SPECTRAL INDICES FOR MULTI-DIMENSIONAL INTERPLANETARY TURBULENCE AT 1 AU

Goldreich & Sridhar (1995) adopt a particular form of the RMHD spectrum. They assume that perpendicular spectral transfer is governed by triple correlations that decay at the nonlinear timescale τ_nl(k). Under this assumption of a high degree of anistropy of the type discussed by Shebalin et al. (1983) this timescale depends only on the perpendicular wavenumber k_⊥ and parallel transfer is fully suppressed. Building these assumptions into an EDQNMA closure calculation, Goldreich & Sridhar (1995) write a steady model spectral density as

$$\begin{equation} E(k_\perp,k_\parallel) = \frac{aV_A^2}{k_\perp ^{10/3} L^{1/3}} f\left(\frac{k_\parallel L^{1/3}}{k_\perp ^{2/3}}\right), \end{equation} \tag{ 2 }$$

where a is a dimensionless number. The fluctuations are assumed small compared to the mean field, δB/B₀ ≪ 1, and the energy-containing length scale is L. The dimensionless function f(ξ) has the property that ∫^+∞_−∞f(ξ) dξ = 1. In keeping with the critical balance separation of timescales, the reduced spectrum along any line with k_∥ = 0 has an infinite Alfvén time and is thus assumed to have the Kolmogorov k^−5/3_⊥ form resulting from τ_nl dynamics. At any k with non-zero k_∥, however, the three-dimensional spectrum is assumed to scale as some function of τ_nl/τ_A ∝ k_∥L^1/3/k^2/3_⊥. The three-dimensional spectrum Equation (2) combines these two assumptions in a general form whose implications can be described and tested, which we now proceed to do.

Integration over the parallel wavenumber k_z yields a "Kolmogorov" spectrum in perpendicular wavenumber

$$\begin{eqnarray} E(k_\perp) &=& \int _{-\infty }^{+\infty } dk_z E(k_\perp,k_\parallel) \nonumber \\ &=& a \int _{-\infty }^{+\infty } \frac{V_A^2}{k_\perp ^3}\frac{1}{(k_\perp L)^{1/3}} f\left(\frac{k_\parallel L^{1/3}}{k_\perp ^{2/3}}\right) \nonumber \\ &=& a \frac{V_A^2}{k_\perp ^3} \frac{1}{(k_\perp L)^{1/3}} \frac{k_\perp ^{2/3}}{L^{1/3}} \int _{-\infty }^{+\infty } dq f(q) \nonumber \\ &=& a V_A^2 L^2 \frac{1}{(k_\perp L)^{8/3}}. \end{eqnarray} \tag{ 3 }$$

The two-dimensional model spectrum has dimensions EL² and one obtains the two-dimensional omnidirectional spectrum from it by integrating over angle, i.e., multiplying by 2πk_⊥ as axisymmetry is assumed. Therefore ${\cal E}(k_\perp) = 2 \pi k_\perp E(k_\perp) = 2 \pi a V_A^2 L (k_\perp L)^{-5/3}$ as expected.

If we integrate the model spectrum over the two k_⊥ directions, we obtain a one-dimensional reduced spectrum that depends on k_z. This is the spectrum that would be observed if the mean magnetic field of the solar wind were radial with the mean super-Alfvénic flow in the same radial direction. The observed (or "reduced") spectrum is

$$\begin{eqnarray} E^r(k_\parallel) &=& 2\pi \int _0^{\infty } dk_\perp k_\perp E({\bf k}_\perp, k_\parallel) \nonumber \\ &=& 2 \pi a V_A^2 L^3 \int _0^\infty dk_\perp k_\perp \frac{1}{(k_\perp L)^{10/3}} f\left(\frac{k_\parallel L}{(k_\perp L)^{2/3}}\right).\nonumber\\ \end{eqnarray} \tag{ 4 }$$

Letting k'_⊥ = k_⊥L and q = k_∥L/k^2/3_⊥ the above integral involves

$$\begin{equation} \frac{3}{2} \frac{1}{k_\parallel L} \int _0^\infty dq \frac{1}{k_\perp ^{2/3}} f(q) = \frac{3}{2} \left(\frac{1}{k_\parallel L}\right)^2 \int _0^\infty dq q f(q). \end{equation} \tag{ 5 }$$

The reduced spectrum is then given by

$$\begin{equation} E^r(k_\parallel) = 3 \pi a^\prime V_A^2 L \left(\frac{1}{k_\parallel L} \right)^2, \end{equation} \tag{ 6 }$$

where a' = a∫^∞₀dqqf(q). Therefore, following the arguments of RMHD and the critical balance assumption, energy is largely absent from quasi-parallel wavevectors due to nonlinear coupling that transports the energy to larger k_⊥. If these arguments apply and under the frozen-in-flow approximation, one expects the observed spectra to approach f⁻² as Θ_BV → 0°.

We now consider the effect of a sharp cutoff in the spectrum at small scales, say at the ion inertial scale wavenumber k_ii = ω_pi/c = Ω_ci/V_A. The observed inertial range of interplanetary magnetic fluctuations is seen to possess a strong spectral break associated with the ion inertial scale, although this normally marks a transition to a steeper power-law form (Leamon et al. 1998; Smith et al. 2001; Hamilton et al. 2008). For simplicity of this demonstration we assume here that E(k_⊥, k_∥) = 0 for k_⊥ > k_ii. This sharp cutoff implies also a cutoff in the reduced spectrum computed above. Using Equation (1) we conclude that the observed spectral density reduced over the two perpendicular directions will cutoff sharply at a parallel wavenumber k_∥ ∼ k^d_∥ where

$$\begin{equation} k^d_\parallel L = C_K \frac{u}{V_A}(k_{ii} L)^{2/3} = C_K \frac{\delta b}{B_0} (\omega _{pi} L/ c)^{2/3}. \end{equation} \tag{ 7 }$$

The second form of the right-hand side emphasizes that the cutoff wavenumber scales as δb/B₀ (assuming approximate equipartition of magnetic and velocity field turbulence energies) and also that the cutoff depends on the ion inertial scale (assuming that the perpendicular dissipation or steepening scale is of this order). Note that the cutoff in k_∥ will be less sharp in all cases in which the kinetic steepening range in the perpendicular direction is less steep than the extreme case considered here. This demonstrates that there should be a change in the spectral slope to steeper than f⁻² at the above parallel wavenumber k^d_∥ when the reduced spectrum is computed along the mean magnetic field.

We note that the above analysis applies to inertial range scales only. We have described the expected cutoff at large k due to dissipation in the Higdon (1984) and Goldreich & Sridhar (1995) theory noting that the observed cutoff k^d_∥ (k^d_⊥) in the parallel (perpendicular) directions is highly anisotropic with k^d_∥ ≪ k^d_⊥. The inertial range also does not extend to k = 0; there is expected to be an energy-containing range at larger scales associated with boundary data or driving that is not yet processed by the in situ turbulent cascade. We assume that the termination of the inertial range at large scales occurs at a wavenumber k ∼ 1/L_C with L_C the correlation scale. Combining these ideas we can create Figure 2 to demonstrate both the computed separatrix between τ_nl- and τ_A-dominated regions of the spectrum. The vertical hatched bars represent the integration necessary to compute the parallel spectrum. Note how this integration spans both regions of quasi-perpendicular wavevectors dominated by τ_nl and quasi-parallel wavevectors where, if the Goldreich & Sridhar (1995) model spectrum correctly and completely describes the turbulence, no additional fluctuation power should exist.

We examine ACE data from day 23 of 1998 through day 56 of 2007. This period spans both solar maximum and solar minimum conditions. We use the merged dataset of thermal ion and magnetic field measurements recorded by the SWEPAM and MAG instruments (McComas et al. 1998; Smith et al. 1998). The data have 64 s resolution because this is the collection rate of the SWEPAM instrument. MAG data are averaged to the same time resolution in a manner that is syncronized to the data collection times of SWEPAM in producing this merged dataset.

The data are recorded in RTN coordinates (R is the radial direction from the Sun to the spacecraft, T is the azimuthal direction coplanar with the Sun's equatorial plane and directed in the sense of rotation, and N is the normal such that R × T = N). The solar wind flows in the +R direction, unless deflected by stream interaction, ejecta, or planetary magnetospheres. Systematic excursions of the wind velocity away from the radial direction are minimal except for the fluctuations that constitute the turbulence. When measurements are recorded within or near the Sun's equatorial plane, as they are here, the N component is directed northward. This has certain advantages as we shall see below.

We take 1 hr intervals of the merged data and perform several tasks: (1) we compute means and uncertainties for each component of the magnetic field and solar wind vector in RTN coordinates; (2) we perform a linear detrend of the data interval in anticipation of obtaining computed variances that are consistent with first-order structure function statistics; (3) we compute variances and uncertainties for each component of the magnetic field and solar wind vector in RTN coordinates; (4) we compute second-order structure functions for each component of the magnetic vector for lags from 1 to 10 points (64 to 640 s), then sum to obtain the trace of the magnetic structure function matrix; (5) we do the same for velocity vectors; and (6) we fit the structure functions for each hour (amplitude and power-law index with uncertainties) assuming a power-law form. This results in ∼74, 000 independent intervals used in this analysis.

The second-order structure function S₂(L) of a variable Z(x) measured at positions x is defined to be

$$\begin{equation} S_2(L) \equiv \langle [Z(x) - Z(x+L)]^2 \rangle, \end{equation} \tag{ 8 }$$

where 〈 ⋅ ⋅ ⋅ 〉 denotes ensemble average. It is common to replace the ensemble average with an average over the data interval and we assume that S₂ is not a function of x. It is then possible to compute the power spectrum of the same time series by transforming S₂ (Smith et al. 1990). However, one can infer directly the spectral index q of the resulting power spectrum (in the sense that the spectrum varies as k^q) if S₂(L) ∼ ALⁿ according to q = −(n + 1). We may therefore fit the computed estimates for S₂ and infer a spectral index together with uncertainty without computing the power spectrum directly.

Figure 3 shows the distribution of computed structure function indices for four measured variables used in this study: the trace of the second-order structure functions for the magnetic vector, the velocity vector, the N component of the magnetic vector, and the N component of the solar wind velocity. We include the statistics for B_N because this component is relatively free of current sheet crossings, which can contaminate the analysis of individual samples. We include the statistics for V_N because variation in wind speed over an hour, as seen in V_R, can dominate the trace analysis for some intervals whereas V_N is largely only a fluctuation due to the turbulence and lacks these large-scale trends. We further subset the data into slow wind (V_SW < 400 km s⁻¹) and fast wind (V_SW > 500 km s⁻¹) in the manner of Dasso et al. (2005) in addition to using all wind conditions. These subsets are based on the mean wind speed for the hour. Several things become readily apparent: first, the most probable structure function index for the magnetic field measurements (both trace and B_N) is centered near 2/3 whereas the most probably index for velocity measurements (also both trace and V_N) is centered near 1/2. In fact, the latter is more nearly 0.4 as we shall see. A structure function index n = 2/3 implies a spectral index q = −5/3 while n = 1/2 is consistent with q = −3/2. Podesta et al. (2006, 2007) report a velocity spectra index that is approximately −3/2 at 1 AU and −3/2 is a prominent prediction in MHD turbulence theory (Kraichnan 1965). Second, the statistics for the N-component fluctuations are in good agreement with the trace values, suggesting that concerns over possible contamination by objects such as current sheet crossings are unwarranted at this level. Third, values of n = 1 implying q = −2 are less common, but not unrepresented in the dataset. Fourth, there is a systematic variation in the fit indices for fast and slow winds in both magnetic and velocity measurements that is beyond the scope of this analysis. Fifth, there are a finite number of events with indices n ∼ 0 implying q ∼ −1, but flatter spectra are not found.

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Before proceeding to the Θ_BR dependence we compute the distribution functions for the standard deviations in the fit indices. These are shown in Figure 4. The key feature is that the uncertainties are generally several times smaller in comparison with the fitted means with most probable values of the uncertainties ⩽0.1. This suggests that fit indices are generally well determined and is a necessary condition to obtaining good statistics in the next stage of the analysis.

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From this point on we compute error-weighted means according to the prescription

$$\begin{equation} \langle n \rangle \equiv \sum _i (n_i / \sigma ^2_i) \bigg/ \sum _i \big(1 / \sigma ^2_i \big), \end{equation} \tag{ 9 }$$

where n_i is an individual fit index and σ_i is the error associated with the fit. We can then compute the variance of the associated distribution

$$\begin{equation} \big\langle \sigma ^2_{\rm stdev} \big\rangle \equiv \sum _i (n_i - \langle n \rangle)^2 \bigg/ \sum _i \big(1/\sigma ^2_i \big) \end{equation} \tag{ 10 }$$

with σ_stdev being the standard deviation. Since there are two sources of error, the individual measurement error and the computed width of the distribution which may arise from many other sources and reflect the natural variation of the parameters, we define the error-of-the-mean σ_err according to

$$\begin{equation} \sigma ^2_{\rm err} \equiv (1/N) \sigma ^2_{\rm stdev} + 1 \bigg/ \sum _i \big(1 / \sigma ^2_i \big), \end{equation} \tag{ 11 }$$

where N is the number of hourly fits. Each Θ_BR bin shown in subsequent plots contains several hundred to several thousand independent estimates of the index, so the computed error-of-the-mean for each binned quantity is considerably smaller than the symbols used in plotting.

Since the widths of the distributions in Figure 3 are greater than the most probable value of the fit uncertainties in Figure 4, we assume there is a natural variation to the distribution of n apart from simple measurement error. This is the motivation for the above definition of σ²_err. We will use standard deviations when plotting uncertainties unless otherwise stated to reflect the broad distribution of values that underly each mean. We do this to demonstrate that spectra with −2 indices are more than one standard deviation off the mean of the distribution, which is an indication that it is relatively rare. If we assume there is a single universal theory governing the distribution of n, then it would be appropriate to compare −2 indices with the error-of-the-mean. As we accumulate estimates the error-of-the-mean becomes progressively smaller and the −2 spectra become an increasing number of uncertainties from the mean, but its occurrence rate in the distribution does not change. Readers are left to ponder which interpretation they favor.

Figure 5 shows the computed average indices according to the above error analysis gathered into 5° bins of the variable Θ_BR. As stated, error bars again represent the computed standard deviation. No statistically significant dependence on Θ_BR is seen in any of the four panels. In each case the mean value of the S₂ index is consistent with the most probable values shown in Figure 3. The computed standard deviations in all Θ_BR bins are comparable to the width of the distribution shown in Figure 3. This means that the predicted structure function index 1, or equivalently the spectral index −2, is consistently more than one standard deviations from the mean for all values of Θ_BR.

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Coverage in each bin varies according to Θ_BR and the quantity in question. For instance, the B_Trace distribution for 40° < Θ_BR < 45° contains 5793 estimates of n_i. The computed 〈n_i〉 is 0.674 while the computed standard deviation is 0.219 and the error of the mean is 0.003. For the bin 0° < Θ_BR < 5° there are 384 estimates of n_i. The computed 〈n_i〉 = 0.702 while the computed standard deviation is 0.22 and the error-of-the-mean is 0.01. If the Goldreich & Sridhar (1995) theory is an occasional truth under carefully prescribed conditions, then its occurrence rate is 1.4 standard deviations from the mean. If it is proposed as a universal theory of MHD turbulence, then its occurrence rate is 30 error estimates from the mean.

It is reported that the wavelet methods of Horbury et al. (2007) better resolve the expected low-amplitude spectra anticipated for the small-Θ_BR spectrum. While this is less of a problem for low-latitude measurements at 1 AU than for the high-latitude Ulysses measurements used by Horbury et al. (2007), since δB/B₀ is smaller in the ACE dataset than for Ulysses measurements, we can address the possibility by limiting our analysis to those intervals with low fluctuation levels so that the instantaneous magnetic field remains within the assigned Θ_BR range. This is also beneficial to comparison with the Goldreich & Sridhar (1995) prediction that is based on the assumption that the mean field is much larger than the fluctuations. To do this, we define Θ_B ≡ arctan(B_⊥,rms/|B₀|) where B_⊥,rms is the root-mean-square variation of the component of the magnetic field perpendicular to the mean field direction. In this way Θ_B becomes a measure of the variability of the field direction and intervals with both small Θ_BR and small Θ_B can be inferred to sample the τ_A-dominated region of the spectrum without contamination from the τ_nl-dominated portion of the spectrum.

Figure 6 shows the resulting mean values of n and q when Θ_B is limited to less than 30°, less than 15°, and less than 5°. To leading order the computed standard deviations do not change with decreasing Θ_B. There is, in fact, a slight decrease in the standard deviations as Θ_B is made small. In the interest of full disclosure, there are only nine intervals for Θ_BR < 10° and Θ_B < 5°. However, their spectra fail to show any trend toward n = 1 or q = −2. The spectral index of magnetic fluctuations continues to be ∼−5/3 and for velocity fluctuations ∼−1.4 for all values of Θ_BR. There is no evidence of spectral indices q = −2 associated with small Θ_BR.

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Hamilton et al. (2008) argue that small-scale fluctuations within the inertial range evolve more rapidly than large-scale fluctuations as a result of the arguments put forward by Kolmogorov (1941) and his assumption that cascade dynamics are local. In an effort to extend these results to smaller spatial scales (higher frequencies) we examine the database used by Smith et al. (2005, 2006a, 2006b) and Hamilton et al. (2005, 2008). This database is built on five years of ACE observations spanning 1998 through 2002 and includes 960 data intervals with 393 intervals from 28 distinct magnetic clouds. Data intervals are from 1 to 3 hr in duration and nonoverlapping. A complete description of the spectral analysis technique and the resulting database parameters is provided in the above references. Unlike the above analysis, we now focus on the fitted results of power spectra and not the inferred spectral characteristics derived from second-order structure functions. The database uses 3 vector s⁻¹ magnetic field data and we focus on frequencies at the high end of the inertial range (typically from 8 mHz to 0.1 Hz). The exact frequency range varies with individual data intervals so as to remain at frequencies smaller than the spectral break and larger than any poorly resolved spectral features that depart significantly from power law. Typically, any such variation in the frequency interval is small.

Figure 7 shows the results of this analysis. The trace of the magnetic power spectral matrix (the spectrum of total power in the magnetic fluctuation) is computed and the spectral index is fitted over the above frequency interval. The indices are averaged in 10° bins of Θ_BR using the same uncertainty-weighted averaging scheme described above. Uncertainties on the plot are again representations of the standard deviation of the underlying distribution. Errors-of-the-means are several to ten times smaller than the standard deviations. As before, the database is subsetted for fast and slow winds of open field conditions and to this we add the above subset of magnetic cloud observations. Results for Θ_BR ⩾ 30° seem perfectly consistent with −5/3 indices seen above. The 20° ⩽ Θ_BR < 30° bin for fast winds and the 0° ⩽ Θ_BR < 10° bin for slow winds may constitute anomalous results due to poor coverage. For Θ_BR < 30° there is some degradation of the results, but no evidence of a trend toward a −2 spectral index. In fact, the cloud set clearly shows a small angle trend toward a flatter than normal spectrum.

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To summarize the analysis using the high time resolution dataset, there is no evidence of a trend toward −2 spectral indices at any value of Θ_BR. Furthermore, this is strong evidence against steepening to spectral indices steeper than −2 that would be due to projection of a perpendicular range cutoff as described by Equation (7).

We should note that despite using 960 data intervals in this analysis, only 23 intervals of open field lines fell in the range Θ_BR < 20° and no intervals of fast wind fell in Θ_BR < 10°. Cloud results were much more likely to exhibit radial mean initial mass function (IMF) conditions. We attempted to increase the number of radial IMF intervals by examining the years 2003–2006 (solar minimum conditions), but found that virtually every interval selected was contaminated by upstream waves from Earth. We do not believe that any existing intervals suffer from this contamination, but most of those additional events selected from solar minimum conditions for possible incorporation into the database were contaminated. We were left with little alternative but to examine data from locations far beyond the Earth–Sun line.

To escape the possible contamination of the analysis in Section 3.2 by upstream waves and to extend the range of validity of our conclusions to include measurements recorded closer to the Sun where the solar wind turbulence is less evolved, we now examine 380 intervals of IMF spectra measured by the Helios 1 spacecraft in the range 0.3 < R_Helios < 1 AU during the years 1974–1976. Intervals are typically several hours in duration and we again employ the same analysis method as used in Section 3.2. We limit our analysis to the frequency range 5 to 20 mHz, which is generally in the middle of the inertial range spectrum. As with the above analysis, every effort was made to cover the broadest possible range of solar wind conditions made available in these years.

Figure 8 shows the results of our analysis. As above, we find no evidence of a consistent shift in the spectral index toward −2 or any steepening of the spectrum associated with nearly radial mean field conditions. Instead, there appears to be a general flattening of the spectrum associated with radial mean field conditions Θ_BR < 40°. To the extent that a greater number of radial events may be found at smaller heliocentric distances, this general trend is reported by Bavassano et al. (1982) where it is shown that the inertial range spectrum of interplanetary magnetic field fluctuations steepens with increasing heliocentric distance from 0.3 to 1 AU. Additionally, our analysis shows that fast wind spectra are generally flatter than slow winds. The distribution of indices as a function of Θ_BR over the range 0.3 < R_Helios < 0.4 AU is statistically equivalent to what is shown in Figure 8. Nowhere in the Helios dataset do we find evidence for −2 spectra in association with radial mean magnetic fields.

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