DIRECTIONAL ALIGNMENT AND NON-GAUSSIAN STATISTICS IN SOLAR WIND TURBULENCE

We analyze the entire 64 s resolution magnetic and velocity field data sets from the Magnetic Field Experiment (MAG; Smith et al. 1998) and Solar Wind Electron, Proton, and Alpha Monitor (SWEPAM) (McComas et al. 1998) instruments on board the Advanced Composition Explorer (ACE) spacecraft. A natural ensemble is obtained by dividing the data into 10 hr intervals. This duration is long enough to contain several correlation times, but short enough to avoid large-scale inhomogeneities. In order to maintain statistical stationarity, intervals are removed from the ensemble if they contain heliospheric current sheet crossings or transient events such as shocks. Intervals are also discarded if the amount of missing or bad data exceeds 5%, since this would reduce the statistical robustness of any computed quantities. The remaining intervals number 156 and contain around 85,000 data points and constitute our solar wind data set for the present study. Each of the data intervals is sector rectified such that positive values of normalized cross helicity and cos θ correspond to outward (anti-sunward) "propagating" Alfvénic fluctuations. This convention is also extended to the sign associated with values of sin θ.

We also carried out numerical simulation of incompressible three-dimensional (3D) MHD turbulence for which the following equations were solved:

$$\begin{eqnarray} \frac{ \partial \mathbf {v}}{\partial t}+\mathbf {v}\cdot \nabla \mathbf {v} & = & {-}\nabla p^{*} + \mathbf {b} \cdot \nabla \mathbf {b} + \mathbf {B}_0 \cdot \nabla \mathbf {b} + \nu \nabla ^2 \mathbf {v} \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \frac{ \partial \mathbf {b}}{\partial t} & = & \nabla \times \left(\mathbf {v} \times \mathbf {b} \right) + \mathbf {B}_0 \cdot \nabla \mathbf {v} + \eta \nabla ^2 \mathbf {b} \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} \nabla \cdot \mathbf {v} &=& 0 = \nabla \cdot \mathbf {b,} \end{eqnarray} \tag{ 5 }$$

where $\mathbf {v}$ is the velocity field, $\mathbf {b}$ is the fluctuating magnetic field, and p* is the total (fluid + magnetic) pressure determined by the incompressibility condition. The kinematic viscosity and magnetic diffusivity are ν and η, respectively. The simulation has unit magnetic Prandtl number (i.e., ν = η).

The fluctuations evolve in the presence of a uniform and static external magnetic field that is taken to be in the z-direction, i.e., $\mathbf {B}_0 = B_0 \hat{\mathbf {z}}$. The non-dimensionality is such that, for the chosen initial conditions, B²₀ is the ratio of the energy density associated with $\mathbf {B}_0$ to that associated with the initial magnetic fluctuations ($\mathbf {b}$ at t = 0).

The magnetic helicity, $H_{m} = \langle \mathbf {a} \cdot \mathbf {b} \rangle / 2$, where $\mathbf {b} = \nabla \times \mathbf {a}$, in the simulation run is approximately zero at all times. However, the run is initially set up with some finite cross helicity, $H_c = \langle \mathbf {v} \cdot \mathbf {b} \rangle /2$. A convenient dimensionless measure is the normalized cross helicity, σ_c ≡ 2H_c/E, where the energy $E = E_{v} + E_{b}=\langle \left|\mathbf {v}\right|^2 + \left|\mathbf {b}\right|^2\rangle /2$. Here, the angle brackets indicate spatial averaging.

Initial conditions for the simulation run are generated in Fourier space. The amplitudes of $\mathbf {v}(\mathbf {k})$ are chosen so that the modal kinetic energy is described by

$$\begin{equation} E^v(k) \equiv \sum _{ \lbrace \mathbf {k}; |\mathbf {k}| =k \rbrace } \frac{1}{2} \left| \mathbf {v}(\mathbf {k})\right|^2 = \frac{C}{1+(k/k_{0})^q}, \end{equation} \tag{ 6 }$$

where C is a normalization constant. In order to determine the phase of each $\mathbf {v} (\mathbf {k})$, the real and imaginary parts are assigned using independent Gaussian random variables. Usually only a subset of the retained Fourier modes is populated initially, that is, only those lying between limiting values, e.g., k_L ⩽ k ⩽ k_H. Initial conditions for $\mathbf {b}(\mathbf {k})$ are chosen in a similar way. However, controlling the degree of correlation between $\mathbf {v}$ and $\mathbf {b}$ allows the cross helicity to be specified. The total energy E in the simulation is initially unity and is equipartitioned between E_v and E_b at all scales.

In Table 1 the parameters for the simulation run are summarized.

Table 1. Parameters Representing the MHD Simulation

Grid	ν, η	B0	$\frac{k_{{\rm max}}}{k_{{\rm diss}}}$	σc(t = 0)	σc(t = 2)
5123	0.001	1.0	1.6	0.23	0.30

Notes. The initially excited Fourier modes have wavenumbers k ∈ [1, 5] and use k₀ = 3 in the spectral shape function. k_max is the maximum retained dynamical wavenumber and k_diss is the dissipation wavenumber (reciprocal of the Kolmogorov scale).

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The 156 intervals that compose our entire solar wind data set are combined in order to analyze their global ensemble characteristics. A normalized cross helicity of 0.29 is obtained, which suggests a dominance of outward propagating fluctuations. This is consistent with expectations and suggests that our data set contains a representative sample of the ecliptic solar wind at 1 AU. Figure 1 shows (solid line) the PDF of cos θ which corresponds to the global behavior of solar wind turbulence. It shows probability enhancements associated with the alignment and anti-alignment of magnetic and velocity field fluctuations. While there is a preference for outward propagating Alfvénic fluctuations, there is a weak minority component of inward propagating fluctuations. This asymmetry is a consequence of the non-zero net cross helicity of solar wind turbulence at 1 AU. For comparison, Figure 1 also shows the PDF (dashed line) of cos θ from the entire spatial domain of the 3D MHD simulation. The normalized cross helicity associated with the simulation is around 0.3, which is comparable to the solar wind case. Indeed, both PDFs behave in a similar manner and are characterized by strong enhancements associated with the alignment of magnetic and velocity field fluctuations. Hence, the PDFs in Figure 1 are consistent with local dynamic alignment and MHD turbulence relaxation toward global evolution of Alfvénic states (Dobrowolny et al. 1980; Pouquet et al. 1986; Ting et al. 1986; Stribling & Matthaeus 1991; Matthaeus et al. 2008; Servidio et al. 2008).

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Figure 2 shows PDFs of sin θ from the solar wind data set (solid line) and MHD simulation (dashed line), further characterizing the global behavior of magnetofluid turbulence. In both cases, the PDFs show significant probability increases linked to induced electric field fluctuations, which are perpendicular to both the magnetic and velocity field fluctuations.

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The global properties of the induced electric field are also examined directly. For the solar wind data set, the induced electric field vector is determined in Radial Tangential Normal (RTN) coordinates. Figure 3(a) shows the PDF of the radial component. The exponential character of the electric field distribution is consistent with previous studies (Breech et al. 2003; Sorriso-Valvo et al. 2004). The induced electric field vector is also computed from the MHD simulation, and Figure 3(b) shows the PDF of the z component e_z. Since the Reynolds number in the solar wind is orders of magnitude greater than that achieved in the MHD simulation, events with large induced electric field values are more likely. Induced electric field strength can be used as a measure of the strength of turbulent couplings in a magnetofluid such as the solar wind. A simple estimate is its standard deviation, equal to the root-mean-square (rms) fluctuation:

$$\begin{equation} \delta e = \langle \left(\mathbf {v} \times \mathbf {b}-\langle \mathbf {v} \times \mathbf {b} \rangle \right)^{2}\rangle ^{1/2}. \end{equation} \tag{ 7 }$$

A heuristic measure of the fluctuating electric field strength can be made by normalizing δe by the rms magnetic and velocity field fluctuations, respectively, δb and δv. Using the entire solar wind data set, δe is about 95% of the product δvδb. This implies a strong induced electric field. Since the electric field helps drive the MHD turbulence cascade, our results indicate the presence of active turbulence dynamics in the solar wind despite the appearance of Alfvénic correlations. Note that the presence of an active cascade in the solar wind has been confirmed by computation of third-order statistics (see Smith et al. 2009; Marino et al. 2011, and references therein), employing the extension of the Kolmogorov–Yaglom law to MHD (Politano & Pouquet 1998).

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Here we examine the cross helicity content derived from smaller samples of solar wind and simulation data. In order to facilitate ease of comparison across these data sets, it is desirable that the 3D simulation emulate the one-dimensional (1D) spacecraft data. Therefore, the magnetic and velocity fields are sampled along a linear trajectory through the MHD simulation domain (Greco et al. 2008). The computation is simplified by aligning the trajectories along a Cartesian axis. Each 1D path through the simulation extends to about 12 correlation lengths (integral scales). This is chosen to match the 10 hr intervals of solar wind data which, assuming an average speed of 400 km s⁻¹ and a correlation length of 1.2 × 10⁶, correspond to around the same spatial scale.

Each 10 hr interval within our solar wind data set is analyzed for evidence of local directional alignment. The left-hand column of Figure 4 shows PDFs of cos θ for three intervals which correspond to localized spatial patches of solar wind turbulence. These particular intervals were selected to illustrate the breadth of σ_c variability within our data set. The solar wind interval associated with the top panel has a normalized cross helicity of 0.75, and consists almost solely of outward-directed Alfvénic alignments and partial alignments. This is reflected in the PDF which has a substantial probability increase linked to alignment of the magnetic and velocity field fluctuations, but has no population of anti-aligned fluctuations. Despite the presence of these strong Alfvénic correlations, the induced electric field associated with this data interval represents about 54% of δvδb, indicating a significant and active turbulence cascade. The middle panel represents a solar wind interval with a normalized cross helicity of 0.17, which contains a mixture of both inward and outward fluctuations. Consequently, the PDF shows probability enhancements associated with both alignment and anti-alignment of magnetic and velocity field fluctuations. An induced electric field is also present and constitutes about 65% of the product δvδb, which suggests the turbulence dynamics are strong. In contrast to the top panel, the bottom panel is linked to a solar wind interval with a normalized cross helicity of −0.73, which consists almost entirely of inward-directed Alfvénic correlations. The PDF has a significant probability increase connected to the anti-alignment of the magnetic and velocity fluctuations, but has no population of aligned fluctuations. However, the presence of an induced electric field within this data interval, which constitutes 49% of the product δvδb, implies a strongly driven turbulence cascade.

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The right-hand column of Figure 4 shows PDFs of cos θ for a selection of linear trajectory samples through the 3D MHD simulation. In order to compare with the results in the left-hand column, the simulation data that best matched the corresponding solar wind σ_c values were chosen. Figure 4 illustrates the good agreement between the simulation data PDFs and their solar wind counterparts. These cos θ distributions are consistent with directional alignment, the emergence of local Alfvénic states as a consequence of rapid turbulence relaxation.

While the PDFs shown in Figure 4 are computed from particular data intervals, they are typical of the variability in both the solar wind and MHD simulation data set. These results are consistent with the idea that MHD turbulence in general, and solar wind turbulence in particular, display a strong tendency to form localized spatial patches of directionally aligned Alfvénic fluctuations (Matthaeus et al. 2008). However, it is interesting to note that a strong induced electric field is maintained (as in Figure 3) despite the local evolution toward Alfvénic correlations. The strength of this electric field does weaken slightly with increasing cross helicity, although it remains significant. Since the electric field contributes to driving the MHD turbulence cascade, the appearance of partial Alfvénic states does not indicate a lack of turbulence dynamics. This is a feature that would not be expected when using a linear wave theory description. Therefore, drawing conclusions regarding the presence or importance of wave activity on the basis of $\mathbf {v}$–$\mathbf {b}$ correlation is problematic.

Phase randomization is a technique used in the study of turbulence (Servidio et al. 2008). Here it is used in order to demonstrate that the observed spatial patches of Alfvénic alignment are associated with phase coherency. Fourier coefficients of the dynamical variables $\mathbf {v}$ and $\mathbf {b}$ are used to form new functions $\mathbf {v}_{{\rm random}}$ and $\mathbf {b}_{{\rm random}}$ that have the same power spectrum as the originals but with random phases. In addition, we require that $\mathbf {v}_{{\rm random}}$ and $\mathbf {b}_{{\rm random}}$ have the same total cross helicity as $\mathbf {v}$ and $\mathbf {b}$. The result is that the new fields $\mathbf {v}_{{\rm random}}$ and $\mathbf {b}_{{\rm random}}$ lack coherency, a property that originates from the nonlinear nature of MHD and is encoded in the phases of the expansion of Fourier coefficients. Hence, this procedure is expected to destroy the coherent spatial patches of locally large positive and negative cross helicities, which correspond to significant alignment and anti-alignment of magnetic and velocity field fluctuations.

Figure 5 shows PDFs of cos θ from the phase-randomized fields (dashed line) and the original fields (solid line). The PDFs of both have a bias toward alignment of magnetic and velocity field fluctuations, which is consistent with their equal values of positive cross helicity. However, the PDF corresponding to the original fields has a greater probability associated with highly aligned and anti-aligned fluctuations. Figure 5 illustrates the non-Gaussian nature of the observed cellularization of MHD turbulence since our results are inconsistent with a superposition of Gaussian fields. Indeed, the requirement of phase coherency implies that there could be a link between local relaxation processes and turbulence intermittency.

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