Kinetic-scale Current Sheets in the Solar Wind at 1 au: Scale-dependent Properties and Critical Current Density

The understanding of turbulence dissipation and plasma heating in a weakly collisionless plasma is of fundamental importance for numerous astrophysical systems (e.g., Matthaeus & Velli 2011). Numerical simulations have shown that turbulence dissipation should be spatially intermittent with substantial plasma heating localized in and around coherent structures, such as current sheets (CSs), which occupy a relatively small volume (Karimabadi et al. 2013; Zhdankin et al. 2013, 2014; Wan et al. 2014, 2016). Numerical simulations have also shown that magnetic reconnection in kinetic-scale CSs produced by turbulence not only results in plasma heating, but also fundamentally affects development of the turbulence cascade at sub-proton scales (e.g., Servidio et al. 2015; Cerri & Califano 2017; Franci et al. 2017; Papini et al. 2019). Modern simulations are still incapable of entirely reproducing the complex dynamics of realistic three-dimensional plasma turbulence, and substantial effort has been directed toward comparing simulation results with observations in the solar wind, a weakly collisional plasma most accessible for in-situ measurements (e.g., Matteini et al. 2020).

Spacecraft measurements have shown that solar-wind heating should continuously occur within a few tens of solar radii of the Sun, as well as further out in the heliosphere, and dissipation of turbulent magnetic field fluctuations should be the dominant solar-wind heating mechanism (e.g., Kohl et al. 1996; Cranmer et al. 2009; Hellinger et al. 2013). The solar-wind observations revealed turbulence to be dominated by Alfénic fluctuations highly oblique (k_⊥ ≫ k_∣∣) to local mean magnetic field, Kolmogorov-like spectrum ${E}_{{k}_{\perp }}\propto {k}_{\perp }^{-5/3}$ at scales larger than proton kinetic scales, and a steeper spectrum ${E}_{{k}_{\perp }}\propto {k}_{\perp }^{-2.8}$ at scales smaller than proton kinetic scales (e.g., review by Chen 2016). Numerical simulations have successfully reproduced these properties of the solar-wind turbulence (e.g., Boldyrev & Perez 2012; Cerri et al. 2017; Franci et al. 2017, 2018; Papini et al. 2019). The crucial unresolved problem is the origin of coherent structures and, specifically, CSs observed in the solar wind, and the very ability of these structures to heat solar wind plasma (e.g., review by Matthaeus et al. 2015). In this Letter we present a statistical analysis of proton kinetic-scale CSs in the solar wind at 1 au, contributing to understanding the origin and dissipation of coherent structures in plasma turbulence.

The presence of CSs on a wide range of temporal scales was established by early spacecraft measurements in the solar wind (Burlaga et al. 1977; Tsurutani & Smith 1979; Lepping & Behannon 1986; Söding et al. 2001). These measurements showed that the magnetic field has more or less constant magnitude, but rotates across a CS through some shear angle. The typical CS thickness was around ten proton inertial lengths, while the occurrence rate at 1 au was a few tens per day. Multispacecraft analysis has shown that the magnetic field component normal to a CS surface is much smaller (if present at all) than local magnetic field magnitude (Horbury et al. 2001; Knetter et al. 2004). The majority of CS studies were limited by magnetic field measurements of relatively low resolution (a few seconds at best) and typically by CSs with shear angles ≳30° (e.g., Söding et al. 2001; Artemyev et al. 2018, 2019). Devoid of these shortcomings, the analysis by Vasquez et al. (2007) included more than 6000 CSs collected at 1 au using magnetic field measurements at 1/3 s resolution. The typical thickness of the CSs was around a few proton inertial lengths, and the occurrence rate was about a few hundred CSs per day. Thus, the higher temporal resolution allowed resolving thinner CSs, which turned out to be much more abundant than larger-scale CSs reported in the early studies. Based on the log-normal distribution of waiting times of the CSs, Vasquez et al. (2007) suggested they are produced by local turbulence, in accordance with earlier theoretical hypothesis (e.g., Matthaeus & Lamkin 1986).

This hypothesis was further supported by observations of similar distributions of magnetic field rotations and waiting times of coherent structures in solar-wind and MHD turbulence simulations (Greco et al. 2008, 2009; Zhdankin et al. 2012). One of the alternatives is that magnetic field rotations through angles ≳30° are boundaries between flux tubes originating at the Sun, while smaller rotations are produced by local turbulence (e.g., Bruno et al. 2001; Borovsky 2008). However, this interpretation can hardly explain the universal log-normal distribution of magnetic field rotations at various temporal scales (Zhdankin et al. 2012; Chen et al. 2015). The ability of coherent structures in heating of solar-wind plasma was questioned by Borovsky & Denton (2011); but there is currently a growing evidence that plasma heating does occur around coherent structures in plasma turbulence (Osman et al. 2011, 2012; Chasapis et al. 2015, 2018).

In this Letter we present the most extensive data set of proton kinetic-scale CSs collected at 1 au using magnetic field measurements at 1/11 s resolution. Due to the higher resolution, CSs with thickness from a few to ten times smaller than in the previous studies could be resolved. We reveal distinct scale dependencies of the current density and shear angle, as well as the critical current density that is not exceeded statistically. The results indicate that proton kinetic-scale CSs in the solar wind are indeed produced by turbulence cascade, and advance understanding of turbulence dissipation.

We use measurements of the Wind spacecraft, located at the L1 Lagrangian point about 200 Earth radii from the Earth (Wilson et al. 2021). We use continuous magnetic field measurements at 1/11 s resolution provided by the Magnetic Field Instrument (Lepping et al. 1995), and proton densities and flow velocities at 3 s cadence provided by the Wind 3DP instrument (Lin et al. 1995). Note that ion densities coincide within a few tens of percent with electron densities provided at about 9 s cadence by the Solar Wind Experiment instrument (Ogilvie et al. 1995). We consider a period of 124 days, from 2010 October 1 to 2011 February 2, except for the first 16 hr on 2010 December 7 when magnetic field measurements at 1/11 s resolution were not available.

CSs were selected using the partial variance increment (PVI) method (e.g., Greco et al. 2018). We computed a PVI index, $\mathrm{PVI}={\left({\sum }_{\alpha }{\rm{\Delta }}{B}_{\alpha }^{2}(t,\tau )/{\sigma }_{\alpha }^{2}\right)}^{1/2}$, where ΔB_α (t, τ) = B_α (t + τ) −B_α (t) are magnetic field increments of three magnetic field components (α = X, Y, Z) and σ_α are standard deviations of ΔB_α (t, τ) computed over 2 hr intervals, that is, over a few outer correlation scales of the solar-wind turbulence (Matthaeus et al. 2005). Coherent structures at various temporal scales correspond to non-Gaussian fluctuations with, for example, PVI > 5. We used only a PVI index computed at τ = 1/11 s to focus on the thinnest resolvable coherent structures. This methodology is essentially equivalent to that of Podesta (2017), who used current density estimates J ∝ ∣Δ B (t, τ)∣/τ with τ = 1/11 s to identify the most intense currents in the solar wind. Analysis of synthetic magnetic field signals with spectra typical of the solar wind showed that J ∝ ∣Δ B (t, τ)∣/τ with τ = 1/11 s provides reasonable current density estimates (Podesta 2017). Note that the instrument noise does not affect the current density estimates and selection of coherent structures in our study, because the magnetometer noise level at frequencies of 0–10 Hz is less than 0.006 nT (Lepping et al. 1995), which is at least six times smaller than the standard deviation, $\sigma ={\left({\sum }_{\alpha }{\sigma }_{\alpha }^{2}\right)}^{1/2}$, characterizing magnetic field increments Δ B (t, τ) at τ = 1/11 s in our interval.

There are different types of coherent structures among the non-Gaussian fluctuations (e.g., Perrone et al. 2016). To select CSs, each of about 2 · 10⁵ continuous clusters of points with PVI > 5 was considered over intervals of ±1 s, ±2 s, ±3 s, and ±4 s around its center and unit vector ${\boldsymbol{x}}^{\prime}$ along the direction of the magnetic field component with the largest variation was computed for each interval using the maximum variance analysis method (e.g., Sonnerup & Scheible 1998). We visually inspected all profiles of ${\boldsymbol{B}}\cdot {\boldsymbol{x}}^{\prime}$ and selected clusters of points with ${\boldsymbol{B}}\cdot {\boldsymbol{x}}^{\prime}$ reversing the sign within at least one of the intervals. We then manually adjusted the boundaries, that is, the regions to the left and to the right of the ${\boldsymbol{B}}\cdot {\boldsymbol{x}}^{\prime}$ reversal, so that each boundary had a duration of at least half a second and excluded events with substantial relative variations of the magnetic field at the boundaries. We also excluded about 10% of events with bifurcated magnetic field profiles (e.g., Gosling & Szabo 2008), to focus on CSs with relatively smooth magnetic field rotation.

The final data set includes 17,043 CSs, ⁵ which is the most extensive data set of proton kinetic-scale CSs at 1 au. Since the CSs were selected using only a PVI index with τ = 1/11 s, the distributions of CS parameters presented in the next section are biased toward the thinnest resolvable CSs, but this bias does not affect the key conclusions of this study (Section 4). Each CS will be considered in a local coordinate system xyz most suitable for describing local CS structure (e.g., Knetter et al. 2004; Gosling & Phan 2013; Phan et al. 2020): unit vector z is along the CS normal determined by the cross-product of magnetic fields at the CS boundaries; unit vector x is along ${\boldsymbol{x}}^{\prime} -{\boldsymbol{z}}\cdot ({\boldsymbol{x}}^{\prime} \cdot {\boldsymbol{z}});$ unit vector y completes the right-handed coordinate system, y = z × x . Note that vectors x and y determine the local CS surface, while vector z is directed across the CS surface.

Figure 1 presents a CS observed on 2011 February 1. Panel (a) presents the magnetic field magnitude and three components in the Geocentric Solar Ecliptic coordinate system. The magnetic field rotates across the CS through shear angle Δθ ≈ 60°, while the magnetic field magnitude remains more or less constant. Note that the shear angle is the angle between the magnetic field vectors at the CS boundaries. We characterize the magnetic field magnitude by 〈B〉, that is, the mean value of magnetic field magnitudes at the CS boundaries, and ${\rm{\Delta }}{B}_{\max }$, that is, the difference of the maximum and minimum values of the magnetic field magnitude within the CS. For the considered CS we have 〈B〉 ≈ 10 nT and ${\rm{\Delta }}{B}_{\max }\approx 1.4$ nT. Panel (b) presents three magnetic field components in the local CS coordinate system xyz . The B_x component varies from about 5 to −5 nT across the CS; B_y has similar values at the CS boundaries and a bit larger value around the B_x reversal. The normal component B_z is around zero at the CS boundaries, in accordance with definition of the CS normal, and remains small within the CS. We characterize the CS asymmetry by 〈B_x 〉, that is, the mean of the B_x values at the CS boundaries, and the CS amplitude by the absolute value of their difference, denoted as ΔB_x . The considered CS is rather symmetric with 〈B_x 〉 ≈− 0.04 nT and ΔB_x ≈ 10.5 nT. The CS's central region, highlighted in panel (b), corresponds to ∣B_x − 〈B_x 〉∣ < 0.2 ΔB_x .

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Panel (c) presents current densities J_x and J_y , estimated as follows:

$$\begin{eqnarray}&&{J}_{y}=-\displaystyle \frac{1}{{\mu }_{0}{V}_{n}}\displaystyle \frac{{{dB}}_{x}}{{dt}},\,\,\,{J}_{x}=\displaystyle \frac{1}{{\mu }_{0}{V}_{n}}\displaystyle \frac{{{dB}}_{y}}{{dt}},\end{eqnarray} \tag{ 1 }$$

where μ₀ is the vacuum permeability and V_n is the normal component of the proton flow velocity at the moment closest to the CS. In this estimate we took into account that solar-wind CSs are locally planar structures (e.g., Knetter et al. 2004; Artemyev et al. 2019), so that only the normal component of the proton flow velocity matters, and the Taylor frozen-in hypothesis is valid at spatial scales larger than electron kinetic scales (e.g., Chasapis et al. 2017). The spatial coordinate along the normal, z = −V_n t, with t = 0 corresponding to B_x = 〈B_x 〉, is shown in Figure 1. We estimate the CS thickness as follows:

$$\begin{eqnarray}&&\lambda =\displaystyle \frac{{\rm{\Delta }}{B}_{x}}{2{\mu }_{0}\langle {J}_{y}\rangle },\end{eqnarray} \tag{ 2 }$$

where 〈J_y 〉 is the absolute value of the current density J_y averaged over the CS's central region, ∣B_x − 〈B_x 〉∣ < 0.2 ΔB_x , highlighted in Figure 1. For the considered CS we have λ ≈ 80 km, or about 1.5λ_p , where λ_p is the proton inertial length. We refer to λ as the CS thickness, but underline that, strictly speaking, λ is a half-thickness, because according to Equation (2) the magnetic field and current density profiles can be approximated as ${B}_{x}\approx \langle {B}_{x}\rangle +0.5{\rm{\Delta }}{B}_{x}\tanh (z/\lambda )$ and ${J}_{y}\approx \langle {J}_{y}\rangle {{\rm{sech}} }^{2}(z/\lambda )$.

Panel (d) presents current densities parallel and perpendicular to the local magnetic field, computed as J_∣∣ =(J_x B_x + J_y B_y )/B and J_⊥ = (J_y B_x − J_x B_y )/B. Since the perpendicular current density is statistically much smaller than the parallel current density (Section 4), we quantify the CS intensity by J₀, that is, the absolute value of the parallel current density J_∣∣ averaged over the CS's central region, and J_peak, that is, the absolute peak value of the parallel current density J_∣∣ within the CS. For the considered CS we have J₀ ≈ 52 nA m⁻² and J_peak ≈ 68 nA m⁻². We compare the CS intensities to local Alfvén current density, J_A , defined as the current density corresponding to the drift between protons and electrons of local Alfén speed V_A :

$$\begin{eqnarray}&&{J}_{A}={{eN}}_{p}{V}_{A},\,\,\,{V}_{A}=\displaystyle \frac{\langle B\rangle }{{\left({\mu }_{0}{N}_{p}{m}_{p}\right)}^{1/2}},\end{eqnarray} \tag{ 3 }$$

where e and m_p are proton charge and mass, and N_p is the closest measurement of proton density. For the considered CS, the proton density was around 21 cm⁻³ and we obtain J₀ ≈ 0.3 J_A and J_peak ≈ 0.4 J_A . In the next section we present results of the statistical analysis performed using the described methodology.

Figure 2 presents statistical distributions of various CS parameters. Panel (a) shows that the magnetic field magnitude across the CSs is almost constant: ${\rm{\Delta }}{B}_{\max }$ is less than 0.15〈B〉 for more than 95% of the CSs, that is, in accordance with the previous studies of larger-scale CSs (e.g., Burlaga et al. 1977; Vasquez et al. 2007; Artemyev et al. 2019). The CSs are typically asymmetric, that is, the values of B_x at the CS boundaries are different: only about 50% of the CSs satisfy 〈B_x 〉 ≲ 0.1 ΔB_x . Panel (b) shows that the current densities J₀ are within 15 nA m⁻² for more than 95% of the CSs. The most probable value of the current densities J_peak is around 5 nA m⁻², that is, in accordance with current density estimates in the solar wind by Podesta (2017). Panel (c) shows that the thickness of the CSs is in a range between about 20 and 1000 km, with the most probable value around 100 km. Note that resolving the CSs with thickness of a few tens of kilometers was possible due to a relatively small normal component, V_n , of the proton flow velocity for those CSs.

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Figure 3 shows that the CSs are proton kinetic-scale structures with some fraction on sub-proton scales. Panel (a) demonstrates that the CS thickness λ is statistically larger in solar-wind plasma with larger proton inertial length λ_p . A similar trend is observed between the CS thickness and thermal proton gyroradius ${\rho }_{p}={\lambda }_{p}{\beta }_{p}^{1/2}$, because the proton beta β_p is in a range between 0.4 and 2 for more than 80% of the CSs in our data set as well as in the solar wind in general (e.g., Wilson et al. 2018). Panel (b) presents a statistical distribution of λ/λ_p and shows that the thickness of the CSs is in a range between about 0.1 and 10λ_p , with the most probable value around λ_p . A similar distribution of λ/ρ_p shows that the most probable value of the CS thickness is around ρ_p . Thus, the typical thickness of the CSs is from a few to ten times smaller than of those in previous statistical studies (e.g., Burlaga et al. 1977; Vasquez et al. 2007; Artemyev et al. 2019). The resolution of these kinetic-scale CSs became possible due to the higher temporal resolution of magnetic field measurements aboard the Wind spacecraft.

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Figure 4 presents analysis of the CS current density and its dependence on the CS thickness. Panel (a) shows a scatter plot of J₀ versus λ. We sorted the CSs into bins corresponding to different spatial scales and computed the median current density value within each bin. Panel (a) shows the median current density profile along with error bars corresponding to the 15th and 85th percentiles of the current density within each bin. The number of CSs within each bin is shown at the bottom of panel (a). The median profile shows that the CSs with smaller thickness tend to be more intense. Least-squares fitting of all the scattered data by a power-law function reveals the following best fit:

$$\begin{eqnarray}&&{J}_{0}=6\,\mathrm{nA}\,{{\rm{m}}}^{-2}\cdot {\left(\displaystyle \frac{\lambda }{100\,\mathrm{km}}\right)}^{-0.56},\end{eqnarray} \tag{ 4 }$$

which is shown in panel (a). Statistical analysis based on the bootstrap method showed that the multiplier (that is, 6 nA m⁻² factor) is uncertain by less than 20%, while the power-law index is between −0.64 and −0.48 at 98% confidence level. Panel (b) presents a comparison between the peak current densities J_peak and local Alfvén current densities J_A . First, the CSs tend to be more intense in solar-wind plasma with larger Alfvén current density. Second, the peak current densities are statistically below local Alfvén current density, J_peak < J_A , for more than 99% of the CSs, and J_peak < J_A /2 for 97% of the CSs.

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Panel (c) shows there is a positive correlation between local Alfvén speed V_A and J_peak/eN_p that is the peak value of the drift velocity between ions and electrons. Thus, the positive correlation between J_peak and J_A in panel (b) is not a trivial effect of plasma density variation. The physics behind the correlation of the two seemingly unrelated quantities in panel (c) will be discussed in the next section. Trends and correlations similar to those in panels (b) and (c) are also observed for the averaged current densities J₀. It is noteworthy there is not any correlation between J_peak/eN_p or J₀/eN_p and local ion-acoustic speed ${({T}_{e}/{m}_{p})}^{1/2}$, where T_e is the local electron temperature. The ion-acoustic speed in our data set is in a range from 40 to 60 km/s for more than 95% of the CSs, while the drift velocities J_peak/eN_p vary over almost two orders of magnitude.

Figure 5 reveals remarkable scale dependencies of the normalized intensity, normalized amplitude, and shear angle of the CSs. To demonstrate the scale dependence of a specific quantity, we sorted the CSs into bins corresponding to different spatial scales, and computed the median as well as 15th and 85th percentile values of the quantity within each bin. The number of the CSs within each bin is shown in the bottom panels. Panel (a) presents the scatter plot of J₀/J_A versus λ/λ_p . Similarly to the trend given by Equation (4) in physical units, the median profile shows that the CSs with smaller normalized thickness have larger normalized current densities. Least-squares fitting of all the scattered data by a power-law function reveals the following best fit:

$$\begin{eqnarray}&&{J}_{0}/{J}_{A}=0.17\,\cdot {\left(\lambda /{\lambda }_{p}\right)}^{-0.51},\end{eqnarray} \tag{ 5 }$$

which is shown in panel (a) and rather well describes the median profile. Analysis based on the bootstrap method showed that the multiplier is uncertain by about 10%, while the power-law index is between −0.6 and −0.4 at 98% confidence level. Panel (b) presents the scatter plot of ΔB_x /〈B〉 versus λ/λ_p . The median profile reveals a clear scale dependence: the CSs with larger normalized thickness have larger normalized amplitudes. Least-squares fitting of all the scattered data by a power-law function reveals the following best fit:

$$\begin{eqnarray}&&{\rm{\Delta }}{B}_{x}/\langle B\rangle =0.33\,\cdot {\left(\lambda /{\lambda }_{p}\right)}^{0.49},\end{eqnarray} \tag{ 6 }$$

which again well describes the median profile. Note that this scaling relation could be foreseen based on Equation (5), because J₀ ≈ ΔB_x /2μ₀ λ and J_A = eN_p V_A = 〈B〉/μ₀ λ_p . The multiplier in Equation (6) is uncertain by about 10%, while the power-law index is between 0.4 and 0.6 at 98% confidence level. Since in a CS with more or less constant magnetic field magnitude, ΔB_x /〈B〉 is unambiguously related to the shear angle, we expect to observe a positive correlation and similar trend between Δθ and λ/λ_p . Panel (c) confirms the scale dependence of the shear angle with the following best fit of the scattered data:

$$\begin{eqnarray}&&{\rm{\Delta }}\theta =0.33\,\cdot {\left(\lambda /{\lambda }_{p}\right)}^{0.5}\approx 19^\circ \,\cdot {\left(\lambda /{\lambda }_{p}\right)}^{0.5},\end{eqnarray} \tag{ 7 }$$

which well describes the median profile. The uncertainties of the best-fit parameters are identical to those of the best-fit parameters in Equation (6). There is a scale-dependent upper threshold on the shear angles, Δθ ≲ 2 λ/λ_p , for more than 99% of the CSs and Δθ ≲ λ/λ_p for about 97% of the CSs.

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We presented the analysis of proton kinetic-scale CSs in the solar wind based on the most extensive data set collected at 1 au. Certainly, this data set does not include all CSs present in the considered interval. We selected the CSs using a PVI index with a minimum time increment τ = 1/11 s and, therefore, our data set and statistical distributions in Figures 2 and 3 are biased toward the thinnest resolvable CSs. The use of PVI indexes at larger increments would be essentially equivalent to the use of magnetic field measurements at a lower resolution. Therefore, the previous studies based on a lower-resolution magnetic field may demonstrate the results of using larger increments in our procedure. The studies based on magnetic field measurements at a few second resolution resolved CSs with typical thickness around 1000 km (Burlaga et al. 1977; Lepping & Behannon 1986; Söding et al. 2001; Artemyev et al.2019), while the use of magnetic field measurements at 1/3 s resolution allowed resolving CSs with typical thickness around a few hundred kilometers; these CSs turned out to be one order of magnitude more abundant than CSs resolved at a few second resolution (Vasquez et al. 2007). We used magnetic field measurements at 1/11 s resolution and resolved CSs with typical thickness around 100 km and averaged occurrence rate (17,043/124 ≈ 140 CSs/day) comparable with that reported by Vasquez et al. (2007). Thus, the higher resolution of the magnetic field measurements allowed us to collect truly kinetic-scale CSs, which are thinner than those reported in previous studies. Although our data set does not include CSs of all possible spatial scales, the collection of which would require repeating our selection procedure with different increments of τ, this data set allows us to address the structure and origin of kinetic-scale CSs in the solar wind.

4.1. Local Structure of Solar Wind Current Sheets

Since solar wind CSs are locally planar structures (e.g., Knetter et al. 2004; Artemyev et al. 2019), we can describe the local magnetic field of the CSs as follows:

$$\begin{eqnarray}&&{\boldsymbol{B}}=B(z)\sin \theta (z)\,{\boldsymbol{x}}+B(z)\cos \theta (z)\,{\boldsymbol{y}}+{B}_{z}\,{\boldsymbol{z}},\end{eqnarray} \tag{ 8 }$$

where θ(z) and B(z) determine, respectively, the magnetic field rotation and magnetic variation within a CS; the normal component B_z is much smaller than B(z). The shear angle Δθ estimated in our experimental analysis corresponds to the difference of θ values at the CS boundaries. Note that Equation (8) provides the most general description of a CS with nonzero B_y . The specific models, such as a Harris CS with a constant B_y or a force-free Harris CS, often used in reconnection simulations and theoretical studies (e.g., Landi et al. 2015) are special cases of Equation (8) corresponding to particular profiles of B(z) and θ(z).

Simple calculations show that current densities parallel and perpendicular to the local magnetic field, J_∣∣ = (J_x B_x +J_y B_y )/B and J_⊥ = (J_y B_x − J_x B_y )/B, determine, respectively, the magnetic field rotation and magnitude variation within a CS:

$$\begin{eqnarray}&&{J}_{| | }=\displaystyle \frac{B}{{\mu }_{0}}\displaystyle \frac{d\theta }{{dz}},\,\,\,\,\,\,{J}_{\perp }=\displaystyle \frac{1}{{\mu }_{0}}\displaystyle \frac{{dB}}{{dz}}.\end{eqnarray} \tag{ 9 }$$

The amplitudes of the parallel and perpendicular current densities can then be estimated as J_∣∣ ≈ (c/4π)〈B〉Δθ/2λ and ${J}_{\perp }\approx (c/4\pi ){\rm{\Delta }}{B}_{\max }/2\lambda$. First, these estimates show that the current-density limitation J_∣∣ ≲ J_A (Figure 4(b)) is equivalent to the scale-dependent upper threshold on shear angles, Δθ ≲ 2λ/λ_p (Figure 5(c)). Second, they show that the relatively small variations of the magnetic field magnitude within the CSs (Figure 2(a)) imply that the current density is dominated by the parallel component. The ratio ${J}_{\perp }/{J}_{| | }\approx {\rm{\Delta }}{B}_{\max }/\langle B\rangle {\rm{\Delta }}\theta$ is less than 0.3 for more than 95% of the CSs (not shown). The small variations of the magnetic field magnitude also imply that the plasma pressure variation across the CSs is much smaller than the magnetic field pressure. The pressure balance $p+{B}^{2}/8\pi =\mathrm{const}$ shows that the maximum variation of the plasma pressure across the CSs is ${\rm{\Delta }}{p}_{\max }\,\approx \,\langle B\rangle {\rm{\Delta }}{B}_{\max }/4\pi$, while its ratio to the typical magnetic field pressure is ${\rm{\Delta }}p/{p}_{B}\approx 2{\rm{\Delta }}{B}_{\max }/\langle B\rangle$. According to Figure 2(a), this ratio does not exceed 0.3 for more than 95% of the CSs. The dominance of the parallel current density and relatively small variations of the plasma pressure show that the kinetic-scale CSs in the solar wind are statistically more or less force-free.

4.2. Current Sheets and Turbulence

The turbulence in the solar wind is dominated by magnetic field fluctuations highly oblique to the mean magnetic field, with a power-law spectrum ${E}_{{k}_{\perp }}\propto {k}_{\perp }^{-\nu }$, where ν ≈ 5/3 in the inertial range, k_⊥ λ_p ≲ 1, and ν ≈ 2.8 at k_⊥ λ_p ≳ 1 (see, e.g., review by Chen 2016). The rms amplitude, δ b_λ, of turbulent fluctuations at spatial scale λ can be then estimated as $\delta {b}_{\lambda }\propto {({E}_{{k}_{\perp }}{\rm{\Delta }}{k}_{\perp })}^{1/2}\propto {\lambda }^{(\nu -1)/2}$, where we take into account that Δk_⊥ ∝ k_⊥ ∝ λ⁻¹. The corresponding current density is j_λ ∝ δ b_λ /λ ∝ λ^(ν−3)/2. In the inertial range, k_⊥ λ_p ≲ 1, we expect

$$\begin{eqnarray}&&\delta {b}_{\lambda }\propto {\lambda }^{1/3},\,\,\,{j}_{\lambda }\propto {\lambda }^{-2/3},\end{eqnarray} \tag{ 10 }$$

while at k_⊥ λ_p ≳ 1 we expect

$$\begin{eqnarray}&&\delta {b}_{\lambda }\propto {\lambda }^{0.9},\,\,\,{j}_{\lambda }\propto {\lambda }^{-0.1}.\end{eqnarray} \tag{ 11 }$$

Thus, the rms amplitudes of the turbulent magnetic field and current density fluctuations should be scale dependent (e.g., Schekochihin et al. 2009; Boldyrev & Perez 2012). Since the magnetic shear angle, δ θ_λ , is proportional to δ b_λ , it should be similarly scale-dependent.

We have found that amplitudes, current densities, and shear angles of proton kinetic-scale CSs are scale dependent in a fashion similar to that of turbulent fluctuations. Moreover, the CS amplitude scales with the CS thickness as ΔB_x /〈B〉 ∝ (λ/λ)^0.49, so that the power-law index is between 1/3 and 0.9, which are expected for turbulent fluctuations at scales above and below proton kinetic scales. Similarly, the CS intensity scales with the CS thickness as ${J}_{0}/{J}_{A}\propto {(\lambda /{\lambda }_{p})}^{-0.51}$ with the power-law index between −2/3 and −0.1. These scale dependencies are strong indications that proton kinetic-scale CSs are produced locally by plasma turbulence. This conclusion is not affected by the uncertainties of the power-law indexes characterizing the scale-dependent properties of the CSs (Section 3).

The critical question is whether the scale dependencies revealed in Figure 5 would be affected if we included all kinetic-scale CSs present in the considered interval, rather than used the data set biased toward the thinnest resolvable CSs. The strong indication that the revealed scale dependencies would not be affected is that they are consistent with the scale-dependent properties of magnetic field rotations in the solar wind (Zhdankin et al. 2012; Chen et al. 2015). These studies showed that statistical distributions of angles α(τ) between magnetic fields B (t) and B (t + τ) in the solar wind behave similarly at various temporal scales. More precisely, α(τ) normalized to its mean value 〈α〉 is described by a universal log-normal distribution independent of τ. In turn, the mean value scales with τ as 〈α〉 ∝ τ^0.46, so that magnetic field rotation angles are larger at larger spatial scales (see also Perri et al. 2012). This is qualitatively, and even quantitatively, consistent with the scale dependence of magnetic shear angles across the CSs, ${\rm{\Delta }}\theta \propto {(\lambda /{\lambda }_{p})}^{0.5}$, revealed in our analysis. Thus, we believe that inclusion of kinetic-scale CSs, which could be selected using PVI indexes at larger increments, would not affect our conclusion that kinetic-scale CSs are produced by turbulence cascade.

4.3. Current Density Limitation Mechanisms

4.4. Our Data Set and Assumptions of the Single-spacecraft Analysis

The results of this Letter can be summarized as follows:

• 1.  
  The proton kinetic-scale CSs in the solar wind are statistically force-free and typically asymmetric.
• 2.  
  The current density within the CSs is scale-dependent, with CSs of smaller thickness being more intense. The magnetic field amplitude and magnetic shear angle are scale-dependent, as well.
• 3.  
  The drift velocity between electrons and ions in the CSs tends to be larger in the solar wind with larger Alfvén speed. The current density does not statistically exceed local Alfvén current density.

Based on these observations, we argue that proton kinetic-scale CSs in the solar wind are produced locally by turbulence, and some mechanism, either CS instability or scattering of electrons by ambient turbulence, should keep the current density below the local Alfvén current density.

The work of I.V. was supported by the Russian Science Foundation, grant No. 21-12-00416. The work of T.P. was supported by a NASA Living With a Star grant #80NSSC20K1781. The work of A.A. was supported by a NASA Living With a Star grant #80NSSC20K1788. The work of K.A. was supported by the National Science Foundation grant No. 2026680. I.V. thanks Ajay Lotekar and Rachel Wang for discussions. The data used in the analysis are publicly available at https://cdaweb.gsfc.nasa.gov/pub/data/wind/. A list of the current sheets considered in this study is publicly available at https://zenodo.org/record/5885741.