Intermittency, nonlinear dynamics and dissipation in the solar wind and astrophysical plasmas

Abstract

An overview is given of important properties of spatial and temporal intermittency, including evidence of its appearance in fluids, magnetofluids and plasmas, and its implications for understanding of heliospheric plasmas. Spatial intermittency is generally associated with formation of sharp gradients and coherent structures. The basic physics of structure generation is ideal, but when dissipation is present it is usually concentrated in regions of strong gradients. This essential feature of spatial intermittency in fluids has been shown recently to carry over to the realm of kinetic plasma, where the dissipation function is not known from first principles. Spatial structures produced in intermittent plasma influence dissipation, heating, and transport and acceleration of charged particles. Temporal intermittency can give rise to very long time correlations or a delayed approach to steady-state conditions, and has been associated with inverse cascade or quasi-inverse cascade systems, with possible implications for heliospheric prediction.

Keywords: intermittency; plasma physics; solar corona; solar wind; turbulence theory.

Figure 1.

Results from a two-dimensional MHD simulation showing the relationship between the spatial distribution of electric current density (a) and various contributions to the PDF of current density (b). In both panels, the current density is normalized to its r.m.s. value. The PDF in (b) is divided into three regions and is compared with a reference unit variance Gaussian distribution. Region (I), the core of the distribution, corresponds to the shaded regions in (c). The sub-Gaussian region designated (II) in (b) corresponds to the shaded areas in (d). (e) The extreme events, i.e. region (III) in (b), suggesting that strong current sheets are located between magnetic islands. Magnetic field lines are superposed in (c)–(e). (Online version in colour.)

Figure 2.

Evidence for intermittency, in the form of multi-fractal scalings of structure functions of increasing order. (a) ζ(p) from hydrodynamic experiments. (Adapted from Anselmet et al. [14].) (b) ζ(p) from an MHD simulation. (Adapted from Biskamp & Müller [15].) When the scaling exponent ζ(p) exhibits anomalous behaviour (i.e. is a nonlinear function of p, the order of the structure function), the scaling is described as multi-fractal [6]. An alternative and more direct approach to characterize intermittency is to compare the PDFs of increments at different spatial lags, finding that fatter non-Gaussian tails appear in the PDFs of the smaller lags. (c) An example from solar wind data, where spatial lag is proportional to time lag. (Adapted from Sorriso-Valvo et al. [16].)

Figure 3.

Out-of-plane electric current density from ideal (a–c) and resistive (d–f) MHD runs started with identical initial data. The current density is shown at three times: t=0, t≈0.2 nonlinear times and at a later time. The early evolution is evidently almost exactly the same in the two cases. In particular, strong sheet-like concentrations form in both ideal and non-ideal cases. (From Wan et al. [32].) (Online version in colour.)

Figure 4.

Examples of cellularization owing to turbulence—consisting of sharp gradients separating relatively relaxed regions. Shown are turbulence simulation examples from two-dimensional (2D) MHD, three-dimensional (3D) isotropic MHD, reduced MHD (RMHD), 3D Hall MHD and 2.5-dimensional (2.5D) hybrid kinetic codes, respectively, from Zhou et al. [33], Mininni et al. [34], Rappazzo et al. [35], Greco et al. [36] and Parashar et al. [37]. (Online version in colour.)

Figure 5.

Distributions of the alignment angles between v and b, etc. (see legend), from an unforced three-dimensional MHD turbulence simulation. As in the text, ∇×b=j is the electric current density, and ∇×v=ω is the vorticity. The distributions are computed less than one nonlinear time into the run. At t=0 the distributions were all flat. The initial conditions contained very low levels of magnetic helicity and cross helicity. (From Servidio et al. [54].) (Online version in colour.)

Figure 6.

Spatial signal PVI computed from simulation versus distance s/λ_c at a spatial lag=0.00625λ_c (solid thin red line), where λ_c is the correlation scale. (b) Time series PVI (normalized to correlation time t_c) computed from ACE data at a time separation of 4 min. The thick dashed blue lines are the values of the thresholds employed for figure 7. (Online version in colour.)

Figure 7.

Waiting times (distances) computed from a three-dimensional Hall MHD simulation and from ACE solar wind data, with the coordinate along the time series normalized by the correlation scale. The distributions are very similar and power-law-like in the inertial range. (Adapted from Greco et al. [63].) (Online version in colour.)

Figure 8.

The dropout phenomenon seen in SEP data by Mazur et al. [74] is explained by a model based on transient trapping of magnetic field lines within magnetic flux tubes. These act as conduits for transport, delaying diffusion [75]. (Online version in colour.)

Figure 9.

Distributions of and computed from solar wind (SW) data and from an MHD turbulence simulation initiated with the same dimensionless cross helicity as the solar wind sample. Simulation results are for a time a few nonlinear times from the initial data. The similarity may be viewed as evidence that the spatial patchiness of correlation seen in the simulations, necessarily associated with non-Gaussian distributions, also occurs in the solar wind. (From Osman et al. [82].) (Online version in colour.)

Figure 10.

Conditional distributions of proton temperature measured at 1 AU by the ACE spacecraft. The different distributions are for different increasing ranges of PVI value. The higher PVI ranges have distinctly higher temperatures. (From Osman et al. [84].) (Online version in colour.)

Figure 11.

Three images from a large 2.5-dimensional PIC simulation of the development of turbulence starting from a proton shear flow. (From Karimabadi et al. [130].) The colour indicates the magnitude of the out-of-plane electric current density where the initial magnetic field is uniform (zero current density). The three phases shown are (a) early phase characterized by small perturbations and linear instabilities; (b) a transitional phase in which turbulence develops; and (c) a strong turbulence phase. It is apparent that small-scale coherent current structures are formed over a range of scales extending fully between proton and electron scales, and also beyond these. (Online version in colour)

Figure 12.

Hybrid Eulerian Vlasov simulation results, showing the concentration of distinctive kinetic features in regions near to current sheets. Colour contours of several quantities in the vicinity of a current sheet (black cross in all panels) with the in-plane magnetic field lines (blue lines): (a) deviation of the proton distribution from an equivalent Maxwellian (in per cent), (b) proton temperature anisotropy T_⊥/T_∥, (c) proton heat flux and (d) kurtosis of the proton velocities. (From Greco et al. [140].) (Online version in colour.)

Figure 13.

Waiting time distributions for the reversal of the dipole moment in a three-dimensional ideal spherical Galerkin MHD model, compared with the record of geomagnetic reversals. When helicity (H_m>0) and rotation (Ω>0) are present, the simulation curve exhibits a power-law distribution similar to the geophysical record. When both are absent (H_m=0, Ω=0), the waiting distribution is very different. (Adapted from Dmitruk et al. [160].) (Online version in colour.)

Figure 14.

Spectral diagram of plasma turbulence suggesting the cascade and intermittency properties summarized here. Large-scale, non-local effects induce temporal variations, in some cases generate 1/f noise, and influence variability at smaller scales, as suggested by Oboukhov [4]. Inertial range cascade is predominantly local in scale but also generates a hierarchy of structures, which in many cases may be viewed as the formation and interaction of a hierarchy of interacting magnetic flux tubes. Significant effects on transport of heat and particles are expected owing to inertial range intermittency. At the smaller scales kinetic processes become important, characteristic coherent small-scale structures (including vortices and current sheets) are formed, secondary instabilities and waves may be in evidence, and ultimately dissipation occurs. (Online version in colour.)