Cross-scale couplings in turbulent space plasmas

1. European Geosciences UnionGeneral Assembly 2006Vienna, Austria, 02 – 07 April 2006 Cross-scale couplings in turbulent space plasmas Z. Vörös (1), W. Baumjohann (1), M. Leubner (2), R. Nakamura (1), (1)Space Research Institute, Graz, Austria, (2)Institute of Astrophysics, Innsbruck, Austria

2. Space plasmas exhibit complex nonlinear interactions over many scales Examples: Shocks System scale coherent structures MHD scale fluid description turbulence Kinetic scale proton scale electron scale Toomre et al., 2000 Turbulence Reconnection Øieroset et al. ,2001 K-H vortices & reconnection Nakamura et al. 2004

3. Cross-scale coupling in turbulence The essence of turbulence is its multi-scale behaviour. Characteristic scales/wavenumber ranges: I: energy containing range; II: inertial range – nonlin. cascade ; III: dissipation range; Local intercation LOCAL : Nonlocal int. • Direct interaction between large scales (velocity field) and small scales • (velocity derivatives, vorticity) (e.g. Tsinober, 2001); • Nonlocal interactions due to scalar gradients, boundaries • (Warhaft, 2000); etc. NONLOCAL :

4. Evidence for nonlinear cascades and cross-scale coupling in the solar wind • Dependence of inertial range spectral index • Dissipation range spectral index on solar • wind proton temperature Typical IMF power spectrum; spectral break at 0.23 Hz. Inertial range scaling (-1.67 ) is in excellent agreement with the Kolmogorov prediction (-5/3 ). Steeper dissipation range spectra are associated with greater heating rates (Leamon et al., 1998 a,b,c).

5. Evidence for nonlinear cascades and cross-scale coupling in the Earth’s plasma sheet I Typical power spectral density in the plasma sheet Uncertain estimations: 1~ 0.5 - 1.5 2 ~ 1.7 - 2.9. [Borovsky& Funsten, 1997, 2003; Volwerk et al., 2003, 2004; Vörös etal., 2003, 2004, 2005; Weygand et al., 2005]

6. Evidence for nonlinear cascades and cross-scale coupling in the Earth’s plasma sheet II Vörös et al., 2004 Spectral index Dissipation (small-) scale vs. large-scale average velocity (Vörös, 2005)  = 2.5  0.2  = 2.5  0.3 Small-scale power Perpendicular velocity Reynolds number (velocity) dependent widening of the inertial range is a generic feature of many turbulent flows with cascades.

7. Nonlocal interactions I Experimental evidence: When a large scale scalar gradient is imposed on a turbulent velocity field, the resultant small scale temperature fluctuations reflect the large scale gradient. The small scales are not universal (Tong & Warhaft, 1994; Warhaft, 2000), the PDFs are skewed. Numerical simulations: Turbulent mixing makes the scalar gradient field patchy. As a consequence, anisotropy induces intermittency (Holzer & Siggia, 1994). Scalar contaminant in a turbulent flow: Skewness and kurtosis plot collapses onto a quadratic curve (Chatwin, Robinson, 1997). Kurtosis Skewness

8. Nonlocal interactions II - analogy Scalar contaminant in a turbulent flow (advection-diffusion equation): Magnetic field fluctuations in MHD • v is the turbulence velocity; • is the magnetic diffusivity. The equation for the magnitude B =Bn (x,t) is the concentration of passive scalar; Kis the molecular diffusivity; Y(x,t) is the random (turbulent) velocity, which satisfies the Navier-Stokes eq. and mass conservation. The statistical properties of the magnitude B in MHD flows resemble, in the inertial range, those of passive scalars in fluid turbulence (Bershadskii & Sreenivasan, 2004)

9. Nonlocal interactions in the Earth plasma sheet: boundaries and gradients • The thickness of the plasma sheet affects the statistical moments during rapid flows – magnetic field gradients are important (Vörös et al. 2004); • BBFs occurrence is the largest in the central plasma sheet  gradients towards flow or plasma sheet boundaries (Nakamura et al., 2004); • Gradients of other quantities: current density, proton number density, proton temperature, pressure (Runov et al., 2006).

10. Multi-point Cluster measurements Possible flow geometry 3700 km

11. Dynamic spectra C4 C1 C3 C2

12. Estimation of Skewness and Kurtosis: Non-stationary processes – error bars. A B A C B C

13. Skewness and Kurtosis 1 1 2 2 3 3 4 4

14. Boundary vs. non-boundary flow statistics in the Earth’s plasma sheet Scales: 1.5 -5 s Kurtosis vs. Skewness plot seems to collapse onto a quadratic curve, resembling passive scalar statistics in fluid turbulence.

15. Comparisons Kurtosis Passive scalar statistics in a fluid flow (Chatwin, Robinson, 1997) Skewness Passive scalar statistics near interplanetary shocks (Vörös et al., 2006) Passive scalar statistics in the Earth’s plasma sheet (this study) Evidence for nonlocal turbulence interactions in space plasmas

16. Conclusions Cross-scale coupling in turbulent space plasmas can occur through local as well as nonlocal interactions in Fourier space. References • Vörös et al., Ann. Geophys., 2003, 21, 1955; • Vörös et al., Phys.Plasmas, 2004, 11, 1333; • Vörös et al., J. Geophys. Res., 2004, 109, A11215; • Leubner & Vörös, Astrophys. J., 2005, 618, 547; • Leubner & Vörös,, Nonlin.Proc.Geophys., 2005, 12, 171; • Vörös et al., in. Multiscale Coupling of Sun-Earth Processes, (Ed. Lui, Kamide, Consolini), 2005, 29; • Vörös et al., Nonlin.Proc.Geophys., 2005, 12, 725; • Vörös et al., J. Geophys. Res., 2006, 111, A02102; • http://iwf.oeaw.ac.at/english/welcome1024_e.html