Modeling the Intermittent Dynamics of Alfvn Waves in
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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 smallamplitude fluctuations at small scales • Power spectrum displays a power-law behavior
Evidence of intermittency in the solar-terrestrial environment • 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)
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 Fast streams Slow streams b = B(t + ) – B(t) Sorriso-Valvo et al. , PSS, 49, 1193 (2001)
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)
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
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
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)
Derivative nonlinear Schrodinger equation Large-amplitude Alfvén wave propagating along the ambient magnetic field in the x direction: b = by+ibz h = dissipation a = 1/[4(1 - )], = c 2 S / c 2 A, = dispersion S(b, x, t) = Aexp(ik ): a circularly-polarized driver wave = x - Vt
Bifucation diagram: global view
Bifurcation diagram: periodic window
Unstable periodic orbits
Interior Crisis: pre- and post-crisis 15
Coupling unstable periodic orbit (p-11 UPO) M
Alfvén crisis-induced intermittency
Characteristic intermittency time
HILDCAA (High Intensity Long Duration Continuous Auroral Activities) BS • 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 !
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. , Ap. J 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
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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