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