Dynamical Systems Approach to Space Environment Research

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Dynamical Systems Approach toSpace Environment
Research

• Chian, A. C.-L.1,2 and Rempel, E. L.1,2
• 1WISER-World Institute for Space Environment
  Research
• U. of Adelaide, Australia
• 2 INPE, Brazil

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Nonlinear plasma waves and intermittent
turbulence play a fundamental role in the
dynamics of space environment

• Evidence of Alfvenic intermittency in the Helios
  data of velocity fluctuations in the inner solar
  wind
• Marsch Liu, Ann. Geophys. (1993)
• Analysis of intermittency in the solar wind
  turbulence via probability distribution functions
  of fluctuations
• Sorriso-Valvo et al., Geophys. Res. Lett.
  (1999)
• Intermittent heating in a model of solar coronal
  loops
• Walsh Galtier, Solar Phy. (2000)
• Signature of intermittency in the SOHO data of
  the transition region and the lower corona of the
  quiet Sun
• Patourakos Vial, AA (2002)

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Two approaches for studying space plasma dynamics

• Low-dimensional dynamical systems
• Stationary solutions of the derivative
• nonlinear Schroedinger equation
• High-dimensional dynamical systems
• Spatiotemporal solutions of the
• Kuramoto-Sivashinsky equation

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Model equations governing the nonlinear dynamics
of Alfven waves

• Nonlinear Schroedinger Equation was used to model
  solar coronal heating by Alfven wave
  filamentation
• Champeaux et al., ApJ (1997)
• Derivative Nonlinear Schroeding Equation was used
  to model
• magnetic holes in the solar wind
• Baumgartel, JGR (1999)
• Derivative Nonlinear Schroeding Equation was used
  to model
• Alfven intermittent turbulence in the solar
  wind
• Chian et al., ApJ (1998), IJBC (2002)

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The Derivative Nonlinear Schroedinger Equation
b byibz ? 1/4(1-?), ? c2S / c2A, cA
(cS ) the Alfven (acoustic) velocity ? the
dispersive parameter ? the dissipative scale
length The external driver S(b, x, t)
Aexp(ik?) a monochromatic left-hand circularly
polarized wave with a wave phase ? x - Vt
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The first integral of DNLS reduces to a system of
ODEs by seeking stationary wave solutions with
b b(?)
The driver amplitude parameter a A/(?b20) The
dissipation parameter ? ?/? The overdot denotes
derivative w.r.t. the phase variable ?
?b20?/? The driver phase variable ? ??, ?
?k/(?b20) ? -1 V/(?b20)
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The Poincare section
? ?p nT (n 1, 2, )
T 2?/? - the period of the external force
S ?p - the initial phase of b(?)
The associated Poincare map
P by(?p), bz(?p) ? by(?p nT), bz(?p nT)
This Poincare map represents the value of b(?) at
each driver period 2?/?.
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The time-T Poincare map
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(a)
(b)
(a) Bifurcation diagram bz(a) for ? 0.02. SNB
denotes saddle-node bifurcation and IC denotes
interior crisis. (b) Maximum Lyapunov exponent as
a function of a for ? 0.02.
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Pomeau-Manneville intermittency for a 0.3213795
for (a) bz(?) (b) bz vs. driver cycles (c)
bz2 vs. f.
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Example of strange attractor of the
Pomeau-Manneville intermittent turbulence for a
0.3213795.
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Interior-crisis-induced intermittency for a
0.33029 for (a) bz(?) (b) bz vs. driver cycles
(c) bz2 vs. f.
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Example of strange attractors for the
crisis-induce intermittent turbulence. (a)
a0.33022 (precrisis), (b) a0.332 (postcrisis),
and (c) superposition of (a) and (b).
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Bifurcation diagram bz(?) for a 0.3. The
dashed lines denote the unstable periodic orbit
of p-2.
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Bifurcation diagram bz(?) for a 0.1. (a)
Attractors (A1, A2, A3) (b) an enlargement of
the middle branch of attractor A3. The dashed
lines denote the unstable periodic orbit of p-9.
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(a)
(b)
bz
bz
by
by
(c)
bz
by
Basins of attraction for (a) two attractors, ?
0.01514 (b) three attractors, ? 0.01746 (c)
four attractors, ? 0.0174771.
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Poincare map of the strange attractor A3 (SA)
near boundary crisis. The cross denotes one of
the Poincare points of the unstable periodic
orbit of period-9 and the light lines represent
its stable manifolds (SM). (a) Homoclinic
tangency of SA with the SM (b) Dynamic
structures just before the boundary crisis.
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Dynamical structures of the boundary crisis of
attractor A2. SA denotes the strange attractor,
the crosses denote the Poincare points of the
middle branch of the unstable periodic orbit of
period-9, the light lines denote the stable
manifolds (SM) of the period-9 saddle. (a) just
before crisis for ?0.01515 (b)
homoclinic tangency of SA with the stable
manifolds of the saddle for ? 0.01514.
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Model equation governing the phase evolution of
nonlinear Alfven waves

• Under weak instability and wave packet limit, the
  derivative Nonlinear Schroedinger equation
  reduces to a complex Ginzburg-Landau equation
  (Lefebvre Hada, 2000).
• The Kuramoto-Sivashinsky equation describes the
  phase evolution of the complex amplitude of the
  Ginzburg-Landau equation (Kuramoto Tsuzuki,
  1976), hence it governs the phase evolution of
  nonlinear Alfven waves.

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The Kuramoto-Sivashinsky equation
The damping parameter ? We assume periodic
boundary conditions u(x,t) u(x2?,t) and
expand u in a spatial Fourier series
obtaining a set of ODEs for the complex Fourier
modes bk
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Odd function solutions
If the initial condition u(x, t 0) is real and
odd (u(x,t) -u(-x,t)), then the solution u
remains odd and real for all time, and the
coeficients bk(t) are purely imaginary. By
setting bk(t) -iak(t)/2, we restrict our
analysis to odd functions to simplify the
computation, where ak are real coefficients
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Truncation used N 16 Fourier modes. Poincare
map the (N-1) dimensional hyperplane defined by

a1 0, with da1/dt gt 0
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(a) Bifurcation diagram of a6 as a function of ?.
I1C denotes interior crisis and SN denotes
saddle-node bifurcation. The dotted lines
represent the p-3 UPO. (b) Variation of ?max with
?. (c) Variation of the correlation length with ?.
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Three-dimensional projection (a1, a10, a16) of
the strong strange attractor SSA (light line)
defined in the 15-dimensional Poincare hyperplane
right after crisis at ? 0.02992020,
superimposed by the 3-band weak strange
attractor WSA (dark line) at crisis (?
0.02992021).
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The spatiotemporal pattern of u(x,t) after crisis
at ? 0.02992006. The system dynamics is
chaotic in time but coherent in space.
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Three-dimensional projection (a1, a10, a16) of
the invariant unstable manifolds of the period-3
saddle (crosses) right after crisis at ?
0.02992020.
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The plots of the strange attractor (dark line)
and invariant unstable manifolds (light lines)
of the saddle before (a), at (b) and after (c)
crisis. The cross denotes one of the saddle
points.
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Relevance of Alfven chaos in the solar atmosphere
and solar wind

• Observation of Alfvenic intermittent turbulence
  in the solar wind shows power-law spectrum
• Marsch Tu, JGR (1990)
• Nonlinear time series analysis of velocity
  fluctuations of the low-speed streams of the
  inner solar wind data indicates that the Lyapunov
  exponent and the entropy are positive, suggesting
  the presence of chaos
• Macek Radaelli, Phys. Rev. E (2000)
• Chaos appears in a theoretical model of solar
  corona heating by a large-amplitude Alfven wave
• White, Chen Lin, Phys. Plasmas (2002)
• Dynamical systems approach to space environment
  turbulence