Local Proton Heating at Magnetic Discontinuities in Alfvénic and Non-Alfvénic Solar Wind

In this work we have used PSP data from Encounter 6 through Encounter 9 (E6–E9), which occurred from 2020 September 9 to 2021 August 15 covering a range of radial distances from the Sun, 0.07 au < R < 0.263 au (we only considered intervals at radial distances less than 50 solar radii, R_s ). We use magnetic field data from the flux-gate magnetometer at 4 samples cycle⁻¹ resolution (FIELDS; Bale et al. 2016). The 3D proton velocity distribution function (L2) and its moments (L3) at 3 s resolution are obtained by the electrostatic analyzer (SPAN-I; Livi et al. 2022), part of the Solar Wind Electrons Alphas and Protons instrument suite (SWEAP; Kasper et al. 2016). Since SPAN-I is partially obstructed by PSP's thermal protection shield, we use the quasi-thermal noise (QTN) measured by the FIELDS Radio Frequency Spectrometer (Moncuquet et al. 2020) to have a more accurate determination of plasma density. The total parallel and perpendicular proton temperatures, T_⊥ and T_∥, are obtained by projecting the temperature tensor in the parallel and perpendicular directions with respect to the magnetic field.

For reference, in Figure 1 we provide an overview of E6. The top four panels show the spacecraft radial distance (R), the proton and electron number density (n_i,e ), and the magnetic ( b and B = ∣ b ∣) and proton bulk velocity ( V ) fields in the instrument frame (x-component pointing toward the Sun). The fifth panel shows the Alfvénic properties of the wind represented by the normalized cross-helicity σ_c and residual energy σ_r of fluctuations given, respectively, by

$$\begin{eqnarray}&&{\sigma }_{c}=\frac{\left\langle \left|{{\bf{z}}}^{+}{|}^{2}\rangle -\langle \right|{{\bf{z}}}^{-}{|}^{2}\right\rangle }{\left\langle \left|{{\bf{z}}}^{+}{|}^{2}\rangle +\langle \right|{{\bf{z}}}^{-}{|}^{2}\right\rangle },\,\text{and}\,{\sigma }_{r}=\frac{2\left\langle \left|{{\bf{z}}}^{+}|\cdot \langle |{{\bf{z}}}^{-}\right|\right\rangle }{\left\langle \left|{{\bf{z}}}^{+}{|}^{2}\rangle +\langle \right|{{\bf{z}}}^{-}{|}^{2}\right\rangle }\end{eqnarray} \tag{ 1 }$$

where brackets 〈...〉 denote the moving average over a time window, and ${{\boldsymbol{z}}}^{\pm }={\boldsymbol{\delta }}{\boldsymbol{V}}\pm {\boldsymbol{\delta }}{\boldsymbol{b}}/\sqrt{{\mu }_{0}\rho }$ are the Elsässer variables defined by the velocity and magnetic field fluctuations with respect to the mean field and by the QTN mass density ρ assuming quasi-neutrality. In this work we fix the averaging window to 2 hr, which corresponds to several times the correlation time of the magnetic field fluctuations in the inner heliosphere (Parashar et al. 2020; Sioulas et al. 2022b). The cross-helicity measures the relative dominance of inward and outward propagating Alfvénic fluctuations ( z ^±), and the residual energy quantifies the partition between the fluctuations' kinetic and magnetic energy. These quantities are not independent and they are geometrically related by the cosine of the angle between the velocity and magnetic field $\cos {\theta }_{\mathrm{vb}}\,=\tfrac{{\boldsymbol{\delta }}{\boldsymbol{V}}\cdot {\boldsymbol{\delta }}{\boldsymbol{b}}}{| {\boldsymbol{\delta }}{\boldsymbol{V}}| | {\boldsymbol{\delta }}{\boldsymbol{b}}| }={\sigma }_{c}/\sqrt{1-{\sigma }_{r}^{2}}$ (Wicks et al. 2013). The bottom panel of Figure 1 shows θ_vb for E6.

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As will be discussed below, fluctuations during E6 are remarkably Alfvénic, with high cross-helicity intervals (defined by ∣σ_c ∣ > 0.75) that last several days (particularly during and after perihelion). In addition, a shorter interval of low Alfvénicity is observed in the vicinity of the heliospheric current sheet (2020 September 25 and 26), a complex structure with dominant magnetic energy and nearly zero cross-helicity. This agrees with the observed non-Alfvénic wind reported near the Heliospheric Current Sheet (HCS) in E4 (Chen et al. 2021).

In Figure 2 we show the statistical analysis of Alfvénic properties and compressibility of solar wind fluctuations from E6 through E9. The top panels show the probability density function (PDF) of σ_c , σ_r , and $\cos {\theta }_{\mathrm{ub}}$ for each Encounter. Although data display a range of values −1 ≤ σ_c ≤ 1, intervals of imbalanced fluctuations with ∣σ_c ∣ > 0.5 can be identified, with peaks of the distribution of σ_c at ∣σ_c ∣ ≥ 0.75 (see also Figure 1). In general, fluctuations are magnetically dominated, a trend which is consistent throughout all Encounters (Chen et al. 2020; Shi et al. 2021), with mean value σ_r ≈ − 0.5 and with only a few intervals being characterized by an excess of kinetic energy. Lastly, the PDF of $\cos {\theta }_{\mathrm{ub}}$ peaks at ±1, corresponding to fluctuations with aligned V and B in all the Encounters. The Alfvénic wind (those intervals with ∣σ_c ∣ ≥ 0.75) present strong alignment, i.e., $| \cos {\theta }_{\mathrm{vb}}| \sim 0.89$, while the non-Alfvénic wind (those intervals with ∣σ_c ∣ ≤ 0.25) present a moderate alignment with a broader distribution of angles, having mean value $| \cos {\theta }_{\mathrm{vb}}| \sim 0.57$. In the latter case, such alignment corresponds with the relaxation and suppression of nonlinearities in standard balanced turbulence that have been previously observed in both simulations and solar wind measurements (Boldyrev 2006; Matthaeus et al. 2008, 2012; Chandran et al. 2015).

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The bottom panels of Figure 2 show the PDF of the amplitude of magnetic field fluctuations,

$$\begin{eqnarray}&&\displaystyle \frac{\delta b}{| \langle {\boldsymbol{b}}\rangle | }:=\sqrt{\displaystyle \frac{\langle {({b}_{x}-\langle {b}_{x}\rangle )}^{2}+{({b}_{y}-\langle {b}_{y}\rangle )}^{2}+{({b}_{z}-\langle {b}_{z}\rangle )}^{2}\rangle }{\langle {b}_{x}{\rangle }^{2}+\langle {b}_{y}{\rangle }^{2}+\langle {b}_{z}{\rangle }^{2}}},\end{eqnarray} \tag{ 2 }$$

and of the ratio of the variances of the magnetic field and magnetic field strength (Villante & Vellante 1982),

$$\begin{eqnarray}&&{C}_{b}:=\sqrt{\displaystyle \frac{\langle {(B-\langle B\rangle )}^{2}\rangle }{\langle {({b}_{x}-\langle {b}_{x}\rangle )}^{2}+{({b}_{y}-\langle {b}_{y}\rangle )}^{2}+{({b}_{z}-\langle {b}_{z}\rangle )}^{2}\rangle }},\end{eqnarray} \tag{ 3 }$$

the latter providing a proxy for magnetic compressibility. In this analysis, we have sorted the solar wind into Alfvénic (intervals with ∣σ_c ∣ > 0.75, corresponding to the blue-shaded areas in Figure 2, top left) and non-Alfvénic wind (intervals with ∣σ_c ∣ < 0.25, corresponding to the yellow-shaded area). For the non-Alfvénic wind we have also removed data corresponding to the HCS crossing, where ∣〈 b 〉∣ ≈ 0 (for the time window considered of 2 hr), to avoid fictitious long tails of the PDF.

From Figure 2, bottom-left panel, it can be observed that the distributions of the normalized fluctuations' amplitude are remarkably different between Alfvénic (blue) and non-Alfvénic (yellow) wind. In the former case, relative fluctuations' amplitudes are bounded roughly by δ b/∣〈 b 〉∣ ≲ 2. This is in line with previous observations of Alfvénic fluctuations by Helios and Ulysses, where saturation of amplitudes (although defined differently) was found, a feature that is consistent with spherically polarized fluctuations (Matteini et al. 2018). In the non-Alfvénic case, instead, relative fluctuations' amplitudes are large and there is no constraint on their value (E9 being the only ambiguous case). We interpret the different PDF of the normalized amplitude as a signature of a higher level of compressibility in the non-Alfvénic wind (at the timescale of 2 hr). The bottom-right panel of Figure 2 shows the distribution of C_b for both types of wind. As expected, C_b in the non-Alfvénic wind is systematically larger than in the Alfvénic wind, with the PDF mean in the non-Alfvénic wind reaching up to five times that of the Alfvénic wind.

To summarize, we have considered the statistical properties of fluctuations over a timescale of 2 hr. From this part of our data analysis, we conclude that fluctuations in the Alfvénic (∣σ_c ∣ > 0.75) and non-Alfvénic (∣σ_c ∣ < 0.25) wind at distances R < 0.25 au display statistical properties similar to those traditionally observed further away from the Sun, namely, almost incompressible and saturated fluctuations in the Alfvénic wind and highly compressible fluctuations in the non-Alfvénic wind.

To study the connection between proton heating and discontinuities or, more generally, intermittent structures in different plasma conditions, we utilized the PVI method by calculating the following quantity (Greco et al. 2008b):

$$\begin{eqnarray}&&\mathrm{PVI}=\displaystyle \frac{| {\rm{\Delta }}{\boldsymbol{B}}(t,\tau )| }{\sqrt{\langle | {\rm{\Delta }}{\boldsymbol{B}}(t,\tau ){| }^{2}\rangle }},\end{eqnarray} \tag{ 4 }$$

where Δ B (t, τ) = b (t + τ) − b (t) is the magnetic field increment vector. The PVI method identifies non-Gaussian features in the magnetic field and events, with values PVI > 3 typically associated with coherent structures representative of intermittent behavior, or discontinuities. Although the PVI defined in Equation (4) captures variations in the magnetic field, it does not distinguish RDs, TDs, or other fluctuations that sometimes display properties of both RDs and TDs (Larosa et al. 2021). For this reason, we also considered the PVI of the magnetic field strength, hereafter called MAG-PVI, by taking the variation of B, i.e., by substituting ΔB(t, τ) ⇒ ΔB(t, τ) = B(t + τ) − B(t) in Equation (4). A comparison of events identified with PVI and MAG-PVI allows us to gain insights on whether compressible or incompressible magnetic structures are statistically associated to local particle energization.

We performed the PVI analysis by using different time lags, τ = 12 s, 3 s, 1 s, to find a compromise between the resolution of the SPAN-i instrument and proton scales. By comparison, the proton gyroperiod has a mean value of τ_i ≈ 0.1 s over the range of radial distances considered (the gyroperiod is τ_i ≈ 0.15 s at a distance of R ≈ 50 R_s, and it decreases down to about τ_i ≈ 0.025 at R ≈ 20 R_s ). By applying the Taylor hypothesis, the lags τ = 12−1 s correspond, on average, to a spatial lag ℓ/d_i ≈ 360–36 and ℓ/ρ_i ≈ 750–75 in units of proton inertial length and gyroradius, respectively.

After constructing the time series of PVI (and MAG-PVI), we computed the normalized PDF of various functions of temperature f(T) conditioned on a given range of PVI (and MAG-PVI) values

$$\begin{eqnarray}&&{ \mathcal P }{ \mathcal D }{ \mathcal F }(f(T)| {\theta }_{1}\leqslant \mathrm{PVI}\leqslant {\theta }_{2}).\end{eqnarray} \tag{ 5 }$$

We have calculated the conditional PDF of (i) the total proton temperature (f(T) = T) at a given point in the PVI time series, (ii) the variation of proton temperature over the same scale that PVI has been calculated over (f(T) = T(t + τ) − T(t) ≡ δ T, with τ the time lag of the PVI), (iii) the parallel and perpendicular temperature (f(T) = T_∥, T_⊥ where ∥ and ⊥ are defined with respect to the local magnetic field direction $\hat{{\boldsymbol{b}}}={\boldsymbol{b}}/B$), and (iv) the temperature anisotropy (f(T) = T_⊥/T_∥). We performed the same PVI analysis with τ = 12 s, τ = 3 s, and τ = 1 s, in the latter case by resampling the PVI data to match the SPAN-i resolution. Results vary within the uncertainty of the temperature measurements (usually about 10%) and, therefore, we only show results with τ = 12 s.

In agreement with other studies, we find that the highest values of T and δ T are associated with the highest PVI values in both winds (not shown here; see also, e.g., Osman et al. 2010; Qudsi et al. 2020; Sioulas et al. 2022b), supporting the idea that heating occurs at localized structures. Results of our PVI analysis are reported in Figure 3 for parallel and perpendicular temperatures.

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Figure 3 shows the conditional PDF with respect to PVI (top panels) and MAG-PVI (bottom panels) of T_∥ (left), T_⊥ (middle), and T_⊥/T_∥ (right). The PDFs have been obtained by combining data from all Encounters (E6–E9) and by using the cross-helicity to sort Alfvénic (∣σ_c ∣ > 0.75, solid lines) and non-Alfvénic (∣σ_c ∣ < 0.25, dashed lines) intervals. Red, blue, and black colors correspond to different ranges of PVI (MAG-PVI), which we split into 0 < PVI ≤ 1.5, 1.5 < PVI ≤ 3.0, and PVI > 3.0, respectively. The legend in each panel reports the mean value of each distribution. Finally, E6 was also analyzed individually, but we found the same trends as those shown in Figure 3 and discussed below.

As can be seen from the PDFs of T_⊥/T_∥ (Figure 3, right panels), the non-Alfvénic wind is more anisotropic than Alfvénic wind and, on average, ${({T}_{\perp }/{T}_{\parallel })}_{\mathrm{nonAlfv}}\lt {({T}_{\perp }/{T}_{\parallel })}_{{Alfv}}\lt 1$. As we will see, such an anisotropy is due to the ubiquitous presence of proton beams. While proton beams have been commonly observed at all radial distances in the Alfvénic wind before PSP (e.g., Marsch 2012), the frequent observation of field-aligned beams in the non-Alfvénic wind (including the so-called "hammerhead" distributions; Verniero et al. 2020, 2022) is one of the unexpected results of PSP. Further inspection of the temperature anisotropy PDFs shows that Alfvénic and non-Alfvénic wind display an opposite correlation between PVI (and MAG-PVI) values and T_⊥/T_∥. For the PVI case (Figure 3, top-right panel), the mean values of the non-Alfvénic wind anisotropy lie in the range 0.71 ≲ T_⊥/T_∥ ≲ 0.73 with increasing PVI (dashed lines). In the Alfvénic wind (solid lines), we find instead 0.88 ≲ T_⊥/T_∥ ≲ 0.97 with decreasing PVI. The MAG-PVI case (Figure 3, bottom-right panel) shows similar values of anisotropy and trends.

Insights on parallel and perpendicular temperatures can be found by inspecting the PDFs of T_∥ and T_⊥ (Figure 3, left and middle panels). Interestingly, we find that the ensembles of events selected through each MAG-PVI value have consistently higher average temperatures than the ensembles of events selected with the traditional PVI values. In Table 1 we report the range of the average T_∥ and T_⊥ of the distributions shown in Figure 3, bottom panels, spanned as MAG-PVI increases. To quantify the trends of perpendicular and parallel temperature with MAG-PVI, the relative average temperature enhancement between the smallest and the largest PVI range is also reported in the table. We find that the Alfvénic wind shows a relative enhancement of parallel temperature larger than in the perpendicular temperature, as larger PVI values are considered. The opposite trend is found in the non-Alfvénic wind. This explains why the anisotropy T_⊥/T_∥ shifts away from (toward) unity as PVI values increase for the Alfvénic (non-Alfvénic) wind. Relative variations of temperature at intermittent structures have been found to be about 10% (Sioulas et al. 2022b), so that we expect parallel and perpendicular relative variations to be of that order.

From this analysis we conclude, first, that structures with the largest variations in B (magnetic compressible structures) contribute to the highest T_∥ and T_⊥ in both types of wind, suggesting that the largest temperature enhancements occur in compressible localized structures also in the Alfvénic wind. Second, analysis of the distribution of T_∥ and T_⊥ suggests that the Alfvénic wind undergoes both parallel and perpendicular local temperature enhancements, with a slightly a preferential enhancement of T_∥, whereas the non-Alfvénic wind undergoes a preferential local enhancement of T_⊥.

To investigate the local energization of protons at discontinuities/intermittent structures and kinetic features of the distribution function in velocity space, we manually inspected 1 hr intervals during quiet solar wind and without switchbacks. We selected two subintervals that characterize the Alfvénic and the non-Alfvénic wind (according to high and low cross-helicity), respectively. The two selected subintervals occurred during E6 at approximately the same radial distance from the Sun (30 R_S ). Different fields and particle quantities are reported in Figure 4, where panels on the left correspond to the non-Alfvénic wind and those on the right to the Alfvénic wind.

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The non-Alfvénic subinterval occurred near the heliospheric current sheet crossing, on 2020 September 25 from 03:10:00 to 04:10:00 UT, with an average cross-helicity σ_c ≃ −0.25. This interval presents many discontinuities with large PVI values. The structures are substantially compressible with variations of B and β enhancements.

The Alfvénic interval occurred on 2020 September 30 between 01:00:00 and 02:00:00 UT with an overall $\left|{\sigma }_{c}\right|\simeq 0.88$. The fluctuations show the typical velocity field is highly correlated with magnetic field that characterizes wave-like fluctuations with rotation of $\cos {\theta }_{\mathrm{ub}}$ near small-scale structures. In general, the variations of the magnitude of the magnetic field and the PVI values are smaller compared to the discontinuities in the non-Alfvénic wind. In both cases, however, the most substantial variations in proton temperature occur within large PVI and MAG-PVI structures and correspond to a net enhancement of T_∥/T_⊥, as can be seen by comparing the fourth or fifth panel and the bottom panel.

The signatures in proton velocity distribution function (VDF) of local heating for these two representative subintervals are given in Figure 5, where we show the reduced VDF, $f(t,{v}_{\parallel }),f(t,{v}_{{\perp }_{\mathrm{1,2}}})$ for the non-Alfvénic (left panels) and for the Alfvénic (right panels) subintervals described above (and reported in Figure 4). We also show the reconstructed 3D VDFs in the plane $\{({v}_{\parallel },{v}_{{\perp }_{1}},{v}_{{\perp }_{2}})\}$, and the VDF projection into the $({v}_{\parallel },{v}_{{\perp }_{\mathrm{1,2}}})$ planes, at the bottom panels for each type of wind. The reduced VDF has been obtained by first transforming the initial 3D energy distribution function from the {E, θ, φ} space (energy, elevation, and azimuth) to velocity space in the field-aligned coordinate system $\{{v}_{\parallel },{v}_{{\perp }_{1}},{v}_{{\perp }_{2}}\}$. Details can be found in Appendix A.

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As can be seen from Figure 5, the proton VDF shows strong deviations from local thermal equilibrium and gyrotropy in the vicinity of large PVI values. We observe that protons have enhanced perpendicular velocities in those regions, and parallel propagating proton beams around the local Alfvén speed are observed near discontinuities in both Alfvénic and non-Alfvénic intervals. This can be seen by inspecting Figure 6, which shows a scatterplot of the beam-to-core drift velocity normalized by Alfvén speed versus ${({T}_{\perp ,b}/{T}_{\perp ,c})}^{* }$, which is the ratio of the perpendicular temperature of the beam and the core (T_⊥,b /T_⊥,c ) normalized by the total (beam+core) T_⊥/T_∥ values. This normalized quantity considers the strength of the beam parallel to the mean field and it provides a measure of the deformation of the distribution function. To help intuition, we have split the scatterplot into four quadrants that we use as a reference to characterize the shape of the VDF. The upper-right quadrant corresponds to distributions with a beam and T_⊥,b > T_⊥,c (i.e., "hammerhead" distributions); the lower-right quadrant corresponds to distributions with a beam and T_⊥,b /T_⊥,c < (T_⊥/T_∥), thus indicating, in the most extreme case, a narrowly focused beam in the v_∥–v_⊥ plane. On the left side, the beam and core are not well separated, and one can attribute those data points to distributions with no beams. A sketch of the shape of the distribution as one moves above and below the line ${({T}_{{\perp }_{b}}/{T}_{{\perp }_{c}})}^{* }=1$ is also presented on the bottom-right corner of Figure 6. Details on the fitting method can be found in Appendix B. For reference, the crosses show data points at t = 03:35:39 UT (non-Alfvénic wind) where ${({T}_{\perp ,b}/{T}_{\perp ,c})}^{* }=2.81$ and at t = 01:22:33 UT where ${({T}_{\perp ,b}/{T}_{\perp ,c})}^{* }=1.16$. Figure 5 suggests that beams in the Alfvénic and non-Alfvénic wind have different origins and display distinct properties.

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We performed hybrid-kinetic simulations to investigate proton heating at different types of magnetic structures under different plasma conditions and compared our results with observations. We considered a quasi-neutral hybrid plasma model with massless and isothermal electrons, while protons are modeled as kinetic particles using the particle-in-cell method (Matthews 1994; Franci et al. 2018). We adopted the following normalization: lengths are normalized to the proton inertial length d_i = c/ω_p with ${\omega }_{p}={(4\pi {{ne}}^{2}/{m}_{i})}^{1/2}$, the proton plasma frequency, the time is normalized to the inverse of the proton gyrofrequency ${\omega }_{{ci}}^{-1}={({{eB}}_{0}/{m}_{i}c)}^{-1}$, and velocities are normalized to the Alfvén speed ${v}_{A}={B}_{0}/{(4\pi {n}_{0}{m}_{i})}^{1/2}$ with B₀, the mean magnetic field. The plasma beta for both ions and electrons is defined as ${\beta }_{p,e}=8\pi {n}_{0}{T}_{p,e}/{B}_{0}^{2}$. To avoid energy accumulation at the grid scale, we have included explicit resistivity with a corresponding dissipation length (l_d ) related to the Reynolds number and the box size (L) through ${R}_{e}\sim {(L/{l}_{d})}^{4/3}$, which is chosen to be greater than the grid size.

We have considered two different setups in 2.5D geometry characterized by two different types of turbulent fluctuations resembling non-Alfvénic and Alfvénic wind, namely, (i) a decaying 2D balanced turbulence simulation, and (ii) a wave simulation with an initial parallel propagating, Alfvénic broadband spectrum. For case (i), hereafter referred to as the "turbulence simulation," we considered an initial large-amplitude fluctuation with σ_c ≈ 0, in energy equipartition and with an out-of-plane guide field ${{\boldsymbol{B}}}_{0}={B}_{0}\hat{{\boldsymbol{z}}}$. The initial condition consists of large-amplitude perturbations of perpendicular wave modes in the range 0.196 < k_⊥ d_i < 0.49 with random phases, similar to previous work (Servidio et al. 2012; Franci et al. 2015; Cerri & Califano 2017). For case (ii), referred to as the "wave-like simulation," we considered an in-plane mean field ${{\boldsymbol{B}}}_{0}={B}_{0}\hat{{\boldsymbol{x}}}$, and the initial perturbation satisfies the Walèn relation in the dispersionless limit δ u = −(ω₀/k₀)δ b . The wave frequency ω₀ is determined from the normalized dispersion relation ${k}_{0}^{2}={\omega }_{0}^{2}/(1-{\omega }_{0})$ for left-handed polarized parallel propagation waves. This initial condition corresponds to a broadband Alfvénic fluctuation composed of parallel modes in the range 0.196 < k_∥ d_i < 0.49 (see also Malara et al. 2000, Matteini et al. 2010, González et al. 2021). In both cases, an initially isotropic and Maxwellian plasma with proton beta β_i = 0.5 and equal ion and electron temperature T_i /T_e = 1 are considered. The guide field is B₀ = 1 and the same rms of the magnetic fluctuations δ b_rms/B₀ = 0.63 is chosen for both simulations. We adopt periodic boundary conditions and fix a box of side L = 128 d_i by using 1024² grid points with mesh size 0.125 d_i and 8000 particles per cell.

Because of the different geometry adopted, interactions between fields and particles are fundamentally different in the two setups (Gary et al. 2020). Naturally, our simulations are not meant to represent realistically the two types of wind (both requiring 3D fields), but rather to isolate different processes that might be dominant in each type of wind and that bear specific signatures in phase space. In Section 3.3 we discuss the correlation between anisotropy and PVI, and the signatures in phase space of proton heating/acceleration in correspondence with large PVI values.

In the turbulence simulation, once the system reaches the fully developed turbulent state, the energy stored in the fields is progressively converted into thermal energy, resulting in an average (over the spatial domain) preferential perpendicular heating with 〈T_⊥/T_∥〉_box = 1.37 at $t=120{\omega }_{{ci}}^{-1}$. The wave-like simulation, by contrast, displays a strong enhancement of the spatially averaged T_∥, and the temperature anisotropy reaches 〈T_⊥/T_∥〉_box = 0.6 at $t=180{\omega }_{{ci}}^{-1}$. ⁴

These different heating processes in the two setups are expected, since the turbulence simulation leads to a well-developed magnetic energy spectrum perpendicular (in k space) to the guide field, allowing for more channels of particle heating, such as stochastic heating and particle scattering at current sheets (Cerri et al. 2021; Sioulas et al. 2022a). Those processes are suppressed in the wave-like simulation due to the different geometry adopted. That said, in the wave-like simulation, the available energy carried by the wave is converted into kinetic and thermal energy via the rapid disruption of the wave packet mediated by wave steepening along the guide field (i.e., through the formation of gradients parallel to B ₀); this leads to both the formation of a field-aligned proton beam at about the local Alfvén speed, and local perpendicular heating at the steepened edges (González et al. 2021). This mechanism is in turn suppressed in the turbulence simulation. Figure 7 shows the contour plot of J_z for the two setups to show the different types of structures that form nonlinearly.

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To make a comparison with PSP data, in Figure 8 we show the time series of fields and plasma quantities in the two setups. The time series have been taken at the location marked by the black dot in Figure 7. From top to bottom, Figure 8 shows the time series of n, B and ∣B∣, u , the electric field e , T_⊥, T_∥ and β, $\cos {\theta }_{\mathrm{vb}}$, σ_r and σ_c , and the PVI values (see Section 3.3 for details on the PVI analysis). The turbulence simulation (left panels) is characterized on average by ∣σ_c ∣ < 0.5 and σ_r < 0 (predominantly). The wave-like simulation (right panel) has σ_r ≈ 0 and ∣σ_c ∣ ≈ 1. As can be seen, temperature enhancements occur in correspondence with large PVI values in both cases. In the turbulence case, those large PVI values are associated with structures such as flux tubes, current sheets, and small-scale plasmoids. These structures have a strong perpendicular electric field, and some also display large out-of-plane electric and magnetic fields (along the z-axis) with strong compressibility, mediating particle acceleration and heating (Dmitruk et al. 2004; Wan et al. 2015; Comisso & Sironi 2022). As discussed above, in the wave-like simulation large PVI values are associated with the steepened edges of Alfvénic fluctuations. Also, rapid and large variations of $\cos {\theta }_{\mathrm{ub}}$ near these structures can be identified (sixth panel), similar to what is observed during the Alfvénic subinterval presented in Section 2.3. In contrast to the magnetic structures in the turbulent simulation, the steepened edges are mostly rotational discontinuities with an embedded compressive component (Matteini et al. 2010; González et al. 2021).

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The PDFs conditioned on PVI and MAG-PVI are shown in Figure 9 in the same format as Figure 3. To obtain the PDFs, we used a similar methodology to that employed for PSP data and we have taken the time series of plasma quantities at a resolution of $0.1{\omega }_{{ci}}^{-1}$ at 10⁴ fixed equidistant points in the simulation domain. Because of computational limitations, our simulations are not long enough to allow us to take a time lag similar to that in the data analysis. That said, we have sufficient time resolution to explore fluctuations around the gyroperiod ($\tau =1{\omega }_{{ci}}^{-1}$). Therefore, we fixed the time lag equal to the proton gyroperiod to select coherent kinetic-scale structures, despite our inability to directly probe this scale with in situ data.

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In general, there is a positive correlation between PVI values with T and δ T (not shown), indicating that the hottest protons are found locally near small-scale structures (in this case represented by PVI > 3 values). Although the turbulence simulation is hotter than the wave-like simulation (this is true both locally at discontinuities and globally over the simulation domain), protons remain close to Maxwellian in the turbulence simulation. By contrast, in the wave-like simulation, particles develop large anisotropies localized at small-scale structures. In that case such a local anisotropy, reaching an average value of T_⊥/T_∥ ≈ 2 for PVI > 3, is caused by strong particle pitch-angle scattering at the steepened fronts, resulting in effective local proton perpendicular energization (González et al. 2021; Malara et al. 2021; Gonzalez et al. 2023). Since this effect is localized at the steepened edges, the background temperature (represented by the lower PVI range) and the spatially averaged temperatures over the entire domain display an opposite anisotropy, T_⊥/T_∥ < 1, as discussed previously, due to the formation of a field-aligned beam that fills the entire simulation domain.

Some differences between PVI (Figure 9, top panels) and MAG-PVI (Figure 9, bottom panels) results can be noticed. In the wave-like simulation, the magnetic structures correspond essentially to rotational discontinuities with relatively small variations in the magnitude of B and, thus, the correlation with the various PVI ranges is reduced in the MAG-PVI case. This can be seen from the relative temperature variations between the smallest and the largest range of PVIs, showing the aforementioned trend (ΔT_∥/T_∥ ≃ 3% and ΔT_⊥/T_⊥ ≃ 59%). For the turbulence simulation, on the contrary, a higher relative temperature increase is found, similar to observations (ΔT_∥/T_∥ ≃ 6% and ΔT_⊥/T_⊥ ≃ 8%).

In the top two panels of Figure 10 we show the gyroaveraged reduced perpendicular and parallel VDF (f(t, v_⊥) and f(t, v_∥), respectively, and in the bottom panels we show the reduced VDF f(v_∥, v_⊥) at the time indicated by the dashed line, marking the occurrence of an event with PVI > 3 in each case. The left column corresponds to the turbulence simulation and the right to the wave-like. As can be seen, magnetic discontinuities/intermittent structures are characterized by non-Maxwellian features, namely temperature anisotropies and beams at the Alfvén speed. However, the turbulence simulation undergoes perpendicular heating in both the core and the beams, so that T_⊥ ≳ T_∥ locally and, on average, and beams are "hot." The wave-like simulation instead displays a strong perpendicular heating of the core, localized at the discontinuities, and a cold beam at the Alfvén speed (more focused in the perpendicular direction than in the turbulence setup).

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To summarize, our simulations show that, in analogy with PSP data, higher temperatures and temperature variations are found at higher PVI values in both setups (both winds), and that colder beams (as found in Alfvénic wind) are associated with the steepening of Alfvénic fluctuations while hot beams (as found in non-Alfvénic wind) are generated in low cross-helicity turbulence. The development of a standard turbulent cascade also favors preferential perpendicular heating due to different mechanisms such as stochastic heating (as in the non-Alfvénic wind). Even though our simulations highlight the different origins and properties of proton beams in the two types of setups/winds and qualitatively reproduce observed properties of non-Alfvénic wind, they do not reflect all the local trends of temperatures at magnetic structures observed in the Alfvénic wind. In particular, the PVI-temperature analysis in the wave-like simulation shows a much larger relative increase of perpendicular temperature than observed (larger than the increase in parallel temperature), and compressibility of the Alfvénic wind at small scales is also not well reproduced numerically. These discrepancies are likely due to the geometry contsraints. However, we also mention the possibility that instrument resolution may also play a role in underestimating strong temperature variations at small scales.

Before concluding we also remark that our simulations do not reproduce differences between the two types of winds in their average properties, such as the fact that the Alfvénic wind is hotter, and that the global (average) anisotropy is T_⊥/T_∥ < 1. This is expected since our setups cannot reproduce fully 3D dynamics. Furthermore, such differences might well be related to different coronal origins and/or a different radial evolution of Alfvénic and non-Alfvénic wind, an aspect that is not addressed in this work.