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

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Outline

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

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• Observation of Alfven intermittent turbulence
• in the heliosphere

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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)

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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

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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)

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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)

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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) gt Lorenz chaotic
  attractor
• Chian, Kamide, et al., JGR (2006) gt Alfven
  chaotic saddle

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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)

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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

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• Model of nonlinear phase dynamics of
• Alfven intermittent turbulence

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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
  Earths bow shock
• Koga and Hada, SSRv 2003

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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

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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)
  ?(x2?,t) and expanding ? in a Fourier series

we obtain a set of ODEs for the Fourier modes ak
We seek odd solutions by assuming ak purely
imaginary
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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)

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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

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Bifurcation Diagram

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

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Attractor Merging Crisis
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Post-Crisis Chaotic Saddles
Rempel and Chian, PRE 71, 016203 (2005)
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Crisis-induced intermittency
n 0.02990
Rempel and Chian, PRE 71, 016203 (2005)
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Characteristic intermittency time
Rempel and Chian, PRE 71, 016203 (2005)
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HILDCAA(High Intensity Long Duration Continuous
Auroral Activities)

• IMP 8
• Gonzalez, Tsurutani, Gonzalez, SSR 1999
• Tsurutani, Gonzalez, Guarnieri, Kamide, Zhou,
  Arballo, JASTP (2004)

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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

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Advanced School on Space Environment (ASSE
2006)10-16 September 2006, LAquila
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.ce
a.inpe.br/wiser)
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• THANK YOU !