Dynamical Systems Approach to Space Environment Research

1. 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

5. 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, A&A (2002)

6. 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

7. 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)

8. The Derivative Nonlinear Schroedinger Equation b = by+ibz  = 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

9. 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)

10. 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/.

11. The time-T Poincare map

12. (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.

13. Pomeau-Manneville intermittency for a = 0.3213795 for (a) bz(); (b) bz vs. driver cycles; (c) |bz|2 vs. f.

14. Example of strange attractor of the Pomeau-Manneville intermittent turbulence for a = 0.3213795.

15. Interior-crisis-induced intermittency for a = 0.33029 for (a) bz(); (b) bz vs. driver cycles; (c) |bz|2 vs. f.

16. Example of strange attractors for the crisis-induce intermittent turbulence. (a) a=0.33022 (precrisis), (b) a=0.332 (postcrisis), and (c) superposition of (a) and (b).

17. Bifurcation diagram bz() for a = 0.3. The dashed linesdenote the unstable periodic orbit of p-2.

18. 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.

19. (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.

20. 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.

21. 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.

22. 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.

23. The Kuramoto-Sivashinsky equation The damping parameter:  We assume periodic boundary conditions: u(x,t) = u(x+2,t) and expand u in a spatial Fourier series: obtaining a set of ODEs for the complex Fourier modes bk:

24. 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 akare real coefficients:

25. Truncation used: N = 16 Fourier modes. Poincare map: the (N-1) dimensional hyperplane defined by: a1 = 0, withda1/dt> 0

26. (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 .

27. 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).

28. The spatiotemporal pattern of u(x,t) after crisis at  = 0.02992006. The system dynamics is chaotic in time but coherent in space.

29. Three-dimensional projection (a1, a10, a16) of the invariant unstable manifolds of the period-3 saddle (crosses) right after crisis at  = 0.02992020.

30. 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.

31. 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