: Chaotic Saddles:
- Set of unstable periodic orbits
Positive maximum Lyapunov exponent
- Attract all initial conditions in a given neighbourhood
- Basin of attraction (continuous stable manifolds, without gaps)
- Responsible for asymptotic chaos
Chaotic Saddles:
- 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 (fractal stable manifolds, with gaps)
- Responsible for transient chaos

10. Evidence of chaos in the solar-terrestrial environment
Chaos in Alfvén 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
Baker et al., GRL (1990)
Sharma et al., GRL (1993)
Pavlos et al., NPG (1999)

11. Derivative nonlinear Schrodinger equation
Large-amplitude Alfvén wave propagating along the ambient magnetic field in the x direction:
b = by+ibz
= dissipation
a = 1/[4(1-)],  = c2S / c2A,
 = dispersion
S(b,x,t) = Aexp(ik): a circularly-polarized driver wave
 = x - Vt

12. Bifucation diagram: global view

13. Bifurcation diagram: periodic window

14. Unstable periodic orbits

15. Interior Crisis: pre- and post-crisis

16. Coupling unstable periodic orbit (p-11 UPO)M

17. Alfvén crisis-induced intermittency

18. Characteristic intermittency time

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

20. CONCLUSIONS Observational evidence of chaos and intermittency in the
Sun-Earth system
Dynamical systems approach provides a powerfull tool to probe the complex nature of solar-terrestrial environment, e.g.,
Alfvén intermittent turbulence in the solar wind
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

21. Books Handbook of Solar-Terrestrial Environment
Y. Kamide and A.C.-L. Chian (Eds.)
Springer, 2006
(ASSE 2006)
Fundamentals of Space Environment Science
V. Jatenco, A. C.-L Chian, J. F. Valdes and M.A. Shea (Eds.)
Elsevier, 2005
(ASSE 2004)
Advances in Space Environment Research
A.C.-L. Chian and the WISER Team (Eds.)
Kluwer, 2003
(WSEF 2002, HPC 2002)
Complex Systems Approach to Economic Dynamics
A.C.-L. Chian
WISER mission: ‘linking nations for the peaceful use of the earth-ocean-space environment’ (

22. THANK YOU !

23. Two approaches to dynamical systems
Low-dimensional chaos:
Stationary solutions of the derivative nonlinear Schroedinger equation
Hada et al., Phys. Fluids 1990
Chian et al., ApJ 1998
Borotto et al., Physica D 2004
Rempel et al., Phys. Plasmas 2006
Chian et al., JGR 2006
High-dimensional chaos:
Spatiotemporal solutions of the Kuramoto-Sivashinsky equation and
the regularized long-wave equation
Chian et al., Phys. Rev. E 2002
He and Chian, Phys. Rev. Lett. 2003
He and Chian, Phy. Rev. E 2004
Rempel and Chian, Phys. Rev. E 2005

24. Unstable periodic orbits & turbulence
UPOS in the Kuramoto-Sivashinsky equation
Christiansen et al., Nonlinearity 1997; Zoldi and Greenside, PRE 1998
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 and Kida, J. Fluid Mech. 2001; Kawahara, Phys. Fluids 2005

25. Chaotic saddles & turbulence
Supertransient in the complex Ginzburg-Landau equation
Braun and Feudel, PRE 1996
Detecting and computing chaotic sadddles in higher dimensions
Sweet, Nusse and Yorke, PRL 1996
Close to the transition from laminar to turbulent flows the turbulent state corresponds to a chaotic saddle
Eckhardt and Mersmann, PRE 1999
Chemical and biological activity in open flows
Tél et al, Phys. Rep. 2005
Dispersion of finite-size particles in open chaotic advection
Vilela, de Moura and Grebogi, PRE 2006
Edge of chaos in a parallel shear flow
Skufca, Yorke and Eckhardt, PRL 2006