# The Radial Variation of the Solar Wind Turbulence Spectra near the Kinetic Break Scale from Parker Solar Probe Measurements, by Lotz et al

The Parker Solar Probe (PSP) has enabled unprecedented opportunity to study the plasma and magnetic field properties of the solar wind from near the Earth (~1 au radial distance r) to very close to the sun (r < 0.1 au). We are interested in the turbulence characteristics of the solar wind and how it evolves from close to the sun to far away. This information is crucial to the understanding of the origin and evolution of the solar wind, turbulence and particle transport, in order to produce better models of solar activity.

Magnetic field fluctuations in the solar wind are commonly observed to follow a power-law spectrum. The inertial range is created by an energy-conserving spectral cascade where interactions between fluctuations can be described by fluid dynamics. Inertial range index is observed to be close to both -5/3 (Kolmogorov), as well as the Iroshnikov–Kraichnan value of -3/2.

This range is followed by a break and the dissipation range — a steepening in the magnetic field power spectrum, where the MHD description breaks down and kinetic effects start playing a role. The dissipation range spectral index is dependent on the type of turbulent fluctuation present and can vary significantly, ranging between -1 and -4.

In this work we determine the spectrum near the break wavenumber and spectral indices using high-cadence PSP magnetic field measurements taken during its  fifth solar encounter between ∼0.1 and ∼0.7 au (2020/05/07 – 2020/06/19), in an extension of the study of Duan et al. (2020). We compare the estimated quantities with their theoretical estimates  to determine which is the dominant turbulence dissipation process in the inner heliosphere. The break wavenumber is then quantitatively compared to previous studies and the radial dependence across the widest range of heliocentric distances yet measured is calculated.

Analysis

The analysis presented in this work is dependent on magnetic field vector ($B_{RTN}$) observations from the FIELDS-MAG instrument (Bale et al. 2016), and solar wind particle density, thermal speed, and temperature measurements from the Solar Probe ANalyzers (SPANs) electrostatic analyzers.

The break frequency is estimated from three-axis magnetic field observations taken by the MAG fluxgate magnetometer (Bale et al. 2016). The 44 day period is divided in to 128 second intervals. After intervals with low quality data was discarded the break frequency is estimated by finding the frequency where the average dissipation and inertial range fits to the log-spaced magnetic field power spectral density cross each other. A novel method calculating the uncertainty of the $f_b$ estimate was developed (see Section 2 of the paper) and $f_b$ where the uncertainty was deemed undesirable were discarded. We find clear radial dependence of the spectral break, closely resembling that found by Duan et al. (2020) for the cruise phase of the second PSP orbit.

The wave number $k_b = 2\pi f_b / Vsw$ was calculated and compared to previous studies that covered mostly higher radial distances. Figure 1 shows the log-log depiction of the mean (red) and median (blue) $k_b$ taken over 10 equal width radial bins. Comparisons with results reported by Bruno and Trenchi (2014) and Smith, et al. (2012) show that the power-law dependence observed by Bruno and Trenchi (2014) holds, albeit with a steeper slope: our fit slope is 1.18, steeper than the 1.08 reported by Bruno and Trenchi (2014)).

Figure 1: The estimated break scale wavenumber  vs. radial distance. Red markers indicate  mean $k_b$ values and the blue markers the medians. The vertical error bars indicate the error derived from the break frequency estimates and horizontal error bars indicate the radial range covered by each bin. Comparison is made with Bruno & Trenchi (2014) for radial distances 0.42–5.3 au (green squares), as well as the average value for this quantity reported at 1 au by Smith et al. (2012; black triangle), where the error bar indicates the standard deviation of the Smith et al. (2012) measurements.

A comparison of the break frequencies calculated here with frequencies corresponding to the proton gyroradius, ion inertial length, and the cyclotron resonance scale, all computed using in situ PSP observations of the various plasma quantities these scales depend on, found that these break frequencies correspond most closely with those corresponding to the cyclotron resonance scale.  This gives a benchmark against which the results of various turbulence transport models can be tested, as well as a valuable input for solar energetic particle and cosmic-ray transport models.

Arranging the dissipation and inertial range indices calculated throughout this orbit by radial distance shows that the inertial index remains relatively constant with radial distance, in contrast to previous studies. The dissipation range index decreases with increasing radial distance.

However, the radial decrease in the dissipation range index is not uniform and corresponds to a marked increase in the observed solar wind speed between about 0.45 and 0.6 au. The implication is that this may be a reflection of the behavior of this quantity at smaller radial distances due to the fact that the solar wind is “younger” here, ie. earlier in the evolution of solar wind turbulence.

To investigate this, we plot the inertial and dissipation range indices versus the age of the solar wind $\tau = r/Vsw$ in Figure 2. The inertial range spectral indices behave in a relatively uniform manner as a function of solar wind age. It varies between approximately -1.65 and -1.45, consistent with either the Kolmogorov (-5/3) or Iroshnikov–Kraichnan (-3/2) values.

Figure 2:  The inertial (top) and dissipation range (bottom) power-law indices as a function of solar wind age.

Future work aims to extend the current analysis: First, by considering longer data intervals, so as to include a portion of the energy-containing range of the turbulence power spectrum, thereby allowing for the calculation of the inertial range outer scale. Secondly, the analysis will be extended to other PSP perihelia, taking into account various additional factors, such as solar wind speed and plasma-$β$, that are known to influence the dissipation range spectral break frequency. Furthermore, future measurements with the MeerKAT radio telescope and the Square Kilometer Array are also planned to get information about solar wind density fluctuations at very small scales inside the Alfvén radius, and thus close the gap between the Sun and $\sim 10 R_{\odot}$ .

Based on a recent paper by S. Lotz et al 2023 ApJ 942 93. DOI: 10.3847/1538-4357/aca903 ; Preprint available at: https://arxiv.org/abs/2212.02441

References

Bale, S. D., Goetz, K., Harvey, P. R., et al. 2016, SSRv, 204, 49

Bruno, R., Trenchi, L., & Telloni, D. 2014, ApJL, 793, L15

Duan, D., Bowen, T. A., Chen, C. H. K., et al. 2020, ApJS, 246, 55

Smith, C. W., Vasquez, B. J., & Hollweg, J. V. 2012, ApJ, 745, 8