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)
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
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
• Alfvén intermittency in the solar wind
Bruno et al., ASR (2005)
Bruno & Carbone, http://solarphysics.livingreviews.org (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)
Evidence of intermittency in the solar-terrestrial environment
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
Non-Gaussian PDF for Alfven intermittency in the solar wind measured by Helios 2
b = B(t + ) – B(t)
Sorriso-Valvo et al., PSS, 49, 1193 (2001)
Fast streams Slow streams
Power-law behavior in the power spectrum of Alfvén intermittency in high-speed solar wind
Helios spacecraft (Marsch & Tu, 1990)
Power spectra of outward (solid lines) and inward (dotted lines) propagating Alfvénic fluctuations in high-speed solar wind,
indicating power-law behavior
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 / ClimateChian et al., JGR (2006):: Alfvén chaotic saddle => Space Weather / Space Climate
Chaotic sets
• Chaotic Attractors:- 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
• 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)
Evidence of chaos in the solar-terrestrial environment
Derivative nonlinear Schrodinger equation
Large-amplitude Alfvén wave propagating along the ambient magnetic field in the x direction:
),,,(22 txbSbiibbb xxt
b = by+ibz = dissipation= 1/[4(1-)], = c2
S / c2A,
= dispersionS(b,x,t) = Aexp(ik): a circularly-polarized driver wave = x - Vt
Bifucation diagram: global view
Bifurcation diagram: periodic window
Unstable periodic orbits
15
Interior Crisis:pre- and post-crisis
Coupling unstable periodic orbit (p-11 UPO)
M
Alfvén crisis-induced intermittency
Characteristic intermittency time
BS
HILDCAA(High Intensity Long Duration Continuous Auroral Activities)
IMP 8• Gonzalez, Tsurutani, Gonzalez, SSR 1999• Tsurutani, Gonzalez, Guarnieri, Kamide, Zhou, Arballo, JASTP (2004)
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
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 Springer, 2006
WISER mission: ‘linking nations for the peaceful use of the earth-ocean-space environment’
(www.cea.inpe.br/wiser)
THANK YOU !THANK YOU !
• 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
Two approaches to dynamical Two approaches to dynamical systemssystems
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
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
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