Ion-scale spectral break of solar wind turbulence at high and low beta

Abstract

The power spectrum of magnetic fluctuations in the solar wind at 1 AU displays a break between two power laws in the range of spacecraft-frame frequencies 0.1 to 1 Hz. These frequencies correspond to spatial scales in the plasma frame near the proton gyroradius ρ_i and proton inertial length d_i. At 1 AU it is difficult to determine which of these is associated with the break, since and the perpendicular ion plasma beta is typically β_⊥i∼1. To address this, several exceptional intervals with β_⊥i≪1 and β_⊥i≫1 were investigated, during which these scales were well separated. It was found that for β_⊥i≪1 the break occurs at d_i and for β_⊥i≫1 at ρ_i, i.e., the larger of the two scales. Possible explanations for these results are discussed, including Alfvén wave dispersion, damping, and current sheets.

1. Introduction

The spectrum of magnetic fluctuations in the solar wind forms a power law over several decades, which is thought to be the inertial range of a turbulent cascade [Alexandrova et al., ²⁰¹³; Bruno and Carbone, ²⁰¹³]. It has long been known that at spacecraft-frame frequencies f_sc∼1 Hz, corresponding to spatial scales in the plasma frame around ion kinetic scales, the spectrum steepens [e.g., Coleman, ¹⁹⁶⁸; Russell, ¹⁹⁷²], although the reason for this remains under debate. At these scales, the turbulent energy is thought to begin to be dissipated, so understanding the spectrum here is crucial to understanding turbulence and heating in collisionless plasmas. In this letter, we present new measurements to investigate the scale associated with the steepening and therefore its cause.

Several ion kinetic scales, related to different physical processes, have been associated with the spectral break, most notably the ion gyroradius ρ_i and ion inertial length d_i (see Appendix A for definitions). For example, Schekochihin et al. [2009] proposed that the break occurs at the transition between Alfvénic turbulence and kinetic Alfvén turbulence, which would happen when the perpendicular scales become comparable to the ion gyroradius, k_⊥ρ_i∼1, under typical solar wind conditions. The framework of incompressible Hall MHD has also been used to explain the break, in which it occurs at k_⊥d_i∼1 [Galtier, ²⁰⁰⁶], and d_i has also been suggested to be relevant as the thickness of current sheets which form in the turbulence [Leamon et al., ²⁰⁰⁰; Dmitruk et al., ²⁰⁰⁴]. Alternative scales have also been suggested: the wave number at which parallel Alfvén waves are cyclotron damped k_c=Ω_i/(v_A+v_th,i) [Leamon et al., ^1998a; Bruno and Trenchi, ²⁰¹⁴], Landau damping of kinetic Alfvén waves at kρ_i∼1 [Leamon et al., ¹⁹⁹⁹], and when 2πf_sc becomes comparable to the ion cyclotron frequency Ω_i [Denskat et al., ¹⁹⁸³; Goldstein et al., ¹⁹⁹⁴].

The difficulty in distinguishing these scales with in situ spacecraft observations at 1 AU is that they typically occur at similar spacecraft-frame frequencies. In particular, because , and typically β_⊥i∼1, ρ_i and d_i are usually not measurably different. Despite this, there have been several attempts to distinguish between the different scales associated with the break. Leamon et al. [1998a] looked at the correlation between the measured break and k_c, found that it was poor and concluded that cyclotron damping was not the cause. Leamon et al. [2000] compared three scales—k_c, d_i, and Ω_i—and, under the assumption of oblique propagation, found the best correlation for d_i. Smith et al. [2001] examined an interval in which β_i dropped significantly and noted that the break in the spectrum moved to lower frequencies, as expected for kd_i∼1, rather than kρ_i∼1. A large statistical study was carried out by Markovskii et al. [2008], in which the measured break was compared to various theoretical scales. A slightly better correlation was obtained for a combination of scale and fluctuation amplitude, although all theoretical scales displayed moderate correlation, making the results somewhat inconclusive.

An alternative approach has been to use the radial variation of the turbulence to investigate the break. Perri et al. [2010] measured the frequency of the break to be independent of distance from the Sun from 0.3 to 4.9 AU, in apparent contradiction to any of the theoretical scales. Bourouaine et al. [2012] also measured no significant change in the break frequency from 0.3 to 0.9 AU but concluded that this could be consistent with a break at k_⊥d_i∼1 if the turbulence is highly anisotropic (k_⊥≫k_∥). Most recently, Bruno and Trenchi [2014], using higher-resolution data within fast streams, instead found the break frequency to decrease almost linearly with distance from 0.42 to 5.3 AU, similar to 1/ρ_i, 1/d_i, and k_c. The value was found to be closer to k_c, so it was concluded that cyclotron damping must be active.

In this letter, we present a different approach, using several intervals of both β_⊥i≪1 and β_⊥i≫1 in which at least two of the relevant scales, ρ_i and d_i, are well separated, so that the difference between them is measurable and therefore physically meaningful. The results are then compared to various theoretical predictions, to investigate the cause of the spectral steepening.

2. Measurements

The solar wind typically has β_⊥i∼1 and only rarely contains periods of β_⊥i≪1 and β_⊥i≫1. To find these exceptional cases, the Wind data set [Acuña et al., ¹⁹⁹⁵] was used, for which 20 years of data are now available. Figure 1 shows the distribution of β_⊥i measured in the free solar wind during the years 1994–2010 (2.16 × 10⁶ data points). The proton densities and temperatures from the SWE instrument [Ogilvie et al., ¹⁹⁹⁵], found by the fitting technique of Maruca and Kasper [2013], were used, along with the magnetic field from the MFI instrument [Lepping et al., ¹⁹⁹⁵] using the calibration of Koval and Szabo [2013]. Cases of β_⊥i<0.1 and β_⊥i#x003E;10 occur rarely, making up 5.3% and 0.46% of the data set, respectively.

Figure 1.

Probability density function (PDF) of log₁₀(β_⊥i) in the solar wind. Intervals used in this letter are from the extreme parts of the distribution with β_⊥i≪1 and β_⊥i≫1 shaded in blue.

The data set was searched for low β_⊥i periods longer than 30 min and high β_⊥i periods longer than 15 min. Thirty-six low β_⊥i and 19 high β_⊥i intervals were selected, which have mean β_⊥i values within the blue shaded regions in Figure 1. The low β_⊥i intervals cover the range of solar wind speeds 300–770 km s⁻¹ and are within interplanetary coronal mass ejections (ICMEs): the majority (31) coincides with events on the ICME list of Jian et al. [2006], and the others also display some of the characteristic signatures. This is expected, since ICMEs make up much of the low β_⊥i solar wind. While the large-scale properties of ICMEs are quite different to the rest of the solar wind, we do not expect the physics at kinetic scales to significantly differ, meaning that these intervals can be used to study the ion break scale of the magnetic spectrum. The high β_⊥i intervals cover a range of speeds 290–510 km s⁻¹ and are due mainly to short periods of low magnetic field strength, which also have a higher than average density. At high β, the solar wind can easily become unstable, and signatures of mirror modes [Winterhalter et al., ¹⁹⁹⁴; Russell et al., ²⁰⁰⁹; Bale et al., ²⁰⁰⁹] are present in some of these intervals.

Figure 2 shows the magnetic power spectra for a low β_⊥i and high β_⊥i interval, calculated from the 92 ms resolution MFI data (with small data gaps linearly interpolated) using the multitaper method [Percival and Walden, ¹⁹⁹³]. The frequencies corresponding to kρ_i=1 and kd_i=1 are marked assuming the Taylor [1938] hypothesis. The individual SWE Faraday cup spectra in these intervals were examined manually to ensure that the automated analysis software correctly determined the ion parameters and a comparison with data from the nearby ACE spacecraft provided additional confirmation. Also shown is the local power law fit α in the range 0.58f_sc to 1.73f_sc. In both cases, α is close to −5/3 for frequencies f_sc<0.3Hz, as is well established in this range [Alexandrova et al., ²⁰¹³; Bruno and Carbone, ²⁰¹³], then becomes smaller before becoming larger once more at the highest frequencies. The flattening of the spectrum at high frequencies is not physical and thought to be due to aliasing of spin tone harmonics [Koval and Szabo, ²⁰¹³]. The steepening at f_sc≈0.5Hz, however, is physical, and is the subject of this letter.

Figure 2.

(a) Magnetic power spectrum and local slope α for β_⊥i=0.010 (19 January 2005 11:00–24:00). (b) Same for β_⊥i=27 (23 September 2001 22:16–22:41). Frequencies corresponding to kρ_i=1 (red dashed line) and kd_i=1 (green dash-dotted line) are marked. In both cases, the break occurs at the larger of the scales.

While the observed break frequency f_b can be found in various ways, here we define it to be the frequency at which α takes a value half way between −5/3 and −8/3. Figure 2 shows that f_b is very close to the frequency corresponding to kd_i=1 in the low β_⊥i example and to kρ_i=1 in the high β_⊥i example. In other words, the break occurs at the larger of the two scales.

To investigate the generality of this finding, spectra were calculated for all of the intervals discussed above. While most display a clear break, some have a more complicated shape or do not significantly steepen, which may be due to instrumental noise and the presence of mirror modes at high β_⊥i. The reason for the lack of a break in several of the low β_⊥i intervals is not known, but it generally occurs when the relative amplitude of fluctuations δB/B₀ is lower than average. The f_b was measurable in 12 of the high β_⊥i spectra and 18 of low β_⊥i spectra, and only these were used in the subsequent analysis. The average plasma parameters for these intervals are given in Table 1 along with their standard deviations; electron temperatures were taken from ground moments of distribution functions measured by the 3DP instrument [Lin et al., ¹⁹⁹⁵].

Table 1.

Mean Interval Parameters

	β⊥i≪1	β⊥i≫1
B (nT)	13.3 ± 4.1	1.14 ± 0.35
ni (cm−3)	3.9 ± 2.9	17.1 ± 7.1
vsw (km s−1)	460 ± 120	389 ± 49
Ti (eV)	1.64 ± 0.86	4.8 ± 2.9
Te (eV)	12.5 ± 4.0	8.6 ± 1.7
fb (Hz)	0.58 ± 0.19	0.223 ± 0.068
(δB/B0)b	0.0229 ± 0.0077	0.349 ± 0.096
sin(θBv)	0.82 ± 0.12	0.68 ± 0.23
βi	0.0122 ± 0.0059	23.2 ± 7.4
βe	0.111 ± 0.054	51 ± 28

Figure 3 shows the β_⊥i dependence of the measured break frequency normalized to the frequencies corresponding to the ion gyroscale and ion inertial length, =v_sw/(2πρ_i) and =v_sw/(2πd_i), where v_sw is the solar wind speed. It is clear that for low β_⊥i the data points cluster around ≈1 but are significantly below ≈1 and at high β_⊥i the reverse is true. It has been pointed out [e.g., Leamon et al.; Bourouaine et al.] that if the turbulence is anisotropic in the sense k_⊥≫k_∥, as it is thought to be at kinetic scales [Chen et al.], then an additional factor of sin(θ_Bv), where θ_Bv is the angle between the mean magnetic field and the solar wind direction, should be included in the definition of and . From Table 1 it can be seen that on average this would make at most a factor of 1.5 difference, so the above result would not be affected. Therefore, it appears generally true that the break from a spectrum of magnetic fluctuations occurs at the larger of the ion gyroradius and ion inertial length. This is the main result of this letter.

Figure 3.

Ratio of measured break frequency f_b to the frequency corresponding to (top) kρ_i=1 and (bottom) kd_i=1, as a function of β_⊥i for all intervals in which a break was measurable.

3. Possible Explanations

3.1. Alfvén Wave Dispersion

The ion-scale spectral break is often attributed to the scale at which the Alfvénic turbulence, thought to exist above the ion scales [Alexandrova et al., ²⁰¹³; Bruno and Carbone, ²⁰¹³], becomes dispersive at the transition to kinetic Alfvén turbulence, thought to exist at smaller scales [Podesta, ²⁰¹³; Chen et al., ^2013a]. While the extent to which linear theory applies to strong turbulence in the solar wind is an open question, the scale at which Alfvén waves become dispersive can be derived from linear kinetic theory at both low and high β_i. It is assumed, based on observations [e.g., Chen et al., ^{2010a, 2010b}], that the turbulence is anisotropic k_⊥≫k_∥ at kinetic scales and each species takes an equilibrium isotropic Maxwellian distribution for simplicity.

For β_i≫1, kρ_i dispersion corrections dominate kd_i corrections, so the latter can be neglected, leading to Alfvén waves with frequency ω ≪ Ω_i and ω ≪ k_∥v_th,i. For k_∥ρ_i≪1 and k_⊥ρ_i≪1, the dispersion relation, with the kρ_i terms retained, becomes

1

Since T_i∼T_e for the β_⊥i≫1 intervals (Table 1) and we are assuming k_⊥≫k_∥, equation 1 indicates that the break should occur at k_⊥ρ_i∼1. This agrees with the measurements in section 2 which show the break at the ion gyroscale at high β_⊥i.

Similarly, the predicted break can be found for β_i≪1. Dispersion corrections related to ρ_i can now be neglected giving k_∥v_th,i≪ω ≪ k_∥v_th,e. This is different to previous studies [e.g., Lysak and Lotko, ¹⁹⁹⁶; Hollweg, ¹⁹⁹⁹], which have assumed ω ≪ Ω_i and have therefore neglected d_i corrections. The dispersion relation can be obtained if it is assumed that . Since β_e≈0.11 in the low β_⊥i intervals (Table 1), this would require wave vector angles θ_kB>71.6°, which is typically satisfied at these scales [Chen et al., ^{2010a, 2010b}]. In this case, the dispersion relation can be shown to reduce to the Alfvén wave for k_⊥d_i≪1 and the kinetic Alfvén wave for k_⊥d_i≫1. The break can be estimated to be where these meet, which occurs at

2

This result holds for T_i≪T_e and corresponds to k_⊥ρ_s∼1if β_e≪1 and k_⊥d_i∼1 if β_e≫1, in agreement with the Hall Reduced MHD dispersion relation [Schekochihin et al., ²⁰⁰⁹]. This scale, however, does not match the measured break in the β_⊥i≪1 intervals: on average, the frequency corresponding to this scale is 4.8 times larger than f_b (it is dominated by ρ_s rather than d_i since β_e≪1). For d_i to be relevant for Alfvén wave dispersion at β_i≪1 would require either β_e≫1, which is not the case (Table 1), or a large k_∥ component to the turbulence, which is not generally observed, either in the free solar wind [Chen et al., ^{2010a, 2010b}] or within ICMEs [Leamon et al., ^1998b].

3.2. Cyclotron Damping

It has been suggested [e.g., Leamon et al., ^1998a, Bruno and Trenchi, ²⁰¹⁴] that the break in the spectrum could be caused by the damping of energy at the ion cyclotron resonance. The condition for this resonance is ω − k·v =± Ω_i, where v is the particle velocity, so assuming parallel Alfvén waves, , the resonance occurs at k_c=Ω_i/(v_A+v_th,i), where the thermal speed has been used for the particle velocity [Leamon et al., ^1998a]. This can also be written as k_c=(d_i+ρ_i)⁻¹, meaning that the break would occur at the larger of ρ_i and d_i, in agreement with the results of section 2. However, since the turbulence is measured to be predominantly anisotropic k_⊥≫k_∥ at these scales [Chen et al., ^{2010a, 2010b}], it is expected to remain low frequency so that cyclotron damping is not efficient at removing energy from the electromagnetic fluctuations [Quataert, ¹⁹⁹⁸; Schekochihin et al., ²⁰⁰⁹]. While such resonances can be broadened in strong turbulence [Quataert, ¹⁹⁹⁸; Lehe et al., ²⁰⁰⁹; Lynn et al., ²⁰¹²], this is only expected to be an order unity effect. If turbulence within ICMEs is significantly less anisotropic, the cyclotron resonance may become significant; however, Leamon et al. [1998b] found it to be more anisotropic at kinetic scales within ICMEs. Therefore, while the cyclotron resonant scale k_c matches the observed break f_b at both low and high β_⊥i, it is not consistent with the generally observed anisotropy of the turbulence.

3.3. Electron Landau Damping

While Alfvén wave dispersion cannot account for the low β_⊥i break and the cyclotron resonance is not thought to be important, could Landau damping explain it? The Landau resonance occurs when the phase speed of a wave matches the particle velocity, so can lead to damping even if the fluctuations are low frequency. Leamon et al. [1999] suggested that electron Landau damping could lead to the observed break at β_i∼1, and Howes et al. [2008] suggested it to be the cause of the break near d_i in the β_i≪1 interval of Smith et al. [2001]. Using the kinetic Alfvén wave electron Landau damping rate derived in Boldyrev et al. [2013], along with the parameters in Table 1 for the β_⊥i≪1 intervals, gives a normalized damping rate γ/ω₀≈−0.066at k_⊥ρ_s=1. The damping rate at k_⊥d_i=1 will be smaller than this, so is unlikely to be the cause of the spectral break in the current data set. Calculating the Landau damping numerically and applying this to a model spectrum [Howes et al., ²⁰¹¹] also shows that the break occurs closer to k_⊥ρ_i=1 than to k_⊥d_i=1 (J. M. TenBarge, private communication, 2014). Therefore, electron Landau damping also appears unlikely to be the cause of the break at kd_i=1 in the β_⊥i≪1 intervals.

3.4. Current Sheets

The break has also been related to the scale of current sheets, which may develop in a turbulent plasma. Leamon et al. [2000] reported a good correlation between the break and d_i, assuming k_⊥≫k_∥, and concluded that a significant fraction of the dissipation occurs in reconnecting current sheets, thought to have thickness d_i. Vasquez et al. [2007] examined the widths of current sheets in the solar wind and found that while there is significant variability, for β_i<0.1 the width scales better with d_i and for β_i>4 it scales better with ρ_i. This would agree with the results of section 2 in which the break was found to occur at the larger of these scales, although a causal relationship is not necessarily implied. On the other hand, simulations and laboratory measurements of reconnection with a large guide field [Cassak et al., ²⁰⁰⁷; Egedal et al., ²⁰⁰⁷], appropriate for the small amplitude turbulent fluctuations at low β_i, show a sudden onset of reconnection when the current sheet thickness becomes ρ_s, where two-fluid effects become important, rather than d_i. As shown in section 3.1, the frequency corresponding to kρ_s=1 is not close to the measured break frequency for β_⊥i≪1.

4. Discussion

We have examined the ion scale break frequency of solar wind turbulence during intervals of β_⊥i≪1 and β_⊥i≫1. While these cases are not typical for the solar wind at 1 AU, they enable the predicted break scales to be measurably different, so that distinguishing between them is physically meaningful. The results of the analysis are summarized in Table 2. The average ratio of the measured break frequency f_b to the theoretical break frequency for each scale f_x is given, which is closest to unity in the β_⊥i≪1 intervals for d_i and in the β_⊥i≫1 intervals for ρ_i. The dispersive scale for low β_i Alfvén waves does not fit the observations and neither does the ion gyrofrequency Ω_i/(2π). While the cyclotron resonant wave number does fit the observations at both high and low β_⊥i within errors, it is not consistent with the observed anisotropy of the turbulence, as discussed in section 3.2. The break is also not at a fixed value of f_sc: the mean break frequency is f_b=0.58Hz in the low β_⊥i intervals and f_b=0.22Hz in the high β_⊥i intervals, a factor of 2.6 different. The linear correlation coefficients r between f_b and f_x, along with their 95% confidence intervals, are also given in Table 2. The uncertainties are large enough that a meaningful distinction based on the correlation coefficients is not possible. Inclusion of the sin(θ_Bv) factor was not found to significantly alter the correlations.

Table 2.

Comparison Between Measured Break Frequency and Theoretical Values

	fb/fx		r
Scale, x	β⊥i≪1	β⊥i≫1	β⊥i≪1	β⊥i≫1
ρi
di
kc
Ωi/(2π)

While the break at ρ_i in the β_⊥i≫1 intervals is consistent with the Alfvén wave dispersion scale, the break at d_i in the β_⊥i≪1 intervals is not consistent with any of the explanations in section 3, assuming that the fluctuations are anisotropic k_⊥>k_∥. This leaves the possibilities that either the turbulence has a significant k_∥ component in the β_⊥i≪1 intervals, or the break is related to a nonlinear process, such as anomalous resistivity, which may manifest in a plasma with β_i≪β_e<1 (S. Boldyrev et al., manuscript in preparation, 2014). The lack of a break in some of the low β_⊥i intervals with small δB/B₀ is consistent with a nonlinear process. Further studies are required to investigate these possibilities.

As well as in the solar wind, identifying the scale associated with the spectral break is important for understanding turbulence and dissipation in other plasma environments. For example, turbulence is thought to heat the interstellar medium (ISM), in which density fluctuations are well measured but magnetic fluctuations are not [e.g., Haverkorn and Spangler, ²⁰¹³]. The spectral break calculated from radio observations has been used to determine which phase of the ISM is responsible for the observed turbulence [Spangler and Gwinn, ¹⁹⁹⁰], so knowing the scale at which the break occurs is important. Spangler and Gwinn [1990] assumed the break to be at the larger of ρ_i and d_i, which is consistent with the results of this letter. Remote measurements of density fluctuations in the solar corona, a low β_i environment, also show a steepening in the spectrum around the ion inertial length [Coles and Harmon, ¹⁹⁸⁹; Harmon, ¹⁹⁸⁹; Harmon and Coles, ²⁰⁰⁵] along with a flattening at slightly larger scales that is consistent with the increased compressibility of kinetic Alfvén turbulence [Harmon, ¹⁹⁸⁹; Hollweg, ¹⁹⁹⁹; Harmon and Coles, ²⁰⁰⁵; Chandran et al., ²⁰⁰⁹; Chen et al., ^{2012, 2013b}]. The upcoming Solar Probe Plus mission will enable further investigation of these features with in situ measurements of the turbulent fields in the corona.

APPENDIX

Appendix A: Definitions

The ion gyroradius is defined as ρ_i=v_th⊥i/Ω_i, where is the perpendicular ion thermal speed, Ω_i=q_iB/m_i is the ion gyrofrequency, T_⊥i is the perpendicular ion temperature, m_i is the ion mass, q_i is the ion charge, and B is the magnetic field strength. The ion inertial length is defined as d_i=c/ω_pi, where is the ion plasma frequency, and n_i is the ion number density. This can also be written d_i=v_A/Ω_i, where is the Alfvén speed and ρis the total mass density. The ratio of ion thermal pressure to magnetic pressure is β_i=n_ik_BT_i/(B²/2μ₀). The sound gyroradius is defined as ρ_s=v_s/Ω_i, where is the ion acoustic speed.

Key Points

• Ion-scale spectral break measured in solar wind with very high and low beta

• Break occurs at the larger of the ion gyroradius and ion inertial length

• Results compared to various theoretical expectations