Session NM10: Mini-Conference on Solar Wind Turbulence I

Generalizations of the theory of Goldreich and Sridhar and the theory of Boldyrev to incompressible MHD turbulence with cross-helicity

A fundamental problem in the theory of incompressible MHD turbulence with cross-helicity, also called imbalanced MHD turbulence, is to understand how the energy cascade rate depends on the normalized cross-helicity $\sigma_c$. Solar wind observations indicate that in the inertial range $\sigma_c$ is approximately constant, independent of wavenumber. For incompressible MHD turbulence with $\mbox{\textit{Pr}}_m \equiv \nu/\eta=1$, it is shown from first principles that if $\sigma_c =$ \emph{const}, independent of wavenumber, then the energy cascade times of the two Elsasser fields are equal to each other: $\tau^+=\tau^-$. Using this result, the theory of Goldreich and Sridhar and the theory of Boldyrev are generalized to imbalanced MHD turbulence in such a way that the inertial range scaling laws of these two theories are both preserved. The functional dependence of the perpendicular energy spectrum on the normalized cross-helicity is determined by requiring that the energy cascade time at a given scale is equal to a particular correlation time determined by the second order Lagrangian structure function, a relationship that is shown to hold for hydrodynamic turbulence and for both the original Goldreich and Sridhar theory and the original Boldyrev theory for which $\sigma_c =0$. The resulting generalizations to imbalanced MHD turbulence are characterized by perpendicular energy spectra that are independent of the normalized cross-helicity.   

Perpendicular Ion Heating by Low-Frequency Alfven-Wave Turbulence in the Solar Wind

A critical unsolved problem in the study of solar wind turbulence is to determine whether low-frequency Alfven-wave (AW) and kinetic-Alfven-wave (KAW) turbulence can explain the perpendicular ion heating that is observed in coronal holes and low-beta fast-wind streams. In linear wave theory, low-frequency AWs and KAWs are incapable of causing perpendicular ion heating. On the other hand, a number of observations suggest that low-frequency AW/KAW turbulence is the primary heating mechanism in the solar wind. This presentation describes recent work that offers a possible solution to this long-standing problem, and which extends previous studies of ``stochastic heating.'' An analytic expression for the stochastic heating rate in low-beta plasmas is derived and tested against simulations of test particles interacting with a spectrum of randomly phased AWs and KAWs. This expression is then used in conjunction with an observationally constrained model of solar-wind turbulence to obtain ion temperature profiles, which agree well with observations from the Ultraviolet Coronagraph Spectrometer.   

A Weakened Cascade Model for Solar Wind Turbulence

A refined cascade model for kinetic turbulence in weakly collisional astrophysical plasmas is presented that includes both the transition between weak and strong turbulence and the effect of nonlocal interactions on the nonlinear transfer of energy. The model describes the transition between weak and strong MHD turbulence and the complementary transition from strong kinetic Alfven wave (KAW) turbulence to weak dissipating KAW turbulence, a new regime of weak turbulence in which the effect of shearing by large scale motions and continued kinetic dissipation play an important role. The inclusion of the effect of nonlocal motions on the nonlinear energy cascade rate in the dissipation range, specifically the shearing by large-scale motions, is proposed to explain the nearly power-law energy spectra observed in the dissipation range of both kinetic numerical simulations and solar wind observations.   

Directional alignments, inhomogeneous heating, and non-Gaussian statistics in solar wind turbulence

The directional alignment of the magnetic and velocity field fluctuations are studied using 33 intervals of ACE data, each 10 hours in duration. The local distributions vary substantially: most intervals have a dominant anti-sunward propagating component, some have an almost flat angular distribution, and very few have a dominant sunward propagating component. These observations are consistent with the localization of directional alignment found in 2D incompressible MHD simulations. In both the local and global cases, the alignment cannot be explained as a superposition of uncorrelated Gaussian distributions, but could be associated with small scale coherent structures. In an independent study, a novel method that relates to intermittency is used in the solar wind to identify small scale coherent structures. These are shown to be associated with enhancements in the electron heat flux, electron temperature, and ion temperature. This is consistent with the presence of inhomogeneous heating in MHD inertial range turbulence. Since the coherent structures represent current sheets that form between flux tubes, the sites of observed localized heating are also candidates for magnetic reconnection. Both the above features of solar wind turbulence can be related to non-Gaussian statistics and therefore may be associated with intermittency.   

The Spaghetti Model of the Turbulent Solar Wind: Implications for the Scaling of Anisotropic Magnetic Fluctuations and Transport

There has been a steady accumulation of observational evidence that the solar wind may be thought of as spaghetti: a network of individual magnetic flux tubes each with its own magnetic and plasma characteristics. As early as 1963, Parker referred to these tubes as magnetic and plasma ``filaments,'' and the picture has undergone several refinements since then [Bartley et al. 1966, Marliani et al. 1973, Tu and Marsch 1990, Bruno et al. 2001], culminating in the recent work of Borovsky [2008] who has suggested that these are fossil structures that originate at the solar surface. We use the weakly compressible MHD turbulence model [Bhattacharjee et al., 1998], which incorporates the effect of background spatial inhomogeneities, to describe such structures. We revisit the model equations, showing their relation to recent work by Hunana and Zank [2010]. For a model of interchange-instability driven turbulence, we then use the 1998 model equations to make predictions for the beta scaling of the anisotropic magnetic fluctuation spectra (the so-called variance anisotropy) observed by ACE, and show that the predictions bracket the observations well. We also predict the scaling of the anisotropic transport coefficients for particles and thermal energy.   

On the generation of coherent structures and nonGaussianity in magnetohydrodynamic turbulence

Numerical simulations of magnetohydrodynamics are used to investigate the production of small scale coherent structures, a feature that is usually associated with enhanced dissipation and the phenomenon of intermittency, and the associated emergence of non-Gaussian statistics. By comparing ideal simulations with well-resolved dissipative simulations with identical initial conditions, non-Gaussianity and characteristic coherent structures are found to initiate almost identically in the two systems. The results suggests that generation of coherent structures and breaking of self-similarity are essentially ideal processes, with dissipation acting only to limit growth of the smallest scale structures. The generation of nonGaussian statistics appears to be related to local rapid relaxation in spatial patches, and the sharp coherent structures that form between them as boundaries. This has important implications for understanding the origin of intermittency in turbulence, as well as to the study of well-resolvedness in spectral method simulations.   

High resolution numerical simulations of steady state imbalanced MHD turbulence

Incompressible MHD equations conserve both energy and cross helicity, which together undergo a turbulent cascade from large to small scales. In the case of nonzero cross helicity, the turbulence is called \emph{imbalanced}. Recent solar wind observations and numerical simulations reveal that at every scale, MHD turbulence consists of regions of positive and negative cross helicity, regardless of the total amount of cross helicity in the system, indicating that MHD turbulence is inherently imbalanced. In this talk, we present results from numerical simulations performed using two different spectral codes that solve MHD and Reduced MHD equations in the steady state over hundreds of Alfv\'en times, massively running on tens of thousands of processor cores and reaching resolutions of up to $2048^2\times 512$. The results from the simulations support the idea that the inertial range scaling of the energy spectra of fluctuations moving in opposite directions along to the background magnetic field is independent of the amount of cross helicity, and it is broadly consistent with phenomenological models based on dynamic alignment that predict a $k_\perp^{-3/2}$ scaling.   

Strong MHD Turbulence

Imbalanced turbulence, a general case of MHD turbulence, is common in nature, it is found in solar wind, which contains perturbations mostly propagating away from the Sun, or in jets where perturbations propagate away from the central object. I will argue that numerics is an efficient tool to constrain theories and present high-resolution direct numerical simulations of MHD turbulence. The shape of the bottleneck effect indicates that MHD energy cascade is less local than hydro cascade. Based on dissipation rates observed in numerics I will argue that there is a smooth transition to the Goldreich-Sridhar model in the limit of small imbalances. It seems that so-called ``dynamic alignment'' saturates and, based on this and the observation of diffuse locality I will argue that numerics support -5/3 spectral slope of strong MHD turbulence, rather than a shallower -3/2 slope. The anisotropy of MHD turbulence is a key to cascading. The subdominant component have stronger anisotropy than the dominant component, which is opposite to what GS critical balance would have predicted. I will explain how to deal with cascading in the imbalanced case.   

Detailed Fit of ``Critical Balance'' Theory to Solar Wind Turbulence

Having first shown in Horbury, et al. (2008) that ``critical balance'' as proposed by Goldreich and Sridar (1995) is consistent with the tendency of the spectral exponent of the power in solar wind fluctuations to approach -2 when the local mean magnetic field was in the radial direction, we now explore the comparison to ``critical balance'' in detail. We have derived the exact power spectrum reduced from the critical balance spectrum in k-space, for any frequency and magnetic angle in the solar wind in terms of certain dimensionless variables combining frequency and angle. Then we fit this to wavelet-derived spectra of magnetic field fluctuations from the Ulysses spacecraft at high latitudes in 1995 solar minimum. The fit is very good. However, in the critical balance theory (even as modified for imbalanced turbulence by Lithwick, et al. 2007) the fit requires a rather small outer scale which implies, according to the same theory, unreasonably large dissipation rates in the solar wind. Until this inconsistency with the theory is resolved, the possibility remains that the f$^{-2}$ spectrum at small angles to the local mean magnetic field has some other origin in the solar wind.   

Dual cascade of kinetic and magnetic energy in MHD turbulence

Using scale-locality of the energy cascade in MHD turbulence, which we have rigorously shown to hold in [1], we prove that the kinetic and magnetic energy budgets statistically decouple beyond a ``conversion'' range. Over the ensuing part of the inertial range, mean kinetic and magnetic energies cascade to smaller scales independently. In other words, we show that magnetic-line stretching acts as a \emph{large-scale} forcing of the magnetic field which vanishes, on average, deep in the inertial range. We present numerical support from $1024^3$ direct numerical simulations. We draw analogies to such a phenomenon from compressible turbulence and geophysical flows and discuss its possible implications on MHD turbulence simulations.\\[4pt] [1] Aluie, H., Eyink, G. L., Phys. Rev. Lett., 104, 8, 081101 (2010)